Numerical modeling of masonry and historical structures : from theory to application 9780081024393, 0081024398, 9780081024409, 0081024401

Table of contents :
Content: Part I: Seismic vulnerability analysis of masonry and historical structures 1. Seismic vulnerability analysis of historical structures 2. Seismic performance-based assessment of masonry structures (Perpetuate guidelines) 3. Nonlinear time history analysis of masonry for seismic assessment 4. Uncertainties in determining the seismic vulnerability of historic masonry buildings 5. A framework for the seismic assessment of existing masonry buildings accounting for different sources of uncertainty 6. Methods and Challenges for the Seismic Assessment of Historic Masonry Structures Part II: Numerical modeling of unreinforced masonry 7. Mechanical properties 8. Macro-modelling technique 9. Micro-modelling technique 10. Homogenization and quasi periodic masonry 11. Limit analysis 12. DEM analysis 13. Equivalent frame modeling/macro-elements Part III: Numerical modeling of FRP- strengthened masonry 14. FRP strengthened masonry. Delamination and numerical issues 15. Homogenization model for pushover analysis of FRP-strengthened masonry 16. Pushover analysis of FRP-strengthened masonry: long-term seismic performance simulation 17. Dynamic analysis of FRP-strengthened masonry Part III: Modeling and analysis of TRM- strengthened masonry 18. Computational modeling of textile reinforced mortars and numerical issues 19. Homogenization model for pushover analysis of TRM-strengthened masonry 20. Macro-modeling approach for pushover analysis of TRM-strengthened masonry 21. Dynamic time-history analysis of TRM- strengthened masonry

Citation preview

Numerical Modeling of Masonry and Historical Structures

Woodhead Publishing Series in Civil and Structural Engineering

Numerical Modeling of Masonry and Historical Structures From Theory to Application

Edited by

Bahman Ghiassi Gabriele Milani

Woodhead Publishing is an imprint of Elsevier The Officers’ Mess Business Centre, Royston Road, Duxford, CB22 4QH, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, OX5 1GB, United Kingdom Copyright © 2019 Elsevier Ltd. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-08-102439-3 (print) ISBN: 978-0-08-102440-9 (online) For information on all Woodhead Publishing publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Matthew Deans Acquisition Editor: Gwen Jones Editorial Project Manager: Peter Llewellyn Production Project Manager: Debasish Ghosh Cover Designer: Alan Studholme Typeset by MPS Limited, Chennai, India

List of Contributors D. Addessi Department of Structural and Geotechnical Engineering, Sapienza University of Rome, Rome, Italy Reza Allahvirdizadeh University of Minho, Guimara˜es, Portugal K. Bagi Budapest University of Technology and Economics, Budapest,Hungary Elisa Bertolesi ICITECH, Universitat Polite`cnica de Valencia, Valencia, Spain S. Caddemi Department of Civil Engineering and Architecture, University of Catania, Catania, Italy I. Calio` Department of Civil Engineering and Architecture, University of Catania, Catania, Italy F. Cannizzaro Department of Civil Engineering and Architecture, University of Catania, Catania, Italy Roberto Capozucca Polytechnic University of Marche, Ancona, Italy G. Castellazzi Department of Civil, Chemical, Environmental, and Materials Engineering (DICAM), University of Bologna, Bologna, Italy S. Cattari University of Genoa, Genoa, Italy; Department of Civil, Chemical and Environmental Engineering (DICCA), University of Genoa, Genoa, Italy A. Chiozzi Department of Engineering, University of Ferrara, Ferrara, Italy A.M. D’Altri Department of Civil, Chemical, Environmental, and Materials Engineering (DICAM), University of Bologna, Bologna, Italy S. de Miranda Department of Civil, Chemical, Environmental, and Materials Engineering (DICAM), University of Bologna, Bologna, Italy K. Ehab Moustafa Kamel Department of Building, Architecture and Town Planning, Universite´ libre de Bruxelles (ULB), Bruxelles, Belgium Bahman Ghiassi Centre for Structural Engineering and Informatics, Faculty of Engineering, The University of Nottingham, Nottingham, United Kingdom; Centre for Structural Engineering and Informatics, Department of Civil Engineering, University of Nottingham, Nottingham, United Kingdom Ernesto Grande Department of Sustainability Engineering, University Guglielmo Marconi, Rome, Italy H. Hernandez Department of Building, Architecture and Town Planning, Universite´ libre de Bruxelles (ULB), Bruxelles, Belgium

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M. Imbimbo Department of Civil and Mechanical Engineering, University of Cassino and Southern Lazio, Cassino, Italy B.A. Izzuddin Department of Civil and Environmental Engineering, Imperial College London, London, United Kingdom S. Lagomarsino University of Genoa, Genoa, Italy; Department of Civil, Chemical and Environmental Engineering (DICCA), University of Genoa, Genoa, Italy F. Lebon Aix-Marseille Univ. CNRS Centrale Marseille LMA, Marseille, France J.V. Lemos National Laboratory for Civil Engineering, Lisbon, Portugal J. Li Tianjin Chengjian University, Tianjin, P.R. China G.P. Lignola Department of Structures for Engineering and Architecture, University of Naples “Federico II”, Naples, Italy P.B. Lourenco ISISE, Department of Civil Engineering, University of Minho, Guimaraes, Portugal L. Macorini Department of Civil and Environmental Engineering, Imperial College London, London, United Kingdom S. Marfia Department of Engineering, University of Roma Tre, Roma, Italy M.J. Masia The University of Newcastle, Callaghan, NSW, Australia T.J. Massart Department of Building, Architecture and Town Planning, Universite´ libre de Bruxelles (ULB), Bruxelles, Belgium N. Mendes ISISE, University of Minho, Guimara˜es, Portugal Gabriele Milani Department of Architecture, Built Environment and Construction Engineering (A.B.C.), Technical University of Milan, Milan, Italy; Department of Architecture, Built Environment and Construction Engineering (A.B.C.), Politecnico di Milano, Milan, Italy E. Minga Department of Civil and Environmental Engineering, Imperial College London, London, United Kingdom Gemma Mininno Polytechnic University of Bari, Bari, Italy Daniel V. Oliveira ISISE, Department of Civil Engineering, University of Minho, Barga, Portugal D. Ottonelli University of Genoa, Genoa, Italy B. Panto` Department of Civil Engineering and Architecture, University of Catania, Catania, Italy B. Pintucchi Department of Civil and Environmental Engineering, University of Florence, Florence, Italy D. Rapicavoli Department of Civil Engineering and Architecture, University of Catania, Catania, Italy A. Rekik INSA CVL, Univ. Orle´ans, Univ. Tours, Orle´ans, France J. Rots Faculty of Civil Engineering and Geosciences, Delft University of Technology, Delft, The Netherlands

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E. Sacco Department of Structures for Engineering and Architecture, University of Naples Federico II, Naples, Italy Matteo Salvalaggio Department of Cultural Heritage, Piazza Capitaniato 7, Padova, Italy V. Sarhosis School of Civil Engineering, University of Leeds, Leeds, United Kingdom Luca Sbrogio` Department of Cultural Heritage, Piazza Capitaniato 7, Padova, Italy Rui A. Silva University of Minho, Guimara˜es, Portugal M.G. Stewart The University of Newcastle, Callaghan, NSW, Australia A. Taliercio Department of Civil and Environmental Engineering (DICA), Politecnico di Milano, Milan, Italy A. Tralli Department of Engineering, University of Ferrara, Ferrara, Italy M. Valente Department of Architecture, Built Environment and Construction Engineering (ABC), Politecnico di Milano, Milan, Italy Maria Rosa Valluzzi Department of Cultural Heritage, Piazza Capitaniato 7, Padova, Italy A.T. Vermelfoort Eindhoven University of Technology, Eindhoven, The Netherlands Xuan Wang University of Macau, Zhuhai, P.R. China

Preface Masonry is one of oldest and most commonly used construction materials around the world. Masonry is highly durable and sustainable and its constituents are readily available around the world, all features making this material a suitable choice for several rural and urban areas. Despite these advantages, structures made of masonry are highly vulnerable to seismic actions due to their low resistance to tensile stresses. Reliable evaluation of the seismic vulnerability of these structures and the subsequent proposal of adequate repair/ conservation strategies have thus been the subject of several studies over the last few decades. Numerical modeling and experimental testing methodologies have been widely used for these purposes. Numerical models are particularly interesting as, once accurately validated and used, they allow performing “virtual tests” at the material, component, and structural levels. Application of numerical models for simulating the performance of masonry structures has always been a challenging task. This is partly due to the complex nature and behavior of this material, lack of sufficient experimental data for validation of the constitutive laws, and lack of sufficient knowledge on the actual condition of existing structures and appropriate boundary conditions. A wide range of modeling strategies, from highly simplified to highly advanced, have been used and developed throughout the years to represent the behavior of masonry structures. As each of these methods has its own advantages and disadvantages, basic knowledge of the capabilities and limitations of each is necessary for selection of the appropriate technique to apply to real problems. Tuning of the input parameters from either phenomenological formulas or experimental tests also has a great influence on the numerical outputs and reflects on the reliability of the simulations. This book, organized in three sections and 21 chapters, presents a comprehensive reference on a wide range of techniques used for numerical modeling of masonry and historical structures. The main objective of the proposed volume is to provide a theoretical background of and practical guidelines for each modeling strategy for unreinforced and reinforced (strengthened) masonry structures. The first section presents an overview of the latest methods and challenges for seismic vulnerability assessment and strengthening of masonry

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and historical structures. This section acts as an opening to the book and the subject at hand. The second section provides an overview of different modeling techniques used for simulation of the performance of unreinforced masonry structures as well as the main mechanical properties required for utilization of such techniques. The third, and last, section deals with different numerical modeling strategies adapted for evaluating the seismic performance of strengthened masonry structures. The main focus is on externally bonded reinforcement techniques using fiber-reinforced polymers and textile-reinforced mortars as the most common methods used in practice for improving the seismic performance of masonry structures. We are convinced that the presented discussions can provide the most up-to-date overview of the numerical modeling strategies used for evaluating the seismic vulnerability of masonry and historical constructions. It can be used as a reference for academic courses, as well as a handbook for practitioners active in both the evaluation of the seismic performance and strengthening of masonry and historical structures. Finally, we would like to thank all the contributors for their valuable efforts in finalizing this volume and all the reviewers for their constructive comments and kind assistance, which had a significant influence on the outcome of this book. The limitless efforts and support of the Elsevier managerial and editorial board are also highly appreciated. Bahman Ghiassi and Gabriele Milani

Chapter 1

A review of numerical models for masonry structures A. M. D’Altri1, V. Sarhosis2, G. Milani3, J. Rots4, S. Cattari5, S. Lagomarsino5, E. Sacco6, A. Tralli7, G. Castellazzi1 and S. de Miranda1 1

Department of Civil, Chemical, Environmental, and Materials Engineering (DICAM), University of Bologna, Bologna, Italy, 2School of Civil Engineering, University of Leeds, Leeds, United Kingdom, 3Department of Architecture, Built Environment and Construction Engineering (A.B.C.), Politecnico di Milano, Milan, Italy, 4Faculty of Civil Engineering and Geosciences, Delft University of Technology, Delft, The Netherlands, 5Department of Civil, Chemical and Environmental Engineering (DICCA), University of Genoa, Genoa, Italy, 6Department of Structures for Engineering and Architecture, University of Naples “Federico II”, Naples, Italy, 7 Engineering Department, University of Ferrara, Ferrara, Italy

1.1 Introduction A significant part of the existing structures in the world are made of masonry. Indeed, most historical buildings consist of monumental masonry structures (churches, mosques, temples, towers, fortresses, etc.). Additionally, most ordinary buildings in are generally made of masonry. Very marked differences can be noted between monumental and ordinary buildings (Fig. 1.1), in terms, for example, of geometry, material, and structural details. The weak seismic performance of masonry structures was highlighted by recent and past earthquakes. Typically, cracks arise in masonry buildings even for low-intensity shocks. Additionally, cracking could also be due to soil differential settlements. Cracks in masonry, comprised of blocks bonded with mortar, generally appear along the mortar joints, even though the block cracking could also appear depending on the block and mortar features. Although several strengthening systems have been developed over the centuries to improve the ductility, dissipation, and strength of masonry, this chapter focuses only on unreinforced masonry structures. Various tools for the prediction and the assessment of the structural behavior of masonry buildings have been developed in recent decades.

Numerical Modeling of Masonry and Historical Structures. DOI: https://doi.org/10.1016/B978-0-08-102439-3.00001-4 Copyright © 2019 Elsevier Ltd. All rights reserved.

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FIGURE 1.1 Examples of (A) monumental and (B) ordinary masonry structures.

Numerical tools have been favorably developed and preferred over analytical approaches, given the complex mechanical response of masonry and the irregular geometries of historic masonry buildings. The idea is that, if a model can simulate the behavior of masonry structures, it can predict the structural response to expected actions, evaluating the safety of a building. These numerical tools have mainly focused on the analysis of the nearcollapse response of existing masonry buildings (rather than new buildings), given their widespread dissemination and their weak structural response. The numerical modeling of masonry structures is still a challenging task, given the deep complexities and uncertainties that characterize the geometry of buildings (especially for the historic ones) and the mechanical response of masonry (highly nonlinear). In this chapter, a thorough review of numerical strategies for masonry structures is presented. Additionally, classification of these approaches is also suggested to logically organize the extensive literature on this topic. Even though a wholly congruent categorization of all the numerical tools is essentially unrealistic given the specific aspects of each solution developed, the numerical strategies are subdivided into four classes (Fig. 1.2): blockbased models (BBMs), continuum models (CMs), macroelement models (MMs), and geometry-based models (GBMs). The mechanical and geometrical issues that arise when dealing with masonry structures are briefly discussed in Section 1.2. Then, the existing analysis approaches that can be used for masonry structures are reviewed in Section 1.3. The classification of numerical strategies for masonry structures is introduced in Section 1.4. Then, each class is thoroughly reviewed: BBM in Section 1.5, CM in Section 1.6, MM in Section 1.7, and GBM in Section 1.8. In Section 1.9, a summary of the fields of utilization of each class is discussed and the open challenges in numerical modeling of masonry structures are critically examined.

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Numerical strategies for masonry structures Chisari et al. (2018)

Block-based models

Milani (2008)

Çakti et al. (2016)

D’Altri et al. (2018b)

Lourenço and Rots (1997)

Baraldi and Cecchi (2017) Serpieri et al. (2017)

Continuum models

RVE

Berto et al. (2002)

Abdulla et al. (2017) Lourenço et al. (1998)

Portioli et al. (2014) Brasile et al. (2007a,b)

Pelà et al. (2013)

Gambarotta and Lagomarsino (1997a,b)

Addessi and Sacco (2012) Bruggi (2014)

Petracca et al. (2016)

Massart et al. (2007)

Milani et al. (2007a,b)

Macroelement models Lagomarsino et al. (2013)

Caliò et al. (2012)

Rinaldin et al. (2016)

Belmouden and Lestuzzi (2009)

Penna et al. (2014)

Roca et al. (2005)

Geometry-based models

Block et al. (2006) Block and Lachauer (2014a,b)

Marmo and Rosati (2017)

Angelillo (2014)

O’Dwyer (1999) Block and Ochsendorf (2007) Chiozzi et al. (2018a,b,c)

Chiozzi et al. (2017)

Fraternali (2010)

FIGURE 1.2 Numerical strategies for masonry structures.

1.2 Mechanical and geometrical challenges The accurate material mechanical characterization and the detailed structural and geometrical survey appear fundamental to the trustworthy simulation of

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PART | I Seismic vulnerability analysis of masonry and historical structures

the structural behavior of existing masonry structures. In this section, the main issues in the mechanical characterization of masonry and in the geometrical definition of historic masonry structures are briefly highlighted. The interested reader is referred to Como (2013) and Hendry (1998) for further details.

1.2.1

Masonry mechanics

The mechanics of masonry is complex, varied, and highly nonlinear. Masonry is comprised of blocks usually bonded with mortar and assembled with a specific texture. Thereby, masonry is evidently an heterogeneous material. Actually, a very wide category of building materials (Fig. 1.3) can be considered with the term “masonry,” with various mechanical characteristics (Como, 2013). The mechanical properties of block, mortar, and the bond between them typically governs the overall masonry response. A quasibrittle behavior in tensile and compressive regimes generally characterize the masonry components (block and mortar). Higher values of strength and fracture energy are observed in the compressive behavior rather than in the tensile one. Usually, the bond between the mortar and the blocks is extremely weak, with a cohesive-frictional response in shear (where both cohesive and frictional contributions depend on the normal stress) and a cohesive response in tension (without cohesion in case of dry stone masonry). Both shear and tensile responses show cohesion softening (Hendry, 1998). Three levels of anisotropy characterize the masonry material (Page, 1981): (1) elastic anisotropy (i.e., in the elastic regime); (2) strength anisotropy (i.e., in the strength properties, beyond the difference between compressive and tensile strengths); and (3) brittleness anisotropy (i.e., in the postpeak response). Periodic brick masonry typically shows significant anisotropic features, whereas anisotropy in irregular stone masonry could be less substantial, given the nonperiodic material structure.

FIGURE 1.3 Examples of masonry: (A) brick masonry, (B) stone masonry, and (C) dry stone masonry.

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FIGURE 1.4 Failure modes and limit domains of masonry for plane stress states: (A) scale of the material and (B) scale of the pier. From Calderini, C., Cattari, S., Lagomarsino, S., 2009. In-plane strength of unreinforced masonry piers. Earthq. Eng. Struct. Dyn. 38 (2), 243 267.

The mechanical behavior of masonry could be deduced from (at least) two different scales, that is, the scale of the material (Page, 1981, 1983; Page et al., 1982) and the scale of the pier (Magenes and Calvi, 1997; Calderini et al., 2009; Beyer, 2012; Petry and Beyer, 2014; Messali and Rots, 2018). For both scales, the mechanical response has to be defined in terms of stiffness, strength, and ductility. The masonry strength domains are shown in Fig. 1.4 at the scale of the material (Fig. 1.4A) and at the scale of the pier (Fig. 1.4B). Fig. 1.5 shows the typical failure modes of masonry at a twoblock masonry assemblage scale.

1.2.2

Masonry experimental characterization

The experimental characterization of the mechanical properties of masonry is still a complex task. Even though various experimental tests and setups have been recently developed, their trustworthiness is still a subject of discussion (Sassoni et al., 2014; Krˇzan et al., 2015).

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FIGURE 1.5 Masonry failure mechanisms (at a two-block masonry assemblage scale): (A) block-mortar bond tensile failure, (B) block-mortar bond shear sliding, (C) diagonal masonry cracking, (D) masonry crushing, and (E) block and mortar tensile cracking. From D’Altri, A.M., de Miranda, S., Castellazzi, G., Sarhosis, V., 2018b. A 3D detailed micro-model for the in-plane and out-of-plane numerical analysis of masonry panels. Comput. Struct. 206, 18 30.

FIGURE 1.6 Experimental characterization of masonry at different scales.

The experimental masonry characterization can be carried out at various scales, as depicted in Fig. 1.6; for example, at the scale of masonry components (block, mortar, and block-mortar bond), wallets (small masonry assemblages), panels (real-scale masonry walls), and buildings (full-scale masonry structures). In situ tests should be used to mechanically characterize existing masonry buildings (Borri et al., 2011; Lumantarna et al., 2014). Nevertheless, in situ tests generally show greater uncertainties on the characterized mechanical properties rather than laboratory tests. Moreover, nondestructive tests are often the only possible approaches in historic monumental buildings to preserve their conservation and authenticity (McCann and Forde, 2001; Bosiljkov et al., 2010). Finally, indirect methods have been proposed (Borri et al., 2015) to specify masonry mechanical properties based on qualitative interpretations of the main morphological features (e.g., status of mortar joints, efficacy of in-plane and transversal interlocking, bond, etc.).

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Structural details

Structural details are fundamental aspects in the mechanical response of masonry constructions. For example, the toothing between perpendicular walls, the junction with horizontal diaphragms, the interaction with adjacent buildings, etc., can sensibly influence the structural response of masonry buildings (Tondelli et al., 2012). Often, the structural details are the results of the complex historical evolution of buildings, which may have altered the original configuration due to restorations, addition of parts, destination changes, damages and repairs, etc. Structural details are, therefore, the result of a subsequent superimposition of modifications along with the centuries and their knowledge could be particularly complicated for historic structures. To improve the knowledge phase in masonry buildings, beyond the standards and guidelines (Guide for the structural, 2010; Direttiva del Presidente, 2011), recent scientific proposals have been developed (Cattari et al., 2015; Haddad et al., 2019).

1.2.4

Geometrical challenges

In historic monumental buildings, which are generally characterized as complex and irregular geometries, even the determination of the geometry of the structure could be complicated. Indeed, precise geometrical and structural surveys are often required when dealing with historic structures. First, identification of the load-bearing system has to be found within the building geometry. The knowledge of the building appears fundamental in this crucial operation. Secondly, the geometry has to be employed for structural analysis purposes. Generally, computer-aided design (CAD) software packages can be used to manually build the geometry. Nevertheless, the usability of this CAD-based geometry in mesh-based numerical analysis is usually controversial. Indeed, the meshing process of these geometries is typically characterized by compatibility problems, mesh errors, excessively refined meshes, etc. Various strategies that use 3D point clouds as input for the automatic mesh generation of historic monumental buildings have been recently proposed (Castellazzi et al., 2015, 2017; Korumaz et al., 2017; D’Altri et al., 2018c) to solve the aforementioned issues. The enhancement of these approaches is still ongoing.

1.3 Analysis approaches Two main types of analysis approaches can be followed to investigate the near-collapse response of masonry structures, that is, incremental-iterative analysis approaches and limit analysis-based solutions. These two analysis approaches are summarily recalled in this section.

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PART | I Seismic vulnerability analysis of masonry and historical structures

1.3.1

Incremental-iterative analysis approaches

Evolution of the equilibrium status of a structure subjected to certain loads is analyzed step-by-step in incremental-iterative analysis procedures. The loading and the structural response are subdivided into a succession of intervals, increments, or “steps.” Iterations are conducted to reach the equilibrium in each step. The mechanical nonlinearity, which is essential for a trustworthy assessment of the collapse and near-collapse behavior of masonry structures, can be accounted for in these procedures. Geometric nonlinearity is often encountered when the structure is subjected to large displacement effects. Even though linear elastic models have been utilized in a few cases for the preliminary assessment of historic masonry structures (Macchi et al., 1993; Cerone et al., 2000), their efficacy in studying the collapse mode and the safety of these structure is greatly limited. Incremental-iterative analyses can be categorized as nonlinear static and nonlinear dynamic analyses: 1. Nonlinear static analysis. In nonlinear static analyses, the structure is step-by-step subjected to certain loads until its collapse and beyond that. The structural response evolves in a pseudo-time that does not stand for any physical characteristic. Analyses can be conducted in either load control or displacement control, as well as in event-by-event damage control (e.g., sequentially linear analysis; Rots et al., 2008; DeJong et al., 2009). Nonlinear differential equations have to be solved due to the mechanical nonlinearity assumed for the material. These equations can be turned into nonlinear algebraic equations and solved within a numerical context. In general, the nonlinear equations are linearized in a stepwise manner and solved using an iterative procedure. Among the most renowned iterative implicit approaches (Reddy, 2004) are the Picard iteration (or direct iteration) method, the Newton Raphson iteration method, and the Riks method. These analysis approaches are usually used to simulate quasistatic experimental tests on masonry structures and to carry out the so-called pushover analysis. Pushover analysis is an ordinary and standardized method to analyze the seismic response of a masonry building. In these analyses, a load pattern of horizontal forces, kept constant in shape along with the simulation, is applied to the structure and the monotonically increasing displacement of a control node is recorded. 2. Nonlinear dynamic analysis. In nonlinear dynamic analysis (also called time history or transient nonlinear analysis), the structure is step-by-step exposed to time-dependent loads and the structural response evolves in the real time, accounting for damping and inertial effects as well. Time integration schemes are employed to nearly fulfill the equations of motion during each time step of the simulation. These simulations could

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exploit explicit and implicit schemes (Clough and Penzien, 2003). In an explicit method, the new response values computed at each step rely only on quantities calculated in the previous step. In contrast, in an implicit method the expressions that generate the new values for a given step include also values that belong to the same step. Therefore, trial values of the unknowns must be assumed and refined by successive iterations. Among the most famous time integration schemes are the following: Newmark beta methods, Euler Gauss procedure, second central difference formulation, and linear acceleration procedures (Clough and Penzien, 2003). The reader is referred to Clough and Penzien (2003) for more details. The effects of dynamic actions (e.g., impacts, explosions, earthquakes, etc.) on masonry structures can be accounted for in nonlinear dynamic analyses. For instance, the response of the structure against a real accelerogram (and therefore shaking table tests on full-scale structures) can be analyzed given the chance to account for time-dependent actions. Finally, time history nonlinear analysis can be used for analyzing quasistatic tests by employing loads in a very slow manner.

1.3.2

Limit analysis-based solutions

The limit theorems of plasticity were first applied by Heyman (1966) to masonry structures using the following three hypotheses: 1. Masonry has no tensile strength. 2. The compressive strength of masonry is infinite. 3. Sliding of one masonry block upon another cannot occur. These three hypotheses, together with insignificant elastic strains, permit the expression of the static theorem (i.e., the lower-bound limit analysis) and the kinematic theorem (i.e., the upper-bound limit analysis) for masonry applications. The rigid no-tension model developed by Heyman was extensively used and successfully employed in studying the stability of masonry structures (Angelillo, 2014). First, these hypotheses permitted simple graphic static solutions for the investigation of the stability of masonry vaults (Huerta, 2001) and kinematic analysis of recurring seismic failure modes of masonry structures (Giuffre´ and Carocci, 1993). Secondly, the Heyman’s model established a trustworthy basis for recent numerical limit analysis-based approaches. These various solutions can be based on either the static (Marmo and Rosati, 2017) or the kinematic theorem (Chiozzi et al., 2017), and the problem can be expressed as the solution of nonlinear differential equations, of an optimization problem (using or not genetic algorithms), of linear or sequential linear programming, etc. The main drawback of limit analysis-based approaches relies on the fact that the collapse multiplier and the collapse mechanism are the only results

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available, and no information is obtainable on the postpeak response, which is essential in displacement-based seismic assessment procedures for masonry structures.

1.4 Modeling strategies In this section, a classification of the numerical strategies for masonry structures is suggested, following the categorization proposed in D’Altri et al. (2019a) and D’Altri (2019). This classification is based on how masonry and/or masonry structures are conceived and modeled. Accordingly, the analysis approaches illustrated in Section 1.3 could be, in theory, employed in each class of numerical strategies. Each numerical approach has some specific attractive characteristics, which, in general, can be optimal for a specific field of utilization. Additionally, different scales of material testing (Fig. 1.6) can be utilized to set the mechanical properties of the model, depending on the scale of representation considered in the modeling approach. Even though a wholly congruent categorization of all the numerical tools is essentially unrealistic given the specific aspects of each solution developed, the classification suggested in the following attempts to organize the extensive scientific literature on this topic. Four main classes of numerical strategies for masonry structures (Fig. 1.2) are proposed: 1. BBMs. A block-by-block definition of the structure is used to model masonry. Accordingly, the real texture of masonry could be accounted for. Each block can be treated as a rigid or deformable body, while the mechanical interaction between blocks can be modeled through various convenient formulations, which are discussed in Section 1.5. 2. CMs. Masonry is conceived as a continuum deformable body, without differentiation between blocks and mortar layers. The masonry material constitutive law could be described either using (1) direct approaches, that is, through constitutive laws tuned, for instance, on experimental tests, or using (2) homogenization procedures and multiscale approaches, where the material constitutive relationship (conceived as homogeneous in the structural-scale model) is deduced from a homogenization procedure which associates the structural-scale model to a material-scale model (which embodies the main heterogeneities of masonry) of a representative volume element (RVE) of the structure. Accordingly, structuralscale problems are formulated and solved in a multiscale approach. CMs are discussed in Section 1.6. 3. MMs. Panel-scale structural components (macroelements) with phenomenological or mechanical-based responses are employed to idealize the structure. Basically, two macroelements (i.e., piers and spandrels) can be recognized. The distinction of macroelements in a masonry structure has

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to be first carried out on the basis of the interpretation of the structural arrangement. The main dissimilarity of MMs with respect to the models in (2) is that the constitutive relationship of macroelements tries to simulate the structural behavior of panel-scale elements (Fig. 1.4B), whereas the constitutive relationship of the models in (2) attempts to simulate the mechanical behavior of the masonry material (Fig. 1.4A). MMs are discussed in Section 1.7. 4. GBMs. A rigid body is employed to model the structure. The only input data needed in these modeling approaches is represented by the geometry of the structure. These approaches typically employ either lower-bound or upper-bound limit analysis-based solutions (see Section 1.3.2). No block-by-block description of masonry is conceived in this class, given that block-based approaches are included in class (1). The GBMs are discussed in Section 1.8. Each class of numerical strategies is thoroughly reviewed in the following. The pros and cons of each approach are also critically examined.

1.5 Block-based models BBMs attempt to interpret the response of the material at the scale of the principal heterogeneity of masonry, which is comprised of blocks assembled with mortar joints. These models can, indeed, account for the actual texture of masonry, which governs the failure mode of the material and its principal overall mechanical properties (e.g., anisotropy). The pioneering work developed by Page (1978) is probably the first example of a nonlinear BBM. In Page (1978), masonry is conceived as an assembly of elastic block linked together by mortar joints elements that have restricted shear strength depending on the bond strength and the normal stress state. From that work, various BBMs have been conceived and suggested. The principal favorable characteristics of the BBMs class can be summarized as: G

G G G

G

Direct description of the actual masonry texture and of the structural details. Setting of the mechanical properties from small-scale experimental tests. Explicit illustration of the crack pattern. The actual texture of masonry directly accounts for the anisotropy of the material. BBMs could simultaneously account for the in-plane and out-of-plane behaviors of masonry (and their interactions; Dolatshahi and Yekrangnia, 2015).

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PART | I Seismic vulnerability analysis of masonry and historical structures

Differently, the principal nonfavorable characteristics of the BBMs class can be summed up as: G

G

G

The principal issue of these models relies in their massive computational effort (Lourenc¸o, 2002; Roca et al., 2010). This problematic typically restricts the application of these numerical approaches to masonry panels. Only a small number of examples of block-based analysis of full-scale structures can be found in the literature (Minga et al., 2018b; D’Altri et al., 2019b). The real texture of existing masonry structures (especially for monumental buildings) is often barely know. Consequently, the block-by-block description of a masonry structure could be coarsely defined in these cases. The assemblage of the blocks is often time-consuming. Therefore, the utilization of these numerical strategies is generally limited to academic research.

In this section, block-based numerical strategies are categorized into various subclasses depending on the manner the interaction between blocks is conceived: 1. 2. 3. 4. 5.

Interface element-based approaches (see Fig. 1.7) Contact-based approaches (see Fig. 1.8) Textured continuum-based approaches (see Fig. 1.9) Block-based limit analysis approaches (see Fig. 1.10) Extended finite element (FE) approaches (see Fig. 1.11) Each subclass is then thoroughly discussed in the following.

FIGURE 1.7 Examples of block-based models: interface element-based approaches.

FIGURE 1.8 Examples of block-based models: contact-based approaches.

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FIGURE 1.9 Examples of block-based models: textured continuum-based approaches.

FIGURE 1.10 Examples of block-based models: block-based limit analysis approaches.

FIGURE 1.11 Examples of block-based models: extended finite element approaches.

1.5.1

Interface element-based approaches

The study reported in Lotfi and Shing (1994) represents one of the first interface element-based approaches to analyze the mechanical response of masonry structures. In Lotfi and Shing (1994), the mortar layers were conceived as zero-thickness interface elements and the blocks were modeled with a smeared crack constitutive law. Other pioneering applications of interface element-based approaches to masonry structures were shown in Rots (1991, 1997). In particular, a procedure to expand the blocks to use zerothickness interface elements for mortar layer was also developed (Fig. 1.12). Additionally, potential cracks within the blocks were also conceived in Rots (1991, 1997). A significant enhancement in the interface element-based approaches was proposed by Lourenc¸o and Rots (1997) through the development of a multisurface interface model where all the nonlinearities were conceived in the zero-thickness interfaces, in the context of softening plasticity. This

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PART | I Seismic vulnerability analysis of masonry and historical structures FIGURE 1.12 Example of a pioneering interfacebased model (Rots, 1991).

enhanced the effectiveness of the model (Lourenc¸o and Rots, 1997), which has been extensively utilized for masonry structures (Sandoval and Arnau, 2016; Caldero´n et al., 2017). This interface model has also been utilized for irregular stone masonry panels (Senthivel and Lourenc¸o, 2009). Additionally, an upgrade of the interface model (Lourenc¸o and Rots, 1997) (fully-based on the plasticity theory) was developed in Oliveira and Lourenc¸o (2004) for the simulation of the cyclic response of masonry panels. A damage mechanics-based cyclic interface model was developed by Gambarotta and Lagomarsino (1997b). Two internal variables representing the frictional sliding and the mortar joint damage were employed in Gambarotta and Lagomarsino (1997b) to formulate the constitutive equations of the interface, which shows a tensile brittle behavior and frictional dissipation together with stiffness degradation in shear (Fig. 1.7). Other cohesive interface models with damage and friction were developed in Formica et al. (2002), Alfano and Sacco (2006), and Parrinello et al. (2009) for the analysis of masonry panels. Various approaches have been developed with the hypothesis of rigid blocks interacting using nonlinear springs. For example, Malomo et al. (2018) developed a model for masonry within the so-called applied element method. Despite akin, in theory, to the rigid body spring model (RBSM) proposed by Casolo (2000) (where the actual texture of masonry is not taken into account), in (Malomo et al., 2018) the analysis of the in-plane cyclic behavior of masonry walls was pursued block-by-block. Most of the numerical strategies discussed so far are aimed at the simulation of 2D structures, limiting the use of these numerical strategies to real structures. Various 3D models have been developed (Baraldi and Cecchi, 2016, 2017; Ordun˜a, 2017) to overcome this issue and thus to deal with

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actual buildings. Two principal interface elements have been proposed in particular for 3D masonry structures. On the one hand, an upgrade of the multisurface interface model developed in Lourenc¸o and Rots (1997) to 3D problems, accounting for geometrical nonlinearity as well, was proposed by Macorini and Izzuddin (2011) through a corotational approach that allows to formulate the geometric nonlinearity at a discrete items level (Fig. 1.7). This modeling approach has been widely used for applications on masonry structures (Chisari et al., 2015, 2018) by also utilizing partitioning strategies (Zhang et al., 2016; Minga et al., 2018a) to optimize the computational cost. Furthermore, this model was further extended to analyze the cyclic behavior of masonry structures (Minga et al., 2018b) through a damage-plasticity framework. On the other hand, a further interface model was proposed in Aref and Dolatshahi (2013) and coupled with a elastoplastic block behavior for the cyclic analysis of 3D masonry walls through explicit solvers. This interface model was widely utilized for the investigation of various features of the mechanical behavior of masonry panels (Dolatshahi and Yekrangnia, 2015; Dolatshahi and Aref, 2016; Wilding et al., 2017; Dolatshahi et al., 2018).

1.5.2

Contact-based approaches

Various examples of BBMs for masonry structures are based on contact mechanics. Frictional and/or cohesive-frictional contact laws are typically used to formulate the interaction between blocks that can be rigid or deformable (linear or nonlinear). Even though various in-house developments have been proposed and validated (e.g., Kuang and Yuen, 2013; Miglietta et al., 2017), three principal categories of contact-based strategies could be identified. First, a broad category of contact-based numerical strategies is based on the distinct element method (DEM), also called the discrete element method (Sarhosis et al., 2016), proposed by Cundall and Strack (1979) in the UDEC code (Cundall, 1980) for the simulation of granular arrangements. DEM strategies are typically based on the contact penalty approach and explicit integration schemes. This formulation allowed several applications on fullscale masonry structures (Papantonopoulos et al., 2002; Lemos, 2007; To´th et al., 2009; Sarhosis and Sheng, 2014; C ¸ akti et al., 2016; Simon and Bagi, 2016; Bui et al., 2017; Forga´cs et al., 2017; Lengyel, 2017; Foti et al., 2018) using elastic or rigid blocks (Fig. 1.8). Secondly, the so-called discontinuous deformation analysis (Shi, 1992) accounts for the deformability of blocks through an implicit integration. No tensile stress between blocks is supposed and penetration of one block into another cannot occur. A Coloumb’s frictional law is supposed at all the contact points (Thavalingam et al., 2001).

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PART | I Seismic vulnerability analysis of masonry and historical structures

Thirdly, a further category of models is based on the nonsmooth contact dynamics (NSCDs) method, developed by Jean (1999) and Moreau (1988) through a straight nonsmooth contact implicit formulation and energy dissipation when impacts between blocks occur. This method has been favorably utilized in various full-scale cases (Rafiee et al., 2008; Rafiee and Vinches, 2013; Lancioni et al., 2016; Beatini et al., 2017), even though it appears limited to dry stone masonry structures, since no cohesive response of the mortar layers has been conceived so far. Even though the contact-based models described so far appear efficient and allow full-scale applications, masonry crushing cannot be suitably accounted for in these models. Since masonry crushing is often a critical aspect in the mechanical response of masonry structures, further recent strategies have been proposed to account also for this aspect. For example, Sarhosis and Lemos (2018) conceived masonry crushing through an arrangement of thickly packed discrete deformable particles linked together by zerothickness contact interfaces for both masonry units and mortar joints. Additionally, in the context of the finite-discrete element method (Munjiza, 2004), a code for the computational analysis of dry stone masonry structures was developed in Smoljanovi´c et al. (2013, 2018). The nonlinear response of blocks simulating masonry crushing was accounted for in Smoljanovi´c et al. (2015). Finally, a damaging 3D BBM with contacting blocks was very recently developed and validated in D’Altri et al. (2018b), where the mortar joints were explicitly modeled in the block 3D discretization [“detailed” according to the definition in Lourenc¸o (2002)]. The model in D’Altri et al. (2018b) uses compressive and tensile damage for the blocks, implicit integration schemes, contact penalty method, and a rigid-cohesive-frictional contact response, and were particularly accurate for the simulation of the in-plane and out-of-plane behavior of masonry panels. Furthermore, an extension of the model developed (D’Altri et al., 2018b) to the cyclic behavior of fullscale masonry structures was proposed in D’Altri et al. (2019b).

1.5.3

Textured continuum-based approaches

The principal conception of textured continuum-based models (Page, 1978) consists of having blocks and mortar layers modeled separately through nonlinear FEs without any interface between them. An illustration of an early mesh discretization of this kind of strategy is depicted in Fig. 1.9 (Ali and Page, 1988), where the elements with block properties are separated from the ones with mortar (or more precisely mortar joint) properties. In particular, a smeared crack approach to account for cracking in blocks and mortar layer was utilized. A textured continuum-based model that meshes both blocks and mortar layers with continuum elements, using a damage model in tension and

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compression, was recently proposed in Petracca et al. (2017b). In particular, the damage model conceived in Petracca et al. (2017b) was enriched to finely reproduce the nonlinear shear behavior of masonry and to control the dilatancy. A further strategy was developed in Addessi and Sacco (2016), where an enriched kinematic damage model was used to model the blocks and the mortar layers. A truly original approach to simulate the nonlinear mechanical behavior of mortar layers was recently developed in Serpieri et al. (2017), where an interphase formulation was proposed on the basis of a multiplane cohesivezone model. According to Serpieri et al. (2017), a multiscale numerical strategy was adopted for the constitutive relationship of mortar joints. This permitted the conduction of a rational and reproducible calibration of the mortar joint properties.

1.5.4

Block-based limit analysis approaches

Block-based limit analysis approaches are trustworthy solutions for the estimation of the critical load and collapse mechanism of masonry structures. Various 2D and 3D strategies based on static or kinematic theorems have been proposed (Fig. 1.10), even though considering friction in the calculations is typically nonconservative within the theorems of limit analysis. A first example of block-based limit analysis approaches for masonry structures is likely the model developed in Baggio and Trovalusci (1998), where the limit analysis problem is formulated in presence of friction at interfaces between rigid blocks (Baggio and Trovalusci, 2000). A further block-based limit analysis approach was proposed in Ferris and Tin-Loi (2001), where the collapse loads of discrete rigid block systems with nonassociative friction and tensionless contact interfaces have been solved through a mathematical program with equilibrium constraints. Additionally, a technique for the calculation of the lower-bound limit load in unreinforced masonry shear walls under conditions of plane strain was developed by Sutcliffe et al. (2001). Later, a procedure for the nonassociated limit analysis of rigid block masonry structures, accounting for nonassociated flow rules and a coupled yield surface, was implemented by Ordun˜a and Lourenc¸o (2005a,b). Furthermore, a linear programming formulation for limit analysis of rigid block assemblages with nonassociative frictional joints was developed in Gilbert et al. (2006), upgraded in Portioli et al. (2013) for 3D structures and torsional effects, and optimized in Portioli et al. (2014) through cone programming. In contrast, the numerical method developed and proposed by Milani (2008) is based on 3D FE upper-bound limit analysis of in- and out-of-plane loaded masonry panels. Particularly, the model in Milani (2008) conceives interfaces with a Mohr Coulomb failure criterion with tension cutoff and cap in compression for mortar joints. Accordingly, mortar joint cohesion and

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PART | I Seismic vulnerability analysis of masonry and historical structures

masonry crushing are also considered in this numerical approach. Further interesting applications of this model can be found in Milani et al. (2007a, 2009), while the applications within homogenization procedures will be discussed in the following section. Even though these numerical strategies have also been used to analyze actual structures, such as masonry bridges in Cavicchi and Gambarotta (2006), their computational effort seems rather expensive.

1.5.5

Extended finite element approaches

Few BBMs (Abdulla et al., 2017; Zhai et al., 2017) have recently been formulated in the context of the extended finite element method (XFEM); see Fig. 1.11. For example, a 3D model with surface-based cohesive response to account for the linear and nonlinear behavior of masonry joints and a Drucker Prager plasticity model to capture crushing of masonry was proposed by Abdulla et al. (2017) (Fig. 1.11). Moreover, the nonlinear response of masonry in infill panels was simulated through XFEM in Zhai et al. (2017), where potential cracks were employed to account for mortar layers and the joints between the frame and the infill (Fig. 1.11). These numerical approaches, even though only two solutions have been proposed so far, could represent robust options for the accurate analysis of masonry buildings.

1.6 Continuum models A continuum deformable body is utilized to simulate masonry in CMs. In this class of models, the mesh does not need to represent the masonry blocks, and, accordingly, the mesh size could be considerably greater than the block size. Consequently, the computational demand of these numerical strategies should be lower than block-based strategies. Nevertheless, the formulation of adequate homogeneous constitutive laws for masonry is a challenging task given the mechanical features of masonry (Section 1.2). Constitutive laws for masonry could be described either using (1) direct approaches (Fig. 1.13), that is, through constitutive laws set on the basis of experimental tests, or using (2) homogenization procedures and multiscale approaches (Fig. 1.14), where the material constitutive law (conceived homogeneous at the structural-scale model) is deduced from an homogenization procedure which links the structural-scale model to a material-scale model (that embodies the main heterogeneities of masonry). Homogenization procedures are generally based on accurate modeling strategies (e.g., BBM) of a RVE.

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FIGURE 1.13 Examples of continuum models: direct approaches.

FIGURE 1.14 Examples of continuum models: homogenization procedures and multiscale approaches.

1.6.1

Direct approaches

In direct CMs, the overall masonry mechanical behavior is, in some way, approximated by continuum constitutive laws. Accordingly, the mechanical properties of these laws could be set using experimental tests or other data (e.g., analytical strength domains derived from experiments), without using, thereby, homogenization processes based on RVEs. Various approaches have been proposed and successfully applied on real case studies. One category of direct approaches adopts a radical hypothesis on the mechanical behavior of masonry, that is, the perfectly no-tension material hypothesis. An isotropic continuum incapable of bearing tensile stresses and linear-elastic elsewhere (Del Piero, 1989) is the general definition of a perfectly no-tension material. The characterization of the mechanical behavior

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PART | I Seismic vulnerability analysis of masonry and historical structures

of masonry in tension is, indeed, very challenging, and this drastic supposition can be seen as a starting point for preliminary analyses (Maier and Nappi, 1990). A piecewise-linear approximate definition of a perfectly no-tension material was proposed in Maier and Nappi (1990). Then, Angelillo (1994) developed a FE approach that uses the complementary energy theorem for no-tension elastic structures. The solution relies on a problem of minimization of a quadratic function with equality and inequality constraints. Starting from an elementary stress field, an optimal approximate solution (safe in the spirit of limit analysis) is reached. Further approaches of the FE simulation of no-tension bodies were proposed in Alfano et al. (2000), Cuomo and Ventura (2000), and Lucchesi et al. (2000). Lately, Bruggi (2014) developed a framework for the FE analysis of no-tension structures by solving a topology optimization problem. Also, Bruggi and Taliercio (2015) developed a energy-based solution to achieve a compression-only state of stress, describing the orientation of an equivalent orthotropic material, by minimizing the potential energy. The application of no-tension approaches to real case studies appears, however, nontrivial. For example, only very recently have no-tension 3D structures been investigated (Bruggi and Taliercio, 2018). It has to be pointed out that these no-tension solutions cannot account for the postpeak response of masonry structures. This aspect consistently limits their application within the framework of seismic assessment of structures. Further direct continuum approaches base their nonlinear constitutive laws on theories of fracture or damage mechanics, and/or plasticity. Various smeared crack (fracture mechanics) (Hillerborg et al., 1976; Rots and De Borst, 1987), plastic (Dragon and Mroz, 1979), damage (Løland, 1980), and plastic-damage (Lubliner et al., 1989; Lee and Fenves, 1998) models have been proposed mainly for the numerical simulation of concrete members. Their use in collapse simulations of masonry structures shows several drawbacks, typically due to the consistent anisotropy of the material. An early assessment of the effectiveness of smeared crack approaches in analyzing masonry is shown in Lotfi and Shing (1991). Although this approach exhibited a trustworthy response in the flexural behavior, some drawbacks emerged in the simulation of the shear behavior of masonry walls. Nevertheless, smeared crack, damage, and plastic-damage models have been widely adopted for analyzing masonry structures (Toti et al., 2015), primarily explained by their performance, their dissemination in commercial software packages, and the rather few mechanical properties needed as input. The use of these models was found to be favorable for the investigation of structural performance historic monumental masonry buildings, especially for the restricted computational demand of these models and their ease in representing complex geometries. Monumental buildings usually show thick and irregular masonries which are very complex to characterize

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mechanically, also due to the stringent restrictions for destructive tests on this kind of buildings (D’Altri et al., 2018a). Inadequate information is, indeed, normally accessible on the mechanical features of historic masonries. These aspects promoted the utilization of isotropic nonlinear models in monumental masonry buildings. Numerous studies that used isotropic smeared crack, damage, and plastic-damage models have been favorably carried out on historic masonry towers (Bartoli et al., 2016; Valente and Milani, 2016; Castellazzi et al., 2018), churches and temples (Betti and Vignoli, 2011; Milani and Valente, 2015; Fortunato et al., 2017; Elyamani et al., 2017), palaces (Betti and Galano, 2012; Tiberti et al., 2016; Castellazzi et al., 2017; Degli Abbati et al., 2019), and bridges (Pela` et al., 2009; Zampieri et al., 2015). In particular, these studies on historic monumental structures are based on 3D models (Fig. 1.15) due to the complexity of the building geometries (Section 1.2). It has to be pointed out that damage models need to conceive a regularization of the fracture energy in order to be consistent and reliable. This is typically pursued through normalization of the fracture energy based on a characteristic length of the element. Nevertheless, a very coarse discretization can result in incorrect outcomes due to the lacks in representing the crack pattern and the stress distribution. To overcome this issue, crack-tracking solutions that guarantee the mesh-bias independence of the numerical results and the reliable propagation of cracks can be used in the simulation of cracks in quasibrittle materials (Saloustros et al., 2015, 2018). Nonetheless, the adoption of, for example, only one value of tensile strength could be too simplistic in periodic masonry. Various orthotropic constitutive laws have been proposed for masonry structures (Rots et al., 2016) to overcome this issue. In Loure´nc¸o et al. (1997), an early example of an orthotropic softening plasticity model was developed, and the capability of the model in representing the mechanical behavior of different masonry structures was shown in

FIGURE 1.15 Examples of direct continuum isotropic approaches applied on historic monumental structures.

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PART | I Seismic vulnerability analysis of masonry and historical structures

Lourenc¸o et al. (1998). Anisotropy was accounted for in Lopez et al. (1999) through a fictitious isotropic space, where the mechanical properties are linked to the real anisotropic space through a fourth-order tensor. Later, Berto et al. (2002) developed an orthotropic damage model especially devoted to the cyclic analysis of in-plane loaded masonry by using distinct linear and nonlinear properties along the two main axes of masonry (head joints and bed joints directions). Lately, an orthotropic damage model for masonry walls, in which the orthotropic response is accounted for using mapped tensors, was developed by Pel´a et al. (2011, 2013). Unilateral effects are also accounted for in Pel´a et al. (2011, 2013) thanks to the stress tensor separation into tensile and compressive parts. This damage model was also coupled with a crack-tracking algorithm in Pel´a et al. (2014). Even though the direct continuum anisotropic models mentioned so far are scientifically worthy, their use on real applications still appears nontrivial due to their high computational demand and the large number of mechanical properties needed to set the model up. Further approaches, even though homogeneous FE models of the structure are conceived, use other solutions to formulate the nonlinear response of masonry, rather than a proper continuum. For instance, a numerical approach for the cracking analysis of masonry, accounting for the anisotropy of the material, was proposed by Reyes et al. (2009) on the basis of the strong discontinuity approach. Other solutions based on FE limit analysis, idealize the structure as an assembly of rigid or deformable elements, connected by nonlinear interfaces where plastic dissipation can take place. FE limit analysis solutions have been efficaciously utilized on historic monumental masonry buildings (Milani et al., 2012; D’Altri et al., 2018c) by adopting averaged material properties, and, so, without resorting to rigorous homogenization procedures. Finally, further solutions based on arrangements of springs (Panto` et al., 2016, 2018) can be entirely set using a proper choice of the linear and nonlinear properties of the springs. FE limit analysis and spring-based solutions could be conceived borderline approaches in the framework of CMs (since they contemplate interfaces or springs between elements rather than of a proper continuum). Nevertheless, their categorization in this class can be believed valid, since they eventually act like a continuum (with nonlinearities concentrated in the springs/interfaces without a block-by-block definition).

1.6.2

Homogenization procedures and multiscale approaches

The constitutive relationship of the structural-scale model could be supposed from RVE-based homogenization procedures. The conception of a suitable RVE is fundamental, given that it should statistically represent the

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Taliercio (2014)

Stefanou et al. (2015)

Anthoine (1995)

Cavalagli et al. (2011)

Milani (2011)

FIGURE 1.16 Examples of RVEs for the homogenization of masonry.

FIGURE 1.17 Example of an a priori homogenization procedure (Bertolesi et al., 2018).

heterogeneity of the masonry under study. Various arrangements of RVEs have been suggested to represent periodic and nonperiodic masonry textures (Fig. 1.16). An extensive category of CMs adopt homogenization procedures and multiscale approaches (Sacco et al., 2018) for the derivation of the constitutive law of masonry. Three principal categories of models can be identified as follows: 1. A priori homogenization approaches (Fig. 1.17), which are generally based on two steps: RVE-based a priori homogenization is used in the first step to define the properties of the material at the structural scale,

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PART | I Seismic vulnerability analysis of masonry and historical structures

while the utilization of the homogenized properties in the structural-scale model is conducted in the second step. 2. Step-by-step multiscale approaches (Fig. 1.18), in which the global structural response is computed step-by-step through the solution, for each integration point of the structural model, of a boundary value problem on the RVE. Accordingly, a step-by-step evaluation of the approximated behavior is calculated in the RVE for the definition of the constitutive relations in the structural-scale model. Given that the masonry heterogeneity is effectively accounted for in the RVE, the structural-scale model does not need to include the material substructure. 3. Adaptive multiscale approaches (Fig. 1.19), where robust coupling between the structural and the material scales is conceived by adaptively inserting the material-scale model into the structural-scale one.

1.6.2.1 A priori homogenization approaches Two steps generally characterize a priori homogenization approaches: the mechanical properties are deduced using an homogenization procedure in the first step, and, then, the homogenized properties are utilized in the structural-

FIGURE 1.18 Example of a step-by-step multiscale procedure (Petracca et al., 2016).

FIGURE 1.19 Example of an adaptive multiscale procedure (Petracca et al., 2016).

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scale model in the second step. Nevertheless, most of the approaches developed in recent decades only conceived the first step, while few solutions accounted for the second step. The definition of homogenized constitutive relationships for heterogeneous materials such as masonry could rely on analytical (closed-form), quasianalytical, and numerical approaches. An early solution for the analytical definition of the overall response of masonry was proposed in Pietruszczak and Niu (1992). Later, Anthoine (1995) obtained the masonry in-plane elastic features using a rigorous homogenization approach. Also, Briccoli Bati et al. (1999) utilized a blockbased approach for the definition of the macroscopic elastic response of a common masonry texture. Masiani and Trovalusci (1996) analyzed 2D periodic rigid block arrangements linked with linear mortar layers in the context of the Cosserat CMs. Accordingly, they obtained the structural description of the equivalent continuum by a power balance. An upgrade to the 3D case was presented in Stefanou et al. (2008). Other solutions for the description of masonry homogenized elastic properties were proposed in Cecchi and Sab (2002, 2007), Cecchi et al. (2005), Mistler et al. (2007), Taliercio (2014), and Drougkas et al. (2015). Further solutions have attempted to also obtain the masonry strength domains (Kawa et al., 2008), beyond the derivation of the elastic properties. For instance, an overall strength criterion for the in-plane behavior of masonry was obtained in De Buhan and De Felice (1997) using a CM. Zucchini and Lourenc¸o (2002, 2004) obtained the elastic properties and strength domain using linear and nonlinear homogenization processes. Wei and Hao (2009) proposed a continuum damage model with strain rate effect for the analysis of masonry through a homogenization theory. Stefanou et al. (2015) developed a direct approach for the evaluation of the in-plane structural-scale strength domain in closed form. Although most of the homogenization procedures adopted periodic masonry textures, Cecchi and Sab (2009) studied nonperiodic historic masonries through a perturbation approach, whereas Cavalagli et al. (2011, 2013) utilized a random media material-scale method. Various methods for the definition of masonry homogenized failure surfaces utilized FE limit analysis (Milani et al., 2006a,b; Cecchi et al., 2007; Cecchi and Milani, 2008; Milani, 2011; Godio et al., 2017). For instance, a simple model for the homogenized limit analysis of shear masonry walls was developed in Milani et al. (2006a) using a linear optimization problem to define the homogenized failure surfaces of the material. These solutions have the benefit that, once the masonry properties are homogenized, they can be immediately utilized to solve real structures (Milani et al., 2006c, 2007b) in structural-scale models. RBSM solutions (Casolo, 2004; Casolo and Pena, 2007; Silva et al., 2017; Bertolesi et al., 2018) have the same advantage. Indeed, the linear and

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PART | I Seismic vulnerability analysis of masonry and historical structures

nonlinear characteristics of the springs between the rigid elements (not representing the actual texture of masonry) can be homogenized a priori and directly adopted for real applications (Bertolesi et al., 2018).

1.6.2.2 Step-by-step multiscale approaches Many step-by-step multiscale solutions have been developed in recent decades. They categorized based on: G

G

G

Formulation of the continuum utilized in the structural-scale model (Cauchy continuum, Cosserat continuum, etc.) Homogenization procedure type (first- or second-order computational homogenizations, transformation field analysis (TFA), etc.) RVE mechanics (i.e., numerical model utilized for the RVE)

These solutions generally use step-by-step and point-by-point passages between the structural and material-scale models, and vice versa. Typically, multiscale computational homogenization approaches are utilized within the FE method (FE2) and based on FE first-order homogenization approaches. In general, structural-scale models have been formulated through homogenized Cauchy continua. An early numerical homogenization approach was developed by Papa (1996), in which a homogenized damage model for masonry was proposed, and by Luciano and Sacco (1997) and Luciano and Sacco (1998), where a homogenized damage model for periodic masonry was proposed. Also, Gambarotta and Lagomarsino (1997a) conceived an equivalent stratified continuum comprised of mortar layers and blocks, using the damage constitutive relationships developed in Gambarotta and Lagomarsino (1997b). Then, a continuum formulation was proposed in Pietruszczak and Ushaksaraei (2003) for the investigation of the nonlinear response of masonry. Such a formulation considered the material anisotropy and accounted for both steps of the deformation process, that is, those linked to homogeneous and localized deformation modes. Calderini and Lagomarsino (2006) derived homogenized constitutive equations for the in-plane response of masonry in terms of mean-stress and mean-strain, conceiving various damage modes. Successively, Zucchini and Lourenc¸o (2009) developed an enhanced material-scale model for masonry homogenization in the nonlinear regime, accounting for plausible deformation modes coupled with plastic-damage constitutive laws. Sacco (2009) developed a multiscale micromechanical-based approach with the assumption of linear elastic blocks. A nonlinear homogenization procedure based on TFA was also proposed in Sacco (2009). An enhancement of this solution was proposed Marfia and Sacco (2012), where an upgrade of the TFA-based homogenization procedure to the condition of nonuniform eigenstrain was implemented with nonlinear blocks in the material-scale model.

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In first-order computational homogenization procedures, the formulation adopts the first gradient of the kinematics field, and, therefore, two principal drawbacks could appear. The first drawback is connected to the principle of separation of scales, which leads to the hypothesis of regularity in the macroscopic fields linked to each RVE. This hypothesis is nonentirely efficacious in macroscopic portions with high deformation gradients in the relative RVE. The second drawback comes from the softening-cohesive response of masonry. Indeed, mesh-sensitivity problems appear with material softening responses, given that the characteristic lengths of the macroscopic and microscopic models is nondirectly accounted for in classical Cauchy continua. Nonlocal approaches, higher-order CMs, and regularization processes have been utilized to overcome such a drawback and to guarantee problem objectivity. A basic way to avoid localization issues consists of using a regularization process, for instance, in terms of fracture energy. A first-order numerical homogenization was developed in Petracca et al. (2016) together with a regularization strategy based on the fracture energy of the material-scale model. In this scheme, the size of the structural-scale element and the size of the RVE are accounted for through a generalized geometrical characteristic length, which guarantees the objectivity of the dissipated energy. Massart et al. (2007) proposed an improved multiscale approach using nonlocal implicit gradient isotropic damage models for block and mortar layers, defining the damage orientations and using at the structural scale an embedded band model. Bacigalupo and Gambarotta (2010, 2012) proposed a second-order computational homogenization of periodic masonry. This scheme was obtained with the assumption of a suitable definition of the microscopic displacement field as the superposition of a local macroscopic displacement field and an unknown microscopic fluctuation field, which takes into account the effects of the heterogeneities. Further solutions utilized Cosserat continua at the macroscopic level, accounting for the internal length of the material and, accordingly, overcoming localization problems (Addessi et al., 2014a). Salerno and de Felice (2009) evaluated the effectiveness of Cauchy and Cosserat continua, obtaining a particularly good performance of the micropolar continuum in the case of nonperiodic deformation modes, given its possibility to account for scale effects. Conversely, Casolo (2006) assumed isotropic linear elastic strategies for block and mortar layers and adopted a numerical approach to find the homogenized tensor of the Cosserat continuum. Furthermore, Addessi et al. (2010) proposed a macroscopic Cosserat medium, which directly conceives the absolute size of the masonry components through a rational homogenization scheme based on TFA. A further homogenization approach for the Cosserat medium was developed by De Bellis and Addessi (2011). Also, Addessi and Sacco (2012) proposed a nonlinear constitutive relationship for

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PART | I Seismic vulnerability analysis of masonry and historical structures

the microscopic model including damage, friction, crushing, and the contact effect for the mortar layers. As the out-of-plane modeling of masonry is also an stimulating issue, Mercatoris and Massart (2011) developed a multiscale scheme for the collapse analysis of periodic quasibrittle shells, using a shear-improved element with a Reissner Mindlin formulation. Another significant computational homogenization approach was developed in Petracca et al. (2017a). The multilevel approach developed by Brasile et al. (2007a,b) was found to be especially efficient. Even though this strategy could be believed borderline in the multiscale context (being somehow a multilevel approach), the modeling scheme proposed in Brasile et al. (2007a,b) uses an iterative procedure with two different (local and global) masonry models adopted simultaneously. The former is a refined BBM which defines the mechanical behavior accounting for damage evolution and friction. The latter is a FE approximation of the previous model, described at the coarse scale of the wall and used to speed up the solution. The iterative approach proposed in Brasile et al. (2007a,b) was found to be efficient and robust for the in-plane analysis of masonry fac¸ades.

1.6.2.3 Adaptive multiscale approaches In adaptive multiscale methods (Greco et al., 2016; Reccia et al., 2018; Leonetti et al., 2018; Lloberas-Valls et al., 2012) (Fig. 1.19), the masonry behavior until the achievement of a threshold criterion is modeled through a first-order homogenized approach. Once the threshold is reached, the portion of interest is replaced by a material-scale formulation that can deal with localized deformations without the mesh-bias of the first-order theory.

1.7 Macroelement models MMs conceive the structure as an arrangement of panel-scale structural components (macroelements) with a mechanical-based or phenomenological behavior. In general, two principal structural components (piers and spandrels) can be recognized. Piers are identified as the vertical bearing elements (that can carry vertical and horizontal loads). Differently, spandrels are the horizontal portions of the structure between two openings aligned along the height. Accordingly, they couple the behavior of adjacent piers when horizontally loaded. These numerical strategies aim at the analysis of the overall seismic behavior of masonry structures. The hypothesis of activation of no local failure mode (typically linked to out-of-plane failures) is generally made in MMs (Quagliarini et al., 2017). Accordingly, the overall seismic behavior is substantially linked to the in-plane panel response and the load redistribution given by the diaphragms. Numerical analyses (static or dynamic

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incremental-iterative) are generally performed on 3D models that take into account the horizontal action redistribution between the piers. In MMs, piers and spandrels elements have to be recognized a priori on the basis of damage surveys in earthquake-stricken areas. Damage observations after seismic shocks highlight, indeed, that cracks and damages in masonry buildings usually arise in piers and spandrels. The recognition of piers and spandrels within a masonry building (Dolce, 1991; Augenti, 2006; Moon et al., 2006; Lagomarsino et al., 2013, 2018; Parisi and Augenti, 2013; Parisi et al., 2013; Calderoni et al., 2015; Berti et al., 2017) could be rather easy for masonry fac¸ades with a even distribution of openings (Fig. 1.1B). However, the component recognition becomes much more complicated for buildings with irregularly arranged openings. The idealization of the structure into piers and spandrels appears essentially unrealistic in historic buildings with complex geometries (Fig. 1.1A). The most utilized numerical strategies, also used by practitioners, are the MMs. The favorable features of these models (e.g., limited computational demand, easy arrangement of the model discretization and simple definition of the mechanical properties) led to their widespread utilization. Nevertheless, MMs are also characterized by some drawbacks. For instance, MMs typically suppose no local (out-of-plane) failure modes and this aspect could lead to excessively conventional assessments of the seismic performance of masonry buildings, given that out-of-plane damage could influence the in-plane damage and vice versa (Dolatshahi and Yekrangnia, 2015). Also, structural details, such as the toothing between perpendicular panels, cannot be finely accounted for in MMs. Finally, the idealization of the structural system into piers and spandrels elements risk to be too conventional is some case (e.g., for buildings with irregular layouts). Even though the majority of MMs belong to the category of equivalent beam-based (Siano et al., 2018), various spring-based solutions have also been recently proposed. Equivalent beam-based (Fig. 1.20) and spring-based (Fig. 1.21) approaches are discussed in the following.

1.7.1

Equivalent beam-based approaches

The masonry building is modeled as an arrangement of nonlinear beams in equivalent beam-based approaches, also called “equivalent frame models.” An early equivalent beam-based model based on simplified elasto-plastic relationships to describe the beam nonlinearity was developed by Tomaˇzeviˇc (1978). The so-called pushover response (POR) method (Tomaˇzeviˇc, 1978) assumed that in-plane damage for horizontally loaded masonry fac¸ades was only caused by shear forces in the piers, while both spandrels and nodal regions were supposed rigid and infinitely resistant. Later improvements were proposed in Dolce (1991), where the flexibility and the limited strength of masonry spandrels were accounted for.

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PART | I Seismic vulnerability analysis of masonry and historical structures

FIGURE 1.20 Examples of macroelement models: equivalent beam-based approaches.

FIGURE 1.21 Examples of macroelement models: spring-based approaches.

Further, more advanced equivalent beam-based models (Calderoni et al., 1987; Magenes and Fontana, 1998; Kappos et al., 2002; Roca et al., 2005; Penelis, 2006; Pasticier et al., 2008; Belmouden and Lestuzzi, 2009) proposed the conception of the masonry structure as an arrangement of pier and spandrel beam elements, connected by rigid links (Fig. 1.20). These models use a phenomenological nonlinear elastoplastic constitutive relationship for the beam elements. Successively, Grande et al. (2011) developed a masonry pier beam FE comprised of three parts: two rigid offsets to account for the stiff response of pier-lintel intersections, and one flexible central portion. Additionally, special shear interfaces have also been implemented in the model to take into account the shear failure. A further force-based beam FE was developed in

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Addessi et al. (2014b), where the stress components were precisely calculated along the beam axis through analytical integration (without, however, a fiber approach). The developed FE model composed of a central flexible element with a no-tension constitutive law, and a concentrated nonlinear shear hinge. Another beam FE was proposed in Addessi et al. (2015), where both flexural and shear plastic lumped hinges were inserted at the two nodes of the beam, following a classical elastic-plastic constitutive relationship. Moreover, Liberatore and Addessi (2015) developed a force-based FE beam comprised of a central linear elastic element, two flexural hinges, and a shear link with elastic-perfectly plastic response, computed through a predictor corrector approach. Lagomarsino et al. (2013) proposed a 2D inelastic beam element with concentrated plasticity and a bilinear law with cut-off in strength and stiffness degradation in the nonlinear regime. This model was formulated in the Tremuri software (Lagomarsino et al., 2012). The implementation of this nonlinear beam was improved by Cattari and Lagomarsino (2013) using a piecewise-linear response. In particular, this constitutive relationship allowed the representation of the nonlinear behavior until extremely serious damage levels (from 1 up to 5), implementing a progressive strength decay dependent on predetermined values of drift. The model also includes an accurate description of the hysteretic response formulated through a phenomenological approach, to capture the differences among the various possible failure modes (flexural type, shear type, or even hybrid) and the different response of piers and spandrels, which revealed to be efficient in performing nonlinear dynamic analyses (Cattari et al., 2018). Recently, an advanced equivalent beam-based macroelement was developed by Raka et al. (2015) for the nonlinear static and dynamic simulation of masonry structures. The beam mechanical description conceived axial, bending, and shear deformation within the Timoshenko beam theory. Particularly, a phenomenological cyclic law was implemented together with a fiber-based model for the axial and bending responses of the beam.

1.7.2

Spring-based approaches

Various MMs were recently developed by utilizing nonlinear springs (Fig. 1.21) within an equivalent frame to simulate the in-plane nonlinear behavior of masonry walls and fac¸ades. An early development of a springbased MM was presented in Chen et al. (2008), modifying a modeling approach with nonlinear shear and rotational springs originally developed for reinforced concrete members. One axial spring, three shear springs, and two rotational springs were conceived in Chen et al. (2008) to analyze the failure modes observed in experiments on masonry piers. In Gambarotta and Lagomarsino (1996) and Brencich and Lagomarsino (1998), a two-node element able to describe the overall cyclic response of

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PART | I Seismic vulnerability analysis of masonry and historical structures

masonry panels was developed. Accordingly, either shear or axial-flexural responses were computed at the two nodes through a bed of springs and two further internal degrees of freedom. In particular, the shear stress strain cyclic law was obtained through an overall integration of the CM proposed in Gambarotta and Lagomarsino (1997a). Some features of this earliest formulation were enhanced in Penna et al. (2014), where the nonlinear decay due to rocking damage was considered. Accordingly, the effect of limited compressive strength has been taken into account. This model was also included within the Tremuri software (Lagomarsino et al., 2012). Calio` et al. (2012) developed a spring-based MM to simulate the in-plane nonlinear response of masonry walls, where piers and spandrels are conceived as equivalent arrangements of nonlinear springs. The primary panel element is described by a quadrilateral element comprised of four rigid edges linked with four hinges and two diagonal nonlinear springs. Each side of the element interacts with adjacent elements through a bed of nonlinear springs. In Calio` et al. (2012) and Calio` and Panto` (2014), the spring-based model was adopted to directly describe piers and spandrels using primary elements. Additionally, it was also utilized in Panto` et al. (2016, 2018) and Cha´cara et al. (2018) to analyze the masonry material behavior (Section 1.6.1). A further spring-based solution was developed in Rinaldin et al. (2016), where piers and spandrels have been modeled using multispring nonlinear elements coupled by rigid links. Nonlinear springs were located at the two ends of the elements to define the flexural response and in the center to describe the shear behavior. Mobarake et al. (2017) developed another primary panel element comprised of six subelements. Each pier, spandrel and node was conceived through a primary panel element. The model proposed in Mobarake et al. (2017) appeared rather efficient for nonlinear static and dynamic simulations of in-plane loaded masonry fac¸ades. Very recently, a simplified approach was developed by Xu et al. (2018), where each masonry fac¸ade was conceived as an integral unit, rather than an arrangement of piers and spandrels. Accordingly, the mechanics of the masonry fac¸ade is described through two vertical springs and a horizontal one that governs the wall shear response.

1.8 Geometry-based models The masonry structure is conceived as a rigid body in GBMs. The geometry of the construction is substantially the only input needed by these numerical strategies, as well as the loading condition. In general, geometry-based approaches aim at the investigation of the structural equilibrium and/or collapse through static theorem (Fig. 1.22) or kinematic theorem (Fig. 1.23) limit analysis-based solutions. Various novel solutions have been formulated

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Marmo and Rosati (2017)

Angelillo (2015)

Block et al. (2006)

Fraternali (2010)

Block and Lachauer (2014a,b)

O’Dwyer (1999) Block and Ochsendorf (2007)

FIGURE 1.22 Examples of geometry-based models: static theorem-based approaches. Chiozzi et al. (2018*)

Chiozzi et al. (2018**)

Chiozzi et al. (2017)

FIGURE 1.23 Examples of geometry-based models: kinematic theorem-based approaches. The symbol (T) refers to Chiozzi et al. (2018b), while the symbol (TT) refers to Chiozzi et al. (2018a).

in the framework of GBMs, typically following the Heyman’s rigid notension hypothesis (Heyman, 1966).

1.8.1

Static theorem-based approaches

The static theorem of limit analysis was employable on actual masonry structures even through simple graphic statics (Heyman, 1966; Huerta, 2001). Static theorem-based solutions (Fig. 1.22) appear particularly appealing for the equilibrium analysis in masonry arches, vaults, and domes. Typically, these strategies could highlight the envelope of admissible equilibrium conditions of vaulted structure, limited within two critical equilibrium states. A first numerical advance in the equilibrium analysis of masonry vaults was developed by O’Dwyer (1999). Once the vault is decomposed into a system of equilibrated arches, the static theorem is utilized for vaults and domes. A further numerical approach (the so-called funicular model) was proposed in Andreu et al. (2007) for the evaluation of the equilibrium of masonry structures. Such an approach was based on the well-known relationship between the equilibrium of arches and the equilibrium of hanging

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PART | I Seismic vulnerability analysis of masonry and historical structures

strings. Moreover, a numerical tool for the real-time limit analysis of vaulted masonry constructions was developed by Block et al. (2006). The so-called thrust network analysis (TNA) approach developed by Block and Ochsendorf (2007) for the equilibrium analysis of vaulted masonry structures represented an interesting advancement in this field. Such a modeling strategy is based on the duality between geometry and in-plane forces in networks, and plausible funicular solutions under gravitational loading within a defined envelope are investigated. Accordingly, compressiononly vaulted surfaces and networks are generated and the range of admissible equilibrium conditions of the vault, limited within a minimum and maximum thrust, could be found. A nonlinear extension of TNA for the application to Gothic masonry vaults was proposed in Block and Lachauer (2014a), whereas TNA was upgraded in Block and Lachauer (2014b) through structural matrix analysis and optimization schemes. Furthermore, an extension of TNA was developed in Fantin and Ciblac (2016), where joints consideration have been implemented. A further thrust network approach was proposed by Fraternali (2010), where the equilibrium of masonry vaults was analyzed using polyhedral stress functions. A no-tension membrane is adopted to idealize the masonry vault. A predictor corrector algorithm is utilized to compute the geometry of the thrust surface and the associated stress field, on the basis of polyhedral approximations of the thrust surface and membrane stress potential. A further approach proposed by Angelillo et al. (2013) and by Angelillo (2015) conceives masonry vaulted structures as unilateral membranes. The discrete network of singular stresses is calculated on the basis of the Airy’s stress formulation (Fraddosio et al., 2019). A reformulation of the TNA was developed by Marmo and Rosati (2017) by abandoning the dual grid and by focusing only on the primal grid, with an improvement of the computational performances. In Marmo and Rosati (2017), horizontal forces in the analyses were also accounted for. This numerical approach was also used to study masonry helical staircases (Marmo et al., 2018). In conclusion, static theorem-based solutions appeared especially indicated for the safety static assessment of masonry vaulted structures. Indeed, the vault will stand if compression-only networks can be found within the boundaries of a vault. Furthermore, any tension (and therefore any hinges) will be present in the section if the solution lie within the middle third of the section. This easy concept for studying the stability and proximity to collapse of vaulted structures was formerly proposed by Heyman (1966). Nevertheless, horizontal actions (such as seismic actions; Marmo and Rosati, 2017) are accounted for only in few solutions, and no one can take into account the interaction with the bearing structures (e.g., bearing walls), whose deformations could produce damage and equilibrium modification in the dome, as highlighted in D’Altri et al. (2017) for earthquakes.

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1.8.2

37

Kinematic theorem-based approaches

Kinematic theorem-based limit analysis solutions have been extensively used in recent decades for the rapid assessment of existing masonry buildings. Giuffre´ (1991) developed a kinematic limit analysis approach based on the decomposition of masonry buildings into rigid blocks, following the collapse mechanisms actually observed in earthquake-stricken areas. Due to the simplicity of the solution developed by Giuffre´, the Italian code (Ordinanza del Presidente, 2005; NTC, 2008; Circolare, 2009; Direttiva del Presidente, 2011) adopted such a procedure for standard verifications. For example, Fig. 1.24 depicts few examples of failure mechanisms to be accounted for in the seismic evaluations of masonry churches using kinematic limit analysis (Direttiva del Presidente, 2011). Kinematic linear and nonlinear (where also the displacement capacity of the structure until collapse is computed) analyses are typically utilized in the professional practice for the safety evaluation of existing masonry buildings (Circolare, 2009). Generally, the collapse mechanisms to be assessed have to be determined a priori, that is, on the basis of recurring collapse mechanisms observed in reality. Nonetheless, the collapse multiplier computed in this way is not necessarily the lowest one and, accordingly, the safety assessment could not be totally conservative. In this regard, advanced computational kinematic theorem-based strategies have been proposed to finely assess the more plausible collapse mechanism of masonry structures (Fig. 1.23). Milani (2015) proposed a discontinuous upper-bound limit analysis tool with sequential linear programming and mesh adaptation to investigate the actual collapse mechanisms of double curvature masonry structures. Chiozzi et al. (2017) recently developed a genetic algorithm-based tool for the upper-bound limit analysis of masonry vaults. Using a nonuniform rational B-spline (NURBS) parametric surface to represent the vault geometry and a NURBS mesh of the given surface, each element of the mesh is conceived as a rigid body. The initial mesh is iteratively modified through a genetic algorithm to investigate the actual failure mechanism. This solution was also validated in Chiozzi et al. (2018a)

FIGURE 1.24 Examples of collapse mechanisms to be accounted for in the seismic assessment of masonry churches through kinematic limit analysis (Direttiva del Presidente, 2011).

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PART | I Seismic vulnerability analysis of masonry and historical structures

for the out-of-plane collapse analysis of masonry walls. Finally, an upperbound adaptive limit analysis tool (the so-called UB-ALMANAC) was developed in Chiozzi et al. (2018c) for masonry churches. A NURBS mesh, prepared within a CAD environment on the basis of the geometry of the whole church, is used to perform limit analysis under the desired horizontal load distribution. Accordingly, a quick evaluation of the first activating collapse mechanism is pursued and the most vulnerable part of the church is identified. Even though these strategies cannot supply the displacement capacity of masonry structures, they are powerful tools for the rapid assessment of the main vulnerable parts of a masonry building.

1.9 Conclusions In this chapter, a thorough review of numerical strategies for masonry structures was presented and a classification of these approaches was suggested to logically organize the extensive literature on this topic. Even though a wholly congruent categorization of all the numerical tools was essentially unrealistic given the specific aspects of each solution developed, the existing numerical strategies have been subdivided into four classes: BBMs, CMs, GBMs, and MMs. From the thorough review of numerical strategies for masonry structures conducted in this chapter, the following conclusions can be drawn: G

G

BBMs appear to be the most accurate approaches to investigate the mechanical behavior of masonry structures. Various solutions highlighted the capacities of BBM to simulate the structural response of large-scale structures (in particular for contact-based strategies). Nevertheless, their large computational effort generally addresses their use on magnificent applications only. However, BBM can be used to obtain in-depth knowledge on specific characteristics of the mechanics of masonry constructions, and to supply reference outcomes for more simplified strategies. CMs are extensively utilized for the numerical simulation of masonry buildings. Regarding direct approaches, isotropic plastic damage and smeared crack constitutive relationships have been extensively utilized for the numerical assessment of historic monumental structures. These approaches, indeed, are often the only convenient approach with this kind of structure. Nevertheless, the numerical outcomes obtained with this kind of model should be scrupulously interpreted. No-tension continuum approaches appear, in general, too simplified for the accurate structural assessment of masonry buildings. Further strategies, such as homogenized FE limit analysis and homogenized discrete models, seem to be

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G

G

39

especially recommended for the numerical analysis of full-scale masonry structures. Even though elegant approaches have been developed in the context of multiscale strategies, they have some drawbacks. For example, the FE2 approaches eventually appeared considerably computational time-consuming. Even though multiscale strategies should be, in theory, more efficient than BBM, their optimization is typically limited given that these models are generally implemented in self-developed codes. For instance, no example of 3D computational homogenization approach can be found in the literature so far, since these strategies are typically limited to 2D case studies. Moreover, the chance of finely represent specific structural details through multiscale strategies, that is, based on the mechanical behavior of a periodic RVE, seems rather limited. MMs substantially appear to be the only numerical strategy for seismic assessments of masonry structures controllable by practitioners. However, further enhancements should be carried out to specifically take into account structural details (e.g., toothing between orthogonal walls), the interaction between out-of-plane and in-plane damage, and specific spandrel macroelements (as the calibration of the models is typically based on piers tests). It has to be noted also that the area of application MM is restricted to the seismic assessment of ordinary masonry structures. GBMs could be very useful for intuitive considerations on historic vaulted and dome structures. Lower bound-based limit analysis computational strategies are generally effective solutions for the analysis of the equilibrium conditions and the safety of masonry vaults. Conversely, upper bound-based limit analysis computational strategies could investigate the collapse mechanism and the collapse multiplier in geometrically complex masonry structures. The results obtainable with these GBMs, even though noncomprehensive and nonusable in displacement-based seismic assessment methods, can provide fundamental information in the structural analysis of masonry arrangements.

As shown in this review, considerable advancements have been carried out in the framework of numerical modeling of masonry structures. In particular, the scientific literature in this field appears decisively broad and currently active. To summarize, each numerical strategy shows a specific context of usability and particular restrictions. Accordingly, the selection of the most appropriate numerical approach should be based on the structural characteristics and on the complexity of the case study, the data accessible, the output required, and the expertise level. On a final note, it should be noted that 3D models should be preferred for the structural assessment of masonry buildings in order to take into account the structural details and the geometric irregularities that generally distinguish ordinary and monumental masonry buildings.

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Baraldi, D., Cecchi, A., 2016. Discrete approaches for the nonlinear analysis of in plane loaded masonry walls: molecular dynamic and static algorithm solutions. Eur. J. Mech.-A/Solids 57, 165 177. Baraldi, D., Cecchi, A., 2017. A full 3D rigid block model for the collapse behaviour of masonry walls. Eur. J. Mech.-A/Solids 64, 11 28. Bartoli, G., Betti, M., Vignoli, A., 2016. A numerical study on seismic risk assessment of historic masonry towers: a case study in San Gimignano. Bull. Earthq. Eng. 14 (6), 1475 1518. Beatini, V., Royer-Carfagni, G., Tasora, A., 2017. A regularized non-smooth contact dynamics approach for architectural masonry structures. Comput. Struct. 187, 88 100. Belmouden, Y., Lestuzzi, P., 2009. An equivalent frame model for seismic analysis of masonry and reinforced concrete buildings. Constr. Build. Mater. 23 (1), 40 53. Berti, M., Salvatori, L., Orlando, M., Spinelli, P., 2017. Unreinforced masonry walls with irregular opening layouts: reliability of equivalent-frame modelling for seismic vulnerability assessment. Bull. Earthq. Eng. 15 (3), 1213 1239. Berto, L., Saetta, A., Scotta, R., Vitaliani, R., 2002. An orthotropic damage model for masonry structures. Int. J. Numer. Methods Eng. 55 (2), 127 157. Bertolesi, E., Milani, G., Casolo, S., 2018. Homogenization towards a mechanistic rigid body and spring model (HRBSM) for the non-linear dynamic analysis of 3D masonry structures. Meccanica 53 (7), 1819 1855. Betti, M., Galano, L., 2012. Seismic analysis of historic masonry buildings: the vicarious palace in Pescia (Italy). Buildings 2 (2), 63 82. Betti, M., Vignoli, A., 2011. Numerical assessment of the static and seismic behaviour of the basilica of Santa Maria all’Impruneta (Italy). Constr. Build. Mater. 25 (12), 4308 4324. Beyer, K., 2012. Peak and residual strengths of brick masonry spandrels. Eng. Struct. 41, 533 547. Block, P., Lachauer, L., 2014a. Three-dimensional (3d) equilibrium analysis of gothic masonry vaults. Int. J. Architect. Herit. 8 (3), 312 335. Block, P., Lachauer, L., 2014b. Three-dimensional funicular analysis of masonry vaults. Mech. Res. Commun. 56, 53 60. Block, P., Ochsendorf, J., 2007. Thrust network analysis: a new methodology for threedimensional equilibrium. J. Int. Assoc. Shell Spat. Struct. 48 (3), 167 173. Block, P., Ciblac, T., Ochsendorf, J., 2006. Real-time limit analysis of vaulted masonry buildings. Comput. Struct. 84 (29 30), 1841 1852. Borri, A., Castori, G., Corradi, M., Speranzini, E., 2011. Shear behavior of unreinforced and reinforced masonry panels subjected to in situ diagonal compression tests. Constr. Build. Mater. 25 (12), 4403 4414. Borri, A., Corradi, M., Castori, G., De Maria, A., 2015. A method for the analysis and classification of historic masonry. Bull. Earthq. Eng. 13 (9), 2647 2665. Bosiljkov, V., Bokan-Bosiljkov, V., Strah, B., Velkavr, J., Cotiˇc, P., 2010. Review of innovative techniques for the knowledge of cultural assets (geometry, technologies, decay). PERPETUATE (EC-FP7 project), Deliverable D6. Brasile, S., Casciaro, R., Formica, G., 2007a. Multilevel approach for brick masonry walls part I: a numerical strategy for the nonlinear analysis. Comput. Methods Appl. Mech. Eng. 196 (49 52), 4934 4951. Brasile, S., Casciaro, R., Formica, G., 2007b. Multilevel approach for brick masonry walls part II: on the use of equivalent continua. Comput. Methods Appl. Mech. Eng. 196 (49 52), 4801 4810.

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Milani, G., Casolo, S., Naliato, A., Tralli, A., 2012. Seismic assessment of a medieval masonry tower in northern Italy by limit, nonlinear static, and full dynamic analyses. Int. J. Architect. Herit. 6 (5), 489 524. Minga, E., Macorini, L., Izzuddin, B., 2018a. Enhanced mesoscale partitioned modelling of heterogeneous masonry structures. Int. J. Numer. Methods Eng. 113 (13), 1950 1971. Minga, E., Macorini, L., Izzuddin, B.A., 2018b. A 3D mesoscale damage-plasticity approach for masonry structures under cyclic loading. Meccanica 53 (7), 1591 1611. Mistler, M., Anthoine, A., Butenweg, C., 2007. In-plane and out-of-plane homogenisation of masonry. Comput. Struct. 85 (17 18), 1321 1330. Mobarake, A.A., Khanmohammadi, M., Mirghaderi, S., 2017. A new discrete macro-element in an analytical platform for seismic assessment of unreinforced masonry buildings. Eng. Struct. 152, 381 396. Moon, F.L., Yi, T., Leon, R.T., Kahn, L.F., 2006. Recommendations for seismic evaluation and retrofit of low-rise URM structures. J. Struct. Eng. 132 (5), 663 672. Moreau, J.J., 1988. Unilateral contact and dry friction in finite freedom dynamics. Nonsmooth Mechanics and Applications. Springer, pp. 1 82. Munjiza, A.A., 2004. The Combined Finite-Discrete Element Method. John Wiley & Sons. NTC, 2008. Norme Tecniche per le Costruzioni, D.M. 14.01.08. O’Dwyer, D., 1999. Funicular analysis of masonry vaults. Comput. Struct. 73 (1 5), 187 197. Oliveira, D.V., Lourenc¸o, P.B., 2004. Implementation and validation of a constitutive model for the cyclic behaviour of interface elements. Comput. Struct. 82 (17 19), 1451 1461. Ordinanza del Presidente del Consiglio dei Ministri (OPCM), 2005. Norme tecniche per il progetto, la valutazione e l’adeguamento sismico degli edifici. Ordun˜a, A., 2017. Non-linear static analysis of rigid block models for structural assessment of ancient masonry constructions. Int. J. Solids Struct. 128, 23 35. Ordun˜a, A., Lourenc¸o, P.B., 2005a. Three-dimensional limit analysis of rigid blocks assemblages. Part I: torsion failure on frictional interfaces and limit analysis formulation. Int. J. Solids Struct. 42 (18 19), 5140 5160. Ordun˜a, A., Lourenc¸o, P.B., 2005b. Three-dimensional limit analysis of rigid blocks assemblages. Part II: load-path following solution procedure and validation. Int. J. Solids Struct. 42 (18 19), 5161 5180. Page, A., 1981. The biaxial compressive strength of brick masonry. Proc. Inst. Civil Eng. 71 (3), 893 906. Page, A., 1983. The strength of brick masonry under biaxial tension-compression. Int. J. Masonry Constr. 3 (1), 26 31. Page, A.W., 1978. Finite element model for masonry. J. Struct. Div. 104 (8), 1267 1285. Page, A., Samarasinghe, W., Hendry, A., 1982. The in-plane failure of masonry. A review. In: Proc. Br. Ceram. Soc., No. 30, p. 90. Panto`, B., Cannizzaro, F., Caddemi, S., Calio`, I., 2016. 3d macro-element modelling approach for seismic assessment of historical masonry churches. Adv. Eng. Softw. 97, 40 59. Panto`, B., Calio`, I., Lourenc¸o, P., 2018. A 3D discrete macro-element for modelling the out-ofplane behaviour of infilled frame structures. Eng. Struct. 175, 371 385. Papa, E., 1996. A unilateral damage model for masonry based on a homogenisation procedure. Mech. Cohes. Friction. Mater. 1 (4), 349 366. Papantonopoulos, C., Psycharis, I., Papastamatiou, D., Lemos, J., Mouzakis, H., 2002. Numerical prediction of the earthquake response of classical columns using the distinct element method. Earthq. Eng. Struct. Dyn. 31 (9), 1699 1717.

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Parisi, F., Augenti, N., 2013. Seismic capacity of irregular unreinforced masonry walls with openings. Earthq. Eng. Struct. Dyn. 42 (1), 101 121. Parisi, F., Lignola, G.P., Augenti, N., Prota, A., Manfredi, G., 2013. Rocking response assessment of in-plane laterally-loaded masonry walls with openings. Eng. Struct. 56, 1234 1248. Parrinello, F., Failla, B., Borino, G., 2009. Cohesive frictional interface constitutive model. Int. J. Solids Struct. 46 (13), 2680 2692. Pasticier, L., Amadio, C., Fragiacomo, M., 2008. Non-linear seismic analysis and vulnerability evaluation of a masonry building using the sap2000 v. 10 code. Earthq. Eng. Struct. Dyn. 37 (3), 467 485. Pela`, L., Aprile, A., Benedetti, A., 2009. Seismic assessment of masonry arch bridges. Eng. Struct. 31 (8), 1777 1788. Pel´a, L., Cervera, M., Roca, P., 2011. Continuum damage model for orthotropic materials: application to masonry. Comput. Methods Appl. Mech. Eng. 200 (9 12), 917 930. Pel´a, L., Cervera, M., Roca, P., 2013. An orthotropic damage model for the analysis of masonry structures. Constr. Build. Mater. 41, 957 967. Pel´a, L., Cervera, M., Oller, S., Chiumenti, M., 2014. A localized mapped damage model for orthotropic materials. Eng. Fract. Mech. 124, 196 216. Penelis, G.G., 2006. An efficient approach for pushover analysis of unreinforced masonry (urm) structures. J. Earthq. Eng. 10 (03), 359 379. Penna, A., Lagomarsino, S., Galasco, A., 2014. A nonlinear macroelement model for the seismic analysis of masonry buildings. Earthq. Eng. Struct. Dyn. 43 (2), 159 179. Petracca, M., Pela`, L., Rossi, R., Oller, S., Camata, G., Spacone, E., 2016. Regularization of first order computational homogenization for multiscale analysis of masonry structures. Comput. Mech. 57 (2), 257 276. Petracca, M., Pela`, L., Rossi, R., Oller, S., Camata, G., Spacone, E., 2017a. Multiscale computational first order homogenization of thick shells for the analysis of out-of-plane loaded masonry walls. Comput. Methods Appl. Mech. Eng. 315, 273 301. Petracca, M., Pel´a, L., Rossi, R., Zaghi, S., Camata, G., Spacone, E., 2017b. Micro-scale continuous and discrete numerical models for nonlinear analysis of masonry shear walls. Constr. Build. Mater. 149, 296 314. Petry, S., Beyer, K., 2014. Influence of boundary conditions and size effect on the drift capacity of urm walls. Eng. Struct. 65, 76 88. Pietruszczak, S., Niu, X., 1992. A mathematical description of macroscopic behaviour of brick masonry. Int. J. Solids Struct. 29 (5), 531 546. Pietruszczak, S., Ushaksaraei, R., 2003. Description of inelastic behaviour of structural masonry. Int. J. Solids Struct. 40 (15), 4003 4019. Portioli, F., Casapulla, C., Cascini, L., D’Aniello, M., Landolfo, R., 2013. Limit analysis by linear programming of 3D masonry structures with associative friction laws and torsion interaction effects. Arch. Appl. Mech. 83 (10), 1415 1438. Portioli, F., Casapulla, C., Gilbert, M., Cascini, L., 2014. Limit analysis of 3D masonry block structures with non-associative frictional joints using cone programming. Comput. Struct. 143, 108 121. Quagliarini, E., Maracchini, G., Clementi, F., 2017. Uses and limits of the equivalent frame model on existing unreinforced masonry buildings for assessing their seismic risk: a review. J. Build. Eng. 10, 166 182. Rafiee, A., Vinches, M., 2013. Mechanical behaviour of a stone masonry bridge assessed using an implicit discrete element method. Eng. Struct. 48, 739 749.

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Chapter 2

Performance-based assessment of masonry churches: application to San Clemente Abbey in Castiglione a Casauria (Italy) S. Lagomarsino, D. Ottonelli and S. Cattari University of Genoa, Genoa, Italy

2.1 Introduction The damage assessment of monumental buildings after seismic events in different countries has systematically highlighted the vulnerability of cultural heritage and the critical importance of reducing risk from economic, cultural, and social perspectives. Palaces, churches, monasteries, convents, towers, and castles are some examples of monumental building typologies. Each typology is characterized by a different response, behavior, and recurring damage modes after the occurrence of an earthquake (Binda et al., 2011; Lagomarsino, 2012; Cattari et al., 2013; Parisi and Augenti, 2013). Monumental buildings are usually made of good quality materials, but their dimensions are significant and they are usually characterized by constructive features, uncommon in ordinary buildings, which tend to increase their vulnerability. Some examples are the presence of wide halls, thin long-span vaults, slender towering or projecting parts, and slender walls with large openings. In the framework of cultural heritage, the chapter focuses on churches. As far as their seismic response is concerned, churches are highly vulnerable to earthquakes when compared to ordinary buildings and also to other monumental structures (Lagomarsino, 2006). The highly seismic vulnerability of churches has been systematically documented after several earthquakes in various countries (e.g., in New Zealand by Leite et al., 2013; in Azores by Guerreiro et al., 2000) and in particular in Italy. Referring to the latter and to Numerical Modeling of Masonry and Historical Structures. DOI: https://doi.org/10.1016/B978-0-08-102439-3.00002-6 Copyright © 2019 Elsevier Ltd. All rights reserved.

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FIGURE 2.1 Examples of emblematic churches hit by seismic events: (A) St. Maria Paganica Church (L’Aquila, 2009); (B) St. Joseph Church (Madonna of the Mill) in San Felice sul Panaro (Emilia, 2012); and (C) St. Benedetto di Norcia (Norcia, 2016/2017). http://www.atg-oxford.co.uk.

the most recent Italian events, the 2009 L’Aquila earthquake (Lagomarsino, 2012), the 2012 Emilia earthquake (Sorrentino et al., 2014), and the last 2016 Amatrice and Norcia earthquake (Penna et al., 2019) are emblematic examples of severe damage, close to complete collapse (Fig. 2.1). This evidence of seismic vulnerability highlights the need for procedures to reliably assess structure behavior and effectively direct compatible strengthening, “compatible” meaning the principle of “minimum intervention” (as stated in the Charter of Venice 1964) under the constraint of an “acceptable safety level.” To this end, the complex configurations of these types of structures as well as the difficulty of adopting a proper modeling strategy have led to a dichotomy in using qualitative or quantitative approaches in the literature. The assessment of cultural heritage assets is considered in some guidelines (ICOMOS, 2005; ISO 13822, 2010; CIB 335, 2010), but they do not specifically address seismic assessment. Rather, these documents focus to all possible causes of damage and deterioration, with the aim of making a diagnosis and designing a strengthening intervention. A common denominator among these recommendations is the importance of the qualitative approach, based on historical analysis, the accurate investigation of structural details, and the interpretation of seismic behavior, on the basis of observed damage in the building (due to previous events, if any) or on similar structures. However, while a preliminary and qualitative assessment is usually sufficient for diagnosis in many critical situations, such as material deterioration or soil settlement, the evaluation of seismic vulnerability without the support of calculations is overambitious, because the qualitative approach can only suggest the expected seismic behavior and historical analysis is not sufficient to ensure building safety. Indeed, the Italian Guidelines for the seismic assessment of cultural heritage (Presidenza del Consiglio dei Ministri (PCM), 2011) clearly state it is not possible to avoid quantitative calculation of structural safety by using models based on accurate knowledge and by eventually adjusting the results on the basis of expert judgment by also considering qualitative evidence. According to the approach also outlined for ordinary

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buildings at international scale (EC8-3 CEN, 2005; ASCE/SEI 41-13, 2014), the Italian Guidelines follow a performance-based assessment (PBA) that aims to check if the construction can meet selected performance levels (PLs) at earthquake hazard levels. The need to check PLs close to the structural collapse strongly recommends the use of nonlinear models and displacement-based procedures for the assessment, as it is not possible to rely on linear analyses using the behavior factor approach, since existing buildings, and in particular monumental ones, are not capacity designed. For churches, evaluating global behavior is in most cases challenging in engineering practice mainly because of their geometric complexity that cannot be modeled through simplified approaches and require more refined models like the finite element (FE) approach, which is more demanding in terms of computational effort (Milani and Valente, 2015; Betti et al., 2017; Elyamani et al., 2017; Ciocci et al., 2018). Indeed, the postearthquake damage assessment of churches is interpreted by referring to the so-called macroelement approach (Doglioni et al., 1994), where macroelements are the portions of an architectural asset characterized by autonomous and unitary structural behavior under seismic action. This approach leads to the common approach of tracing back the seismic assessment of the church to that of the single macroelements that compose it. The Italian Guidelines (Presidenza del Consiglio dei Ministri (PCM), 2011) also highlight the fact that for global verification of the seismic response of a church the use of a global 3D model is not strictly required, since it is possible to proceed with the decomposition of the structure into the macroelements, providing that the distribution of seismic actions among them is evaluated to assess the effect of different stiffness and connection effectiveness. Then, the seismic evaluation can be performed systematically on each macroelement of the construction. Recently, within the PERPETUATE project (Lagomarsino and Cattari, 2015), PBA concepts have been specifically targeted to monumental buildings by introducing, firstly, some specifications on the definition of PLs to consider conservation principles and, secondly, to develop quantitative procedures for various architectural typologies, including churches. This chapter presents this PBA procedure which, starting from the assessment of nonlinear response of single macroelements, combines them to define a fragility curve representative of the seismic behavior of the church as a whole. Recognizing the importance of the seismic action distribution among macroelements, in particular in churches subjected to strengthening interventions in the roof, the proposed procedure takes advantage of the combined use of nonlinear analyses performed on single macroelements and linear analysis carried out on an FE model of the whole church. After a general illustration of the procedure (Section 2.2), the latter is then applied to the Abbey of San Clemente in Castiglione a Casauria (Pescara, Italy), an important monument hit by the L’Aquila earthquake in 2009 and then restored in 2011 (de Felice, 2018), as described in Section 2.3.

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2.2 PERPETUATE performance-based assessment procedure for churches 2.2.1

General principles for complex architectonic assets

As introduced in Section 2.1, a methodology for the evaluation and mitigation of seismic risk to cultural heritage assets has been proposed in the PERPETUATE project (Lagomarsino and Cattari, 2015). In line with the main standards applied to ordinary buildings based on the PBA approach (EC8-3 CEN, 2005; ASCE/SEI 41-13, 2014), the basic steps of the PERPETUATE procedure are: (1) to define the capacity of the historical building; (2) identify the PLs through proper thresholds; (3) the definition of the seismic demand and selected intensity measure (IM); and (4) compute the outcome of the assessment, that is, the IMPLk , which is the maximum value of the IM that is compatible with the fulfilment of each target PLk (with k 5 1.4). The variable IMPLk is directly computed, without any iterative procedure (Lagomarsino and Cattari, 2015), by the capacity spectrum method based on the overdamped spectra originally proposed by Freeman (1998). Such general steps are then particularized in the PERPETUATE procedure taking into account the specific features of cultural heritage assets as a function of six architectural typologies as classified by Lagomarsino et al. (2011) in relation to the different types of seismic behavior that can be activated, the building morphology (architectural shape and proportions), and technology (masonry type, horizontal diaphragms, and effectiveness of wallto-wall and floor-to-wall connections). In particular, class A—assets subjected to prevailing in-plane damage—collects structures with a box behavior (e.g., palaces, castles), while class B—assets subjected to prevailing out-ofplane damage—collects buildings that can be studied by independent macroelements (e.g., churches, mosques). The classification proposed for architectonic assets is based on “mechanical” criteria and should be not used in a strict way because of complexity and variety of buildings. The prevailing seismic behavior and not the end use plays the fundamental role in the assignment of a certain class: as a consequence, for example, as a function of the roof stiffness and effectiveness of the wall-to-wall connection a palace and a church can alternatively belong to classes A and B. The class then addresses the choices for the most suitable modeling approach and how to deal with the PBA procedure. In the case of class A, a global 3D model of the whole building is used and is well described by a single capacity curve. But one of the critical issues in the PBA is the availability of reliable criteria to define the PLs on the pushover curve (step 2). To this aim, a multiscale approach was proposed in the PERPETUATE project (Lagomarsino and Cattari, 2015) and then recently revised by Marino et al. (2018). It takes into account the asset response at different scales: structural elements scale (local damage, E);

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architectonic elements scale (damage in macroelements such as walls and diaphragms, M); and global scale (pushover curve, G). Through the evolution of these variables it is possible to define the displacement of the overall pushover curve corresponding to a certain damage level (DL) as the minimum among displacements corresponding to the attainment of those conditions. In particular, the PERPETUATE procedure considers four DLs, which are then related to the corresponding PLs with the aim of passing from the structural performance (assumed to be measurable by proper engineering demand parameters assessed through the models) to safety, usability, and conservation principles. While Lagomarsino and Cattari (2015) defined more refined criteria to pass from DLs to PLs, they are herein assumed to be coincident; thus IMPLk coincides with IMDLk and this notation will be adopted in the following in an equivalent way. In the case of class B, the building is subdivided and modeled by a set of Nm independent macroelements. The seismic action can be assigned by considering for each macroelement its own inertial mass and the tributary area from diaphragms or by a proper redistribution, if some interaction among macroelements is expected. Eventually a 3D global model is adopted to support the identification of macroelements and the seismic action redistribution as clarified in Section 2.2.3. Once the IMPLk;m for each macroelement that composes the asset is evaluated, the assessment of the whole asset (IMPLk ;G ) is then made through proper combination of results achieved in each macroelement. Also, in this case, a multiscale approach is proposed that aims to define a fragility curve of the whole asset by combining the contribution offered by each macroelement. In particular, it is computed as: PPLk ðIMÞ 5

Nm X

ρm HðIM 2 IMPLk;m Þ

ð2:1Þ

m51

where H is the Heaviside function (0 if IM , IMPLk,m; 1 otherwise) and ρm is the weight that has to be assigned to each macroelement. Finally, the value of IMPLk;G is obtained as the minimum of the following two conditions: (1) the lower value of IM for which the fragility curve has PPLk(IM) $ 0.5 and (2) the value of IM for which the fragility curve of the PL (k 1 1) is greater than 0. In particular, Fig. 2.2 shows the construction of the global fragility curves in the case of class B (through the use of Eq. 2.1) for two different DLs (assumed to correspond to two PLs), where the red dot identifies the final value of IMDLk;G (or IMPLk;G ) after applying the abovementioned rules. A final fragility curve for each PL and main direction of analysis (if longitudinal or transversal) has to be defined. In the next section further details on the modeling strategy and the assessment of IMPLk;G are provided for the specific case of churches, when they are ascribable to class B.

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DL3

PPL(IM)

DL4 0.5

0 0

2

4

6

8

10

12

IMDL3, G FIGURE 2.2 Example of the fragility curves representative of the seismic behavior of the whole church for two DLs and identification of IMDL3;G .

2.2.2

Identification and modeling of single macroelements

As already mentioned, the macroelements are architectonic elements characterized by a proper seismic behavior, almost independently from the rest of the structure. They can be determined by surveying the geometry, the connections to the rest of the asset, and the evidence from the recurring damage modes that occurred after past earthquakes (Lagomarsino and Podesta`, 2004; Sorrentino et al., 2014; Penna et al., 2019). The latter ones together with a deep knowledge of the church are essential tools to address their identification. The knowledge of the asset requires: G

G

G

G

G

the geometric and technological survey, that is, a description of the structure’s geometry, the construction techniques, and the actions involved; the historical analysis to define the different construction phases of the monumental building; the analysis of construction details: the quality of the connections between vertical walls, inner walls, and also between different masonry typologies; the effectiveness of antiseismic measures (e.g., tie-rods); the damage survey: in the case of seismic damage, it is particularly useful to understand historic seismic events; and the investigations by diagnostic techniques to determine the parameters for the characterization of the structural typology and the mechanical properties of its materials.

As noted in Section 2.2.3, the final choice of the macroelements and their damage mechanisms may also be suggested/confirmed on the basis of the results of a linear analysis performed on a 3D model of the asset.

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Once each macroelement is defined, the choice of the most suitable structural model depends on: 1. the geometry of the macroelement, in particular the distribution of the openings in the masonry walls and 2. the prevailing seismic response (the in-plane or out-of-plane response), which in turn depends on the macroelement form, the external constraints, and the analysis direction. Among possible modeling approaches, macroblock model (MBM) is usually the main tool for the seismic assessment of these portions, but is not the only possible choice, especially for the in-plane response. In fact, in this latter case, the continuum constitutive laws model (CCLM) and the structural elements model (SEM), like the equivalent frame approach, could be employed. The CCLM is suitable in the case of both irregular geometry configuration and uneven disposal of the openings in the front, since the idealization of the equivalent frame would be not appropriate or not trivially established a priori, or vice versa; in the case of regular geometry of the macroelement, SEM can also be used. In some cases, although rarely in engineering practice due to the increasing computational effort, the discrete interface model may be used. It is noted that the same macroelement, depending on the seismic direction (if transversal or/and longitudinal), can be analyzed with two different models since different possible collapse mechanisms should be assigned. Fig. 2.3 shows two examples of the macroelement “fac¸ade” that can be studied with two different types of models according to their prevalent seismic response and geometry. The first case (Fig. 2.3A) is analyzed with an MBM: this is due to its geometry, in particular for the presence of the pillars, arches, and vaults, and to the hypothesized collapse mechanism, with the formation of four hinges at the top and four at the base of the pillars. The second is instead assessed through a SEM model (Fig. 2.3B): in fact, the presence of regular piers and openings is the ideal equivalent frame for this model. According to the prevailing failure mode expected for the macroelement and the modeling approach adopted, the seismic assessment and consequently the verification is then performed through a nonlinear static analysis (NLSA) or a nonlinear kinematic analysis (NLKA). The latter has to be performed for all mechanisms identified as possible for the macroelements under investigation. Once the capacity curves are obtained by such analyses then proper DLs (and corresponding PLs) have to be defined. Possible criteria are proposed by Lagomarsino (2015), for the case of NLKA, and in Lagomarsino and Cattari (2015), for the case of NLSA.

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PART | I Seismic vulnerability analysis of masonry and historical structures

FIGURE 2.3 Two examples of macroelements fac¸ade analyzed with two different models: (A) MBM and (B) SEM.

2.2.3

Information from 3D elastic modeling

In the case of churches belonging to class B, the support provided by a 3D model of the whole church is useful for: 1. Understanding the structural behavior of the asset and ensuring it is rational to adopt a macroelements approach, when this is not obvious from qualitative analysis of the structure (Cattari et al., 2013,a,b). 2. Supporting the definition of the macroelements and addressing the choice of the plausible mechanisms to be examined. The latter can be supported by the analysis of the stress state, because hinges will form where peaks of tensile stresses occur. In fact, despite the usefulness of the recurring damage modes found from past seismic events, for configurations of macroelements that are not standard it can turn out difficult the a priori choice of most probable mechanisms. 3. Defining the loading redistribution among the macroelements that compose the asset. The loading redistribution is an important aspect that often is handled in a conventional way, as a function of the masses that weigh directly on each macroelement (on the basis of its volume and the tributary area deriving from diaphragms). However, this choice can be arbitrary and, sometimes not on the safe side, especially when the structure

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has undergone a strengthening intervention, making the connections between the structural elements more efficient and the horizontal diaphragms rigid. In general, the complexity of the geometrical configurations of churches requires the adoption of FE models, particularly those suitable for modeling complicated geometry (e.g., characterized by curved surfaces) in general conditions of loading and constraints. The 3D model can be realized by introducing some simplifications of the actual geometry (e.g., by referring to the midplane of walls), although recent generation procedures that start from detailed laser scanner survey facilitate the implementation of a very accurate FE model with limited effort (Castellazzi et al., 2015). The results obtained in the linear analysis of the 3D model are influenced by three factors: the choice of the element types, its stiffness properties, and the connections between the structural elements. Indeed, in the case of linear models, the results in terms of stress concentrations are very sensitive to such modeling choices and their interpretation must take into account this drawback. The element types are a function of the characteristics of each structure. For example, in the case of massive structures it will be possible to recur to 3D solid elements, while often the masonry walls, vaults, and domes should be modeled also through shell elements (with bending and membrane capabilities), for which both in-plane and normal loads are permitted. The stiffness properties of the structural vertical elements are calibrated as a function of the masonry types identified. Vice versa for the horizontal structures, the detailed modeling of vaults and domes allows us to take into account the stiffening contribution due to their shape and geometrical proportion (e.g., rise-to-span ratio); the roofs and the other types of floors, as an alternative to the modeling of each structural element, could be modeled like a membrane with equivalent parameters of stiffness.

2.2.4 Combined use of 3D elastic model and 2D analyses on single macroelements As introduced in Section 2.1, the proposed procedure is based on the use of combined linear and nonlinear analyses, once the Nm macroelements that might be activated (m 5 1,. . ., Nm) by an earthquake have been identified. In particular, the linear analyses conducted on a 3D FE model are used to determine accurately the distribution of loads among the macroelements of the church, defining two coefficients αm that will be applied to the capacity curves for each macroelement. To compute αm two different load patterns are applied to the 3D model, that is, the one proportional to the masses and the one proportional to the

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main vibrational modes, which give rise to the computation of the coefficients α1,m and α2,m, respectively. The adoption of two sets of lateral load distributions—analogously to the approach currently adopted in the case of NLSA—is recommended, since a priori the effects of seismic actions cannot be assessed. In particular, the coefficient α1,m, relating to the load distribution proportional to the masses, is achieved through the execution of: (1) a static analysis where only the vertical static loads are applied (the loads derive from both the masonry weight and the roof) and (2) a static analysis where the model is subjected to horizontal acceleration equal to gravity, along the transversal (x) and longitudinal (y) direction of the church. From (1) the static vertical forces (Fz,m) associated with each macroelement (Nm) are obtained and consequently the coefficient β m are computed. These express the weight of each macroelement (Fz,m) compared to the total mass of the structure (Mtot): βm 5

Fz;m Mtot

ð2:2Þ

From step (2) the horizontal static forces (F1,x(y),m) associated with each macroelement are evaluated. At the end of these analyses, the coefficient α1, m is defined as the ratio of the static lateral force (F1,x(y),m) to the static vertical force (Fz,m): α1;m 5

F1;xðyÞ;m Fz;m

ð2:3Þ

The computation of the coefficient α2,m is based on the principles of modal spectral analysis. It is obtained by the following phases: (1) execution of a modal analysis to define the main vibration modes of the church, selected as those that activate a significant mass (e.g., .5%) or that are important for a particular macroelement and (2) execution of a spectral analysis (along the transversal, x, and longitudinal, y, directions) carried out on the main vibration modes. From step (2), the base shear of each macroelement is computed, selecting the competent horizontal reaction forces (F2,x(y),m). Finally the result in terms of reaction forces is combined according to the most suitable combination law, that is, through the SRSS (square root of the sum of the squares) or CQC (complete quadratic combination) rules, as a function of the relation among the main vibration modes (i.e., if the model values are statistically independent or not). Furthermore, the total shear force (F2,x(y)) is calculated as the sum of the shear force components resulting from each macroelement. Last, the coefficient α2,m is computed as the ratio of the spectral shear of each macroelement over the total shear force multiplied for the coefficient β m: α2;m 5

F2;xðyÞ;m N P m51

F2;xðyÞ;m

1 βm

ð2:4Þ

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The application of such coefficients produces a variation of the collapse multiplier α in the case of NLKA or the equivalent mass (m ) of the single degree of freedom system in the case of NLSA. In the first case, α1,m and α2,m divide α while in the second they multiply m . If the coefficients are less than one, the macroelement benefits from the load redistribution. Fig. 2.4 shows the capacity curves resulting from the application of such coefficients in both cases. In particular, the black curve refers to the response of a macroelement in which the seismic actions are computed considering only its weight (function of its volume) and the gravity loads transferred by diaphragms insisting on it; the gray solid curve is obtained for a α1,m (or α2, m) greater than one (that reduces the seismic capacity), and vice versa the gray dashed curve for a α1,m (or α2,m) lower than one (that increases the seismic capacity, decreasing the loads on the macroelement). Once the capacity curves are defined and eventually modified as illustrated earlier, the PBA procedure described in Section 2.2.1, has to be applied. In the literature, an approach that presents some analogies with the procedure herein presented is that originally proposed by Mele and De Luca (1999) and then applied in several case studies, the last of which is described by Brandonisio et al. (2013). This proposal, called the “two-steps procedure” by the authors, consists of (1) the execution of 3D static and spectral linear analyses on the structural 3D FE model of the church and (2) the execution of 2D nonlinear pushover analysis on the models of each single macroelement. The latter are finalized to obtain the ultimate overall base shear capacity of each macroelement that is then compared to the collapse multiplier derived from the application of limit analysis. Then, the seismic demand on each single structural macroelement, resulting from the linear analyses executed on the 3D model (step 1), is compared to the strength capacity resulting from step (2). This allows a direct, though approximate, verification in terms of strength. In both procedures, the use of a 3D linear model is proposed. However, while in the case of Mele and De Luca (1999) it is used to assess the seismic demand of each macroelement in terms of required base shear, here it is adopted to extract the coefficients to alter the seismic capacity and explicitly (B)

α∗0

Fy / Γm∗

(A)

d ∗0

d/Γ

FIGURE 2.4 Effects of the seismic distribution for the capacity curves in the case of: (A) a nonlinear kinematic analysis and (B) a nonlinear static analysis.

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include the effect of the seismic actions redistribution among portions of the church. Thus the main difference is that while in the Mele and De Luca (1999) proposal a strength approach is followed to define the seismic safety level of the church, here a PBA approach—based on the explicit evaluation of the displacement capacity—is adopted. Finally, here for the first time a procedure to combine the outcomes resulting from various macroelements is proposed in order to integrate the common approach of adopting the minimum safety index provided by the most vulnerable macrolement as representative of the whole church.

2.3 Case study of San Clemente Abbey in Castiglione a Casauria The procedure illustrated in the previous sections was applied to the Abbey of San Clemente in Castiglione a Casauria (Pescara, Italy), hit by the L’Aquila earthquake (2009) and then restored in 2011 (de Felice, 2018). The PBA of the asset is herein presented by focusing on some specific aspects of the procedure: the role of the seismic action redistribution among macroelements (resulting in the computation of α1,m and α2,m coefficients) and the definition of the final fragility curve representative of the whole church. Within the context of a research study carried out by the authors under the “Sisma Abruzzo Project” promoted by the Regional Directorate for Cultural Heritage and Landscape of Abruzzo (Lagomarsino et al., 2012), an in-depth knowledge phase on the church was performed by the authors in 2012 that included a historical, architectural, and technological analysis; a laser scanner survey addressed to integrate the geometrical data available (made by Arch. Carlo Battini); a set of nondestructive diagnostic techniques (including sonic tests and sclerometer test on stones and mortar); and a damage interpretation consequent to the seismic response of the church after the L’Aquila 2009 earthquake, deduced by the activated mechanisms. All these data have supported the accuracy in the modeling and analysis phases described in the following section.

2.3.1

Brief description

The medieval Abbey of San Clemente is located in Castiglione a Casauria (Pescara, Italy) and was founded in 871 by Emperor Ludovico II, after a vow made during his imprisonment in the Duchy of Benevento. Initially entitled to the Holy Trinity, it was dedicated to St. Clement when the latter’s remains were brought here in 872. The Abbey soon became a powerful and tempting landmark (Fig. 2.5) and now is used as state museum. Over the centuries the abbey was plundered several times: by the Saracens in 920 and, repeatedly, by the Norman count Malmozzetto between 1076 and 1097. After this destructive episode, the Benedictine abbot Grimoald promoted the rebuilding of the church, which was reconsecrated in

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FIGURE 2.5 San Clemente Abbey in its current state: (A) the fac¸ade; (B) the nave; (C) the lateral front; (D) the apse; and (E) and (F) the r.c. beams in the transept designed by Gavini.

1105. However, the works ended only in the late 12th century under abbot Leonate (115282, cardinal from 1170). In 1348 the church was struck by a terrible earthquake and was partially restored in 1448. In 1775 the whole building came under royal patronage and underwent a period of further damage and degradation. The church was restored in 1891, then in the early 1900s, after the 1915 Avezzano earthquake, and again in 1970 and 1980 when it became a state museum.

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In particular, after the Avezzano earthquake some significant interventions were carried out as follows. The insertion of tie-rods on the fac¸ade was done to prevent the activation of its out-of-plane response, that caused the collapse of the vaults in the loggia. The latter was rebuilt by the architect C. I. Gavini (from 1920 to 1923), who is especially remembered for the structural works of the transept. In fact, the arms of the transept were originally formed by two sacristies that interrupted the spatial continuity of the presbytery. Thus the architect decided to demolish them, replacing the walls of the sacristy that provided the support of the roof with two reinforced concrete beams connected to each other with a third beam that simulated a triumphal arch (Fig. 2.5E and F). This work was one of the first interventions in reinforced concrete on monumental landmarks in the area. Moreover the abbey of S. Clemente in Castiglione a Casauria hosts various artistic assets well conserved such as the ambo, the candelabrum, the urn, and the altar. The floor plan of the church has a Latin cross shape (Fig. 2.6A), divided into three naves with a single semicircular apse covered by an half-dome (Fig. 2.5D). At the end of the side naves there are two small stone stairways built in the 800s that lead down into the crypt. The crypt is divided into two transverse naves, each with nine spans, and has two apse enclosures. It is covered by massive cross-vaults, of different widths and shapes. In front of the fac¸ade there is a very fine portico with three arches and cross-vaults (Fig. 2.5A); the central one is a round arch and the side ones are pointed. In the section above the portico there are four double lancet windows that were originally in the ancient monastery and were inserted into the facade in 1448. To the left of the portico the remains of the primitive bell tower can still be seen. The nave is 16.5 m wide, 48.1 m long, and has a maximum height of 18.0 m in the first four bays of the nave and a minimum height of 12.6 m in the remaining bays. The aisles are 4.0 m wide and 9.5 m high. The masonry

FIGURE 2.6 (A) Plan of the church and the museum (in gray); (B) lateral front; and (C) longitudinal section (made by Arch. Carlo Battini, Lagomarsino et al., 2012).

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FIGURE 2.7 The wooden trusses in the nave and the half-wooden trusses in the aisles.

FIGURE 2.8 The damage due to the L’Aquila earthquake: (A) collapse of the tympanum; (B) longitudinal response of the colonnade; and (C) and (D) transversal response of the nave.

wall thickness varies between 0.85 and 1.3 m, and the columns of the nave arcade have rectangular (0.95 m 3 1.25 m), square (0.95 m 3 0.95 m), and cruciform sections. The structural walls are made of different masonry typologies as identified and detailed in Table 2.3, but the prevailing ones are “cut stone masonry with good bonding” and “irregular stone masonry.” The columns of the first four bays are built with “dressed rectangular stone masonry with nonsoft stones,” and the columns of the remaining bays with “soft stone masonry.” The structural system of the roof is today composed of wooden trusses in the nave and half-wooden trusses in the aisles (Fig. 2.7). As a result of the L’Aquila earthquake (April 6, 2009), severe damage occurred to the church and the complete collapse of the tympanum increased by the altimetric irregularities of the church (Fig. 2.8A); significant cracks associated with the transversal (Fig. 2.8C and D) and longitudinal response of the nave (Fig. 2.8B); the overturning of the end walls of the transept and the apse; and damage in the roof elements of the nave and cracks in the vault of the apse. In particular, concerning the transversal response of the nave the following main damage occurred: severe cracks in the triumphal arch, and activation of the transversal response of the upper part of the nave (called in the following “front nave”) and of the transept. Vice versa for the longitudinal mechanisms: minor cracks for the in-plane response in the lateral wall of the nave, with more significant diagonal cracks corresponding to the discontinuity in the masonry and the left transept. The colonnade was also heavily damaged with permanent out-of-plumb and crushing phenomena. Two years after the L’Aquila 2009 earthquake, the Abbey was fully restored (de Felice, 2018). The main interventions were mortar injection in

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FIGURE 2.9 The tie-rods of the church: in blue the historical ones and in red the new ones added after the 2009 L’Aquila earthquake.

the cracked walls; insertion of new tie-rods (red in Fig. 2.9) in the transept to improve the transversal response of the nave and for preventing the outof-plane of the gable; and bracing of the timber roof. The new tie-rods (red in Fig. 2.9) strengthened the structural system in addition to the historical ones (blue in Fig. 2.9) already present in the transept and for the longitudinal response of the fac¸ade. It is important to note that the modeling and analyses presented herein are limited to the San Clemente Church, and do not include the “P.L. Calore” Museum that completes the Abbey (Fig. 2.6A). This is due to the fact the museum is connected only to the lower part of the fac¸ade and contro-fac¸ade of the church, as can be seen in Fig. 2.6A.

2.3.2

Macroelement identification

Before moving to the modeling and analyses execution, the Nm macroelements that compose the church were defined, distinguishing the transversal and longitudinal response as illustrated in Fig. 2.10 through a plan view.

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FIGURE 2.10 Macroelements of the church: (A) transversal and (B) longitudinal direction.

TABLE 2.1 Macroelements in the transversal direction. Acronym

Macroelements

Transversal response

F

Fac¸ade

MBM/kinematic

CF

Contro-fac¸ade

SEM/static

FN

Transversal section of the front nave

MBM/kinematic

TA

Triumphal arch

MBM/kinematic

BN

Transversal section of the back nave

MBM/kinematic

T

Transversal section of the transept

MBM/kinematic

Some macroelements were considered for the sake of completeness in the FE model for correctly computing the seismic actions redistribution (dark gray in Fig. 2.10), but then for simplicity were not been analyzed in the nonlinear field since they were not vulnerable in the specific direction of the analysis (e.g., the transversal response of the apse) or not relevant for the global seismic response of the church (as the sacristy). Tables 2.1 and 2.2 summarize the identified macroelements, clarifying also the modeling type (if based on the MBM or SEM approach) and the nonlinear analysis (according to the kinematic or static approach) adopted. They have been differentiated according to their prevalent seismic response and the direction of the analysis (Figs. 2.11 and 2.12). According to the geometry of the church and the bays of the nave, each “transversal section of the front nave” and “transversal section of the back nave” is constituted of three different macroelements as shown in Fig. 2.10. The plane belfry (gray element in Fig. 2.12) is not considered a macroelement itself, since it is a projection of the apse; however, for complete verification of the church, it should be studied as a local mechanism.

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TABLE 2.2 Macroelements in the longitudinal direction. Acronym

Macroelements

Longitudinal response

F

Fac¸ade

MBM/kinematic

LWL

Left lateral wall

SEM/static

CL

Left colonnade

MBM/kinematic

LWR

Right lateral wall

SEM/static

CR

Right colonnade

MBM/kinematic

TL

Lateral wall of the left transept

SEM/static

TR

Lateral wall of the right transept

SEM/static

A

Apse

MBM/kinematic

FIGURE 2.11 Sketch of the main macroelements considered for the transversal response.

FIGURE 2.12 Sketch of the main macroelements considered for the longitudinal response.

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2.3.3

73

Information from the 3D modeling

Once the macroelements were identified, the 3D model of the San Clemente Church was developed taking into account the data acquired from the knowledge phase, in particular those resulting from the geometric survey, the load analysis, and the definition of mechanical properties of the materials. The model reflects the state of the church after the strengthening interventions were executed in 2011. The model was realized by using the ANSYS 8.1 software that works with the FE approach identifying the mean plane of the structural elements, both in regard to the vertical and horizontal components (ANSYS, 2004). The walls, the vaults, and the roof coating were modeled using shell elements, whereas the wooden elements of the roof were modeled using beam elements, and the ties were modeled as link elements analogously to the bracing elements adopted in the stiffening intervention of the roof. The mesh was generated in an automatic but controlled way (i.e., by defining a priori for each macroelement a set of subarea), in order to achieve a model with regular mesh. Thus the FE model consists of 43,724 joints and 42,850 elements. Elastic-isotropic or elastic-orthotropic behavior were assumed for the materials of beams (constituted by traditional timber), links (stainless steel), and shell elements (masonry). Table 2.3 clarifies the assignment of the various masonry typologies to the different parts of the church according to the data acquired during the knowledge phase, while Table 2.4 summarizes the values adopted for their corresponding main mechanical properties.

TABLE 2.3 List and identification of the masonry types in the church. ID.

Masonry types

1

Irregular stone masonry with good mortar

2

Irregular stone masonry

3

Cut stone masonry with good bonding

4

Soft stone masonry

5

Dressed rectangular stone masonry with nonsoft stones

6

Solid brick with lime mortar

7

Cut stone masonry with good bonding

Legend

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TABLE 2.4 Mechanical properties of the masonry material. Masonry typologies 1

2

37

4

5

6

Properties for the linear response Density (kg/m3)

1900

1900

2100

1600

2200

1800

Ey (MPa)

1305

870

1740

1080

2800

1500

Ex 5 Ey (MPa)

1434

956

1914

1188

3920

1650

ν

0.2

0.2

0.2

0.2

0.2

0.2

Gxy (MPa)

435

290

580

360

933

500

Gyz 5 Gxz (MPa)

479

319

638

396

1306

550

Properties for the nonlinear response τ 0 (MPa)

0.039

0.026

0.065

0.035

0.105

0.076

fm (MPa)

2.10

1.40

3.20

1.90

7

3.20

The 3D FE model was used to (1) assess the dynamic characteristics of the church (Section 2.3.3.1); (2) confirm the macroelements identification described in Section 2.3.2; (3) evaluate the seismic action redistribution among the macroelements (Section 2.3.3.1); and (4) support the identification of potential failure mechanisms of the macroelements (Section 2.3.3.2).

2.3.3.1 Evaluation of the redistribution among macroelements The execution of static and dynamic analyses constituted the tool to define the distribution coefficients αm according to the procedure described in Section 2.2.4. Static analysis consists of two phases: in the first phase only the vertical static loads are applied, while in the second phase the model is also subjected to a horizontal acceleration equal to the gravity (set to 9.81 m/s2) in both the x and y directions. The coefficients α1,m (proportional to static loads) are then defined through the use of Eq. (2.3). From Figs. 2.14 and 2.15, it is evident that the contro-fac¸ade, the triumphal arch, the transept, and the longitudinal walls have a coefficient α1,m greater than 1, which means they could be subjected to a seismic force higher than that estimated without explicitly considering the redistribution among macroelements, that is, simply on the basis of their volume and tributary area of diaphragms. After the static analysis, the modal and spectrum analyses were executed. The spectrum was assumed constant within the frequency range of 210 Hz. The main vibration modes resulting from the modal analysis are summarized in Table 2.5. The modal shapes (as illustrated in Fig. 2.13) show

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TABLE 2.5 Main modes in the x and y directions. Modes in x direction

Frequency (Hz)

M (%)

Modes in y direction

Frequency (Hz)

M (%)

3

2.35

38.66

5

2.93

44.80

6

2.97

4.32

6

2.97

2.62

12

3.50

5.60

11

3.44

1.97

13

3.66

5.07

30

7.23

3.28

16

4.63

2.45

43

8.95

5.81

20

5.30

9.49

23

5.83

3.43

58.48

69.02

FIGURE 2.13 Main modes of the church resulting from the modal analysis: (A) mode 3 and (B) mode 5.

significant deformation of the nave in the transversal direction, especially in the front (upper part); torsional deformations affect mainly the portico; a behavior of the apse stiffer than the other structural elements; the plane belfry more flexible in particular as regards the longitudinal response. Then the values of the reaction forces (in the x, y, and z directions) were extracted from the spectral analysis considering the base nodes of each macroelement as identified in Section 2.3.2. The final values of the α coefficients (Figs. 2.14 and 2.15) were obtained by applying the SRSS combination law considering five and seven modes for the longitudinal and transversal responses, respectively. For the church under examination the contro-fac¸ade, the transversal section of the front nave, the triumphal arch, and the lateral walls are the macroelements for which α2,m results in values greater than 1. Such evidence, although rigorous only in the linear field, is also significant for understanding the actual role played by the roof in terms of

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PART | I Seismic vulnerability analysis of masonry and historical structures 2.5

α1 α2

Transversal macroelements

2

α

1.5 1 0.5 0 F

CF

FN_1

FN_2

FN_3

TA

BN_1

BN_2

BN_3

T

FIGURE 2.14 The coefficients for the seismic action distribution for the transversal response of the macroelements.

2.5

α1 α2

Longitudinal macroelements

2.0

α

1.5 1.0 0.5 0.0 LW,L

C,L

C,R

LW,R

T,L

T,R

A

F

FIGURE 2.15 The coefficients for the seismic action distribution for the longitudinal response of the macroelements.

connection among macroelements and force distribution. It is a variable that, in most cases, is very difficult to exactly determine during the knowledge phase so it is interesting to figure out how it potentially can influence the church seismic behavior (Cattari et al., 2013,a,b) and above all to compare the performance of different retrofit techniques. In fact, insertion of rigid diaphragms (a technique widely used) entails a concentration of strength demand in the stiffest macroelements, so that the seismic capacity of the building is not necessarily increased, as also shown by the results illustrated by Mele et al. (2003). A sensitivity analysis of the stiffness of the roof was performed in order to understand and quantify how the coefficients associated with the seismic action redistribution may vary. Three different conditions were analyzed (Fig. 2.16): (1) the as-built condition (the one after the strengthening interventions were added following the L’Aquila 2009 earthquake) representative of a light and well-connected timber roof; (2) a very flexible and not connected timber roof; and (3) a rigid and massive diaphragm made by reinforced concrete. The graphs in Fig. 2.16 show, by way of example in the case of macroelements in the transversal direction, that the coefficients associated to the main vibrational modes are much more sensitive to the stiffness variation of

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Coefficients proportional to masses 2.5 2.0

α

1.5 1.0 0.5 0.0

F

CF

FN_1 FN_2 FN_3

TA

BN_1 BN_2 BN_3

T

Coefficients proportional to modes 2.5

α

2.0 1.5 1.0 0.5 0.0 F

CF

FN_1 FN_2 FN_3 Actual

TA

Flexible

BN_1 BN_2 BN_3

T

Stiff

FIGURE 2.16 The coefficients for the seismic action distribution for the transversal response of the macroelements according to different levels of diaphragm stiffness.

the roof and, as mentioned earlier, the stiffer macroelements (fac¸ade and contro-fac¸ade in the transversal response) are those mainly affected by the insertion of a heavy and stiff roof. In contrast, the actual roof exhibits an action redistribution closer to that provided by a flexible one. It is worth noting that, even in the case of flexible diaphragms, the seismic action redistribution differs from that resulting from the simplest computation based on the volume of each macroelement and the tributary area of diaphragms for each of them. This is because the 3D model also takes explicitly into account the redistribution effect consequent to the effective connection among the macroelements, as in the examined case (resulting from a good interlocking in the corners or artificially improved through the tie-rod insertion). This result highlights the potential usefulness of the 3D model in the case of flexible diaphragms but with good macroelements connection, especially if they are different in terms of stiffness. Indeed, such a redistribution effect could be questionable when a strong nonlinear response is activated and severe cracks propagate leading to the actual independent behavior of macroelements.

2.3.3.2 Identification of failure mechanisms of macroelements As introduced earlier, the principal stresses quantified by the execution of the linear analysis on the FE model can constitute a powerful support to

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PART | I Seismic vulnerability analysis of masonry and historical structures

FIGURE 2.17 The stress distribution for the identification of the plastic hinges for the colonnade (A) and the triumphal arch (B).

suggest the position of the hinges to be considered in the mechanisms identified for the given macroelement. For the San Clemente Abbey, this support has been very useful to reduce the uncertainties inherent in the definition of the collapse mechanisms analyzed by the nonlinear kinematic approach. Fig. 2.17 clarifies such use in the case of the colonnade and the triumphal arch, where the light blue areas highlight a main stress close to the tensile strength value of the masonry assumed approximatively equal to the 10% of the compressive strength (that varies from 1.4 to 7 MPa according to the masonry typologies of the asset as given in Table 2.3).

2.3.4

Seismic assessment of single macroelements

The NLKAs were executed by adopting the MBM approach, employing the software Mc4 Loc (distributed by Mc4software and applied in different case studies in Podesta`, 2012). It is based on a data graphic input that requires the 3D geometry of the bodies involved in the kinematism, realized in the CAD software (an example is shown in Fig. 2.18A for the triumphal arch). Once the model has been imported into the software, it is possible to assign to each block the value of the density (coherently to that summarized in Table 2.4); the software proceeds then to the automatic computation of the values related to the weight. NLSAs were also done by referring to the equivalent frame approach (as illustrated in Fig. 2.18B for the contro-fac¸ade), employing the software

Performance-based assessment of masonry churches Chapter | 2

79

FIGURE 2.18 The triumphal arch modeled with MB approach (A) and the contro-fac¸ade modeled with the equivalent frame approach (in orange the piers, in green the spandrels, in light blue the rigid nodes) (B).

3Muri (distributed by STADATA s.r.l.). The shear strength of the masonry panels was computed according to the criteria recommended in the Italian Building Code (NTC 2008 and MIT 2009). In particular, the shear failure mode was interpreted according to the diagonal shear criterion proposed by Turnˇsek and Sheppard (1980), while the flexural response was calculated on the basis of the beam theory, neglecting the tensile strength of the material and assuming a stress block for the normal stress distribution at the compressed toe; then the failure was associated with the attainment of the compressive strength of masonry normal to the bed joints. For both criteria the strength mechanical parameters were assumed coherently with the values proposed by the MIT (2009) for the masonry typologies identified for the church in Table 2.3, reduced by a confidence factor equal to 1.2. Further details on the principles the 3Muri program is founded on are described by Lagomarsino et al. (2013). The pushover curves obtained by the execution of these analyses were then converted into the corresponding capacity curves according to the principles stated by NTC 2008 and EC8-3 CEN (2005). The capacity curves were then modified through the redistribution coefficients α1,m and α2,m obtaining for each macroelement three curves. Fig. 2.19A shows the curves of the transversal response of the fac¸ade, analyzed through the MB approach: since both coefficients are less than 1, the macroelement benefits from the seismic action redistribution. For simplicity perfect rigid behavior was assumed; this assumption could be refined as discussed by Lagomarsino (2015), for example. Instead, Fig. 2.19B illustrates the capacity curve representative of the transversal response of the contro-fac¸ade, studied with the equivalent frame model: conversely, in this case both coefficients are greater than 1.

PART | I Seismic vulnerability analysis of masonry and historical structures

Sa (m/s2)

(A) 8

Base Prop. to masses - α1, m Prop. to modes - α2, m DL3 DL4

7 6 5 4 3 2 1 0

(B) 3.5 Base Prop. to masses - α1, m Prop. to modes - α2, m DL3 DL4

3

Sa (m/s2)

80

2.5 2 1.5 1 0.5 0

0

0.1

0.2

0.3

0.4

Sd (m)

0.5

0.6

0.7

0

0.02

0.04

0.06

0.08

0.1

Sd (m)

FIGURE 2.19 The capacity curves of the transversal response of the fac¸ade (A) and the contro-fac¸ade (B).

On the curves, the position of the PLs was assumed to be coincident with the corresponding DLs, as shown in Fig. 2.19. In particular, in the application only the Life Safety and the Near Collapse LSs, corresponding respectively to DL3 and DL4, were considered. These were computed by codes criteria (EC8-3 CEN, 2005), in the case of SEM models, and on basis of the criteria proposed by Lagomarsino (2015), in the case of MBM models. Thus the DLs were defined for each macroelement. In particular, in the case of NLKA, the displacements associated with the DL3 and DL4 were assumed respectively to equal 25% and 40% of the ultimate displacement corresponding to the multiplier value equal to zero. In the case of NLSA, the displacement associated with DL4 was computed in correspondence to the overall base shear decay equal to 20% of the maximum value in the pushover curve, while the displacement associated with DL3 was assumed as three-fourths of the latter. The definition of the DLs on the capacity curves preludes the application of the PBA procedure used in the computation of the IMDLk;m values of each single macroelement as introduced in Section 2.2.1. The peak ground acceleration was assumed to be the reference IM and the procedure adopted for the comparison between the capacity and the seismic input was based on the adoption of overdamped spectra, as proposed by Freeman (1998), with the additional specifications proposed for masonry buildings by Lagomarsino and Cattari (2015). Tables 2.6 and 2.7 summarize the resulting values of IMDLk;m (m/s2) for all macroelements identified in Section 2.3.2 in the two directions of analysis.

2.3.5 Performance-based assessment of San Clemente Abbey as a whole In the following the IM values obtained at the scale of single macroelements are combined according to the procedure illustrated in Section 2.2.2 in order to pass to the global scale and evaluate the final value of IMDLk;G . To this aim a proper weight (ρm) was assigned to each macroelement based on the use of Eq. (2.2) and computed on the basis of the FE model. The resulting values are summarized in Tables 2.6 and 2.7.

Performance-based assessment of masonry churches Chapter | 2

81

TABLE 2.6 IMDLk;m and ρm of the transversal response. m

ρm

IMDL3,

IMDL3,

IMDL3,

IMDL4,

IMDL4,

IMDL4,

Original

Masses

Modes

Original

Masses

Modes

F

0.119

3.479

4.400

5.514

4.501

5.693

7.134

CF

0.251

2.495

2.134

2.450

2.947

2.450

2.889

FN_1

0.089

2.338

3.396

2.063

3.024

4.394

2.669

FN_2

0.073

2.338

2.797

1.681

3.024

3.619

2.175

FN_3

0.079

2.338

2.784

1.671

3.024

3.601

2.163

TA

0.119

3.942

3.546

2.636

5.100

4.588

3.410

BN_1

0.055

2.246

2.325

1.915

2.906

3.008

2.478

BN_2

0.053

2.246

2.609

2.263

2.906

3.375

2.928

BN_3

0.055

2.246

2.928

3.014

2.906

3.788

3.899

T

0.107

4.859

4.094

5.652

6.286

5.297

7.312

IMDL4,

TABLE 2.7 IMDL,m and ρm of the longitudinal response. m

ρm

IMDL3,

IMDL3,

IMDL3,

IMDL4,

IMDL4,

Original

Masses

Modes

Original

Masses

Modes

F

0.136

4.608

3.066

3.248

4.786

3.632

3.677

LFL

0.160

5.085

6.597

5.291

6.579

8.536

6.846

CL

0.162

3.998

4.940

3.834

5.173

6.391

4.960

LFR

0.150

3.657

2.739

2.892

3.841

3.221

3.292

CR

0.058

3.871

3.454

7.765

3.956

3.765

7.765

TL

0.077

4.476

4.108

4.909

4.636

4.267

5.150

TR

0.167

5.258

5.763

7.044

6.803

7.456

9.113

A

0.089

1.481

2.573

1.679

1.916

3.328

2.172

The following figures show the resulting fragility curves of San Clemente Abbey for the two DLs considered by explicitly taking into account (Figs. 2.21 and 2.22) or not (Fig. 2.20) the effect of the seismic action redistribution among the macroelements derived by the FE model. These curves are drawn by a piecewise linear connection link between the characteristic points of fragility for each macroelement (see Fig. 2.2). In the fragility curves the red dots identify the final values of IMDL3,G for the different cases; it results from the worst condition between the two

82

PART | I Seismic vulnerability analysis of masonry and historical structures (A)

DL3

P

1

8. 7. 6. 5. 4. 3. 2. 1.

0.5

DL4 8. A 7. CL 6. CR 5. LWL 4. TR 3. TL 2. LWR 1. F

A CL LWL TR CR TL LWR F

0 0

2

4 6 PGA (m/s2)

(B)

DL3 1

P

8

DL4

10. T 9. TA 8. F 7. CF 6. FN_3 5. FN_2 4. FN_1 3. BN_3 2. BN_2 1. BN_1

0.5

10

10. T 9. TA 8. F 7. FN_3 6. FN_2 5. FN_1 4. CF 3. BN_3 2. BN_2 1. BN_1

0 0

2

4

6

8

10

PGA (m/s2) FIGURE 2.20 Fragility curves of the DL3 and DL4 for the (A) longitudinal and (B) transversal response considering the macroelement response without seismic action distribution.

considered, as explained in Section 2.2.1. In all cases, apart from the transversal response when the seismic actions on macroelements are computed neglecting the redistribution effect, the resulting IMDL3,G value derives from the condition for which the fragility curve of the damage level DL4 is greater than 0. Furthermore, the figures show also the list of macroelements ordered for increasing vulnerability (i.e., from the higher to the lower value of the IMDLk;m ). It is interesting to observe that the order changes according to: (1) the DLs of reference (as in the case of Fig. 2.20) and (2) the seismic action redistribution, above all for the transversal response of the church (from Fig. 2.20 to Fig. 2.22). The overall view of the seismic response of the church provided by the fragility curve is very useful also to plan, if necessary (i.e., when the IMDL3,G is lower than the target seismic demand), the strengthening interventions. For example, it is evident from Fig. 2.20 that, in the case of fragility curves similar to those of the longitudinal response, acting on specific macroelements can improve significantly the overall response (moving to the left of the beginning of the fragility curve), while in the case of fragility

Performance-based assessment of masonry churches Chapter | 2

83

(A)

P

1

DL3

DL4

8. 7. 6. 5. 4. 3. 2. 1.

8. 7. 6. 5. 4. 3. 2. 1.

0.5

CL A CR F TR TL LWL LWR

CL A CR F TR TL LWL LWR

0 0

2

4 6 PGA (m/s2)

(B)

DL3

1

P

8

10

DL4

10. F 9. T 8. TA 7. FN_1 6. BN_3 5. FN_2 4. FN_3 3. BN_2 2. BN_1 1. CF

0.5

10. F 9. T 8. TA 7. FN_1 6. BN_3 5. FN_2 4. FN_3 3. BN_2 2. BN_1 1. CF

0 0

2

4 PGA (m/s2)

6

8

10

FIGURE 2.21 Fragility curves of the DL3 and DL4 for the longitudinal and transversal response considering the seismic action distribution “proportional to masses.”

curves with a trend similar to those of the transversal response it is necessary to act on a set of macroelements to significantly increase the final value of IMDL3,G. By comparing Fig. 2.20 with Figs. 2.21 and 2.22 it is evident, in particular in the transversal response, how the redistribution effects and the load patterns considered can affect the seismic assessment (see also Fig. 2.23). Apart from the final outcome of the verification (summarized in Table 2.8), what is interesting is the different capabilities of the examined approaches to differentiate the potential vulnerability of different macroelements. More specifically and focusing by way of example on the transversal response, from Fig. 2.20B—related to the outcome obtained for the case in which the redistribution effect among macroelements assessed by the 3D model is neglected—it can be seen that all macroelements belonging to the nave (i.e., the six transversal sections of the back and front nave) exhibit almost the same seismic response. In contrast, in the case of Figs. 2.21B and 2.22B, it is possible to observe as the response of different transversal sections of the nave varies that each one is affected by the interaction with adjacent macroelements or by the actual dynamic response differently. In fact, coherently

84

PART | I Seismic vulnerability analysis of masonry and historical structures (A)

P

1

0.5

DL3

DL4

8. 7. 6. 5. 4. 3. 2. 1.

8. 7. 6. 5. 4. 3. 2. 1.

TL A CL TR CR LWL F LWR

A TL CL TR CR F LWL LWR

0 0

2

4

6

8

10

PGA (m/s2)

P

(B)

DL3

1

0.5

DL4

10. T 9. F 8. BN_3 7. TA 6. CF 5. BN_2 4. FN_1 3. BN_1 2. FN_2 1. FN_3

10. T 9. F 8. BN_3 7. TA 6. BN_2 5. CF 4. FN_1 3. BN_1 2. FN_2 1. FN_3

8

10

0 0

2

4

6

PGA (m/s2) FIGURE 2.22 Fragility curves of the DL3 and DL4 for the longitudinal and transversal response considering the seismic action distribution “proportional to modes.”

with the results of the modal analysis (see also Fig. 2.13A), in the case of application of α2,m coefficients all central transversal sections of the nave are strongly aggravated. Such effect is less exacerbated in the case of load pattern proportional to masses where the contro-fac¸ade attracts significant actions due to its significant stiffness. As a first attempt to provide some preliminary considerations on the actual reliability of the different approaches considered, a comparison with the evidence from the actual seismic response of the church after the L’Aquila 2009 earthquake is discussed in the following. First, it is important to highlight that indeed the analyses carried out refer to the actual state of the church after the interventions made in 2011. However, as briefly illustrated in Section 2.3.1, such strengthening was mainly addressed to prevent the activation of local mechanisms of standing elements (i.e., the out-ofplane response of the tympanum in the central nave through the integration of new tie-rods) to solve local critical issues (such as the poor connections among the structural elements of the roof) and to improve the connections among macroelements (through tie-rods). In other words, the main global

Performance-based assessment of masonry churches Chapter | 2 (A)

85

1

P

Proportional to modes 0.5 Proportional to masses

0 0

2

4

6

8

10

PGA (m/s2) (B) 1

P

Proportional to modes 0.5 Proportional to masses

0 0

2

4

6 PGA

8

10

(m/s2)

FIGURE 2.23 Fragility curves of the DL3 for the longitudinal and transversal response considering the different conditions of seismic action distribution.

TABLE 2.8 The IMPL,G for both directions of analysis. Direction of analysis

IMDL

Macroelement

Prop. masses

Prop. modes

Transversal response

IMDL3,50%

2.416

3.246

2.405

IMDL4,min

2.906

2.450

2.163

IMDL3,G

2.416

2.450

2.163

IMDL3,50%

4.483

4.911

4.359

IMDL4,min

3.644

3.221

3.292

IMDL3,G

3.644

3.221

3.292

Longitudinal response

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PART | I Seismic vulnerability analysis of masonry and historical structures

behavior of the church was not substantially altered by such an intervention, justifying the comparison made with the actual response after the seismic 2009 event. The latter showed better agreement with the results achieved by considering the seismic effect redistribution using the load pattern proportional to modes. This outcome seems consistent in this case where the quality of connection among macroelements is good. Table 2.8 summarizes the final values of IMDL3,G (m/s2). Being the target seismic demand of the site with a return period of 475 years equal to 2.74 m/ s2, it can be noted that the church is almost verified with a minimum safety coefficient equal to 0.8 in the worst case (i.e., for the longitudinal response in the case of load pattern proportional to modes).

2.4 Conclusion In this chapter a performance-based procedure developed for masonry churches within the context of the PERPETUATE project was illustrated. The procedure proposed highlights the role of nonlinear analyses in assessing the seismic vulnerability of macroelements and the potential of the combined use of a 3D FE method to improve the evaluation by quantifying in particular the redistribution effects among macroelements that are usually neglected in common engineering practices. The evaluation of the redistribution effect is particularly important in the case of good quality connections among macroelements and to assess the potential repercussion of the roof stiffening interventions, one of the most common. Moreover, as an original contribution, the response of macroelements was then combined in order to define a fragility curve representative of the whole response of the church. The application of the proposed procedure to the Abbey of San Clemente in Castiglione a Casauria (Pescara, Italy) provided interesting results and confirmed that the use of different load patterns (as commonly adopted in the case of NLSAs performed on ordinary buildings) can also be beneficial in improving the reliability of the assessment in the case of churches. Of course, such results have to be proven by a more extensive application of the procedure to other case studies and with additional validation (e.g., provided by the actual response on real structures or numerical simulations through nonlinear dynamic analyses).

Acknowledgments The results shown have been achieved in the “Sisma Abruzzo Project” promoted by the Regional Directorate for Cultural Heritage and Landscape of Abruzzo, thanks to the outcomes of the research project PERPETUATE (www.perpetuate.eu) funded by the Seventh Framework Programme (FP7/200713) of the European Commission, under grant agreement no. 244229.

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References ANSYS, 2004. Elements, theory, and optimization. User’s Manual, Version 8.1. ANSYS, Canonsburg, PA. ASCE/SEI 41-13, 2014. Seismic Evaluation and Retrofit of Existing Buildings. ASCE/SEI 41-13, Reston, VA. Betti, M., Borghini, A., Boschi, S., Ciavattone, A., Vignoli, A., 2017. Comparative seismic risk assessment of Basilica-type churches. J. Earthq. Eng. 22, 6295. Binda, L., Modena, C., Casarin, F., Lorenzoni, F., Cantini, L., Munda, S., 2011. Emergency actions and investigations on cultural heritage after the L’Aquila earthquake: the case of the Spanish Fortress. Bull. Earthq. Eng. 9, 105138. Brandonisio, G., Lucibello, G., Mele, E., De Luca, A., 2013. Damage and performance evaluation of masonry churches in the 2009 L’Aquila earthquake. Eng. Failure Anal. 34, 693714. Castellazzi, G., D’Altri, A.M., Bitelli, G., Selvaggi, I., Lambertini, A., 2015. From laser scanning to finite element analysis of complex buildings by using a semi-automatic procedure. Sensors 15 (8), 1836018380. Available from: https://doi.org/10.3390/s150818360. Cattari, S., Lagomarsino, S., Ottonelli, D., 2013a. Simulazione tramite analisi lineari del danno sismico della chiesa di santa maria paganica (l’aquila). In: Proc. of Anidis 2013, l’Ingegneria Sismica in Italia. 30 Giugno-4 Luglio, Padova, Italia. Cattari, S., Degli Abbati, S., Ferretti, D., Lagomarsino, S., Ottonelli, D., Tralli, A., 2013b. Damage assessment of fortresses after the 2012 Emilia earthquake (Italy). Bull. Earthq. Eng. Available from: https://doi.org/10.1007/s10518-013-9520-x. CEN, 2005. EN 1998-3: 2005. Eurocode 8: design of structures for earthquake resistance—part 3: assessment and retrofitting of buildings. Comite´ Europe´en de Normalisation, Brussels, Belgium. CIB 335, 2010. Guide for the structural rehabilitation of heritage buildings, CIB Commission W023Wall Structures. ISBN: 978-90-6363-066-9. Ciocci, M., Sharma, S., Lourenc¸o, P., 2018. Engineering simulations of a super-complex cultural heritage building: Ica Cathedral in Peru. Meccanica 53 (7), 19311958. de Felice G (a cura di), 2018. San Clemente a Casauria. Il restauro dopo il sisma del 6 aprile 2009, Ianieri Edizioni. Doglioni, F., Moretti, A., Petrini, V., Angeletti, P., 1994. Le Chiese e il Terremoti: Dalla Vulnerabilita` Constatata nel Terremoto del Friuli al Miglioramento Antisismico nel Restauro, Verso una Politica di Prevenzione. Edizioni Lint, Trieste, Italy. Elyamani, A., Roca, P., Caselles, O., Clapes, J., 2017. Seismic safety assessment of historical structures using updated numerical models: the case of Mallorca cathedral in Spain. Eng. Failure Anal. 74, 5479. Freeman, S.A., 1998. The capacity spectrum method as a tool for seismic design. In: Proceedings of the 11th European Conference of Earthquake Engineering, Paris, France. Guerreiro, L., Azevedo, J., Proenc¸a, J., Bento, R., Lopes, M., 2000. Damage in ancient churches during the 9th of July 1998 Azores earthquake. In: XII World Conference on Earthquake Engineering, January 30February 4, Auckland, New Zealand. ICOMOS, 2005. Recommendations for the analysis, conservation and structural restoration of architectural heritage. In: International Scientific Committee for Analysis and Restoration of Structures and Architectural Heritage (ISCARSAH). Document approved on June 15, 2005 in Barcelona, Spain. ISO 13822, 2010. Bases for Design of Structures  Assessment of Existing Structures, second ed. ISO International Standard, Switzerland.

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Lagomarsino, S., 2006. On the vulnerability assessment of monumental buildings. Bull. Earthq. Eng. 4, 445463. Lagomarsino, S., 2012. Damage assessment of churches after L’Aquila earthquake (2009). Bull. Earthq. Eng. 10, 7392. Lagomarsino, S., 2015. Seismic assessment of rocking masonry structures. Bull. Earthq. Eng. 13 (1), 97128. Lagomarsino, S., Podesta`, S., 2004. Seismic vulnerability of ancient churches: I: damage assessment and emergency planning. Earthq. Spectr. Available from: https://doi.org/10.1193/ 1.1737735. Lagomarsino, S., Cattari, S., 2015. PERPETUATE guidelines for seismic performance-based assessment of cultural heritage masonry structures. Bull. Earthq. Eng. 13 (1), 1347. Lagomarsino, S., Abbas, N., Calderini, C., Cattari, S., Rossi, M., Ginanni Corradini, R., et al., 2011. Classification of cultural heritage assets and seismic damage variables for the identification of performance levels. In: Proceedings of the 12th International Conference on Structural Repairs and Maintenance of Heritage Architecture (STREMAH), 57 September 2011, Chianciano Terme (Italy). WIT Trans. Built Environ. 118, 697708. Available from: https://doi.org/10.2495/STR110581. Lagomarsino, S., Cattari, S., Ottonelli, D., Rossi, M., Battini, C., Cosso, T., et al., 2012. Final report of the research study CRUIE 5/2010. “Abbazia di San Clemente a Castiglione a Casauria (Pe): Final Report” (in Italian). Lagomarsino, S., Penna, A., Galasco, A., Cattari, S., 2013. TREMURI program: an equivalent frame model for the nonlinear seismic analysis of masonry buildings. Eng. Struct. 56, 17871799. Leite, J., Lourenco, P.B., Ingham, J.M., 2013. Statistical assessment of damage to churches affected by the 20102011 Canterbury (New Zealand) earthquake sequence. J. Earthq. Eng. 17, 7397. Marino, S., Cattari, S., Lagomarsino, S., 2018. Use of nonlinear static procedures for irregular URM buildings in literature and codes. In: Proceeding of the 16th European Conference on Earthquake Engineering, 1821 June 2018, Thessaloniki, GR. Mele, E., De Luca, A., 1999. Behavior and modeling of masonry church buildings in seismic regions. In: Proceedings of the Second International Symposium on Earthquake Resistant Engineering Structures, ERES’99, Catania, Italy. Mele, E., De Luca, A., Giordano, A., 2003. Modeling and analysis of a basilica under earthquake loading. J. Cult. Herit. 4 (4), 355367. Milani, G., Valente, M., 2015. Failure analysis of seven masonry churches severely damaged during the 2012 Emilia-Romagna (Italy) earthquake: non-linear dynamic analyses vs conventional static approaches. Eng. Failure Anal. 54, 1356. Parisi, F., Augenti, N., 2013. Earthquake damages to cultural heritage constructions and simplified assessment of artworks. Eng. Failure Anal. 34, 735760. Penna, A., Calderini, C., Sorrentino, L., Carocci, C., Cescatti, L., Sisti, R., et al., 2019. Damage to churches in the 2016 Central Italy earthquakes. Bull. Earthq. Eng. (accepted). Podesta`, S., 2012. Verifica sismica di edifici in muratura. Aggiornato a NTC e Linee guida per la visualizzazione e riduzione della vulnerabilità sismica. Flaccovio Editore (ISBN 978-88579-0119-0), Palermo, p. 176. Presidenza del Consiglio dei Ministri (PCM), 2011. Recommendations for the assessment and mitigation of seismic risk of cultural heritage with reference to the Italian Building Code (NTC2008). Directive of the Prime Minister, February 9, 2011. G.U. no. 47, February 26, 2011 (suppl. ord. no. 54) (in Italian).

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Sorrentino, L., Liberatore, L., Decanini, L.D., Liberatore, D., 2014. The performance of churches in the 2012 Emilia earthquakes. Bull. Earthq. Eng. 12, 22992331. STADATA, 2012. 3Muri Program, Release 5.0.4 (www.3muri.com). Turnˇsek, V., Sheppard, P., 1980. The shear and flexural resistance of masonry walls. In: Proceedings of the International Research Conference on Earthquake Engineering, Skopje, Japan, pp. 517573.

Further reading Grunthal, G., 1998. European Macroseismic Scale 1998: EMS-98. Chaiers du Centre Europe´en de Ge´odynamique et de Se´ismologie, vol. 15, Luxembourg.

Chapter 3

Probabilistic modeling of unreinforced masonry walls subjected to lateral out-of-plane loading J. Li1, M.G. Stewart2 and M.J. Masia2 1

Tianjin Chengjian University, Tianjin, P.R. China, 2The University of Newcastle, Callaghan, NSW, Australia

3.1 Introduction Masonry has traditionally been the main form of construction in many parts of the world. It remains popular due to its many advantages, such as its durability, ease of construction, insulation characteristics, and aesthetic appeal. Being a composite material of brick and mortar, masonry has its own weaknesses when subjected to forces creating tension in the structure, such as wind or earthquake load. For example, a Richter scale 5.6 earthquake occurred in the city of Newcastle in Australia in 1989, leading to the loss of 13 lives, injuries to more than 100 people, and 50,000 buildings damaged to varying degrees. Elsewhere, a Richter scale 6.2 earthquake on December 23, 1972 killed 5000 people in Managua, Nicaragua, due to the collapse of masonry walls. However, it has been shown that if designed and constructed to appropriate loading and design codes, unreinforced masonry (URM) structures should be able to withstand moderate earthquake and wind loads (Page, 1992; Scrivener, 1993; Potter, 1994; Page, 1995). The capacity problem of URM is compounded by the fact that its strength properties are highly variable. Masonry is a complex material consisting of brick units set in a more flexible mortar matrix. The mortar joints act as planes of weakness due to the inherently low bond strength between bricks and mortar. High unit-to-unit spatial variability is also observed, particularly for flexural tensile and torsional shear bond strengths, due to variations in the quality of the workmanship, the weather during construction, and the materials used in various locations. Numerical Modeling of Masonry and Historical Structures. DOI: https://doi.org/10.1016/B978-0-08-102439-3.00003-8 Copyright © 2019 Elsevier Ltd. All rights reserved.

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Some research has focused on the probabilistic behavior of masonry, mainly the effects of the workmanship on the strength of the masonry walls and structures (e.g., Lawrence, 1991; Fyfe et al., 2000). However, there are still no suitable models available for predicting the effect that unit-to-unit spatial variability has on masonry structural capacity. This is principally due to the lack of probabilistic information for brick and mortar joint unit strengths, unit-to-unit spatial variability, and model errors (the degree of accuracy of the predictive strength models, that is, the ratio between test strength and predicted strength) for wall capacities. Also, unlike steel and concrete, masonry is a nonisotropic material and so its behavior is significantly more difficult to predict. Stewart and Lawrence (2002) developed a preliminary technique for computing the structural reliabilities of URM walls subject to vertical bending. To the best of our knowledge, this is the first paper to describe a model to predict the structural reliabilities of a masonry wall subject to lateral loads (such as wind or earthquake loads). This work involved considering a masonry wall as a system of individual units (bricks) using simplifying assumptions about its spatial variability (i.e., statistically independent unit material properties). An analytical solution was developed for this structural configuration, yet even this relatively simple problem formulation posed significant theoretical and computational challenges. It showed the important effect that the spatial variability of material properties can have on wall capacity. The present chapter aims to consider not only one-way vertical bending but also one-way horizontal and two-way bending (the interaction of horizontal and vertical bending) caused by lateral out-of-plane loading on walls with various support arrangements. Analytical solutions are not possible for these structural configurations—this will necessitate the development of nonlinear finite element analysis (FEA) models that have the capability to incorporate the spatial variability of the material properties, including the tensile strength of mortar joint and brick; the tensile fracture energy of mortar joint and brick; and the cohesion (shear bond strength) and shear fracture energy of the mortar joints. A stochastic FEA can then be developed to predict the strength of masonry walls. Random field analysis is a modeling method used to represent the spatial variability of variables. The structure is divided into k elements (brick units or mortar joints) and a random variable is used to represent the random field over each element. The statistical correlations between the random variables (correlation coefficient ρ; see more detail in Section 3.2.3) for different elements are based on the correlation characteristics of the random field (e.g., see Vanmarcke, 1983). Once the stochastic random field is defined, Monte Carlo simulation methods can be used to randomly generate parameter values for each of the k discretized values. In the present chapter, the spatially varying parameters are (1) mortar joint flexural tensile strength; (2) brick flexural tensile strength; (3) tensile fracture energy of mortar joints;

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(4) tensile fracture energy of brick; and (5) mortar joint torsional shear strength. The midpoint method will be used to discretize the random fields into random variables. The k values for each parameter can be included in the nonlinear FEA of masonry subjected to one-way vertical, one-way horizontal, and two-way bending, and the variability of wall strength can then be predicted from a spatially variable stochastic FEA where Monte Carlo simulation is the computational tool. This chapter will focus on URM walls subject to out-of-plane lateral loading, resulting in flexural actions in walls (one-way and two-way bending). This accounts for most structural masonry walls subjected to severe out-of-plane loads in countries where historically seismic activity has been considered low and masonry remains a major construction material due to its economic benefits.

3.2 Spatial analyses of unreinforced masonry walls in one-way vertical bending 3.2.1

Introduction to spatial and nonspatial analysis

Generally, there are two kinds of stochastic analysis methods applied in the numerical and mathematical models studying the behavior of masonry walls, these being spatial and nonspatial analysis. Nonspatial analysis is a scenario that considers a stochastic analysis with the full-sized wall with nonspatially varying material properties (unit flexural bond strength in this case) for each realization, that is, where wall bond strength is fully correlated (i.e., uniform bond strength in the wall). This means the flexural bond strength is identical for each unit in the wall. Most existing analyses of masonry structures assume this scenario. The spatial analysis is the scenario considering a stochastic analysis with the full-sized wall with spatially varying unit flexural bond strengths. The correlation coefficient (ρ) can characterize how correlation may exist between the units (see details in Section 3.2.3). Spatial analysis with ρ 5 0 assumes that there is no spatial correlation in flexural bond strength existing between each unit in the wall. In other words, each unit has statistically independent flexural bond strength. In this chapter, spatial analysis with ρ 5 0.4 assumes that the unit flexural bond strengths are correlated at a level of ρ 5 0.4 along each course of masonry (see more details in Section 3.2.3.3). This chapter compares the results obtained from spatial analysis with ρ 5 0, spatial analysis with ρ 5 0.4, and nonspatial analysis. Heffler et al. (2008) recommended that each unit is statistically independent of its neighbors. However, the mean correlation coefficient within a course was 0.4 in six walls tested by Heffler et al. (2008); therefore the spatial analysis with correlation coefficient of 0.4 in a course is carried out here as well. Different from what Heffler (2009) did, the spatial cases of ρ 5 0.4 are carried out,

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FIGURE 3.1 Diagram illustrating base cracking and mid-height cracking in one-way vertical bending.

performing a comprehensive comparison, showing how the wall strengths change when the correlation coefficient of flexural bond strength changes. The means and coefficient of variation (COV)s are compared, in terms of base cracking load (the load at which tensile cracking first occurs in the base region of the wall on the loaded side) (see Fig. 3.1); the mid-height cracking load (the load at which tensile cracking appears in the mid-height region of the wall on the unloaded side); and the peak load. Finally, the failure modes obtained from those spatial and nonspatial analysis models are compared.

3.2.2

Finite element analysis modeling strategies

3.2.2.1 Modeling approach The behavior of masonry is complex as it is a composite material of bricks and joints. The overall behavior of masonry is determined by the properties of the masonry components (unit, mortar, and unit/mortar interface) and the orientation of the unit/mortar interfaces (Sutcliffe et al., 2001). Depending on the degree of accuracy required versus simplicity desired, there are two different approaches used to model masonry, namely the micromodeling approach and the macromodeling approach (see Fig. 3.2). Macromodeling The macromodeling approach deems masonry as a one-phase material, and all of the components of the masonry assemblage (the brick units, mortar joints, unit/mortar interface) are smeared into a homogeneous continuum

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FIGURE 3.2 Modeling strategies for masonry structures (Lourenc¸o, 2008): (A) detailed micromodeling; (B) simplified micromodeling; (C) macromodeling.

(see Fig. 3.2C). The macromodeling technique is suitable for modeling large sections of masonry where only a simplified representation of composite behavior is required, and local failure modes are not so important. However, this technique is unable to model local failure modes. Micromodeling The micromodeling approach models the masonry components separately. Depending on the level of accuracy and simplicity desired, detailed and simplified micromodeling strategies may be used (Lourenc¸o, 2008). Detailed micromodeling: The units and the mortar in the joints are represented by continuum elements and the unit/mortar interface is represented by discontinuum elements (see Fig. 3.2A). Simplified micromodeling: The expanded units are represented by continuum elements and the behavior of the mortar joints and the unit/mortar interface is lumped into discontinuum elements (see Fig. 3.2B). In this approach, the mortar joint and the mortar/brick unit interface are lumped into a zero thickness interface element, meaning that the brick units are expanded to maintain the overall geometry of the masonry (see Fig. 3.4). Micromodels incorporate all of the failure mechanisms of masonry including joint tensile cracking and sliding, unit cracking, and crushing of the masonry. These models can also reproduce crack patterns and the complete load
displacement path of a masonry structure up to, and beyond, the peak load (Lourenc¸o, 2008). A simplified micromodel, giving a better understanding of the local behavior of masonry structures, is adopted in the current study. As mentioned previously, a macromodel cannot capture detailed local crack patterns and wall behavior. The micromodel also allows material strengths to differ from “unit” to “unit” in the spatial analyses.

3.2.2.2 Interface element nonlinear behavior The FEA was performed using the commercial software package TNO DIANA 9 (de Witte and Kikstra, 2007). The interface material model

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used herein is the combined cracking
shearing
crushing model (de Witte and Kikstra, 2007), also known as the composite interface model. This composite interface model was developed by Lourenco and Rots (1997) and Van Zijl (2004), and it is appropriate to simulate tensile and shear fracture, frictional slip as well as crushing along material interfaces (de Witte and Kikstra, 2007). The software was used to implement a simplified micromodeling strategy in which the brick units are modeled as linear elastic continuum elements, while the mortar joints are modeled with interface elements, which obey the nonlinear behavior described by this combined cracking
shearing
crushing model (Lourenco and Rots, 1997). For the potential crack interface element at the mid-length of the brick unit, a nonlinear tension softening model is used. All simulations reported in the remainder of this study are performed using a 3D composite interface model (see more details in de Witte and Kikstra, 2007).

3.2.3

Probabilistic models

3.2.3.1 Vertical bending wall structural configuration A deterministic model is generated before the establishment of the nonspatial and spatial probabilistic analysis models. In this section, a 3D nonlinear FEA model of a full-sized, single-leaf clay brick URM wall of dimensions 2.43 m (28 courses) (height) 3 2.46 m (10 brick units) (width) 3 0.112 m (one brick unit) (thickness) is generated (to be short, 2.5 m 3 2.5 m is used in the remaining part of the chapter), considering the brick units are solid. This wall was chosen because it is realistic for both the height (equal to the height of a single storey between lateral supports) and the length of the wall. Also, based on the context of assuming the load redistribution system hypothesis of failure (Stewart and Lawrence, 2002), a 2.5 m-long wall may have potentially a greater chance of a weak joint at which cracking could initiate, compared to a wall with shorter length, as there are more joints across the length of the wall. The same principle holds true in the height direction. The structural configuration considered here is a non-load-bearing single skin panel simply supported at the top and bonded at the bottom (one-way vertical bending) (see Fig. 3.3). The boundary conditions for the full-sized wall model are the same as in many practical situations. That is, the bottom course of the wall is bonded by a layer of mortar to an underlying floor or footing, and at the top, the wall is restricted from lateral (out-of-plane) displacement but can move in the vertical direction and is not rotationally restrained. A uniform out-of-plane lateral pressure load is applied over the full face of the wall.

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FIGURE 3.3 Boundary conditions of the 3D full-sized FEA wall.

FIGURE 3.4 Adopted modeling strategy (Lourenco, 1996a).

The two important parameters for the “joint” interface elements, linear normal and tangential stiffness moduli (kn and ks respectively), can be calculated using the following equations (see Fig. 3.4): kn 5

Eu E m h m ð Eu 2 Em Þ

ð3:1Þ

ks 5

Gu Gm hm ðGu 2 Gm Þ

ð3:2Þ

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Gu 5

Eu 2ð1 1 ν Þ

ð3:3Þ

Gm 5

Em 2ð 1 1 ν Þ

ð3:4Þ

where Eu and Em are the Young’s moduli of the brick unit and mortar; Gu and Gm are the shear moduli of the brick unit and mortar; hm is the thickness of the mortar (normally 10 mm); and ν is the mortar’s Poisson ratio. Based on various values reported in the literature, the average values of Young’s moduli for brick and mortar were used. To this end, a Young’s modulus of 20,000 MPa for bricks and 3000 MPa for mortar were used. The computational intensity associated with the treatment of multiple random variables necessitates restricting the number of random variables to only those to which the wall response under one-way vertical bending is sensitive. The many other parameters required for the numerical modeling of the wall behavior can be treated using representative average values, with small changes in these parameters having negligible effect on the system behavior. In other words, the one-way vertical bending behavior is dominated by the flexural bond strength and tensile fracture energy of the unit-to-mortar bond, which are therefore treated as variables in the model, and the less sensitive parameters (e.g., linear normal stiffness modulus) are treated as deterministic. The values of the material parameters to be used in the 3D FEA of the full wall are listed in Table 3.1. In this model, the tensile fracture energy of the unit-to-mortar interface is related to the tensile strength by the following expression (Heffler, 2009): Gf I 5 0:01571 3 ftm 1 0:0004882 GfI

ð3:5Þ

where ftm is the direct tensile strength (MPa) and is tensile fracture energy (Nmm/mm2). The model error of the tensile fracture energy is considered as unity, and the variance of the model error is zero. There are several experiments for one-way vertical bending of URM walls (Doherty, 2000, 2002; Griffith et al., 2004; Ismail and Ingham, 2012). In a recent study, Heffler (2009) demonstrated that the 3D FEA model described herein reproduced experimental wall results in Doherty (2000) with good agreement, confirming the suitability for the model to be applied in the full-sized wall with spatial variability in flexural bond strength. Heffler et al. (2008) found that the distribution of unit flexural bond strengths for full-sized clay brick URM walls is best represented by the truncated normal probability distribution (truncated at zero strength) and this was adopted for the current study.

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TABLE 3.1 Summary of material parameters to be used in the 3D FEA model. Brick/mortar

Property

Value

Horizontal and vertical mortar joint interface elements

Linear normal stiffness modulus

353 N/mm3 (Heffler, 2009)

Linear tangential stiffness modulus

146 N/mm3 (Heffler, 2009)

Direct tensile strength

Variable (McNeilly et al., 1996; Heffler et al., 2008)

Tensile fracture energy

Variable (Heffler, 2009)

Cohesion

0.65 N/mm2 (Heffler, 2009)

Tangent friction angle

0.75 (Heffler, 2009)

Tangent dilatancy angle

0.6 (Heffler, 2009)

Tangent residual friction angle

0.75 (Heffler, 2009)

Confining normal stress

21.2 N/mm2 (Heffler, 2009)

Exponential degradation coefficient

5 (Heffler, 2009)

Capped critical compressive strength

20 (25)a N/mm2 (Heffler, 2009)

Shear traction control factor

9 (Heffler, 2009)

Compressive fracture energy

15 Nmm/mm2 (Heffler, 2009)

Equivalent plastic relative displacement

0.012 (Heffler, 2009)

Shear fracture energy factor

0.15 (Heffler, 2009)

Brick Young’s modulus

20,000 N/mm2 (Heffler, 2009)

Brick’s Poisson’s ratio

0.15 (Heffler, 2009)

Brick density

1800 kg/m3 (Heffler, 2009)

Linear normal stiffness modulus

1000 N/mm3 (Heffler, 2009)

Linear tangential stiffness modulus

1000 N/mm3 (Heffler, 2009)

Direct tensile strength

2 N/mm2 (Heffler, 2009)

Fracture energy

0.5 Nmm/mm2 (Heffler, 2009)

Expanded brick elements

Potential brick cracks (all values are artificially high to force cracking in mortar joints and not bricks)

a


25 N/mm2 is used in the FEA model with ft 5 1.4 MPa for the parameter consistency in the analysis.

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The mean flexural bond strength and COVs provided by McNeilly et al. (1996), for 19 building sites in Melbourne, range from 0.22 to 0.85 with an average of 0.53 MPa, and COV from 0.16 to 0.49 with a mean of 0.3. For the current study, the chosen mean value of direct tensile strength (0.4 MPa), based on the ratio of 1.5 between flexural bond strength and the direct tensile strength (Van Der Pluijm, 1997; Petersen, 2009; Konthesingha Chaminda, 2012), is representative for many sites, and a COV of 0.3 is selected to characterize common sites. Several mesh densities were trialed, however; the mesh density decided on and listed in Table 3.2 was chosen for its compromise between accuracy and computational speed. The element types and mesh density are shown in Table 3.2 and Fig. 3.5.

TABLE 3.2 Summary of 3D FEA element type and mesh selection for the full-sized wall. Brick/mortar bodies

Element types and mesh density

Masonry units (bricks)

HE20 CHX60 (2 3 4 3 1): A 20-node, 3D isoparametric solid brick element based on quadratic interpolation and Gauss integration

Mortar joints

IS88 CQ48I (2 3 4 3 1): A 16-node, 3D interface element based on quadratic interpolation

Mid-length brick interface element

IS88 CQ48I (1 3 4 3 1): The same as above

FIGURE 3.5 Mesh density of 3D FEA model showing a single masonry unit.

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3.2.3.2 Nonspatial analysis The nonspatial analysis model is generated making use of the 3D nonlinear FEA full-wall model and Monte Carlo computer simulation techniques. The main procedure for nonspatial analysis model generation is as follows: 1. Establish the expressions describing the relationships between the unit direct tensile bond strength and the wall cracking loads, and between the unit direct tensile bond strength and the wall peak loads, respectively. This is done by considering fully correlated spatial simulations using a range of masonry unit direct tensile strengths, then fitting cubic spline interpolations to the results for wall cracking loads and wall peak loads (see Fig. 3.6). Although compared with (A) and (C) in Fig. 3.6, the cubic regression in (B) does not show a perfect match, it will not have a significant effect on the results. This is because it is always the lower tail of the resulting histogram of wall strengths that are of more interest in the calculation of the probability of wall failures, meaning that the quality of the cubic spline fit in the range of higher strength values is less important than that in the lower range of unit strengths. 2. Generate a random variable as the masonry unit direct tensile strength, following a truncated normal distribution, and find the corresponding cracking loads and peak load of the wall using the best fit curve interpolation of the nonspatial runs. 3. Repeat step(2) for 100,000 runs. 4. Calculate the mean and COV of wall cracking loads and peak loads.

3.2.3.3 Spatial analysis with ρ 5 0.4 The experimental results shown in Heffler et al. (2008) indicate that the correlation coefficient between adjacent joints along the courses in walls is 0.4, and between the courses is 0.2, indicating only a weak correlation between courses. Therefore, in the current study, the correlation coefficient of 0.4 between adjacent units within each course of masonry is adopted, while the correlation coefficient is ignored between the courses. Every first unit (direct tensile bond strength) in a course of the wall is randomly generated due to leaving out the correlation between the courses, and the remaining units in that course are correlated one by one starting from the first one. In the spatial analysis with ρ 5 0.4, 150 FEA simulations are deemed as the reasonable number based on the convergence check of the mean and COV of the wall peak load (see Li et al., 2014). The model considering unit direct tensile bond strength (μft 5 0.4 MPa, COV 5 0.3) being correlated by ρ 5 0.4 is obtained in the following way: 1. Following the truncated normal probability distribution, generate a set of random numbers, as the direct tensile strengths for every first unit in a

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PART | I Seismic vulnerability analysis of masonry and historical structures (A) Wall base cracking load (kPa)

6 5 4

R 2 = 0.9999

3 2 1 0

0

0.2 0.4 0.6 0.8 1 1.2 Masonry unit direct tensile strength (MPa)

1.4

Wall mid-height cracking load (kPa)

(B)

Wall peak load (kPa)

(C)

6 5 4 3

R 2 = 0.9718

2 1 0

0

0.2 0.4 0.6 0.8 1 1.2 Masonry unit direct tensile strength (MPa)

1.4

6 5 4

R 2 = 0.9999

3 2 1 0

0

0.2 0.4 0.6 0.8 1 1.2 Masonry unit direct tensile strength (MPa)

1.4

FIGURE 3.6 Cubic spline interpolation fit to 3D nonlinear FEA results for nonspatial simulations: (A) base cracking; (B) mid-height cracking; (C) peak loads for uniform tensile strengths ranging from 0.01 to 1.4 MPa.

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course of the masonry, then generate the remaining direct tensile strengths in that course being correlated (ρ 5 0.4) one by one starting from the first one. 2. Assign the strengths, and the associated fracture energy values according to Eq. (3.5), to horizontal (bed) and vertical (perpend) mortar joints in the FEA model. In this process a unique strength is assigned to the mortar bed joint along the complete length of each masonry unit (brick) and a unique strength is assigned to the mortar perpend joint over the complete height of each unit. 3. Run the FEA model and record its failure progression (i.e., the base cracking load, mid-height cracking load, peak load, and load versus deflection curve). 4. Repeat steps (1), (2), and (3) for 150 runs. The resulting histograms are shown in Fig. 3.7.

3.2.4

Comparison of the spatial and nonspatial analysis

The results, including the base cracking, mid-height cracking, and peak loads obtained from nonspatial analysis and spatial analysis with ρ 5 0.4, are compared for the mean direct tensile strength of 0.4 MPa in Table 3.3. Table 3.3 shows that, for the base cracking, mid-height cracking, and peak loads, the mean values obtained from the spatial analysis with ρ 5 0.4 are lower than those obtained from a nonspatial analysis for a COV of 0.3, and the standard deviation values obtained from spatial analysis with ρ 5 0.4 are lower than those from nonspatial analysis. They show that ignoring the spatial variability of the unit flexural bond strength (nonspatial analysis) overestimates the mean of base cracking load, mid-height cracking load, and peak load of the wall, while at the same time underestimates the wall strength (peak load) in the lower tail of the histograms. This observation is intuitive for the base cracking and mid-height cracking loads. The base cracking load and mid-height cracking load are the loads at which cracking first occurs in the base region and mid-height region of the wall. Therefore these will occur at a lower wall pressure load for the spatial case, due to the presence of units of lower than average strength, than in the nonspatial case, for which all units have equal strength. The differences between the nonspatial and spatial analyses estimates for mean peak loads are less pronounced. This presumably results from load redistribution and hence load sharing between joints of different strength along any given course of mortar bed joints, as peak load is approached. Despite the smaller differences for peak loads compared to the cracking limit states, the presence

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Probability density

4 3 2 1 0

0

1

4

5

0

1 2 3 4 Mid-height cracking load (kPa)

5

0

1

5

3 2 Base cracking load (kPa)

Probability density

4 3 2 1 0

Probability density

4 3 2 1 0

2 3 Peak load (kPa)

4

FIGURE 3.7 Simulation histograms of cracking and peak loads for the spatial analysis with ρ 5 0.4; μft 5 0.4 MPa, COV 5 0.3.

of weaker units in the spatial analyses still results in earlier crack initiation and propagation to failure and so smaller mean peak loads than observed for the nonspatial simulations. This load redistribution and load sharing also leads to another fact that the COVs of the wall strengths are always lower than the COV of direct tensile bond strength for both the nonspatial and spatial analyses. Table 3.3 also shows that, in all the spatial analysis and nonspatial analyses, the standard deviations of the base cracking load, mid-height cracking load, and peak load provided by the spatial analysis with ρ 5 0.4 are lower

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TABLE 3.3 Summary of base cracking, mid-height cracking, and peak loads for nonspatial and spatial analyses. Direct tensile strength (MPa)

Sample size

Analysis

Load

Mean (kPa)

σ (kPa)

COV

μft 5 0.4, COV 5 0.3

100,000

Nonspatial

Base cracking load

1.24

0.32

0.26

Mid-height cracking load

2.10

0.46

0.22

Peak load

2.27

0.48

0.21

Base cracking load

0.75

0.18

0.24

Mid-height cracking load

0.62

0.26

0.42

Peak load

2.09

0.13

0.07

μft 5 0.4, COV 5 0.3

150

Spatial ρ 5 0.4

than the standard deviations obtained from the nonspatial analysis. For every spatial analysis, it is always the weak bonds lower than mean direct tensile strength (0.4 MPa) that influence the base cracking load and mid-height cracking load. For the nonspatial analysis simulation, with the uniform bond strength of the wall, the randomly generated unit direct tensile strength can all be a smaller value, or can also all be a larger value than the mean direct tensile strength (0.4 MPa). Obviously, a smaller direct tensile strength for all units would lead to a smaller cracking load and a higher direct tensile strength would lead to a higher cracking load for the nonspatial analysis. Hence, the standard deviation in the spatial analysis is lower than in the nonspatial analysis. In the two cases considered here, the nonspatial simulation provides the upper limits for the mean and COV, in terms of the peak loads (this is of most concern during the design work). The spatial analyses offer the lower limits for the mean and COV of the wall strength, respectively. When it comes to peak load, load sharing between mortar joints plays an important role. The probability of a weak joint surrounded by higher strength joints for the spatial analysis is higher compared with nonspatial analyses, causing the crack propagation to be slower, and the peak appears in the relatively later load step, hence the lower COV. There is no such load sharing in nonspatial analysis. The failure behavior of the walls for the nonspatial and spatial analyses also differ greatly. Examining a propped cantilever in linear beam theory, the bending moment at the fixed end (the base in the current case) is greater than the bending moment in the middle area. Therefore when a pressure load

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is applied to the wall, the maximum moment in the wall will develop at the base of the wall. It is expected that the mortar bed joints along the course at the base of the wall will crack first. For the nonlinear finite element analyses used in the current study, the postpeak strength associated with the mortar joints allows the cracked joint to continue resisting the moment as it softens. During this process, the moment is redistributed to the mid-height region of the wall and the total load on the wall is able to further increase. When the moment in the wall in the mid-height region reaches the moment capacity of the wall cross-section, cracking occurs at mid-height (mid-height cracking) typically over multiple courses. For the majority of simulations conducted during the current study, mid-height cracking occurred prior to peak load. However, for some of the nonspatial analyses with high unit direct tensile strength, the peak load occurred prior to mid-height cracking, with the cracking occurring during the postpeak branch of the load versus wall displacement response. During the pushover analysis (monotonically increasing displacement), the failure behavior of the model with uniform direct tensile bond strength always acts symmetrically with respect to the vertical center line of the wall elevation. Doherty (2000) reported experimental results for vertical-spanning walls including the associated crack patterns. A typical result from Doherty (2000) shows a wall specimen in a static push test, subjected to one-way vertical bending. Doherty (2000) found that the crack does not develop symmetrically, illustrating the importance of considering the spatial variability of masonry properties in the structural analysis if the failure behavior is to be accurately captured. A comparison is made between the results from spatial analysis with ρ 5 0.4 and Standards Australia (2011) to examine if the code is conservative or not. Generally, the nominal capacity is expected to be a lower percentile of the simulation results. The comparison results show, when the model error of FEA results is unity, the wall strength provided by the design model in Standards Australia (2011) tends to be conservative; the probability of exceeding the nominal capacity is close to 100%. Therefore there is a need to collect more data to determine the model error, and hence a further assessment if the design model in Standards Australia (2011) is conservative or not. For more detail of the stochastic analysis of masonry walls in vertical bending see Li (2016).

3.3 Spatial analyses of unreinforced masonry walls in one-way horizontal bending 3.3.1

Introduction

Two distinct failure modes (see Fig. 3.8), or combinations thereof, are possible when walls are subjected to horizontal bending (i.e., bending stresses acting parallel to the mortar bed joints). The supporting and loading details for horizontal bending as discussed in this section are shown in Fig. 3.10 in

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FIGURE 3.8 Possible failure modes under horizontal bending (plan view of the bottom side): (A) line failure and (B) step failure.

FIGURE 3.9 Horizontal beam configuration (plan view of the bottom side).

Section 3.3.2. The failure mode observed depends on the relative strengths of the masonry constituents. Lawrence (1983, 1991, 1995) obtained the strength parameters of horizontal beams from flexure tests dominated by line failure, as well as the corresponding Monte Carlo simulations using simple FEAs (four elements, two bricks and two perpends) considering the tensile strengths of perpend joints and brick units as variables. During the simulations, Lawrence considered only three crack locations (3), (4), (5) (see Fig. 3.9) in the middle region of the beam. In addition, the tensile strengths of the perpend joints and brick units were assumed to follow a truncated normal distribution, rather than log normal and Weibull distributions as suggested in the literature (Lawrence, 1983; Lawrence and Cao, 1988; Lawrence and Lu, 1991), and are applied in the present study. Vaculik and Griffith (2010) developed a stochastic analysis method to calculate the ultimate strength of URM subjected to horizontal bending with combined failure modes using the mathematical models for horizontal bending proposed by Willis et al. (2004), in which the tensile strengths of the mortar joints and brick units are considered as Weibull distributed random variables. The current chapter presents the development and application of a stochastic FEA model, allowing a comparison of peak load distribution between stochastic simulation and experimental results from Lawrence (1983) for a four-course beam subjected to horizontal bending. The mean and COV of the peak load of a four-course beam subject to horizontal bending are calculated by spatial analysis with ρ 5 0.4 (which considers the partial correlation of ρ 5 0.4 for mortar joints and zero correlation for brick properties) and nonspatial analysis (which assumes the beam has uniform material

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properties), making a comparison of the peak loads and failure modes between the numerical simulations and experimental results. The 310 experimental results of the horizontal beams provided by Lawrence (1983) are used to check the results of the stochastic simulation.

3.3.2

Description of experimental program

Lawrence (1983) performed horizontal bending tests on 310 identical (same brick, same mortar type, and same configuration) URM beams (see Fig. 3.10). The horizontal beams were four courses wide by four bricks long and were subjected to four-point bending, resulting in one-way horizontal bending (see Fig. 3.11). The load and central deflection were recorded for each specimen.

FIGURE 3.10 A photo of the four-course beam specimen subjected to out-of-plane bending (Lawrence, 1983).

FIGURE 3.11 Four-course beam configuration subjected to horizontal bending.

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FIGURE 3.12 The vertical beam test (Lawrence, 1983).

The large number of 10-hole extruded bricks required for the tests necessitated the use of separate mortar batches at different times during the test program. All batches of the bricks were of the same type. By using a constant source of supply throughout the test program and carefully batching the ingredients by weight, the mortar properties were kept as consistent as possible from batch to batch. Finally, the whole experimental program was separated into seven batches, but the material properties and experimental results were summarized as a whole. The joint flexural bond strengths in Lawrence (1983) were derived from the flexural behavior of 311 vertical beams (see Fig. 3.12). Lawrence (1983) used 155 test specimens from seven batches to characterize the brick modulus of rupture using three different methods (standard test, brick-on-edge test, and three-brick-long test); however, the results from three bricks long glued end to end, which is the test given in Standards Australia (2003), were adopted because of the negative skew, indicating a long “tail” of results below the mean is significant in most cases for the three-brick-long Lawrence test (Lawrence, 1983). The final Young’s modulus of the brick was corrected, based on the gross cross-sectional dimensions, with good agreement with a 3D FE model including modeling of the perforations in the brick. The Poisson’s ratio was determined as the ratio of lateral strain to longitudinal strain obtained in the modulus of elasticity test.

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TABLE 3.4 Test results of the peak load (P) of four-course beam in horizontal bending. Minimum (kN)

Maximum (kN)

Mean (kN)

COV

4.2

17.4

10.7

0.22

TABLE 3.5 Material properties provided by Lawrence (1983). Joint flexural tensile strength (MPa)

Brick modulus of rupture (MPa)

Strength of shear triplet (MPa)

Mean

1.82

4.70

1.02

COV

0.30

0.34

0.27

Suggested distribution

Normal

Weibull

Normal

A triplet shear test was conducted in Lawrence (1983) as well (this was used to derive the cohesion in the later FEA simulation). The basic principle of this test, which is essentially identical to that described by the subsequently developed CEN.EN1052-3 (2002), is to subject two mortar joints to shearing load simultaneously, producing a symmetrical arrangement with consequent simplification of the apparatus and testing procedure. The final mean and standard deviation of joint strength were corrected from the shear triplet results. Table 3.4 shows the results for peak load and Table 3.5 shows the material parameters relating to the experimental study (Lawrence, 1983). Failures observed in these tests invariably followed line failure (see Fig. 3.8A), which means the peak load is highly dependent on the lateral modulus of rupture (tensile strength) of the brick units (Lawrence, 1991).

3.3.3

Finite element analysis model

A 3D nonlinear FEA model of a four-course beam with dimensions (960 mm length 3 340 mm width 3 112 mm thickness) nominally identical to those tested by Lawrence (1983) is described. The element type here is the same as that used in Section 3.2 (see Tables 3.2 and 3.6). Potential cracks were assumed at the mid-length of bricks, meaning seven possible crack locations from left to right (see Figs. 3.8 and 3.9). The mortar joint and the mortar/brick unit interface were lumped into a zero thickness interface element; in other words, the size of the bricks is expanded to account for the

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zero thickness of the mortar joints in the model. Depending on the mode of failure (line or step) the horizontal bending behavior is expected to be dominated by the flexural tensile behavior (strength and fracture energy) of the mortar perpend joints and the bricks and the torsional shear behavior (cohesion, shear fracture energy, friction) of the mortar bed joints.

3.3.3.1 Mesh density study Two four-course beam models were subjected to horizontal bending, with material parameters and configuration being the same except for the FE mesh density, to study the mesh sensitivity of the FE model. These two mesh densities, coarse mesh and fine mesh, are 2 3 4 3 1 (i.e., number of elements long 3 number of elements thick 3 number of elements high) and 8 3 8 3 6 for each half brick unit (see Fig. 3.13). The element types and mesh densities are summarized in Table 3.6. It was found that the coarse mesh may result in a slight overestimation (1.7%) for the cracking load compared to the fine mesh. The failure mode was identical for both mesh sizes. As the overall response of the two meshes was very similar, it was decided to adopt the coarse mesh in the interest of computational efficiency due to the high CPU times for Monte Carlo simulation (e.g., more than 24 hours CPU for one simulation run when using the fine mesh). All

FIGURE 3.13 Two densities of finite element mesh for half masonry unit used in the mesh density study: (A) coarse mesh and (B) fine mesh.

TABLE 3.6 Summary of 3D FEA element type and mesh selection for the full-sized wall. Brick/mortar bodies

Element types

Course mesh

Fine mesh

Masonry units (bricks)

HE20 CHX60

23431

83836

Mortar joints

IS88 CQ48I

23431

83831

Mid-length brick interface element

IS88 CQ48I

13431

83631

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simulations reported in the remainder of this study for the horizontal bending specimens were performed using the coarse mesh, namely 2 3 4 3 1.

3.3.3.2 Deterministic values in the model Representative average values for the deterministic parameters (those with a fixed numerical value in Table 3.10) are used as input parameters in the model and are shown in Table 3.10. Parameters derived from tests in Lawrence (1983) include the linear normal stiffness modulus, linear tangential stiffness modulus, tensile strength of mortar and brick, cohesion (shear bond strength), brick Young’s modulus, and Poisson’s ratio. Parameters obtained from the literature include the tensile fracture energy of mortar and brick and the shear fracture energy factor (Van Der Pluijm, 1997; Lourenco et al., 2005; Heffler, 2009; Petersen, 2009). The remaining deterministic parameters in Table 3.10 are from the literature (Lourenco, 1996b; Heffler, 2009; Petersen, 2009). 3.3.3.3 Random variables There are six material properties considered as the random variables in the FEA model, as they can influence the beam behavior of one-way horizontal bending to different degrees. The many other parameters required for the numerical modeling of the wall behavior are treated using representative average values, with small changes in these parameters having negligible effect on the system behavior. The mean elastic modulus and Poisson’s ratio for brick were tested as 30.07 GPa and 0.16 (Lawrence 1983), and the mean elastic modulus for mortar is considered as 3 GPa. Random variable 1: Tensile strength of the mortar joint The model here makes use of the joint strengths obtained by Lawrence (1983), as stated in Section 3.3.2. The mean and COV of the flexural strength of the mortar joints are 1.82 MPa and 0.30, respectively. The ratio of 1.5 between flexural bond strength and direct tensile strength for mortar joints is applied here (Van Der Pluijm, 1997; Petersen, 2009; Li et al., 2014), see Table 3.9. A log normal distribution function was suggested for the tensile strength of the mortar joints (Lawrence and Cao, 1988; Lawrence and Lu, 1991). The statistics for the model error of tensile strength of mortar joint are mean of one and COV of zero, because these tensile strength of mortar joints have been corrected by Lawrence (1983) from the small vertical beam tests. Random variable 2: Tensile fracture energy of mortar joints 1. Model error for tensile fracture energy of mortar joints The empirical predictive model for the tensile fracture energy of mortar joints assumes a linear relationship with the tensile strength of the mortar joint (Heffler, 2009) (see Eq. 3.5 in Section 3.2.3 and Fig. 3.14).

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FIGURE 3.14 Fitting a linear relationship between tensile bond strength versus tensile fracture energy for clay brick masonry with general-purpose mortar (1:1:6 1 air entrainer) using Figure 4.11 from Heffler (2009) and Van Der Pluijm (1997).

Based on the data analysis, both linear and exponential functions can adequately represent the relationship between tensile strength and tensile fracture energy of mortar joints, but using an exponential function poses the danger of overestimating tensile fracture energies for higher bond strengths for which there is no data available in Van Der Pluijm (1997). Therefore the linear function provided by Heffler (2009) is adopted here. A database of test results was applied from Van Der Pluijm (1997) due to no available experimental data in Lawrence (1983) and analyzed to derive statistics of the model error (MEm) using Eq. (3.6). MEm 5

Experimental capacity Predicted capacity

ð3:6Þ

A total of 43 sets of test data (Van Der Pluijm, 1997) were selected to calculate the mean and COV of the model error MEm for the tensile fracture energy of mortar joints. The histogram and various fitted probability distributions (Gamma, Log normal, Normal, Weibull and Gumbel) for the tensile fracture energy of the mortar joints are shown in Fig. 3.15. Fig. 3.15B shows that a range of probability distributions are fitted to the model error of the tensile fracture energy of the mortar joints using an inverse cumulative distribution function (CDF21) plot. When the CDF21 of a particular probabilistic model sits on the 1:1 line this indicates that the probabilistic model is a good fit for the data (see Fig. 3.15B). The Kolmogorov
Smirnov test found that no probability models were rejected at the 5% significance level. Fig. 3.15B shows that all the distributions of model error overestimate the lower tail of the histogram with the log normal distribution being the best fit, and so it is recommended for the model error of the tensile fracture energy of mortar joints.

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FIGURE 3.15 (A) Probability distributions of model error for the tensile fracture energy of mortar joints and (B) inverse CDF plot for the model error of the tensile fracture energy of mortar joints.

TABLE 3.7 Statistics for model error of tensile fracture energy of mortar joint.

I

Gf for mortar joints

Sample size

Mean

VM

VME

Distribution

43

1.04

0.67

0.67

Log normal

2. Model error statistics The statistics of the model error for the tensile test in masonry, using Eq. (3.7), assuming that Vtest 5 0.02 and Vspec 5 0.04, are shown in Table 3.7.
0:5 VME 5 VM 2 2Vtest 2 2Vspec 2 ð3:7Þ

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where VM is the coefficient of variation obtained directly from a comparison of the measured and predicted data; Vtest represents the coefficient of variation in the measured data due to the accuracy of the test measurements and the definitions of failure; and Vspec represents the uncertainties due to differences between the strength of the test specimen and the control specimens, the variation in actual specimen dimensions from those measured, etc. Ellingwood et al. (1980) suggested that for testing reinforced concrete (RC) beams and columns Vtest varies from 0.02 to 0.04, and Vspec is about 0.04. Random variable 3: Cohesion (shear bond strength) of the mortar joints The cohesion was obtained by correcting the test results of the shear triplet tests from Lawrence (1983) (see Table 3.5). The first correction, based on order statistics, was to allow for the fact that the triplet test returns the weaker of two joint strengths. The second correction involved multiplying the triplet test results by 1.5. This allowed for the observation that the nonuniform distribution of the shear stress in the triplet test results in the mean failure stress being approximately 1.5 times lower than the true local shear bond strength at a material point (Masia et al., 2012). Therefore the mean and coefficient of variation of the cohesion used in the FEA analyses were 1.81 and 0.27 MPa, respectively. The best fit distribution can be obtained for the strengths of the shear triplets according to the 251 shear triplet test results in Lawrence (1983), and the log normal distribution is assumed to express the cohesion, which is the same as that for the strengths of the shear triplets. The cohesion of masonry is correlated to the flexural strength of the mortar joints based on Standards Australia (2011) and Masia et al. (2012). Masia et al. (2012) found that the flexural strengths of mortar joints are correlated with cohesion by doing the bond wrench tests and triplet tests. This section determines the relationship between those two properties under the specific situation and circumstances of the experiments in Lawrence (1983), assuming they are fully correlated. Both cohesion and flexural strength follow the log normal distribution with a different mean and COV. It is assumed that cohesion and flexural strength are fully correlated. Random variable 4: Shear fracture energy of the mortar joints A linear relationship (Petersen, 2009) is adopted between the shear fracture energy (GfII) of the mortar joints and the normal compressive stress (negative σn): Gf II 5 0:035
0:49 3 σn

ð3:8Þ

where σn is the normal compressive stress (MPa) and GfII is shear fracture energy (Nmm/mm2). This relationship fits between the lower and upper

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bound relationships used for the URM walls (Masia et al., 2007). Due to a lack of available experimental data, the model error of the shear fracture energy is considered as unity, and the variance of model error is zero. Random variable 5: Tensile strength of the brick The three-brick-long modulus of rupture of brick test results were used to obtain the flexural strength of the brick. The direct tensile strength was estimated as being equal to the flexural tensile strength divided by 1.5 (Petersen, 2009; Petersen et al., 2012). This reduction is similar for concrete (a similar material to brick unit) in Standards Australia (2001). The tensile strength of the brick unit used in the brick crack interface elements was 4.7 MPa/ 1.5 5 3.1 MPa (see Table 3.9). In principle, the model error statistics for the tensile strength of brick should be adjusted to account for the three-brick-long test procedures. The distribution statistics of brick tensile strength have been summarized by Lawrence (1983) based on the mean and COV of the experimental data. Hence, the mean and COV of the brick tensile strength will change if combined with the model error statistics. Hence, to keep the input parameters consistent with the experimental results by Lawrence (1983), the model error of the brick tensile strength is ignored, that is, the mean of model error is one, and COV of model error is zero. Random variable 6: Tensile fracture energy of the brick 1. Importance of the fracture energy of the brick Fracture energy is a very important property for the brick unit, especially in the current study, as it affects line failure, which means both the peak load and failure mode are dominated by a failure of the brick units. The fracture energy of the brick unit plays a more important role than expected during the analysis. Since there is little evidence in the literature of the effect of the fracture energy of the brick on the horizontal bending behavior of URM, two FEA analyses were carried out with all identical parameters, except the fracture energy of the brick units, to illustrate the significance of fracture energy. One fracture energy value is 0.5 Nmm/mm2, the other is 0.025 Nmm/mm2. The line failure location and load versus displacement curve are shown in Figs. 3.16 and 3.17. Fig. 3.16 shows that the main cracks appear in different potential crack lines. Also, the load versus displacement curve changes significantly (see Fig. 3.17), indicating that the results are sensitive to fracture energy. 2. Model Error for tensile fracture energy of the brick Due to no available experimental data on the tensile fracture energy of bricks in Lawrence (1983), a database of 20 test results for solid bricks under uniaxial tensile testing carried out by Lourenco et al. (2005) were used to

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FIGURE 3.16 Failure modes for when (A) fracture energy value is 0.5 Nmm/mm2 (B) fracture energy value is 0.025 Nmm/mm2.

FIGURE 3.17 Comparison of load versus displacement curves for different fracture energy values.

calculate the mean and COV of the model error. The empirical predictive model for the tensile fracture energy of brick assumes a linear relationship with the tensile strength of the brick (see Fig. 3.18). The expression is: Gf 5 0:0097 3 ftb 1 0:0277

ð3:9Þ

where Gf (N mm/mm2) is the tensile fracture energy of the brick unit and ftb (MPa) is the tensile strength of the brick. It should be noted that the linear model for the brick tensile fracture energy is selected among linear, exponential, logarithmic, and power functions. The leave-one-out cross-validation is used to estimate how accurately the predictive model can express the data. It was found that the logarithmic function can best express the 20 test database with the lowest root-meansquare error of prediction of 0.017 and highest correlation value of 0.58. The COV of model error is 0.28, which is slightly lower than 0.31 provided by the linear model. But the fracture energy is estimated smaller than zero when the brick tensile strength is lower than 0.204 MPa, causing the additional data analysis for the negative values. Therefore the linear model, with rootmean-square error of prediction and correlation values of 0.018 and 0.54, respectively, placing it second among the four functions and selected as the adopted model for brick fracture energy.

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FIGURE 3.18 Fitting a linear relationship between tensile fracture energy versus tensile strength for brick.

3. Model error statistics The log normal distribution is adopted for the model error of the tensile fracture energy of the brick (see Fig. 3.19). The statistics of model error for the tensile test in masonry, assuming that Vtest 5 0.02 and Vspec 5 0.04 (see Section 3.3.3.3, Random Variable 2, and Eqs. 3.6 and 3.7), are shown in Table 3.8.

3.3.3.4 Stochastic analyses Spatial analysis with ρ 5 0.4 In this spatial analysis with ρ 5 0.4, the correlation coefficient for the tensile strength of mortar joints along each course of masonry is 0.4, while the correlation is ignored between the courses based on the test results from Heffler (2009). The cohesion is assumed to have the same correlation coefficient as the tensile strength of the mortar joints due to being fully dependent with the tensile strength of the mortar joint. The brick tensile strength is considered as a statistically independent variable in this analysis. The material parameters to be used in the four-course beam 3D FEA model are listed in Tables 3.9 and 3.10. Nonspatial analysis The nonspatial probabilistic model used the same FEA software and modeling strategy as the spatial analyses. However, for each simulation only one randomly generated value for each material parameter was assigned to all units in the beam (homogenous properties).

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FIGURE 3.19 (A) Probability distributions of model error of brick tensile fracture energy and (B) inverse CDF plot for the model error of brick fracture energy.

TABLE 3.8 Statistics for model error of brick fracture energy.

Gf for brick

3.3.4

Sample sizes

Mean (Nmm/mm2)

VM

VME

Distribution

20

1.00

0.31

0.31

Log normal

Discussion of the variation in the experimental results

The coefficient of variation for the peak load obtained directly from experimental tests is 0.22 (see Table 3.4). However, this COV includes (1) Vtest, the coefficient of variation in the measured test data due to the accuracy of

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TABLE 3.9 Statistics of material properties adopted to be used in the FEA model. Variables Tensile strength of mortar ftm (MPa) Tensile fracture energy of mortar, (Nmm/mm2)

GfI

COV

Distribution

1.21

0.30

Log normal

0.66

Log normal

0.27

Log normal

Eq. (3.5)

Model error for GfI Shear fracture energy of mortar

Mean

1.04 GfII

2

(Nmm/mm )

Cohesion, Co (MPa)

Eq. (3.8) 1.81

Correlated with ftm Tensile strength of brick, ftb (MPa)

3.13

Tensile fracture energy of brick, Gf (Nmm/mm2)

Eq. (3.9)

Model error for Gf

1.00

0.34

Weibull

0.31

Log normal

the test measurements and the definitions of failure; and (2) Vspec, the uncertainties due to the variation in the actual specimen dimensions from those specified. Ellingwood et al. (1980) suggests that for testing RC beams and columns Vtest varies from 0.02 to 0.04, and Vspec is about 0.04. However, Vtest and Vspec for the four-course masonry beam here would tend to be higher due to a more difficult test setup and the fabrication of the masonry panels. Engineering judgement and statistical knowledge are applied to broadly estimate Vtest (e.g., Li et al., 2016). Assuming a normal distribution, Vtest is estimated as 0.10
0.16. The value of Vspec is considered to be slightly higher for masonry than for RC elements, but to be conservative we assume a lower limit of Vspec 5 0.04. Therefore based on the conditions above, the COV of actual horizontal beam strength is 0.15
0.19.

3.3.5

Comparison of results

Table 3.11 shows the comparison of the peak load statistics for the 310 experimental test results and the 150 FEA simulation results for the spatial analysis with ρ 5 0.4 and 150 FEA simulation results for the nonspatial analysis. Table 3.11 indicates that the mean peak load of the experimental test results lies within the bounds covered by the spatial simulation results (ρ 5 0.4) and the nonspatial simulation results. The spatial simulation underestimated the experimental mean by 0.9%, and the nonspatial simulation overestimated the experimental mean by 10.3%. The COV of the peak load from the test (0.15
0.19) is nearly in the middle of the spatial simulation

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TABLE 3.10 Summary of material parameters to be used in the 3D FEA model. Brick/mortar

Property

Value

Horizontal and vertical mortar joint interface elements

Linear normal stiffness modulus

333 N/mm3 (Lawrence, 1983)

Linear tangential stiffness modulus

148 N/mm3 (Lawrence, 1983)

Tensile strength

Variable (Lawrence, 1983; Heffler et al., 2008)

Tensile fracture energy

Variable (Heffler, 2009)

Cohesion

Variable (Lawrence, 1983; Masia et al., 2012)

Tangent friction angle

0.75 (Heffler, 2009)

Tangent dilatancy angle

0.6 (Heffler, 2009)

Tangent residual friction angle

0.75 (Heffler, 2009)

Confining normal stress

21.2 N/mm2(Heffler, 2009)

Exponential degradation coefficient

5 (Heffler, 2009)

Capped critical compressive strength

25 N/mm2 (Heffler, 2009)

Shear traction control factor

9 (Heffler, 2009)

Compressive fracture energy

15 Nmm/mm2 (Heffler, 2009)

Equivalent plastic relative displacement

0.012 (Heffler, 2009)

Shear fracture energy

Variable (Masia et al., 2007)

Brick Young’s modulus

30,700 N/mm2 (Lawrence, 1983)

Brick’s Poisson’s ratio

0.16 (Lawrence, 1983)

Brick density

1800 kg/m3 (Heffler, 2009)

Linear normal stiffness modulus

1000 N/mm3 (Heffler, 2009)

Linear tangential stiffness modulus

1000 N/mm3 (Heffler, 2009)

Tensile strength

Variable (Lawrence, 1983)

Tensile fracture energy

Variable (Lourenco et al., 2005)

Expanded brick elements

Potential brick cracks

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TABLE 3.11 Comparison of the peak load of four-course beam in horizontal bending. Sample size

Minimum (kN)

Maximum (kN)

Mean (kN)

σ (kN)

COV

Test (Lawrence, 1983)

310

4.2

17.4

10.7

2.35

0.15
0.19

Spatial simulation with ρ 5 0.4

150

6.8

14.3

10.6

1.29

0.12

Nonspatial simulation

150

4.5

21.2

11.8

2.23

0.19

TABLE 3.12 Distribution of joint failure locations in horizontal beams (%). Failure line 1

2

3

4

5

6

7

Test (Lawrence, 1983)

0

9%

24%

29%

27%

11%

0

Spatial simulation with ρ 5 0.4

0

2%

20%

53%

22%

3%

0

Nonspatial simulation

0

0

0

100%

0

0

0

with ρ 5 0.4 (0.12) and the nonspatial (0.19) results, rather than closer to the spatial result as may have been expected. This indicates that the values of the tensile strengths of brick from unit-to-unit in the four-course beam may not be completely statistically independent, as assumed in the spatial analyses with ρ 5 0.4. The model error for FEA simulations with ρ 5 0.4 can be obtained by 10.7/10.6 5 1.01. Apparently, the spatial simulation with ρ 5 0.4 is considered as a method closer to test results with the model error statistics of mean 5 1.01 and VME 5 0.18. Table 3.12 shows the distribution of the failure crack locations in the horizontal beams. The number of times that the main failure crack was located in the 4th failure line, which is the middle of the horizontal beam (see Fig. 3.9), for the spatial simulation with ρ 5 0.4 is nearly 1.8 times that in the test, with correspondingly fewer failures further from midspan in the simulation compared to the tests. This is consistent with the lower COV of the peak failure loads for the spatial simulation with ρ 5 0.4 results compared to the experimental test results. For the nonspatial simulations, failure occurred in the midspan of the beam in every realization, as the maximum moment occurs in the midspan when considering the influence of the masonry self-weight. From Tables 3.11 and 3.12 it is clear that the spatial simulation

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with the ρ 5 0.4 FEA model can better represent the experimentally observed mean peak load and wall performance, when compared to the nonspatial simulations. Note that the failure mode changes from the midspan of the beam to the third or fifth failure crack location when considering the possible asymmetrical loading (see Section 3.3.4). This phenomenon certainly makes the failure mode in the spatial analysis closer to the experimental test results. Fig. 3.20 shows the load versus displacement curves generated in the FEA simulations with the dotted line being a typical experimental test result provided by Lawrence (1983). This is another way to illustrate the variability of the peak loads. It is evident that the spatial simulation analyses with ρ 5 0.4 result in less variable wall performance, compared with the results from the nonspatial analyses. However, the comparison of those results shows that the predicted displacement of the wall at the peak load is approximately two-thirds of the experimental value. This may be due to the fact that the boundary conditions applied in the model assume no movement in the four-course beam for all the support nodes (e.g., the roller and pin). However, this is an assumption regarding the supports of the experiment, because in the experimental tests this may not be the case. Possible wall support movement in the experiment may lead to a lower apparent stiffness. The

FIGURE 3.20 Load versus displacement of the horizontal beam. (A) Spatial simulation with ρ 5 0.4 and (B) nonspatial simulation.

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Probability density

(A)

0.3 0.25 0.2 0.15 0.1 0.05 0 5

10

15

20

Peak load (kN)

(C) 0.3

0.3

0.25

0.25

Probability density

Probability density

(B)

0.2 0.15 0.1 0.05

0.2 0.15 0.1 0.05

0 5

10

15

Peak load (kN)

20

0 5

10

15

20

Peak load (kN)

FIGURE 3.21 Peak load histograms for the test data and simulation results. (A) Test results; (B) spatial simulation with ρ 5 0.4; (C) nonspatial simulation.

histograms of peak load obtained from the test and simulations are shown in Fig. 3.21. For more details see Li et al. (2016).

3.4 Spatial analyses of unreinforced masonry walls in two-way bending 3.4.1

Introduction

To assess the ability of the FEA stochastic model to duplicate URM wall behavior in two-way bending, the 2.5 m 3 2.5 m wall tests reported by Lawrence (1983) were modeled with two types of support conditions, four sides simply supported (support category 1 in the Lawrence test) and the two sides and bottom edge all simply supported but the top edge unsupported (support category 5 in the Lawrence test). Lawrence (1983) performed extensive laboratory tests of wall panels subjected to two-way (biaxial) bending. The walls were supported in a stiff frame, and uniform lateral loads were applied using air bags, which were supported by a reaction frame.

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These tests included various wall dimensions, including 6 m (length) 3 3 m (height), 2.5 m 3 2.5 m, 3.75 m 3 2.5 m, 5 m 3 2.5 m, and 6 m 3 2.5 m. As stated in Section 3.3, the experimental work was separated into seven mortar batches, but the materials were kept as consistent as possible from batch to batch. Only one test was performed for a given wall dimension, batch number, and support category, except for one case in which two nominally identical panels were tested with 6 m 3 3 m, batch number 1, and four sides with built-in support (BI). The smallest wall dimensions of 2.5 m 3 2.5 m are selected for the Monte Carlo simulations in the current study due to the simulation running time. Along with the full-sized wall test, several small specimen tests were also conducted by Lawrence (1983) to obtain material properties for the mortar and brick as stated in Section 3.3.2. Note that only seven panels with 2.5 m 3 2.5 m were tested by Lawrence (1983), including four from batch 2 (two sides with built-in support but top and bottom edges simply supported; three and four sides simply supported; and four sides with built-in support); one from batch 3 (two sides with built-in support and bottom edge simply supported and top edge unsupported); and two from batch 7 (two sides with built-in support but top and bottom edges simply supported). Therefore only the two panels with 2.5 m 3 2.5 m (four and three sides simply supported) from batch 2 are selected. For consistency with the experimental results, the material parameters to be used in the 3D FEA model Monte Carlo simulation in this chapter are the test results (Lawrence, 1983) of batch 2. Tables 3.13 and 3.14 show the material properties to be used in the two-way bending TABLE 3.13 Statistics of the material properties adopted to be used in the FEA model (two-way bending). Variables

Mean

COV

Distribution

Direct tensile strength of mortar joint, ftm (MPa)

0.87

0.25

Log normal

Tensile fracture energy of mortar joint, GfI (Nmm/mm2)

Eq. (3.5)

Model error for GfI

1.04

0.66

Log normal

Shear fracture energy of mortar joint, GfII (Nmm/mm2)

Eq. (3.8)

Cohesion, Co (MPa)

1.34

0.38

Log normal

Correlated with ftm Direct tensile strength of brick, ftb (MPa)

3.82 2

Tensile fracture energy of brick, Gf (Nmm/mm )

Eq. (3.9)

Model error for Gf

1.00

0.18

Weibull

0.31

Log normal

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TABLE 3.14 Summary of the material parameters to be used in the 3D FEA model (two-way bending). Brick/mortar

Property

Value

Horizontal and vertical mortar joint interface elements

Linear normal stiffness modulus

338 N/mm3 (Lawrence, 1983)

Linear tangential stiffness modulus

145 N/mm3 (Lawrence, 1983)

Tensile strength

Variable (Lawrence, 1983; Heffler et al., 2008)

Tensile fracture energy

Variable (Heffler, 2009)

Cohesion

Variable (Lawrence, 1983; Masia et al., 2012)

Tangent friction angle

0.75 (Heffler, 2009)

Tangent dilatancy angle

0.6 (Heffler, 2009)

Expanded brick elements

Potential brick cracks

Tangent residual friction angle

0.75 (Heffler, 2009)

Confining normal stress

21.2 N/mm2 (Heffler, 2009)

Exponential degradation coefficient

5 (Heffler, 2009)

Capped critical compressive strength

25 N/mm2 (Heffler, 2009)

Shear traction control factor

9 (Heffler, 2009)

Compressive fracture energy

15 Nmm/mm2 (Heffler, 2009)

Equivalent plastic relative displacement

0.12 (Heffler, 2009)

Shear fracture energy

Variable (Masia et al., 2007)

Brick Young’s modulus

26,400 N/mm2 (Lawrence, 1983)

Brick’s Poisson’s ratio

0.18 (Lawrence, 1983)

Brick density

1800 kg/m3 (Heffler, 2009)

Linear normal stiffness modulus

1000 N/mm3 (Heffler, 2009)

Linear tangential stiffness modulus

1000 N/mm3 (Heffler, 2009)

Tensile strength

Variable (Lawrence, 1983)

Tensile fracture energy

Variable (Lourenco et al., 2005)

simulations. The material properties in Tables 3.13 and 3.14 are different from those in Section 3.3, because the material in this section is from batch 2 in Lawrence (1983), but the material in Section 3.3 was summarized from all seven batches.

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Han (2007) simulated the two-way bending of the 2.5 m 3 2.5 m masonry wall of Lawrence (1983) using deterministic material properties using Abaqus FEA software. However, the results overestimated test results by 18%, due to the fact that the variability of brick and mortar material properties in the experimental program were not accounted for in the deterministic numerical model. By contrast, the study reported in the current section predicts the strength of URM walls subject to two-way bending including consideration of the spatial variability of all the masonry material properties considered to significantly influence the behavior. These properties are tensile strength of mortar joints, tensile fracture energy of mortar joints (correlated with tensile strength of mortar joints), cohesion of mortar joints (shear bond strength, correlated with tensile strength of mortar joints), shear fracture energy of mortar, tensile strength of brick, and tensile fracture energy of brick (correlates with tensile strength of brick), which are the same as those mentioned in Section 3.3.3 for horizontal bending. A 3D FEA combined with stochastic Monte Carlo simulation is used. The simulation results are compared to the experimental results by Lawrence (1983) in terms of the wall failure progression and capacity. The two support categories (four sides simply supported and three sides simply supported) carried out in this section are simulated based on the spatial analyses with ρ 5 0.4 and the nonspatial analysis. The methods of model establishment and material property generation are the same as those in Section 3.3.3.3, meaning that the spatial variability of material property, including the mortar joints and brick units, are under consideration. The mesh type and density adopted here are the same as those in Section 3.3.3.1 (see Table 3.6). The way in which the wall panels were loaded (uniform pressure acting normal to the wall surface) is the same as described for the one-way vertical bending in Section 3.2.

3.4.2

Case 1—Modeling of a wall simply supported on four sides

The first case modeled here was for a 2.5 m 3 2.5 m wall that was simply supported on four edges (support category 1 in the experimental work (Lawrence, 1983)), and was observed to exhibit failure pattern C in the laboratory tests (see Fig. 3.24D). Fig. 3.22 shows the FE mesh and boundary conditions of the 3D FEA wall with four sides simply supported. Table 3.15 summarizes the Monte Carlo simulation results of the peak load for the four sides simply supported 2.5 m 3 2.5 m wall with spatial simulation (ρ 5 0.4) and nonspatial simulation. Fig. 3.23 illustrates the histograms of the peak load for the spatial simulations (ρ 5 0.4) and the nonspatial simulation. It is shown in Table 3.15 that the nonspatial simulation offers 20.8% higher mean peak loads compared with the spatial simulations with ρ 5 0.4.

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FIGURE 3.22 Mesh and boundary conditions of the wall used in the finite element model— four sides simply supported preventing the displacement in z direction.

TABLE 3.15 Simulation results of peak load for 2.5 m 3 2.5 m wall with four sides simply supported. Mean (kPa)

σ (kPa)

COV

11.1

9.34

0.85

0.09

6.7

25.1

11.28

2.41

0.21

8.6

8.6

8.6

Sample size

Minimum (kPa)

Maximum (kPa)

Spatial simulation ρ 5 0.4

150

6.4

Nonspatial simulation

150

Test result (Lawrence, 1983)

1

Table 3.15 also shows that the nonspatial simulation returns the larger range between the minimum and maximum peak loads, the higher mean peak load, and the highest COV value. The reason is that there is a higher likelihood of walls all having low or high mortar strength, resulting in high variability of wall strengths. The test result by Lawrence (1983) is listed in Table 3.15 as well. The experimental ultimate pressure of 8.6 kPa is lower than both of the mean

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129

0.6

Probability density

0.5 0.4 0.3 0.2 0.1 0 (B)

10

15 20 Wall peak load (kPa)

25

10

20 15 Wall peak load (kPa)

25

0.6

Probability density

0.5 0.4 0.3 0.2 0.1 0

FIGURE 3.23 Simulation histograms of peak load for 2.5 m 3 2.5 m wall with four sides simply supported. (A) Spatial simulation with ρ 5 0.4 and (B) nonspatial simulation.

simulation values in Table 3.15, and represents the 19.2% and 13.3% percentiles of the spatial simulations with ρ 5 0.4 and the nonspatial simulations, respectively, assuming the FEA results following a normal distribution. Although the FEA results seem to overestimate the ultimate pressure of the wall based on the mean values, they display a range covering the single experimental result very well. Further, the spatial correlation in the brick, which was not considered in the present study, may lead to a lower wall pressure similar to the role of ρ values in the mortar joint strength. Alternatively, it is possible that the single experimental result just happens to be lower than average. Experimentally, it was observed that for this type of support, the first crack was usually horizontal, through the bed joint near the mid-height of the panel. However, when the length-to-height ratio was low, vertical cracking near the mid-length occurred first. A cracking pressure of 7.6 kPa and ultimate pressure of 8.6 kPa were recorded for the wall undergoing the test.

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Note that the last scan of the instrumentation (data logger) was at 8 kPa (see Fig. 3.25), but visual observation of the meter showed failure at 8.6 kPa. Fig. 3.24A
C show a typical failure mode from the spatial simulation with ρ 5 0.4, which illustrates the simulated crack formation at the early stage and the final stage of cracking. At first, there are some horizontal cracks in the mortar joints near the mid-length of the panel. Then, the first vertical crack, defined as the first crack of the brick, occurs in the middle region of the wall (see Fig. 3.24A). It was found that the first crack does not necessarily happen in the middle of the wall among the 150 simulations due to the random brick strengths in the wall. Comparison of the final crack pattern shown in Fig. 3.24C with the experimentally observed pattern (Fig. 3.24D) shows that the two are generally in good agreement. The diagonal crack formation at the lower part of

FIGURE 3.24 Crack progression for 2.5 m 3 2.5 m wall from spatial simulation with ρ 5 0.4 and experiment for four sides simply supported: (A) first vertical cracks (simulation); (B) before full crack pattern (simulation); (C) full crack pattern (simulation); (D) experimental result (Lawrence, 1983).

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the wall in the simulation is not as distinct as that at the upper part, which may be due to the self-weight effect of the bricks that strengthens the lower part of the wall. Fig. 3.25 shows the comparison of the pressure
deflection curve between a typical simulation randomly selected from spatial simulation with ρ 5 0.4 and the experiment from Lawrence (1983) (Fig. 3.26, test). The same scales of the x and y axes are used in the simulation (see Fig. 3.25A) and experimental results (see Fig. 3.26) to make a good comparison between them.

FIGURE 3.25 A typical pressure
deflection for 2.5 m 3 2.5 m wall from spatial simulation with ρ 5 0.4 and experiment for four sides simply supported: (A) a typical prediction results and (B) prediction results in detail.

10

FEA simulation Test (Lawrence 1983)

9 8

Pressure (kPa)

7 6 5 4 3 2 1 0

0

10

20 30 40 Displacement (mm)

50

60

FIGURE 3.26 Pressure
deflection for 2.5 m 3 2.5 m wall from spatial simulation with ρ 5 0.4 and experiment for four sides simply supported.

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Fig. 3.25B is the magnified graph presenting a clearer view of the behavior of the simulated wall panel in the lower displacement range. The model provides a value of 6.02 kPa for the pressure at the first obvious vertical cracks, which is a small difference from the experimental cracking pressure of 7.6 kPa due to the randomly generated masonry materials in the wall. In addition, the model shows that there is excellent agreement on the ultimate pressure (8.61 kPa) in the model, whereas the experiment has given the same value of 8.6 kPa. Although Fig. 3.25A and B are only typical examples from the 150 simulations, the minimum and maximum peak loads of the 150 walls ranging from 7.8 to 10.7 kPa cover the experimental result very well. Fig. 3.26 shows the wall behavior of the lateral pressure versus deflection in the middle point of the wall for the mean of 150 spatial simulations with ρ 5 0.4 and the experimental result by Lawrence (1983). Comparison of the two results shows that there is a large difference between the predicted deflection of the wall at the peak load and the experimental value. This may be due to the fact that the boundary conditions applied in the model assume no movement is allowed normal to the wall plane for all the support nodes. However, in the experiment this may not be the case. Possible slight wall support movement in the experiment may lead to lower apparent stiffness. The nominal capacity of the wall with four sides simply supported is calculated as 8.9 kPa based on Standards Australia (2011). The nominal capacity (8.9 kPa) represents the 29% percentile of the ρ 5 0.4 simulations. Generally, the nominal capacity is expected to be a lower percentile of the simulation results. If the spatial simulations with ρ 5 0.4 is deemed the most realistic, then the wall strength provided by the design model in the code Standards Australia (2011) represents a 30th percentile, which likely overestimates the wall strength leading to nonconservative strength predictions. For more details see Li (2016) and Li et al. (2017).

3.4.3 Case 2—Modeling of a wall simply supported on both vertical sides and the bottom edge URM walls simply supported on both sides and the bottom edge with the top edge unsupported (support category 5 in the experimental work (Lawrence, 1983)) lead to the type A failure mode as mentioned in Lawrence (1983) (see Fig. 3.29C), that is, the failure follows immediately as soon as the first crack forms without any progression of failure. Han (2007) simulated this support category wall without considering the variability of the material parameters. She predicted good agreement in terms of the wall failure mode, and an 18% higher failure pressure (9.18 kPa) compared to the experimental value (7.8 kPa). She claimed that the overestimation in the load capacity of the wall was quite likely due to the fact that the numerical model did not include the possibility of the flexural failure of bricks. The wall modeled

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here includes the possibility for flexural failure of the bricks as well as the spatial variability of the material properties of the brick and mortar. Fig. 3.27 shows the FE mesh and boundary conditions of the 3D FEA wall with three sides simply supported. The mesh types, density, and material properties to be used in this category are the same as that in Section 3.4.2, Case 1 (see Tables 3.13 and 3.14). Table 3.16 shows the simulation results of the peak load for the three sides simply supported (both sides and the bottom edge) 2.5 m 3 2.5 m wall,

FIGURE 3.27 Mesh and boundary conditions of the 2.5 m 3 2.5 m wall used in the finite element model, three sides simply supported preventing the displacement in the z direction.

TABLE 3.16 Simulation results for 2.5 m 3 2.5 m wall with three sides simply supported. Mean (kPa)

σ (kPa)

COV

7.8

6.80

0.40

0.06

4.1

10.4

7.03

1.24

0.18

7.8

7.8

7.8

Sample size

Minimum (kPa)

Spatial simulation ρ 5 0.4

150

5.6

Nonspatial simulation

150

Test result (Lawrence, 1983)

1

Maximum (kPa)

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PART | I Seismic vulnerability analysis of masonry and historical structures (A)

Probability density

1 0.8 0.6 0.4 0.2 0

4

6 8 Wall peak load (kPa)

10

4

6 8 Wall peak load (kPa)

10

(B)

Probability density

1 0.8 0.6 0.4 0.2 0

FIGURE 3.28 Simulation histograms of peak load for 2.5 m 3 2.5 m wall with three sides simply supported: (A) spatial simulation with ρ 5 0.4 and (B) nonspatial simulation.

with spatial simulation (ρ 5 0.4) and nonspatial simulation. Fig. 3.28 shows the histograms of the peak load for the spatial simulations (ρ 5 0.4) and the nonspatial simulation. The predicted mean pressure load from the spatial analysis shown in Table 3.16 is lower than that observed in the experiment (7.8 kPa). It is shown that the experimental result nearly reaches the upper limits of peak pressure in the spatial analysis. The FEA spatial simulation provides a mean 12.8% lower than the experimental result. However, comparing one experimental result with the mean from FEA simulation is not particularly informative because the single experimental result could easily be lower or higher than average. It is worth highlighting that the single experimental result falls within a range which, based on the variability of wall strengths, and is likely to occur. Table 3.16 clearly shows that the FEA results display a range covering the single experimental result very well.

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As shown in Table 3.16, the nonspatial simulation provides a 2.9% higher mean peak load than the spatial analyses. This is because the weaker brick tensile strength in the upper part of the wall could dominate the peak pressure, and the possibility of having weaker brick strength in the spatial analysis is obviously higher than that in the nonspatial analysis. Moreover, the stronger brick strength in the spatial analysis cannot definitely lead to the higher peak load due to the crack starting from the weaker, while in the nonspatial analysis, the stronger brick strength would undoubtedly result in higher peak pressure. Therefore the mean peak pressure obtained in the spatial analysis is lower than that in the nonspatial analysis. It also can be seen in Table 3.16 that the deviation between the minimum and maximum wall strengths in the nonspatial simulation is nearly three times higher than the spatial simulation, and that the standard deviation and COV of the peak load obtained from the nonspatial simulation are nearly three times higher than the spatial simulation. This phenomenon can be explained by the load sharing between the brick strengths. It is evident that the probability that the weaker brick is surrounded by strong bricks in the spatial analyses is much higher than that in the nonspatial analysis. That means a weaker (or stronger) brick in the spatial analysis cannot lead to a lower (or higher) peak pressure, while in the nonspatial case, the weaker (or stronger) brick will lead directly to a lower (or higher) peak pressure due to no load sharing between bricks because of identical strength. Hence, the significant difference of variation between spatial and nonspatial analyses. The experimental results by Lawrence (1983) show that the first crack was always vertical, usually near the mid-length of the panel. The final failure pattern observed in the laboratory test for this type of wall is shown in Fig. 3.29C. It is observed, experimentally, that the cracking pressure, full crack pattern pressure, and ultimate pressure had the same value of 7.8 kPa for the three sides simply supported wall. Note that the last scan of the instrumentation (data logger) was at 7 kPa (see Fig. 3.30), but visual observation of the instruments showed failure at 7.8 kPa.

FIGURE 3.29 Full crack patterns for 2.5 m 3 2.5 m wall with three sides simply supported: (A) first vertical cracks (simulation); (B) full crack pattern (simulation); (C) experimental result (Lawrence, 1983).

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FEA simulation Test (Lawrence 1983)

6

Pressure (kPa)

5

4

3

2

1

0

0

10

20

30

40

50

Displacement (mm) FIGURE 3.30 Pressure
deflection for 2.5 m 3 2.5 m wall from spatial simulation with ρ 5 0.4 and experiment for three sides simply supported.

The crack development provided by the numerical analysis, as the lateral pressure was increased, is shown in Fig. 3.29. The first cracks were seen to appear in the perpend joints between the bricks on the top several courses of the wall panel, rather than in the mid-length, due to the inclusion of material variability (see Fig. 3.29A). Thereafter, vertical cracks formed in the upper part of the wall extending through perpend joints and brick units followed by the formation of the diagonal cracks (see Fig. 3.29B). The deformation in those figures has been enlarged 1000 times to make them obvious for the comparison with the test results (Fig. 3.29C). The main full crack pattern is shown in Fig. 3.29B, which agrees very well with the crack pattern observed in the test. There are still other minor cracks in Fig. 3.29B because of the variability of the masonry properties, but only one of them develops further and connects together, forming a vertical crack on the upper part of the wall. Fig 3.30 shows the curves of lateral pressure versus deflection in the middle point of the wall for the mean of 150 spatial simulations with ρ 5 0.4 and the experimental result by Lawrence (1983). It is shown in Fig. 3.30 that the wall behaviors between the FEA simulation and experimental result agree very well, unlike the large difference in deflection of the wall at the peak load in the four sides simply supported condition (see Fig. 3.26). Maybe the reason that there is closer match between experiment and FEA for case 2 (three sides simply supported) than case 1 (four sides simply supported) is as follows: For case 2 the behavior in the top portion of the wall

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where most deflection occurs is essentially one-way horizontal bending so the model does a good job of matching the experiment. For case 1, which is truly two-way behavior, the model assumes perfect contact at the supports on all four edges. But in the experiment, it is very difficult to ensure the equal contact around all the edges at the start of the test. As load is applied, one of the spanning directions could be favored over the other, until some damage occurs and loads redistribute to the true two-way support assumed in the FEA. This effect probably does not influence the final damage pattern too much, but it could influence the deflections required to achieve the damage pattern. The nominal capacity of the wall with three sides simply supported is predicted using Standards Australia (2011) as 6.8 kPa, and represents the 51% percentile of the ρ 5 0.4 spatial simulations. Hence, the observations that the Standards Australia (2011) code prediction corresponds to a 51th percentile for spatial simulations with ρ 5 0.4 strongly suggests that the design model in Standards Australia (2011) code predictions overestimates the wall strength and so is nonconservative. For more details see Li (2016) and Li et al. (2017).

3.5 Concluding remarks A combined Monte Carlo and FEA simulation was conducted in this chapter to investigate how the spatial variability of the masonry material properties affects the failure progression and strength for URM walls subjected to outof-plane lateral loading. The work carried out here contributes to the discussion on the wall strength and wall failure mode between the two analyses, with and without considering the spatial variability of the masonry material properties (the spatial and nonspatial analyses), for walls subject to one-way vertical, one-way horizontal, and two-way bending. The specific contributions made by the chapter include: 1. The chapter developed methods for stochastic FEA for the prediction of wall strength considering the unit-to-unit spatial variability and improved probabilistic material models for masonry unit strength and unit-to-unit spatial variability. 2. The chapter conducted the modeling of the spatial analyses coupled with a Monte Carlo FEA simulation with correlation coefficient (ρ) being 0.4 in the mortar joints in a course of the URM, which is adapted for the spatial correlation of the experimental results. In summary, the main findings of the study are as follows: 1. For one-way vertical bending, a nonspatial analysis coupled with a Monte Carlo simulation overestimates the mean of the base cracking load, mid-height cracking load, and peak load of the wall, and

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underestimates the peak load of the wall in the lower tail of the distribution of the wall strengths compared with a spatial analyses with ρ 5 0.4. Hence, a nonspatial analysis will overestimate the probability of wall failure compared to a more accurate spatial analysis. 2. For one-way vertical bending, the failure mode in the nonspatial analysis is symmetrical due to the uniform direct tensile strength of all units in the wall, while for the spatial analyses, the failure processes are random and asymmetrical, which is considered as the more realistic failure mode. 3. In the four-course beam subject to one-way horizontal bending, very reasonable agreement was found on the peak load prediction and the failure location between the spatial simulation with ρ 5 0.4 and the experimental masonry wall tests by Lawrence (1983). The COV of the test results lies in the middle of the COV values provided by the spatial and nonspatial analyses. Therefore the spatial simulation with ρ 5 0.4 is the more realistic compared with the nonspatial simulation. 4. In two-way bending, the FEA model can capture the failure patterns quite well and predict both the cracking and ultimate loads quite well compared with the experimental tests for walls with four sides simply supported and three sides simply supported.

Acknowledgment The authors gratefully acknowledge the financial support of the Australian Research Council and Think Brick Australia for Linkage Project LP0669538, the National Natural Science Foundation of China under Grant 51708391, as well as test data results from Paulo B. Lourenco and technical assistance provided by Dr. Robert Petersen.

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Masia, M.J., Simundic, G., Page, A.W., 2012. Assessment of the AS3700 relationship between shear bond strength and flexural tensile bond strength in unreinforced masonry. In: 15th IBMAC, Florianopolis, Brazil. McNeilly, T., Scrivener, J., Lawrence, S.J., Zsembery, S., 1996. A site survey of masonry bond strength. Australian Civil/Structural Engineering Transactions. IEAust, pp. 103
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the lessons from the Newcastle earthquake. Research Report No. 073.03.1992. Department of Civil Engineering and Surveying, the University of Newcastle. Page, A.W., 1995. Unreinforced Masonry Structures
An Australian Overview in the Pacific Conference on Earthquake Engineering, Parkville, VIC, Australia. Petersen, R.B., 2009. In-plane Shear behaviour of unreinforced masonry panels strengthened with fibre reinforced polymer strips. PhD thesis, The University of Newcastle, Australia. Petersen, R.B., Ismail, N., Masia, M.J., Ingham, J.M., 2012. Finite element modelling of unreinforced masonry shear wallettes strengthened using twisted steel bars. Constr. Build. Mater. 33, 14
24. Potter, R.J., 1994. Earthquake and Masonry
An Overview in the 3rd National Masonry Seminar, Brisbane, Queensland, Australia. Scrivener, J.C., 1993. Masonry Structures
Earthquake Resistant Design and Construction, in the Earthquake Engineering and Disaster Reduction Seminar. Australian Earthquake Engineering Society, Melbourne, Australia, pp. 39
44. Standards Australia, 2001. AS 3600 2001: Concrete structures, 2011. Standards Association of Australia, Sydney. Standards Australia, 2003. AS/NZS 4456.15 2003: Masonry units, segmental pavers and flags
Methods of test, Method 15: Determining lateral modulus of rupture, 2003. Standards Association of Australia, Sydney. Standards Australia, 2011. AS 3700 2011: Masonry structures, 2011. Standards Association of Australia, Sydney. Stewart, M.G., Lawrence, S.J., 2002. Structural reliability of masonry walls in flexure. Masonry Int. 15 (2), 48
52. Sutcliffe, D.J., Yu, H.S., Page, A.W., 2001. Lower bound limit analysis of unreinforced masonry shear walls. Comput. Struct. 79, 1295
1312. Van Der Pluijm, R., 1997. Non-linear behaviour of masonry under tension. Heron 42 (1), 25
54. Vanmarcke, E.H., 1983. Random Field: Analysis and Synthesis. The LIT Press, Cambridge, Massachusetts, London. Vaculik, J., Griffith, M.C., 2010. Probabilistic approach for calculating the ultimate strength of unreinforced masonry in horizontal bending. In: 8th International Masonry Conference, Dresden. Van Zijl, G.P.A.G., 2004. Modeling masonry shear-compression: role of dilatancy highlighted. J. Eng. Mech. 130 (11), 1289
1296. Willis, C.R., Griffith, M.C., Lawrence, S.J., 2004. Horizontal bending of unreinforced clay brick masonry. Masonry Int. 17 (3), 109
121.

Chapter 4

Seismic assessment of historic masonry structures: out-ofplane effects N. Mendes and P.B. Lourenc¸o ISISE, University of Minho, Guimara˜es, Portugal

4.1 Introduction Natural hazards such as tsunamis, earthquakes, heat waves, cyclones, and volcanic eruptions have caused severe losses globally. Earthquakes have a high contribution to these losses; here, the Haiti earthquake of 2010 that resulted in 159,000 deaths (Fig. 4.1). It is also estimated that in this earthquake 230,000 buildings collapsed or were severely damaged. In general, the losses caused by earthquakes are mainly associated with the poor seismic performance of the buildings. Among the different building typologies, it is believed that the existing masonry buildings without box behavior are the most vulnerable buildings. The seismic behavior of existing masonry buildings can be divided into two main types of behavior (Fig. 4.2): (1) in-plane behavior and (2) out-of-plane Earthquake, tsunami (Thailand, December 26, 2004)

2,20,000

Earthquake (Haiti, January 12, 2010)

1,59,000

Cyclone Nargis, storm surge (Myanmar, May 2–5, 2008)

1,40,000

Tropical cyclone, storm surge (Bangladesh, April 29–30,…

1,39,000

Earthquake (Pakistan, October 8, 2005)

88,000

Earthquake (China, May 12, 2008)

84,000

Heat wave, drought (Central Europe, July–August 2003)

70,000

Heat wave (Russia, July–September 2010)

56,000

Earthquake (Iran, June 20, 1990)

40,000

Earthquake (Iran, December 26, 2003)

26,200 0

50,000

1,00,000

1,50,000

2,00,000

2,50,000

FIGURE 4.1 Most severe natural hazards by death toll from 1980 to 2016 (Statista, 2018). Numerical Modeling of Masonry and Historical Structures. DOI: https://doi.org/10.1016/B978-0-08-102439-3.00004-X Copyright © 2019 Elsevier Ltd. All rights reserved.

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FIGURE 4.2 Typical out-of-plane response of existing masonry buildings. Adapted from Bruneau, M., 1994. Seismic evaluation of unreinforced masonry buildings—a state-of-the-art report. Can. J. Civil Eng. 21(3), 512 539 (Bruneau, 1994).

behavior. In-plane behavior has been very well studied in recent decades, and a high number of numerical and experimental research works have been carried out. Thus, there is already significant knowledge on the in-plane behavior of masonry structures, including on the types of failures, namely (1) flexuralrocking failure with possible toe-crushing; (2) shear failure with sliding along the bed joints (horizontal plane); and (3) shear failure with diagonal cracking (Calderini et al., 2009). However, the out-of-plane behavior of masonry buildings is complex and is still a challenge (Lourenc¸o et al., 2011). This type of behavior depends mainly on the following aspects: (1) material properties of the masonry (high specify mass, low tensile, and shear strength, low ductility); (2) stiffness of the horizontal diaphragms (flexible floors and roofs); (3) efficiency of the connections between orthogonal walls and between the horizontal diaphragms and the masonry walls; and (4) in-plane and in-elevation irregularity. Motivated by the challenging issue of the out-of-plane behavior of masonry buildings, about 25 world experts on the seismic behavior of masonry structures were invited to submit blind predictions of the u-shaped masonry structures tested in the LNEC (National Laboratory for Civil Engineering, Lisbon, Portugal) shake table. Then, the experts met for a 1-day workshop on the outof-plane assessment of existing masonry buildings, including a discussion on the results of the blind predictions. Finally, six groups of researchers were invited to present an evaluation a posteriori of the seismic performance of the structures (postdictions). This chapter presents the main results of the shake table tests and the main outcomes of the blind predictions and postdictions.

4.2 Seismic assessment of masonry structures Masonry is a heterogeneous material composed by units and joints with distinct directional properties, in which the mortar joints are planes of

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weakness, in the case of regularly dressed masonry (Lourenc¸o, 1996). Geometrical parameters, such as unit and joint dimensions or the arrangements of bed and head joints, increase the complexity of masonry structural analysis. According to Lourenc¸o (2002), three main modeling approaches can be adopted for masonry, namely (1) detailed micromodeling, in which the units and mortar of joints are represented by continuum elements whereas the unit/mortar interface is represented by discontinuous elements; (2) simplified micromodeling, in which the expanded units are represented by continuum elements whereas the behavior of the mortar joints and unit/ mortar interface is lumped in discontinuous elements; and (3) macromodeling, in which units, mortar, and the unit/mortar interface are smeared out as a homogeneous continuum material. Besides the different modeling approaches in terms of material, several structural analysis techniques can be adopted for masonry structures, such as limit analysis, pushover analysis, and nonlinear dynamic analysis with time integration. Furthermore, four main methods for numerical modeling can be adopted (Fig. 4.3), namely (1) finite element method; (2) discrete element method (DEM); (3) methods based on rigid blocks; and (4) methods based on structural components.

FIGURE 4.3 Examples of models used to evaluate the behavior of masonry structures.

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From the inverted catenary principle, published by Robert Hooke in the 17th century, to the development of graphical catenary-based methods by La Hire in the 18th century, and Rankine and Moseley in the 19th century, rational approaches have been considered. After classic limit analysis and the relation between limit analysis and thrust line was reported by Kooharian (1952), several simplified but more sophisticated analysis methods were developed (Nielsen, 1999). For the assessment of arches two main approaches were developed (Kooharian, 1952; Heyman, 1969), which were later applied to other types of masonry elements, namely (1) the static approach, based on the principles of thrust lines; (2) and the kinematic approach, based on the analysis of failure mechanisms with rigid macroblocks. In general, these approaches correspond to simplified analyses assuming that the masonry has no tensile strength along the block interfaces, has infinite compressive strength and that sliding failure is not permitted. Different assumptions have been implemented for limit analysis with macroblocks (e.g., see Gilbert et al., 2006). The use of graphic methods became outdated due to advances in computer technology (Lourenc¸o, 2002), and the analysis of historical construction using the thrust line approach is difficult to solve, such that the kinematic approaches are more practical and effective. Models based on the rocking motion of monolithic walls, that is, kinematic approach with macroblocks, allow good estimations of the collapse load factor and of the displacement capacity (Doherty et al., 2002; Ordun˜a and Lourenc¸o, 2005; Mendes, 2014; Lagomarsino, 2015). Nonlinear dynamic analyses of rocking systems have been proposed as well for the interpretation of field and laboratory observations (Papantonopoulos et al., 2002; DeJong, 2012; Sorrentino et al., 2014a,b). Mechanisms can be proposed on the basis of the knowledge obtained from postearthquake surveys of similar buildings using the crack patterns obtained from experimental research and on the basis of practitioner experience. Thus, a bad evaluation of the possible mechanisms can lead to the nonconsideration of the mechanism with the lowest load factor and, consequently, to a failure load higher than the real maximum capacity of the structure (Mendes, 2014; Mauro et al., 2015). The numerical models based on the finite element method (FEM) allow several materials and types of elements (beam, shell, solid, etc.) to be easily combined. The nonlinear seismic analysis of masonry buildings through FEM numerical models has been performed using discrete models (simplified micromodeling approach) (Lourenc¸o, 1996), continuous and anisotropic models (macromodeling approach) (Lourenc¸o et al., 1997; Lourenc¸o, 2000) and, mainly, continuous and isotropic models (macromodeling approach) (Pen˜a et al., 2010; Roca et al., 2013; Mendes and Lourenc¸o, 2014). The detailed micromodeling approach has not often been used for masonry buildings, mainly due to the difficulty of mesh preparation using FEM software solutions and the long time needed to run the nonlinear analyses. FEM numerical models based on the macromodeling approach present in general

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several simplifications, in terms of geometry and material properties, with respect to the real nonlinear dynamic behavior mainly for complex masonry buildings. The DEM has two types of formulation for masonry structures: (1) discrete models in which the blocks with polyhedral shapes can be assumed as rigid or deformable, and the discontinuities are treated as boundary conditions between blocks; and (2) discrete models with spherical particles, which are not yet practical for larger structures. The DEM models allow realistic representations of complex structures (e.g., monuments composed by domes, vaults, arches, and columns), including detailed block arrangements (Azevedo et al., 2000; De Felice and Giannini, 2001; Lemos et al., 2011) and typical dynamic rocking motion (Pen˜a et al., 2010). DEM is also appropriate for modeling the outof-plane collapse of multileaf masonry walls taking into account the real unit arrangement (De Felice, 2011). Although most FEM codes allow the development of models using the micromodeling approach, only the general contact formulations implemented in DEM and combined FEM DEM (Munjiza, 2004) allow for the development of analyses in the large displacement range. The assessment of the stability of masonry structures may be carried out by using three types of approaches, namely force-based (FBA), displacementbased (DBA), and energy-based (EBA or rigid body-based as named in Ferreira et al., 2014) approaches. According to these approaches the stability is evaluated comparing the demand and capacity of the structure in terms of maximum load capacity/strength (FBA), maximum displacement/deformation (DBA), and energy balance (EBA). For more information on seismic assessment of masonry structures see Sorrentino et al. (2017) and Penna (2015) for the case of stone masonry buildings.

4.3 Shake table tests Two u-shaped masonry structures were built and tested in the shake table of the LNEC. Both specimens are comprised of a gable wall with an opening, a return wall without openings, and a return wall with an opening, which results in an asymmetric configuration and causes torsional effects. The first specimen (Fig. 4.4) corresponds to an idealized stone house (Stone House) with three-leaf walls, made of irregular granite stones and lime-based mortar. The length of the gable wall and the return walls are equal to 4.15 and 2.50 m, respectively. The maximum high is equal to 3.00 m (middle of the gable wall) and thickness of the walls is equal to 0.50 m. Over the door of the gable wall and the window of the return wall, a stone lintel was placed. The second specimen (Fig. 4.5) was made with fired clay bricks and prebatched hydraulic lime mortar (Brick House). The walls were built with English bond arrangement and thickness equal 0.235 m. The length of the gable wall and the return walls are equal to 3.50 and 2.50 m, respectively. The maximum high, at middle of the gable wall, is equal to 2.75 m. This

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FIGURE 4.4 Stone House. (A) general view and (B) detail of the return wall with opening.

FIGURE 4.5 Brick House. (A) General view and (B) detail of the return wall with opening.

specimen was also timber lintels over the windows of the gable and return walls (Candeias et al., 2017). Six wallets for each masonry type were built and tested under vertical and diagonal loading, aiming at obtaining the material properties. The brick masonry wallets present a specific mass of 2360 kg/m3 and a Young’s modulus of 2080 MPa. The compressive and tensile strength of these wallets are equal to 5.4 MPa and 224 kPa, respectively. The stone masonry presents a specific mass of 1890 kg/m3 and the Young’s modulus is equal to 5170 MPa. The compressive and tensile strength are equal to 24.8 MPa and 102 kPa, respectively. The shake table tests were carried out by applying a unidirectional seismic action orthogonal to the gable wall, in which the N64E component of the February 21, 2011 Christchurch (New Zealand) earthquake was adopted as seismic input of reference. Several tests were performed on each specimen by stages of increasing amplitude up the collapse. In the Stone House and Brick House, six and eight seismic tests were carried, in which the maximum peak

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ground acceleration (PGA) of 1.07 and 1.27 g were reached, respectively. In the instrumentation setup, 20 accelerometers and 6 linear variable displacements transducers were used. The sensors were placed in several levels of the specimen’s height, aiming at measuring the in-plane and out-plane response. Besides the seismic tests, dynamic identification tests were also done, before the first and after each seismic tests, aiming at evaluating the reduction of the frequencies as a function of the applied seismic action (Candeias et al., 2017). The shake table tests showed that, in addition to the fact that the full collapse did not occur for the last tests (PGA equal to 1.07 g), the collapse mechanism of the Stone House (Fig. 4.6) involves the severe damage in the return wall due to the in-plane shear and flexure behavior. Stones at the right-top corner of this wall fell out. The return wall without openings presented low damage. If the test structure was subjected to a further test of increasing amplitude, the gable wall would have likely fallen out and substantial damage to the return wall with opening would have occurred. Outof-plane collapse of the gable wall would likely be a result of diagonal cracking of the left pier and horizontal cracking at midheight of the right pier. The idealized collapsed mechanism can be observed in Fig. 4.9. The

FIGURE 4.6 Still images of the behavior of the Stone House in the last shake table test (PGA 5 1.07 g).

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maximum displacement observed in this specimen is equal to 219 mm and occurred at the top of the gale wall. The last seismic tests (PGA 5 1.27 g) carried out on the Brick House allowed observing the collapse mechanism (Fig. 4.7) that involves the outof-plane failure of the tympanum of the gable wall. Furthermore, the lintel and pier of the return wall with an opening collapsed due to the in-plane behavior, and the right corner rotated due to the torsional effects. Once again, the return wall without openings presents very low damage. The idealized collapsed mechanism can be observed in Fig. 4.10. The maximum displacement occurred at the top of the gable wall (136 mm). The results of dynamic identification tests showed that the reduction of the first natural frequency is equal to 29% (10.2 7.2 Hz) and 54% (21.2 9.7 Hz) for the Stone House and Brick House, respectively. It is noted that the Brick House collapsed and the last frequencies correspond to the dynamic identification test before the last seismic test (after seismic tests with PGA equal to 0.84 g) (Candeias et al., 2017).

4.4 Blind predictions Several experts on the seismic behavior of masonry structures were invited to predict the seismic capacity, including the collapse mechanism, of the

FIGURE 4.7 Still images of the behavior of the Brick House in the last shake table test (PGA 5 1.27 g).

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Stone House and Brick Houses tested in the LNEC shake table. The geometric and material properties (specific mass, Young’s modulus, compressive and tensile strength), and the normalized acceleration response spectrum and accelerogram envelope for both specimens were provided to the experts. No specific requirements on the type of parameter to estimate the collapse and the analytical tools to be used were given to the experts. The experts used several modeling approaches, types of structural analysis, and assessment criteria for predicting the dynamic response of the specimens (Mendes et al., 2017). Three modeling approaches were adopted: (1) modeling approach based on rigid blocks based on the predicted collapse mechanism (total of 23 models of rigid blocks were defined); and (2) modeling approach based on the FEM (10 FEM models were prepared, in which 7 models were developed using the macromodeling approach and 3 models were based on the simplified micromodeling approach). One of the latter used a combined FEM DEM strategy. Fig. 4.8 presents examples of the models used by the experts to predict for both specimens. Three types of structural analysis were also used: (1) limit analysis based on the kinematic approach; (2) static nonlinear analysis (pushover analysis), mainly assuming a lateral load pattern proportional to mass (in some analyses a load pattern proportional to the first mode shape was also adopted); and (3) nonlinear dynamic analysis with time integration. In this type of analysis, the experts generated artificial accelerograms based on the normalized response spectrum and accelerogram envelope of the seismic action measured in the shake table.

FIGURE 4.8 Examples of models developed by the experts for the blind predictions: (A) Stone House and (B) Brick House.

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Most of experts used the limit analysis with rigid blocks and forcedbased and displacement-based criteria to predict the PGA that caused the collapse of the specimens. The collapse mechanisms were defined based on the results of pushover analysis and eigenvalues analysis, for FEM and DEM models, and personal judgment. Furthermore, several numerical tools for structural analysis were used, such as 3DEC, Abaqus Unified FEA, ANSYS, DIANA, LS-DYNA, and Strand7, as well as tools developed by the experts for limit analysis. The collapse mechanisms of the models with rigid blocks and the idealized collapse mechanisms from the FEM and DEM models proposed by the experts were compared with the experimental results (Figs. 4.9 and 4.10). Furthermore, the PGAs of collapse were also evaluated. It is noted that some experts used different assessment methods for the same proposed collapse mechanism, which resulted in more than one PGA for the same collapse mechanism. In the blind predictions of the Stone House, 13 different collapse mechanisms were proposed by the experts (Fig. 4.9), which can be divided into the following types of damage (Mendes et al., 2017): G

G

G

G

G

Partial collapse of the gable wall (Mechanisms 1 4): Out-of-plane overturning of the lintel of the door with diagonal cracks from the top corners of the door to the top of the tympanum. Out-of-plane overturning of tympanum with horizontal cracks at the top of the door. Partial out-of-plane overturning of the gable wall with diagonal cracks from the top corners of the gable wall to the base of the door. Total collapse of the gable wall (Mechanism 5): Out-of-plane overturning of the gable wall with vertical cracks between orthogonal walls and without any collapse of the return walls. Total collapse of the gable wall and partial collapse of the return walls (Mechanisms 6 and 7): Out-of-plane overturning of the gable wall with partial in-plane collapse of one or both return walls. Collapse of the tympanum and partial collapse of the return walls (Mechanism 8): Out-of-plane overturning of tympanum with horizontal cracks and partial in-plane collapse of both return walls. Partial collapse of the gable and return walls (Mechanisms 9 13): Partial out-of-plane overturning of the gable wall, involving diagonal and horizontal cracks, and partial in-plane collapse of the return wall with opening.

The PGA of collapse predicted by the experts for the Stone House ranges from 0.22 to 2.50 g (Fig. 4.11), in which the average PGA is equal to 0.91 g (experimental PGA equal to 1.07 g). The wide dispersion on the results (coefficient of variation (COV) equal to 63%) is mainly associated with predicting the correct collapse mechanism. When considering only the collapse mechanisms similar to the damaged observed in the shake table tests (good

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FIGURE 4.9 Idealized collapse mechanisms for the Stone House in the blind predictions.

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FIGURE 4.10 Idealized collapse mechanisms for the Brick House in the blind predictions.

predictions: Mechanisms 9 13), the estimated PGA of collapse ranges from 0.53 to 1.42 g and dispersion of the results decreases (COV 5 31%). The error between the average PGA of the good predictions and the experimental PGA is equal to 28%. Within the good predictions, 67% of the estimated PGAs are lower than the experimental PGA and only two predictions presented the displacement of collapse, namely 0.16 and 0.25 m at the top of the tympanum. It is noted that the maximum displacement obtained in the shake table tests is equal to 0.22 m, which corresponds to about half of the wall thickness. The good prediction of the House Stone closer to the experimental results presents a PGA equal to 1.11 g (Mechanism 11), with an error of about to 4% with respect to the experimental PGA. Furthermore, this prediction presents also the lower error in terms of displacement of collapse (0.25 m, assuming that the displacement of collapse is equal to half of the wall

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FIGURE 4.11 Seismic acceleration capacity (PGA) obtained by blind predictions: (A) Stone House and (B) Brick House (green and yellow bars correspondent to good and fair mechanism predictions).

thickness). The collapse mechanism proposed in this predictions was based on the damage obtained through a pushover analysis with a FEM model prepared using the simplified micromodeling approach. The maximum load capacity (PGA) was estimated using the limit analysis based on the kinematic approach and the stiffness of the FEM model. In the limit analysis, a flexural tensile strength of masonry parallel and orthogonal to the bed joints equal to 0.10 and 0.20 MPa was assumed, respectively, for the crack at the gable wall (out-of-plane behavior). In the return wall with opening (in-plane behavior), the tensile strength of the masonry obtained from the diagonal

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compression tests (0.22 MPa) was used for the predicted cracks. Furthermore, the dynamic effects were considered using the frequency of first mode and the linear spectral acceleration. In the blind predictions of the Brick House, eight different collapse mechanisms were proposed by the experts (Fig. 4.10). These collapse mechanisms can be organized into the following groups (Mendes et al., 2017): G

G

G

Partial collapse of the gable wall (Mechanisms 1 4): Out-of-plane overturning of the lintel of the door with diagonal cracks from the top corners of the door to the top of the tympanum or to the top corners of the gable wall. Out-of-plane overturning of the tympanum with a horizontal crack. Partial out-of-plane overturning of the gable wall with diagonal cracks from the top corners of the gable wall to the base of the window. Collapse of the gable wall (Mechanisms 5 and 6): Total or partial out-ofplane overturning of the gable wall with vertical cracks between orthogonal walls and without any collapse of the return walls. Partial collapse of the gable wall and returns walls (Mechanisms 7 and 8): Partial out-of-plane overturning of the gable wall, involving diagonal cracks and partial in-plane collapse of the return wall with opening. Partial out-of-plane overturning of the gable wall, involving a vertical crack at the connection between the gable wall and the return wall without openings and partial in-plane collapse of the return wall with opening.

In the assessment of the seismic capacity of the brick, 17 predictions were presented. The PGA of collapse ranges from 0.30 to 1.00 g (Fig. 4.11), with a COV of 39%. The average PGA is equal to 0.64 g, and all the predictions present a PGA lower than the experimental PGA (1.27 g). The collapse mechanism of the Brick House was the most difficult to predict, which can be associated with the slenderness of the structure and the torsional effects clearly observed in the shake table tests. Thus, the experts only presented fair predictions with respect to the collapse mechanism obtained in the shake table tests (Fig. 4.10). The collapse mechanisms classified as fair (Mechanisms 2 and 7) present damage associated with the out-of-plane behavior of the tympanum of the gable wall and damage due to in-plane behavior of the return wall with opening. When only the fair predictions are considered, the average PGA is equal to 048 g (error equal to 262%). Furthermore, the collapse displacement at the top of tympanum predicted by the experts ranges from 0.12 to 0.31 m. It is noted that, based on the shake table tests, the collapse displacement of the tympanum occurred for a displacement less than or equal to 0.14 m (Candeias et al., 2017).

4.5 Postdictions After the presentation of the experimental results and discussion of the blind predictions in the workshop, six groups of researchers were invited to carry

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out a numerical study a posteriori on the seismic behavior of the specimens tested in the shake table. Several modeling approaches and methods were adopted, such as models with rigid blocks, FEM models, DEM models, and models with combined finite discrete elements (Fig. 4.12). Furthermore, different types of structural analysis were used, mainly the pushover analysis and the dynamic analysis with time integration (de Felice et al., 2017). Derakhshan et al. (2017) evaluated the seismic performance of both specimens based on the limit analysis with rigid macroblocks. The failure modes were not simulated and were defined based on the collapse mechanisms observed in the shake table tests. The collapse (or near collapse) was evaluated using a displacement-based criteria and the elastic-displacement spectra. The results showed that this methodology is a good simplified approach to estimating the peak displacement demands for the rocking failure of masonry walls. However, the procedure can overestimate the displacement for low seismic amplitudes, when the walls do not present rocking and the behavior remains almost elastic. AlShawa et al. (2017) developed a study on the seismic behavior of the specimens using models with combined finite discrete elements. The response of the specimens was evaluated through dynamic analysis with time integration based on the explicit formulation. The numerical models were able to simulate correctly the collapse mechanisms observed in the shake table tests. The results of the analyses showed that, in general, the

FIGURE 4.12 Examples of models developed by the experts for the postdictions: (A) Stone House and (B) Brick House.

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displacements are in reasonable agreement with the experimental results, mainly for the last dynamic analysis with the highest seismic amplitudes. In the first dynamic analyses, when the behavior is almost elastic, the numerical displacements are overestimated. In this study, a sensitivity analysis was also carried out, in which the damage accumulation, seismic amplitude, discretization of the mesh, and mechanical parameters were considered. The results allowed us to conclude that the very refined meshes (blocks and finite elements) have low influence on the response, whereas meshes with large elements can lead to inappropriate collapse mechanism predictions. It was also concluded that the shear strength plays a secondary role in the response, with exception when it is very low in comparison with the tensile strength. Gams et al. (2017) evaluated the seismic behavior of the specimens using FEM models based on the simplified micromodeling approach. The methodology involved two main phases: (1) pushover analyses (inwards and outwards direction) to determine the collapse mechanisms; and (2) nonlinear dynamic analysis with time integration to evaluate the damage. The analyses showed that the results are satisfactory only when using predefined mechanisms with enough number of blocks. When using idealized mechanisms with large blocks the results are not satisfactory. Ch´acara et al. (2017) performed a study using two FEM modeling approaches (simplified micromodeling and macromodeling) to simulate the dynamic behavior of the specimens. Furthermore, two types of structural analysis were used, namely the nonlinear static analysis and the nonlinear dynamic analysis with time integration. The results showed that the adopted tools simulate correctly the damage due to in-plane behavior. Furthermore, the collapse mechanism of the Stone House is in good agreement with the experimental results. However, the out-of-plane behavior of the Brick House was not correctly simulated. Lemos and Campos-Costa (2017) developed a DEM model of the Stone House in the 3DEC software. The model was prepared using rigid blocks and interface elements. The stiffness of the interface elements was calibrated based on the experimental frequencies estimated from the dynamic identification tests. Pushover analysis and nonlinear dynamic analysis with time integration were carried out. The numerical analyses were able to reproduce the collapse mechanism and deformation of the model is similar to that observed in the shake table tests. Finally, Cannizzaro and Lourenc¸o (2017) used a macroelement approach to prepare models and to evaluate the out-of-plane behavior of the specimens. Two types of modeling approaches were adopted, namely macromodeling and micromodeling with mesh coherent with the masonry arrangement. The models were able to simulate the experimental in-plane

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FIGURE 4.13 Idealized collapse mechanisms for the Stone House in the postdictions. Adapted from De Felice, G., Santis, S., Lourenc¸o, P.B., Mendes, N., 2017. Methods and challenges for the seismic assessment of historic masonry structures. Int. J. Architect. Herit. 11(1), 143 160, doi:10.1080/15583058.2016.1238976.

behavior of the return walls. However, the out-of-plane response of the gable wall was not perfectly simulated. In general, the postdictions were mainly focused on the simulation of a response of the specimens similar to the behavior observed in the shake table tests. Thus, the PGAs that cause the collapse of the structures were not the main parameter of discussion in these studies and the responses were mainly evaluated in terms of displacements and damage. In general, the damage observed in the models developed in the postdictions are in reasonable and good agreement with collapse mechanisms observed in the shake table tests (Figs. 4.13 and 4.14). In the Stone House, the diagonal cracks at the gable wall of Mechanisms 1 and 2 and the cracks of the return wall with opening of Mechanisms 3 6 are highlighted. In the idealized collapse mechanism for the Brick House, the damage observed in Mechanism 2 is in good agreement with experimental results and is highlighted. However, in general the displacement of collapse was not perfectly simulated, in which an average error of about 117% and 252% was obtained for the Brick House and Stone House, respectively (de Felice et al., 2017).

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FIGURE 4.14 Idealized collapse mechanisms for the Brick House in the postdictions. Adapted from De Felice, G., Santis, S., Lourenc¸o, P.B., Mendes, N., 2017. Methods and challenges for the seismic assessment of historic masonry structures. Int. J. Architect. Herit. 11(1), 143 160, doi:10.1080/15583058.2016.1238976.

4.6 Conclusions The out-of-plane behavior of masonry structures with box behavior is a challenging issue, and there is not a clear definition on the most appropriate assumptions and approaches for modeling this type of structure. Thus, two ushaped masonry structures (Stone House and Brick House) were built and tested in the LNEC shake table. Several experts on the seismic behavior of masonry buildings were invited to present blind predictions on the seismic capacity of the two structures tested in the LNEC shake table. Then, the results of the blind predictions were evaluated, compared with the experimental results, mainly in terms of collapse mechanism and PGA of collapse, and discussed in a workshop. Finally, the six groups of researchers were invited to present a posteriori a study on the seismic assessment of the structures. In the blind predictions, several types of modeling approaches and structural analysis were adopted by the experts. The collapse mechanisms were defined based on FEM models, DEM models, and personal judgment. In general, the assessment of the collapse was carried out using the FBA and the DBAs, in which the PGA was the most common parameter to discuss the collapse. The predictions for the Stone House presented good results, either

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in terms of collapse mechanism or PGA of collapse. In the predictions classified as good, the minimum error obtained was equal to 4%. The predictions on the response of the Brick House presented more difficulties and only fair results were obtained for the collapse mechanism. Furthermore, the minimum error obtained for the PGA of collapse was 21%. These aspects can be associated with the slenderness of the Brick House and the torsional effects observed in the shake table tests. In the postdictions, when the experimental results are known, the researchers also used different approaches to evaluate the seismic performance of the structures, such as models with rigid blocks, FEM models, DEM models, and models with finite discrete elements. The pushover analysis and mainly the nonlinear analysis with time integration were the most common types of structural analysis. In this stage, the main objective was to simulate a response (damage and deformation) similar to the response observed in the shake table tests. In general, the damage of the numerical models are in reasonable and good agreement with the collapse mechanisms observed in the shake table tests. However, in general the deformation was not perfectly simulated and significant errors were obtained. Finally, the blind predictions and postdictions allowed us to conclude that there are several powerful tools available to evaluate the complex out-ofplane behavior of masonry structures without box behavior. However, high dispersion of the results occurred. Furthermore, limit analysis with predefined collapse mechanism can lead to wrong evaluations of the seismic behavior, showing that experience and appropriate engineering judgment are needed for this purpose. Thus, more efforts on the evaluation of the out-ofplane behavior of masonry structures are needed.

References AlShawa, O., Sorrentino, L., Liberatore, D., 2017. Simulation of shake table tests on out-ofplane masonry buildings. Part (II): combined finite-discrete elements. Int. J. Architect. Herit. 11 (1), 79 93. Available from: https://doi.org/10.1080/15583058.2016.1237588. Azevedo, J., Sincraian, G., Lemos, J.V., 2000. Seismic behavior of blocky masonry structures. Earthq. Spectr. 16 (2), 337 365. Available from: https://doi.org/10.1193/1.1586116. Bruneau, M., 1994. Seismic evaluation of unreinforced masonry buildings a state-of-the-art report. Can. J. Civil Eng. 21 (3), 512 539. Calderini, C., Cattari, S., Lagomarsino, S., 2009. In-plane strength of unreinforced masonry piers. Earthq. Eng. Struct. Dyn. 38 (2), 243 267. Available from: https://doi.org/10.1002/eqe.860. Candeias, P.X., Campos-Costa, A., Mendes, N., Costa, A.A., Lourenc¸o, P.B., 2017. Experimental assessment of the out-of-plane performance of masonry buildings through shaking table tests. Int. J. Architect. Herit. 11 (1), 31 58. Available from: https://doi.org/ 10.1080/15583058.2016.1238975. Cannizzaro, F., Lourenc¸o, P.B., 2017. Simulation of shake table tests on out-of-plane masonry buildings. Part (VI): discrete element approach. Int. J. Architect. Herit. 11 (1), 125 142. Available from: https://doi.org/10.1080/15583058.2016.1238973.

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Ch´acara, C., Mendes, N., Lourenc¸o, P.B., 2017. Simulation of shake table tests on out-of-plane masonry buildings. Part (IV): macro and micro FEM based approaches. Int. J. Architect. Herit. 11 (1), 103 116. Available from: https://doi.org/10.1080/15583058.2016.1238972. DeJong, M.J., 2012. Seismic response of stone masonry spires: Analytical modeling. Engineering Structures 40, 556 565. Available from: https://doi.org/10.1016/j.engstruct.2012.03.010. Derakhshan, H., Nakamura, Y., Ingham, J.M., Griffith, M.C., 2017. Simulation of shake table tests on out-of-plane masonry buildings. Part (I): displacement-based approach using simple failure mechanisms. Int. J. Architect. Herit. 11 (1), 72 78. Available from: https:// doi.org/10.1080/15583058.2016.1237590. De Felice, G., 2011. Out-of-plane seismic capacity of masonry depending on wall section morphology. Int. J. Architect. Herit. 5 (4 5), 466 482. Available from: https://doi.org/10.1080/ 15583058.2010.530339. De Felice, G., Giannini, R., 2001. Out-of-plane seismic resistance of masonry walls. J. Earthq. Eng. 5 (2), 253 271. Available from: https://doi.org/10.1080/13632460109350394. De Felice, G., Santis, S., Lourenc¸o, P.B., Mendes, N., 2017. Methods and challenges for the seismic assessment of historic masonry structures. Int. J. Architect. Herit. 11 (1), 143 160. Available from: https://doi.org/10.1080/15583058.2016.1238976. Doherty, K., Griffith, M.C., Lam, N., Wilson, J., 2002. Displacement-based seismic analysis for out-of-plane bending of unreinforced masonry walls. Earthq. Eng. Struct. Dyn. 31 (4), 833 850. Available from: https://doi.org/10.1002/eqe.126. Ferreira, T.M., Costa, A.A., Costa, A., 2014. Analysis of the out-of-plane seismic behavior of unreinforced masonry: a literature review. Int. J. Architect. Herit. Available from: https:// doi.org/10.1080/15583058.2014.885996. Gams, M., Anˇzlin, A., Kramar, M., 2017. Simulation of shake table tests on out-of-plane masonry buildings. Part (III): two-step FEM approach. Int. J. Architect. Herit. 11 (1), 94 102. Available from: https://doi.org/10.1080/15583058.2016.1237589. Gilbert, M., Casapulla, C., Ahmed, H.M., 2006. Limit analysis of masonry block structures with nonassociative frictional joints using linear programming. Comput. Struct. 84 (13 14), 873 887. Available from: https://doi.org/10.1016/j.compstruc.2006.02.005. Heyman, J., 1969. The safety of masonry arches. Int. J. Mech. Sci. 11 (4), 363 385. Available from: https://doi.org/10.1016/0020-7403(69)90070-8. Kooharian, A., 1952. Limit analysis of voussoir (segmental) and concrete arches. J. Am. Concr. Inst. 24 (4), 317 328. Available from: https://doi.org/10.14359/11822. Lagomarsino, S., 2015. Seismic assessment of rocking masonry structures. Bull. Earthq. Eng. 13 (1), 97 128. Available from: https://doi.org/10.1007/s10518-014-9609-x. Lemos, J.V., Campos-Costa, A., 2017. Simulation of shake table tests on out-of-plane masonry buildings. Part (V): discrete element approach. Int. J. Architect. Herit. 11 (1), 117 124. Available from: https://doi.org/10.1080/15583058.2016.1237587. Lemos, J.V., Campos Costa, A., Bretas, E.M., 2011. Assessment of the seismic capacity of stone masonry walls with block models. In: Papadrakakis, M., Fragiadakis, M., Lagaros, N.D. (Eds.), Comput. Methods Earthq. Eng. Springer, pp. 221 235. , ISBN: 978-94-007-0053-6. Lourenc¸o, P.B., 1996. Computational strategies for masonry structures. PhD thesis, Delft University of Technology, Delft, The Netherlands. Lourenc¸o, P.B., 2000. Anisotropic softening model for masonry plates and shells. J. Struct. Eng. 126 (9), 1008 1016. Available from: https://doi.org/10.1061/(ASCE)0733-9445(2000)126:9(1008). Lourenc¸o, P.B., 2002. Computations on historic masonry structures. Prog. Struct. Eng. Mater. 4 (3), 301 319. Available from: https://doi.org/10.1002/pse.120.

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Lourenc¸o, P.B., De Borst, R., Rots, J.G., 1997. A plane stress softening plasticity model for orthotropic materials. Int. J. Numer. Methods Eng. 40 (21), 4033 4057. Lourenc¸o, P.B., Mendes, N., Ramos, L.F., Oliveira, D.V., 2011. Analysis of masonry structures without box behavior. Int. J. Architect. Herit. 5 (4-5), 369 382. Available from: https://doi. org/10.1080/15583058.2010.528824. Mauro, A., de Felice, G., DeJong, M., 2015. The relative dynamic resilience of masonry collapse mechanisms. Eng. Struct. 85, 182 194. Available from: https://doi.org/10.1016/j. engstruct.2014.11.021. Mendes, N., 2014. Masonry macroblock analysis. In: Beer, M., Kougioumtzoglou, I.A., Patelli, E., Au, I.S.-K. (Eds.), Encyclopedia of Earthquake Engineering. Springer-Verlag, Berlin Heidelberg, doi:10.1007/978-3-642-36197-5_154-1. ISBN: 978-3-642-36197-5 (Online). Mendes, N., Lourenc¸o, P.B., 2014. Sensitivity analysis of the seismic performance of existing masonry buildings. Eng. Struct. 80 (1), 137 146. Available from: https://doi.org/10.1016/j. engstruct.2014.09.005. Mendes, N., Costa, A.A., Lourenc¸o, P.B., Bento, R., Beyer, K., Felice, G., et al., 2017. Methods and approaches for blind test predictions of out-of-plane behavior of masonry walls: a numerical comparative study. Int. J. Architect. Herit. 11 (1), 59 71. Available from: https:// doi.org/10.1080/15583058.2016.1238974. Munjiza, A., 2004. The Combined Finite-Discrete Element Method. John Wiley and Sons, Chichester, ISBN: 0470841990. Nielsen, M., 1999. Limit Analysis and Concrete Plasticity, second ed. CRC Press LLC, ISBN: 9780849391262. Ordun˜a, A., Lourenc¸o, P.B., 2005. Three-dimensional limit analysis of rigid blocks assemblages. Part I: torsion failure on frictional interfaces and limit analysis formulation. Int. J. Solids Struct. 42 (18 19), 5140 5160. Available from: https://doi.org/10.1016/j.ijsolstr.2005.02.010. Papantonopoulos, C., Psycharis, I.N., Papastamatiou, D.Y., Lemos, J.V., Mouzakis, H.P., 2002. Numerical prediction of the earthquake response of classical columns using the distinct element method. Earthq. Eng. Struct. Dyn. 31 (9), 1699 1717. Available from: https://doi.org/ 10.1002/eqe.185. Pen˜a, F., Lourenc¸o, P.B., Mendes, N., Oliveira, D.V., 2010. Numerical models for the seismic assessment of an old masonry tower. Eng. Struct. 32 (5), 1466 1478. Available from: https://doi.org/10.1016/j.engstruct.2010.01.027. Penna, A., 2015. Seismic assessment of existing and strengthened stone-masonry buildings: critical issues and possible strategies. Bull. Earthq. Eng. 13 (4), 1051 1071. Available from: https://doi.org/10.1007/s10518-014-9659-0. Roca, P., Cervera, M., Pela`, L., Clemente, R., Chiumenti, M., 2013. Continuum FE models for the analysis of mallorca cathedral. Eng. Struct. 46, 653 670. Available from: https://doi. org/10.1016/j.engstruct.2012.08.005. Sorrentino, L., Liberatore, L., Decanini, L.D., Liberatore, D., 2014a. The performance of churches in the 2012 Emilia earthquakes. Bull. Earthq. Eng. 12 (5), 2299 2331. Available from: https://doi.org/10.1007/s10518-013-9519-3. Sorrentino, L., Liberatore, L., Liberatore, D., Masiani, R., 2014b. The behavior of vernacular buildings in the 2012 Emilia earthquakes. Bull. Earthq. Eng. 12 (5), 2367 2382. Available from: https://doi.org/10.1007/s10518-013-9455-2. Sorrentino, L., D’Ayala, D., de Felice, G., Griffith, M.C., Lagomarsino, S., Magenes, G., 2017. Review of out-of-plane seismic assessment techniques applied to existing masonry buildings. Int. J. Architect. Herit. 11 (1), 2 21. Available from: https://doi.org/10.1080/15583058.2016.1237586.

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Statista, 2018. The statistics portal (www.statista.com). The 10 most significant natural disasters worldwide by death toll from 1980 to 2016. Munich Re.

Further reading Marques, R., Lourenc¸o, P.B., 2011. Possibilities and comparison of structural component models for the seismic assessment of modern unreinforced masonry buildings. Comput. Struct. 89, 2079 2091. Mendes, N., Zanotti, S., Lourenc¸o, P.B., Lemos, J.V., 2016. An´alise s´ısmica da Igreja de Kun˜o Tambo. 10o Congresso Nacional de Sismologia e Engenharia S´ısmica (SI´SMICA 2016). % Ac¸ores (in Postuguese). Ordun˜a, A., 2003. Seismic assessment of ancient masonry structures by rigid blocks limit analysis. PhD thesis, University of Minho, Portugal.

Chapter 5

Seismic assessment of historical masonry structures through advanced nonlinear dynamic simulations: applications to castles, churches, and palaces M. Valente and G. Milani Department of Architecture, Built Environment and Construction Engineering (ABC), Politecnico di Milano, Milan, Italy

5.1 Introduction The seismic vulnerability assessment of historical masonry constructions is a topic of relevant importance, as evidenced in recent earthquakes around the world, especially in Italy (Binda et al., 2011; D’Ayala and Paganoni, 2011; Lagomarsino, 2012; Lucibello et al., 2013; Penna et al., 2014; Crespi et al., 2016; Dal Cin and Russo, 2016; Clementi et al., 2017; Coisson et al., 2017; Valente et al., 2017a; D’Altri et al., 2018). Masonry constructions are typically complex structures that require a thorough and detailed knowledge and information concerning the behavior of their structural systems (Ceroni et al., 2012; Altunisik et al., 2017; Ascione et al., 2017; Barbieri et al., 2017; Dall’Asta et al., 2018). In particular, for many historical masonry structures located in moderate-to-high seismicity regions, a good comprehension of their seismic response is of paramount importance due to their limited earthquake resistance capacity (Lourenc¸o et al., 2012; Masciotta et al., 2016; Jorquera et al., 2017; Milani et al., 2018a). Moreover, appropriate numerical modeling of a historical masonry structure is a fundamental preliminary step for effective strengthening interventions (Casolo and Sanjust, 2009; Formisano and Marzo, 2017; Milani et al., 2017; Bayraktar et al., 2018; Valente and Milani, 2018a). On the other hand, the numerical simulation of the seismic response of historical masonry constructions is a very difficult, complex, and computationally demanding task. Advanced finite element (FE) Numerical Modeling of Masonry and Historical Structures. DOI: https://doi.org/10.1016/B978-0-08-102439-3.00005-1 Copyright © 2019 Elsevier Ltd. All rights reserved.

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analyses make it possible to combine refined modeling strategies and sophisticated material constitutive models, providing accurate results based on micromodeling and macromodeling approaches (Betti et al., 2010; Sandoval et al., 2017; Milani et al., 2018b). In the case of micromodeling approach, all the masonry components are separately discretized, leading to models with a larger number of degrees of freedom and requiring high computational efforts. Conversely, the macromodeling approach considers masonry as a continuum and homogeneous material in which isotropic or anisotropic behavior can be assumed: such an approach is adopted in this study because it is particularly suitable for large-scale seismic analyses of complex historical masonry constructions, as shown in the literature (Roca et al., 2010; Endo et al., 2015; Castellazzi et al., 2017; Valente et al., 2017b; Clementi et al., 2018). In May to June 2012 a large part of the Po Valley between the cities of Ferrara, Modena, and Mantua (Northern Italy) was struck by a damaging seismic sequence. The first major earthquake (magnitude 5.9) occurred on May 20 with the epicenter between Finale Emilia and San Felice sul Panaro: two aftershocks of magnitude 5.2 followed and seven people were killed. The second major earthquake (magnitude 5.8) occurred on May 29 with the epicenter in Medolla that caused extensive damages, particularly to buildings already weakened by the first seismic event. A great number of historical masonry constructions were seriously damaged by the seismic sequence in the southeast Lombardia. In particular, several masonry churches located in the southern part of the province of Mantua suffered extensive damage (Valente and Milani, 2018b) and required important structural interventions. Significant damage was also observed in several historical masonry constructions located in Mantua, recently declared an UNESCO World Heritage Site with the nearby city of Sabbioneta (2007). This chapter examines the damage state and the seismic behavior of three monumental masonry constructions located in Mantua, presenting the results of advanced numerical investigations performed on detailed threedimensional (3D) FE model considering different peak ground accelerations (PGAs). The three masonry constructions are some of the most important symbols of the outstanding cultural heritage in Mantua: the damage observed after the earthquake provided strong motivations for deeply investigating the seismic vulnerability of the structures. In spite of the unquestionable importance of these case studies, before the present work the structures were never studied with advanced numerical simulations. Detailed 3D FE models of the structures were created and a damage plasticity model with different softening behavior in tension and compression was used for masonry. As already mentioned, sophisticated FE approaches using refined constitutive laws for masonry require a high computational demand and advanced skills in the model implementation and in the interpretations of the numerical results, but they provide a thorough and

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comprehensive understanding of the seismic behavior of the structure (Betti and Vignoli, 2011; Barbieri et al., 2013; Clementi et al., 2016; Tiberti et al., 2016; Valente and Milani, 2018c,d). Full 3D nonlinear dynamic analyses are carried out to investigate the seismic behavior of the structures under earthquake motions with different PGA values. The main objectives of the study are (1) to deeply investigate the seismic performance of the structures under study, identifying the most vulnerable elements, and (2) to obtain important response parameters and damage patterns for different levels (PGA) of seismic action. It is worth mentioning that this study provides precious indications about the seismic response and damage distribution of the structures for higher PGAs than those registered during the 2012 Emilia earthquake.

5.2 Numerical simulations and concrete damage plasticity model Three monumental masonry constructions located in Mantua are analyzed in this chapter through advanced numerical simulations. The three historical structures are invaluable symbols of the outstanding cultural heritage in Mantua: the Castle of St. George, Sant’Andrea church, and Palazzo Te. Eigenfrequency analyses were conducted on the 3D FE models in order to obtain preliminary insight into the dynamic behavior of the structures under study, identifying the main vibration modes, the corresponding periods, and the participating mass ratios (PMRs). The seismic response of the three structures was investigated through nonlinear dynamic analyses using the real accelerograms registered on May 29 during the 2012 Emilia seismic sequence. The same accelerograms, presenting equal intensity in the two orthogonal directions, were used for the numerical simulations of all the structures. Fig. 5.1 shows the two horizontal components of the acceleration time histories with PGA 5 0.15 g applied in the X and Y directions and the corresponding acceleration response spectra. The duration of the accelerograms was assumed equal to 10 s because of the high computational demand required by the analyses. Three different PGA values, ranging between 0.05 and 0.25 g, were used in the nonlinear dynamic analyses. The nonlinear dynamic analyses with PGA 5 0.05 g aim at both simulating the seismic response of the structures to earthquakes of small magnitudes and comparing the results with the damage observed after the 2012 earthquake. The nonlinear dynamic analyses with PGA 5 0.15 and 0.25 g provide useful information about the seismic response and damage distribution of the structures with higher PGA values than those registered in Mantua during the 2012 Emilia earthquake. It is worth mentioning that the seismic response of the church was investigated considering lower PGA values (PGA 5 0.1 and 0.15 g) because of its high vulnerability.

FIGURE 5.1 Horizontal components of the accelerogram (Mirandola, May 29, 2012) used in the nonlinear dynamic analyses and corresponding acceleration response spectra: E
W component (blue) applied in the Y direction, N
S component (red) applied in the X direction.

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The tensile damage contour plots obtained at the end of the numerical simulations are shown for each structure; then, the energy density dissipated by tensile damage (EDDTD) and the maximum normalized displacements are reported for the main macroelements of each structure. The main aims of the numerical simulations are: (1) to identify the most vulnerable elements for each structure and (2) to assess the damage evolution and the main response parameters variations for different levels of seismic action.

5.2.1

Concrete damage plasticity model

The concrete damage plasticity (CDP) model has been adopted to simulate the nonlinear behavior of masonry. Although originally developed to describe the nonlinear behavior of concrete (Lubliner et al., 1989; Lee and Fenves, 1998), the utilization of such a model for masonry is commonly accepted in the literature after an appropriate adaptation of the main parameters. The CDP model is a continuum plasticity-based damage model that allows for different tensile and compressive strength, as is the case of masonry, with distinct damage parameters in tension and compression. The model assumes that the uniaxial tensile and compressive response is characterized by damaged plasticity (see Fig. 5.2). Under uniaxial tension the stress
strain response follows a linear elastic relationship until the value of the failure stress σto is reached. The failure stress corresponds to the onset of microcracking in the material. Beyond the failure stress the formation of microcracks is represented macroscopically with a softening stress
strain response. Under uniaxial compression the response is linear until the value of the initial yield σco is reached. In the plastic range the response is typically characterized by stress hardening followed by strain softening beyond the ultimate stress σcu . Such a representation, although somewhat simplified, captures the main features of the response of masonry. When the material is unloaded from any point on the strain softening branch of the stress
strain curve, the unloading response is characterized by

FIGURE 5.2 Representation of the masonry constitutive behavior in tension and compression.

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a reduced elastic stiffness. The degradation of the elastic stiffness is different in tension and compression; in both cases, the effect is more pronounced as the plastic strain increases. The degradation of the elastic stiffness is characterized by two damage variables, denoted as dt and dc in tension and compression, respectively, which are increasing functions of the equivalent plastic strains: their values range between zero and one, representing a zerodamage state and complete damage state. The following standard relationships define the uniaxial tensile σt and compressive σc stresses: σt 5 ð1 2 dt ÞE0 ðεt 2 εpl t Þ σc 5 ð1 2 dc ÞE0 ðεc 2 εpl c Þ

ð5:1Þ

where E0 is the initial elastic modulus, dt and dc are the scalar damage variables in tension and in compression, εt and εc are the total strain in tension pl and in compression, and εpl t and εc are the equivalent plastic strain in tension and in compression. The CDP model describes the postfailure behavior in tension as a function of the cracking strain εck t , which can be expressed as follows: el εck t 5 εt 2 ε0t

ð5:2Þ

where εt is the total tensile strain and εel0t 5 ðσt =E0 Þ is the elastic tensile strain. The tensile equivalent plastic strain εpl t can be obtained as follows: ck εpl t 5 εt 2

dt σ t ð1 2 dt Þ E0

ð5:3Þ

Similarly, the postfailure behavior in compression is related to the inelastic strain εin c , which can be expressed as follows: el εin c 5 εc 2 ε0c

ð5:4Þ εel 0c

5 σc =E0 is the elastic comwhere εc is the total compressive strain and pressive strain. The compressive equivalent plastic strain εpl c can be evaluated using the following equation: in εpl c 5 εc 2

dc σ c ð1 2 dc Þ E0

ð5:5Þ

In addition, the CDP model takes into account the effect of closing of previously formed cracks under cyclic loading conditions, which results in the recovery of the compression stiffness. In fact, experimental observations in most quasibrittle materials indicate that the compressive stiffness is recovered upon crack closure as the load changes from tension to compression. On the other hand, the tensile stiffness is not recovered as the load changes from compression to tension once crushing microcracks have developed.

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In uniaxial stress conditions the loss of elastic stiffness is computed as follows: ð1 2 dÞ 5 ð1 2 st dc Þð1 2 sc dt Þ

ð5:6Þ

where st and sc are functions of the stress state and are introduced to model the stiffness recovery effects due to stress reversal. They are computed using the following equations:  st 5 1 2 wt HðσÞ ð5:7Þ sc 5 1 2 wc ð1 2 HðσÞÞ where wt and wc are the weight factors (assumed as material properties) that control the recovery of tensile and compressive stiffness upon load reversal: they can range from zero, which represents no stiffness recovery, to one, which represents a total stiffness recovery. H(σ) is the Heaviside function that is assumed equal to 1 if σ . 0 and equal to 0 if σ , 0. Fig. 5.3 illustrates a uniaxial load cycle assuming the default behavior adopted in Abaqus, which corresponds to wt 5 0 and wc 5 1. The CDP model uses a Drucker
Prager strength criterion, modified through a parameter, Kc, which represents the ratio between the second stress invariant on the tensile meridian and the one on the compressive meridian, and assumes a nonassociated potential flow rule. The value of Kc is set equal to 0.666, as suggested by the user’s guide (ABAQUS). A regularization of the tensile corner has been performed using a correction parameter called eccentricity: such a parameter defines the rate at which the plastic flow potential approaches the asymptote, that is, the flow potential tends to a straight line as the eccentricity tends to zero. A value equal to 0.1 is adopted for the eccentricity parameter. It is worth mentioning that smaller values of the eccentricity parameter may cause convergence problems when the

FIGURE 5.3 Uniaxial load cycle (tension
compression
tension) assuming default values for the stiffness recovery factors: wt 5 0 and wc 5 1.

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material is subjected to low confining pressures because of the very tight curvature of the flow potential (Abaqus). The dilation angle ψ, which is the angle due to a variation in volume of the material following the application of a shear force, is set equal to 10 degree, in agreement with experimental evidences available in the literature (Van Der Pluijm, 1993). The strength ratio fb0 =fc0 , which is the ratio of initial equibiaxial compressive yield stress to initial uniaxial compressive yield stress, is assumed equal to 1.16, as suggested in Page (1981). Material models exhibiting softening behavior and stiffness degradation may lead to severe convergence difficulties in implicit analysis programs, such as Abaqus/Standard. Some of these convergence difficulties can be overcome by using a viscoplastic regularization of the constitutive equations. The CDP model can be regularized using viscoplasticity with a small value for the viscosity parameter that usually helps improve the convergence rate of the model in the softening branch, without compromising results. The viscoplastic strain rate component ε_ pl v and the viscous stiffness degradation variable dv are expressed as: 1 pl ðε 2 εpl v Þ μ 1 d_v 5 ðd 2 dv Þ μ

ε_ pl v 5

ð5:8Þ

where μ is the viscosity parameter representing the relaxation time of the viscoplastic system, εpl is the plastic strain component, and d is the degradation variable. The stress
strain relationship of the viscoplastic model becomes: σ 5 ð1 2 dv ÞE0 ðε 2 εpl v Þ

ð5:9Þ

If the viscosity parameter is different from zero, output results of the plastic strain and stiffness degradation refer to the viscoplastic values εpl v and dv . A value of the viscosity parameter equal to 0.002 has been assumed in this study. In the absence of available results from experimental tests for the case studies, the mechanical parameters of the material are selected referring to Table C8A.2.1 in Circolare 02/2009 (DM, 2008; Circolare, 2009; DPCM, 2011). A masonry typology with quite regular texture constituted by clay bricks and lime mortar is considered. The parameters used in the nonlinear dynamic analyses are the following: (1) the density and the elastic modulus are equal to 1800 kg/m3 and 1500 MPa, respectively; (2) the compressive strength is equal to σcu 5 2.4 MPa. The tensile strength is set equal to σto 5 0.15 MPa, obtaining a ratio between the tensile and compressive strength equal to about 0.06. The compressive (dc) and tensile (dt) scalar damage variables, representative of the stiffness degradation of the material,

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are assumed to vary linearly: the values range from zero, for the strain corresponding to the stress peak, to 0.95, for the ultimate strain value of the softening branches.

5.3 Castle of St. George 5.3.1

Description of the structure

The Castle of St. George is one of the most emblematic monuments in Mantua. It was built between 1395 and 1406 on the project by Bartolino da Novara, one of the most renowned military architects of the time, under commission by Francesco I Gonzaga. Originally designed for the defense of the city, it was transformed into the family residence by Ludovico I and served for many years as the home of Isabella d’Este, Francesco Gonzaga’s wife. In 1459, architect Luca Fancelli, acting on Marquis Ludovico III Gonzaga’s instructions, renovated the building, which thus lost its original military and defensive function. After the renovation, the castle became one of the most important examples of the Italian Renaissance, especially thanks to the extraordinary Camera Picta or Bridal Chamber, frescoed by Mantegna from 1465 to 1474. The castle presents an almost square-shaped plan with four massive corner towers, surrounded by a moat, three gates and their drawbridges, and an internal square courtyard. The whole complex occupies a large almost square area with sides equal to about 46 m. The four corner towers present a square plan with sides equal to about 10 m, but different heights: the northeast and southeast towers (TNE/TSE) are 23.5 m high, the northwest tower (TNW) is about 25 m high and the southwest tower (TSW) is about 29 m high (hereinafter, the heights are indicated from the end of the escarpment to the base of the covering). The towers exhibit a batter base and then, from the ground level, maintain the same cross-section: in the case of the northeast, southeast and southwest corner towers, the upper part presents a larger cross-section supported by corbels. Moreover, the towers exhibit characteristic dual (interior and exterior) battlements presenting an average height equal to about 2.5 m: the northwest tower does not present either battlements or a protruding upper part. In correspondence with the southeast corner tower, there are two smaller countertowers that are 12.7 and 17 m high, respectively: the smallest one does not present battlements. In correspondence with the southwest corner tower, there is another small countertower that is 19.5 m high. The walls of the central body among the corner towers present a series of openings of different sizes, which are generally arranged uniformly: on the contrary, the corner towers exhibit a limited number of openings for defensive purposes. The external north and east walls present an enlargement of

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FIGURE 5.4 Plan of the different storeys of the castle with indication of the main geometrical dimensions.

the cross-section in the upper part, supported by corbels. The internal courtyard is surrounded by four window-walls: a block with portico (6.4 m high) is adjacent to the north and east walls, while the north wall presents large arcades at the ground level. The internal spaces are covered by vaults or timber floors. The main structures of the castle (walls and vaults) consist of masonry made by solid bricks and lime mortar, with the exception of some wooden slabs. The coverings of both the four corner towers and the central body are composed of wooden frameworks with tiles roof coating. Fig. 5.4 shows the plan of the different storeys of the castle with an indication of the main geometrical dimensions.

5.3.2

Finite element model and eigenfrequency analysis

In order to better understand the seismic response of the castle, a detailed 3D FE model is created using the drawings and the data collected from existing available documentations and during the survey phase. Fig. 5.5

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FIGURE 5.5 Castle of St. George. Geometrical and FE models with indication of the sides and corner towers.

FIGURE 5.6 Castle of St. George. Indication of the main macroelements (towers, countertowers, walls) under study. Notation: TNE, northeast tower; TNW, northwest tower; TSE, southeast tower; TSW, southwest tower; CTE, east countertower; CTS, south countertower; CTW, west countertower; WEE, external east wall; WEC, east wall of the courtyard; WNE, extemal north wall; WNC, north wall of the courtyard; WSE, extemal south wall; WSC, south wall of the courtyard; WWE, extemal west wall; WWC, west wall of the courtyard.

shows the geometrical and FE models of the castle. The discretization of the castle model consists of about 400,000 four-node tetrahedral elements with an average size ranging between 20 and 40 cm. Fig. 5.6 highlights the most relevant macroelements (towers, countertowers, external walls, and walls of the courtyard) of the model that are investigated in detail in the following sections.

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An eigenfrequency analysis is performed on the 3D FE model in order to obtain a preliminary assessment of the dynamic behavior of the castle, identifying the frequencies and mode shapes of the structure. Fig. 5.7 shows the deformed shapes and the corresponding periods of the main vibration modes of the castle, characterized by a PMR larger than 4%: moreover, the distribution of the first 300 modes in the two orthogonal directions is presented. It can be noted that the first (T 5 0.263 s) and second (T 5 0.260 s) modes involve mainly the upper part of the southwest corner tower with the largest PMR equal to about 37% and 38.5% in the X and Y directions, respectively. The third mode (T 5 0.243 s) concerns mainly the north side of the castle, including the northeast and northwest corner towers, and the upper part of the southwest tower with a PMR equal to 19.5% in the Y direction and a significant torsional component. The fourth mode (T 5 0.214 s) concerns mainly the east side of the castle, including the northeast and southeast corner towers, with a PMR equal to 22.9% in the X direction. The other relevant modes are the sixth (T 5 0.190 s) and the thirteenth (T 5 0.136 s) modes involving mainly the southwest and northwest corner towers. Considering the first 300 modes, a cumulative PMR of about 84% in each horizontal direction is obtained. From the preliminary results of the eigenfrequency analysis it can be concluded that the upper parts of the corner towers are the most flexible parts as well as the most relevant elements for the dynamic behavior of the castle, while the other parts are involved by high-order modes with lower PMR values.

5.3.3

Nonlinear dynamic analyses

Nonlinear dynamic analyses are performed to investigate the seismic response and the damage distribution of the castle for different PGA levels. Fig. 5.8 shows the tensile damage contour plots of the castle at the end of the nonlinear dynamic analyses with three different PGA values. The nonlinear dynamic analysis with PGA 5 0.05 g shows an onset of damage that is limited to only a few parts of the castle. A good correlation is registered between the numerical results and the damage surveyed after the 2012 seismic sequence, as in the case of the cracks observed in correspondence with the openings of the northeast tower. The numerical analysis highlights a crack in the connection region between the barrel vault and the wall of the corridor near the northwest tower, as emerged from the field survey. An onset of damage, which is consistent with what was observed after the earthquake, is also registered at the base of the battlements and near the enlargement of the upper section of the corner towers: in particular, the northeast tower and the southwest tower present evident damage at the base of the battlements and in correspondence with the corbels supporting the upper part. Moreover, vertical cracks can be observed in the connection regions between the corner towers and the perimeter walls.

FIGURE 5.7 Castle of St. George. Deformed shapes, corresponding periods and participation mass ratios of the main vibration modes, with reference to the response spectra of the accelerograms used in the nonlinear dynamic analyses.

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FIGURE 5.8 Castle of St. George. Tensile damage contour plots of the castle at the end of the nonlinear dynamic analyses with different PGA.

The results of the nonlinear dynamic analysis with PGA 5 0.15 g highlight a significant damage in the towers, mainly in correspondence with the battlements and the corbels. It can be noted that the southwest tower presents the most severe damage, exhibiting severe cracks along the walls, mainly in correspondence with the openings. An onset of damage can be detected near the openings of the external walls and in some walls overlooking the internal courtyard. Some cracks can be observed in the connection regions between the internal partitions and the perimeter walls. It has to be noticed that significant damage occurs in the coverings, mainly in the connection region with the adjacent walls; in some cases the cracks propagate in the vault toward the keystone. The castle presents a considerable increase of damage at the end of the nonlinear dynamic analysis with PGA 5 0.25 g: in particular, a more marked damage is registered along the height of the towers, with vertical cracks passing through the openings. Severe damage can be observed at the base of

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the towers battlements: severe cracks concentrate in correspondence with the section enlargement and then propagate in the covering elements. Significant damage occurs in the walls and in the corbels of the countertowers, in the connection regions between the towers and the walls and between the towers and the countertowers. In the internal courtyard, severe cracks concentrate in the connection regions between the covering elements and the walls: moreover, some cracks are registered in the upper part, near the battlements. Finally, considerable damage is observed in the vaults, with a clear concentration in the connection regions with the perimeter walls. Fig. 5.9 shows the evolution of the total EDDTD during the nonlinear dynamic analyses with different PGA values. It can be noted that negligible values of EDDTD are registered until about 1.2 s for all the analyses. Moreover, it is important to observe that the castle experiences small values of EDDTD at the end of the nonlinear dynamic analyses with PGA 5 0.05 g; conversely, a considerable increase of EDDTD is registered for higher values of PGA, especially in the case of PGA 5 0.25 g. Figs. 5.10 and 5.11 show the EDDTD for the main macroelements and vaults of the castle, respectively, at the end of the nonlinear dynamic analyses with different PGA. The numerical results highlight that the vaults and the towers are the most damaged elements of the castle. In detail, the maximum values of EDDTD are registered for the vaults of the southwest tower (VTSW), which present considerable values even under PGA 5 0.15 g. High values of EDDTD can be observed also for the vaults of the northeast tower (VTNE) and for the vaults of the south side (VS). Among the towers, the highest value of EDDTD under PGA 5 0.25 g is registered for the TSE and the TNE, while the smallest value is observed for the TNW. It has to be noticed that the countertowers present smaller

FIGURE 5.9 Castle of St. George. Total energy density dissipated by tensile damage for the castle during the nonlinear dynamic analyses with different PGA.

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FIGURE 5.10 Castle of St. George. Energy density dissipated by tensile damage registered for the main macroelements (towers, countertowers, walls) at the end of the nonlinear dynamic analyses with different PGA. (For notation, see Fig. 5.6.)

FIGURE 5.11 Castle of St. George. Energy density dissipated by tensile damage registered for the main vaults of the castle at the end of the nonlinear dynamic analyses with different PGA. Notation: VTNE, vaults of the northeast tower; VTNW, vaults of the northwest tower; VTSW, vaults of the southwest tower; VTSE, vaults of the southeast tower; VN, vaults of the north side; VS, vaults of the south side; VE, vaults of the east side; VW, vaults of the west side.

EDDTD values than the corner towers: an exception is represented by the west countertower (CTW). As regards the walls, the highest value of EDDTD is registered for the north wall of the courtyard (WNC), which presents a considerable increase under PGA 5 0.25 g. It can be noted that the north wall of the courtyard (WNC) exhibits several openings and large arcades at the base. In any case, as already mentioned, the EDDTD values of the walls are much smaller than those of the towers and vaults. Fig. 5.12 shows the maximum normalized displacements (top displacement/height) registered in the X and Y directions for the different towers, countertowers and walls. It is worth mentioning that, for the evaluation of the maximum normalized displacements, the control point of each macroelement may be different in the two orthogonal directions. As expected, the normalized displacements of the towers are much larger than those of the walls and countertowers.

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FIGURE 5.12 Castle of St. George. Maximum normalized displacements registered in the X and Y directions for the main macroelements (towers, countertowers, walls) during the nonlinear dynamic analyses with different PGA.

The largest normalized displacement (about 1.55%) in the X direction is registered for the TSE: such a result is consistent with the damage concentration observed in the upper part of the tower. High normalized displacements (about 1.15%) are computed also for the TNE. It can be noted that a large increase of normalized displacements is registered for the TSE and the TNE in the case of PGA 5 0.25 g. It is worth mentioning that the south countertower (CTS) presents a significant normalized displacement equal to about 0.8%. The highest normalized displacement (about 1.45%) in the Y direction is registered for the TSE: significant normalized displacements within the range 0.9%
1% are observed for the TNE, the TSW, and the CTW. As regards the walls, it can be noted that the normalized displacements of the walls of the courtyard are generally smaller than those of the external walls, despite their smaller thickness. The highest normalized displacements (larger than 0.45%) in the X direction are computed for the external north wall (WNE), the edge of the east wall of the courtyard (WEC) near the TSE and the external south wall (WSE). The highest normalized displacement (about 0.6%) in the Y direction is registered for the external west wall (WWE).

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FIGURE 5.13 Castle of St. George. Maximum vertical displacements registered for the main vaults of the castle during the nonlinear dynamic analyses with different PGA.

Fig. 5.13 shows the maximum vertical displacements registered for the main vaults of the castle during the nonlinear dynamic analyses with different PGA. It has to be noticed that a large increase of vertical displacements is generally observed under PGA 5 0.25 g. The largest displacements (about 5 cm) are registered for the vaults of the southwest tower (VTSW) and for the vaults of the northeast tower (VTNE). Among the vaults of the main structure, the largest displacements are observed for the vaults of the north side (VN).

5.3.4

Discussion of results

The results provided by the eigenfrequency analysis show that the first main modes with high PMR involve mainly the towers, and in particular the southwest tower that exhibits the maximum height. Moreover, the main modes of the castle present low values of period: such a result indicates that the structure may experience high amplifications of spectral accelerations. The damage contour plots obtained at the end of the nonlinear dynamic analyses with different PGA provide a clear picture of the tensile damage distribution in the different macroelements of the castle. It can be noted that damage is mainly concentrated in the towers and in the vaults rather than in the walls and the countertowers. The nonlinear dynamic analysis with PGA 5 0.05 g shows that damage is limited only to a few parts of the structure, while it results very widespread at the end of the nonlinear dynamic analysis with PGA 5 0.25 g. A good correlation between numerical results and real damage is found at the end of the nonlinear dynamic analysis with PGA 5 0.05 g. Evident damage has been registered in the TNE, in the barrel vault of the corridor at the second storey of the west side and in the vaults of the Prison of “Martiri di Belfiore.” The results obtained show that the damage patterns experienced by the castle during the earthquake can be satisfactorily simulated by the numerical approach adopted.

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All the towers present pseudo-vertical damage along the height, mainly near the openings. In particular, significant cracks are observed in the main body of the towers that are well connected to the perimeter walls and in the upper part of the tower, above the top level of the perimeter walls. The maximum normalized horizontal displacements of the towers are much larger than those of the walls for all the PGA values considered. The TSE and the TNE present the most severe damage and the highest normalized horizontal displacements: in particular, the maximum normalized displacement (1.55%) is computed for the TSE in the X direction. A relevant damage concentration is observed in the upper part, above all in the battlements and in correspondence with the enlargement of the cross-section. More specifically, severe damage is registered at the base of the battlements of the TNE, indicating an onset of overturning mechanism; moreover, extensive damage can be observed in the corbels supporting the protruding upper parts of the TSE, especially at the corners. The vaults of the towers exhibit severe damage at the end of the nonlinear dynamic analysis with PGA 5 0.25 g. It is important to highlight that the highest value of EDDTD is registered for the vaults of the southwest tower (VTSW), which present considerable damage even under PGA 5 0.15 g: moreover, the VTSW exhibit the maximum vertical displacement (about 5 cm). In the central body of the castle the major damage is observed in the covering elements. In fact, the highest values of EDDTD are registered for the vaults of the north (VN) and south (VS) sides: moreover, the maximum vertical displacement is computed for the vaults of the north side (VN). The nonlinear dynamic analysis with PGA 5 0.25 g shows extensive damage in the external and internal walls: severe cracks are observed mainly in the connection regions between the orthogonal walls and near the openings. Moreover, vertical cracks are registered in the perimeter walls near the connection regions with the tower: it is evident that they are caused by the interaction between two elements characterized by different geometric and dynamic characteristics. The highest value of EDDTD is computed for the north wall of the courtyard (WNC), with a considerable increase in the case of PGA 5 0.25 g, probably due to the presence of several openings and large arcades at the ground level. The highest normalized horizontal displacements are registered for the external north wall (WNE) in the X direction and for the external east wall (WEE) in the Y direction.

5.4 Sant’Andrea church 5.4.1

Description of the structure

Sant’Andrea church is the largest church in Mantua and one of the major examples of the 15th-century Renaissance architecture in Northern Italy: it is located in Piazza Mantegna, in the historical city center of Mantua.

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The construction of the church started in 1472 according to designs by Leon Battista Alberti on a site occupied by a Benedictine monastery, but the building was completed only in the 18th century. In 1497 the vestibule, the major and minor chapels were erected and the central nave was partially covered by the large barrel vault. In 1597, the lateral arms were added and the crypt was finished. Between 1732 and 1765, the large late-Baroque dome, designed by Filippo Juvarra and covering the space between the nave and the transept, was erected. The church presents a Latin cross-plan with a single nave covered by a barrel vault and several side chapels: the overall dimensions of the plan are about 100 3 66 m. The main entrance of the church, overlooking Piazza Mantegna, is located along the longitudinal axis leading to the central nave, while the side one, overlooking Piazza Alberti, is placed along the transversal axis of the transept north arm. The fac¸ade, which was built near a preexisting gothic bell tower erected in 1414, presents four large pilasters with Corinthian capitals supporting an entablature and a pediment: these elements recall the front of ancient temples. In the central part of the fac¸ade there is also a large arch that is supported by two shorter fluted pilasters: the arch extends deep into the facade, creating a recessed barrel vault that frames the main entrance to the church. The large nave covered by a barrel vault is intersected by a transept with two arms covered by a barrel vault. Along the two sides of the nave there are a series of chapels (six for each side), exhibiting sides equal to about 6 m. The large dome, located in the intersection region between the nave and the transept, is supported by a tambour with openings and is surmounted by a lantern: the tambour is strengthened by four buttresses. The base of the dome is at a height of about 47 m: the overall height, including the lantern, is 67 m. The church ends with a presbytery closed by a semicircular apse that is about 26 m high, including the semidome. On the left side of the church, adjacent to the fac¸ade, there is the bell tower presenting a square section with side equal to 7.75 m: overall it is 62.6 m high and is surmounted by an octagonal tambour with a conical pinnacle. Fig. 5.14 shows the plan and two sections of Sant’Andrea church, with indication of the main geometrical dimensions.

5.4.2

Finite element model and eigenfrequency analysis

A detailed 3D FE model of the church under study is developed through the computer code Abaqus 6.14 using four-node tetrahedral elements (Abaqus). The complex geometry of the church is reproduced accurately using the drawings and the data collected from existing available documentations and during the survey phase. Fig. 5.15 shows the geometrical and FE models of the church. The numerical model is created considering the vertical bearing elements and the masonry vaults and domes. In the FE model, the main

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FIGURE 5.14 Sant’Andrea church. Plan, two sections (longitudinal and transversal) and main geometrical dimensions.

FIGURE 5.15 Geometrical and FE models of Sant’Andrea church.

macroelements, indicated in Fig. 5.16, are highlighted and investigated in detail in the following. Fig. 5.17 shows the modal deformed shapes of the first six main vibration modes of Sant’Andrea church characterized by PMRs larger than about 3.5% and the corresponding periods. The first mode (T 5 0.979 s) concerns the upper part of the bell tower with a small PMR equal to about 4% in the longitudinal direction. The third (T 5 0.518 s), fifth (T 5 0.393 s) and sixth (T 5 0.363 s) modes involve the dome and in one case (mode 5) also the central nave with a PMR equal to about 29%, 22% and 9% in the transversal direction, respectively. The fourth (T 5 0.494 s) and tenth (T 5 0.325 s)

FIGURE 5.16 Sant’Andrea church. Indication of the different macroelements in the FE model. Notation: AD, right arcades; CD, right buttresses; LD, right side; AS, left arcades; CS, left buttresses; LS, left side; T, transept; TA, tambour; CU, dome; L, lantern; C, bell tower; FS, south fac¸ade; PrF, fac¸ade pronaos; F, fac¸ade; PrT, transept pronaos; FN, north fac¸ade; A, apse; CA, apse semidome; P, presbytery; VP, presbytery vault; VC, chapels vaults; CC, lateral chapels small domes; VPr, pronaos vaults; VT, transept vaults; VNC, central nave vault.

FIGURE 5.17 Sant’Andrea church. Deformed shapes, corresponding periods and participation mass ratios of the main vibration modes, with reference to the response spectra of the accelerograms used in the nonlinear dynamic analyses.

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modes concern the dome and the central nave with a PMR equal to about 17% and 33%, respectively, in the longitudinal direction. It is worth mentioning the high PMR (18.3%) of mode 23 (T 5 0.221 s) in the vertical direction involving the dome and the vaults. The other relevant modes involve mainly the barrel vaults of the central nave, the transept and the bell tower. It can be noted that the dome with the lantern, the barrel vault of the central nave and the bell tower may be preliminarily indicated as the critical elements of the church.

5.4.3

Nonlinear dynamic analyses

Fig. 5.18 shows the tensile damage contour plot of Sant’Andrea church at the end of the nonlinear dynamic analyses with different PGA.

FIGURE 5.18 Sant’Andrea church. Tensile damage contour plot (different views) at the end of the nonlinear dynamic analyses with different PGA.

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The results of the nonlinear dynamic analysis with PGA 5 0.05 g show a good correlation with the damage survey. In fact, extensive damage can be observed in the lantern, at the base and near the openings: such a result is consistent with what was observed after the 2012 earthquake. Moreover, the south fac¸ade presents some vertical cracks, especially in correspondence with the oculus: a similar damage was registered during the field survey, highlighting the reopening of preexisting cracks that were restored in the past. This result indicates that such cracks would have probably originated also without preexisting cracks, as indicated by the numerical model. Several marked cracks are registered in the dome and in correspondence with the four spurs; such a damage increases considerably in the case of higher PGA levels. Moreover, an onset of damage can be observed in the upper part of the bell tower, in the fac¸ade, in the buttresses, and in the different vaults of the church. The results of the nonlinear dynamic analysis with PGA 5 0.1 g highlight an increase of damage especially in the coverings. In particular, a considerable damage concentration can be observed in the dome and in the lantern above; widespread damage is also detected in the semidome of the apse, mainly in the connection regions with the adjacent barrel vault and over the whole surface. The vaults covering the transept and the central nave exhibit an onset of damage along the connection regions with the walls and in the middle part of the bottom surface. Severe damage is registered in the buttresses and in the vaults and domes of the side chapels. The upper part of the bell tower shows several marked cracks, mainly near the openings. An evident damage enlargement can be observed in the fac¸ade and in the pronaos. The results of the nonlinear dynamic analysis with PGA 5 0.15 g show a large increase of damage in several parts of the church: it is worth noting that the coverings present more significant damage than the vertical elements. In particular, very widespread damage is registered over the whole surface of the dome and at the base and near the openings of the lantern: consequently, the collapse of this part of the church may be expected. Moreover, an evident increase of damage is observed in the vaults covering the transept, the presbytery, and the central nave, especially in the connection regions with the walls: in particular, a damage concentration is clearly visible at the two edges of the barrel vault of the central nave. The vaults of the side chapels undergo severe damage over the majority of their surface, while the small domes of the minor chapels present minor cracks: furthermore, it can be noted that the buttresses are seriously damaged. The fac¸ade presents considerable damage in the central part near the openings and in the connection regions with the nave: some cracks are also registered in the left side of the pronaos. The bell tower exhibits widespread damage in the upper part near the openings and vertical cracks along the body and in the connection regions with the fac¸ade and pronaos. The south

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fac¸ade, the north fac¸ade, the pronaos of the transept and the longitudinal walls show several vertical cracks, especially in correspondence with the openings. The apse presents extensive damage over the whole surface of the semidome and vertical cracks involving the three openings. Fig. 5.19 shows the values of EDDTD for the different macroelements of Sant’Andrea church at the end of the nonlinear dynamic analyses with different PGA. It is important to highlight that the lantern (L) presents very high values of EDDTD even under PGA 5 0.05 g, as emerged from field surveys. Under PGA 5 0.15 g, the lantern (L), the dome (CU), the semidome of the apse (CA), the tambour (TA), and the vaults of the side chapels (VC) present the highest values of EDDTD, as already seen in the damage contour plot. The dome (CU), the tambour (TA), and the semidome of the apse (CA) show a large increase of EDDTD for higher PGA. It is worth mentioning that the covering elements present higher values of EDDTD than the vertical elements: the left and right buttresses (CS and CD) exhibit the highest values of EDDTD among the vertical elements. Fig. 5.20 shows the maximum normalized displacements (displacement/ height) in the longitudinal (Y) and transversal (X) directions for the main macroelements of Sant’Andrea church during the nonlinear dynamic analyses with different PGA. Under PGA 5 0.15 g, in the longitudinal (Y) direction, the tambour (TA), the lantern (L), the presbytery (P), and the bell tower (C) present the highest values of normalized displacements: the maximum value (0.5%) is registered for the tambour (TA). It is important to observe that the lantern shows high values of normalized displacements even under PGA 5 0.05 g. Under PGA 5 0.15 g, in the transversal (X) direction, the pronaos of the transept (PrT), the tambour (TA), the south fac¸ade (FS), and the transept (T) present the highest values of normalized displacements: the maximum value (0.55%) is registered for the pronaos of the transept (PrT).

FIGURE 5.19 Sant’Andrea church. Energy density dissipated by tensile damage (EDDTD) for the main macroelements at the end of the nonlinear dynamic analyses with different PGA.

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FIGURE 5.20 Sant’Andrea church. Maximum normalized displacements (top displacement/ height) in the transversal (X) and longitudinal (Y) directions for the main macroelements during the nonlinear dynamic analyses with different PGA.

FIGURE 5.21 Sant’Andrea church. Maximum vertical displacements for the main macroelements during the nonlinear dynamic analyses with different PGA.

Fig. 5.21 shows that the highest vertical displacements are observed for the tambour (TA) and the lantern (L): high values of vertical displacements are also registered for the chapels vaults (VC) and the central nave vaults (VNC). The other macroelements exhibit small values of vertical displacements.

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5.5 Palazzo Te 5.5.1

Description of the structure

Palazzo Te is located in the suburbs (south part) of Mantua and is universally considered a masterpiece of the late Renaissance. It was built by Giulio Romano between 1525 and 1535, as a suburban residence for Federico II Gonzaga: the site chosen was that of the family’s stables. The main block of Palazzo Te presents a large square plan with sides equal to about 70 m including a large inner courtyard, recalling an ancient Roman villa. The maximum height of the building is about 13 m in correspondence with the tympanum of the Loggia of David. The walls are about 11 m high and present an average thickness of about 65 cm: the openings are generally uniformly arranged. The north, east, and west external sides present one or more arcades in the middle, while the south side is without entrance arcades and the arrangement of the windows does not follow a regular distribution, as observed for the other sides. The east side overlooking the large garden presents a high openings percentage due to the presence of large central arcades and a series of side Serlian windows. The sides delimiting the courtyard present a series of openings that are vertically and horizontally aligned. The east and west walls overlooking the courtyard are characterized by a series of niches that are 40 cm thick, both at the ground level and in correspondence with the mezzanines, and present a limited number of openings. The bearing walls of the building are built in regular course bricks with lime mortar and plastered to simulate a coursed rubble. The covering elements are masonry (barrel, cloister, and ribbed) vaults and wooden coffered slabs. The pitched roof consists of wooden truss beams and clay tiles. Fig. 5.22 presents the plan and two sections of the building along with the main geometrical dimensions.

FIGURE 5.22 Palazzo Te. Plan, two sections and indication of the main geometrical dimensions.

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5.5.2

191

Finite element model and eigenfrequency analysis

A detailed 3D FE model of the building under study is created using the drawings and the data collected from existing available documentations and during the survey phase. Fig. 5.23 shows the geometrical and FE models of the building. It is worth mentioning that the wooden structures of the coverings were not considered in the FE model. Four-node tetrahedral elements (about 430,000) with a size ranging between 20 and 40 cm were used in the discretization of the models. In each FE model, the most relevant macroelements, indicated in Fig. 5.24, are highlighted and investigated in detail in the following. Fig. 5.25 shows the deformed shapes and the corresponding periods of the main vibration modes with PMR larger than about 5% for Palazzo Te: moreover, the distribution of the first 300 modes in the two orthogonal directions is presented. The first two main modes are Mode 3 (T 5 0.253 s), involving the upper part of the wall of the Loggia and the external east wall with PMR equal to

FIGURE 5.23 Geometrical and FE models of Palazzo Te.

FIGURE 5.24 Palazzo Te. Indication of the different macroelements in the FE model. Notation: WL, loggia wall; WEE, external east wall; WEC, courtyard east wall; WNE, external north wall; WNC, courtyard north wall; WWE, external west wall; WWC, courtyard west wall; WSE, external south wall; WSC, courtyard south wall; VE, east vaults; VW, west vaults; VN, north vaults.

Spectral acceleration (g)

0.6 0.5 0.4 0.3 0.2 0.1 0

Mode 29 T = 0.120 s PMRx = 2.85% PMRy = 6.48%

0.2

0

0.2

0.4 0.6 Period (s) 0.4 0.6

0.8

1

0.8

1

0 5 PMR (%)

Mode 32 T = 0.117 s PMRx = 0.04% PMRy = 7.86%

0

Mode 3 T = 0.253 s PMRx = 0% PMRy = 5.52%

10 15 20 25 30

Mode 25 T = 0.128 s PMRx = 0.05% PMRy = 15.36%

X direction Y direction

Mode 23 T = 0.134 s PMRx = 7.35% PMRy = 0.06%

Mode 15 T = 0.153 s PMRx = 7.6% PMRy = 0%

Mode 17 T = 0.148 s PMRx = 14.64% PMRy = 0.04%

FIGURE 5.25 Palazzo Te. Deformed shapes, corresponding periods and participation mass ratios of the main vibration modes, with reference to the response spectra of the accelerograms used in the nonlinear dynamic analyses.

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5.52% in the Y direction, and Mode 15 (T 5 0.153 s), involving the north wall with PMR equal to 7.6% in the X direction. It is interesting to observe that there is a large difference between the periods of the first two main modes. The most relevant modes are Mode 17 (T 5 0.148 s), involving the east and north sides with the largest PMR (14.6%) in the X direction, and Mode 25 (T 5 0.128 s), involving the north side with the largest PMR (15.3%) in the Y direction. The first 300 modes correspond to a total PMR of 83% in the X direction and 82% in the Y direction. It can be noted that the main vibration modes present a period corresponding to high amplifications of the spectral accelerations, above all in the Y direction: the critical parts of the building result the east and north sides.

5.5.3

Nonlinear dynamic analyses

Figs. 5.26 and 5.27 show the tensile damage contour plots of Palazzo Te at the end of the nonlinear dynamic analyses with different PGA. The results of the nonlinear dynamic analysis with PGA 5 0.05 g show the onset of damage in the critical elements of the building. It can be noted that the vaults present an evident damage in the connection regions with the walls. A good correlation can be observed between the numerical results and the real damage surveyed after the 2012 earthquake: in fact, there is an onset of cracks along the corners of the walls, as registered in the “Hall of the

FIGURE 5.26 Palazzo Te. Tensile damage contour plot at the end of the nonlinear dynamic analyses with different PGA: axonometric views.

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FIGURE 5.27 Palazzo Te. Tensile damage contour plot at the end of the nonlinear dynamic analyses with different PGA: top and bottom views.

Horses” and in the “Room of Eros and Psyche”: such cracks can be enlarged by the thrusts of the vaults on the walls. A moderate damage can be observed in different parts of the building at the end of the nonlinear dynamic analysis with PGA 5 0.15 g. An onset of damage is registered in correspondence with the corners of the walls. The damage is widespread not only in the external walls and in the walls overlooking the courtyard, but also in the orthogonal partition walls subdividing the internal spaces. The masonry vaults of the south, east, and west sides present cracks in the connection regions with the perimeter walls: damage increases in correspondence with the vaults covering long spans, causing a stiffness reduction of such vaults. A significant increase of damage is observed in the case of nonlinear dynamic analysis with PGA 5 0.25 g. Damage is considerably widespread in the vaults and in the external and internal walls. The onset of damage visible in the external walls in the case of smaller PGA increases significantly. Horizontal cracks can be seen in the spandrels in correspondence with the openings: moreover, shear cracks can be observed along the piers (in-plane mechanism), then reducing their strength. The thrusts of the vaults on the walls enlarge the cracks, triggering possible overturning mechanisms. The partition walls and the vaults covering long spans exhibit widespread damage. Fig. 5.28 shows the EDDTD for the main macroelements of Palazzo Te at the end of the nonlinear dynamic analyses with different PGA. It has to be pointed out that the vaults are the most damaged elements of the building.

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FIGURE 5.28 Palazzo Te. Energy density dissipated by tensile damage (EDDTD) for the different macroelements (walls, transversal walls, vaults) at the end of the nonlinear dynamic analyses with different PGA.

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Under PGA 5 0.25 g, the maximum value of EDDTD is registered for vault V7 in the west side and vault V17 in the northeast corner. High values are computed also for vaults V1 and V2 in the south side, vault V4 in the southeast corner, vaults V13, V15, and V16 in the north side. As regards the transversal walls, the highest values of EDDTD are registered for walls T13
T17 in the north side, walls T4
T5 in the south side and walls T20
T17 in the east side. Among the external and internal walls, the wall of the Loggia (WL) presents the highest EDDTD value. It can be noted that there is a large increase of EDDTD for all the macroelements in the case of PGA 5 0.25 g. Fig. 5.29 shows the maximum normalized displacements (top displacement/height) registered in the X and Y directions for the main macroelements of Palazzo Te during the nonlinear dynamic analyses with different PGA. The external north wall (WNE) presents the largest normalized displacement (larger than 2%) in the X direction: high values (larger than 1%) are registered also for the courtyard north wall (WNC), the external south wall (WSE), and the courtyard south wall (WSC). The wall of the Loggia (WL) presents the largest normalized displacement (larger than 2%) in the Y direction: high values (larger than 1.35%) are registered also for the external west wall (WWE) and the courtyard west wall (WWC). As regards the transversal walls, the highest normalized displacements are registered for walls T13
T16 (north side) and walls T4
T5 (south side) in the Y direction and walls T20
T17 (east side) in the X direction. Fig. 5.30 shows the maximum vertical displacements registered for the main vaults during the nonlinear dynamic analyses with different PGA. It is important to highlight that vaults V7 (west side), V1 (east side), and V17 (northeast corner) present considerable vertical displacements (larger than 20 cm), indicating an onset of a probable collapse.

5.6 Concluding remarks This chapter presented an investigation of the seismic response and damage distribution of three masterpieces of the outstanding cultural heritage in Mantua. Detailed 3D FE models with an elastoplastic damage constitutive law for masonry have been used to simulate the seismic behavior of the three monumental masonry constructions. The numerical simulations carried out have provided a deep insight into the seismic performance of the structures, identifying the damage patterns and the most vulnerable parts for different seismic intensity levels. Moreover, the results obtained in this study may provide useful information to address future strengthening interventions aimed at reducing vulnerability to higher seismic actions than those registered in Mantua during the 2012 Emilia earthquake.

FIGURE 5.29 Palazzo Te. Maximum normalized displacements (top displacement/height) registered for the main macroelements in the X and Y directions during the nonlinear dynamic analyses with different PGA.

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FIGURE 5.30 Palazzo Te. Maximum vertical displacements registered for the main vaults during the nonlinear dynamic analyses with different PGA.

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139. Dall’Asta, A., Leoni, G., Meschini, A., Petrucci, E., Zona, A., 2018. Integrated approach for seismic vulnerability analysis of historic massive defensive structures. J. Cult. Herit. Available from: https://doi.org/10.1016/j.culher.2018.07.004. DM, 2008. Nuove norme tecniche per le costruzioni. Ministero delle Infrastrutture (GU n.29 04/ 02/2008), Rome, Italy [New technical norms on constructions]. DPCM, 2011. Linee guida per la valutazione e la riduzione del rischio sismico del patrimonio culturale con riferimento alle Norme tecniche delle costruzioni di cui al decreto del Ministero delle Infrastrutture e dei trasporti del 14 gennaio 2008 [Italian guidelines for the evaluation and the reduction of the seismic risk for the built heritage, with reference to the Italian norm of constructions]. Endo, Y., Pela`, L., Roca, P., Da Porto, F., Modena, C., 2015. Comparison of seismic analysis methods applied to a historical church struck by 2009 L’Aquila earthquake. Bull. Earthq. Eng. 13 (12), 3749
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430. Masciotta, M.G., Roque, J.C., Ramos, L.F., Lourenc¸o, P.B., 2016. A multidisciplinary approach to assess the health state of heritage structures: the case study of the Church of Monastery of Jero´nimos in Lisbon. Constr. Build. Mater. 116, 169
187. Milani, G., Shehu, R., Valente, M., 2017. Possibilities and limitations of innovative retrofitting for masonry churches: advanced computations on three case studies. Constr. Build. Mater. 147, 239
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2980. Valente, M., Barbieri, G., Biolzi, L., 2017b. Seismic assessment of two masonry Baroque churches damaged by the 2012 Emilia earthquake. Eng. Fail. Anal. 79, 773
802. Valente, M., Milani, G., 2018a. Damage assessment and partial failure mechanisms activation of historical masonry churches under seismic actions: three case studies in Mantua. Eng. Fail. Anal. 92, 495
519. Valente, M., Milani, G., 2018b. Damage survey, simplified assessment, and advanced seismic analyses of two masonry churches after the 2012 Emilia earthquake. Int. J. Architect. Herit. Available from: https://doi.org/10.1080/15583058.2018.1492646. Valente, M., Milani, G., 2018c. Effects of geometrical features on the seismic response of historical masonry towers. J. Earthq. Eng. 22 (Suppl. 1), 2
34. Valente, M., Milani, G., 2018d. Seismic response and damage patterns of masonry churches: seven case studies in Ferrara, Italy. Eng. Struct. 177, 809
835. Van Der Pluijm, R., 1993. Shear behaviour of bed joints. In: Proceedings of 6th North American Masonry Conference, Philadelphia, pp. 125
136.

Chapter 6

Repair and conservation of masonry structures Maria Rosa Valluzzi, Matteo Salvalaggio and Luca Sbrogio` Department of Cultural Heritage, Piazza Capitaniato 7, Padova, Italy

6.1 Introduction Interventions on existing masonry buildings aim at improving the deteriorated conditions of materials and at rehabilitating reduced structural capacity (strength and/or deformation limit states). The causes of precarious scenarios can be found in environmental exposure and use without regular maintenance, and the existence of construction vulnerability (e.g., poor materials and inadequate construction details; irregular layouts; inaccurate design; miscalculations in effects of changes, including past interventions). In such conditions, the structural safety of existing masonry is particularly threatened by phenomena that cause tension in the material and/or the disaggregation of structural components, as in the worst cases of dynamic actions (i.e., earthquakes) or large-scale settlement. Preservation of architectural heritage also requires to comply with restoration criteria (e.g., minimum intervention; compatibility and durability; “reversibility,” understood as substitutability and removability; durability; respect of authenticity) (ICOMOS, 2001), so that interventions can improve mechanical behavior without excessive structural alterations and provide overall collaboration among components to avoid local or more extended brittle collapses. To allow for a uniform response of the building, the current seismic codes (NTC, 2008; Eurocode 8, 2005) indicate regular distribution of interventions, alert to solutions inducing mass increases (especially at the tops of buildings) and possible alterations in stiffness due to local repair works. In particular, masonry buildings fall into categories of interventions, listed in order to match the capacity design rule: primarily, techniques aimed at establishing box-like behavior (e.g., improvements in wall-to-wall and wall-tofloor/roof connections, compensation of the thrusts of arches and vaults/ roofs, stiffening of floors/roofs), and then those reinforcing masonry piers

Numerical Modeling of Masonry and Historical Structures. DOI: https://doi.org/10.1016/B978-0-08-102439-3.00006-3 Copyright © 2019 Elsevier Ltd. All rights reserved.

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and their integrity (e.g., improvement of monolithic behavior, strengthening of opening borders), and, lastly, all other elements (e.g., stairwells, nonstructural components). However, according to current building codes, both methods of analysis and design parameters for calculations (before and after interventions) depend to a great extent on the knowledge level (KL) achieved for the building and the consequent confidence factors (CFs), which affect (i.e., reduce) the mechanical properties of masonry (compression and shear strengths, elastic moduli). Nevertheless, intervention techniques commonly used in the past on masonry components and sufficiently validated by experimental analyses are recognized for their effectiveness through enhancing coefficients, which increase the above-mentioned material properties. Such coefficients refer to the most favorable conditions, that is, the techniques in question are intended to work effectively with the components, as this is an aspect which is implicitly assumed in the design phase but which must be very carefully checked in practice (i.e., correct applications with suitable workforces and tools). Starting from the basic assumptions for numerical modeling of masonry, this chapter evaluates the effects of intervention techniques gradually added to the structure, with software well-known for their reliability, that is, DIANA FEA 10.2 (DIANA FEA BV, 2017) and STA Data TREMURI 11.4 (S.T.A. Data, 2018), which refer to the finite element method (FEM) and equivalent frame method (EFM), respectively. This study quantifies the seismic improvements provided by strengthening, taking into account techniques enhancing overall box-like behavior (e.g., ties, ring-beams, confining rings, stiffening of floors/roofs and vaults) and then single masonry components increasing masonry properties (e.g., mortar repointing, grout injection, jacketing), as in real-life restoration works. In addition, the incremental effects of interventions are evaluated, adding new elements step by step, and in combination, in order to observe their influence on the overall behavior of the structure. This analysis was performed on some ordinary buildings selected from historical town centers in central Italy (Campi Alto di Norcia and Montesanto di Sellano, both in the province of Perugia), which have been struck by earthquakes for centuries and indeed also recently (2016
18). These case studies constitute a significant example of continual situation of damage, rebuilding and repair/strengthening, still readable today, and the effectiveness of which has often been revealed as insufficient or even worsening with respect to their expected behavior (Valluzzi and Sbrogio`, 2019). In Campi Alto di Norcia, houses were built in rows following the slope of the hills, and are separated from one another by streets and interrupted by flights of steps linking the various levels. A typical house in Campi Alto di Norcia is rectangular in plan, the longer side being perpendicular to the slope; it lays against the rock behind and often has three storeys, but only the last storey may be considered totally out-of-ground. Masonry barrel vaults perpendicular to the fac¸ade are only found at ground level and, at the

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end of each row, one or more buttresses, built against the thrust of the vaults, are visible. There are also gaps between the fac¸ade and the vaults, which have been interpreted as seismic-proof details to preserve the vaults once the fac¸ade collapses (Cardani, 2003). At higher levels, floors and roofs have simpler and lighter timber structures, with joists laid on transversal walls. Instead, Montesanto di Sellano lays on a relatively level ridge, and is therefore mainly composed of isolated houses (except for two aggregates), larger than those in Campi Alto di Norcia, and following a more regular pattern, with three main parallel streets and many perpendicular steep alleys linking the streets. Although multilayer (i.e., two or three layers) cross-sections characterize the masonry walls of these two town centers, their types of masonry are different: chaotic arrangements of yellowish porous limestone units with many brick fragments to reduce mortar nuclei in Campi Alto di Norcia, and roughly dressed but compact pink or white limestone blocks commonly arranged in regular courses in Montesanto di Sellano. The building stock of these town centers clearly demonstrate the unfavorable consequences of misconceived and ill-applied interventions and their resulting seismic behavior. In the following, a small house in Campi Alto di Norcia (CA24) (Fig. 6.1A and B), the last surviving in its row, was chosen as a case study due to an unusual mechanism, that is, the wall overturning at the ground floor. A typical example of a house in Montesanto (MS27) (Fig. 6.1C and D), showing slight damage, was also analyzed. The simple structure of these houses allowed testing the effects of interventions in computer-simulated analyses, thus limiting the spurious effects induced by their geometrical or structural irregularities.

6.2 Strategies for modeling masonry buildings Structural modeling is one of the key steps during the design process of interventions. Modeling strategies usually fall into two main groups: analytical and numerical. Both have their strengths and weaknesses, in terms of time costs, precision of results, and modeling complexity. On one hand, numerical models can produce a wide range of analyses, but they require considerable computational time and user practice to process data; on the other hand, analytical models involve faster calculations and modeling approaches, but their range of results is narrower. Improvement in computational efficiency during the last few years has led to more numerical codes and in the engineering field to simulate structural interventions. The continuum FEM and EFM are taken here as examples, since they are the most common approaches for predictive simulations of masonry structures. In this chapter, the two methods are called FEM and EFM, although FEM contraction is not limited to the continuum approach but to all modeling strategies involving finite element discretization. As the

FIGURE 6.1 Photos of CA24 (A) and (B) and MS27 (C) and (D) after 2016 earthquake in central Italy.

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modeling assumptions of both FEM and EFM follow different paths, the results are not expected to be equivalent. However, although the material models, calculation methods, and data management are all quite different, each approach can capture more interesting aspects than the others, and vice versa. In the following, the peculiarities of the two main modeling strategies are described.

6.2.1

Finite element continuum model assumptions

FE modeling is an affordable numerical method in which the analyzed system is discretized by finite elements connected to each other by nodes. Modeling strategies for masonry structures exploit different approaches: (1) detailed micromodeling (both units and mortar are discretized by continuum elements and the interface between them by discontinuous elements); (2) simplified micromodeling (units are represented by continuum elements and the properties of mortar joints and mortar-units interfaces are lumped in discontinuous elements); and (3) macromodeling (units and mortar are merged in the continuum media) (Lourenc¸o, 1996; Rots, 1997; Lourenc¸o et al., 1998). The macromodeling approach is usually adopted to simulate masonry buildings and structures, thanks to its capacity for discretizing large walls and taking their overall behavior into account. Masonry composite shows quasibrittle behavior, with different responses under compressive and tensile stresses, so that different constitutive laws must be implemented. One of the most suitable material models for masonry is the “total strain-based crack model”, which relies on fracture energies for both compression and tension. Masonry properties can also be described according to the Mohr
Coulomb (or Drucker
Prager) shear-compression law, an isotropic plastic model based on cohesion and friction angle.

6.2.2

Equivalent frame model assumptions

Masonry buildings with regular (aligned) openings and effective lintels can be modeled as equivalent frames of piers and beams connected by rigid nodes, provided that box-like behavior can be activated and inhibit “first mode” (out-of-plane) collapses. This kind of modeling is also applied to more irregular buildings, since it is inexpensive in terms of automatic calculation resources and offers an overall assessment of its safety levels. EFM approach derives from the former POR method (Tomaˇzevic, 1978) and became of widespread use in its many implementations (Dolce, 1989; Tomaˇzevic and Weiss, 1990) due to: (1) allowing deformations and failures in piers and lintels, and (2) considering different failure mechanisms in masonry (Magenes and Calvi, 1996; Brencich and Lagomarsino, 1998; Lagomarsino et al., 2013). Both POR and EFMs enter the nonlinear static analysis of a building with calculation of macroscale forces (M, N, V) only

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(rather than stresses). Walls are subdivided into piers, bearing vertical and horizontal loads, and lintels, coupled to the piers at different degrees of continuity. Actual 2D masonry panels are represented by 1D elements the strength limits of which, depending on panel dimensions, materials, and vertical loading, are expressed by three possible failure modes: (1) rocking, (2) diagonal cracking, and (3) sliding. However, implementation of internal DOFs in each pillar allows for still considering it as a 2D element (Magenes et al., 2000). The limit strengths of lintels are expressed only by previous cases (1) and (2), although a different definition of the shear strength—parallel instead of perpendicular to the bed joints—is required. Rigid nodes are obtained as buffer areas around the junctions among equivalent beams and equivalent pillars, which keeps their structural integrity throughout the analysis. This procedure permits “second-mode” (in-plane) failures only. In pushover analysis, collapse is caused by an ultimate “soft storey” mechanism, which occurs when the number of collapsed piers (i.e., they have reached or passed one of the (1)
(3) limit strengths) makes the masonry structure hypostatic at a certain level.

6.3 Modeling of components and related interventions Intervention techniques are conceived to enhance the seismic behavior of existing buildings by reducing both overall (e.g., lack of connections among components, low stiffness of floors and roofs) and local vulnerability (e.g., poor quality of masonry, deterioration of materials). The question is how to model these interventions and what their effect will be on overall structural behavior, provided that the modeling approaches (FEM and EFM) are suitable for this aim. In addition, as more than one strengthening measure can be applied to a building to solve concomitant damage or weaknesses, it is important to estimate both the single and multiple contributions of the above techniques. Of the various techniques aimed at rehabilitating the overall behavior of existing masonry buildings in seismic areas, the most common interventions are slab reinforcements, ties, ring-beams, and confining rings. These solutions were also often applied in the past with substitutions/retrofitting of timber floors/roofs and vaults with heavy r.c. elements; later, the unfavorable effects demonstrated by subsequent earthquakes indicated more compatible solutions (ICOMOS, 2001), for example, superimposition of timber boarding and/or diagonal tying for floors, as well as composite materials, e.g., fiberreinforced polymer (FRP) and, later, textile-reinforced mortars (TRM) (Papanicolaou et al., 2011; de Felice et al., 2014; Valluzzi et al., 2014; Valluzzi, 2016). In the present study, results from experimental research as in the literature are examined, to evaluate design and assessment parameters related to the interventions mentioned above.

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In regards to the techniques aimed at improving the mechanical behavior of masonry, the Italian seismic code (NTC, 2008) and its application guide (Circolare 2 Febbraio, 2009) can be an useful tool in identifying the mechanical properties of various types of masonry (11 categories are described) and their increases (all-inclusive multiplying coefficients are provided) according to the most frequent intervention techniques adopted for existing masonry buildings, for example, mortar repointing, transverse connections (ties or shear stones), grout injection, and jacketing. As composite materials are not included in this list, data from the research literature should be adopted (see part III of the book). Table 6.1 lists the mechanical parameters of the various masonry types (i.e., density ρ, Young’s modulus E, shear modulus G, compressive strength fc, and shear strength τ 0) according to Circolare 2 Febbraio (2009) and the corresponding coefficients of efficiency of intervention techniques. The seismic code requires the minimum or mean values of the ranges indicated for compression and shear parameters to be adopted, according to the KL acquired for the building in question, depending on the accuracy of information collected on geometrical survey, structural details, and material properties. The ratios of intervention effectiveness are then applied to both compression and shear strengths and elastic moduli, and these values are reduced in the assessment phase by the respective CF, as shown in Table 6.2. In the case studies discussed in this chapter, a KL of 1 and a corresponding CF of 1.35 were adopted, as the survey was complete and the in situ investigations limited. The tensile strength ft, which is required to describe the total strain crack model, was derived from shear strength with the ˇ coviˇc (1971) equation: Turnˇsek and Caˇ ft 5 1:5τ 0

6.3.1

ð6:1Þ

Interventions to increase wall strength

Wall structures,in FEM simulations, are usually discretized by isotropic 2D shell finite elements, capable of being loaded in both in-plane and out-ofplane directions, and working as a combination of plane stress and platebending elements. They are suitable for simulating wall elements, due to their small thickness compared with the other dimensions. Alternatively, walls can be discretized by means of solid elements, in order to capture second-order effects arising through the thickness of the wall, and thus increasing the number of nodes and computational effort. In EFM, masonry piers and lintels are modeled as beam elements, whereas the intersections between them are represented by nondeformable nodes. In particular, in TREMURI (Galasco et al., 2002; Penna et al., 2004), the wall and lintel elements are provided by elements which have no resistance to out-of-plane loads. These macroelements are described as panels

TABLE 6.1 Properties of existing masonry types and coefficients of effectiveness of interventions. Masonry type

Masonry properties fm (MPa)

τ 0 (MPa)

E (MPa)

G (MPa)

Coefficients of effectiveness of interventions ρ (kN/m3)

ft (MPa)

Mortar repointing

Grout injection

Jacketing

Irregular stone masonry

1.0
1.8

0.020
0.032

690
1050

230
350

19

0.030
0.048

1.3

2

2.5

Multileaf masonry with thin outer layers of rough-worked ashlars and rubble core

2.0
3.0

0.035
0.051

1020
1440

340
480

20

0.520
0.076

1.2

1.7

2

Rough-cut stone masonry with good texture

2.6
3.8

0.056
0.074

1500
1980

500
660

21

0.084
0.111

1.1

1.5

1.5

Soft-stone ashlar masonry

1.4
2.4

0.028
0.042

900
1260

300
420

16

0.042
0.063




1.7

2

Stonework with squared blocks

6.0
8.0

0.090
0.120

2400
3200

780
940

22

0.135
0.180




1.2

1.2

Brickwork with solid units and lime mortar

2.4
4.0

0.060
0.092

1200
1800

400
600

18

0.090
0.138




1.5

1.5

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TABLE 6.2 Conditions required to identify knowledge levels and confidence factors for masonry. Knowledge levels

Geometric survey

Structural details

Material properties

Confidence factors

KL1

Survey of masonry walls, vaults, floors, and stairs

Limited in situ tests

Limited in situ tests (strength: minimum as from Table 6.1 ranges; elastic modulus: average as from Table 6.1 ranges)

1.35

Extensive and exhaustive in situ tests

Extensive in situ tests (strength: average as from Table 6.1 ranges; elastic modulus: average of tests or average as from Table 6.1 ranges)

1.20

Exhaustive in situ tests

1.00

KL2

KL3

Detection of loads applied by walls and type of foundations Survey of damage and deformation status

1. Three or more experimental strength values available (strength: average of experimental values; elastic modulus: average of tests or average of Table 6.1) 2. Two experimental strength values available (strength: if average of experimental values is lower than or between range of Table 6.1: average of Table 6.1; if average of experimental values is higher than range of Table 6.1: average of experimental values; elastic modulus: average of tests or average of Table 6.1) 3. One experimental strength value available (strength: if average of experimental values is higher than or between range of Table 6.1: average of Table 6.1; if average of experimental values is lower than range of Table 6.1: average of experimental values; elastic modulus: average of tests or average of Table 6.1)

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PART | I Seismic vulnerability analysis of masonry and historical structures

subdivided into three subelements: the two extreme ones (top and bottom), of infinitesimal thickness, have axial deformability (but no shear deformability), whereas the central element has shear deformability (but no axial deformability). Overturning of the panel is represented by contact interfaces, placed at its extreme sections, which work as plastic hinges. Design of strengthening interventions for existing and historic masonry walls first requires evaluation of the units and the state of mortar conservation. In particular, in historic multileaf masonry walls with rubble core, the quality of the inner mortar (if present) is usually precarious. As this is the most vulnerable type of masonry, interventions aimed at ensuring cohesion should be designed first, for example, grout injections, which are applicable in network of interconnected voids (Tomaˇzevic and Apih, 1993; Binda et al., 1997; Valluzzi et al., 2004, 2018). Alternatively, or even in combination, other interventions can also be applied, according to masonry types and their condition state, for example, jacketing, reinforced repointing, installation of FRP textiles, or TRM meshes (da Porto et al., 2018). Numerical simulation of strengthening interventions on walls can be carried out adjusting the linear and nonlinear masonry properties, according to experimental tests or the scientific literature (Onsiteformasonry, 2005). Otherwise, Table 6.1 may be used, if no other data are available. An alternative method of simulation, according to the FE micromodeling approach, is discretization of all masonry and reinforcements elements. However, both modeling and computational effort is dramatically increased, making this approach undesirable.

6.3.2

Interventions to increase slab stiffness

Slab structures in masonry buildings (i.e., floors, roofs) are usually discretized with 2D shell elements (membrane) in both FEM and EFM approaches. Orthotropic properties can also be implemented, simulating the different degrees of stiffness in floor structures with preferential framework direction (e.g., r.c. and timber slabs with monodirectional beams). Strengthening interventions in seismic conditions mainly aim at increasing the in-plane stiffness of slabs, so that they can convey horizontal forces to the piers. The most common techniques include adding nondeformable (rigid) slabs, made of r.c. or timber, to existing floors. In particular, timber double-planking has been shown to be a compatible choice for the preserving existing wooden floors in masonry buildings, where the stiffness required is not as high as that for new buildings (Valluzzi et al., 2010, 2013, 2015; Gattesco and Macorini, 2014; Gubana, 2015). These interventions can be implemented in numerical models by increasing membrane elasticity and shear moduli. Proper values can be obtained by experimental tests from the scientific literature or from codes, if any. Another way of simulating nondeformable slabs, in the FEM approach, is to

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211

use tying or rigid links elements. A slab is simulated as one translational point mass, located in the centroid of the floor with its mass, working along the principal directions. This is then connected to the floor-walls intersection nodes by rigid links/tying. These elements have the function of forcing some slave nodes (i.e., floor
wall intersection nodes) to have the same displacements and/or rotations of another master node (slab point mass). Thus, all the nodes of the model at floor height are constrained to move together.

6.3.3

Intervention to increase vault stability

Vaults are building components that are analyzed by a specialized approach. Their structural behavior depends on the geometric configuration of their elements and, only secondarily to material strength (Heyman, 1995). Thus, structural assessment of vaults cannot be limited to FEM and EFM analyses. Vaults, in FEM, can be modeled with curved shell elements (i.e., plate/shell elements, the nodes of which are not necessarily in the same plane). The EFM, allows for vaults being described in the model only in terms of equivalent stiffness and mass, so that only its contribution to global behavior is assessed (Cattari et al., 2008). In general, the horizontal stiffness given to a building by masonry vaults is not sufficient to consider it as a diaphragm capable of conveying seismic forces to vertical structures. Stiffening interventions should thus be provided, for example, the addition of nondeformable slabs (r.c. or double-planked flooring) relying on vault infill. Nevertheless, stiffening interventions do not prevent vaults from collapsing. In fact, they may fail due to: (1) progressive growth of plastic hinges (due to local tensile failures) until the system is longer stable; and (2) overturning of supporting walls, due to destabilizing thrusts of the vault itself. Strengthening interventions, therefore, aim at neglecting the formation of hinges or increasing the stability of piers against horizontal thrusts. A recent practice considers application of composite systems applied with inorganic matrix (i.e., TRM) covering the areas where tensile failures may occur and contributing to resettling acting forces and infill in a more balanced configuration, so that a pseudo-ductile behavior can be activated (Borri et al., 2009; Corradi et al., 2014; Valluzzi et al., 2014; Giamundo et al., 2015; Cescatti et al., 2018); see also part III of the book.

6.3.4

Intervention to improve connections among components

Several techniques can be adopted to improve collaboration among structural components, so that box-like behavior can be ensured against brittle out-ofplane collapses. This strategy is of primary importance in the design of preservation measures for existing buildings in seismic prone areas and, together with stiffening of horizontal elements, must precede any strengthening of

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PART | I Seismic vulnerability analysis of masonry and historical structures

wall components. Building and seismic codes (Eurocode 8, 2004, 2005; NTC, 2008) stress the importance of improving the box-like behavior of a masonry building through stiffening of floors and roof, ties, and ring-beams or confining rings, the latter being usually made of r.c., reinforced masonry, or steel. Ring-beams are the most common in Italian buildings, because they have been prescribed by seismic codes since the 1990s. Otherwise, r.c. has proved to be undesirable in masonry strengthening because of its high weight and obtrusiveness, so that, today, interventions with reinforced masonry or steel beams are preferable (da Porto et al., 2018). These works, according to both FEM and EFM, can be simulated through beam elements, implementing the proper parameters and sections of r.c. or steel-ring beams. Their material models can be kept linear-elastic, in view of the higher strength of these materials compared with existing masonry. More traditional interventions to improve wall-to-wall collaboration are the application of metal ties (e.g., bars or cables) or confining rings (e.g., strips, laminates, belts). Ties connect opposite walls at floors/roof levels and are usually anchored outside fac¸ades to ensure suitable constraints. Confining rings also embrace the whole building at floors/roof levels, and can constrain the walls to overall behavior thanks to their shape factor. In both cases, composites (e.g., carbon or glass FRP bars as ties, FRP or TRM for rings), can also be adopted (see part III of the book). These elements can be modeled by truss elements or rigid links in the FEM method. Truss elements, being monodimensional, can only bear axial forces, so they are appropriate to simulate ties; nevertheless, the adoption of rigid link constraining two nodes (of different walls), in order to have the same displacement, could also be a good choice. Although the previous approaches are often suitable to ties modeling, an unrealistic rise in stress peaks at truss/rigid link nodes may appear, compromising structural analyses. In these cases, an alternative to the previous approaches is to use bar reinforcement elements, that is, peculiar tools that can be embedded in all plane stress and shell elements and allow the properties of the plates to be modified in some location points (Fig. 6.2A). The latter approach can be used to model both ties and reinforcement rings, avoiding any increase in local stresses.

FIGURE 6.2 (A) Bar reinforcement in shell element; (B) seismic response of lintels without ties; and (C) seismic response of lintels with ties connecting opposite walls.

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As regards EFM, based on the assumption of box-like behavior (like TREMURI), pier overturning is neglected. Ties are therefore modeled as elements with no bending or compression reactions, embedded in the piers. The ties can activate the inclined strut response of lintels, thus improving the overall seismic response of the structure (Fig. 6.2B and D).

6.4 Incremental effect of interventions To evaluate the effect of interventions applied to existing masonry buildings in seismic prone areas, the two modeling strategies (FEM and EFM) were adopted and compared. Linear modal analyses and pushover nonlinear static analyses were carried out on two case studies involved in the seismic events of 2016 in central Italy, that is, CA24 and MS27, at Campi Alto di Norcia and Montesanto di Sellano, respectively. Various proposals for interventions were designed and tested according to an incremental approach, taking into consideration both single and combined techniques, as shown in Table 6.3. In particular, two categories were defined and then crossed: stiffening of horizontal slabs, key players in the activation of box-like behavior, and interventions aimed at improving the seismic capacity of piers (i.e., strengthening of walls and improvement of their connections). Ties and confining rings are TABLE 6.3 Matrix of numerical testing with configuration labels (slab stiffness increases from A to E, vertical structures properties from 1 to 4; gray gradient: strengthening level achieved). Interventions on vertical structures –

Slab stiffening

Wall strengthening

1

2

3

None

Ties

R.c. ring beams

Traditional timber floor

A1

A2a

–

A4

B

Iron girders and flooring blocks

B1

–

B3

–

C

Double-planking timber floor

C1

C2

–

–

D

R.c. floor

D1a

–

D3

D4a

Eb

R.c. floor + r.c. slab on vault

E1a

–

E3

E4

A

a

Connection improvement

4

Repointing

Grout injections

Jacketing

Additional parametric study on effects of limited to extensive interventions (nonuniform). Applied only to CA24.

b

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PART | I Seismic vulnerability analysis of masonry and historical structures

FIGURE 6.3 CA24: (A) FEM and (B) EFM model; MS27: (C) FEM and (D) EFM model.

necessary when slabs are not stiff enough to convey forces to load-bearing piers (e.g., single-planked floor or iron girders with flooring blocks), forcing them to work as consistent system (Fig. 6.2C). The analyses started from a pilot case, A1, with poor floors stiffness and no interventions on vertical components. To this, step by step, single interventions were applied to improve pier behavior, keeping the other elements in condition A1. Later, the same approach was repeated for various roof stiffnesses, whether they belonged to built heritage (iron girders and floating blocks, r.c. floor) or to stiffening interventions (double-planked timber floors, r.c. slab additions to vaults). Three parametric studies (cases A2, D4, D1, E1) were also scheduled, based on variations of the number and location of interventions in the structure: starting from the strengthening/stiffening at one floor, this was gradually extended to the entire building, so that the effects of local interventions on overall seismic behavior could be assessed. The numerical models of CA24 and MS27 were arranged according to in situ geometric, structural, and critical surveys. According to the FEM approach, CA24 and MS27 models were discretized by means of 8-noded regular curved shell elements CQ40S and CT30S (Fig. 6.3A and C). According to the EFM approach, the structures were modeled as a system of piers, lintels, and rigid nodes, as shown in Fig. 6.3B and D. Foundation constraints were modeled by two approaches: the inclined foundation ground of CA24 was kept with its own site inclination in the FEM, whereas, in the EFM, the corresponding masonry piers were subdivided into four segments to approximate the foundation plane; in MS27, horizontal foundations were modeled. Material properties were derived from Tables 6.1 and 6.2 (reference KL1), as listed in Table 6.4. Before analysis of all combinations of interventions, a model updating procedure based on the damage survey was applied, in order to calibrate the numerical models to the current state of structures and simulate their actual seismic behavior (Fig. 6.4). The corresponding configurations are D3 for

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215

TABLE 6.4 Masonry properties of CA24 and MS27 case studies. Case

Masonry

E

G

fc

Gfc

τ0

ft

Gft

ρ

study

type

(MPa)

(MPa)

(MPa)

(N/mm)

(MPa)

(MPa)

(N/mm)

(kg/m3)

CA24

Irregular stone masonry

690

230

1

3

0.020

0.03

0.015

1900

MS27

Multileaf masonry with thin outer layers of roughworked ashlars and rubble core

1020

340

2

3

0.035

0.052

0.015

2000

FIGURE 6.4 CA24: (A) diagram of tensile strain for E1 configuration (FEM) and (B) deformation and damage at ultimate state (EFM). Both software captured the wall overturning at the base (Fig. 6.1B).

CA24 and D4 for MS27 (Table 6.3). Several intervention hypotheses were then modeled, removing or adding structural parts such as slabs and ties to assess their effects on the structural behavior of the buildings. Results were processed in terms of capacity curves, obtaining the α vs. displacement relationship law (where α is the seismic ratio between base shear and total mass; the displacement is the average of top nodes displacements). The Newton
Raphson method was adopted for EF models, and the Secant method (with arc-length algorithm), faster than the former, for FE

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PART | I Seismic vulnerability analysis of masonry and historical structures

models. The convergence criterion was based on energy, with a 0.01 tolerance. For brevity, the most significant results obtained for uniform (mass proportional) load distribution along the direction X are shown, as they were the most onerous in terms of seismic verification and those corresponding to current damage status. The state-of-fact first mode shapes and capacity curves are shown in Fig. 6.5. Natural frequencies of FE models were higher than those of EF, resulting in higher overall stiffnesses of the continuum models. The capacity curves calculated from pushover analyses also showed more conservative values for EFM, with lower seismic coefficient (α) and displacement capacity. Better performance of the continuum FE model with smeared cracking was expected, as vertical and horizontal structures had perfect continuity and also out-of-plane resistance. Therefore, the two modeling approaches revealed several differences, which must be taken into consideration in preliminary modeling choices. The EF approach is suitable for regular masonry buildings, with piers (continuous from building top to bottom) and aligned openings. For example, in the CA24 case, the subdivision of the base longitudinal wall into four piers of different height led to underestimation of the actual stiffness of the wall.

6.4.1

Strengthening interventions on floors and roofs

As the base for the modeling of horizontal components in existing masonry buildings, four common floor types were examined, that is: (1) traditional timber floor (monoplanked timber with preferential framework direction); (2) floor with iron girders and flooring blocks (I-shaped steel beams, brickwork flooring blocks and a thin r.c. slab usually detached from walls); (3) double-planked timber floor (superposition of a double-planked slab, with high stiffness in both main directions to existing plank floor); and (4) r.c. slab. Table 6.5 lists the characteristics of the various slab types; the symbols are as follows: G1 5 permanent structural load; G2 5 nonpermanent structural load; Qk 5 accidental loads on floors; Qk,roof 5 accidental loads on roofs; t 5 seismic resistant slab thickness; Ex 5 axial elasticity modulus along the main framework direction; Ey 5 axial elasticity modulus along secondary framework direction; GXY 5 shear elasticity modulus in plane XY. In the FEM, slabs were modeled as orthotropic plate with thickness equal to those of r.c. or timber-planked floors, without out-of-plane stiffness. Thus, the floor masses were modeled by point mass elements directly at slab-wall intersection nodes, avoiding improper out-of-plane effects on floors. In the EFM, slabs were also modeled as orthotropic membranes. When conservation status and connection to vertical structures are scarce, traditional timber floors can only be modeled as point masses distributed along load-bearing walls, in order to neglect their stiffness contribution completely. Instead, slabs with high stiffness, such as those in r.c., can be

FIGURE 6.5 Results of eigenvalue and pushover analyses for CA24 and MS27.

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PART | I Seismic vulnerability analysis of masonry and historical structures

TABLE 6.5 Floor typologies and related loads and properties. Traditional timber floor

Floor with iron girders and flooring blocks

Double-planked timber floor

r.c. floor

G1 5 1.0 kN/m2

G1 5 3.5 kN/m2

G1 5 1.0 kN/m2

G1 5 3.5 kN/m2

2

2

2

G2 5 1.5 kN/m2

Loads

G2 5 1.0 kN/m

G2 5 1.5 kN/m

Qk 5 2.0 kN/m2 Qk,roof 5 1.5 kN/m

G2 5 1.0 kN/m

Qk 5 2.0 kN/m2 2

Qk,roof 5 1.5 kN/m

Qk 5 2.0 kN/m2 2

Qk,roof 5 1.5 kN/m

Qk 5 2.0 kN/m2 2

Qk,roof 5 1.5 kN/m2

Structural properties t 5 4 cm

t 5 5 cm

t 5 7 cm

t 5 5 cm

Ex 5 9240 MPa

Ex 5 9247 MPa

Ex 5 16,280 MPa

Ex 5 43,200 MPa

Ey 5 0 MPa

Ey 5 0 MPa

Ey 5 11,000 MPa

Ey 5 27,000 MPa

Gxy 5 0 MPa

Gxy 5 11,250 MPa

Gxy 5 750 MPa

Gxy 5 11,250 MPa

modeled with rigid links (see Section 6.3.2). Orthotropic shell elements, with implementation of experimental elasticity and shear moduli, usually contribute some degree of stiffness to overall behavior, between the previous approaches. Fig. 6.6 compares the results in terms of overall stiffness (1st mode frequency), horizontal coefficient α and displacement capacities of the different slab modeling strategies. According to the FEM, the capacity curves diverged slightly, meaning that the previous methods are successful. Instead, according to the EFM, not modeling timber floors caused an unrealistic decrease in overall behavior. Therefore, in the following, models in both software were modeled adopting orthotropic shells. The contributions of structural roof types to overall behavior of buildings, such as CA24 and MS27, were evaluated by generating different models with increasing horizontal stiffness of slabs (i.e., A1, B1, C1, D1, E1), in which improvements in vertical structures behavior were not implemented. The results are shown in Fig. 6.7. The models with traditional single-planed floors were revealed as the most vulnerable by both FEM and EFM in both case studies. Apart from this, EFM did not show great differences between the remaining structural slabs, probably due to the box-like behavior assumption of the approach, independently of floor/roof type. In addition, FEM revealed the highest stiffnesses and capacities for r.c. and double-planked slabs, with a major ultimate displacement capacity of the latter. Therefore, the double-planking

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219

(A)

α coefficient

0.25 0.2 FEM

0.15

CA24_A1_0 STIFF. CA24_A1_ORTOTR.

0.1

CA24_E1_ORTOTR.

0.05

CA24_E1_NON DEFOR.

0 Displacement (mm) (B)

0.1

α coefficient

0.08

EFM

0.06

CA24_A1_ORTOTR.

0.04

CA24_A1_0 STIFF.

0.02

UMI24_E1_ORTOTR.

CA24_E1_NON DEFOR.

0 0

5

10

15

20

25

Displacement (mm)

FIGURE 6.6 Comparison of slab modeling strategies for (A) FEM and (B) EFM.

interventions were shown, numerically, to be good approaches for floor/roof strengthening, proving that high-stiffness elements, such as r.c., need not to be implemented to increase structural seismic capacity. Lastly, it should be recalled that the good numerical results for r.c. slabs models were due to the assumption of perfect continuity between walls and diaphragms in the FEM, not taking into account the poor connections between masonry and r.c., which usually occurs on-site due to the obtrusiveness of such a technique.

6.4.2

Interventions on buildings with traditional timber slabs

The preservation of traditional structural components in historic buildings, such as single-planked timber slabs, requires the adoption of techniques characterized by low obtrusiveness and high removability, such as application of ties and reinforcement rings, able to constrain piers to a consistent response. A parametric study on the contribution of ties applied step by step (starting from the third top floor to the second one, up to all floors), gave the capacity curves shown in Fig. 6.8. FEM was capable to detect the increasing benefits of using ties, particularly clear-cut for MS27 (Fig. 6.8A), which shows the increasing effect on ultimate displacement capacity, compared to case A1.

(A) 0.25

(B)

EFM - CA24 CA24_A1 CA24_B1 CA24_C1 CA24_D1 CA24_E1

0.2

α coefficient

α coefficient

0.2

0.25

0.15

FEM - CA24 0.1

CA24_A1 CA24_B1 CA24_C1 CA24_D1 CA24_E1

0.05

0.15 0.1 0.05

0

0

0

5

10 15 Displacement (mm)

20

(C) 0.6

0

5

10 15 Displacement (mm)

20

(D)

0.6

0.5

EFM - MS27

α coefficient

α coefficient

0.5

0.4

FEM - MS27

0.3

MS27_A1

0.2

MS27_B1

MS27_A1 MS27_B1

0.4

MS27_C1 MS27_D1

0.3 0.2

MS27_C1

0.1

0.1

MS27_D1

0 0

5 10 Displacement (mm)

15

0 0

5 Displacement (mm)

10

FIGURE 6.7 Floor strengthening interventions according to FEM approach on (A) CA24 and (C) MS27; according to EFM approach (B) CA24 and (D) MS27.

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As expected, there were no benefits in terms of overall stiffnesses, as ties worked as constraints to overturning. As the TREMURI assumption of box-like behavior is not compatible with pier overturning, the effects of ties are not due to a retaining effect but to strut activation in lintels. This is shown by the slight increase in coefficient α and the displacement capacity (Fig. 6.8C and D). The design of tie interventions assumes that piers behave like monolithic macroblocks during overturning mechanism. Nevertheless, this hypothesis requires good masonry quality, which is not always the case in historical buildings, especially for multileaf walls with rubble cores. Therefore, reinforcements on walls (e.g., grout injections, reinforced repointing, or jacketing), when allowed, should be considered and their effectiveness carefully checked. The advantages of these techniques were assessed by comparison of models A1, A2, and A4 (Table 6.3). As shown in Fig. 6.9A and B, grout injections, according to FEM, influence only the overall stiffness (increase). Instead, according to TREMURI, in which box-like behavior is assumed, wall strengthening improved both the overall stiffness and displacement capacity. In the case of single-planked floors, the two software showed quite different results. In particular, FE models, in CA24 for cases A1 and A4, without ties/ring beams, showed brittle behavior, which TREMURI avoids, due the assumption of box-like behavior. Brittle performance is due to the overturning at third floor, which can be prevented by the addition of ties (A2). Once the ties are applied and overall behavior ensured, results between the two numerical approaches becomes more similar. This behavior is clearly detected for the CA24 case study (Fig. 6.10).

6.4.3 Interventions on buildings with iron girders and flooring block slabs The main issue of existing floors and roofs made of iron girders and flooring blocks is represented by the poor connections of the r.c. covering slab to the masonry walls, which makes floor/roof assume significant horizontal stiffness only in the direction of steel beams. Interventions aim at increasing the slab-to-wall connection by a r.c. ring-beam. Fig. 6.11 compare cases B1 and B3 (Table 6.3). Interventions clearly influence results in FE model with respect to EF, both in terms of overall stiffness and ultimate displacement. Otherwise, interventions cannot be completely implemented with EF software, in which it has a damaging effect. Fig. 6.12 shows, as an example, the collapse configurations for the MS27 case study.

(B)

0.16

0.16

0.14

0.14

0.12

0.12

0.1 0.08

FEM - CA24 CA24_A2_FL. 1, 2, 3

0.06

CA24_A2_FL. 2, 3

0.04

0.08 CA24_A2_FL. 2, 3 CA24_A2_FL. 3

0.02

CA24_A1

CA24_A1

0 0

5

10 15 Displacement (mm)

20

0

(D)

0.35 0.3

5

0.25 0.2 FEM - MS27 MS27_A2_FL. 1, 2, 3

0.15

MS27_A2_FL. 2, 3

0.1

10 15 Displacement (mm)

20

0.35 EFM - MS27

0.3 α coefficient

α coefficient

CA24_A2_FL. 1, 2, 3

0.06

0

(C)

EFM - CA24

0.1

0.04

CA24_A2_FL. 3

0.02

α coefficient

α coefficient

(A)

MS27_A2_FL. 1, 2, 3 0.25

MS27_A2_Fl. 2, 3

0.2

MS27_A2_FL. 3 MS27_A1

0.15 0.1

MS27_A2_FL. 3

0.05

0.05

MS27_A1 0 0

5

10

Displacement (mm)

15

0 0

2

4 6 Displacement (mm)

8

10

FIGURE 6.8 Effect of application of ties at local and overall scale according to FE approach on (A) CA24 and (B) MS27; according to EF approach on (C) CA24 and (D) MS27.

Repair and conservation of masonry structures Chapter | 6

A4

0.16

0.14

0.14

0.12 0.1 0.08

FEM-CA24

0.06

CA24_A1

0.04

CA24_A2

0.02

A2

A4

0.12 0.1 0.08 0.06

EFM - CA24 CA24_A1

0.04

CA24_A2

0.02

CA24_A4

CA24_A4

0

0 0

5

10

15

20 Displacement (mm)

Displacement (mm)

(E)

(F)

0.35

0.35

0.3

0.3 α coefficient

α coefficient

A1

EFM

(D)

0.16 α coefficient

α coefficient

(C)

5.07 4.56

4.51

FEM

5.06 4.56

A2

MS27

(B) f1 (Hz)

3.20 2.49

A1

EFM

3.41

3.19 2.49

f1 (Hz)

FEM

6.59 5.83

CA24

(A)

223

0.25 0.2

FEM-MS27

0.15

MS27_A1

EFM - MS27 MS27_A1

0.25

MS27_A2

0.2

MS27_A4

0.15 0.1

0.1

MS27_A2

0.05

0.05 MS27_A4

0

0 0

5 10 Displacement (mm)

15 Displacement (mm)

FIGURE 6.9 Comparison of ties and injections application in terms of (A) and (B) mode frequency; in terms of capacity curves according to FE approach on (C) CA24 and (D) MS27; according to EF approach on (E) CA24 and (F) MS27.

6.4.4 Interventions on buildings with double-planked timber floors The effectiveness of the double-planked flooring depends on its connection with the already existing timber floor and, overall, to load-bearing walls. Once the connection has been ensured, the orthotropic timber membrane can be modeled according to the thickness of the triple-board slab and the shear modulus of timber. In the following, the validity of double-planked flooring intervention was assessed by the addition of ties at each floor. If the floors are sufficiently stiff to be nondeformable, not much additionally seismic capacity can be expected from the structure. This situation occurred in both CA24 and MS27, according to the FE method (Fig. 6.13). The contribution of ties to the EF model changes lintels collapse from combined compression and bending to shear, as reported by Magenes et al. (2000), (Fig. 6.2B and D). In

224

PART | I Seismic vulnerability analysis of masonry and historical structures

FIGURE 6.10 Traditional timber floor collapse states in CA24 according to FEM: (A) A1; (C) A2; and (E) A4; according to EFM: (B) A1; (D) A2; and (F) A4.

addition, the two software produced similar collapse configurations in terms of damage patterns; results for CA24 are shown in Fig. 6.14.

6.4.5

Interventions on buildings with r.c. slabs

In the case of r.c. slabs used in masonry buildings, critical aspects are the positioning of such heavy horizontal components on poor-quality masonry walls, and their effective connection with the main walls to ensure collaborative box-like behavior. The effect of r.c. ring-beams at floor levels and grout injections in walls was therefore evaluated. However, the use of r.c. slabs to retrofit existing masonry buildings has frequently triggered severe damage and even collapse in city centers struck by earthquakes. Therefore, the common practice of substituting traditional floors with r.c. elements has

Repair and conservation of masonry structures Chapter | 6

(D)

0.3

α coefficient

α coefficient

0.25 0.2

FEM-CA24

0.15 0.1

CA24_B1

0.05

CA24_B3

A2

A4

0.3 0.25

EFM - CA24 CA24_B1

0.2 CA24_B3

0.15 0.1 0.05 0

0 0

5

10

15

20

25

Displacement (mm)

Displacement (mm)

(E)

(F) 0.6

0.6

0.5

0.5

α coefficient

α coefficient

A1

EFM

6.59 5.83

4.51 A4

5.07 4.56

A2

f1 (Hz)

A1

EFM

MS27 FEM

3.41

3.20 2.49

f (Hz)

3.19 2.49

FEM

(C)

(B)

CA24

5.06 4.56

(A)

225

0.4

FEM-MS27

0.3 0.2

MS27_B1

0.1

MS27_B3

EFM - MS27 MS27_B1

0.4 MS27_B3

0.3 0.2 0.1

0

0

0

5

10

15

Displacement (mm)

20

25

Displacement (mm)

FIGURE 6.11 Comparison of iron girders and flooring blocks without and with r.c. ring beam in terms of (A) and (B) mode frequency; in terms of capacity curves according to FE approach on (C) CA24 and (D) MS27; according to EF approach (E) CA24 and (F) MS27.

“hybridized” many masonry constructions, meaning that defining proper modeling strategies for their study has become topical. Ring-beams are usually built in continuity with r.c. slabs, although connections with walls remain critical. According to the FE model, the orthotropic membranes or rigid links of r.c. slabs already account for the stiffness of r.c. beams, since they are essential to floor-to-wall connections. Therefore, there is no further need to model beam elements, as this could lead to overestimation of overall structural stiffness or other numerical issues. Instead, interventions on wall structures (e.g., grout injections) can only have positive effects, since they were modeled by the corrective coefficients applied to masonry properties (Table 6.1). Fig. 6.15 shows the remarkable increase in acceleration and ultimate displacement capacity due to the addition of beam elements in the FE model; a reduction in displacement capacity occurred in the EF model for the CA24 case study, probably due to a numerical issue. However, wall strengthening ensured good results in terms of both

FIGURE 6.12 Iron girders and flooring blocks collapse configurations in MS27 according to FEM: (A) B1 and (C) B3; according to EFM: (B) B1 and (D) B3.

Repair and conservation of masonry structures Chapter | 6 (B)

(C)

6.50 6.18

C2

EFM

MS27 6.49 6.18

C1

FEM

f1 (Hz)

3.50 2.83

CA24 3.50 2.83

f1 (Hz)

(A)

227

C1

C2

FEM

EFM

(D)

0.25

0.25

0.2

0.2

α coefficient

α coefficient

EFM-CA24

0.15

FEM-CA24 0.1 CA24_C1

0.05

CA24_C1

0.15

CA24_C2

0.1 0.05

CA24_C2

0

0 0

5

10

15

20

0

5

Displacement (mm)

10

15

20

Displacement (mm)

(D)

(E)

0.6

0.6

0.5

0.5

α coefficient

α coefficient

EFM-MS27 0.4

FEM-MS27

0.3 0.2

MS27_C1

0.1

MS27_C2

MS27_C1

0.4 MS27_C2

0.3 0.2 0.1

0

0 0

5

10

15

20

Displacement (mm)

0

5

10

15

20

Displacement (mm)

FIGURE 6.13 Comparison of double-planked timber floor without (C1) and with (C2) ties in terms of mode frequency, (A) and (B); in terms of capacity curves according to the FE approach on (C) CA24 and (D) MS27; according to the EF approach (E) CA24 and (F) MS27.

stiffness and capacity: the first mode frequency increased, and both modeling strategies revealed enhanced capacity (Fig. 6.15). The effects of nonuniform wall injections were also checked by comparing them with interventions uniformly distributed throughout the buildings (Fig. 6.16). For MS27, analyses showed linear increases in capacity curves, those of overall uniform interventions at all floors being higher than the others. The most significant results were obtained for CA24: the greater stiffness measured in floors 2 and 3, as expected, caused a sudden decrease in displacement capacity, due to the concentration of forces at the top of the building. This confirms that nonuniform stiffening interventions may worsen the seismic response.

6.4.6

Interventions on masonry vaults

CA24 was taken as a reference case study to validate the intervention techniques applied to masonry vaults: it has nondeformable diaphragms (i.e., r.c.

FIGURE 6.14 Double-planked timber floor collapse configurations in CA24 according to FEM: (A) C1 and (C) C2; according to EFM: (B) C1 and (D) C2.

Repair and conservation of masonry structures Chapter | 6

6.87 5.46

D4

EFM

6.81 5.56

4.98

f1 (Hz)

D3

3.88

3.48 2.70

D1

FEM

(C)

D1

D3

D4

(D)

0.3

0.3

0.25

0.25

α coefficient

α coefficient

MS27

(B)

EFM

3.56 2.77

f1 (Hz)

FEM

8.60 7.02

CA24

(A)

229

0.2

FEM-CA24

0.15

CA24_D1

0.1

CA24_D3

0.05

EFM-CA24 CA24_D1 CA24_D3

0.2

CA24_D4

0.15 0.1 0.05

CA24_D4

0

0 0

5

10

15

20

25

0

5

Displacement (mm)

(E)

15

20

(F) 0.7

0.7

0.6

0.6

0.5

0.5

MS27_D3

0.4

MS27_D4

0.4

α coefficient

α coefficient

10

Displacement (mm)

FEM-MS27

0.3

MS27_D1

0.2

MS27_D3

0.1

EFM-MS27 MS27_D1

0.3 0.2 0.1

MS27_D4

0

0 0

5

10

15

Displacement (mm)

20

0

2

4

6

8

10

Displacement (mm)

FIGURE 6.15 Comparison of r.c. floors model (D1) with ties (D2) and with wall injections (D4) in terms of mode frequency, (A) and (B); in terms of capacity curves according to the FE approach on (C) CA24 and (D) MS27; according to the EF approach (E) CA24 and (F) MS27.

slabs) in its second and third floors, but a masonry vault on the first floor. However, as the equivalent stiffness of the vault does not allow it to work as a rigid element (with respect to the r.c. slabs), intervention techniques are required. Although the addition of a thin r.c. slab above the infill of the vault is not considered to be compatible with respect to more recent TRM techniques, that storey may be allowed to behave as a rigid plane. Thus, from the modeling viewpoint, an orthotropic shell with r.c. properties 4 cm thick was added at the height of the vault infill (E1). Later, r.c. ring-beams (E3) and injections (E4) were also added (Fig. 6.17). Both FE and EF software revealed that the r.c. slab caused stiffening of the first storey and shifted damage from there to the upper floors (Fig. 6.18).

6.4.7

Incremental effects of addition of rigid slabs

A parametric study on the influence of local addition of rigid slabs was also carried out on the E case for CA24 and the D case for MS27 (Fig. 6.19).

PART | I Seismic vulnerability analysis of masonry and historical structures

230 (A)

(B) 0.16

0.3

0.14

α coefficient

α coefficient

0.25 0.2

FEM-CA24

0.15

CA24_D4_FL. 1, 2, 3

0.1

CA24_D4_FL. 1, 2 CA24_D4_FL.. 2, 3

0.05

10

15

EFM-CA24 CA24_D4_FL. 1, 2, 3

0.06

UMI24_D4_FL. 1, 2

0.04

UMI24_D4_FL. 2, 3 UMI24_D1

0

0 5

0.1 0.08

0.02

CA24_D1

0

0.12

20

0

25

5

Displacement (mm)

(C)

15

20

25

(D) 0.7

0.4

0.6

0.35 0.3

0.5

α coefficient

α coefficient

10

Displacement (mm)

0.4

FEM-MS27

0.3

MS27_D4_FL. 1, 2, 3

0.2

MS27_D4_FL. 1, 2

0.1

MS27_D4_FL. 2, 3

0.25

EFM-MS27

0.2

MS27_D4_FL. 1, 2, 3

0.15

MS27_D4_FL. 1, 2

0.1

MS27_D4_FL. 2, 3

0.05

MS27_D1

MS27_D1

0

0 0

5

10

0

15

2

Displacement (mm)

4

6

8

10

12

Displacement (mm)

FIGURE 6.16 Incremental effects of local and overall wall injections on (A) and (B) CA24, and (C) and (D) MS27.

E3

3.92

3.58 2.73

E1

(B)

E4

(C)

0.35

0.35

0.3

0.3

α coefficient

α coefficient

EFM

3.63 2.80

f1 (Hz)

FEM

5.08

CA24

(A)

0.25 0.2

FEM-CA24

0.15

CA24_E1

0.1

CA24_E3

0.05

EFM-CA24 UMI24_E1

0.25

UMI24_E3

0.2

UMI24_E4

0.15 0.1 0.05

CA24_E4

0

0 0

5

10

15

20

Displacement (mm)

25

30

0

5

10

15

20

25

30

Displacement (mm)

FIGURE 6.17 Comparison of r.c. slab and injections interventions in the CA24 vault model in terms of (A) mode shapes, and (B) and (C) capacity curves.

Overall stiffness increased linearly with the addition of rigid slabs, from the top to the bottom of the buildings. Good results started to appear after the reinforcement of at least two floors. Fig. 6.20 shows the development of damage configurations matching both FEM and EFM, as the number of slabs increased.

FIGURE 6.18 Vault with r.c. slab collapse configurations in CA24 according to FEM: (A) E1; (C) E3; and (E) E4, according to EFM (B) E1; (D) E3; and (F) E4.

(B)

0.25

0.25

0.2

0.2

EFM - CA24

0.15

FEM - CA24 0.1

CA24_E1_FL. 1, 2, 3

α coefficient

α coefficient

(A)

CA24_E1_FL. 2, 3

0.05

CA24_E1_FL. 1, 2, 3 CA24_E1_FL. 2, 3

0.15 CA24_E1_FL. 3

0.1 0.05

CA24_E1_FL. 3

0

0 0

5

10

15

20

0

Displacement (mm)

10

15

20

Displacement (mm)

(D)

(C) 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

0.5

EFM - MS27

0.45

MS27_D1_FL. 1, 2, 3

0.4 α coefficient

α coefficient

5

FEM - MS27 MS27_D1_FL. 1, 2, 3

0.35

MS27_D1_FL. 2, 3

0.3

MS27_D1_FL. 3

0.25 0.2 0.15

MS27_D1_FL. 2, 3

0.1 MS27_D1_FL. 3

0

5 10 Displacement (mm)

0.05 15

0 0

5 10 Displacement (mm)

15

FIGURE 6.19 Incremental effects of local and overall stiffening of slabs on (A) and (B) CA24, and (C) and (D) MS27.

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PART | I Seismic vulnerability analysis of masonry and historical structures

FIGURE 6.20 Collapse configurations of MS27 at rigid slabs increasing: (A) and (B) third floor; (C) and (D) second and third floors; (E) and (F) all floors.

6.5 Concluding remarks This study aimed at assessing the effect of strengthening interventions by numerical methods applied with common FEM and EFM software; it also allowed assessment of numerical methods by means of validated structural techniques. The main conclusions are: G

G

Although currently available numerical codes are powerful tools, they have advantages and disadvantages that must be taken into account to prevent erroneous interpretations of results. The EFM, in particular the TREMURI approach, is not suitable for overall analysis of historic buildings with deformable floors, as box behavior is assumed; EFM can be generated also inside finite element environments (e.g., DIANA), by means of beam elements: in this case, the boxbehavior can be neglected.

Repair and conservation of masonry structures Chapter | 6 G

G

G

G

G

G

233

Although the continuum FE approach proved to be effective in highlighting the major effects of strengthening interventions, the computational effort is much higher than that of EFM. Modal analysis is a fast, powerful tool to assess the overall stiffness of a structure, and can also identify the variable effects of interventions. Numerical analysis confirms that double-planked slabs provide effective strengthening for historic timber floors, theoretically capable of providing rigid diaphragms and thus avoiding the need to use r.c. elements. Modeling wall strengthening techniques by applying corrective coefficients to the properties of masonry is a straightforward method. However, it should be recalled that effectiveness ratios proposed according to codes (e.g., NTC, 2008) quantify successful application of techniques, the real effectiveness of which must be assessed in situ. Numerical modeling may misrepresent the effectiveness of interventions, for example, for r.c. ring-beams that, numerically, can increase structural stiffness, although their damaging effects have been proven by several postearthquake scenarios. Nonuniform strengthening of buildings should be avoided (as recommended in codes, e.g., NTC, 2008), as it may alter the dynamic behavior of the structure and reduce its seismic capacity.

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75. Borri, A., Casadei, O., Castori, G., Hammond, J., 2009. Strenghtening of brick masonry arches with externally bonded steel reinforced composites. J. Compos. Constr. 13 (6), 468
475. Cardani, G., 2003. La vulnerabilita` sismica dei centri storici: il caso di Campi Alto di Norcia. Linee guida per la diagnosi finalizzata alla scelta delle tecniche di intervento per la prevenzione dei danni. PhD thesis, Politecnico di Milano, Milano, Italy (in Italian). Cattari, S., Resemini, S., Lagomarsino, S., 2008. Modeling of vaults as equivalent diaphragms in 3D seismic analysis of masonry buildings. Structural Analysis of Historic Construction. Taylor & Francis, London, pp. 517
524. Cescatti, E., da Porto, F., Modena, C., 2018. Analysis and comparison of EBR techniques applied to masonry vault. In: de Felice, G., Sneed, L.H., Nanni, A. (Eds.), Book SP-324: Composites with Inorganic Matrix for Repair of Concrete and Masonry Structures. ACI American Concrete Institute. Corradi, M., Borri, A., Castori, G., Sisti, R., 2014. Shear strengthening of wall panels through jacketing with cement mortar reinforced by GFRP grids. Composite B 64, 33
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Dolce, M., 1989. ‘Schematizzazione e modelazione per azioni nel piano delle pareti’, Corso sul consolidamento degli edifici in muratura in zona sismica, Ordine degli Ingegneri, Potenza (in Italian). de Felice, G., De Santis, S., Garmendia, L., Ghiassi, B., Larrinaga, P., Lourenc¸o, P.-B., et al., 2014. Mortar-based systems for externally bonded strengthening of masonry. Mater. Struct. 47 (12), 2021
2037. da Porto, F., Valluzzi, M.R., Munari, M., Modena, C., Areˆde, A., Costa, A.A., 2018. Strengthening of stone and brick masonry buildings. In: Costa, A., Areˆde, A., Varum, H. (Eds.), Strenghtening and Retrofitting of Existing Structures, Series ‘Building Pathology and Rehabilitation, vol. 9. Springer Nature Singapore Pte Ltd, pp. 59
84. Available from: https://doi.org/10.1007/978-981-10-5858-5_3. Eurocode 8: Design of structures for earthquake resistance, 2004. Part 1: general rules, seismic actions and rules for buildings—EN 1998-1:2004 1 A1:2013. Eurocode 8: Design of structures for earthquake resistance, 2005. Part 3: assessment and retrofitting of buildings—EN 1998-3:2005. Galasco, A., Lagomarsino, S., Penna, A., 2002. Analisi sismica non lineare a macroelementi di edifici in muratura. In: Atti del X Convegno Nazionale ANIDIS “L’ingegneria sismica in Italia,” Potenza-Matera, Italy (in Italian). Giamundo, V., Lignola, G.P., Maddaloni, G., Balsamo, A., Prota, A., Manfredi, G., 2015. Experimental investigation of the seismic performances of IMG reinforcement on curved masonry elements. Composites B 70, 53
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652. Lourenc¸o, P.B., 1996. Computational strategies for masonry structures. PhD thesis, TU Delft. Magenes, G., Bolognini, D., Braggio, C., 2000. Metodi semplificati per l’analisi sismica non lineare diedifici in muratura. CNR-Gruppo Nazionale per la Difesa dai Terremoti, Roma, pp. 22
23 (in Italian). Magenes, G., Calvi, G.M., 1996. Prospettive per la calibrazione di metodi semplificati per l’analisi sismica di pareti murarie. In: Atti del Convegno Nazionale “La Meccanica delle Murature tra Teoria e Progetto,” Messina, 18
20 settembre 1996, Pitagora Ed. Bologna (in Italian). NTC, 2008. Decreto Ministeriale 14/01/2008, Norme Tecniche per le Costruzioni (in Italian). Onsiteformasonry, 2005. On-site investigation techniques for the structural evaluation of historic masonry buildings. EC Project FP5 EESD, EVK4-CT-2001-00060 (CD-ROM). Papanicolaou, C.G., Trinantafillou, T.C., Lekka, M., 2011. Externally bonded grids as strengthening and seismic retrofitting materials of masonry panels. Constr. Build. Mater. 25 (2), 504
515.

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Penna, A., Cattari, S., Galasco, A., Lagomarsino, S., 2004. Seismic assessment of masonry structures by nonlinear macroelement analysis. In: Modena, C., Lourenco, P.B., Roca, P. (Eds.), IVth International Seminar on Structural Analysis of Historical Construction-Possibilities of Numerical and Experimental Techniques, Padova URI, November 2004, 2, pp. 1157
1164, ISBN: 0415363799. Rots, J., 1997. Structural Masonry: An Experimental/Numerical Basis for Practical Design Rules. Balkema, Rotterdam, Netherlands. S.T.A. Data, 2018. 3Muri V. 11.4—Manuale d’Uso, Turin, Italy (in Italian). Tomaˇzevic, M., 1978. The computer program POR, report ZRMK. Tomaˇzevic, M., Apih, V., 1993. The strengthening of stone-masonry walls by injecting the masonry
friendly grouts. Eur. Earthq. Eng. 6, 10
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156. Valluzzi, M.R., 2016. Challenges and perspectives for the protection of masonry structures in historic centers: the role of innovative materials and techniques. RILEM Tech. Lett. 1, 45
49. Available from: https://doi.org/10.21809/rilemtechlett.2016.10. Valluzzi, M.R., Sbrogio`, L., 2019. Vulnerability of architectural heritage in seismic areas: constructive aspects and effect of interventions. ‘Cultural landscape in practice. Conservation vs. Emergencies’. In: Amoruso, G., Salerno, R. (Eds.), Lecture Notes in Civil Engineering, 26. Springer Cham, pp. 203
218. ISBN: 978-3-030-11421-3 (print), 978-3-030-11422-0 (online). Available from: https://doi.org/10.1007/978-3-030-11422-0_1. Valluzzi, M.R., da Porto, F., Modena, C., 2004. Behavior and modeling of strengthened threeleaf stone masonry walls. J. Mater. Struct. 37, 184
192. Available from: https://doi.org/ 10.1007/BF02481618. Valluzzi, M.R., Cescatti, E., Cardani, G., Cantini, L., Zanzi, L., Colla, C., Casarin, F., 2018. Calibration of sonic pulse velocity tests for detection of variable conditions in masonry walls. Construction and Building Materials 192, 272
286. Available from: https://doi.org/ 10.1016/j.conbuildmat.2018.10.073. Valluzzi, M.R., Modena, C., De Felice, G.M., 2014. Current practice and open issues in strengthening historical buildings with composites. Mater. Struct. 47 (12), 1971
1985. Available from: http://dx.doi.org/10.1617/s11527-014-0359-7. Valluzzi, M.R., Garbin, E., Dalla Benetta, M., Modena, C., 2010. In-plane strengthening of timber floors for the seismic improvement of masonry buildings. In: Proceeding of 11th World Conference on Timber Engineering—WCTE, Riva del Garda, Italy. Valluzzi, M.R., Garbin, E., Dalla Benetta, M., Modena, C., 2013. Experimental characterization of timber floors strengthened by in-plane improvement techniques. In: Proceedings of SHATIS, September 4
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11, Wroclaw, Poland.

Chapter 7

Masonry mechanical properties B. Ghiassi1, A.T. Vermelfoort2 and P.B. Lourenc¸o3 1

Centre for Structural Engineering and Informatics, Faculty of Engineering, The University of Nottingham, Nottingham, United Kingdom, 2Eindhoven University of Technology, Eindhoven, The Netherlands, 3ISISE, Department of Civil Engineering, University of Minho, Guimaraes, Portugal

7.1 Introduction Accurate modeling of the performance of masonry structures requires a thorough experimental description of its behavior and development of adequate constitutive laws that can represent its heterogenous behavior. The extent of knowledge required on materials properties depends on the modeling strategy and analysis method of interest. Elastic properties of the units and mortar or masonry as a composite is sufficient for linear analysis. However, masonry cracks at very low stress levels, which makes the reliability of linear analyses methods doubtful. For nonlinear analysis, in addition to the elastic properties, nonlinear mechanical properties should be established. Special attention should also be given to the overall aspects of the structure: unit dimensions, type and quality of mortar joint, and unit surface conditions (perforations or indentations). The details of the experimental method used for establishing the material properties, especially boundary conditions, are of critical importance and should be carefully considered in interpretation of the experimental results. The large variety of units and mortar types and construction methods used worldwide do not allow development of unified constitutive laws. For this reason, several codes and standards prioritize experimental characterization of material properties for design and numerical simulations. However, in some cases either performing experimental tests is not possible at all (such as in historical structures) or a limited set of experimental tests are performed (usually compressive and flexural tests for characterization of strength) due to economical reasons or lack of suitable testing equipment. This chapter discusses the way these mechanical properties are established and gives some suggestions for establishment of material properties for numerical simulations when limited experimental results are available. Numerical Modeling of Masonry and Historical Structures. DOI: https://doi.org/10.1016/B978-0-08-102439-3.00007-5 Copyright © 2019 Elsevier Ltd. All rights reserved.

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Other mechanical properties like creep, moisture movement and thermal expansion, although extremely important, are beyond the scope of this chapter.

7.2 Masonry in general perspective 7.2.1

Type of units and materials

Masonry is made by piling units on top of each other in a certain pattern using mortar to allow for dimensional tolerance. In some cases, the units are connected with interlocking mechanisms without application of any mortar joint. The units are available in several dimensions and types (Fig. 7.1), with the smallest brick size being approximately 210 3 100 3 55 mm3. These units can be laid using one hand. The next size are blocks, in sizes ranging from 200 3 100 3 100 mm3 to 300 3 600 3 100 mm3 and sometimes even larger depending on the volume mass of the material they are made of. The dimension is often limited by rules concerning the weight 14
15 kg max. Blocks are laid using both hands, in some cases by two persons. The largest are element-sized units of approximately 600 3 900 3 100
300 mm3. This means only four layers are generally needed to build a wall of 2.5 m height. Besides dimensions, the other main distinction in masonry units is the materials they are made of. Fired clay and natural stone are the most common types all over the world. Other contemporary materials are calcium silicate, aerated concrete, and normal concrete. Units can be made quite easily by filling a prismatic mold with the paste. Small indentations (so called frogs) or perforations can also be made relatively easily. Some clay bricks and blocks are made with an extrusion process that allows for highly perforated units,

FIGURE 7.1 Some typical masonry units. Photographs from producers brochuers.

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with perforations up to 50% of their section. Concrete blocks can be made with webs of 6 20 mm allowing for voids large enough to apply reinforcement with grouting. Mortars are a mixtures of sand and a binder, historically lime. After cement became available it was often used in combination with lime resulting in the so-called bastard mortars. According to Eurocode 6 three types of mortars may be used: general-purpose mortar, thin-layer mortar, and lightweight mortar. The joint thickness depends on the dimensional tolerances and the unit size. The so-called general-purpose mortar allows a joint thickness between 10 and 15 mm for bricks and up to 20 mm for larger blocks. Moreover, calcium-silicate elements are made with smooth and level surfaces and with small dimensional tolerances that allow the use of thin mortar joints (2
3 mm thick). The properties of a mortar before and after curing depend on its constituents like sand, cement, and/or lime and additives, and the amount of water used during building and available during curing. In general, lime-based mortars are more flexible (lower elastic modulus) and cement-based mortars are more brittle. Additionally, cement-based mortars harden faster than limebased mortars. It is clear that the type of masonry must be considered for numerical simulation. The modeling details and required parameters differ depending on the modeling strategy followed (see next section). The properties of units and mortars are the main parameters to be characterized. In modeling of the brick
mortar interaction, the surface properties (and perforations/indentations) of the units must also be considered. Depending on the production procedure, surfaces may be smooth or sanded. Sand is often used to allow for easy demolding. The surface structure is also clearly dependent on the unit type (e.g., surface characteristics of concrete are different than those of extruded clay). These characteristics are in principle considered in the constitutive law and properties of the interface. A detailed discussion on the properties of units and mortars can also be found in the literature (Hendry, 1990; Drysdale et al., 1993; Hendry et al., 2004; Klingner, 2010).

7.2.2

Modeling at different levels of detail

The set of mechanical properties required for simulation of the structural performance of masonry depends both on the range of interest (elastic or inelastic) of material’s response and on the adopted modeling approach. For simplified modeling purposes (elastic range), knowing the modulus of elasticity (E) and the compressive strength (fc) is usually sufficient. When linear elastic behavior is assumed, the secant E value for the part of the stress
strain diagram to one-third of the ultimate stress can be used. When masonry cracking occurs, nonlinearity effects become apparent. Modeling the cracking and failure processes in masonry structures requires additional

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mechanical properties like shear strength, tensile strength, and fracture energies (Gf). The fracture energy is defined as the amount of energy required for creating a unit crack area, and it is obtained as the area under the stress
displacement diagram in the softening regime (postpeak). This definition is straightforward for tensile tests, but not for compressive tests. Aspects such as whether the compressive fracture is a material property or a structural property or the compressive fracture energy represents formation of several cracks and not a single crack are still under discussion. Depending on the constitutive law and the finite element (FE) package used, this value might be defined differently, but in most cases it is defined as the area under the stress
displacement diagram in the postpeak regime. To obtain these parameters, detailed measurements in dedicated test setups must be performed. In these tests the main problem is applying the appropriate boundary conditions. As for the modeling approaches, depending on the level of accuracy and the simplicity desired, the following three different strategies are usually used: G

G

G

Detailed micromodeling—units and mortar in the joints are represented by continuum elements, whereas the unit-mortar interface is represented by discontinuous elements. In this approach, Young’s modulus, Poisson’s ratio and, optionally, inelastic properties of both unit and mortar are required. The interface represents a potential crack/slip plane with initial dummy stiffness to avoid interpenetration of the continuum. The nonlinear properties of the interface are also required. Simplified micromodeling—expanded units are represented by continuum elements, whereas the behavior of the mortar joints and unit-mortar interface is lumped into discontinuous elements. In this approach, each joint, consisting of mortar and the two unit-mortar interfaces, is lumped into an average interface while the units are expanded in order to keep the geometry unchanged. Masonry is thus considered as a set of elastic blocks bonded by potential fracture/slip lines at the joints. Accuracy is lost in this modeling method since the Poisson’s effect of the mortar is not included. Discrete element modeling (DEM), an approach that has recently received increasing attention, is also followed based on similar assumptions. Macromodeling—units, mortar, and the unit-mortar interface are smeared out in the continuum. This approach does not make a distinction between individual units and joints but treats masonry as a homogeneous continuum. Here, the elastic and inelastic properties of the masonry as a continuum are necessary. In this case, the damage of masonry is usually modeled following a smeared crack or damage plasticity approach.

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7.3 Unit properties The variability of the masonry units used around the world for construction of masonry and historical structures together with the different deterioration processes due to environmental action makes the proposal of unique relations for prediction of mechanical properties a complicated task. Old masonry units present large variation in their mechanical properties even if the samples are chosen from the same building. In contrast, new masonry units usually have a much lower variation due to the controlled construction procedures. A problem in evaluation of the mechanical properties of materials in existing and historical structures is that performing destructive tests is usually prohibited. This has led to the development of a range of nondestructive techniques for the evaluation of the mechanical properties. Estimation of the mechanical properties based on visual aspects is another technique proposed by the New Zealand Society for Earthquake Engineering (NZSEE) (Table 7.1). Another important issue is that masonry units are usually assumed isotropic, although extruded units can present different mechanical properties along different directions. The main elastic properties of bricks are the elastic modulus, E, and Poisson’s ratio, ν. If the nonlinear response of the units is also considered the compressive strength, fc, the peak strain, εu, the postpeak fracture energy in compression, Gcf , the tensile strength, ft , and the tensile fracture energy Gtf are required. The compressive strength is probably the simplest parameter for experimental characterization and for this reason other mechanical properties are usually correlated with this value. The ratio between the fracture energy and the compressive strength is sometimes called the ductility index, duc , and can be used as an indication for the brittleness of the response (see Chiaia et al., 1998).

TABLE 7.1 NZSEE recommendations for bricks mechanical properties based on visual aspects. Type

Description

fc (MPa)

E (GPa)

ν

Soft

Weathered, pitted, distinct color variation with depth, bright orange, possibly under fired

1.0
5.0

4

0.35

Stiff

Common brick, can be scored with a knife, red

10.0
20.0

13

0.2

Hard

Dense, hard surface, well fired, dark reddish brown

20.0
30.0

18

0.2

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PART | II Modeling of unreinforced masonry

Elastic properties

The elastic modulus can be obtained from measurements on pieces of units loaded in compression. The specimens have to be capped at their loaded ends to allow for a smooth force transfer for reducing peak stresses from the steel load platens into the specimen. Deformation sensors (e.g., LVDTs) are mounted on the specimen to measure the deformation. The deformation should be measured in an area sufficiently far away from the loaded ends due to the three axial stress states in these areas. The specimens may be assumed to be in a uniaxial condition far from the loaded ends. Fig. 7.2 shows LVDTs with a measuring length of 185 mm (three mortar joints, unit height 5 52 mm and joint height 5 13 mm) on a calcium silicate masonry prism. Stresses are obtained by dividing the measured forces by the loaded gross sectional area and the strains by dividing the measured displacement of the end points of the measuring device by their distance. Usually the samples are loaded in a range of stresses far from the compressive strength (to avoid damage in the specimens) for several cycles. The average slope of the loading regimes is then accounted as the elastic modulus. The specimens are either made while building the real masonry or afterward from the same constituents. At the end of compression testing, the measuring devices may fall off the specimen, which makes it complicated to record the postpeak behavior. Then the results of the load platen movement may be used. A correction factor, however, may be needed in these cases. Fig. 7.2 shows an almost ideal situation where LVDTs measured reliably until the end of the test. When experimental results are not available, the elastic modulus can be estimated from the compressive strength of the unit. However, it should be noted that a wide range of relations can be found in the literature for this purpose showing that units properties vary significantly with type and

FIGURE 7.2 Example of a stress
strain diagram from a compression test on a 70 3 70 cm Wallette with cyclic loading showing the difference between measuring load platen displacement and deformation of the central part of the specimen.

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production method. The NZSEE proposes (Table 7.1) a range of 125
1400 fc with an average value of 350 fc for historical bricks. This average is very similar to the relation proposed for North Indian bricks in Kaushik et al. (2007). Caporale et al. (2015) reported an elastic modulus of 160 fc for adobe bricks and Vasconcelos (2005) found a value of 380 fc for granite stone. In the Dutch study (CUR 171) a ratio of 420 fc is given for solid clay, soft mud, and extrusion bricks, with E values of 6 GPa for soft mud units ranging to 16.7 GPa for the strongest extrusion units. These values were obtained by measurements on piles of five units in height bonded with a two-componenthigh strength material with thin joints (Vermeltfoort, 2005).

7.3.2

Compressive behavior

Compressive strength tests are easy to perform. A large database of experimental compressive test results can be found in the literature. However, most of these studies are focused on derivation of the compressive strength and little information is available on the postpeak behavior necessary for extraction of the nonlinear parameters. Although it can also be argued that compressive testing is not suited to establish the compressive fracture energy as even under compression, the failure is due to exceeding the local tensile stresses. Among the available experimental results, different test conditions including specimens’ dimensions and moisture content, boundary conditions, and temperature have been adopted which can significantly affect the reported results. These differences make comparisons between different experimental results complicated. If the test conditions are known, it is suggested to convert the results to an equivalent compressive strength relevant to the air-dry conditioning regime. For example, for calcium silicate, this value can be obtained by multiplying the compressive strength by 0.8 if the units are oven dried and by 1.2 if they are submerged in water for over 15 hours. The compressive strength of well-fired clay bricks is generally not sensitive to moisture content, but nonwell-fired bricks and stone are sensitive to moisture content. The boundary conditions may also affect the results. The friction between the load platen and the specimen produces a triaxial stress state near the boundaries and prevents free expansion of the specimen. This, depending on the specimens’ shape and size, can affect the observed strength and the postpeak response (thus affecting the fracture energy). The use of two sheets of Teflon reduces this friction considerably. For comparison of results of specimens of various sizes, shape factors can be used. A shape factor gives the ratio between a compression test result on a specimen of a certain dimension and a reference specimen presumably made of the same material. Several equations exist, from which the equation of Khalaf and Hendry (1989) is quite convenient for evaluating this correction factor: C100 5 ðh=A0:5 Þ0:37

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where C100 is the correction factor with a cube with ribs of 100 mm as reference, h is the height of the specimen, and A is the loaded area. The common range of compressive and tensile strength values for different types of units are presented in Tables 7.2 and 7.3. It can be observed that the compressive strength of the units varies significantly with the type. In general, ancient masonry units present lower mechanical properties with larger variability compared to new units. It should be noted that in fired clay bricks the compressive strength is highly dependent on the firing temperature. If the experimental data on the postpeak compressive response is not available, the fracture energy can be estimated from the compressive strength

TABLE 7.2 NZSEE reference values for compressive and tensile strength of masonry units and mortar. Material

Compressive strength (MPa)

Tensile strength (MPa)

Natural stone

10
200

1
5

Clay brick

10
70

1

Calcium silicate

10
35

1
2

Concrete

10
35

3

Aerated concrete

3
7.5

0.5
2

Gypsum

5
10

1
2

Loam (soft mud)

2
3

0
0.33

Mortar

3
25

0.25
0.33 3 fc

TABLE 7.3 Typical ancient masonry unit types. Materials

fc (MPa)

Stone Igneous—granite

40
150

Sedimentary—limestone

10
100

Metamorphic—marble

30
150

Metamorphic—schist

5
60

Clay

1.5
30

Adobe

0.5
3

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of the unit. Due to the lack of sufficient literature data on the nonlinear behavior of masonry units, comprehensive data on the correlation of the fracture energy with the compressive strength are not available yet. However, the formulation proposed for concrete in the Model Code 90 may be used. This code proposes a peak strain of 0.2% and a compressive fracture energy of Gf 5 15 1 0.43fc 2 0.0036fc2. It should be noted that this formula is only applicable for compressive strength values between 12 and 80 MPa. For fc values lower than 12 MPa, a ductility index, du,c, equal to 1.6 mm (MC 90 value for fc 5 12 MPa) and for fc values higher than 80 MPa, a du,c value equal to 0.33 mm is suggested (MC 90 value for fc 5 80 MPa). The fracture energy in granite stones can be obtained as Gf 5 2.54fc0.58 as suggested in Vasconcelos (2005).

7.3.3

Tensile behavior

Tensile response of units can be obtained by performing direct tensile tests. Due to the complexities related to preparation of dog-bone shape specimens with these materials, the tests are usually performed on notched prisms. These tests are complex and the results, especially in the postpeak range, depend on the boundary conditions. At low load levels, it is assumed that stresses are uniformly distributed and the line of action of the load coincides with the center of gravity of the loaded area. At higher levels, stresses become larger than the strength of the material and (micro) cracking starts. Consequently, stress distribution will no longer be uniform and bending effects increase, locally leading to larger stresses and a brittle failure. These issues with direct tensile testing has lead to development of test setups in which the rotation of the specimen’s ends can be prevented, for example, by Hordijk (for concrete) and Van der Pluijm (for masonry). As direct tensile tests are tricky to perform and usually large variations are obtained in the results, tensile splitting or flexural tests that require less preparation are preferred. However, it should be noted that the tensile strength obtained from these tests is, in principal, different than that of obtained from direct tensile tests. Extensive information exists in the literature regarding the tensile strength and less on tensile fracture energy of masonry units (Van der Pluijm, 1999; Lourenc¸o et al., 2005; Vasconcelos, 2005). Again, unified predictive relations are complicated due to the large variety of the unit types, manufacturing processes, specimens’ dimensions, and testing conditions and procedures. Detailed studies show a dependency of tensile strength on production methods. In an extrusion process, the soft clay is squeezed through an opening, which brings the flat-shaped clay particles into layers, leading to the units having different properties in different directions (Vermeltfoort, 2005). The way clay particles are oriented in a brick, both soft mud—randomly—and extruded—more ordered—influences the strength, stiffness, and

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failure pattern of a specimen. This is why the base material of extruded perforated units has almost linear behavior until fracture and is relatively strong, but brittle. As a correlation with the compressive strength, it has been found that the brick tensile strength is in the range of 12%
14% of its compressive strength. For unreinforced adobe bricks a value of 36% of the compressive strength was proposed (Caporale et al., 2015). NZSEE proposes a tensile strength equal to 10% of the compressive strength for historical bricks. In The Netherlands a ratio of 18 between compressive (5.5%) and tensile splitting strength is often used for soft mud units and a ratio of 14 (7%) for extruded bricks. The ratio between the tensile fracture energy, Gf, and tensile strength, ft (defined as the ductility index, du,t), is around 0.018
0.040 mm for bricks (Van Der Pluijm, 1997, 1999; Lourenc¸o et al., 2005). In the absence of specific information regarding the properties of the unit, using the average value, 0.029, is recommended. For stone granites, the ductility index is proposed as a function of tensile strength as 0.239 ft21.138 as suggested in Vasconcelos (2005). Assuming a tensile strength of 3.5 MPa, this relation gives a value of 0.057 mm for the ductility index, being the double of the value reported for the bricks. The Model Code 90 recommends Gf 5 0.025 (fc/10)0.7 for the tensile fracture energy of concrete with a maximum aggregate size of 8 mm. Assuming that the tensile strength of masonry units is around 5% of the compressive strength on average, this expression can be rewritten as Gf 5 0.04ft0.7. This gives a fracture energy of 0.0976 N/mm and a ductility index of 0.028 mm for brick with an average tensile strength of 3.5 MPa.

7.4 Mortar properties The most common binders used in masonry structures are cement-based or lime-based or a mix of the two. Lime-based mortars are either aerial lime or hydraulic lime. The aerial lime needs air for hydration and hardening, while hydraulic limes harden with water. Cement-based mortars can be found in both new and historical masonry structures, but lime-based mortars are usually found in historical masonry or in repair mortars used for historical structures. The mechanical properties of mortars are dependent on several factors including the type and proportion of the mortar constituents as well as the curing and the testing (e.g., specimen geometry and test setup) conditions. The moisture transfer between the mortar and brick and the level of degradation can also significantly affect the mechanical properties of mortars. Often the quality (and therefore the mechanical properties) of mortar varies even within the same structure.

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Experimental testing is therefore highly recommended for characterization of mortar mechanical properties. However, it should be noted that the experimental tests are usually performed on specimens with geometrical details and curing conditions that are different from those of mortar joints in real structures. Differences in the obtained mechanical properties should therefore be expected. If evaluation of material properties in an existing structure is of concern, often taking samples is not possible at all or the obtained samples are highly irregular and therefore tests need to be performed with consideration of these irregularities. Mortars are usually classified in different categories depending on their expected mechanical properties. For example, ASTM 270 classifies mortars into four groups (M, S, N, and O mortars); see Table 7.4. The NZSEE provides benchmark mechanical properties for historical mortars; see Table 7.5. In The Netherlands, the graph shown in Fig. 7.3 is often used to explain the behavioral differences between the three main mortar types. It can be clearly observed that the lime-based mortar has a very slow hydration rate and lower compressive strength compared to that of cement and bastard mortars. Despite the existence of a large database of experimental results in the literature, the above-mentioned issues make proposal of uniform relations or constitutive laws for predicting the in situ mechanical properties of mortars complicated. Additionally, the experimental specimens are generally made in laboratory molds and thus the results are not representative of real structural conditions. The experimental results are also often reported without detailed explanations on influencing factors including the test methods and curing conditions. Hence only a few studies can be found with sufficient details and direct use of the experimentally measured values in numerical simulations.

TABLE 7.4 ASTM C270 mortar types. Type

Volume proportions

fc (MPa)

Portland cement

Hydrated lime

Sand

M

1

Min: 0, max: 0.25

S

1

Min: 0.25, max: 0.5

Not less than 2.25 nor more than three times some of the total volume of cement and lime

N

1

Min: 0.5, max: 1.25

5.2

O

1

Min: 1.25, max: 2.

2.4

17.2 12.4

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TABLE 7.5 Recommended mortar properties in NZSEE. Type

Description

c (MPa)

μ

fc (MPa)

Non cohesive

Lime-based, heavily leached and weathered, sand like, easily raked out by hand, unbound aggregate

0.0

0.0

0.0

0.0

0.0

Soft

Lime binding mildly leached, can be raked out, punch test ,30 mm

0.1

0.4

1.0

7.0

0.05

Firm

Lime-based, not weathered, punch test ,20 mm

0.2

0.6

4.0

9.0

0.07

Stiff

High Portland cement content, punch test ,10 mm

0.4

0.8

8.0

12.0

0.11

20

100

16

Strength development (%)

Cement

Stress (MPa)

ν

E (GPa)

Bastard Lime

12 8 4 0

80 60 40

Cement Bastard

20

Lime

0 0.0

0.5

1.0

1.5

Strain (mm/m)

2.0

2.5

0

60

120

180

240

300

360

Time (days)

FIGURE 7.3 Characteristics of cement mortar, cement-lime mortar, and lime mortar.

Similar to the units, the main elastic mechanical properties of mortars required for numerical simulations are elastic modulus, E, and Poisson’s ratio, ν. As for the nonlinear properties, the compressive strength, fc, peak strain, εu, postpeak fracture energy in compression, Gfc, the tensile strength, ft, and the tensile fracture energy Gft are required.

7.4.1

Elastic properties

Testing of elastic properties of mortars can be done following the same procedure explained for the units. The available studies in the literature usually

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lack characterization of elastic modulus of mortars. If performing experimental tests is not possible, the use of common formulations for concrete is recommended.

7.4.2

Compressive behavior

As for the masonry units, most of the available literature reports on the compressive strength of mortars and little information is available on the postpeak compressive behavior. Compressive tests are easy, but usually performed on half prisms left from three-point bending tests with the main focus on characterization of mechanical strength. Surface hardness measurements have also been extensively reported for in situ mechanical characterization of mortars. In cases where experimental results are not available, the proposal of Model Code 90 for concrete with maximum aggregate size of 8 mm can be used (Gf 5 15 1 0.43fc 2 0.0036fc2).

7.4.3

Tensile behavior

Tensile tests are also mainly limited to characterization of the tensile strength. Similar to units, usually flexural tests or tensile splitting tests are preferred due to the complexities involved in performing direct tensile tests. Again these values should be converted to direct tensile strength with appropriate factors. Measurement of the postpeak response is also usually neglected due to the brittle behavior of the mortar. In such cases, the formulation proposed by Model Code 90 (Gf 5 0.025(fc/10)0.7) can be used.

7.4.4

Leveling mortar

In experiments, sample specimens and walls are usually prepared and built outside the test setup. For this reason, a filling material is usually used to allow for a smooth load transfer during the tests and to make sure that the contact between the (steel) load platen and the masonry is snug and tight. In this way, the dimensional tolerances and the usually rough and grooved specimen’s surface are bridged. In some situations, soft board serves this purpose quite well. A stiffer contact is obtained by using mortar. However, mortar takes some time to cure. Thus, gypsum could be an alternative. This mortar, depending on its properties and the measurement details, might affect the obtained results. When simulating the response of experimental tests, the presence of leveling mortar should, therefore, be considered. The properties of this mortar can be established in a similar way as done for mortars to build specimens.

7.5 Mortar
brick interface properties The interface bond between units and mortar is usually the weakest link in masonry structures and controls the failure mode in many cases. Its behavior is highly dependent on the brick properties (composition, strength, size,

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surface with or without sanding, etc.); mortar properties (composition, water content, curing conditions, etc.) and workmanship (condition of joints, etc.); and the distribution of mortar over the contact surface and the consequent edge effects (Vermeltfoort, 2005). The interface properties are usually required in micromodeling or DEM approaches where the interface elements at the unit-to-mortar or unit-to-unit interfaces. The input parameters are mainly the stiffness and nonlinear properties of the interface in the normal (mode I) and the shear (mode II) directions. These properties might be given as discrete values that are used internally by the FE package for generation of the bond-slip behavior. It is also possible in most cases to directly provide the interface laws. The main nonlinear properties in the normal (tensile) direction are the bond tensile strength, ft, and the bond fracture energy, Gf. Meanwhile, the bond strength, c (also called cohesion), the friction angle, tan ϕ, and the dilatancy angle, tan ψ, are generally needed in the shear direction. More advanced material models may require variable friction and dilatancy angles yielding to a nonconstant mode II fracture energy, however, these parameters can be assumed constant for general applications.

7.5.1

Elastic properties

The stiffness of interface elements in the shear, Ks, and normal, Kn, directions can be obtained from the elastic modulus (Eu and Em) and shear modulus (Gu and Gm) of unit and mortar as Ks 5 GuGm/tm(Gu 2 Gm) and Kn 5 EuEm/tm(Eu 2 Em). Here tm is the mortar thickness. It should be noted that the interface is often a weak/soft spot. The above formulation might lead to negative or extremely low E-values in some cases (CUR 173) when subtracting the calculated brick deformation from the measured masonry deformation obtained from the experimental tests.

7.5.2

Tensile behavior

Direct tensile tests can be performed to obtain the bond properties under tensile loading (mode I) as shown in Fig. 7.4. Deformation controlled tests need to be carried out on masonry specimens to obtain the tension softening curve for calculation of the fracture energy. This fracture energy is defined as the amount of energy needed to create a unit area of a crack along the brick
mortar interface. It is equal to the area under the tension softening curve for a discrete crack. The values result from an extrapolation of the measured net bond surface of the specimen to the assumed net bond surface of the wall, neglecting any influence of the vertical joints. It should, however, be noted that the tensile bond response is highly brittle and measurement of the postpeak response is a complicated task even if the tests are run under displacement-controlled conditions.

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FIGURE 7.4 Test setup for mode I characterization of brick
mortar interface (Lourenc¸o, 1994).

No specific recommendation can be given for obtaining these parameters based on the unit and the mortar properties. As a rough estimation, EC6 states that the tensile bond strength is generally in the range of 0.1
0.4 MPa. In Rots et al. (1997), the bond fracture energy was obtained in the range of 0.006
0.08 Nmm/mm2 for a range of mortar and unit combinations. This value on average was about c/10 (0.012 Nmm/mm2) and can be used as a reference when experimental results are not available. Note that c is the bond strength (cohesion) and is explained in the next section.

7.5.3

Shear behavior

Triplet tests can be used to obtain the bond properties under shear loads (Fig. 7.5). As the shear response is dependent on the normal stress, the specimens should be tested at least under three different pre-compression levels. The shear and accompanying precompression loads are then plotted in a graph. The initial shear strength, that is, the shear strength at zero prestress can then be found by a linear regression of the experimental results to obtain the Mohr-Coulomb relationship. For repetitive loading, the residual stresses can be used to establish the coefficient of friction. When the specimen is loaded repetitively, a stick and slip behavior may occur when the direction of movement changes. It should be noted that a triplet shear test is actually a four-point bending configuration and the stresses are not uniformly distributed over the height of the specimens during the tests. However, the relatively small spanto-height ratio of the specimens limits the bending moments. In such conditions, the simplifying assumption of uniform stress distribution can be reasonable. Alternatively, the shear test setup proposed by Van der Pluijm (1999) that was developed for minimization of the bending component can be used.

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FIGURE 7.5 Load time diagram of shear test. Triplet shear test set up (Lourenc¸o, 1996).

Van der Pluijm (1999) found that the ductility index, du,s, defined as the ratio between the fracture energy GfII and the bond strength c is in the range of 0.062
0.147 mm for different combinations of units and mortars. The recommended ductility index in the absence of more information is the average of 0.093 mm. It is noted that the Mode II fracture energy is clearly dependent on the normal stress level and the above values hold for a zero normal stress. The bond shear strength, c, is recommended to be in the rage of 0.1
0.4 MPa in EC6. The formula proposed by Lumantarna (2012) that correlates the bond shear strength with compressive strength (c 5 0.055 fc) for historical masonry assemblages in New Zealand might also be used as an estimation. The tangent of friction angle, tan ϕ, can be assumed equal to 0.75 independent of the type of unit and mortar (Atkinson et al., 1989; Rots et al., 1997). The recommendation of EC6 is 0.4 and Augenti and Parisi (2011) proposed a value of 0.29 for the interface between tuff stones and hydraulic lime mortar. In some cases, the residual friction angle is also necessary and can be assumed equal to the friction angle. The dilatancy angle can also be assumed equal to zero as it diminishes with increasing axial stresses and slip at the interface.

7.6 Masonry properties The main mechanical properties of masonry as a composite material are elastic modulus, Poisson’s ratio, compressive strength, tensile strength, shear

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strength, and corresponding fracture energies. Depending on the damage models and constitutive laws adapted, additional parameters might be needed. These parameters should be usually identified and selected based on the theoretical backgrounds of the selected constitutive law.

7.6.1

Elastic properties

The procedure for characterizing the elastic modulus of masonry is similar to that explained for units. The measuring area in this case should cover two or more bed joints and have a minimum length equal to n 3 layer height, with n $ 2 where layer height is equal to the height of a unit plus the height of a joint. Again, an extensive effort has been devoted to propose formulations for correlating this parameter to the compressive strength of masonry prisms. These formulations might be used when experimental results are not available. However, this should be done with special care as the range of proposed factors is very wide. EC6 proposes to obtain the elastic modulus as the secant modulus from testing or directly from the compressive strength as 1000fk. The National Annex to EC6 by NL proposes a value of 700fk. Kaushik et al. (2007) obtained a range of 250
1100fk with an average of 550fk for masonry prisms made of four different North Indian bricks and three mortar grades. Similarly, Drysdale et al. (1993) reported a wide range of 210
1670fk. As the result of compressive tests on masonry prisms made of historic and new clay bricks obtained from existing buildings, Vermeltfoort and Martens (2016) obtained a ratio of the elastic modulus of 240fk.

7.6.2

Compressive behavior

The compressive stress
strain diagrams are one of the main inputs for numerical simulations. Linear, parabolic-rectangular, rectangular, idealised or parabolic stress
strain diagrams have been frequently used (Fig. 7.6). For masonry made of (highly) perforated units, it is recommended to use a linear elastic no-tension diagram due to the brittle behavior of such masonry. Although a bilinear diagram is not usually suggested, using this diagram makes calculations much easier compared to the adopted parabolicrectangular diagrams. Although, with the recent advancements in the available FE programs implementation of all these constitutive laws can be done with little effort. The shape of the stress
strain curve or its main characteristics (compressive strength and fracture energy) can be directly obtained from experiments. These experiments can be easily done for new masonry where the unit and mortars are available. Historic masonry (or existing buildings) can be examined by cutting pieces of masonry from existing buildings or by using in situ test procedures.

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FIGURE 7.6 Parabolic rectangular stress
strain diagram for masonry.

For example, the tests performed by Vermeltfoort and Martens (2016) on pieces of masonry cut from existing walls (Fig. 7.7). When only units are available, which is often the case for historic lime mortars with relatively small strength, specimens can be made using these old bricks and a replica of the original mortar. Several analytical formulas have been proposed for prediction of the masonry compressive strength based on the properties of units and mortar. These formulas are generally in the form of fk 5 Kfbαfmβ where fb is the units’ compressive strength and fm is the mortar compressive strength. EC6, for example, proposes this formula with different sets of parameters for different combinations of masonry and mortars. As an example, these are proposed as parameters K 5 0.6, α 5 0.65, and β 5 0.25 for solid units and general-purpose mortars. The values proposed by PIET-70 for compressive strength of stone masonry depends on the stone type and are in the range of 0.5
8 N/mm2 (see Table 7.6). Sarhat and Sherwood (2014) modified this formula as fk 5 0.886fb0.75fm0.18 for hollow concrete block masonry systems made of concrete blocks with compressive strength in the range of 9
50 MPa. It should be noted that these relations are only valid for compressive strength of masonry prisms in the perpendicular direction to the horizontal joints. As masonry is an orthotropic (or anisotropic) material, it has different mechanical properties under different directions. The available experimental results

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FIGURE 7.7 Stack-bonded prisms for detailed observation of behavior under compression and (right) wall piece after testing.

on the compressive strength of masonry under different directions is still limited. In the case of lack of experimental data, the compressive strength of solid unit masonry in the direction of horizontal joints can be assumed as 0.75fk (Glitzka, 1988). For the compressive fracture energy, the values proposed for concrete in the Model Code 90 (CEB-FIP, 1993) may again be used (Gf 5 15 1 0.43fc 2 0.0036fc2). EC6 proposes a ratio of fracture energy to elastic modulus of Gf/E 5 0.4 for clay and lightweight concrete masonry systems that can be used when experimental results on the elastic modulus are available.

7.6.3

Tensile behavior

For tensile loading, theoretically three failure mechanisms can be observed in masonry prisms perpendicular to the bed joints: (1) tensile failure of the brick or mortar; (2) tensile failure at the mortar
brick interface; and (3) tensile failure at the brick
mortar interface in horizontal and vertical joints. When the first failure mechanism occurs, the tensile strength of the masonry can be assumed equal to the brick’s or mortar’s tensile strength considering the material that has failed. The third failure, failure at the brick
mortar interface, is the most common failure mode. In this case, the masonry tensile strength can be assumed to be equal to the tensile bond strength (Klingner, 2010). Some experimental results show that this value is in the range of 10%
20% of the masonry compressive strength (Hamid and Heidebrecht, 1979; Klingner, 2010). This is actually a very rough

TABLE 7.6 Design compressive strengths recommended by PIET-70 for stone masonry. Stone type

Stone strength (N/mm2)

Ashlar masonry

Other masonry

Dry, with good adjustment between faces

Ashlars h .0.30 mMortar . M8

Ashlars h , 0.30 mMortar . M4

With welldefined coursesMortar . M4

IrregularMortar . 0.5

Dry

GraniteBasalt

.100

8.0

6.0

4.0

2.5

1.0

0.7

Quartz sandstoneHard limestoneMarble

.30

4.0

3.0

2.0

1.2

0.8

0.6

Lime sandstoneSoft limestone

.10

2.0

1.5

1.0

0.8

0.6

0.5

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estimation and may not be accurate in all cases. As for the fracture energy, the formulation proposed for the units and mortar may be adapted in the lack of experimental results.

7.6.4

Shear behavior

Shear is a complex phenomenon and several methods of testing are available for its characterization. The masonry shear strength can be obtained as (fvk 5 fvk0 1 0.4σd, fvk , 0.065fb, fvk , fvlt), where fvk0 is the masonry cohesion or initial shear strength (takes a value of 0.2
0.6 MPa depending on the joint type); 0.4 is the tangent of frictional angle (for safety reasons, a relatively small value is taken for this parameter); σd is the compressive stress on the masonry; and fvlt is a limit value that can be taken from EC6. The masonry cohesion, fvk0, is generally in the range of 0.15
0.3 MPa for new masonry and 0.05
0.1 MPa for old masonry with weak mortars. The tangent of frictional angle is recommended to be taken as 0.3 (for irregularly coursed stone) and 0.2 (for rubble masonry) in old masonry structures (Vasconcelos, 2005).

7.7 Overview of the input parameters for numerical simulations As an overview, the main mechanical properties required as the input of numerical simulations and suggestions for when experimental results are not available are presented in the following tables (Tables 7.7
7.9).

TABLE 7.7 Unit and mortar main mechanical properties. Parameter

Description

Recommendation

E

Elastic modulus

350 fc for historical bricks 380 fc for stone 160 fc for adobe bricks

fc

Compressive strength

Tables 7.1
7.3

ft

Tensile strength

7%
10% fc for bricks 5% fc for soft mud

Gf

c

Gft

Fracture energy in compression

15 1 0.43fc 2 0.0036fc2

Fracture energy in tension

0.025(fc/10)0.7 or 0.029 ft

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TABLE 7.8 Interface properties. Parameter

Description

Recommendation

Ks

Normal stiffness

GuGm/tm(Gu 2 Gm)

Kn

Shear stiffness

EuEm/tm(Eu 2 Em)

c

Bond strength (joint cohesion)

0.055 fc, 0.1
0.4 MPa

tan ϕ

Tangent of friction angle

0.4
0.75

tan ϕr

Tangent of residual friction angle

tan ϕ

tan ψ

Joint dilatancy

0

ft

Joint tensile strength

0.1
0.2 MPa

Mode I fracture energy

0.1c Nmm/mm2

Mode II fracture energy

0.1c Nmm/mm2

Gf

I

GfII

TABLE 7.9 Masonry mechanical properties. Parameter

Description

Recommendation

E

Elastic modulus

300
700 fc

fc

Compressive strength

0.6fb0.7fm0.3

ft

Tensile strength

10%
20% fc

fv

Shear strength

fvk 5 fvk0 1 0.4σd

Gfc

Fracture energy in compression

15 1 0.43fc 2 0.0036fc2

Gft

Fracture energy in tension

0.025(fc/10)0.7 or 0.029 ft

References Atkinson, R.H., Amadei, B.P., Saeb, S., Sture, S., 1989. Response of masonry bed joints in direct shear. J. Struct. Eng. ASCE 115 (9), 2276
2296. Augenti, N., Parisi, F., 2011. Constitutive modelling of tuff masonry in direct shear. Constr. Build. Mater. 25 (4), 1612
1620. Caporale, A., Parisi, F., Asprone, D., Luciano, R., Prota, A., 2015. Comparative micromechanical assessment of adobe and clay brick masonry assemblages based on experimental data sets. Compos. Struct. 120, 208
220. CEB-FIP, 1993. Model Code 90. Thomas Telford Ltd, UK. Chiaia, B., Van Mier, J.G.M., Vervuurt, A., 1998. Crack growth mechanisms in four different concretes: microscopic observations and fractal analysis. Cem. Concr. Res. 28 (2), 103
114. Drysdale, R.G., Hamid, A.A., Baker, L.R., 1993. Masonry Structures, Behaviour and Design. Prentice Hall, Englewood Cliffs, NJ.

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Glitzka, H., 1988. Druckbeanspruchung parallel zur Lagerfuge. Mauerwerk Kalender 1988. Ernst & Sohn. Hamid, A.A., Heidebrecht, A.C., 1979. Tensile strength of concrete masonry. J. Struct. Div. ASCE 105 (7), 1261
1276. Hendry, A.W., 1990. Structural Masonry. Macmillan Press Ltd, London. Hendry, A.W., Sinha, B.P., Davies, S.R., 2004. Design of Masonry Structures. E & FN SPON, London. Kaushik, H.B., Rai, D.C., Jain, S.K., 2007. Stress-strain characteristics of clay brick masonry under uniaxial compression. J. Mater. Civil Eng. 19 (9), 728
739. Khalaf, F.M., Hendry, A.W., 1989. Effect of bed-face preparation in compressive testing of masonry units. Proceedings of the Second International Brick Masonry Conference London. Klingner, R.E., 2010. Masonry Structural Design. McGraw-Hill Professional. Lourenc¸o, P.B., 1994. Analysis of Masonry Structures with Interface Elments. Delft University of Technology, The Netherlands. Lourenc¸o, P.B., 1996. Computational strategies for masonry structures. PhD thesis. ISBN: 90407-1221-2. Lourenc¸o, P.B., Almeida, J.C., Barros, J.O.A., 2005. Experimental investigation of bricks under uniaxial tensile testing. Masonry Int. 18 (1), 11
20. Lumantarna, R., 2012. Material Characterisation of New Zealand’s Clay Brick Unreinforced Masonry Buildings. The University of Auckland. Rots, J.G., Van der Pluijm, R., Vermeltfoort, A.T., 1997. Structural masonry: an experimental/ numerical basis for practical design rules, AA Balkema. Sarhat, S.R., Sherwood, E.G., 2014. The prediction of compressive strength of ungrouted hollow concrete block masonry. Constr. Build. Mater. 58, 111
121. Van Der Pluijm, R., 1997. Non-linear behaviour of masonry under tension. Heron 25
48. Van der Pluijm, R., 1999. Out-of-Plane Bending of Masonry: Behaviour and Strength. Eindhoven University of Technology. Vasconcelos, G., 2005. Experimental investigations on the mechanics of stone masonry: characterization of granites and behaviour of ancient masonry shear walls. PhD thesis, University of Minho, Portugal. Vermeltfoort, A.T., 2005. Brick
mortar interaction in masonry under compression. PhD thesis, TU Eindhoven, The Netherlands. Vermeltfoort, A.T., Martens, D.R.W., 2016. Compression and shear properties of masonry cut from walls, three to ninety-five years old, compared with laboratory made masonry. In: 16th International Brick and Masonry Confenrece, Padova, Italy.

Further reading CEN. Eurocode 6, 2005. Design of Masonry Structures, Part 1
1: General Rules for Reinforced and Unreinforced Masonry Structures. European Committee for Standardization, Belgium. Lourenc¸o, P.B., Barros, J.O.A., Oliveira, J.T.T., 2004. Shear testing of stack bonded masonry. Constr. Build. Mater. 18, 125
132. PIET-70, 1971. Masonry work. Prescriptions from Instituto Eduardo Torroja. Consejo Superior ˜ ŋficas, Madrid (in Spanish). de Investigaciones CientA Pluijm, R., Vermeltfoort, A.T., 1999. CUR 193 Materiaal Parameters Voor Constructief Metselwerk. CUR Gouda, The Netherlands.

Chapter 8

Macromodeling E. Minga1, L. Macorini1, B.A. Izzuddin1 and I. Calio’2 1

Department of Civil and Environmental Engineering, Imperial College London, London, United Kingdom, 2Department of Engineering and Architecture, University of Catania, Catania, Italy

8.1 Introduction Unreinforced masonry (URM) has been used in the construction of buildings, bridges and monuments for centuries. Historical masonry structures form an important part of the world’s cultural and engineering heritage whose structural health conditions have to be assessed in order to guarantee their conservation. In general, old masonry structures have been designed following basic structural principles and empirical rules and have high vulnerability and poor performance when subjected to earthquake loading (Bruneau, 1994; Vicente et al., 2018), mainly due to the quasibrittle nature of masonry and the substandard structural detailing. As a result, there is a crucial need for realistic assessment of the structural capacity and integrity of URM components and structures and the implementation of efficient strengthening solutions to enhance their seismic performance. This requires the development of accurate modeling strategies, enabling realistic numerical predictions of the complex behavior under earthquake loading, as well as simplified but reliable approaches suitable for most engineering applications. The seismic behavior of masonry structures is governed by a complex interaction between the in-plane and out-of-plane response of masonry walls subjected to cyclic loading. This behavior is further complicated by the interaction between masonry elements and other structural components made of different materials (e.g., slabs and floor systems in masonry buildings). Masonry components can be investigated considering different scales of representation. Accurate detailed modeling of URM behavior at structural scale is not a simple task. This is because URM can be viewed as a heterogeneous material with distinct constituents, which include the block units, mortar, and block
mortar interface. The last two form the mortar joints, which generally constitute weak zones within the mesostructure, where most cracks

Numerical Modeling of Masonry and Historical Structures. DOI: https://doi.org/10.1016/B978-0-08-102439-3.00008-7 Copyright © 2019 Elsevier Ltd. All rights reserved.

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are localized. The presence of weak zones induces a strongly nonlinear quasibrittle behavior at structural scale, which directly depends on the mesostructure and the distinct properties of the individual constituents. To explicitly represent this effect, detailed modeling approaches that explicitly account for the actual masonry bond have been developed. This class of models includes micro- (Lourenc¸o and Pina-Henriques, 2006) and mesoscale finite element (FE) representations of masonry with regular bonding patterns (Lourenc¸o and Rots, 1997; Shieh-Beygi and Pietruszczak, 2008; Spada et al., 2009; Macorini and Izzuddin, 2011), as well as discrete element method mesoscale modeling techniques (Asteris et al., 2015). Such approaches can offer an accurate description of the response of URM components. However, they are associated with often prohibitive computational cost, as well as an excessively time-consuming preprocessing stage, which restrict their scope of applicability. More efficient and hence less detailed strategies are therefore necessary when the nonlinear seismic analysis of full-scale URM buildings is considered. For the modeling of such structures, URM is typically represented as a homogeneous material, at structural scale, and its behavior is described in a macroscopic way (Lourenc¸o, 1997). Two main approaches can be identified within this framework. The first one consists of the use of shell or solid FEs with specific 2D or 3D constitutive laws to represent material nonlinearity in the masonry. These laws might be developed on a phenomenological basis, founded on experimental observations on the behavior of URM components (Lourenc¸o et al., 1998; Berto et al., 2002; Saloustros et al., 2017), or they might be based on micromechanical considerations and homogenization procedures (Lourenc¸o et al., 2007). Alternative strategies to deal with the macroscopic representation of masonry are based on the use of macroelements, which represent large parts of URM components allowing for specific deformation modes. The response under each mode is generally calibrated based on phenomenological laws. The complexity of a macroelement approach can vary, depending on the kinematics of the element (i.e., the deformation modes that can be reproduced), the type of interaction with adjacent elements, and the constitutive laws employed. Classical 1D macroelements with concentrated or distributed plasticity (Magenes and Della Fontana, 1998; Roca et al., 2005; Lagomarsino et al., 2013; Penna et al., 2014; Raka et al., 2015; Rinaldin et al., 2016) generally offer a very efficient way to model regular URM buildings with reduced computational cost, but they ignore important interactions between structural members, especially in the case of complex geometries, and they do not account for the out-of-plane response. In contrast, certain recent macroelement formulations based on 2D or 3D geometries achieve a better balance between accuracy and efficiency and constitute a tangible alternative to continuum FE modeling approaches (Casolo and Pena, 2007; Calio` et al., 2012; Bertolesi et al., 2016; Panto` et al., 2016; Minga et al., 2018b).

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The present chapter is dedicated to the presentation of macroscale models, which can be used for the seismic assessment of existing masonry structures. At first, the principal features of the macroscopic response of URM components are briefly summarized. An overview of existing macroscale modeling strategies for URM structures follows in Section 8.3. These include macroscopic constitutive laws for continuum FE models and alternative macroelement formulations that can potentially offer a comparable level of accuracy in the response predictions. It is noted that this chapter is concerned with methods that can simulate the full nonlinear response of masonry up to collapse and can thus be used not only for the assessment of the ultimate capacity of URM structures, but also for directing the development of strengthening solutions that prevent and mitigate damage related to different limit conditions. Consequently, alternative approaches based on limit analysis—which can offer a computationally efficient way to accurately estimate the collapse load and failure modes of masonry structures but do not simulate the nonlinear response before and after failure—will not be discussed. In the last part of the chapter, a novel 3D macroelement approach for the FE modeling of URM structures under cyclic loading is presented. Its features and advantages with respect to classical macroscopic continuum FE approaches are discussed.

8.2 Macroscopic nonlinear behavior of masonry components The modeling of URM at a macroscale requires the identification of the main failure modes characterizing the ultimate response of URM components and the corresponding macroscopic material parameters that describe the onset of each specific mode. Regarding the in-plane response of masonry walls, the damage and failure modes usually observed include flexural cracking along one of the principle directions of the bonding pattern, as shown in the example of Fig. 8.1A, which is associated to the exceedance of the tensile strength of masonry in the specific direction. Shear sliding also develops in masonry components, as shown in Fig. 8.1B, when the shear sliding

FIGURE 8.1 In-plane failure modes of URM components: (A) diagonal shear cracking; (B) shear sliding; (C) flexural cracking; (D) crushing.

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PART | II Modeling of unreinforced masonry

strength of the joints is exceeded. When masonry components are loaded under in-plane shear, diagonal cracking might also develop, as shown in Fig. 8.1C. From a micromechanical point of view, this mode is associated with tensile and shear stresses within the masonry joints that exceed the respective strength values. The diagonal cracks can also traverse the blocks, when excessive tensile principal stresses develop within the block material. The rocking failure mechanism is characterized by a distribution of tensile and compressive loadings, which may lead to the development of cracks induced by Poisson’s effects along the direction of the compressive force, as shown in Fig. 8.1D. The concentrated damage in the compressive zone is associated with masonry crushing and to a macroscopic compressive strength of the masonry material. Regarding the out-of-plane response of URM components, flexural failure develops when a component is loaded in one-way bending, as in the example of Fig. 8.2A, and the flexural strength in the specific direction is exceeded. Similar to the in-plane response, shear sliding can develop along the joint in the out-of-plane direction, as depicted in Fig. 8.2B. Finally, out-of-plane, two-way bending often appears in masonry components that are adhered to other parts of the structure along consecutive boundaries and leads to a flexural out-of-plane response with the formation of pseudodiagonal cracking, as shown in Fig. 8.2C. Note that in many cases the failure mechanism of a masonry component is more complex and involves the interaction between two or more basic modes. These cases can be described as mixed-mode damage and failure at

FIGURE 8.2 Out-of-plane failure modes of URM components: (A) flexural cracking; (B) shear sliding; (C) diagonal out-of-plane cracking.

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structural scale. Both in-plane and out-of-plane, the macroscopic strength properties of masonry, including the flexural and shear strength as well as the compressive strength, depend on the direction of the loading with respect to the main axes of the material (parallel and perpendicular to the bonding pattern). Hence, any realistic constitutive model in a continuum framework should account for the anisotropic behavior of the material, especially in the case of masonry with a regular bond pattern.

8.3 Macroscale modeling strategies In this section, existing macroscale models that can be used for the modeling of URM structures in earthquake engineering applications are discussed. Here, we emphasize models that account for the cyclic response of masonry, though certain approaches not explicitly developed for cyclic loading but that have other novel attractive features are also mentioned. In general, approaches developed for the representation of either in-plane or out-ofplane behavior of URM and the interaction between in-plane and out-ofplane response are included. URM structures can be modeled using a continuum FE representation. In this case, masonry is considered a homogeneous material at structural scale that can be described by an isotropic or orthotropic constitutive law. The main advantages of this strategy are the consistent FE description of the structure, the simple implementation of URM-specific constitutive models in the frame of commercial or research FE software, the flexibility in the description of complex geometries, and the relative computational efficiency when compared to micromodeling approaches, which potentially allows the representation of full-scale structures under cyclic loading conditions. One of the earliest examples of URM macroscopic material descriptions is the orthotropic model by Lourenc¸o et al. (1998) for 2D continuum FE analysis. The model is based purely on plasticity, with a Rankine-type criterion in tension and a Hill-type criterion in compression that represent the distinct strength and softening behavior of the material under different stress states. It also takes into account the difference in the macroscopic tensile strength in any direction moving from the one parallel to the one normal to the bed joints. The model shows the ability to predict the global monotonic in-plane behavior of masonry components. Furthermore, it has been extended to describe the out-of-plane behavior of URM modeled using shell elements (Lourenc¸o, 2000). In this version of the model, an orthotropic failure criterion with different flexural strength along each of the main directions of the material is adopted. Additionally, a variable strength under two-way bending is assumed. However, this model does not include the effect of compressive stress in the flexural strength of masonry, which might significantly affect the accuracy of the predictions.

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PART | II Modeling of unreinforced masonry

Damage mechanics has also been employed for the macroscopic description of URM in 2D, setting a framework more suitable for cyclic analysis. Papa (1996) proposed an orthotropic model based on micromechanical considerations, employing separate damage variables for shear, tension, and compression in the different phases of the material. A macroscopic 2D damage model was also developed by Berto et al. (2002) considering distinct damage variables in tension and compression for each direction with respect to the bed joints. By employing separate damage in tension and compression both models account for the stiffness recovery upon load inversion. This class of orthotropic damage models reproduces certain important characteristics of the behavior of URM components subjected to in-plane cyclic loading, such as the initial stiffness and the peak resistance. On the other hand, the hysteretic effects are not perfectly captured, as pure damage mechanics does not allow for permanent strains. A potentially more serious limitation of these models is related to the representation of damage as a smeared property of the material within a certain volume, which does not reflect the actual crack localization and propagation, typical of masonry. Pela` et al. (2013) proposed an orthotropic damage model and the use of a local crack-tracking technique for the representation of damage in quasibrittle materials that was previously proposed by Cervera et al. (2010). The model aims to reduce the inaccuracies associated with the smeared damage and better capture and track the damage localization in such materials; mesh independence is also ensured. Recently, the model was tested in a large-scale wall under monotonic in-plane loading (Saloustros et al., 2017). It was shown to accurately reproduce the failure modes at piers and spandrels, providing a significant improvement compared to classical macroscale smeared approaches, as shown in Fig. 8.3 where indicative results obtained with the orthotropic model of Berto et al. (2002) are compared with results of the localized orthotropic model from Saloustros et al. (2017). It is noted that most URM-specific macroscale material descriptions are developed for 2D analysis or for shell elements in 3D analysis, as in Lourenc¸o (2000). Though it is possible to extend the orthotropic formulation in a 3D continuum space considering uniform behavior along the thickness, this development does not necessarily contribute to a more accurate description of the response, since it increases the complexity and computational cost of the models. In general, the modeling of masonry with isotropic laws developed for other quasibrittle materials such as concrete (Lee and Fenves, 1998) might be acceptable for certain types of applications (e.g., when relatively small deformations and early stages of damage are considered). However, it should not be considered realistic and should not be used for modeling ultimate states and the estimation of collapse loads and failure mechanisms. The macroscale models described so far are not explicitly concerned with the microstructure of the material, so, formally, they should be calibrated

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FIGURE 8.3 (A) Compressive (top) and tensile (bottom) damage in URM facade analyzed under cyclic loading (Berto et al., 2002); (B) tensile damage in URM facade analyzed under monotonic loading with coarse (top) and fine (bottom) mesh (Saloustros et al., 2017).

with structural scale experimental testing for each individual type of URM. In fact, identification of the anisotropic yield surfaces requires a series of tests with various angles of loading with respect to the main directions of the material. This involved parameter identification process is an important weakness of the phenomenological macroscopic constitutive models. A more consistent approach to the derivation of macroscopic constitutive laws for heterogeneous materials such as masonry is the use of the homogenization theory. This is based on the assumption of periodicity in the URM mesostructure, which allows for the identification of a representative volume element (RVE) of the material, as shown in the examples of Fig. 8.4. The RVE can be described based on micromechanical properties (i.e., material properties of the individual constituents). Homogenization theorems can be used to derive a homogenized stress
strain relationship, which can essentially be used as a macroscopic constitutive law of URM within a continuum structural model. It is obvious that the rigorous application of this concept in arbitrary URM structures is not a simple task, especially in the nonlinear softening range. However, important steps have been made so far toward this direction mainly for simple regular bonding patterns. Early work by Pande et al. (1989) estimated the elastic moduli of the homogenized material based on a two-step closed-form solution. Masonry

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PART | II Modeling of unreinforced masonry

FIGURE 8.4 Alternative RVE definitions for running bond masonry: (A) Mistler et al. (2007); (B) Lourenc¸o et al. (2007).

units and mortar joints are considered as isotropic materials. At the first step, homogenization is applied in the direction parallel to the bed joints. At the second step, the procedure is repeated along the direction vertical to the initial one. Simple analytical solutions are obtained for the elastic moduli in each direction. Obviously, this simple approach has limitations, including the fact that the bonding pattern is not considered and that different elastic characteristics are obtained if the direction of each step is inverted. Closed-form solutions for the homogenized elastic properties were also obtained by Lourenc¸o et al. (2007); in this work, which has been used to analyze running bond masonry, the mortar joints are reduced to zero-thickness interfaces and the block units are assumed rigid. The previous strategies, though useful for the relatively simple derivation of the elastic properties, are not sufficiently accurate in the nonlinear range (Lourenc¸o et al., 2007). An alternative approach that can be more successfully extended to describe nonlinearity is the use of FE methods for the solution of the homogenization problem, first proposed by Anthoine (1995) for the elastic range. In this strategy, the homogenized stress
strain relation is derived within the discretized cell unit at each step and used in the macroscale FE mesh, creating a multiscale computational method. Within this framework, Anthoine (1995) used a continuous unit cell mesh in with distinct isotropic materials for the block units and the mortar, thus accounting for the thickness of the joints and the real bonding pattern. The approach has been extended to inelastic behavior, considering isotropic damage (Anthoine, 1997; Pegon and Anthoine, 1997; Massart et al., 2007) or plasticity (Lopez et al., 1999) for the URM constituents. Significantly, the homogenized multiscale strategy has also been extended to represent the nonlinear out-of-plane behavior of URM (Mistler et al., 2007; Mercatoris and Massart, 2011).

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The above multiscale FE methods based on homogenization are conceptually promising, since any type of periodic masonry can be described based on the bonding pattern and the properties of its constituents, which can be identified in a much simpler way than the macroscopic properties. However, they involve a two-level FE solution—at unit cell level for each integration point and at macroscale for the entire structure—at each step of the analysis. Thus, they are associated with increased computational cost in the nonlinear range, partially defying the purpose of the macroscale representation. Also, it should be noted that the cyclic response of masonry has not yet been represented using this class of models. More recently, a new type of macroelement approach has been explored, which allows a 2D or full-3D representation of the macroscopic response of URM with computational efficiency and increased accuracy compared to traditional 1D macroelement models. They are based on 2D or 3D discrete FEs interacting along their boundaries through springs, which reproduce the macroscopic behavior of a unit cell of the URM assembly. These include the Rigid Body Spring Model proposed by Casolo and Pena (2007) utilizing 2D rigid blocks and uncoupled axial and shear springs, as shown in Fig. 8.5. The springs are calibrated based on assumed failure mechanisms within a unit cell of URM, which is considered a heterogeneous periodic material. The model, which has been conceived for dynamic analysis, shows computational benefits due to its simplicity and ability to reproduce the in-plane flexural, diagonal shear, and compressive crushing failure modes of masonry. However, the adopted simplified constitutive laws overestimate the hysteretic energy dissipation in the case of rocking. Moreover, mechanisms that follow other than the assumed failure patterns might not be well represented. A similar strategy was employed in the work developed by Bertolesi et al. (2016) for investigating the in-plane URM behavior and the modeling approach put forward by Silva et al. (2017) for the out-of-plane URM response. In this case, homogenization principles are employed to derive holonomic

FIGURE 8.5 RBSM: definition of “unit cell” described by four rigid elements interacting through axial and shear springs (Casolo and Pena, 2007).

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PART | II Modeling of unreinforced masonry

constitutive laws for the springs connecting the rigid elements. These laws are derived in an independent step and then used within standard commercial FE software providing efficiency in the structural analysis. However, they do not account for the cyclic response of masonry. Calio` et al. (2012) proposed a plane element, implemented within a discrete element framework, which also includes shear deformation modes for the homogeneous block, thus allowing the representation of diagonal shear cracking with a reduced number of elements. The macroelement is based on a quadrilateral of rigid edges connected by four hinges. The shear diagonal behavior is governed by one degree of freedom only, while zero-thickness interfaces along the four edges determine the interaction with the adjacent elements. This approach was extended to represent the 3D behavior (Panto` et al., 2016, 2017) employing macroelements with seven degrees of freedoms (DOFs) to represent the in-plane and out-of-plane response of masonry components. In the recent work by Minga et al. (2018b) a novel 3D macroelement was proposed. It adopts the discrete representation of URM through homogeneous deformable blocks, introducing significant enhancements toward a more accurate description of the cyclic nonlinear response of URM components under both in-plane and out-of-plane loading. In particular, the element adopts a shear deformation mode of the inner block, but also a novel outof-plane diagonal bending mode, which allows the simulation of two-way bending and diagonal out-of-plane failure within the volume of a single element. It is noted that when out-of-plane-rigid discrete elements are used to represent masonry blocks as in Panto` et al. (2017), diagonal out-of-plane failure cannot be represented unless a large number of elements is used for the simulation of an individual masonry component. A cyclic constitutive law with strength softening governs the behavior of the internal nonlinear springs allowing for a realistic description of the hysteretic energy dissipation capacity of masonry components. Moreover, the macroelement blocks interact with adjacent elements through nonlinear interfaces with a cyclic cohesivefrictional constitutive law, which explicitly accounts for the coupling between the normal and tangential direction (i.e., the coupling between flexural and shear damage and the effect of confinement in the shear response). Consequently, both flexural cracking and frictional sliding are reproduced in a realistic way. The 3D macroelement is developed in a finite element framework with flexible connectivity with adjacent elements through its four boundary edges, which allows its combination with different types of FEs (i.e., beam or shell elements representing the floor systems) for the modeling of entire URM buildings. In the following section, the novel 3D macroelement is presented pointing out the potential of this modeling strategy, which leads to accurate results with significant computational efficiency due to the discrete representation of large parts of URM by one single element.

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8.4 A 3D macroelement approach The approach described in this section was proposed by Minga et al. (2018b) as an efficient strategy for the nonlinear analysis of URM structures. In the following, the modeling assumptions are explained and the principal features of the macroelement formulation are briefly outlined. Subsequently, numerical examples are provided to illustrate the characteristics of the URM response that can be reproduced by the element. Finally, two validation examples are presented, where the macroelement strategy is employed for the simulation of physical tests on URM components under cyclic loading.

8.4.1

Assumptions

The macroelement described herein is developed for the modeling of rectangular parts of URM assemblies, comprising several masonry units connected through joints. It is therefore designed to reproduce failure patterns appearing in URM components in a macroscopic, phenomenological way. In fact, the element is capable of representing all the in-plane and out-of-plane failure patterns described in Section 8.2 and depicted in Figs. 8.1 and 8.2, that is, in- and out-of-plane flexural failure, in- and out-of-plane sliding failure, inplane diagonal shear failure, and out-of-plane failure due to two-way bending and masonry crushing. In this strategy, URM is modeled through 3D homogeneous rectangular blocks interacting with adjacent elements through cohesive interfaces, as shown in Fig. 8.6A. Each block has two deformation modes: in-plane shear deformation, illustrated in Fig. 8.6B, and out-of-plane diagonal bending, as shown in Fig. 8.6C. The block is otherwise rigid. These two deformation modes are described by two springs, also sketched in Fig. 8.6B and C. The response of the in-plane spring reproduces in a phenomenological way the diagonal shear damage and the failure mode of a URM block, while the

FIGURE 8.6 (A) URM macroelement; (B) in-plane shear deformation mode of inner block; (C) out-of-plane diagonal bending mode of inner block.

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PART | II Modeling of unreinforced masonry

FIGURE 8.7 Areas of influence of the interfaces between the inner block and the external edges.

response of the out-of-plane spring reproduces the macroscopic response of a URM block under two-way bending with diagonal cracking. Zero-thickness cohesive interfaces are defined along the four boundaries of each macroelement, as illustrated in Fig. 8.6A. All the normal elastic inplane deformation is concentrated along these interfaces, since the macroelement exhibits only in-plane shear deformation and out-of-plane diagonal bending. Each interface iði 5 1:4Þ accumulates the deformation within the volume Vi of the respective influence area, as depicted in Fig. 8.7. In a similar way, tensile damage and shear sliding within the influence area is represented by tensile or shear damage accumulated along the corresponding interface. In addition, the limit strength of masonry under compression is taken into account through a cap imposed on the normal compressive stresses that can be developed along the interfaces. When the compressive strength of masonry is exceeded, negative plastic deformations develop reproducing in a phenomenological way the crushing mode of masonry under compression. It is noted that the assumed deformation of the external faces of the inner block does not include in- or out-of-plane bending. Therefore, the twonoded external edges that achieve the connectivity between two faces of adjacent macroelements can be assumed linear.

8.4.2

Macroelement formulation

The connectivity of the 3D macroelement is defined by four 2-noded edges, as shown in Fig. 8.8A. As explained in Section 8.4.1, the displacements along the edges are assumed linear. Three translational DOFs uX ; uY ; uZ are defined at each node of the macroelement, where XYZ is the local coordinate system of the macroelement, sketched in Fig. 8.8A with respect to the nodal

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275

FIGURE 8.8 (A) External edge DOFs and GPs of cohesive interface; (B) additional DOFs of macroelement.

connectivity. In addition, each node holds a rotational DOF θt that defines the twisting rotation of the specific external edge (θt 5 θX for edge 1 and 3 and θt 5 θY for edge 2 and 4). The rotational DOFs θt are necessary to represent relative displacements at the interface under out-of-plane bending. The interpolation of the nodal DOFs gives the displacement and twisting rotation along the each external edge iði 5 1:4Þ. Hence, the displacement at each point of the external surface of the cohesive interface illustrated in Fig. 8.8A can be derived. This external surface is, in fact, a fictional extension of the linear external edge, which in the initial configuration is coincident to the face of the inner block. Besides the nodal DOFs, each macroelement has eight additional DOFs, as illustrated in Fig. 8.8B, that govern the deformation modes of the inner block. The additional DOFs d1 to d4 define the in-plane displacement of each rigid face of the block, while d5 to d8 define the out-of-plane displacement at each corner of the block. A linear interpolation of these displacements is assumed within the block domain.

8.4.2.1 Cohesive boundaries The interfaces along the four macroelement boundaries have a cohesivefrictional behavior governed by a 3D plasticity-damage constitutive law. This law defines the relation between the relative displacements εi and the interface tractions σi at each Gauss point (GP) within the 2D domain of the zerothickness interface. The relative displacement between the face of the block and the external surface defined by the two-noded edge is a function of the additional DOFs of the block U a and the nodal DOFs of the edge U s : εi 5 uint;i 2 uext;i 5 N a;i U a 2 Ns;i U s ð8:1Þ

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PART | II Modeling of unreinforced masonry

where N s;i is the matrix containing the linear shape functions for the interpolation of the nodal DOFs for edge iði 5 1:4Þ and Na;i is the matrix containing the linear shape functions for the interpolation of the additional DOFs for block face i. The strain measure εi is a vector consisting of the normal and two tangential components of the relative displacement. The vector of the respective stress measure σi contains the normal and the two tangential components of the interface traction. The vector σi is estimated at each GP of the boundary interfaces, based on the plasticity-damage constitutive law developed by Minga et al. (2018b). This law reproduces the main characteristics of the cyclic behavior of cohesive-frictional interfaces: stress softening in tension and shear, stiffness degradation depending on the level of damage, and recovery of normal stiffness in compression and residual (plastic) strains at zero stresses when the interface is damaged. Additionally, the effect of masonry crushing in compression is taken into account through negative plastic normal strain in the interfaces of the crushed area. The elastic yield domain is described by three surfaces in the stress space, which are depicted in Fig. 8.9. Surface F2 defines the limit strength under shear, controlled by the cohesion c and the friction angle tan ϕ of the interface between URM blocks. The tensile cap F1 is fully defined by the tensile strength ft of masonry in the direction normal to the interface, while fc is the compressive strength of masonry assumed as a homogeneous material and it defines the compressive cap of the elastic yield domain. When the yield domain is exceeded, plastic deformation and damage develop, producing a softening behavior in the σi 2 εi response in the normal and tangential directions. When the damage under tension or shear is fully developed, the normal tensile stress drops to zero, while the tangential stresses follow a Mohr
Coulomb friction law, that is, their residual value depends on the compressive stress. On the other hand, when the compressive cap is exceeded, the negative plastic deformation and the damage under compression that develop reproduce in a phenomenological way the crushing of masonry within the area of influence of the specific GP. Fig. 8.10A and B shows examples of the cyclic behavior obtained with the interface constitutive law in the normal and tangential directions. FIGURE 8.9 Multisurface yield domain of interface constitutive law (Minga et al., 2018a).

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FIGURE 8.10 Illustrative examples of the cyclic constitutive law employed at the cohesive interfaces (Minga et al., 2018a): (A) normal direction; (B) tangential direction.

Since the elastic deformation of the URM block is concentrated in the interfaces, as outlined in Section 8.4.1, the elastic stiffness values Kn and Ks represent the Young’s modulus and elastic shear modulus of masonry per unit length of the area of influence of the specific interface, respectively. Let EX and EY be the Young’s modulus of masonry in the direction of the local element axes X and Y, respectively, and G be the elastic shear modulus of masonry. Then: Kn;1 5 Kn;3 5

2EY H

K n;2 5 K n;4 5

2EX L

ð8:2Þ

2G H

K s;2 5 K s;4 5

2G L

ð8:3Þ

Ks;1 5 Ks;3 5

8.4.2.2 In-plane shear spring The springs that govern the deformation modes of the block are associated with the additional DOFs of the element. In particular, the deformation of the in-plane shear spring is obtained by d1 , d2 , d3 ; and d4 through the equation: ud 5 2 cosα d1 1 sinα d2 2 cosα d3 1 sinα d4

ð8:4Þ

The force Fd developed in the spring is obtained based on a constitutive law that reproduces in a phenomenological way the global response of a masonry block that fails due to diagonal cracking under in-plane shear. The piecewise-linear law illustrated in Fig. 8.11 is employed, as it approximately captures the main characteristics of the specific cracking pattern, discussed, for example, in Magenes and Calvi (1997). The model parameters for the

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PART | II Modeling of unreinforced masonry

FIGURE 8.11 Constitutive law employed for the macroelement springs.

diagonal shear spring will be noted as Fyd ; Frd ; Ked ; and Kpd . They are estimated as functions of the macroscopic shear properties of masonry, namely the elastic shear modulus Ge , the shear strength at zero confinement τ Y;0 ; and the parameter μd defining the influence of the confinement to the shear strength. In addition, a postpeak shear modulus Gp is considered for the estimation of Kp . Gp is the property defining the effective stiffness of the postpeak branch of the response of a URM block under in-plane shear. All the above macroscopic material properties of masonry can be obtained experimentally or through numerical tests on detailed models, such as mesoscale models. The values of τ Y;0 and μd can also be estimated based on the cohesion and friction angle of the masonry joints, as proposed by Mann and Muller (1982). Following the same strategy already adopted by Calio` et al. (2012), the relation between the constitutive model parameters Fyd ; Frd ; Ked ; and Kpd and the shear properties of masonry is based on the equivalence of the inner block to a homogeneous masonry plate under pure shear, as illustrated in Fig. 8.12. This equivalence results in the following relations:

FYd 5

Ked 5

Ge LW Hcos2 α

ð8:5Þ

Kpd 5

Gp LW Hcos2 α

ð8:6Þ

ðτ Y;0 1 μd σn;m ÞLW Hcosα

ð8:7Þ

where σn;m is the mean compressive normal stress applied to the URM block.

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279

FIGURE 8.12 Calibration of in-plane shear spring.

8.4.2.3 Out-of-plane diagonal bending spring The capability to represent, in a phenomenological way, the out-of-plane diagonal bending mode is an original aspect of this macroelement. This further mode allows a better representation of the out-of-plane response of masonry walls with respect to the model already proposed by Panto` et al. (2017). Since this mode is related to a single degree of freedom, the corresponding force
displacement law has been conventionally attributed to a nonlinear spring centered on the element. The length of the out-of-plane spring coincides with the lateral distance between the central points of the two diagonals of the inner block mid-surface. Initially, its length is zero. In a deformed configuration, the deformation of the spring is obtained as a function of the additional DOFs d5 , d6 , d7 ; and d8 as follows: uout 5 0:5ðd5 2 d6 1 d7 2 d8 Þ

ð8:8Þ

The force Fout developed in the spring is assumed based on the same constitutive law employed for the diagonal spring. The calibration of this addition spring is not straightforward, since in this case, there is little experimental evidence on which the calibration of the model parameters could be based. Hence, a calibration of the stiffness and strength parameters (Fyout ; Frout ; Keout ; Kpout ) based on detailed numerical models is proposed here. Specifically, numerical tests on URM blocks under out-of-plane diagonal bending modeled using a detailed 3D mesoscale approach developed by Macorini and Izzuddin (2011) and Minga et al. (2018b) are employed. For details on the calibration procedure, the reader is referred to Minga (2018).

8.5 Numerical examples To examine the main response characteristics that can be reproduced by the proposed macroelement, numerical tests performed on single elements are presented in the following sections. These tests are designed to illustrate the

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PART | II Modeling of unreinforced masonry

activation of typical failure modes individually and show relevant features of the element behavior, such as the effect of compressive stress. The validation of the macroelement modeling approach through comparison with experimental results is discussed. All the analyses presented here were performed in ADAPTIC (Izzuddin, 1990), a general FE software for nonlinear structural analysis, where the macroelement has been implemented.

8.5.1

In-plane failure modes

The in-plane response of the element depends both on the shear spring and the interfaces between the block and the external edges. Any of the in-plane failure modes (shear sliding, flexural failure, and diagonal shear cracking) can develop within a macroelement under in-plane loading. The actual failure mode depends on the boundary conditions applied to the element and the material parameters employed for the shear spring and the interfaces. The numerical tests described here are performed on one single element with dimensions L 5 1000 mm, H 5 1350 mm, and W 5 250 mm. The element is fully restrained at the bottom edge and a horizontal displacement is applied on top giving rise to shear and flexural bending deformation. Different levels of normal compressive stress σy are imposed on the element. A schematic representation of the test is shown in Fig. 8.13A. Two different sets of material parameters are considered to illustrate the different response the element can develop in-plane. In both cases, the parameters for the interfaces are given in the table in Fig. 8.13B. The shear modulus and the slope of the postpeak branch for the diagonal shear spring are Ge 5 500 MPa and Gp 5 2 10 MPa; respectively. In contrast, different values are employed for the shear strength under zero compression and for

FIGURE 8.13 (A) Geometry and boundary conditions of in-plane numerical test; (B) material parameters for macroelement interfaces.

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281

FIGURE 8.14 Case 1: force
displacement response for different levels of normal stress σy .

the friction coefficient of the diagonal shear spring to account for different relative resistance against diagonal shear damage with respect to the resistance against flexural damage. In all the simulations shear sliding was prevented. In Case 1, τ Y;0 5 0:25 MPa and μ 5 0:5 respectively, while in Case 2, τ Y;0 5 0:2 MPa and μ 5 0:36. Fig. 8.14 shows the base shear
displacement response of the element for Case 1 at different levels of compressive stress. In this case, the failure mode that prevails is the flexural bending, while the diagonal shear spring remains elastic. It can be observed that the capacity of the element increases with the level of confinement, since the top and bottom interfaces are initially under compression. Fig. 8.15 shows the base shear
displacement response of the element for Case 2 at different levels of compressive stress. At zero or low confinement levels, the element develops mainly flexural damage, as in Case 1. However, when the confinement increases the shear diagonal cracking mode prevails. The linear softening branch that corresponds to the postpeak branch of the diagonal spring characterizes this failure mode. This behavior is consistent with what is expected theoretically, since the increased vertical compression increases the tensile stresses along the diagonal under shear, which eventually results in the diagonal failure of the wall. The typical deformed shapes of the element at the end of the numerical test for both the considered cases and for σy 5 0:6 MPa are shown in Fig. 8.16A and B.

8.5.2

Out-of-plane failure modes

The out-of-plane response of the element is determined by the out-of-plane spring and the contribution of the interfaces between the block and the

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PART | II Modeling of unreinforced masonry

FIGURE 8.15 Case 2: force
displacement response for different levels of normal stress σ y .

FIGURE 8.16 Deformed shape of element under in-plane loading at ux 5 20 mm for σy 5 0:6 MPa: (A) Case 1; (B) Case 2.

external edges. The actual response in each case depends on the boundary conditions and the material parameters. This will be illustrated through verification tests in one single element with dimensions L 5 2000 mm, H 5 1250 mm, and W 5 110 mm. The material parameters for the interfaces are the ones reported in the table in Fig. 8.13B. The parameters for the outof-plane spring are Ee;m 5 1000 N=mm, Ep;m 5 2 50 N=mm, py 5 1 kPa, pres 5 0:05 kPa, and μ 5 0:004. Initially, a test with the boundary conditions presented in Fig. 8.17A is performed. In this case, the bottom and left edges of the element are pinned and an out-of-plane displacement is imposed on the two adjacent nodes of the free corner. The free rotation of the restrained edges results in negligible relative displacements at the corresponding interfaces. Hence, the deformation of the element is assumed to be entirely governed by the out-of-plane

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283

FIGURE 8.17 Numerical test for out-of-plane deformation mode: (A) two pinned edges; (B) two fully restrained edges.

spring, which deforms up to complete damage. This mode is depicted in Fig. 8.18. The out-of-plane force
displacement response of the element for different levels of compressive stress is shown in Fig. 8.19. The trilinear response is directly linked to the constitutive law of the out-of-plane spring (see Fig. 8.11). The direct effect of the confinement on the capacity of the element under the out-of-plane diagonal bending mode can also be observed. Subsequently, a test with the boundary conditions described in Fig. 8.17B is performed. In this case, both the out-of-plane displacement and rotation of the bottom and left edges are restrained. Thus, the interfaces contribute to the out-of-plane response, partially restricting the rotation of the block faces. At the maximum out-of-plane displacement of 40 mm, the strength of the out-of-plane spring has reached its residual values, similar to the test with the pinned edges. However, the force
displacement response of the element, plotted in Fig. 8.20, is not uniquely governed by the spring constitutive law. In contrast, it is produced by the combination of out-of-plane failure with diagonal cracking—through the out-of-plane spring damage—and interface out-of-plane bending and cracking. The capacity of the wall in this case is higher and it is increased with a higher rate when the level of compressive stress increases. This is due to the combined effect of the confinement on both the spring and the interfaces’ ultimate capacity.

8.5.3

Validation of macroelement representation

In this section, the proposed macroelement is validated based on some experimental tests on URM masonry walls subjected to in- and out-of-plane loadings. The results are also compared against some simulations obtained by other macroscale models reported in the literature. The latter comparison

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PART | II Modeling of unreinforced masonry

FIGURE 8.18 Out-of-plane loading of element with two adjacent pinned edges: (A) deformed shape; (B) strength degradation of out-of-plane spring at final step.

FIGURE 8.19 Out-of-plane loading of element with two adjacent pinned edges: force
displacement response for different levels of normal stress.

aims at highlighting the advantages of the proposed strategy in terms of accuracy and efficiency. The analyses were performed in ADAPTIC (Izzuddin, 1990).

8.5.3.1 In-plane loading validation of unreinforced masonry panels The first validation example concerns the in-plane response up to the collapse of masonry piers. The experimental tests performed by Anthoine et al.

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285

FIGURE 8.20 Out-of-plane loading of element with two adjacent fully restrained edges: force
displacement response for different levels of normal stress.

(1995) are numerically simulated. Namely, two URM wall specimens characterized by different aspect ratios (wall height over length) are considered. The “short wall” possesses an aspect ratio of 1.35, while the “tall wall” is characterized by an aspect ratio of 2.0. The specimens were tested under inplane shear loading with a fixed value of vertical load. In this section, the ability of the proposed macroelement modeling approach to predict the monotonic and cyclic response of the tested piers and their distinct failure modes is investigated. The tested wall specimens were connected to a rigid base through a mortar bed joint. Both the piers were subjected to a uniform compressive stress of 0.6 MPa through a stiff beam, which was located at the top of the piers and constrained to remain horizontal during the test. Horizontal displacement cycles of increasing magnitude were imposed to the top beam. Each specimen developed a distinct cracking pattern. As expected, the short wall developed diagonal shear cracking at failure, while the tall wall specimen developed horizontal cracking, due to flexural bending, and a rocking cyclic behavior, without suffering strength degradation. The simulation performed with the proposed approach is based on a mesh of 3 3 3 macroelements. The element edges along the bottom of the wall are fully restrained, simulating a rigid boundary. The bottom interfaces corresponding to the restrained edges represent the frictional cohesive surface between the wall specimens and the assumed rigid base. The edges along the top of the walls are constrained to remain horizontal by coupling the translations of their nodes along the Y direction. Horizontal displacement cycles along the X direction are imposed on the same nodes. All loads are applied

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TABLE 8.1 Material parameters for macroelement cohesive interfaces. Young’s modulus, E (N/mm2)

Tensile strength, ft (N/ mm2)

Cohesion, c (N/mm2)

Horizontal

2500

0.10

0.23

Vertical

1500

0.68

1.56

Friction angle, tan ϕ

Fracture energy, Gf (N/mm) Mode I

Mode II

0.58

0.05

0.10

0.80

0.05

0.10

TABLE 8.2 Material parameters for macroelement diagonal shear spring. Elastic shear modulus, Ge (N/ mm2)

Postpeak shear modulus, Gp (N/mm2)

Shear strength at zero confinement, τ Y,0 (N/mm2)

Coefficient of friction, μd

580

2200

0.17

0.43

in-plane and the top edges have common X and Y displacements, thus simulating a rigid slab support. The interfaces corresponding to the top edges represent the frictional cohesive surface between the wall and the steel beam. The compressive stress σn 5 0:6 MPa is applied through the application of nodal forces at the top edges. Tables 8.1 and 8.2 report the material properties adopted in the numerical simulations. For more details on the identification of the parameters, the reader is referred to Minga et al. (2018b) and Minga (2018). The experimental global response curve of the short-wall specimen superimposed on the numerical results obtained with the macroelement model are reported in Fig. 8.21. A very close agreement with the experimental test in the force
displacement curve can be observed. The maximum shear capacity of the wall and the corresponding drifts are very well captured. Furthermore, the rate of strength and stiffness degradation are reproduced in a very accurate way and, consequently, the amount of hysteretic energy dissipation, at each cycle, appears in good agreement with the experimental evidence. Fig. 8.22A qualitatively shows the deformed shape of the short-wall model at the maximum displacements attained on each side during the analysis. In Fig. 8.22B the level of stiffness degradation, developed in the diagonal shear springs, is represented according to a colored scale. This latter representation is indicative of the level of diagonal shear cracking in the masonry pear. It is worth noting that the cracking pattern predicted by the numerical model is similar to that obtained experimentally. The damage scenario is characterized by tensile damage, in the interfaces along the top and

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FIGURE 8.21 Model of short wall under in-plane monotonic and cyclic loading: numericalexperimental comparison of global response.

FIGURE 8.22 Model of short wall under in-plane monotonic and cyclic loading: (A) deformed shape at edges of largest cycle; (B) strength degradation of diagonal shear springs at the end of the analysis.

bottom of the wall, and diagonal shear cracking in the central area of the wall, extending toward the corners. Fig. 8.23A and B illustrate the results for the tall wall specimen. Fig. 8.23A shows the numerical-experimental comparison of the global response curves. A satisfactory representation of the rocking behavior, characterizing the response of the tall wall specimen, can be observed, although the experimental results exhibit greater hysteretic dissipation at each cycle. The envelope of the experimental cyclic behavior is well captured, meaning

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PART | II Modeling of unreinforced masonry

FIGURE 8.23 Model of tall wall under in-plane monotonic and cyclic loading: (A) numericalexperimental comparison of global response; (B) deformed shape at edges of largest cycle.

that the capacity of the wall in the case of flexural failure mode is accurately predicted. Fig. 8.23B qualitatively reports the deformed shape of the specimen at the maximum drifts in both directions. Flexural cracking appears at the top and bottom interfaces; a small shear deformation can also be observed although the diagonal springs, in this case, remain elastic. The results obtained provide a very good representation of the experimental cracking pattern. These results confirm the ability of the macroelement description with correct calibration of the material parameters to reproduce the influence of the wall geometry and the actual failure mode for different wall aspect ratios. As already observed, the numerical cyclic response prediction does not capture the increase in the amount of energy dissipation as the drift increases as observed in the experimental tests. This could be partly due to the assumption of elastic unloading
reloading in the employed constitutive model (Minga et al., 2018a), which is a simplification of the real unloading
reloading path that might involve a certain level of hysteresis. Additionally, the proposed numerical description cannot capture the actual asymmetric response, as it does not allow for non-uniform material properties in the bricks and mortar joints, and potential asymmetric bond and geometry in the physical masonry specimen.

8.5.3.2 Two-way bending of unreinforced masonry The second validation example concerns the two-way bending of URM components, which develops when out-of-plane loads are applied to a wall connected along the vertical edges to return walls. Aiming at investigating the ability of the proposed macroelement representation to accurately predict this type of response, the experimental tests, performed by Griffith et al. (2007), are simulated. Namely, the solid wall specimens 1 and 2 of the experimental

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program are modeled herein. The specimens consist of a solid main wall of 4000 3 2500 mm2 and 480 mm long return walls on both the vertical sides. The walls were built in a running bond pattern, overlapping at the intersections between perpendicular panels. Simply supported constraints were imposed along the top and bottom edges, while full moment connections were guaranteed along the vertical edges of the return walls. Uniform pressure was applied at the two faces of the main wall resulting in cyclic outof-plane response. In one of the specimens, a uniform compressive stress was initially applied at the top to examine the influence of confinement due to a constant vertical load. In the numerical simulations reported herein a mesh of 8 3 4 macroelements is employed, with the main wall represented by 6 3 4 macroelements of equal size and each lateral wall represented by 1 3 4 equal-sized elements (Fig. 8.26). To be consistent with the experiment, the external edges of the macroelements along the bottom surface of the model are fully restrained while the top edges of the main wall are restrained only in the out-of-plane direction, simulating pinned supports. The vertical edges of the return walls are restrained in the direction of the X and Z axis and are not allowed to rotate around the vertical Y axis. Nodal forces are applied to the top edges representing the compressive stress, where necessary. The uniform lateral pressure along the surfaces of the main wall are applied through nodal forces with values that correspond to the area of influence of each node. Table 8.3 reports the material parameters for the horizontal and vertical interfaces. In Figs. 8.24 and 8.25, the results of the numerical simulations are compared to the experimental ones for σn 5 0:0 MPa and σn 5 0:1 MPa; respectively. In both cases, the envelope of the cyclic response in the positive quadrant is in very close agreement with the experimental envelope. Furthermore, the load and out-of-plane drift capacity of the wall as well as the rate of strength degradation are accurately reproduced. In the experimental curves reported in Griffith et al. (2007), the strength upon load reversal appears significantly reduced, which is attributed to the precracking of the wall up to 130 mm. This response characteristic is reproduced only to a

TABLE 8.3 Material parameters for macroelement cohesive interfaces. Young’s modulus, E (N/mm2)

Tensile strength, ft (N/ mm2)

Cohesion, c (N/mm2)

Horizontal

3540

0.163

0.75

Vertical

2124

1.08

2.43

Friction angle, tan ϕ

Fracture energy, Gf (N/mm) Mode I

Mode II

0.24

0.05

0.10

0.56

0.05

0.10

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FIGURE 8.24 Model with σn 5 0.0 MPa: numerical-experimental comparison of global response.

FIGURE 8.25 Model with σn 5 0.1 MPa: numerical-experimental comparison of global response.

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FIGURE 8.26 Model with σn 5 0.0 MPa: (A) strength degradation of out-of-plane springs; (B) deformed shape at maximum displacement.

small extent in the numerical analyses. These differences can be attributed to the simplified constitutive law employed for the out-of-plane springs (Fig. 8.6). According to the adopted material description, strength degradation develops separately in the positive and negative quadrants. Thus, the softening behavior in one direction does not influence the response in the other direction (this assumption will be reexamined in future work). A qualitative representation of the cracking pattern, for the case of zero compressive stress, is presented in Fig. 8.26A and B. Fig. 8.26A displays the strength degradation in the out-of-plane springs expressed as level of outof-plane diagonal cracking. It can be observed that the damage of the outof-plane springs of the main wall governs the response of the structure. The degraded springs are basically the ones close to the corners, corresponding to the area where diagonal cracks appeared during the experiments. Fig. 8.26B shows the deformed shape of the model at maximum positive displacement; the interfaces that develop significant damage are also identified in the figure. The flexural damage distributed along the horizontal interface in the top-center of the wall corresponds to the horizontal cracks along the bed joints in the center of the experimental specimen. Moreover, sliding occurs along the intersection of the main and the lateral walls. This scenario is also in good agreement with the actual experimental results.

8.6 Conclusions Macroscale modeling represents an effective strategy to describe the response of masonry components and structures under different loading conditions including extreme loading like an earthquake. This chapter surveyed some of the main modeling approaches at the macroscale. They include continuum FE descriptions, where URM is assumed as a homogeneous material and specific constitutive laws are employed to represent material nonlinearity; and efficient 1D equivalent frame models and discrete 2D or 3D

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PART | II Modeling of unreinforced masonry

macromodels, where rigid or deformable blocks interact along their boundaries through nonlinear springs to represent large portions of masonry components. Moreover, a novel 3D masonry macroelement formulation, which can represent the main in-plane and out-of-plane collapse mechanisms of URM panels, was presented. It is characterized by significant enhancements on the kinematic and material descriptions compared to previous macroscale macromodels. According to the proposed macroscale description, flexural cracking, shear sliding, and toe crushing are represented through damage concentrated along the element boundaries. The in-plane shear cracking and out-of-plane diagonal cracking modes are described in a phenomenological way by nonlinear springs associated with the respective deformation modes of the inner block. The constitutive behavior of the springs is coupled with the mean normal stress in the boundary interfaces, which provides the level of confinement. The enhanced kinematics, as well as the detailed cohesive-frictional constitutive law along the boundaries, allow the accurate representation of the in-plane and out-of-plane nonlinear behavior of URM components and the reliable prediction of failure modes with a reduced number of elements. This was shown in numerical examples, including comparisons against the results of physical tests on masonry walls under in-plane and out-of-plane cyclic loading.

References Anthoine, A., 1995. Derivation of the in-plane elastic characteristics of masonry through homogenization theory. Int. J. Solids Struct. 32 (2), 137
163. Anthoine, A., 1997. Homogenization of periodic masonry: plane stress, generalized plane strain or 3D modeling? Int. J. Numer. Methods Biomed. Eng. 13 (5), 319
326. Anthoine, A., Magonette, G., Magenes, G., 1995. Shear-compression testing and analysis of brick masonry walls. In: Proceedings of the 10th European Conference on Earthquake Engineering, pp. 1657
1662. Asteris, P.G., Sarhosis, V., Mohebkhah, A., Plevris, V., Papaloizou, L., Komodromos, P., 2015. Numerical modeling of historic masonry structures. In: Asteris, P., Plevris, V. (Eds.), Handbook of Research on Seismic Assessment and Rehabilitation of Historic Structures. IGI Global, Hershey, PA, pp. 213
256. Berto, L., Saetta, A., Scotta, R., Vitaliani, R., 2002. An orthotropic damage model for masonry structures. Int. J. Numer. Methods Eng. 55 (2), 127
157. Bertolesi, E., Milani, G., Lourenc¸o, P.B., 2016. Implementation and validation of a total displacement non-linear homogenization approach for in-plane loaded masonry. Comput. Struct. 176, 13
33. Bruneau, M., 1994. State-of-art report on seismic performance of unreinforced masonry buildings. J. Struct. Eng. ASCE 120 (1), 230
251. Calio`, I., Marletta, M., Panto`, B., 2012. A new discrete element model for the evaluation of the seismic behavior of unreinforced masonry buildings. Eng. Struct. 40, 327
338. Casolo, S., Pena, F., 2007. Rigid element model for in-plane dynamics of masonry walls considering hysteretic behavior and damage. Earthq. Eng. Struct. Dyn. 36 (8), 1029
1048. Cervera, M., Pela`, L., Clemente, R., Roca, P., 2010. A crack-tracking technique for localized damage in quasibrittle materials. Eng. Fract. Mech. 77 (13), 2431
2450.

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Griffith, M.C., Vaculik, J., Lam, N., Wilson, J., Lumantarna, E., 2007. Cyclic testing of unreinforced masonry walls in two-way bending. Earthq. Eng. Struct. Dyn. 36 (6), 801
821. Izzuddin, B.A., 1990. Nonlinear Dynamic Analysis of Framed Structures. Imperial College London (University of London). Lagomarsino, S., Penna, A., Galasco, A., Cattari, S., 2013. TREMURI program: an equivalent frame model for the seismic analysis of URM structures. Eng. Struct. 56, 1787
1799. Lee, J., Fenves, G.L., 1998. Plastic-damage model for cyclic loading of concrete structures. J. Struct. Mech. 124 (8), 892
900. Lopez, J., Oller, S., Onate, E., Lubliner, J., 1999. A homogeneous constitutive model for masonry. Int. J. Numer. Methods Eng. 46 (10), 1651
1671. Lourenc¸o, P.B., 2000. Anisotropic softening model for masonry plates and shells. J. Struct. Eng. 126 (9), 1008
1016. Lourenc¸o, P.J.B.B., 1997. Computational Strategies for Masonry Structures. Technische Universiteit Delft. Lourenc¸o, P.B., Milani, G., Tralli, A., Zucchini, A., 2007. Analysis of masonry structures: review of and recent trends in homogenization techniques. Can. J. Civil Eng. 34 (11), 1443
1457. Lourenc¸o, P.B., Pina-Henriques, J., 2006. Validation of analytical and continuum numerical methods for estimating the compressive strength of masonry. Comput. Struct. 84 (29
30), 1977
1989. Lourenc¸o, P.B., Rots, J.G., 1997. Multisurface interface model for analysis of masonry structures. J. Eng. Mech. 123 (7), 660
668. Lourenc¸o, P.B., Rots, J.G., Blaauwendraad, J., 1998. Continuum model for masonry: parameter estimation and validation. J. Struct. Eng. 124 (6), 642
652. Macorini, L., Izzuddin, B., 2011. A non-linear interface element for 3D mesoscale analysis of brick-masonry structures. Int. J. Numer. Methods Eng. 85 (12), 1584
1608. Magenes, G., Calvi, G.M., 1997. In-plane seismic response of brick masonry walls. Earthq. Eng. Struct. Dyn. 26 (11), 1091
1112. Magenes, G., Della Fontana, A., 1998. Simplified non-linear seismic analysis of masonry buildings. Proc. Br. Masonry Soc. 8, 190
195. Mann, W., Muller, H., 1982. Failure of shear-stressed masonry. An enlarged theory, tests and application to shear walls. Proc. Br. Ceram. Soc. 30, 223
235. Massart, T., Peerlings, R., Geers, M., 2007. An enhanced multi-scale approach for masonry wall computations with localization of damage. Int. J. Numer. Methods Eng. 69 (5), 1022
1059. Mercatoris, B., Massart, T., 2011. A coupled two-scale computational scheme for the failure of periodic quasi-brittle thin planar shells and its application to masonry. Int. J. Numer. Methods Eng. 85 (9), 1177
1206. Minga, E., 2018. 3D Meso- and Macro-Scale Models for Nonlinear Analysis of Masonry Systems. Imperial College London. Minga, E., Macorini, L., Izzuddin, B.A., 2018a. A 3D mesoscale damage-plasticity approach for masonry structures under cyclic loading. Meccanica 53, 1591
1611. Minga, E., Macorini, L., Izzuddin, B.A., Calio`, I., 2018b. Macroelement representation for URM components under cyclic loading. In: 16th European Conference on Earthquake Engineering, Thessaloniki, Greece. Mistler, M., Anthoine, A., Butenweg, C., 2007. In-plane and out-of-plane homogenisation of masonry. Comput. Struct. 85 (17
18), 1321
1330. Pande, G., Liang, J., Middleton, J., 1989. Equivalent elastic moduli for brick masonry. Comput. Geotech. 8 (3), 243
265.

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Panto`, B., Cannizzaro, F., Caddemi, S., Calio`, I., 2016. 3D macro-element modeling approach for seismic assessment of historical masonry churches. Adv. Eng. Soft. 97, 40
59. Panto`, B., Cannizzaro, F., Calio, I., Lourenc¸o, P., 2017. Numerical and experimental validation of a 3D macro-model for the in-plane and out-of-plane behavior of unreinforced masonry walls. Int. J. Architect. Herit. 11 (7), 946
964. Papa, E., 1996. A unilateral damage model for masonry based on a homogenisation procedure. Mech. Cohes. Frict. Mater. 1 (4), 349
366. Pegon, P., Anthoine, A., 1997. Numerical strategies for solving continuum damage problems with softening: application to the homogenization of masonry. Comput. Struct. 64 (1
4), 623
642. Pela`, L., Cervera, M., Roca, P., 2013. An orthotropic damage model for the analysis of masonry structures. Constr. Build. Mater. 41, 957
967. Penna, A., Lagomarsino, S., Galasco, A., 2014. A nonlinear macroelement model for the seismic analysis of masonry buildings. Earthq. Eng. Struct. Dyn. 43 (2), 159
179. Raka, E., Spacone, E., Sepe, V., Camata, G., 2015. Advanced frame element for seismic analysis of masonry structures: modelformulation and validation. Earthq. Eng. Struct. Dyn. 44 (14), 2489
2506. Rinaldin, G., Amadio, C., Macorini, L., 2016. A macro-model with nonlinear springs for seismic analysis of URM buildings. Earthq. Eng. Struct. Dyn. 45, 2261
2281. Roca, P., Molins, C., Marı`, A.R., 2005. Strength capacity of masonry wall structures by the equivalent frame method. J. Struct. Eng. 131 (10), 1601
1610. Saloustros, S., Pela`, L., Cervera, M., Roca, P., 2017. An enhanced finite element macro-model for the realistic simulation of localized cracks in masonry structures: a large-scale application. Int. J. Architect. Herit. (in press). Shieh-Beygi, B., Pietruszczak, S., 2008. Numerical analysis of structural masonry: mesoscale approach. Comput. Struct. 86, 1958
1973. Silva, L.C., Lourenc¸o, P.B., Milani, G., 2017. Nonlinear discrete homogenized model for out-ofplane loaded masonry walls. J. Struct. Eng. 143 (9), 04017099. Spada, A., Giambanco, G., Rizzo, P., 2009. Damage and plasticity at the interfaces in composite materials and structures. Comput. Methods Appl. Mech. Eng. 198 (49
52), 3884
3901. Vicente, R., Lagomarsino, S., Ferreira, T.M., Cattari, S., Mendes da Silva, J.A.R., 2018. Cultural heritage monuments and historical buildings: conservation works and structural retrofitting. In: Costa, A., Areˆde, A., Varum, H. (Eds.), Strengthening and Retrofitting of Existing Structures. Building Pathology and Rehabilitation, vol. 9. Springer, Singapore.

Chapter 9

Micromodeling A. Rekik1 and F. Lebon2 1

INSA CVL, Univ. Orle´ans, Univ. Tours, Orle´ans, France, 2Aix-Marseille Univ. CNRS Centrale Marseille LMA, Marseille, France

9.1 Introduction Masonry is one of the oldest construction materials and is still commonly used today to build houses or structures because of its strength, solidity, durability, resistance, its elegant appearance, etc. However, masonry, which is not generally thought to be a highly technological material, shows highly complex behavior, due in particular, to the interactions between the components (mortar, bricks) and the anisotropy induced by the direction of the joints, which are a source of weakness. Masonry structures were classically designed on the basis of empirical rules. Modern virtual methods of design have been developed only quite recently. Structures built long ago were extremely stable because they were massive. In modern masonry buildings, the walls are very thick, requiring the stability to be studied from a theoretical point of view, especially when wind or earthquakes are a concern. The strength of the masonry is thus critical and it is necessary to study the solidity of the structure using fine models and numerical simulations as in the case of concrete and steel structures. Other problems such as cracks also require more detailed studies on the design of masonry structures. Mortar joints are usually weaker than masonry units, which explains the existence of planes of weakness along which cracks can propagate. Several models have been developed and presented in the literature for studying and predicting the behavior of masonry structures. Depending on the level of accuracy and simplicity required, either macro- or micromodeling strategies can be used for this purpose. In continuum structural and macromodels, bricks, mortar, and brick
mortar interfaces are smoothed out into a homogeneous continuum, the average properties of which are identified at the level of the constituents, taking their geometric arrangement into account. This approach is applicable when the dimensions of a structure are sufficiently large for the ratio between the

Numerical Modeling of Masonry and Historical Structures. DOI: https://doi.org/10.1016/B978-0-08-102439-3.00009-9 Copyright © 2019 Elsevier Ltd. All rights reserved.

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average stresses and average strains to be acceptable such as the macromodels (classical no-tension models; Di Pasquale, 1992; Lourenco, 1998; Marfia and Sacco, 2005) that have been widely developed in the past. During the last few decades, other models have been developed such as micropolar Cosserat continuum models (Masiani and Trovalusci, 1996; Sulem and Muhlhaus, 1997) as well as applications of the mathematical theories of homogenization to periodic (Anthoine, 1995; Luciano and Sacco, 1997; Ushaksarei and Pietruszczak, 2002) and nonperiodic media (Alpa and Monetto, 1994). To describe the inelastic behavior of structural masonry, some authors have combined homogenization techniques with a continuum damage mechanics approach (Pegon and Anthoine, 1997; Zucchini and Lourenc¸o, 2004; Chengqing and Hong, 2006). Other authors such as Alpa and Monetto (1994) and de Buhan and de Felice (1997) have defined suitably macroscopic yield failure surfaces. Macroapproaches obviously require a preliminary mechanical characterization of the model, based on experimental laboratory or in situ tests (Gabor et al., 2005, 2006). In studies based on microanalysis, two main approaches have been used: the simplified approach and the detailed micromodeling approaches. Simplified methods consist of modeling the bricks, mortar, and interface separately by adopting suitable constitutive laws for each component. This approach gives highly accurate results, especially at a local level. A simplified micromodel is an intermediate approach, where the properties of the mortar and the mortar interface unit are lumped into a common element, while expanded elements are used to model the brick units. Although this model reduces the computational cost of the analysis, some accuracy is obviously lost. Several authors (Lotfi and Shing, 1994; Lourenc¸o and Rots, 1997; Pegon et al., 2001; Pelissou and Lebon, 2009) have established that the interface elements reflect the main interactions occurring between bricks and mortar. Several methods have been presented for modeling the behavior of interfaces with zero thickness and predicting their failure modes. Giambanco and Di Gati (1997), for example, expressed the constitutive law at the interface in terms of contact traction and the relative displacements of the two surfaces interacting at the joint. The fracture of the joint and the subsequent sliding are associated with the interface yield condition. A method based on limit analysis combined with a homogenization technique was recently shown to be a powerful structural analysis tool, giving accurate collapse predictions: de Buhan and de Felice (1997), for example, presented a homogenized model of this kind that can be used for the limit analysis of masonry walls. The units are assumed in this model to be infinitely resistant and the joints are taken to be interfaces with zero thickness having a friction failure surface. In addition, the brittle damage model developed in Luciano and Sacco (1997) and Pelissou and Lebon (2009)

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involves an elementary cell composed of units, mortar, and a finite number of fractures at the interfaces. This chapter summarizes recently developed models based on micromechanics (linear and nonlinear homogenization methods) and the coupling of this approach with structural analysis and/or brittle fracture theory and creep of masonry components to predict local and overall behavior of masonry and also to reproduce creep or prevent collapse of these structures.

9.2 Coupling between homogenization techniques and damage theory 9.2.1

Accounting for damaged brick
mortar interface

Interface models for assessing the safety of civil and historical masonry constructions have attracted considerable attention, since their resistance depends to a large extent on the brick
mortar interfacial properties. In fact, mortar joints are usually less strong than masonry units, which explains the existence of planes of weakness along which cracks can propagate. Several models have been developed and presented in the literature for studying and predicting the behavior of masonry structures. Depending on the level of accuracy and simplicity required, either macro- or micromodeling strategies can be used for this purpose. This section aims to identify the crack-length evolution laws governing a recently proposed constitutive equation (Rekik and Lebon, 2010, 2012), generalized in Raffa et al. (2016, 2017) with a small number parameters for microcracked interfaces of masonry structures. It also aims to study the effect of the masonry structure size and the load type on these identified parameters. Experimental tests (Gabor et al., 2006; Fouchal et al., 2009) on small and large masonry panels have been used to estimate the small number of parameters describing the microcrack evolution law and leading to the best fit between the numerical and experimental tests. In the case of a masonry structure under a compression load, the evaluation of the local numerical fields requires us to add a unilateral contact condition to avoid the overlap between the bricks and the joints constituents. In our first approach and for the sake of simplicity, we do not introduce friction between the brick and mortar units.

9.2.1.1 Effective properties of the brick
mortar lamina Due to the fact that damage occurs mostly at the interface between brick and mortar materials, we assume the existence of an extremely thin layer of material between each brick unit and its mortar joint. The mechanical properties of this layer are obtained by applying an asymptotic limit analysis procedure (Raffa et al., 2017). For this purpose, it is proposed first to obtain the mechanical properties of the 3D material obtained by homogenizing those of

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brick and mortar. Assuming brick and mortar to be isotropic and linear elastic materials, the homogenization of the brick
mortar lamina can be carried out exactly using an analytical homogenization formulation, as described in Rekik and Lebon (2010, 2012). The homogeneous equivalent undamaged material, denoted hereafter by HEMu, is transversally isotropic and characteru ized by the effective compliance tensor S~ written in the form of Eq. (9.1) with respect to the classical Voigt notation. In what follows, exponents h and v correspond to bed and head joints, respectively; and e3 and e1 represent the HEMuh and HEMuv revolution axis, respectively, as shown in Fig. 9.1. 1 0 1 ν~ 013 ν~ 012 C B 2 0 2 0 C B E~ 0 E~ 1 E~ 1 1 C B C B 0 0 C B ~ 1 ν ~ ν 13 C B 2 12 2 0 C B 0 0 ~ ~ ~ C B E E E uh 1 1 1 C ð9:1Þ S~ 5 B C B 0 0 1 C B ν~ ν~ 13 C B 2 130 1 2 0 0 C B E~ 3 E~ 1 E~ 1 0 1 C B G~ 23 C B 1 0 C B ~ G 23 A @ 0 ~ G 12

h For further details about the method of obtaining the components of S~u see Rekik and Lebon (2010).

FIGURE 9.1 Determination of the elastic properties of the third material (a brick/mortar lamina) located at bed (A) and head (B) joints.

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9.2.1.2 Effective properties of the microcracked material HEMc In the previous step, in the case of bed joints, an uncracked homogeneous material HEMuh was defined, based on the known properties of brick and mortar. Now assuming the presence of parallel microcracks to the e1 axis in this material, it is necessary to determine its effective properties. Many studies have dealt with assessing the effective elastic properties of damaged materials with defects of various kinds (holes and/or cracks). The choice of modeling method depends here mainly on the interactions between cracks. For the sake of simplicity, we started to model the degradation of the brick
mortar interface taking only the interactions between microcracks and neglecting the interactions with the matrix of the HEMu material. Moreover, we assume the existence of a small number of rectilinear cracks 21(k) in length. To solve this 2D problem it is proposed to apply the method proposed by Tsukrov and Kachanov (2000) to determine the equivalent properties of the damaged HEMu material. The accuracy of this model, which generally depends on the density of the cracks, is satisfactory up to quite small distances between cracks (distances much smaller than the crack width). Rectilinear cracks are assumed to be located on the plane (e1 , e3 ) in a representative area A 5 L0 e, where L0 is the bed mortar length and e is the thickness of the microcracked HEMu material. In the case of the present 2D problem, the Kachanov model includes a global parameter called the crack density, which is defined by the number and the length of all the cracks given by: 1 X ðkÞ 2 ρ5 ðl Þ ð9:2Þ A k The main result obtained with the Kachanov model is that the average value of the crack opening displacement (COD) vector “b” is colinear with the average stress σ as follows: hbi 5 nUσUB

ð9:3Þ

where n is a vector normal to the crack. The components of the symmetric B second-order tensor depend on those of the uncracked homogeneous HEMu h material, that is, on the components of S~u and on the orientation of the crack with respect to the matrix anisotropy: 8 < Btt 5 Cð1 2 D cosð2φÞÞ B 5 Cð1 1 D cosð2φÞÞ ð9:4Þ : nn Bnt 5 CDðsinð2φÞÞ where l is the length of the half-representative rectilinear microcrack in the HEMu material, as shown in Fig. 9.2.

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PART | II Modeling of unreinforced masonry

FIGURE 9.2 Assessment of the effective properties of the microcracked bed (A) and head (B) joints using the Kachanov model.

FIGURE 9.3 Local crack vectors and the principal axis of the masonry.

We recall that φ is the angle between the vector t tangential to the crack and the principal axis e1 , as illustrated in Fig. 9.3. C and D are scalars that are independent of the representative microcrack half-length parameter l, and are given by: 8 11 qffiffiffiffiffiffiffi qffiffiffiffiffiffiffi 0 > 2 > uh uh > ~ 1 1 E~ 3 uh > E π > ~ B C ν > 2 > qffiffiffiffiffiffiffiffiffiffiffiffiffiffi @ 1uh 22 13uh 1 qffiffiffiffiffiffiffiffiffiffiffiffi C5 A > > uh uh ~ 13 ~1 uh uh 4 < E G ~ ~ ~ ~ E E1 E3 1 E3 ð9:5Þ qffiffiffiffiffiffiffi qffiffiffiffiffiffiffi > uh uh > > E~ 1 1 E~ 3 > > > D 5 qffiffiffiffiffiffiffi qffiffiffiffiffiffiffi > > > uh uh : E~ 1 1 E~ 3 uh uh ~ uh where E~ 1 ; E~ 3 ; ν~ uh 13 ; G13 are the elastic engineering constants of the crackfree HEMuh material. On the principal axes, the effective engineering moduli

Micromodeling Chapter | 9

301

of HEMc denoting the homogeneous material equivalent to the damaged HEMu are given by: 8 c E~ 1 > > > 1u 5 u > 2 2 > ~ E1 1 1 2ρ sin φðBtt cos φ 1 Bnn sin2 φ 2 Bnt sinð2φÞÞE~ 1 > > > > c > > E~ 3 1 > > > u5 u > 2 2 < E~ 1 1 2ρ cos φðBtt sin φ 1 Bnn cos2 φ 1 Bnt sinð2φÞÞE~ 3 3 ð9:6Þ c > G~ 13 1 > > > u 5 u > > 1 1 ρðBnn sin2 ð2φÞ 1 Btt cos2 ð2φÞ 2 Bnt sinð4φÞÞG~ 13 G~ 13 > > > > > ν~ c13 ν~ u13 > > > 5 c u > : E~ E~ 1

1

In the bed masonry joints, the cracks are assumed to run parallel to the principal axis e1 , that is, with the crack orientation φ 5 0. Under plane stress c conditions, the components of the compliance tensor S~ in the (e1, e3) plane read: 1 0 1 ν~ uh 13 2 uh 0 C B uh ~1 C B E~ 1 E C B 0 1 C B C B ~ uh 1 ν C B ch c @ C ð9:7Þ 2 13 1 2ρBnn ð0ÞA 0 S~ 5 S~ ð0Þ 5 B uh uh C B E~ ~ E3 C B 1 B 0 1C C B C B 1 @ A @ 0 0 1 ρBtt ð0Þ A uh G13 where

8 < Btt ð0Þ 5 Cð1 2 DÞ B ð0Þ 5 Cð1 1 DÞ : nn Btn ð0Þ 5 0

ð9:8Þ

As shown in relations (9.7), the effective properties of the cracked lamina h are sensitive to the effective properties of the uncracked lamina S~ and to the u ch

representative crack length. Inverting the compliance tensor S~ gives the ch corresponding stiffness tensor C~ associated with the properties of HEMch .

9.2.1.3 Interface constitutive law It has been assumed that cracks exist only in the plane (e1, e3) parallel to either the principal axis e1 (in the case of bed joints) or to the e3 vector (in the case of head joints). We have therefore focused only on the pair of

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PART | II Modeling of unreinforced masonry

 ch   cv  ch cv components C~ 3333 ; C~ 3131 and C~ 1111 ; C~ 1313 corresponding to the bed and head interface stiffness, respectively. Now focusing on the head interface ch stiffnesses, the inversion of the compliance tensor S~ leads to expressing the ch ch components (C~ 3333 ; C~ 3131 Þ as a function of the microcrack density parameter ρ and the angle φ is null: ch C~ 3333 5

α0h 33

αh33 1 ρβ h33 2 0h 1 ρβ 0h 33 1 ρ γ 33

1 αh13 1 ρβ h13 ch and C~ 1313 5 0h 2 0h 2 α13 1 ρβ 0h 13 1 ρ γ 13

ð9:9Þ

0h where αhij ; β hij ; α0h ij ; β ij are scalars that are independent of the crack density parameter ρ. The normal and tangential stiffness of the bed interfaces are determined as follows:

CNh 5

ch C~ 3333 ðeyields0Þ and e

CTh 5

ch C~ 3131 ðeyields0Þ e

ð9:10Þ

Replacing ρ by the term l2 =eLh0 in expressions (9.9), we obtain: CNh 5

β h33 Lh0 Lh0 5 2 2Bnn ð0Þl2 γ 0h 33 l

and CTh 5

β h13 Lh0 Lh0 5 2 4Btt ð0Þl2 γ 0h 13 l

ð9:11Þ

As the components Bnn and Btt depend on the half crack length l (see relation (9.2)), the expressions for the interface stiffness CN and CT at the bed position read: CNh 5

Lh0 2Cð1 1 DÞl2

and

CTh 5

Lh0 4Cð1 2 DÞl2

where dl $ 0

ð9:12Þ

dl is the increment of crack length, assumed to be positive during the shear loading. It is worth noting that the properties of the material HEMcv , which is transversally isotropic with e1 as the revolution axis, are deduced from those of the material HEMch by making a simple 90 degree rotation. Therefore, the normal and tangential stiffness of the head joints read: CNv 5

cv C~ 1111 Lv ðeyields0Þ 5 0h CNh e L0

and CTv 5

cv C~ 1313 Lv ðeyields0Þ 5 0h CTh e L0 ð9:13Þ

where Lh0 is the bed mortar joint length. These defined stiffnesses can be clearly seen to decrease as the crack length increases with respect to the applied load F (or shear stress τ). In addition, they are closely related to the law of microcrack evolution l 5 f ðForτ Þ; which will be identified in the case of masonry structures of various sizes under loads of various kinds in the following section. The crack-length evolution is assumed to show a similar tendency at the head and bed interfaces.

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9.2.1.4 Estimation of the representative law of microcrack evolution based on experimental tests In view of Eq. (9.10), one of the most important steps consists of defining, testing, and validating a law governing the crack-length evolution. An alternative simpler solution consists of defining directly by choosing crack lengths at several points on experimental diagrams. Hereafter, it is necessary to distinguish between the case of quasibrittle failures, with which the “stress
strain” diagram shows a “plateau” in the postpeak load part (in the case of nonconfined masonry) and those showing a softening and sliding parts after the peak in the load. In fact, numerical tests carried out on nonconfined (Rekik and Lebon, 2010) and confined masonry panels have shown that the laws of crack-length evolution available so far in the case of nonconfined masonry are not able to reproduce the softening and sliding parts seen in the case of the confined masonries. Hereafter, for numerical computations, the geometry and boundary conditions are given in Fig. 9.4 (with the confining pressure σ) for the case of seven bricks. Table 9.1 lists the mechanical properties of the bricks and mortar constituting the prism (Gabor et al., 2006). Because of the symmetry of the prism problem, only half-structures will be used in the computations. In what follows, bricks and mortar joints will be modeled using Q4 quadrangular finite elements.

FIGURE 9.4 Initial geometrical configuration and loading conditions imposed on a small confined wall (A and B); deformation of the small wall in a shear test (C).

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PART | II Modeling of unreinforced masonry

TABLE 9.1 Mechanical properties of the prism and wall masonry constituents. Young’s modulus (MPa) of full brick

12,800

Poisson’s ratio of full brick

0.2

Young’s modulus (MPa) of mortar

4000

Poisson’s ratio of mortar

0.2

Source: From Gabor, A., Ferrier, E., Jacquelin, E., Hamelin, P., 2005. Analysis of the inplane shear behavior of FRP reinforced hollow brick masonry walls. Struct. Eng. Mech. 19, 237
260; Gabor, A., Bennani, A., Jacquelin, E., Lebon, F., 2006. Modelling approaches of the in-plane shear behaviour of unreinforced and FRP strengthened masonry panels. Comput. Struct., 74, 277
288.

FIGURE 9.5 Effect of the confining pressure: Experimental and numerical “shear stress
displacement” diagrams of a small confined wall under shear loading conditions.

Simulation of a confined medium-sized masonry panel under shear loading conditions In the case of confined masonry panels subjected to shear loads with various confining stresses (σ 5 0:4; 0:6; 0:8, and 1.2 MPa), the joint response differs from that observed under nonconfined conditions, as shown in Fig. 9.5. Experimental results are plotted in dashed lines. In the “stress
displacement” diagrams, the distinction will be made between three stresses, τ c , τ u , and τ cr (see Fig. 9.6; Rekik and Lebon, 2010, available for nonconfined masonry structures), where τ cr denotes the end of the softening phase.

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FIGURE 9.6 Function describing the evolution of the crack half-length with respect to the shear stress applied: the case of a triplet of hollow bricks (Rekik and Lebon, 2010), a nonconfined seven brick structure and that of a wall.

Additional confining pressure was found to increase the cohesion between mortar and hollow bricks and thus to induce the occurrence of softening and sliding processes after the peak load has been reached. These softening and sliding parts cannot be modeled in the framework of a cracklength evolution law similar to that used for a nonconfined masonry panel (Figs. 9.5 and 9.6; Rekik and Lebon, 2010). In this case, a nonlinear piecewise increasing representative crack length from the peak load up to failure gives better predictions. To obtain a better fit between the numerical and experimental data, the crack lengths were identified at several points on the experimental diagram. At various confining stresses, the changes in the crack lengths given in Fig. 9.7 show that it is necessary to include a bilinear or trilinear function in the postpeak load part to account for the set of the softening and sliding parts. As shown in Fig. 9.7, these functions describe the increase in the crack length, while the shear stress decreases, in line with the properties of cohesive cracks (Park et al., 2008; Chaimoon and Attard, 2009). In the identified functions l 5 f ðτÞ corresponding to confining stresses σ 5 0:8and1:2 MPa, note the existence of a first positive slope describing the increase in the crack length with the increase in the shear stress occurring before the peak of load is reached. This first linear evolution of l is not included in the description of the crack-length evolution in the softening and sliding parts given by the “stress
displacement” diagrams. The numerical “stress
displacement” curves corresponding to the cracklength functions depicted in Fig. 9.7 are in line with experimental data as can be seen from Fig. 9.5 with each of the confining stresses. Table 9.2 lists the ultimate crack lengths obtained at the various confining pressures tested. Note that the crack length lu varies slightly with the confining pressure. Its 22 main value is lcp µm. The relative errors er between lu and the u 5 6:46 3 10 cp average value lu do not exceed 11%.

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PART | II Modeling of unreinforced masonry

FIGURE 9.7 Identified crack-length laws giving the best fit between experimental and numerical data on confined small walls under shear loads and various confining pressures.

TABLE 9.2 Identified ultimate representative crack length and the corresponding relative errors obtained on small confined walls under shear loading and different confining pressures. Confining stress, σðMPaÞ

lu ðµmÞ

er ðlu Þð%Þ

0.4

22

6.22 3 10

2.5

0.6

5.98 3 1022

5.0

0.8

22

11.1

22

5.0

1.2

7.63 3 10 5.98 3 10

Fig. 9.8 gives the local shear stress distribution with a 0.4 MPa confined small wall, which shows a local stress concentration at the longest vertical interface v1 , where the decohesion between brick and mortar mainly occurs, as in the experimentally tested specimen (Fig. 9.9). Fig. 9.8 gives the local shear stress distribution with a 0.4 MPa confined small wall, which shows a local stress concentration at the longest vertical interface v1 , where the decohesion between brick and mortar mainly occurs, as in the experimentally tested specimen (Fig. 9.9).

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FIGURE 9.8 Local shear stress snapshot of a confined small wall (σ 5 0.4 MPa) under shear loading conditions at failure (the identified ultimate crack length is l 5 6.22 3 1022 µm).

FIGURE 9.9 Experimental deformation of a small confined wall under shear loading conditions. From Gabor, A., Bennani, A., Jacquelin, E., Lebon, F., 2006. Modelling approaches of the in-plane shear behavior of unreinforced and FRP strengthened masonry panels. Comput. Struct. 74, 277
288.

Discussion of the results Table 9.3 recapitulates the identified ultimate crack lengths giving a best fit between the numerical and experimental results at the failure of the wall with and without the unilateral contact condition. The relative difference

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TABLE 9.3 Identified ultimate representative crack length and the corresponding relative errors obtained on a diagonally compressed wall with and without a unilateral contact condition. lu ðµmÞ

Unilateral contact condition With

er ðlu Þð%Þ 22

3

22

4

6.46 3 10

Without

7.18 3 10

TABLE 9.4 Relative errors in the identified (average) ultimate representative crack lengths and stiffnesses in the case of masonries of various sizes under shear loading or diagonal compression conditions. er ð lu Þ ð % Þ

er ðCN Þð%Þ

22

5.8

1 17.4

Confined prism

22

6.46 3 10

0.6

2 2.0

Wall (with/without u.c.c.)

6.76 3 1022

3.7

2 11.0

lu or average of lu ðµmÞ Nonconfined prism

5.86 3 10

between these values is taken to be negligible (about 7%). It was therefore proposed to calculate the mean ultimate crack length from the values available on wall interfaces at failure. The relative errors er between the identified crack lengths lu and the mean value lwu 5 6.64 3 1022 µm obtained in the case of the wall were negligible (below 4%). Table 9.4 gives the identified (average) ultimate crack lengths obtained with masonry structures of various sizes under shear loads (with and without confining pressure) or diagonal compression loads (with and without the unilateral contact condition). Due to the negligible differences existing between these values, we will assume that failure occurs when the crack length reaches the average value of this set of identified crack lengths, that is, lu 5 6.4 3 1022 µm. In the case of masonry composed of constituents with the properties given in Table 9.1, comparisons between the stiffnesses of the interfaces obtained with masonry of various sizes (see the values for the stiffnesses) give a mean stiffness value per mm, with upper and lower bounds for the properties thus identified: 0 1 0 1 8 > > av N < 11 @ N A 11 A CN 5 2:52 3 10 and CTav 5 1:42 3 10 @ mm3 mm3 ð9:14Þ > > : ð1 2 11%ÞCNav # CN # ð1 1 17%ÞCNav

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The discrepancies between the individual interface stiffnesses and the mean value obtained (maximum of 17%) can be explained by the fact that masonry mortar joints are manmade materials.

9.2.1.5 Conclusions The identification of the crack-length evolution law for masonry structures with various sizes subjected to shear and diagonal compression (Rekik and Lebon, 2010) loads showed the ability of a recently presented model (Rekik and Lebon, 2010, 2012) to provide estimations for the stiffness of masonry interfaces. At failure, the discrepancies between the identified crack lengths were almost negligible (below 6%). The interface stiffnesses are inversely proportional to the square of the ultimate crack length lu ; which explains the maximum discrepancy of about 17%. An experimental campaign in which the joint mortar is consistently prepared and laid (constant thickness, regular rate of cover between brick and mortar) will help to reduce the discrepancies between the stiffnesses of interfaces at failure. To obtain a good fit between experimental and numerical data on loaded nonconfined masonry structures in which the “stress
strain” diagrams show the occurrence of a “plateau” after the peak load (or stress), it is necessary to adopt a linearly increasing crack length up to the failure, corresponding to the ultimate load applied. The number of parameters is reduced to 4 in this case: lc , lu , c, and u. In the case of confined masonry structures under shear loading conditions, the present model gives good agreement with the experimental data, thanks to the introduction of a bilinear or trilinear function describing the increase in the crack length with the decrease in the shear stress in the postpeak part (softening and sliding parts). The number of parameters increases in this case to 6 or 8. In the postpeak part of the “stress
displacement” diagram, a single linear function describing the increase in the crack length with the decrease in the shear stress does not suffice to reproduce accurately the softening and sliding parts. 9.2.2

Accounting for creep of masonry components

The recent collapse of famous historical constructions (e.g., middle-age masonry buildings) was mainly attributed to the creep behavior of the masonry (Binda et al., 1992; Shrive et al., 1997; Papa and Taliercio, 2005). Recent experimental findings have shown that the accumulation of creepinduced damage in time under sustained loads is a possible reason for this collapse. Thus, in order to increase the performance and safety of refractory linings and ancient masonry buildings subjected to heavy sustained loading, the development of theoretical models of creep evolution and creep-induced damage is of crucial importance. In Choi et al. (2007), an experimental study was carried out to investigate the creep of masonry. Different rheological

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PART | II Modeling of unreinforced masonry

models are considered to assess their ability to predict the creep of masonry. Accordingly, it was found that the Modified Maxwell (MM) model is the most accurate one. On the other hand, these materials (e.g., refractory linings, masonries) are generally heterogeneous and composed of bricks and mortar joints. Therefore, the evaluation of their response requires homogenization approaches. In this connection, the so-called hereditary approaches based on Stieltjes convolution in the time domain has been used by many authors for modeling linear nonaging viscoelastic composites. Two steps are performed. First, through the use of the Laplace
Carson (LC) transform with the correspondence principle (Mandel, 1966), the time-dependent constitutive relations of the local phase properties are converted into symbolic elastic-like relations in the LC domain. Then, the symbolic macroscopic elastic moduli of the fictitious elastic material are derived by using classical elastic micromechanical schemes such as the self-consistent (SC) scheme (Hashin, 1969; Rougier et al., 1994), the Mori
Tanaka estimate (Li and Weng, 1994; Pichler and Lackner, 2009), or the Hashin
Shtrikman bounds (De Botton and Tevet-Deree, 2004). Finally, the overall properties of the viscoelastic composites in the physical domain are obtained by LC inversion, which can be performed either analytically or numerically. However, apart from some particular cases (Rougier et al., 1994), the inversion of the LC transform is usually performed numerically (see, e.g., the collocation method; Schapery, 1962). Moreover, the analytical method based on the Bromwich integral defined in the complex plane as shown in Beurthey and Zaoui (2000) leads most of the time to integral equations over the whole loading path even if the different phases of the heterogeneous composite exhibit limited memory effects. This last point makes difficult direct extensions to more general situations (e.g., thermomechanical loading, aging viscoelasticity). Moreover, these methods require the complete past history of stress and strain. To overcome these limitations, a number of theories have been proposed in the past aiming to formulate incremental constitutive equations for the linear viscoelastic behavior. Among them, researchers Dubois et al. (1999), Kim and Sing Lee (2007), and Chazal and Moutu Pitti (2009) proposed the incremental formulation and constitutive equations in the finite element (FE) context. In fracture mechanics of viscoelastic materials, Dubois et al. (2002) and Nguyen et al. (2010) applied the incremental formulation in order to evaluate the crack growth process in wood and concrete, respectively. Concerning combined damage and creep effects for masonry within the framework of homogenization, it is worth noting that in the literature there are few works devoted to these studies. For instance, Brooks (1990) obtained the creep coefficients of brickwork according to the properties of the individual constituents. Cecchi and Tralli (2012) adopted an asymptotic homogenization procedure for the derivation of the creep behavior of uncracked periodic masonry cell with joints of finite dimensions. For uncracked masonry, Cecchi and Taliercio (2013) compared predictions given

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by a simplified analytical model and a more accurate FE model, both based on homogenization procedures. Nguyen et al. (2011) derived the effective behavior of microcracked linear viscoelastic concrete obeying the Burgers model by performing a combination of Griffith’s theory (Huy Duong, 1978) and the Eshelby-based homogenization scheme (Bornert and Suquet, 2001; Deude´ et al., 2002). This model does not rest on a series expansion such as the widely used Prony
Dirichlet series or the collocation method and its extensions (the multidata method (Cost and Becker, 1970) or the optimized collocation method (Rekik and Brenner, 2011)) for the required temporal functions. Indeed, as the uncracked concrete, the microcracked concrete was assumed to obey the Burgers model. The FE homogenization method classically used for uncracked elastic or viscoelastic masonries is extended here to microcracked viscoelastic masonry.

9.2.2.1 Main objective and hypothesis The objective of this section is to evaluate at each time t the effective and local behavior of masonries exhibiting nonlinear behaviors, mainly viscoelastic at short and/or long times especially when subjected to severe or longterm loading such as historical monuments or refractory masonry linings working under high temperatures. For the sake of simplicity, it can be assumed that only the mortar is a microcracked viscoelastic material (Luciano and Sacco, 1997; Sacco, 2009). Its behavior (at the uncracked state) obeys the MM rheological model. Blocks or bricks are assumed to be uncracked and to have either rigid or elastic isotropic behavior. In the mortar, the cracks are assumed to be penny-shaped and to have an isotropic distribution. The proposed approach is based on three main steps. First, the homogenization technique is applied in order to assess the effective behavior of the nonaging microcracked mortar. The results of brittle fracture mechanics—Griffith’s theory—could be useful if we move from the real temporal space to the symbolic one due to the LC transform. In the latter space, the apparent behavior of the mortar is linear elastic. This procedure allows the use of expressions available in the literature for the displacement’s jump induced by the crack (Nguyen et al., 2011). Assuming again that the displacement jump field depends linearly on the macroscopic stress, it is possible to define an effective linear behavior for the microcracked mortar in the symbolic space. To determine the global behavior in the real space time, it is possible to apply the inverse of the LC transform in some simple cases. It is then interesting to approach in the symbolic space, at least in short and long terms, the symbolic effective stiffness (or compliance) by an existing rheological model. For example, if the uncracked mortar behaves as the MM model, we can try to approach the symbolic effective behavior of the corresponding microcracked mortar by the same model. After validation of this approximation at short and long terms, the inversion of the apparent effective

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PART | II Modeling of unreinforced masonry

FIGURE 9.10 Main steps of the proposed FE model: The first one (s1) relies on the coupling between Griffith’s brittle fracture theory and stress-based dilute homogenization scheme defining the homogeneous material HEM-1 (C) equivalent to the microcracked linear nonaging viscoelastic mortar (B)-(i) joints present in the periodic masonry cell (A). At each time and for every crack density dc, the second step (s2) provides the effective stiffness of the homogeneous material HEM-2 (E) equivalent to the masonry’s periodic cell (D). Here, the rheology of the mortar with penny-shaped microcracks follows the Modified Maxwell (B)-(ii) model.

stiffness will be straightforward. Therefore, the effective creep behavior of the microcracked viscoelastic mortar could be expressed in the real space time. This first step permits us to determine fast and easily temporal bulk and shear moduli of mortar as explicit functions of the crack density parameter (Budiansky and O’Connell, 1976; Dormieux et al., 2006). For the proposed model in this section, the second step relies either on FE homogenization of the periodic masonry cell (see step s2) in Fig. 9.10 when considering the FE “direct” method. Basic steps followed by the proposed FE model are summarized in Fig. 9.10.

9.2.2.2 Creep model for microcracked mortar (step 1) The rate-dependent mechanical behavior of mortar is often approximated by a linear viscoelastic model (Choi et al., 2007; Ignoul et al., 2007). For the sake of simplicity, only nonaging formulation will be considered in this work. The practical interest of this simple formulation is that it allows us to

Micromodeling Chapter | 9

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transform a time-dependent boundary value problem into a linear elastic one using the well-known correspondence theorem based on the LC transform. Among the simplest formulations used to model the nonaging linear viscoelastic mortar’s behavior, it is possible to quote the Ross, Feng, Burgers, and MM models (Choi et al., 2007; Cecchi and Tralli, 2012) mainly based on connections in parallel and/or in series of Maxwell (M) and Kelvin
Voigt (KV) parts. Each element (spring and dashpot) of the M or KV model is characterized by an isotropic fourth-order tensor related to its elasticity or viscosity: e e v CKV 5 3kKV J 1 2μeKV K; CKV 5 ηsKV J 1 ηdKV K e e 5 3kM J 1 2μeM K; CM

v CM 5 ηsM J 1 ηdM K

ð9:15Þ

where kα and μα (α 5 KV or M) denote the bulk and shear moduli and ηsα and ηdα represent the bulk and shear viscosities, respectively. The tensors J and K 5 I
J are the usual projectors on the subspaces of purely spherical or deviatoric second-order tensors, and i and I are second- and fourth-order identity tensors. In the following, only the MM model is considered since it has been demonstrated in Choi et al. (2007) and Rekik et al. (2016) that this rheological model is relevant at short and long terms for the masonry. The constitutive law of the MM’s model (see Fig. 9.10B-(ii)) is given by: SvM σ 1 SeM :σ 5 SvM CRe ε 1 ðI 1 SeM CRe Þ:ε

ð9:16Þ

where for isotropic mortar material, the elastic and viscous compliances of the Maxwell part are given respectively by: SeM 5

1 1 1 1 v e J 1 2μe KandSM 5 ηs J 1 d K 3kM η M M M

ð9:17Þ

The elastic stiffness of the spring reads CRe 5 3kRe J 1 2μeR K. The LC transform applied to the behavior law (9.16) leads to: ðSvM 1 pSeM Þσ 5 ðSvM CRe 1 pðI 1 SeM CRe ÞÞε

ð9:18Þ

and allows the definition of the following symbolic MM elastic compliance:   21 v  SMM 5 SvM CRe 1p I1SeM CRe ððSM 1 pSeM Þ ð9:19Þ RecallÐ that the LC transform of a temporal function f(t) is given by N FðpÞ 5 p 0 e2pt f ðtÞdt. p is the variable that replaces time t in the symbolic LC space. Assuming the isotropy of the mortar behavior, the symbolic compliance (9.19) reads: 

SMM 5

1 1 J1 K 3ks 2μs

ð9:20Þ

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PART | II Modeling of unreinforced masonry

The associated apparent creep function is then given by: 

JMM 5

1 1 1 1 1 1 5  1  5 0 11 0  pkM ηsM EMM 9ks 3μs pμM ηdM B C B C 3 2 C 9@ k R 1 3B μR 1 pηsM A d A @ pη M kM 1 2 μM 1 2 ð9:21Þ

The analytical direct inversion of (9.21) leads to the MM real creep function: JMM ðtÞ 5

1 1 kM μM s d e2t=τ MM 2 e2t=τ MM 1 2 9kR 3μR 9kR ðkR 1 kM Þ 3μR ðμR 1 μM Þ ð9:22Þ

with the characteristic times τ sMM 5 ηsM ðkR 1 kM Þ=3kR kM and τ dMM 5 ηdM ðμR 1 μM Þ=2μR μM for the spherical and deviatoric parts of the MM viscous behavior, respectively.

9.2.2.3 Microcracked mortar: Modified Maxwell model parameters This section provides elastic and viscous coefficients for a microcracked mortar following the MM rheological model. The identification procedure of these parameters, which represents step 1 of the proposed FE model, is detailed in Rekik et al. (2016). kR ðdc Þ 5

kR ; ð1 1 dc Q00 Þ

μR ðdc Þ 5

μR 1 1 dc M00

e kM ðdc Þ 5

ðkM 1 kR Þ kR 2 ; ð1 1 dc QN Þ ð1 1 dc Q00 Þ 0

μM ðdc Þ 5

μM 1 μR μR 2 1 1 dc M0N 1 1 dc M00

ηsM ðdc Þ 5

ðηsM 1 dc ðηsM Q00 2 3kRe Q10 ÞÞ ; ð11dc Q00 Þ2

ηdM ðdc Þ 5

ηdM 1 dc ðηdM M00 2 3μR M01 Þ ð11dc M00 Þ2 ð9:23Þ

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where Q00 5

4kRs ð3kR 1 4μR Þ ; 3μR ð3kR 1 μR Þ

Q10 5

2 ð2ηsM μR 2 3kR ηdM Þð9kR2 1 4μ2R 1 6kR μR Þ 9 3μ2R ð3kR 1μR Þ2

M00 5

16 ð3kR 1 4μR Þð9kR 1 4μR Þ ; 45 ð3kR 1 μR Þð3kR 1 2μR Þ

8 ð3kR ηdM 2 2ηsM μR Þð63kR2 1 16μ2R 1 60kR μR Þ 45 3μ2R ð3kR 1μR Þ2   4 1 3 N Q0 5 ðkM 1 kR Þ 1 3 μM 1 μR 3ðkM 1 kR Þ 1 ðμM 1 μR Þ   16 9ðkM 1 kR Þ 6ðkM 1 kR Þ N ðkM 1 kR Þ 2 M0 5 45 3ðkM 1 kR Þ 1 ðμM 1 μR Þ 3ðkM 1 kR Þ 1 2ðμM 1 μR Þ ð9:24Þ M01 5

The approximate creep function of a microcracked mortar matrix that follows the MM model reads:   2τ s t ðd Þ 1 kM ðdc Þ app c 12 e MM JMM ðt; dc Þ 5 k R ð dc Þ ðkR ðdc Þ 1 kM ðdc ÞÞ ! ð9:25Þ 2d t 1 μM ðdc Þ τ ðdc Þ MM   1 12 e 3μR ðdc Þ μR ðdc Þ 1 μM ðdc Þ where here the characteristic times of the spherical and deviatoric parts of the MM model are, respectively: τ sMM ðdc Þ 5 ηsM ðdc ÞðkR ðdc Þ 1 kM ðdc ÞÞ= 3kR ðdc ÞkM ðdc Þ and τ dMM ðdc Þ 5 ηdM ðdc ÞðμR ðdc Þ 1 μM ðdc ÞÞ=2μR ðdc ÞμM ðdc Þ.

9.2.2.4 Principle of the finite element homogenization of a microcracked viscoelastic masonry periodic cell (step 2) Instead of differentiating the mortar’s constitutive law as it can be done when considering an incremental homogenization approach (Nguyen et al., 2010), it is easier and more practical to consider the approximate mortar’s creep function (9.25) identified at the short and long terms, which is an explicit function of time and crack density parameter. Therefore, there is no prestress in the considered viscoelastic mortar. At each time t, the behavior of the viscoelastic phase r can be considered to be “purely elastic” with a

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PART | II Modeling of unreinforced masonry

FIGURE 9.11 Boundary and symmetry conditions for the considered quarter cell subjected to axial normal (A) or tangential (B) compression or shear (C) loadings.

constant Poisson’s ratio and Young’s modulus Er ðt; dc Þ 5 1=J r ðt; dc Þ if damaged or Er ðtÞ 5 1=J r ðtÞ otherwise. When applying a constant macroscopic stress and assuming that the per phase localization tensor Ar is timeindependent following the hypothesis of Deude´ et al. (2002) then the average strain εr per phase r and the masonry overall behavior reduce, respectively, to εr 5 Ar :ε and σ 5 C~ : ε, where the overall tangent stiffness is given by C~ 5 , C : A . and the average strain localization over the periodic cell reads , A . 5 I. It is then important to determine components of the localization strain tensor Arijkl . Since regular masonry presents periodic microstructure, it is possible to consider only a periodic cell as shown in Fig. 9.11A. Moreover, as the periodic cell presents two axes of symmetry, normal and tangential directions along the unit vectors n and t, respectively, only its quarter (see Fig. 9.11B) will be retained for computation. To assess the effective “elastic engineering constants,” it is proposed to subject the unit cell to three types of loadings: axial compression along n, axial compression along t, and shear loading, as shown in Fig. 9.10. In this case, strain localization components Arijkl are given by the following equations: εrxx 5 Arxxyy εyy ;

εryy 5 Aryyyy εyy

for

ε 5 εyy ey  ey

εrxx 5 Arxxxx εxx ; εryy 5 Aryyxx εxx for ε 5 εxx ex  ex εrxy 5 2Arxyxy εxy ; for ε 5 εxy ðex  ey 1 ey  ex Þ

ð9:26Þ

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Note that the localization strain tensor Ar is assumed to be orthotropic. Since the symmetry of the Cauchy strain tensor both in the anisotropic and isotropic spaces is required, it follows that Arijkl5 Arjikl5 Arjilk (minor symmetry). The major symmetry of Ar is also necessary Arjikl5 Arklji . It follows that only the componentsAxxxx ; Ayyyy , Axyxy 5 Axyyx and Ayxxy 5 Ayxyx are not null. According to the classical Voigt notation, the constitutive behavior law ~ of the unit cell reads: σ 5 C:ε 0 1 0 10 1 σyy εyy 0 C~ nnnn C~ nntt @ σxx A 5 @ C~ nntt C~ tttt ð9:27Þ 0 A@ εxx A ~ σxy 2εxy 0 0 Cntnt where σ 5 hσiV is the overall applied stress on the periodic cell. The software Cast3M has been used to provide local mechanical fields and mainly average mechanical fields such as strain εr , stress σr over each P phase r (r 5 b for bricks, m for mortar), and macroscopic strain ε 5 r 5 m;b f r εr calculated in order to deduce components of the effective tangent stiffness C~ (Eq. 9.27). The five engineering “constants” are then given by: C~ nnnn 1 ; 5 ~ ~ ~ Ett ðt; dc Þ C tttt Cnnnn 2 C~ ttnn C~ nntt μ~ nt ðt; dc Þ 5 C~ ntnt ;

C~ tttt 1 5 ~ ~ ~ Enn ðt; dc Þ C tttt Cnnnn 2 C~ ttnn C~ nntt C~ nntt C~ ttnn ν~ nt ðt; dc Þ 5 ; ν~ tn ðt; dc Þ 5 C~ tttt C~ nnnn ð9:28Þ

Recall that for an isotropic material (brick and mortar), components of the stiffness tensor Cr (r 5 b, m) read: 4 r r Cxxxx 5 Cyyyy 5 k r 1 μr ; 3

2 r r Cxxyy 5 Cyyxx 5 kr 2 μr ; 3

r Cxyxy 5 2μr

ð9:29Þ where kr 5 ðE=3ð1 2 2νÞÞ and μr 5 ðE=2ð1 1 νÞÞ are the bulk and shear moduli, respectively.

9.2.2.5 Time-dependent crack density and first application of the proposed model Time-dependent crack density Various damage models are described in the literature (Lemaitre, 1996; Garavaglia and Lubelli, 2002; Sukontasukkul et al., 2004). Here, for the sake of simplicity and as a first approach we have chosen for the microcracked masonry a simple damage evolution model following

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PART | II Modeling of unreinforced masonry

Reda Taha and Shrive (2006) and Shrive and Reda Taha (2008). According to these papers, accumulated damage is assumed to follow Weibull’s failure rate function such that:   t1 X 100η t n dc ðtÞ 5 ð9:30Þ τD τD t0 where τ D is a constant damage time that refers to the time where most damage would occur. This damage not related to externally applied loads can be induced by external or internal effects such as freeze
thaw, alkali
silica reaction, sulfate
attack, etc. This load-independent model is consistent with Verstrynge et al. (2009) and Garavaglia et al. (2004) who showed that the Weibull failure rate function could be used successfully to predict the failure of masonry. As a first approach and according to a damage scenario considered by Shrive and Reda Taha (2008), the coefficients are taken here as τ D 5 800 (days), η 5 0.3 (days), and n 5 10. dc(t) represents the level of damage accumulated from the time at which damage starts, t0, to the time of evaluation. In the calculations here, damage is assumed to begin at 400 days. The rate of damage accumulation with this model is slow initially, but accelerates over time, as shown in Fig. 9.12 reporting Figure 4.4 in Shrive and Reda Taha (2008). Quite considerable damage is assumed to occur in a relatively short time in this example. Here, the damage factor attains about 0.33 after 1000 days with the damage starting at 400 days. Other possible damage scenarios or sophisticated accumulated damage functions accounting for both external applied loading and time parameters as that available for rockslat material developed by Chan et al. (1992) could be used in future investigations. As shown hereafter, Eq. (9.30), the only time-dependent model, is a starting point allowing first assessments of the proposed FE model.

FIGURE 9.12 Nonlinear evolution of damage ratio with time. From Shrive, N.G., Reda Taha, M.M., 2008. Effects of creep on new masonry structures. In: Binda, L. (Ed.), Learning from Failure
Long-Term Behaviour of Heavy Masonry Structures. WIT Press, pp. 83
108.

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Case of a periodic unit cell: comparisons at short and long terms In this section, it is proposed to investigate trends of evolutions with the time of overall predictions of a periodic masonry cell provided by the proposed FE model and their sensitivity to mortar joint thickness and brick dimensions (height and width). Microcracked mortar is assumed to follow the generalized Maxwell (GM) model. Here, for the sake of simplicity, only one term is considered for the GM model. Accordingly, the rheological model followed by the mortar’s behavior coincides with the MM’s model. In this study, bricks are assumed to be either rigid (Eb 5 1000Em (t 5 0)) or elastic (Eb 5 2.22Em (t 5 0)) and uncracked with a Poisson’s ratio ν b 5 0.15. Bricks are 250 mm thick. Their dimensions in the plane (x, y) are the following: height a 5 55 mm and width b 5 120 mm. The mortar joint’s thickness is th 5 10 mm. For the viscous rheological model, since the instantaneous Young’s modulus E0 for the MM model is given by Em (t 5 0) 5 ER 1 EM, where the relaxation modulus is set equal to EM 5 ei E0, the spring’s Young’s modulus ER reads ER 5 (1 2 ei)E0. Here, ei is a dimensionless parameter. All the ensuing computations have been carried out under the plane stress assumption by using a quadratic element “QUA8” with eight nodes and a refined mesh comprised of 10,336 elements using the software Cast3M. This fine mesh is chosen because it provides accurate effective results. Since we are studying the masonry creep phenomenon, we apply instantaneously a constant force at selected points of the boundary (i.e., a sustained macroscopic stress) as shown in Fig. 9.10. Hereafter, for a mortar joint thickness th 5 10 mm and properties identified at short (Table 9.5) and long terms (Table 9.6), time evolutions of effective tangent creep coefficients

TABLE 9.5 Elastic and viscous moduli of a mortar identified at the short term and tested by Brooks (1990) and Cecchi and Taliercio (2013).

Mortar

E0 ðMPaÞ

νm

ei

er ðlu Þð%Þ

7700

0.2

0.7602

7.1

TABLE 9.6 Elastic and viscous moduli of hybrid mortar gathered at the long term. EM ðMPaÞ 4000

τ M ðsÞ 8

2 3 10

ER ðMPaÞ

τ R ðs Þ

νm

2112

300,000

0.29

Source: From Verstrynge, E., Ignoul, S., Schueremans, L., Gemert, V.D., 2008. Modelling of damage accumulation in masonry subjected to a long-term compressive load. In: d’Ayala, D., Fodde, E. (Eds.), Structural Analysis of Historic Construction. CRC Press, pp. 525
532 (Verstrynge et al., 2008).

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PART | II Modeling of unreinforced masonry

provided by the FE model for masonries with rigid bricks and cracked mortar with a crack density evolving according to (Eq. 9.30) are reported in Fig. 9.13. Table 9.7 shows the decrease with increasing time and crack density of mortar’s Young’s moduli either for short- or long-term identified properties. FE predictions: As a whole, it is observed that FE predictions for masonry’s effective tangent moduli decrease with the increase of time. This can be explained by the increase of the damage level with time as illustrated in Fig. 9.12. For masonry with short-term mortar properties and either elastic (Fig. 9.14, Rekik et al., 2016) or rigid (Fig. 9.4) bricks, effective moduli

FIGURE 9.13 FE predictions for effective tangent moduli of masonry with rigid bricks (Eb 5 1000Em (t 5 0)), joints thickness th 5 10 mm, and mortar parameters identified at short (A) and long terms (B).

TABLE 9.7 Mortar Young’s moduli for different crack densities evolving due to the law (Eq. 9.30). t (days)

dc

0

0

Ej (short term) (MPa)

Ej (long term) (MPa)

7700

6112

214

2148

6027

29

1855

5866

26

1846

5575

450

24

1.19 3 10

1846

5441

650

4.70 3 1023

1831

5159

750

22

1784

4923

22

1645

4454

1517

4074

1345

3581

50 150 350

850 900 950

3.41 3 10

2.01 3 10 9.63 3 10

1.96 3 10 6.87 3 10

21

1.217 3 10

21

2.10 3 10

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FIGURE 9.14 Comparisons at time t 5 900 days of CTR (A and C) and FE (B and D) predictions for stress snapshots (σyy (A and B), σxy (C and D)) in the compressed wall with rigid bricks Eb 5 1000Em (t 5 0) and mortar’s properties identified at the long term.

decrease significantly during the first 50 days. This is consistent with the significant decrease of the mortar’s Young’s modulus as illustrated in Table 9.7 (column “short term”). After almost 100 days, the decrease of effective moduli is slow as observed for the case of masonry with long-term mortar properties throughout the whole period considered ([0, 950] (days)). Moreover, masonries with rigid bricks (Eb 5 1000Em (t 5 0)) show pronounced anisotropy compared to those with elastic bricks (Eb 5 2.22Em (t 5 0)) for which effective Young’s moduli E~ xx and E~ yy are close mainly at the long term, see Fig. 9.14B (Rekik et al., 2016). Hereafter, only the time range [600, 950] (days) is considered since the crack density is almost negligible for the time period [0, 600] (days) (see Table 9.7). According to Fig. 9.15, the decrease of the mortar thickness from 10 mm to 4 mm for masonries with rigid bricks almost double the masonry effective moduli. On the other hand, it can be seen in Figs. 9.16 and 9.17 that E~ xx and E~ yy are almost nonsensitive to the change of the brick height and width. However, note that the increase of brick height a (width b) causes the increase of the effective moduli E~ yy (E~ xx ) and μ~ xy . Also note that the shear effective moduli μ~ xy is more sensitive to the brick’s height a than to the brick’s width b. Similar trends are observed for time evolutions of masonry’s effective tangent moduli with elastic bricks and long-term mortar’s properties (see Figs. 9.18 and 9.19 in Appendix B; Rekik et al., 2016). Quantitatively, the decrease of the mortar’s thickness only slightly affects the masonry’s effective tangent properties with elastic bricks in contrast to the rigid ones. Table 9.8 summarizes the trends of evolutions of the microcracked masonry’s effective moduli with variation of the parameters mortar thickness, brick height, or width. These results allow us to conclude that effective FE predictions are as a whole more sensitive to the change of brick height “a” and also to the decrease of morta thickness “th” for both elastic and rigid bricks. Indeed, the lowest value of brick height gives the lowest masonry stiffness. It is then more beneficial to dispose of the highest possible value for “a.” Moreover,

FIGURE 9.15 Masonry with rigid bricks (Eb 5 1000 Em (t 5 0)) and mortar’s parameters identified at the long term: sensitivity of the FE predictions for Young’s E~yy (A), E~ xx ; (B) shear μ~ xy ; and (C) moduli to mortar joint’s thickness.

FIGURE 9.16 Masonry with mortar’s parameters identified at the long term and rigid bricks (Eb 5 1000Em (t 5 0)): sensitivity of FE predictions for Young’s E~ yy (A), E~xx (B); shear μ~ xy ; and (C) moduli to the brick’s height a (mm).

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FIGURE 9.17 Masonry with mortar’s parameters identified at the long term and rigid bricks (Eb 5 1000Em (t 5 0)): sensitivity of the FE predictions for Young’s E~yy (A), E~xx ; (B) shear μ~ xy ; and (C) moduli to the brick’s width “b” (mm).

FIGURE 9.18 Comparisons at time t 5 900 days of CTR (A and C) and FE (B and D) predictions for strain snapshots (εyy (A and B), εxy (C and D)) in the compressed wall with rigid bricks Eb 5 1000Em (t 5 0) and mortar’s properties identified at the long term.

the lowest value of the mortar thickness provides the stiffest masonry mainly in the case of rigid bricks. Lastly, there is no great profit in increasing the brick width “b,” which induces little increase of E~ yy and μ~ xy moduli. Case of a compressed masonry panel In this subsection, it is proposed to investigate FE predictions allowing the assessment of the relevance of the CTR model (Rekik et al., 2016) at the

324

PART | II Modeling of unreinforced masonry

FIGURE 9.19 Comparisons at time t 5 900 days of evolutions with abscise x of CTR and FE predictions for stress components (σyy (A) and σxy (B)) at the middle height’s of the compressed wall with rigid bricks Eb 5 1000 Em (t 5 0) and mortar’s properties identified at the long term.

TABLE 9.8 Sensitivity to various parameters (mortar thickness th, brick dimensions) of time evolutions of FE predictions for masonry effective tangent moduli with microcracked mortar and viscous parameters identified at the long term (Table 9.6). Parameter Mortar’s thickness

Brick’s height, a

Brick’s width, b

Bricks

̃E xx

̃E yy

̃μ xy

Rigid

m for th k

m for th k

m for th k

Elastic

m for th k (small effect)

m for th k

m for th k

Rigid

No effect of a

m for a m

m for a m

Elastic

m for a m

m for a m

m for a m (small effect)

Rigid

m for b m

No effect of b

Small m for b m

Elastic

m for b m

B no effect of b

Small m for b m

local level. For this purpose, we study the case of a masonry panel of dimensions L 5 1560 mm (length) and H 5 1040 mm (height) treated in Cecchi and Tralli (2012) and subjected to boundary conditions BC-2 with three distributed loads at the top and two lateral edges and an additional concentrated load F applied on the top as shown in Fig. 9.20A. Here, according to the results obtained in the “Case of a periodic unit cell: comparisons at short and long terms” section and for the sake of brevity, only the case of rigid bricks is treated (Eb 5 1000Em (t 5 0)). The mortar inside joints are assumed to be microcracked with a matrix that obeys linear viscoelastic behavior following the MM model. Microcrack is assumed to evolve with time following the nonlinear law (9.30). On the other hand, as the arrangement of the bricks is

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FIGURE 9.20 Equivalent problem (B) for the masonry panel submitted to boundary conditions BC-2 (A).

FIGURE 9.21 Periodic masonry cell (A) and its quarter part (B) considered for the modeling.

regular, the effective behavior of the panel is assumed to be well estimated by that of a periodic cell (see Fig. 9.21A). The panel can then be modeled as a homogeneous material with properties that coincide with those of the equivalent material HEM-2 (Fig. 9.20B). The mortar data used to compute this problem are those gathered at the long term as shown in Table 9.6. Qualitatively, under BC-2, distribution of the stress field σyy either for the FE or CTR model is symmetric (Fig. 9.14B) by reference to the axis of symmetry of the panel x 5 L/2 unlike that of the stress σxy which is antisymmetric (Fig. 9.14D). Similar qualitative aspects are observed for snapshots of strains εyy (symmetric; see Fig. 9.19B) and εxy (antisymmetric according to Fig. 9.19D). Snapshots of strain (Fig. 9.18) and stress (Fig. 9.18) fields show similar localization areas at the vicinity of the application’s point of the concentrated load F under condition BC-2. Quantitatively, FE and CTR estimates for stress components are close under BC-2 as shown in Fig. 9.22, illustrating evolutions of stress components along the x axis located at the middle height of the wall (x 5 H/2). For both the FE and CTR models, as shown in the maps of the stress component σyy, except for area surrounding the application’s point of load F, which is subjected to compression (σyy # 0), the wall is subjected locally to tensile stress (σyy $ 0). In this area, it can be noted that the absolute values

326

PART | II Modeling of unreinforced masonry

FIGURE 9.22 Comparisons at time t 5 900 days of evolutions with abscise x of CTR and FE predictions for strain components (εyy (A) and εxy (B)) at the middle height’s of the compressed wall with rigid bricks Eb 5 1000Em (t 5 0) and mortar’s properties identified at the long term.

of CTR estimates for σyy and σxy are stiffer than the FE ones. Moreover, for this area, the FE and CTR estimates for shear stress σxy are close compared to local predictions for stresses σyy. In contrast, at the middle height of the wall, it can be observed that the FE and CTR estimates are close either for σyy or σxy. Moreover, CTR predictions for shear stress are slightly softer than FE ones. However, CTR estimates for σyy are slightly stiffer when x-L/2; otherwise, they are almost the same. Globally, under this boundary condition, it is observed that the MM model predicts small strains. Moreover, the CTR model seems to overestimate strain localization by comparison to FE predictions. Indeed, in the area at the vicinity of the application’s point of load F, local strains (εyy and εxy) derived from the CTR model are almost three to four times greater than those provided by the FE model. The evolutions of strain components εyy and εxy (Fig. 9.18) at the middle height of the wall confirm that the CTR model overestimates local strains. However, away from the area at the vicinity of the application’s point of load F, the CTR and FE estimates for strain components are closer since CTR predictions are around 1.2
1.5 times greater than the FE ones.

9.2.2.6 Conclusions and perspectives This section extends the FE homogenization method for regular microcracked viscoelastic masonries. It provides accurate orthotropic overall tangent properties for this masonry in the short and long terms. The accuracy of this model is based on similar in-plane stress hypotheses for constitutive functions in joints and bricks in contrast to the analytical model. Moreover, this accuracy is a function of both factors: numerical error function of the mesh refinement and the choice of the mean-field homogenization scheme used to assess the behavior of the microcracked mortar. Moreover, in this work, there is no recourse to the LC transform when assessing the creep behavior of the mortar. This work, which rests on the computation of the

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strain localization tensors in each phase constituting the masonry (brick and mortar), proposes an alternative to an incremental homogenization approach that requires additional parameters such as the time increment and polarization tensors in viscoelastic phases. Estimates provided by the proposed numerical homogenization model serve to assess the accuracy of the recently proposed extension of the Cecchi and Taliercio’s model for microcracked masonry—the CTR approach (Rekik et al., 2016)—at the local and global levels for different parameters (mortar thickness and brick dimensions). In a future work, it could be interesting to investigate the effects of more sophisticated damage evolution law functions of both time and external loading (Taliercio and Papa, 2008) on FE predictions and the accuracy of the CTR model. Moreover, the choice of the mean-field homogenization scheme could influence the overall and local results of the proposed numerical model. Indeed, a mean-field homogenization model accounting for crack interactions such as the Ponte
Castan˜eda and Willis model (Bornert, 2001) could be more appropriate to assess the creep behavior of microcracked mortar and to account for higher crack densities (more than 20%). At last, taking into account the creep of bricks and crack propagation as proposed in Nguyen and Dormieux (2015) for homogeneous material could improve and enrich the proposed numerical model.

9.3 Nonlinear homogenization methods for masonry For reasons of durability and resistance to harmful factors (fire, water, chemical products, etc.), conventional bonded masonry is sometimes replaced by mortarless masonry systems such as interlocking mortarless hollow concrete block systems (Thanoon et al., 2008a); dry-stack mortarless sawn stone constructions (such as the Egyptian pyramids and the Zimbabwe ruins; Senthivel and Lourenco, 2009); and refractory linings of industrial furnaces including vessels of steel industry where the ceramic bricks are laid in direct contact with each other (Andreev et al., 2012). In contrast to conventional mortared masonry structures, for mortarless masonry, there have been limited analytical and numerical studies, and these depend mainly on the type of blocks used to assemble the walls. Among these studies, a FE model was proposed by Oh (1994) to simulate the behavior of interlocking mortarless block developed in Drexel University. Such a procedure is useful to simulate the contact behavior of mortarless joints including geometric imperfection but is suitable only for modeling small masonry assemblies. Material nonlinearity is not considered to account for the behavior of the masonry near the ultimate load and to predict the failure mechanism. Alpa and Monetto (1998) suggested a macromodel based on homogenization techniques to model the joint and block as a homogenous material. That model focuses on the joint movement mechanism assuming a perfect joint. This model ignores significant issues such as material

328

PART | II Modeling of unreinforced masonry

nonlinearity, joint imperfection, and progressive material failure. Recently, Thanoon et al. (2008a,b) proposed an FE model and developed an incremental iterative program to predict the behavior and failure mechanism of the system under compression. The nonlinear progressive contact behavior of mortarless joint that takes into account the geometric imperfection of the block-bed interfaces is included based on experimental testing. The developed contact relations for dry joints within specified bounds can be used for any mortarless masonry system efficiently with less computational effort. On the other hand, Senthivel and Lourenco (2009) developed a nonlinear FE analysis based on experimental data to model deformation characteristics such as load
displacement envelope diagrams and failure modes of drystack masonry shear walls subjected to combined axial compression and lateral shear loading. This analysis is based on a multisurface interface model where bricks and joints are assumed elastic and inelastic, respectively. More recently, Andreev et al. (2012) investigated the compressive closure of dry joints in two classes of refractory bricks: Magnesia
Carbon and Magnesia
Chromite bricks. The general aim of the investigation was to obtain data on the compressive joint closure behavior to get better insight into the masonry stress state and the joint condition during the service cycle of the furnace. To this end, the process of joint closure was measured indirectly by compressing samples with and without joints in a wide temperature range. At room temperature, direct optical measurements were also performed. FEM computer analysis was used to interpret the measurement results. For both conventional mortared or mortarless masonry structures, a continuum model based on micromechanical considerations is preferable. Indeed, recently, especially in the case of regular masonry, efficient and reliable models based on periodic homogenization have been created to allow nonlinear analysis of large-scale structures at low numerical cost. The present work is closely connected with the latter kind of analysis. Its relevance is based on its dependence on nonlinear homogenization methods sustaining mean-field theories classically applied to nonlinear composites. In this section, it is then proposed to assess the accuracy of predictive schemes belonging to the class of secant methods (the classical; Hutchinson, 1976; Berveiller and Zaoui, 1979) and its modified approach (Ponte Castan˜eda, 1991; Suquet, 1995, 2001)) to the particular case of refractory mortarless masonry. At room temperature, the nonlinear behavior of the mortarless ceramic joint was identified experimentally based on the digital image correlation (DIC) method (Rekik et al., 2015; Allaoui et al., 2017). The behavior of the brick unit was assumed to be linear elastic. Linearization procedures defining a linear comparison composite (LCC) were then applied only for the head and bed dry joint behaviors. The linear homogenization of the LCC behavior was performed using the FE method. Therefore, the approximations on the macroscopic level are related to the sole linearization procedure.

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The results of nonlinear homogenization sustaining mean-field theories are compared at global and local scales to the results of the nonlinear reference solution. Furthermore, it is proposed to improve the results of the classical secant scheme to better estimate local and global behaviors of mortarless masonry. Note that the methodology proposed in this part can be enlarged to the more general case of mortared masonry or eventually for masonry at high temperatures.

9.3.1

Experimental characterization of mortarless joint behavior

In many furnaces (e.g., converters of the steel industry) Magnesia
Carbon (MaC) bricks are laid on dry joints, without mortar. Quantitative knowledge of the compressive behavior of dry joints is an essential design parameter. As an example, consider the superposition of the stress-reducing effect of the joint. For these reasons and in order to support optimization of refractory masonry structures, only the compressibility of dry joints will be investigated. Compressive tests on a stack of two MaC bricks (without mortar) were carried out. Commercially available MaC bricks were used. Their composition is shown in Table 9.9. Because of their high resistance against chemical and mechanical wear the bricks are used in the insulating linings of steel-making vessels. The morphology of the brick is bigger grains of magnesia and graphite in the matrix of small magnesia grains. The maximal grain size is 5 mm. The bricks are resin bonded.

TABLE 9.9 Chemical composition and physical properties of MaC bricks. Material type 3

MaC

Density (g/cm )

2.93

Open porosity (%)

10

MgO (%)

98

Cr2O3 (%)




CaO (%)

1

FeO3 (%)

0.5

Al2O3 (%)




SiO2 (%)

0.5

Total C (%)

14

Source: From Andreev, K., Sinnema, S., Rekik, A., Allaoui, S., Blond, E., Gasser, A., 2012. Compressive behavior of dry joints in refractory ceramic masonry. Constr. Build. Mater. 34, 402
408.

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PART | II Modeling of unreinforced masonry

Tests were performed at atmospheric conditions on a mechanical frame Instron 4507 with a load cell of 200 kN (Fig. 9.23). The load accuracy was about 0.2% of the reached load. The samples were cut from bricks with dimensions of 100 3 50 3 50 mm3 and the faces were not polished. The compression tests were performed with a constant displacement rate of 0.033 mm/min. Two-dimensional DIC (Sutton et al., 1983; Vacher et al., 1999) was used to measure the compressive behavior of the dry joint with 7D correlation software (Vacher et al., 1999). The DIC is an optical method based on gray value digital images. The plane surface of the specimen was observed by a CCD camera with a resolution of 1380 3 1024 pixels in our case. Then, the images on the specimen surface, one before and others after deformation, were recorded, digitized, and stored in a computer as digital images. These images were compared to detect displacements by searching a matched point from one image to another using a series of mathematical mapping and cross-correlation functions. Once the location of this point in the deformed image was found, the local strain tensor was determined from the spatial distribution of the displacement field for each image. As it is almost impossible to find the matched point using a single pixel, an area with multiple pixel points is used to perform the matching process. This area, usually called a subset, should contain several clear features, but it is often a compromise between resolution and accuracy. As a general rule, larger subset sizes will increase the accuracy, whereas smaller subsets will increase the resolution. However, realistically, the size of a subset is determined by the quality of the image and the speckle pattern. In our case, another criterion is added for the subset size. Indeed, in order to evaluate the joint behavior, the grid must be put in place on the joint and must have only a small overlap onto the bricks. For this reason, the grid steps were optimized before using the DIC analysis on joints. The chosen subset

FIGURE 9.23 Experimental setup, compression test on brick-dry joint-brick laminate.

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was 6 3 6 pixels, which corresponds to an area width of about 0.5 mm. The accuracy of the DIC reached 0.01 pixels, which represents, in our case, a resolution of 0.001 mm on the displacement. In order to perform this process, a grayscale random pattern that allows matching the subset was needed on the surface of the specimen. In our case, the natural pattern of the bricks is enough to produce a suitable pattern. Due to roughness, shape defaults, and nonparallelism of faces, the dry joint was not horizontally aligned and its thickness was not constant. It was difficult to contain the joint in the same line of subsets. For this, measurements were performed at different locations along a joint (Fig. 9.24). For each location, the DIC method allowed the measurement of the evolutions of the local normal εnn , tangential εtt , and shear εnt strains. These strain components were averaged over each grid area and led to the dry joint compressive stress
strain curves shown in Fig. 9.25A for the third selected area, for example. Note that the DIC method does not provide the local stress in the dry joint. Moreover, as the bricks and dry joints were disposed in series, it is possible to assume that σðxÞ is set equal to the imposed normal stress σnn n  n. In Fig. 9.25A, it can be seen that at the beginning, the intensive joint strain develops at relatively low stresses. With progressive loading, the reaction to the compaction increases. At a certain stress level the joint appears to be closed completely as the closure curve aligns itself parallel to the compressive stress axis. After the joint closure, the compressive behavior of the sandwich brick/dry-joint/brick will be approximatively linear. Fig. 9.25B presents an example of measurements taken at different locations of a MaC dry joint. We note that the compressive strains are different according to the place where they were determined, but the dispersion remains correct. The fluctuation of the obtained data is due to the pattern size, which is function of the microstructure size of the MaC material. The bad contact resulting from natural roughness or from the fact that the contacting surfaces were not perfectly parallel

FIGURE 9.24 Optical measurement areas during a two-brick compression test (MaC).

332

PART | II Modeling of unreinforced masonry

FIGURE 9.25 “Stress
strain components” curves (A) and “σnn 2 εnn ” (B) evolutions at different areas selected around the mortarless joint of MaC material.

is also a parameter that influences the fluctuation and the dispersion of the measured strains. In the following, the subscripts b and j denote the bricks and joints, respectively. The properties of the dry joint were evaluated in terms of the average over all the selected areas Ai (i 5 1, N) of the local normal stress and strain componentsεnn , εtt , εnt , and εzz . Indeed, the latter component is not null under the adopted assumption of plane stress. Moreover, the shear strain components εlz (l 5 t or n) are null and the strain components εtt and εzz are assumed to be equal in the (t, z) plane orthogonal to the direction of the compressive loading.

Micromodeling Chapter | 9

9.3.2

333

Nonlinear homogenization of refractory mortarless linings

Since refractory mortarless linings present periodic microstructure, it is possible to consider only a periodic cell as shown in Fig. 9.25A. Note that the MaC bricks were assumed to follow an isotropic linear elastic behavior. The behavior of the dry joints is nonlinear as identified previously by the DIC method. The lining’s periodic microstructure enables a FE computation of the local and global responses. The FE result is regarded as a reference solution and denoted hereafter by NL. Note that the local and overall behavior of the mortarless masonry can also be estimated or approximated using nonlinear mean-field homogenization theories such as the classical secant procedure and its modified extension. Other “stress
strain” linearization schemes (e.g., the affine formulation) or potential-based approaches (e.g., the tangent second-order formulation) are to be addressed in the future since they need many more material parameters such as the polarization (or prestress) and the prestrain for thermoelastic “stress
strain” formulations or the potential strain energy for “potential-based” approaches. For mortarless refractory linings, in order to assess the accuracy of the existing secant linearization schemes known to provide predictions that are too stuff for usual viscoplastic power-law composites (see, e.g., Rekik et al., 2015), it is proposed to compare their predictions at global and local scales by referencing the NL solution. Moreover, in order to evaluate the sole effect of the linearization scheme without any bias or ambiguity, it is proposed to avoid any approximation related to the linear homogenization step. The main idea relies on the adoption of an LCC with an identical microstructure to that of the original problem and to perform FE linear homogenization on this LCC using the FE method. Moreover, as the periodic cell presents two axes of symmetry—the normal and the tangential directions along the unit vectors n and t, respectively—only its quarter (see Fig. 9.25B) will be retained for computation. In this section, note that the term “exact” is set in quotation marks since the accuracy of the reference solution depends on the numerical errors and mainly on the accuracy of the adopted functions fitting the experimental data.

9.3.2.1 Reference solution: finite element nonlinear homogenization Reference material properties of the constituents The following power-law relation between the local normal stress σnn and normal strain εnn is identified using the experimental data for the MaC mortarless joint (see Fig. 9.26): 0 σnn ðεnn Þ 5 Eej εnn 1 σ0 εm nn

where the scalars Eej , σ0 (MPa) and m0 are given in Table 9.10.

ð9:31Þ

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PART | II Modeling of unreinforced masonry

FIGURE 9.26 Evolutions of the experimental data: the linear part of the MaC mortarless joints “σnn 2 εnn ” relation (A) and “ðσnn 2 ðEje εnn ÞÞ 2 εnn ” evolution (B) functions of the local normal strain εnn .

Note that the scalar Eej can be considered as the initial Young’s modulus of the interphase since it is determined by the linear part of the curves σnn
εeq (see Fig. 9.26A). Moreover, by analogy with the usual (concave) power-law viscoplastic materials, the constant 0 can be assumed to represent the flow stress parameter. Note that, in the current study, the exponent m0 is superior to 1, which is not the case for the usual viscoplastic (concave) power-law composites for which it is well known that the work-hardening exponent m is less than 1. This is due to the convex qualitative trend of the σnn 2 εnn constitutive law. The local normal compressive behavior of the dry joint can then be defined by the nonlinear convex power-law “hσnn ij 2 hεnn ij ” relationship given by Eq. (9.31). However, the transversal behavior of the considered interphase can be defined by the evolution of the ratio 2ðhεtt ij =hεnn ij Þ between the tangential and normal strain field components over the interphase, denoted hereafter by the parameter ν j , as a function of the interphase

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TABLE 9.10 Parameters of the “normal stress
normal strain” relation for the MaC mortarless joint. Ee

σ0

m0

0.489

2.11 3 106

4.6

j

local normal strain εnn . This evolution depicted in Fig. 9.27A can be fitted by the ensuing polynomial second-order evolution: ν j ðεnn Þ 5 c2 ε2nn 1 c1 εnn 1 c0

ð9:32Þ

The scalars ci ði 5 0; 2Þ are given by Table 9.11. A linear approximation of the evolution of “ν j 2 εnn ” was avoided because it presents more than one slope (two different slopes) and the accuracy for each linear approximation is less than the 0.5 shown in Fig. 9.27A. Moreover, since this evolution (see Fig. 9.27A) is very fluctuant, a polynomial approximation of the parameter j with a degree greater than 2 was also avoided. Indeed, in practice, such polynomial approximation does not necessarily improve the accuracy shown in Fig. 9.27A—it is either inferior or not much higher (e.g., around 0.6 instead of 0.5 for a polynomial function of degree 3 or 4). For the isotropic linear elastic behavior of the MaC bricks, the Young’s modulus and Poisson’s ratio were taken, respectively, and set equal to Eb 5 10 GPa and ν b 5 0:1 (see Andreev et al., 2012). Reference local and global behaviors of the nonlinear mortarless masonry For the considered nonlinear problem, the local stress σ and strain ε fields in the periodic unit cell, assumed to have the volume V and to be submitted to the macroscopic strain ε, are solutions of the following set of equations (Bornert, 2001): 8 uðxÞ 5 ε:x 1 u ðxÞ; ’ xAV and u on @V > > > 1 > t  > > < εðuðxÞÞ 5 2 ðruðxÞ 1 ruðxÞÞ 5 ε 1 εðu ðxÞÞ; ’xAV ð9:33Þ > divðσÞ 5 0;X ’ xAV and σ:n 2 on @V > > > r > χ ðxÞgr ðεðxÞÞ; ’ xAV σðxÞ 5 > : r 5 j; b

where u is the local displacement vector and u is its fluctuating part; χr ðxÞ is the characteristic function of phase r (set to 1 if xAV r and 0 otherwise); and gr is the nonlinear constitutive law σ 5 gr ðεÞ followed by this phase. The general notations and 2 # mean that the fluctuating part of the displacement vector u and the surface compression σ:n (n being the outer normal) are

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FIGURE 9.27 Evolution of the dry joint’s parameter ν j as function of (A) the local normal strain εnn and (B) the spherical part traceðεÞ of the local strain.

TABLE 9.11 Parameters of the evolution law of ν j as a function of the MaC mortarless joint’s local normal strain. c2

c1

c0

29.16

2 3.313

0.131

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periodic and antiperiodic on the cell boundary, respectively. Note that the average hε ðu Þij over the periodic unit cell of the strain field of the fluctuating part u of the displacement vector is null (Michel et al., 2001). The local and effective behaviors of the mortarless refractory unit cell were computed using the software Cast3M (http://www-cast3m.cea.fr/ cast3m/indeIX.jsp) under the assumption of a plane stress field. In the unit periodic cell, the joints and bricks were assumed to be perfectly bonded. To determine the effective behavior of the cell, three types of loading were applied to the periodic mortarless unit cell. Since the behavior of the dry joints can be assumed to be piecewise linear, it is possible to define at each strain increment the following macroscopic law (Eq. 9.34) where L~ denotes the instantaneous “secant” effective stiffness of the reference unit cell. According to the classical Voigt notation, the constitutive behavior law of the unit cell reads: 0 1 0 10 1 σnn εnn L~nnnn L~nntt 0 @ σtt A 5 @ L~nntt L~tttt ð9:34Þ 0 A@ εtt A ~ σnt 2εnt 0 0 Lntnt where σkl 5 fj σ jkl 1 fb σbkl fr is the volume fraction of the phase r defined by fj 5 V r =V and ar 5 hair is the average over phase r of the stress or strain field component a. Note that the software Cast3M provides the reference local strain and stress fields inside each phase (bricks and mortarless joints). Moreover, it allows the calculation of the average fields over each phase. For computation purposes, note that the components εtt and εzz inside the dry joint are not assumed to be equal, as is the case in the previous section, but they are given due to the FE method. The relations between the components of the effective stiffness L~ijkl and the overall elastic engineering constants (normal E~ n and tangential E~ t Young’s modulus, Poisson’s ratios ν nt and ν tn , and shear modulus μ~ nt ) under plane stress assumption read: 8 E~ n > > > L~nnnn 5 > > 1 2 v~nt v~tn > > > > > > E~ t > > > L~tttt 5 > 1 2 v~nt v~tn > < ~ ð9:35Þ ~nntt 5 En v~tn > L > > > ~ ~ 1 2 v v nt tn > > > > > ~ ~ Lntnt 5 Gnt > > > v~nt v~tn > > > > : E~ n 5 E~ t

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To assess the effective elastic engineering constants, it is proposed to subject the unit cell to three types of loading: compression along t, compression along n; and shear loading. In the following, as we have only experimental data related to the compressive behavior of the MaC mortarless joint, we will consider only two types of loading (axial compression along n, axial compression along t). The case of shear loading is left for future work.

9.3.2.2 Secant linearization schemes for assessing global behavior of masonry It is worth noting that a nonlinear mean-field homogenization approach relies on two steps: linearization and linear homogenization. The first step consists of applying one of the numerous available linearization schemes in order to linearize the nonlinear behavior and thus to define a LCC. For secant linearization schemes, the original nonlinear problem (9.36) can then be rewritten as: 9 8 uðxÞ 5 ε:x 1 u ðxÞ; ’ xAV and u # on @V > > > > > > 1 > > t  > > > ðruðxÞ 1 εðuðxÞÞ 5 ruðxÞÞ 5 ε 1 εðu ðxÞÞ; ’ xA > = > > 2 > > < > divðσÞ 5 0; ’X xAV and σ:n 2 # on @V > > local linear problem > r r > > χ ðxÞL ðεðxÞÞ σðxÞ 5 > > ; > > > r 5 j; b > > r > > L 5 Lr ðεr Þ > : r nonlinear relations r ε 5 , ε . r ðfor SECÞ or ε ðfor VARÞ ð9:36Þ where Lr ðεÞ are known functions whose exact expressions depend on the chosen linearization scheme. The procedure followed to solve this system of equations is described below (see “The linearization step” section). The second step of a nonlinear mean-field homogenization evaluates the effective properties of the LCC defining thus the homogeneous equivalent material (HEM). The effective properties of the HEM were assessed by applying one of the available approximative linear homogenization schemes such as the Hashin
Shtrikman (HS) bounds or the SC model (Bornert, 2001). Frequently, since such approaches induce differences between the microstructure of the nonlinear composite and that of the LCC, it is proposed in this chapter—as in Rekik et al. (2007)—to carry out an “exact” linear homogenization step by considering an LCC with an identical microstructure to that of the nonlinear composite—that is, the periodic unit cell—and using the FE method to compute the effective properties of the LCC. Accordingly, the sole effect of the linearization

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step can be evaluated without any bias related to a change of microstructure or other hypothesis adopted by a classical linear homogenization scheme. For the linearization step, it is possible to adopt one of the several linearization schemes available in the literature. Nevertheless, since the experimental data in our disposal, related to the dry joint’s properties are limited, only secant linearization schemes that means the classical secant model (Hutchinson, 1976; Berveiller and Zaoui, 1979), referred to as by SEC and the modified secant method (Ponte Castan˜eda, 1991; Suquet, 1995, 2001) noted in the following by VAR will be treated in this chapter. The VAR method accounts for both the inter- and intraphase strain fluctuations unlike its original version, SEC, which considers only the interphase fluctuations. In the following, we propose also to test an empirical version of the classical secant method referred to as SECα . The linearization step Interphase properties in the LCC: Since a secant linearization scheme attributes to each phase r in the LCC a secant shear moduli μ~ sct defined by the equation (Suquet, 1995; Bornert and Suquet, 2001) σeq ðεeq Þ ð9:37Þ 3εeq where the von Mises stress (respectively strain) measures the deviatoric part of the stress (respectively strain) tensor as done in Ponte Castan˜eda (1991), Gilormini et al. (2001), and Rekik et al. (2007), it is useful to define the interphase behavior in terms of the “σeq 2 εeq” evolution as shown in Fig. 9.28 provided by the experimental data. According to the definition (9.37), the secant shear modulus of the interphase in the LCC defined by a secant linearization scheme reads: μrsct ðεeq Þ 5

1 μj ðεeq Þ 5 μje 1 μ1 εm eq

ð9:38Þ

where the scalars μej and μ1 (MPa) and the exponent m1 are given in Table 9.12. Note that, in this study, there is no use of the von Mises plasticity criterion since the deviatoric part of the dry joint’s behavior is assumed to be nonlinear elastic following a power-law type relation. The constant μej can be considered as the elastic shear modulus of the dry joint since it is provided by the linear part (see Fig. 9.28A) of the “σeq 2 εeq” evolution (i.e., μej 5 σeq ðεeq Þ=3εeq for εeq # 0:012). It is worth noting that a polynomial approximation of the shear modulus evolution was avoided as it could lead to aberrant (negative) values for j for some ranges of the local equivalent strain. An exponential approximation was also avoided since such function overestimates μj with the increase of the local equivalent strain. For this step, we chose to not linearize the spherical part of the joint’s behavior but

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PART | II Modeling of unreinforced masonry

FIGURE 9.28 Periodic mortarless masonry cell under compression along n (A and B) or compression along t (C and D): effective mechanical properties versus the macroscopic strain.

TABLE 9.12 Parameters of the evolution of the dry joint’s secant shear modulus versus the local equivalent strain. j

μe 0.208

μ1

m1 7

10

4.05

to use the “exact” expression of the parameter j as a function of the spherical part traceðεÞ of the strain field in the joint. It reads: ν j ðtrðεÞÞ 5 b2 ðtrðεÞÞ2 1 b1 ðtrðεÞÞ 1 b0

ð9:39Þ

where the scalars bi ði 5 1; 3Þ are provided in Table 9.13 and tr ðεÞ 5 ε: i. The secant Young’s modulus of the interphase can then be deduced as j follows: Esec 5 2j ð1 1 ν j Þ. Its bulk modulus reads kj 5 Ej =3ð1 2 ν j Þ. In the LCC, the MaC interphase is then assumed to be an isotropic linear elastic j phase characterized by the secant Young’s modulus Esec and the “exact”

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TABLE 9.13 Parameters used to approximate the MaC dry joint’s parameter ν j 5 2 ðhεtt ij =hεnn ij Þ as a function of the spherical part of the strain field in the joint. b2

b1

b0

33.23

23:46

0.127

joint’s parameter ν j (see formulae (9.39)). Recall that the term “exact” is set in quotation marks since it is related to the accuracy of the approximative function used to fit the fluctuant evolution of the parameter ν j as a function of the spherical part of the local strain. For the interphase, it was also possible to linearize the (convex powerlaw) spherical part of the MaC joint’s behavior by evaluating the joint’s secant bulk modulus as kj 5 σm ðεm Þ=3εm and therefore to deduce the j Poisson’s ratio ν j 5 ð3ksec 2 2μ jsec Þ=2ð3k jsec 1 μ jsec Þ. Nevertheless, the latter j secant bulk modulus ksec risks coming to aberrant (negative) values for the Poisson’s ratio if the adopted (or chosen) function fitting the “σm 2 εm” evolution provided by the DIC method is not so accurate. Resolution of the nonlinear problem (Eq. 9.36): To define the LCC for each loading step, the reference strain εr for the SEC and VAR procedures needs to be assessed. Since there is no experiment carried out on the periodic mortarless masonry cell using the DIC method as is the case for the laminate elementary structure (see Section 9.2), we do not have experimental data allowing the deduction of the reference strains εj for the mortarless linings. Accordingly, we propose to use an iterative method (e.g., the fixed point) in order to resolve the nonlinear set Eq. (9.36). For this nonlinear system, it is recalled that Lr ðεÞ are known functions whose exact expressions depend on the chosen linearization procedure. Moreover, to ensure numerical accuracy in these investigations, the convergence criterion adopted for the iterative fixed-point method in this work was set equal to 1026 ððpr1 1 pr0 Þ=2Þ, where pr1 denotes the new evaluation of the reference strain εr and pr0 is its initial value in each phase r. More details about this iterative method are given in Rekik et al. (2007). Results and discussion This section provides insight into the influence of the secant linearization procedures on the global and local behavior of MaC regular mortarless masonry. With this aim, we consider a periodic cell made of bricks of dimensions 100 3 50 mm2 and mortarless joint with 0:104 mm thickness. This cell is discretized through a mesh relying into 50 3 25; 4 3 50; and 2 3 25 four nodes quadrilateral finite elements inside the brick, the bed, and the head joints in the quarter cell, respectively. The choice of such discretization instead of a

342

PART | II Modeling of unreinforced masonry

more refined mesh with eight-node quadrilateral finite elements was motivated by the fact that the former allows the fixed point to converge faster and due to negligible differences between results provided by both meshes. For the simulated results, it is noted that the computations are run until εnn 5 2 3 1025 ðεtt 5 1:75 3 1025 Þ for unit cells under compression along n (along t). Effective properties Evolutions of the computed effective stiffnesses ðL~nnnn ; L~tttt ðMPaÞÞ and Poisson’s ratios (ν nt and ν tn ) with respect to the imposed macroscopic strain are depicted in Fig. 9.21. For the mortarless periodic cell submitted to compression along n, the secant estimates (see Fig. 9.21A and B) reproduce qualitatively well the evolutions of the reference solutions. Moreover the VAR method provides good estimates for the effective stiffness L~nnnn and Poisson’s ratio ν tn of the MaC mortarless masonry. Unlike for usual viscoplastic (concave) power-law composites, the classical secant model leads to too soft overall estimates for the mortarless masonry. The SECαn empirical model where the scalar n is found to be set to 1.3 improves the overall estimates of the classical secant procedure. Note that αn is superior to 1. This amplification of the reference strain ε jeq for the classical secant model allows then the definition of an improved LCC more relevant than that defined by the SEC scheme. Note that the reference strain αn ε req almost coincides with the secondr order moment of the strain field εeq (see Fig. 9.10 in Rekik et al., 2016). This argues the quasiequality between the overall predictions of SECαn and VAR. Note that, even though the VAR model is a sophisticated model accounting for both the inter- and intraphase strain field fluctuations, the empirical model SECαn accounting only for the interphase field fluctuations could be a satisfactory alternative for the VAR scheme as it is easier to implement and requires less theoretical investigation and numerical expense. However, it requires the implementation of an automatized inverse identification procedure not yet done in this work. For computations carried out under compression along t (Fig. 9.21C and D), it is observed that the secant (SEC and VAR) schemes (highly) overestimate the overall reference response. The SECαt estimates, with a scalar αt 5 0:85 less than 1, softens the SEC estimates. Indeed, as shown in Fig. 9.10B in Rekik et al. (2016), the reference αt εreq is softer than the secr ond moment ε eq and obviously softer than the first moment ε req with r αn ε req # ε req # ε eq : Accordingly and due to the convex qualitative trend of the “σeq 2 εeq” curve for the mortarless MaC joint, the scheme SECαt leads to better global estimate than those provided by VAR and SEC. The inequality r ε req # ε eq justifies also that VAR overall estimate is stiffer than that provided by SEC in the current study unlike for results obtained for usual viscoplastic (concave) power-law composites. The different general trends observed for the SEC and VAR predictions at the global scale for mortarless masonry under compression along n and

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that along t can be justified by the hypotheses adopted in this study. Indeed, for compression along t, the dry joint was assumed to behave as a joint submitted to compression along n. Moreover, the strain field components εtt and εzz were assumed to be equal, which is not necessarily true. The third hypothesis was related to the plane stress assumption for the nonlinear problem and the linear problems associated with the LCC defined by the secant schemes. Accordingly the overall trends observed for a mortarless unit under compression along n should be more rigorous. Those obtained for the mortarless unit cell under compression along t to be checked or confirmed by the investigation of the real dry joint’s behavior under compression along t using DIC or another appropriate experimental technique. This idea is left for future work.

9.3.3

Conclusions and perspectives

In this section, the dry joint was assumed to be an interphase perfectly bonded with MaC bricks. Accordingly it was possible to apply mean-field homogenization theories to the mortarless masonry. A convex power-law behavior was identified for the dry joint using the DIC method for an elementary mortarless specimen under compression orthogonal to the plane of the joint. A rigorous assessment of the existing secant linearization schemes for a mortarless periodic masonry with reference to the FE solution demonstrated the superiority of the VAR model compared to the SEC scheme for mortarless unit cells under normal compression. This result confirms again— as is the case for the usual viscoplastic (concave) power-law materials—the relevance of the VAR model since it accounts for both the inter- and intraphase strain fluctuations instead of the SEC model, which considers only the interphase fluctuations. Unusually, the SEC estimates are softer than the VAR and nonlinear responses. This is due to the convex qualitative trend of the deviatoric part of the dry joint behavior instead of the usual concave trend of viscoplastic power-law composites. For mortarless unit cells under tangential compression, different trends were observed. The secant estimates, especially the VAR predictions, were found to be too stiff. To improve these results, an empirical variant SECα of the SEC scheme was proposed. It relies on the adjustment of a scalar in order to reduce (amplify) the reference strain ε req if the SEC overall estimate is stiffer (softer) than the nonlinear solution. The appropriate value of the parameter led to global and local estimates in well agreement with the reference solution. Even though the proposed model is not based on theoretical investigations and accounts only for interphase field fluctuations, it could be a satisfactory alternative for the secant schemes (SEC and VAR) if these models lead to too stiff or soft estimates. The evaluations and comparisons carried out in the current study can be extended to mortarless refractory linings submitted to loading
unloading compressive cycles at room and high temperatures. They can also be carried

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PART | II Modeling of unreinforced masonry

out under other mechanical tests (shear or biaxial loading) at various ranges of temperatures. However, it is important to have reliable reference solutions provided, for instance, by experiments based on the DIC method. These perspectives are left for future work. The empirical parametrical model proposed in this section for the classical secant scheme can also be applied for the VAR model. But these parametrical models require a reference solution provided by experiments or FE or FFT method. A computational inverse procedure could facilitate the determination of the tuning parameter. This approach can also be extended either for other types of brick materials or more generally for conventional mortared masonry at room or high temperatures.

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347. Rekik, A., Lebon, F., 2010. Identification of the representative crack length evolution for a multi-level interface model for quasi-brittle masonry. Int. J. Solids Struct. 47, 3011
3021.

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Rekik, A., Brenner, R., 2011. Optimization of the collocation inversion method for the linear viscoelastic homogenization. Mech. Res. Commun. 38, 305
308. Rekik, A., Lebon, F., 2012. Homogenization methods for interface modeling in damaged masonry. Adv. Eng. Soft. 46, 35
42. Rekik, A., Auslender, F., Bornert, M., Zaoui, A., 2007. Objective evaluation of linearization procedures in nonlinear homogenization: a methodology and some implications on the accuracy of micromechanical schemes. Int. J. Solids Struct. 44 (10), 3468
3496. Rekik, A., Allaoui, S., Gasser, A., Andreev, K., Blond, E., Sinnema, S., 2015. Experiments and nonlinear homogenization sustaining mean-field theories for mortarless masonry : The classical secant and its improved variants. European Journal of Mechanics - A/Solids 49, 67
81. Rekik, A., Nguyen, T.T.N., Gasser, A., 2016. Multi-level modeling of microcracked viscoelastic masonry. Int. J. Solids Struct. 81, 63
83. Rougier, Y., Stolz, C., Zaoui, A., 1994. Self-consistent modeling of elasticviscoplastic polycrystals. C. R. Acad. Sci. 318, 145
151. Sacco, E., 2009. A nonlinear homogenization procedure for periodic masonry. Eur. J. Mech. A/ Solids 28, 209
222. Schapery, R.A., 1962. Approximate methods of transform inversion for viscoelastic stress analysis. In: Proceeding of the 4th U.S. National Congress of Applied Mechanics, 2. ASME, pp. 1075
1085. Senthivel, R., Lourenco, P.B., 2009. Finite element modeling of deformation characteristics of historical stone masonry shear walls. Eng. Struct. 31, 1930
1943. Shrive, N.G., Reda Taha, M.M., 2008. Effects of creep on new masonry structures. In: Binda, L. (Ed.), Learning from Failure
Long-Term Behaviour of Heavy Masonry Structures. WIT Press, pp. 83
108. Shrive, N.G., Sayed-Ahmed, E.Y., Tilleman, D., 1997. Creep analysis of clay masonry assemblages. Can. J. Civil Eng. 24 (3), 367
397. Sukontasukkul, P., Nimityongskul, P., Mindess, S., 2004. Effect of loading rate on damage of concrete. Cem. Concr. Res. 34, 2127
2134. Sulem, J., Muhlhaus, H.B., 1997. A continuum model for periodic two-dimensional block structures. Mech. Cohes. Frict. Mater. 2, 31
46. Suquet, P., 1995. Overall properties of nonlinear composites: a modified secant moduli theory and its link with Ponte Castan˜eda’s nonlinear variational procedure. C.R. Me´canique 320, 563
571. Suquet, P., 2001. Nonlinear composites: secant methods and variational bounds. Lemaitre Handbook of Materials Behaviour Models. Academic Press, pp. 968
983, Section 10.3. Sutton, M.A., Wolters, W.J., Peters, W.H., Ranson, W.F., McNeill, S.R., 1983. Determination of displacements using an improved digital correlation method. Image Vision Comput. 1 (3), 133
139. Taliercio, A., Papa, E., 2008. Modelling of the long-term behavior of historical masonry towers. In: Binda, L. (Ed.), Learning from Failure
Long-Term Behaviour of Heavy Masonry Structures. WIT Press, pp. 153
173. Thanoon, W.A., Alwathaf, A.H., Noorzaei, J., Jaafar, M.S., Abdulkadir, M.R., 2008a. Finite element analysis of interlocking mortarless hollow block masonry prism. Comput. Struct. 86, 520
528. Thanoon, W.A., Alwathaf, A.H., Noorzaei, J., Jaafar, M.S., Abdulkadir, M.R., 2008b. Nonlinear finite element analysis of grouted and ungrouted hollow interlocking mortarless block masonry system. Eng. Struct. 30, 1560
1572.

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Tsukrov, I., Kachanov, M., 2000. Effective moduli of an anisotropic material with elliptical holes of arbitrary orientational distribution. Int. J. Solids Struct. 37, 5919
5941. Ushaksarei, R., Pietruszczak, S., 2002. Failure criterion for structural masonry based on critical plane approach. J. Eng. Mech. ASCE 128 (7), 769
778. Vacher, P., Dumoulin, S., Arrieux, R., 1999. Determination of the forming limit diagram from local measurement using digital image analysis. CIRP Ann. Manuf. Technol. 48 (1), 227
230. Verstrynge, E., Ignoul, S., Schueremans, L., Gemert, V.D., 2008. Modelling of damage accumulation in masonry subjected to a long-term compressive load. In: d’Ayala, D., Fodde, E. (Eds.), Structural Analysis of Historic Construction. CRC Press, pp. 525
532. Verstrynge, E., Schueremans, L., Van Gemert, D., Wevers, M., 2009. Monitoring and predicting masonry’s creep failure with the acoustic emission technique. NDT E Int. 42 (6), 518
523. Zucchini, A., Lourenc¸o, P.B., 2004. A coupled homogenization-damage model for masonry cracking. Comput. Struct. 82, 917
929.

Further reading Andreev, K., Zinngrebe, E., Heijboer, W., 2009. Compressive behavior of ACS torpedo bricks. In: 11th Biennial Worldwide Conference on Unified International Technical Conference Refractories. UNITECR, Salvador (Brazil). Bagi, K., 1993. A quasi-static numerical model for micro-level analysis of granular assemblies. Mech. Mater. 16 (1
2), 101
110. Benveniste, Y., 1986. On the Mori-Tanaka method in cracked solids. Mech. Res. Commun. 13, 193
201. Brulin, J., Roulet, F., Rekik, A., Blond, E., Gasser, A., Mc Nally, R., et al., 2011. Latest evolution in Blast Furnace Hearth thermo-mechanical stress modeling. In: 4th International Conference on Modelling and Simulation of Metallurgical Processes in Steelmaking, Dusseldorf, Germany. Cecchi, A., Barbieri, A., 2008. Homogenisation procedure to evaluate the effectiveness of masonry strengthening by CFRP repointing technique. Appl. Theor. Mech. 1 (3), 12
27. Chaboche, J.L., Kanoute, P., 2003. Sur les approximations “isotrope” et “anisotrope” de l’ope´rateur tangent pour les me´thodes tangentes incre´mentale et affine. C. R. Me´canique 331, 857
864. Cluni, F., Gusella, V., 2004. Homogenisation of non-periodic masonry structures. Int. J. Solids Struct. 41, 1911
1923. Cost, T.L., Becker, E.B., 2007. A multidata method of approximate Laplace transform inversion. Int. J. Numer. Methods Eng. 2, 207
219. Deng, H., Nemat-Nasser, S., 1992. Microcrack arrays in isotropic solids. Mech. Mater. 13, 15
36. Ferber, M.K., Weresczak, A.A., Hemrick, J.G., 2006. Compressive creep and thermo-physical performance of refractory materials. Final Technical Report of Oak Ridge National Laboratory. Gasser, A., Terny-Rebeyrotte, K., Boisse, P., 2004. Modelling of joint effects on refractory lining behavior. J. Mater. Des. Appl. 218, 19
28. Gasser, A., Spangenberg, J., Blond, E., Rekik, A., Andreev, K., 2011. Comparison of Different Designs of Bottom Linings with Dry Joints. UNITECR’11, Kyoto, Japan. Giambanco, G., Rizzo, S., Spallino, R., 2001. Numerical analysis of masonry structures via interface models. Comput. Methods Appl. Mech. Eng. 6494
6511.

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Hashin, Z., 1988. The differential scheme and its application to cracked materials. J. Mech. Phys. Solids 36, 719
734. Hoenig, A., 1979. Elastic moduli of a non-randomly cracked body. Int. J. Solids Struct. 15, 137
154. Jin, S., Harmuth, H., Gruber, D., 2014. Compressive creep testing of refractories at elevated loads
device, material law and evaluation techniques. J. Eur. Ceram. Soc. 34, 4037
4042. Laws, N., Dovrak, G.J., 1987. The effect of fiber breaks and aligned penny-shaped cracks on the stiffness and energy release rates in unidirectional composites. Int. J. Solids Struct. 23, 1269
1283. Levin, V.M., 1967. Thermal expansion coefficients of heterogeneous materials. Mekh. Tverd. Tela. 2, 83
94. Lourenco, P.B., 1996. Computational strategies for masonry structures. Delft University of Technology, The Netherlands, Ph.D. thesis. Lourenco, P.B., Rots, J., 1993. On the use of macro-models for the analysis of masonry shear walls. In: Proceeding of the 2nd International Symposium on Computer Methods in Structural Masonry, pp. 14
26. Masson, R., Bornert, M., Suquet, P., Zaoui, A., 2000. An affine formulation for the prediction of the effective properties of nonlinear composites and polycrystals. J. Mech. Phys. Solids 48, 1203
1227. Molinari, A., Toth, L.S., 1994. Tuning a self-consistent visco-plastic model by finite element results, part I: modeling. Acta Metall. Mater. 42, 2453
2458. Molinari, A., Ahzi, S., Kouddane, R., 1997. On the self-consistent modeling of elastic-plastic behavior of polycrystals. Mech. Mater. 26, 43
62. Nguyen, T.M.H., Blond, E., Gasser, A., Prietl, T., 2009. Mechanical homogenisation of masonry wall without mortar. Eur. J. Mech. A/Solids 28 (3), 535
544. Nguyen, T.T.N., Rekik, A., Gasser, A., 2014. A multi-level approach for micro-cracked masonry. In: 11th World Congress on Computational Mechanics (WCCM XI), Barcelona. Parvanova, S., Gospodiniv, G., in press. Development of “event-to-event” nonlinear technique to lightly reinforced concrete beams by simplified constitutive modeling. Int. J. Solids Struct. Ponte-Castan˜eda, P., Willis, J.R., 1995. The effect of spatial distribution on the behavior of composite materials and cracked media. J. Mech. Phys. Solids 43, 1919
1951. Rekik, A., 2006. Une me´thodologie pour une e´valuation pre´cise des proce´dures de line´arisation en homoge´ne´isation non line´aire. Ecole Polytechnique, Ph.D. thesis.

Chapter 10

Homogenization and multiscale analysis D. Addessi1, S. Marfia2 and E. Sacco3 1

Department of Structural and Geotechnical Engineering, Sapienza University of Rome, Rome, Italy, 2Department of Engineering, University of Roma Tre, Roma, Italy, 3Department of Structures for Engineering and Architecture, University of Naples “Federico II”, Naples, Italy

10.1 Introduction Masonry constructions are a large part of the world architectural and historical heritage and are also widely spread in civil buildings. The accurate assessment of the safety of these constructions and the correct evaluation of their stress states is even now an interesting and, at the same time, hard research task. In fact, masonry is obtained by assembling natural or artificial bricks using mortar layers; thus, it is a composite material whose constituents are characterized by a strongly nonlinear (cohesive) mechanical response. A recent overview of different modeling approaches aimed at describing the response of masonry material and structural elements can be found in the Special Issue of Meccanica entitled “New trends in mechanics of masonry” (Sacco et al., 2018). The response of the masonry can be investigated at different scales, as illustrated, for instance, in Addessi et al. (2014). To study and predict the behavior of a masonry structural element, taking into account its specific microstructure and the nonlinear phenomena arising in the constituents, multiscale techniques can be conveniently used. A comprehensive review of the homogenization techniques developed for masonry elements has been presented by Lourenc¸o et al. (2007). According to the multiscale method, the structural analysis is performed considering two different scales: the macroscale at the continuum mechanics structural level, and the microscale at the material level, where a detailed description of the single heterogeneity present in the masonry material is made. The development of multiscale procedures is a complex task, as it requires solving the micromechanical problem and adopting the obtained results to determine the overall mechanical response of the Numerical Modeling of Masonry and Historical Structures. DOI: https://doi.org/10.1016/B978-0-08-102439-3.00010-5 Copyright © 2019 Elsevier Ltd. All rights reserved.

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composite and, hence, to perform the structural analysis. The micromechanical problem can be solved by using homogenization techniques that study a representative volume element (RVE), taking into account the nonlinear phenomena. At this level, bricks have been modeled as rigid blocks or deformable continuum, characterized by linear or nonlinear responses, while interface or continuum models with linear or nonlinear responses have been adopted for the mortar. At the macroscale, Cauchy, Cosserat, or higher order continua have been considered. In this framework, different studies have been proposed considering regular arrangement, that is, periodic composite material, or quasiperiodic and irregular texture of the masonry. In particular, several homogenization techniques available in literature deal with the analysis of periodic masonry. In this case, a unit cell (UC) can be introduced to derive the overall response. Many studies have been developed for walls subjected to in-plane loading conditions. Among others, Anthoine (1995) studied the homogenization of a periodic masonry with brick and mortar characterized by elastic behavior. Luciano and Sacco (1998b) proposed numerical approaches based on the variational formulation to solve the homogenization problem of a twodimensional (2D) masonry UC characterized by linear elastic constitutive laws for both the constituents. Milani and Cecchi (2013) developed a simplified kinematic procedure at the cell level for the evaluation of the in-plane elastic moduli of herringbone masonry. Moreover, they also determined the macroscopic masonry strength domains. Concerning the nonlinear models, Luciano and Sacco (1997, 1998a) proposed a brittle damage law for masonry, characterized by the coalescence and growth of a finite number of fractures developing only in the mortar. More complex models introduced the nonlinear response of both brick and mortar. Among others, Gambarotta and Lagomarsino (1997) considered an equivalent medium made of mortar joints and brick unit layers and adopted damage constitutive laws both for bricks and mortar joints. Massart et al. (2007) presented an enhanced multiscale model based on nonlocal implicit gradient isotropic damage formulations for both brick and mortar. They were able to describe the damage preferential orientations and employ an embedded band model at the macroscopic scale. Sacco (2009) developed a special nonlinear constitutive law for mortar joints, which accounts for the coupling of damage and friction phenomena occurring during the loading history; the nonlinear micromechanical problem was solved through a transformation field analysis (TFA) technique. In Chettah et al. (2013) the TFA was developed to derive the localization direction. Zucchini and Lourenc¸o (2009) presented a micromechanical model for the masonry homogenization, adopting damage and plasticity models. Wei and Hao (2009) proposed a homogenization approach adopting a continuum damage model for masonry, taking into account the strain rate effect. Marfia and Sacco (2012) developed a multiscale analysis of periodic

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masonry walls considering nonlinear constitutive laws, based on damage and friction models, for mortar and blocks. The micromechanical problem of the UC was solved using a numerical homogenization technique. Spada et al. (2015) and Giambanco et al. (2018) presented a multiscale approach for masonry, adopting the meshless method to solve the boundary value problem for the evaluation of the overall response of the UC. Investigations concerning the response of regular masonry, characterized by viscoelastic behavior, have also been developed. Cecchi and Tralli (2012) proposed a rigorous homogenization procedure to evaluate the in-plane response of masonry walls, considering rigid or elastic blocks bonded by viscoelastic mortar. Taliercio (2014) developed a method based on a cells-type approach for the determination of closed-form expressions of the macroscopic elastic constants, accounting for the creep of brick masonry with regular pattern. Rekik and Gasser (2016) provided the effective tangent properties of regular microcracked viscoelastic masonry, performing a twostep homogenization. Initially, the response of the mortar is deduced through homogenization of a viscoelastic microcracked material; then, the orthotropic overall properties of masonry is obtained by performing the second homogenization using the finite element method for periodic masonry. Very few models have been developed for nonperiodic microstructure masonry. Cecchi and Sab (2009) studied nonperiodic masonry material, typical of historical buildings, adopting a perturbation approach. Cavalagli et al. (2011), for instance, developed a micromechanical approach for random media with the aim of evaluating the strength domain for nonperiodic masonry. Cluni and Gusella (2018) analyzed a sequence of quasiperiodic wall portions numerically generated with the objective of evaluating the size of the RVE to ensure convergence of the elastic coefficients. The problem of the evaluation of the failure surface of quasiperiodic masonry, adopting a homogenization procedure applied to a statistically equivalent periodic UC, was discussed in Cavalagli et al. (2018). Several models have been proposed in the framework of the Cosserat and couple stress continua. Among others, Masiani et al. (1995) and Masiani and Trovalusci (1996) presented a study of the masonry made of 2D periodic rigid blocks and elastic mortar. They obtained the macroscopic characterization of the equivalent medium by equating the virtual stress power of the coarse model to the virtual power of the internal actions of the discrete fine model. Casolo (2006) proposed a study where the homogenized elastic tensor of the equivalent Cosserat medium was identified by considering isotropic linear elastic models both for brick and mortar. Salerno and de Felice (2009) investigated the accuracy of various schemes in the framework of Cauchy and Cosserat continua, showing that, in the case of nonperiodic deformation states, micropolar continuum better reproduces discrete solutions, due to its capability to take into account scale effects. Sab and Pradel (2009) developed

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homogenization procedures for Cosserat materials with periodic microstructure. Addessi et al. (2010) evaluated the Cosserat continuum mechanical properties at the macroscale by considering at the microscale the nonlinear response of the mortar joints, using a cohesive-friction constitutive model. De Bellis and Addessi (2011) proposed a Cosserat-based multiscale model considering nonlinear behavior for both brick and mortar. Addessi et al. (2012) proposed a TFA based multiscale approach for periodic composite materials, considering the Cosserat continuum at the macrolevel and the Cauchy continuum at the microlevel. The two levels were linked by a kinematic map based on a third-order polynomial expansion. Leonetti et al. (2018) developed a multiscale strategy for the analysis of damage propagation in masonry structures modeled as periodic composites with the aim of reducing the typically high computational cost that characterized the fully microscopic numerical analyses. Some efforts have also been devoted to study the response of walls subjected to out-of-plane loading conditions. Mercatoris et al. (2009) proposed a multiscale failure analysis for in-plane masonry thin shell based on computational homogenization. A limit analysis approach was adopted by Milani and Taliercio (2016) to investigate the bearing capacity of laterally loaded masonry elements. Upper bounds to the macroscopic strength domain of the wall were obtained by applying the kinematic theorem of limit analysis within the framework of homogenization theory for periodic media. Silva et al. (2017) presented a mechanical model that couples rigid elements with interfaces, whose response is derived by a homogenization procedure, for the analysis of out-of-plane loaded masonry panels. Petracca et al. (2017) presented a multiscale approach based on computational homogenization, developing an efficient technique where both macroscale and microscale were described by the same shell theory. Kirchhoff
Love and Mindlin
Reissner homogenized models were presented in Silva et al. (2018) for the evaluation of the out-of-plane response of regular masonry. Mortar joints were modeled using interface elements characterized by a nonlinear response able to reproduce fracture, frictional slip and crushing of the mortar. The out-of-plane response of masonry walls in cylindrical bending can be derived, adopting a beam model. In this framework, Addessi and Sacco (2018) proposed a beam multiscale approach considering the nonlinear response of the mortar joints and developing an analytical approach for solving the micromechanical problem. Di Re et al. (2018) proposed a Timoshenko beam finite element in the framework of a force-based formulation for nonlinear analysis of curved planar elements for masonry arches. A two-scale archto-beam homogenization procedure was adopted to reproduce the nonlinear response of periodic masonry materials made of linear elastic brick and nonlinear mortar layers. Coupling between micro- and macroscale have been proposed in the framework of the homogenization of periodic masonry, for instance, by Brasile et al. (2007a,b), who presented an in-plane multilevel strategy for the

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static and dynamical multiscale analysis of masonry walls. Greco et al. (2016) proposed a multiscale model for the study of the overall mechanical behavior of several masonry structures, taking into account damage growth and other nonlinear phenomena with the aim of reducing the computational cost of fully microscopic models. It was based on a multiscale/multidomain model with an adaptive capability able to zoom in on the zones incipiently affected by damage onset. This chapter presents a survey of the homogenization and multiscale analysis of regular, that is periodic, masonry walls subjected to in-plane loading conditions. The masonry is considered as a composite obtained by assembling bricks connected by mortar joints. Two scales are introduced: the structural level, i.e., the macroscale, and the material level, i.e., the microscale. It is worth noting that in this context the microscale is the so-called meso-scale in mechanics of materials. It is assumed that the two introduced scales are very far each other so that the principle of separation of the scales holds true. At the macroscale, 2D continuum models are introduced, that is, Cauchy and Cosserat formulations are adopted. At the microscale, because of the assumed periodicity, a UC is selected. The UC is analyzed introducing the Cauchy model and, in particular, three-dimensional (3D), two-dimensional (2D), and enriched plane state (EPS) micromodels are adopted. Different nonlinear constitutive laws are proposed for mortar and brick. A very important role is played by the constitutive model adopted for the mortar joints, as these are mostly responsible for the failure of masonry walls. The response of the mortar joints is assumed to be governed by a damaging process due to opening and shearslip mechanisms. Moreover, the unilateral effect due to reclosure of the mortar joint is accounted for to capture the restiffening for compressive loading. Frictional forces emerging in the mortar joints are also modeled in the constitutive law. Failure of the bricks is generally due to compressive stresses, while masonry often fails because of the presence of tensile stresses. Thus, a linear elastic behavior could be assumed for the bricks. Nevertheless, a damage model is introduced capable of properly describing their compressive failure. Once the constitutive laws for mortar and bricks have been introduced, the piecewise transformation field analysis (PWTFA) homogenization technique is described. This technique permits to derive the overall response of the masonry UC without performing finite element analysis (FEA) at the microlevel. Several numerical applications are presented to assess the material modeling, as well as the effectiveness of the adopted homogenization technique and multiscale approach. Initially, the response of the UC is numerically studied to check the capability of the proposed constitutive laws for brick and mortar of describing the masonry mechanical behavior and verifying the performance of the homogenization technique. Then, structural analyses of masonry elements are presented, comparing the results obtained using

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different modeling approaches among them and with available experimental evidences. Finally, concluding remarks are reported. In the following, the analysis is developed in the framework of small strain and displacement assumption, Voigt notation is adopted so that stress and strain tensors are arranged in vectors, and fourth-order constitutive tensors are represented as matrices.

10.2 Multiscale analysis of masonry walls A masonry wall is a 3D heterogeneous structural element made of blocks or bricks in direct contact with each other or joined by mortar. Thus, two different scales can be introduced for masonry structures: the structural scale and the heterogeneity scale. The distance between the two scales is a relevant parameter for the analysis of masonry structures. Schematically, three possible cases can be considered: G

G

G

When the two scales are almost close, such as for Cyclopean masonry, structural analyses should be performed, modeling the single block with its proper size and position; thus, computations should be based on discrete or finite element techniques that account for the exact position of each block in the texture. If there is significant distance between the scale of the structure and that of heterogeneity, the multiscale approach can be adopted, introducing a macroscopic homogeneous model at the structural level and a lower scale model at the heterogeneity level, accounting for the geometric and constitutive detailed aspects of the masonry texture. In this case, the kinematics description at the two scales is coupled. When the distance between the two scales is large enough, an uncoupled multiscale approach can be used that considers the two scales, the structural and the material scale, so far that there is no kinematic coupling. Hence, the constitutive law at the macrostructural scale is deduced by applying the homogenization approach at the microscale, that is, the heterogeneity scale.

This chapter presents the study of the mechanical response of a masonry wall loaded in its plane, adopting an uncoupled multiscale approach that relies on the homogenization theory. Masonry can be organized according to different textures. Most of them are characterized by a regular, periodic, arrangement that results in very high mechanical performance. Conversely, quasiperiodic or even irregular masonry arrangements are common in old and poor constructions. In the following, the case of periodic masonry is considered. Fig. 10.1A contains an example of a homogenized 2D masonry wall modeled at the macroscale; Fig. 10.1B shows the UC for the classic running bond texture, whose regular repetition forms the entire heterogeneous wall.

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FIGURE 10.1 Schematic of (A) masonry wall loaded in its plane and (B) unit cell characterizing the regular periodic texture.

10.3 Macromodels Different continuum models can be adopted at the macroscale for the 2D medium M, modeling the middle plane of the masonry wall. The classic Cauchy model could be employed, but this fails in the presence of constitutive response with strain softening. Indeed, masonry usually undergoes damaging during loading histories, characterized by a softening response of the material. As is known, softening causes the loss of ellipticity of the structural governing equations, which leads to localization and strong mesh dependency of the solution in FEA. To overcome these drawbacks, generalized continua (higher order, nonlocal, Cosserat) can be introduced to describe the structural problem. This has the effect of introducing a materiallength scale into the constitutive description, providing a natural way to make the structural response depending on the material’s internal scale.

10.3.1 Cauchy model The Cartesian coordinate system ðX1 ; X2 Þ is introduced in the 2D medium M, as shown in Fig. 10.1A, and the kinematics of the Cauchy model is described by the two-component displacement vector UðXÞ 5 fU1 U2 gT , where X 5 fX1 X2 gT denotes the location of the typical point on the surface M. The macroscopic strain vector E 5 fE11 E22 Γ 12 gT is defined as: @U1 @X1 @U2 E22 ðXÞ 5 : @X2 @U1 @U2 Γ 12 ðXÞ 5 1 @X2 @X1 E11 ðXÞ 5

ð10:1Þ

The stress vector work conjugate to the introduced strain is denoted as Σ 5 fΣ11 Σ 22 Σ 12 gT , where Σ 11 and Σ22 are the normal components and

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Σ 12 is the shear stress. Hence, vectors E and Σ collect the membrane strains and stresses. The equilibrium equations are: @Σ11 @Σ12 1 1 F1 5 0 @X1 @X2 @Σ12 @Σ22 1 1 F2 5 0 @X1 @X2

ð10:2Þ

where F1 and F2 are body forces along X1 and X2 directions, respectively. Nonlocal strain measures can be introduced in the Cauchy model in a straightforward way. To this end, the standard Gaussian weight function is introduced as:
 jX2Yj2 ΨðX 2 YÞ 5 exp 2 ; ð10:3Þ L2c where X is the point where the nonlocal strain is evaluated and Y is the generic point lying in the medium M. The quantity Lc denotes the radius of the nonlocal domain at the structural level. The nonlocal strain is, then, introduced as: ð 1 EðXÞ 5 EðYÞΨ ðX 2 YÞdA ð10:4Þ A M Herein, the nonlocal strain E is only used to drive the evolution of the damage internal variables, responsible for the softening of the material constitutive response, while the classic Cauchy continuum is adopted to describe kinematics and equilibrium of the structural model.

10.3.2 Cosserat model The 2D Cosserat continuum introduces the rotational kinematic degree of freedom in addition to the two translation components of the Cauchy model. Then, the displacement vector U 5 fU1 U2 ΦgT contains three independent kinematic fields, representing the translations U1 and U2 and the rotation Φ, respectively, at each point X of the medium M. ^ is expressed as: The Cosserat strain E   ^ 5 E11 E22 Γ 12 Θ K1 K2 T E ð10:5Þ where E11 , E22 and Γ 12 are the Cauchy strains introduced in Eq. (10.1), while K1 and K2 are the curvatures and Θ is the unsymmetric shear strain, that is, the rotational strain, defined as: K11 5

@Φ @X1

K22 5

@Φ @X2

Θ5

@U1 @U2 2 2 2Φ @X2 @X1

ð10:6Þ

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The dual quantities in the virtual work sense associated with the strain ^ which is components in Eq. (10.5) are collected in the stress vector Σ, expressed as:   ^ 5 Σ 11 Σ 22 Σ 12 Z M1 M2 T ð10:7Þ Σ The equilibrium equations for the 2D Cosserat model are: @Σ 11 @Σ 12 @Z 1 1 1 F1 5 0 @X2 @X1 @X2 @Σ 12 @Σ 22 @Z 1 2 1 F2 5 0 @X1 @X1 @X2

ð10:8Þ

@M1 @M2 1 1 2Z 1 C 5 0 @X1 @X2 where C is the body couple.

10.4 Micromodels The UC sizes in the plane of the wall are 2 a1 3 2a2 , while the wall thickness is t. The UC is denoted by Ω, its volume by V, and its midsurface by A. A Cartesian coordinate system ðx1 ; x2 ; x3 Þ is introduced, with the origin in the center of the UC. At the microscale the Cauchy model is considered and the displacement field u is represented as the sum of two parts: uðxÞ 5 uðxÞ 1 u ðxÞ;

ð10:9Þ

The term u, the kinematic map in the multiscale procedure, is the displacement occurring in the homogenized UC subjected to prescribed average strains, while u is the displacement perturbation field due to heterogeneity of the UC.

10.4.1 Three-dimensional micromodel Considering the Cauchy 2D model at the macroscale and a full 3D Cauchy formulation at the microscale, the kinematic map is written as: u1 ðxÞ 5 u2 ðxÞ 5 u3 ðxÞ 5

2D map 1 E11 x1 1 Γ 12 x2 2 1 Γ 12 x1 1 E22 x2 2

3D effect

perturbation 1 u1 ðxÞ 1 u2 ðxÞ

x3 e



1 u3 ðxÞ

;

ð10:10Þ

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where the displacement along the wall thickness is due to the Poisson effect under the condition of free stress on the two surfaces at x3 5 6 t=2. The quantity e does not depend on x, that is it is constant in the UC, and cannot be prescribed as this information is not known at the macroscale. The assumed regular texture of the masonry implies that the perturbation field is periodic, so that this has to satisfy the following periodicity and continuity conditions in the x1 2 x2 plane: u ð 2a1 ; x2 ; x3 Þ 5 u ða1 ; x2 ; x3 Þ u ðx1 ; 2a2 ; x3 Þ 5 u ðx1 ; a2 ; x3 Þ

’x2 ; x3 Að 2a2 ; a2 Þ 3 ð 2a3 ; a3 Þ ’x1 ; x3 Að 2a1 ; a1 Þ 3 ð 2a3 ; a3 Þ:

ð10:11Þ

By applying the compatibility condition, the strain field is derived as:

ε11 ε22 ε33 γ 12 γ 23 γ 13

5 5 5 5 5 5

2D map E11 E22

3D effect

e Γ 12

periodic strain 1 ε11 1 ε22 1 ε33 1 γ 12 γ 23 γ 13

with

ε11 5 ε22 5 ε33 5 γ 12 5 γ 23 5 γ 13 5



u1;1  u2;2  : u3;3   u1;2 1 u1;2   u2;3 1 u3;2   u1;3 1 u3;1 ð10:12Þ

The following points are noted: G

Because of the periodicity conditions (Eq. 10.11), the in-plane components of the periodic part of the strain ε have a null average, that is: Ð  ð10:13Þ Ω uα;β dV 5 0 with α; β 5 1; 2;

G

The displacement field has to be symmetric with respect to the wall midplane. Thus, the displacement components have to satisfy the relationships: u1 ðx1 ; x2 ; x3 Þ 5 u1 ðx1 ; x2 ; 2 x3 Þ u2 ðx1 ; x2 ; x3 Þ 5 u2 ðx1 ; x2 ; 2 x3 Þ

ð10:14Þ

u3 ðx1 ; x2 ; x3 Þ 5 2 u3 ðx1 ; x2 ; 2 x3 Þ:

G

In other words, the in-plane components are even functions of x3 and the out-of-plane component is an odd function of x3 .   The periodic part of the out-of-plane shear strain γ 13 and γ 23 are characterized by null average, that is: Ð  Ð  ð10:15Þ Ω uα;3 dV 1 Ω u3;α dV 5 0 α 5 1; 2; Indeed, the first integral in Eq. (10.15) is zero as u1 and u2 are even functions of x3 , so that their derivatives with respect to x3 are odd

Homogenization and multiscale analysis Chapter | 10

G

361

functions of x3 and the integral along x3 is zero. The second integral is even zero because of the in-plane periodicity of u3 , according to Eq. (10.11). To ensure that the quantity e represents the average of the strain component ε33 , the condition ð  u3;3 dV 5 0 ð10:16Þ Ω

has to be enforced. It can be noted that in Eq. (10.12) the terms due to perturbation, u3 , and 3D effect, x3 e, can be considered as one single contribution. Indeed, as the quantity e is not known, it is more convenient to introduce only the unknown field u3 , simultaneously accounting for x3 e 1 u3 . Moreover, as x3 e is constant with respect to x1 and x2 , while u3 is periodic, u3 is also a periodic function in the x1 2 x2 plane. Eventually, displacements and compatible strains in Ω can be written as: 1 u1 5 E11 x1 1 Γ 12 x2 1 u1 ðx1 ; x2 ; x3 Þ 2 u2 5

1 Γ 12 x1 1 E22 x2 1 u2 ðx1 ; x2 ; x3 Þ 2

u3 5 u3 ðx1 ; x2 ; x3 Þ

ε11 5 E11 1 u1;1 ε22 5 E22 1 u2;2 ε33 5 u3;3 γ 12 5 Γ 12 1 u1;2 1 u1;2

;

γ 23 5 u2;3 1 u3;2 γ 13 5 u1;3 1 u3;1 ð10:17Þ

with the further periodicity conditions: u3 ð2 a1 ; x2 ; x3 Þ 5 u3 ða1 ; x2 ; x3 Þ ’x2 ; x3 Að2 a2 ; a2 Þ 3 ð2 a3 ; a3 Þ u3 ðx1 ; 2 a2 ; x3 Þ 5 u3 ðx1 ; a2 ; x3 Þ ’x1 ; x3 Að2 a1 ; a1 Þ 3 ð2 a3 ; a3 Þ:

ð10:18Þ

10.4.2 Two-dimensional
enriched plane state micromodel The full 3D micromodel is generally too complex and requires significant computational effort. Simplified approximated models can be suitably derived, still allowing a 3D description of the problem at the microlevel. In this section, an effective model for describing the in-plane response of masonry walls, accounting for the 3D effect, is presented. This is the 2D kinematic EPS model proposed in Addessi and Sacco (2014, 2016b) for linear and nonlinear mechanical response, respectively.The 2D-EPS model is able to satisfactorily describe the mortar
brick interaction and can be seen as simplified kinematics for the 3D formulation or enriched kinematics for the classic 2D framework.

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PART | II Modeling of unreinforced masonry

The crucial point is the introduction of a suitable representation for the displacement components in the thickness direction. Recalling that the inplane displacement components are even functions of x3 and the out-of-plane component is an odd function of x3 , the displacement components along the x3 direction can be represented as a power series, with even functions for the in-plane components and odd functions for those out-of-plane. Thus, the following approximated form is introduced: 1 u1 ðxÞ 5 E11 x1 1 Γ 12 x2 1 u1 ðx1 ; x2 Þ 1 x23 v1 ðx1 ; x2 Þ 2 1 u2 ðxÞ 5 Γ 12 x1 1 E22 x2 1 u2 ðx1 ; x2 Þ 1 x23 v2 ðx1 ; x2 Þ ; 2

ð10:19Þ

u3 ðxÞ 5 x3 e1 ðx1 ; x2 Þ 1 x33 e2 ðx1 ; x2 Þ where u1 and u2 are the periodic parts of the in-plane components evaluated in the midplane of the wall, while v1 and v2 are additional components enriching the in-plane displacement description; e1 and e2 represent the outof-plane displacement components. All the functions u1 , u2 , v1 , v2 , e1 , and e2 have to satisfy the in-plane periodicity conditions. By applying the 3D compatibility operator to the displacement fields in Eq. (10.19), the components of the strain vector ε result in: 







ε11 5 E11 1 u1;1 1 x23 v1;1 ε22 5 E22 1 u2;2 1 x23 v2;2 ε33 5 e1 1 3x23 e2 γ 12 5 Γ 12 1 ðu1;2 1 u2;1 Þ 1 x23 ðv1;2 1 v2;1 Þ

:

ð10:20Þ

γ 23 5 x3 e1;2 1 x33 e2;2 1 2x3 v2 γ 13 5 x3 e1;1 1 x33 e2;1 1 2x3 v1 Denoting by σ the conjugate stress vector, the equilibrium equations associated with the introduced kinematics can be deduced through the virtual displacement theorem: ð 05

Ω

½δeðuÞT σ dV;

ð10:21Þ

which leads to: 2 3

   ð
2 2 2 t t t     0 5 4 δu1;1 1 δv1;1 σ11 1 δu2;2 1 δv2;2 σ22 1 δe1 1 δe2 σ33 5dA 12 12 4 A 2 3 :
 
ð 3 t     1 4t δu1;2 1 δu2;1 1 δv1;2 1 δv2;1 5σ12 dA 12 A ð10:22Þ

Homogenization and multiscale analysis Chapter | 10

363

A simplified version of the 2D-EPS model can be obtained considering v1 5 v2 5 e2 5 0 in Eq. (10.19). In such a case, the fiber orthogonal to the wall midplane remains straight after deformation, but its length can change. To be noted is that the wall transverse deformation is not uniform in the UC but is a function of the x1 ; x2 coordinates.

10.4.3 Two-dimensional micromodel The simplest micromodel can be obtained by considering the classic 2D kinematics description only governed by the two displacement components, u1 and u2 , for which the macro-micro map results in: u1 5 E11 x1 1 u2 5

1 Γ 12 x2 1 u1 ðx1 ; x2 Þ 2

; 1 Γ 12 x1 1 E22 x2 1 u2 ðx1 ; x2 Þ 2

ð10:23Þ

and the compatible strains are: ε11 5 E11 1 u1;1 ε22 5 E22 1 u2;2

:

γ 12 5 Γ 12 1 ðu1;2

ð10:24Þ

1 u2;1 Þ

10.4.4 Two-dimensional micromodel for Cosserat macromodel When considering the 2D Cosserat model at the macroscale, the UC is subjected to the macroscopic strain components collected in vector E^ defined in Eq. (10.5). The displacement field for the 2D Cauchy micromechanical model is accordingly expressed as (Addessi et al., 2010):   1 ^ 2 x1 x2 K1 2 1 x2 K2 1 u ðx1 ; x2 Þ u1 5 E11 x1 1 Γ 12 x2 2 α x32 2 3ρ2 x21 x2 Θ 1 2 2 2   1 ^ 1 1 x2 K1 1 x1 x2 K2 1 u ðx1 ; x2 Þ u2 5 Γ 12 x1 1 E22 x2 2 ρ2 α ρ2 x31 2 3 x1 x22 Θ 2 2 2 1 ð10:25Þ where α5

5 a21 1 a22 4 a41

ρ5

a2 ; a1

ð10:26Þ

and 2 ^ 5 Θ 1 1 ρ 2 1 Γ 12 : Θ 2 ρ2 1 1

ð10:27Þ

;

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PART | II Modeling of unreinforced masonry

Definition (Eq. 10.27) ensures that the Cauchy deformation modes can be activated independently from the Cosserat modes. The stress component ^ is denoted with Z. ^ work conjugate to Θ The compatible strain fields for the Cauchy medium subjected to the prescribed overall Cosserat strains result in: ^ 2 x2 K1 1 u ε11 5 E11 1 6αρ2 x1 x2 Θ 1;1 ^ 1 x1 K2 1 u ε22 5 E22 1 6αρ2 x1 x2 Θ

:

2;2

ð10:28Þ

^ 1 ðu 1 u Þ γ 12 5 Γ 12 1 3αðρ2 2 1Þðx22 2 ρ2 x21 ÞΘ 1;2 2;1 



Furthermore, the following additional conditions with respect to the Cauchy homogenization are required to ensure that the displacement perturbation fields produce zero average strain: Ð a2  Ð a1  ð10:29Þ 2a1 u1 ðx1 ; 2 a2 Þ dx1 5 0; 2a2 u2 ð2 a1 ; x2 Þ dx2 5 0:

10.5 Masonry constitutive law Masonry is a cohesive material characterized by a compressive strength much higher than the tensile strength. As matter of fact, the tensile response is mostly governed by the cohesion arising between the masonry constituents, mortar and bricks, while failure in compression mainly occurs due to the cracking of the bricks that are subjected to a dilatation in the direction of the wall thickness accelerated by the interaction with the soft mortar. At the mortar
brick interface frictional stresses can arise, playing a significant role in the overall response of the masonry, as these activate energy dissipation mechanisms under seismic actions. To reproduce the mentioned mechanisms, different constitutive laws are proposed for mortar and brick. In particular, the mortar constitutive law also accounts for the mortar
brick interaction. It has to be noted that for the 3D and 2D-EPS micromodels full 3D constitutive laws have to be specified, whereas for the 2D micromodel plane stress, plane strain or generalized plane state is assumed.

10.5.1 Mortar constitutive law A constitutive model accounting for damage, friction and unilateral contact is considered for the mortar material. As mentioned previously, the mortar failure is indeed due to opening and/or sliding of the mortar from the joined brick. The presented model, illustrated in Addessi and Sacco (2016b), is based on the coupled damage-plastic constitutive relationship proposed in

Homogenization and multiscale analysis Chapter | 10

365

Sacco (2009), Addessi et al. (2010), and Marfia and Sacco (2012). Specifically, this considers the following nonlinear mechanisms: G

G

G

G

Damage due to shear and tensile opening of the joints, introducing the variable Dm , denoted in the following as the primary damage variable. Damage due to the presence of tensile strains occurring in the mortar plane, related to transverse compressive stress, introducing the variable Dm c , denoted in the following as the secondary damage variable. Reclosure of the joint opening, introducing the inelastic strain c accounting for the unilateral effect. Frictional slip of the joint, introducing the inelastic shear strain p.

A local coordinate system ðxT ; xN ; x3 Þ is set in the typical with xT and xN denoting directions parallel and orthogonal respectively, while x3 is the direction of the wall thickness, Fig. 10.2. Then, two damage mechanisms are considered in the mortar

mortar joint, to the joint, as shown in joint:

1. Dm denotes damage due to possible opening and sliding, arising because of the dilatation εNN and shear strains γ TN and γ 3N . 2. Dm c is the damage activated by the presence of tensile strains εTT and ε33 , occurring in the plane of the mortar joint. Concerning the first damage mechanism, the special constitutive law proposed in Sacco (2009) and based on the interface mechanical model discussed in Alfano and Sacco (2006) and Uva and Salerno (2006) is introduced. At the typical point of the mortar joint shown in Fig. 10.3A, it is assumed that fractures can nucleate and evolve in the RVE along the plane xT 2 x3 , as illustrated in Fig. 10.3B. A simple scheme of the RVE is contained in Fig. 10.3C, where the total volume is split into undamaged and damaged parts. The total, undamaged, and damaged areas in the plane xT 2 x3 are denoted with A, Au , and Ad , respectively. Hence, the damage parameter is defined as the ratio between the damm aged and total area of the RVE, that is,  D 5 Ad =A. Then, the average stress T in the RVE, defined as the vector σ 5 σTT σNN σ33 τ TN τ 3N τ T3 , is deduced as: σ 5 ð1 2 Dm Þσ u 1 Dm σ d ;

ð10:30Þ

where σ and σ are the stresses arising in the undamaged and damaged parts of the RVE. u

d

FIGURE 10.2 Local coordinate system for the mortar joint.

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PART | II Modeling of unreinforced masonry

FIGURE 10.3 Schematic of the opening-sliding damage model; (A) mortar joint; (B) representative volume element of the mortar joint; and (C) simplified representative volume element.

The two stress vectors σ u and σ d are related to the strain vector in the mortar, e, by the following constitutive equations: m σu 5 ð1 2 Dm c ÞC ε;

m σ d 5 ð1 2 Dm c ÞC ðε 2 ðc 1 pÞÞ;

ð10:31Þ

which involve the second type of damage mechanism. In Eq. (10.31), Cm denotes the isotropic elasticity matrix of the undamaged mortar, while the strain vectors c and p account for possible unilateral opening effect and friction sliding, respectively. Substituting Eq. (10.31) into Eq. (10.30), the stress at the typical point of the mortar joint is, eventually, obtained in the form: m ^ σ 5 ð1 2 Dm c ÞC ðε 2 πÞ;

ð10:32Þ

where π^ 5 Dm ðc 1 pÞ is the vector that at the same time accounts for damage Dm , unilateral contact c, and friction sliding p. Eq. (10.32) can be written in the equivalent form: σ 5 Cm ðε 2 πÞ;

ð10:33Þ

^ where π 5 π^ 1 Dm c ðε 2 πÞ. The sliding in the mortar joint is assumed to occur on the plane parallel to mortar midplane xT 2 x3 , so that this is described by the inelastic strain components γ pTN and γ p3N ; the components of the unilateral contact and sliding vectors, then, result: 9 9 8 8 0 > εTT > > > > > > > > > > > > εNN > > 0 > > > > > > > > > = = < < 0 ε33 ; p5 ; ð10:34Þ c 5 hðεNN Þ p γ > 0 > > > > > > > > > > > TN p > > > > 0 > γ > > > > > > ; ; : : 3N > γ T3 0 where hðεNN Þ is the Heaviside function, which assumes the following values: hðεNN Þ 5 0 if εNN # 0 and hðεNN Þ 5 1 if εNN . 0.

Homogenization and multiscale analysis Chapter | 10

367

The friction effect is modeled as a classic plasticity problem. The evolution law of the inelastic slip strain components γ pTN and γ p3N is governed by the Coulomb yield function: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϕðσ d ; ζÞ 5 μðζÞ σdNN 1 ðτ dTN Þ2 1 ðτ d3N Þ2 ; ð10:35Þ where μ is the friction parameter, evolving according to the following exponential law:   μðζÞ 5 ðμf 2 μi Þ 1 2 e2ωζ 1 μi ; ð10:36Þ μi and μf being the initial and final friction values, respectively; ω is the exponential rate parameter; and ζ is the accumulated plastic slip strain at the current time τ^ , defined as: ð τ^ ζ5 jγ_ pTN jdτ: ð10:37Þ 0

A non associated flow rule is considered for the slip: τ dTN γ_ pTN 5 λ_ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðτ dTN Þ2 1 ðτ d3N Þ2

τ d3N γ_ p3N 5 λ_ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ðτ dTN Þ2 1 ðτ d3N Þ2

ð10:38Þ

with the following loading
unloading Kuhn
Tucker conditions: λ_ $ 0

ϕðσ d Þ # 0; λ_ ϕðσ d Þ 5 0;

ð10:39Þ

where λ is the inelastic multiplier. A model that accounts for the coupling of fracture modes I and II is considered to evaluate the primary damage parameter Dm . The two quantities ηε and ηγ , which depend on axial and shear first cracking strains ε0 and γ 0 , peak values of the stresses σ0 and τ 0 , and fracture energies GcI and GcII are introduced in the form: ε0 σ 0 γ τ0 ηε 5 ; ηγ 5 0 ð10:40Þ 2 GcI 2 GcII : The equivalent strain measures Yε and Yγ are defined as: Yε 5 ðhεN i1 Þ2 ;

Yγ 5 γ 2TN 1 γ 23N ;

ð10:41Þ

where the operator hi1 gives the positive part of the number. To overcome the localization phenomenon due to the strain-softening branch in the constitutive relationship, the nonlocal definition Y ε and Y γ of the equivalent strain measures Yε and Yγ are introduced as: ð 1 Ð Y ε=γ 5 Y ε=γ ðyÞ ψðx 2 yÞ dV ð10:42Þ Ωm ψðx 2 yÞ dV Ωm

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PART | II Modeling of unreinforced masonry

where Ωm is the mortar domain, x is the point where the nonlocal strain is evaluated, and y is the generic point in Ωm . The weight function ψ is chosen as: ! x2y 2 ψðx 2 yÞ 5 exp 2 ; ð10:43Þ lc 2 lc being the radius of the nonlocal domain at microlevel. Then, the following quantities are introduced on the basis of the nonlocal equivalent strain measures: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uY ε Yγ 1 η 5 2 ðY ε ηε 1 Y γ ηγ Þ; Y 5 t 2 1 2 ; N 5 Y ε 1 Y γ : ð10:44Þ N ε0 γ0 Finally, the primary damage is evaluated according to the following law:

 m m ~ ð10:45Þ D 5 max 0; min fD ; 1g ; history

where

history

Y 21 m : D~ 5 ð1 2 ηÞY

ð10:46Þ

Note that, for Dm 5 1 and εNN . 0, π 5 c 1 p results. From the constitutive Eq. (10.32) it arises that stress components σTT , σNN , σ33 , and τ T3 are zeros. Moreover, the stress admissibility condition in (10.35) enforces that τ TN 5 τ 3N 5 0, which implies γ pTN 5 γ TN and γ p3N 5 γ 3N . Instead, for Dm 5 1 and εNN . 0, π 5 ε results, and all the stress components are zero. This means that when the full damage is reached and the mortar joint is open, the whole stress state in the mortar is zero. The evolution law of the secondary damage parameter Dm c is assumed to be governed by tensile strains arising in the plane of the mortar joint. The equivalent strain measure Yc is defined as: Yc 5 ðhεT i1 Þ2 1 ðhε3 i1 Þ2 :

ð10:47Þ

Denoting by Y c the nonlocal counterpart of Yc , evaluated analogously to Eq. (10.42), the following quantity can be determined: vffiffiffiffiffi u ^21 uY c Y m ^ ~ with Y 5 t 2 : Dc 5 ð10:48Þ ^ ε0 ð1 2 ηN ÞY on the basis of which the secondary damage variable, associated with the tensile strains arising in the mortar plane, is computed as:

 m m ~ ð10:49Þ Dc 5 max 0; min fDc ; 1g : history

history

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369

10.5.2 Brick constitutive law The following stress
strain relationship based on the isotropic elasticdamage model presented in Addessi and Sacco (2016a,b) is adopted for the bricks: σ 5 ð1 2 Db Þ Cb ε ;

ð10:50Þ

where Cb denotes the 6 3 6 isotropic constitutive matrix of the brick and Db the damage variable, with 0 # Db # 1. The damage limit function is expressed in terms of the nonlocal equivalent strain εeq , in the form: F 5 εeq 2 εt ;

ð10:51Þ

where εt is the tensile strain threshold. As for εeq , this is computed as: ð 1 εeq 5 Ð εeq ðyÞ ψðx 2 yÞ dV ð10:52Þ Ωb Ωb ψðx 2 yÞ dV where the local equivalent strain εeq is introduced as: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + u* 3 3 X 3 X u X ð12δ Þ ij 2 hεi i2 hεj i2 2 εo ; hεi 1εo i1 2κ εeq 5 t 2 i51 i51 j51

ð10:53Þ

1

with hi1=2 denoting the positive/negative part of the variable and δij the Kronecker’s symbol. The material parameter κ regulates the reduction of the equivalent tensile strain due to the presence of compressive strains, while εo is a small regularization parameter ensuring the convexity of the elastic domain. The current value of the damage variable in the brick is then computed as: Db 5 max f0; minfD~ ; 1gg: b

history

ð10:54Þ

where variable D~ evolves driven by the nonlocal equivalent strain εeq according to the following exponential law: b

b D~ 5 1 1

1 e2βðεeq 2εt Þ ðεeq 2εu Þ2 ðεeq εu 1 εeq εt 2 2ε2t Þ; εeq ðεt 2εu Þ3

ð10:55Þ

Eq. (10.55) contains the ultimate value of the tensile equivalent strain corresponding to the full damage εu and a parameter governing the softening branch of the stress
strain relationship, β, set as: 1 β 5 βt 1 ðβ 2 β t Þ; ð10:56Þ 1 1 expðαðI1 2 I1m ÞÞ c where β c and β t are the values of β for a mostly contracted or elongated strain state, respectively; α rules the rate of variation of β from β c to β t and

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PART | II Modeling of unreinforced masonry

viceversa; and I1 is the value of the strain first invariant and I1m its value corresponding to β 5 ðβ c 1 β t Þ=2. As detailed in Addessi and Sacco (2016a), the material parameters can be set on the basis of the experimental data, given in terms of compressive strength σ0c and damage energy Gc .

10.6 Piecewise transformation field analysis procedure ^ is prescribed in the UC, the homogenizaOnce the macroscopic strain E or E tion problem consists of determining: G G G G

the the the the

displacement field uðxÞ; total strain field εðxÞ; b set of internal variables Dm , Dm c , D , c and p; and stress field σðxÞ:

which satisfies the classic governing equations of continuum mechanics, that is, compatibility, equilibrium, and constitutive relationships with suitable boundary conditions. Several numerical techniques can be used to solve the introduced micromechanical problem. Commonly, the finite element method is employed as it is simple, effective, and very popular. Other approaches based on finite differences, meshless methods, and, more recently, the virtual element method (Artioli et al., 2018) can also be conveniently adopted. These numerical methods are commonly very expensive from a computational point of view, when the homogenization procedure is introduced into a multiscale framework. Standing on this, reduced order models (ROMs) have recently received great attention, as these permit us to determine a suitable approximate solution of the micromechanical problem and satisfactorily describe the overall response of the composite material. In particular, referring to the 2D Cauchy macromodel, on the basis of the prescribed macroscopic strain E, it is possible to determine the macroscopic stress Σ satisfying the Hill
Mandel equation: ð 1 T E Σ5 eT σ dV: ð10:57Þ V Ω The same relation holds in the case of the 2D Cosserat model considered ^ ^ and Σ. at the macroscale, where E and Σ are substituted by E A very interesting ROM technique is TFA, initially proposed by Dvorak (1992) and modified, generalized, and improved in a series of papers: Dvorak and Bahei-El-Din (1997), Michel and Suquet (2003), Sepe et al. (2013), Fritzen and Bohlke (2014), Covezzi et al. (2017), Marfia and Sacco (2018). Herein, the so-called piecewise TFA technique is described for the 3D micromechanical problem. The case of the 2D micromodel can be easily derived by the 3D formulation. The UC is partitioned in n homogeneous subsets, with the sth S subset Ωs characterized by the volume Vs . These satisfy the relationships ns51 Ωs 5 Ω

Homogenization and multiscale analysis Chapter | 10

371

T and ns51 Ωs 5 [. The TFA procedure follows the steps discussed as follows. These refer to the Cauchy model at the macroscale, but can be easily generalized for the Cosserat formulation. The constitutive Eqs. (10.33) or (10.50) is written in s-th subset Ωs as: σs 5 Cs ðεs 2 πs Þ;

ð10:58Þ

where C denotes the elasticity matrix of the material of subset Ω and π s is the inelastic strain assumed as uniform in Ωs . The effects of the prescribed macroscopic strain E in each subset Ωj ðj 5 1; . . .; nÞ are determined through the introduction of n localization 6 3 3 matrices LjE ðxÞ that permit us to evaluate the local strain and, hence, the local stress at point xAΩj due to E by the relationships: s

s

ej ðxÞ 5 LjE ðxÞ E sje ðxÞ 5 Cj ej ðxÞ

:

ð10:59Þ

The effects of the uniform inelastic strain πs acting in each subset Ωj ðj 5 1; . . .; nÞ are determined by introducing the n2 localization matrices Ljπs ðxÞ, with dimensions 6 3 6, able to give the total pjs and elastic ηjs strains and the stress σjπs at xAΩj due to πs , as: pjs ðxÞ 5 Ljπs ðxÞ πs hjs ðxÞ 5 pjs ðxÞ 2 δjs π s ;

ð10:60Þ

σ jπs ðxÞ 5 Cj ηjs ðxÞ where δjs is the Kronecker symbol (δjs 5 1 if j 6¼ s, δjs 5 0 otherwise). The solution is determined by superimposing the effects of the prescribed macroscopic deformation E with those due to inelastic strains πs with s 5 1; . . .; n. In particular, the total strain in the subset Ωj is evaluated as the sum of the contributions given by Eqs. (10.59) and (10.60): εj ðxÞ 5 ej ðxÞ 1

n X

pjs ðxÞ 5 LjE ðxÞE 1

s51

n X

Ljπs ðxÞπs :

ð10:61Þ

s51

The average total strain and stress, εj and σ j , in the jth subset are evaluated as: s n X X j j ε j 5 ej 1 pjs 5 LE E 1 Lπs πs ; ð10:62Þ i51

s51

σ j 5 Cj ðεj 2 πj Þ:

ð10:63Þ Lje ðxÞ

Ljπs ðxÞ

where the average of the localization matrices and set are computed as: ð ð 1 1 j j LE 5 LjE ðxÞ dV Lπs 5 Lj s ðxÞ dV: Vj Ωj Vj Ω j π

in each sub-

ð10:64Þ

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PART | II Modeling of unreinforced masonry

FIGURE 10.4 State of stress evaluated at two points characterized by opposite x3 coordinate.

The average total strain and stress are assumed to govern the evolution of the inelastic strain in each subset. In other words, the constitutive and evolution equations for mortar and bricks are written in terms of averages in each subset. It is worth noting that for a wall loaded in its plane, it is expected that the solution results be symmetric with respect to the wall midplane. In particular, considering the stress state on the midplane of the mortar joint evaluated at two points characterized by opposite x3 coordinates, as illustrated in Fig. 10.4, the following conditions have to be satisfied: σ11 ðx1 ; x2 ; x3 Þ 5 σ11 ðx1 ; x2 ; 2 x3 Þ σ22 ðx1 ; x2 ; x3 Þ 5 σ22 ðx1 ; x2 ; 2 x3 Þ σ33 ðx1 ; x2 ; x3 Þ 5 σ33 ðx1 ; x2 ; 2 x3 Þ τ 12 ðx1 ; x2 ; x3 Þ 5 τ 12 ðx1 ; x2 2 x3 Þ τ 31 ðx1 ; x2 ; x3 Þ 5 2 τ 31 ðx1 ; x2 ; 2 x3 Þ

;

ð10:65Þ

τ 32 ðx1 ; x2 ; x3 Þ 5 2 τ 32 ðx1 ; x2 ; 2 x3 Þ so that the strain field according to Eq. (10.14) is characterized by: ε11 ðx1 ; x2 ; x3 Þ 5 ε11 ðx1 ; x2 ; 2 x3 Þ ε22 ðx1 ; x2 ; x3 Þ 5 ε22 ðx1 ; x2 ; 2 x3 Þ ε33 ðx1 ; x2 ; x3 Þ 5 ε33 ðx1 ; x2 ; 2 x3 Þ γ 12 ðx1 ; x2 ; x3 Þ 5 γ 12 ðx1 ; x2 2 x3 Þ γ 31 ðx1 ; x2 ; x3 Þ 5 2 γ 31 ðx1 ; x2 ; 2 x3 Þ

ð10:66Þ

γ 32 ðx1 ; x2 ; x3 Þ 5 2 γ 32 ðx1 ; x2 ; 2 x3 Þ The stress components τ 31 and τ 32 are odd functions of x3 , and these are essentially related to the brick
mortar interaction arising from the different elastic properties of the two masonry components. For this reason, it is expected that their values are significantly lower than the other stress components; moreover, the average values of τ 31 and τ 32 in the masonry thickness are zeros. Analogously, the shear strains γ 31 and γ 32 are negligible and these have zero average in the thickness.

Homogenization and multiscale analysis Chapter | 10

373

Relying on these considerations, it is reasonably supposed that stress and strain components have a small variation along the thickness coordinate, assumed as uniform. Thus, in the PWTFA technique here the UC is only partitioned in the 2D framework, considering the subsets involving the whole thickness of the masonry wall.

10.7 Numerical applications This section contains some numerical applications aimed at: G G

G

G

validating the presented micromodels (3D, 2D-EPS, 2D); assessing the capability of the proposed PWTFA procedure of reproducing the response of the UC; investigating the influence of the Cosserat modes on the structural behavior of masonry elements; verifying the effectiveness of the multiscale approach.

The section is organized in two subsections. First, some micromechanical and homogenization problems are illustrated; then, some structural applications based on the multiscale technique are reported.

10.7.1 Unit Cell response The micromodels presented in Sections 10.4 and 10.5 are used to analyze the response of the running bond UC shown in Fig. 10.1B subjected to simple loading conditions. First, Cauchy macroscopic strain components are applied to the UC, that is, uniaxial compression, compression
tension, and compression
symmetric shear tests are performed. Then, the additional Cosserat macroscopic strains are considered, applying pure flexural loading to the UC, as well as combining this with the compression
symmetric shear test. Micromechanical results are also compared with those obtained by applying the presented PWTFA procedure. The geometrical parameters are set as a1 5 130 mm, a2 5 130 mm, b 5 250 mm, h 5 120 mm, s 5 10 mm, and t 5 55 mm, where the brick dimensions are b 3 h 3 t. The mechanical parameters in Tables 10.1 and 10.2 are adopted for mortar joints and bricks, respectively. It has to be noted that all the applications assume the bricks as linear elastic, with Young’s modulus and Poisson ratio as reported in Table 10.2 except for the uniaxial compression test, where the damage model presented in Section 10.5.2 is adopted, considering the relevance of the brick failure mechanisms in this case. The material nonlocal radius lc is set equal to 15 and 25 mm for mortar and bricks, respectively. Quadrilateral four-node and brick eight-node finite elements are adopted for the 2D and 3D micromechanical analyses, using 2 3 2 and 2 3 2 3 2 Gauss integration rules, respectively.

374

PART | II Modeling of unreinforced masonry

TABLE 10.1 Material parameters adopted for the mortar. Em ðMPaÞ

1000

νm

0.15

ε0

5.0 3 1024

γ0

1.0 3 1023

GcI ðMPaÞ

1.25 3 1023

GcII ðMPaÞ

5.0 3 1023

μi

0.5

μf

0.5

ω

0.0

TABLE 10.2 Material parameters adopted for the bricks. Eb ðMPaÞ

18,000

νb

0.15

εt

1.11 3 1024

κ

0.03

εo

1 3 1025

εu

1.11 3 1022

βt

10,000

βc

1000

α

8000

10.7.1.1 Uniaxial compression test The UC is first subjected to a monotonically increasing vertical compressive macrostrain E22 . Fig. 10.5 contains the plot of the homogenized macroscopic normal stress Σ 22 versus the applied macroscopic compression strain E22 . The results computed through the 2D-EPS model (line with full circles) are compared with those evaluated by the full 3D model (line with empty circles), adopting a regular mesh of 338 finite elements in the first case and 338 3 3 finite elements for the 3D computations.

Homogenization and multiscale analysis Chapter | 10

375

40 35 30

–Σ22 (MPa)

UC EPS 25

UC 3D

20 15 10 5 0

0

0.2

0.6

0.4

–E22

0.8

1.0 x10–2

FIGURE 10.5 Uniaxial compression response of the UC: 2D-EPS model (line with full circles), 3D model (line with empty circles).

After the initial linear elastic phase, the damage mechanism starts in the vertical joints due to the horizontal tensile strains. Subsequently, the bricks are also significantly damaged due to the tensile state emerging both inplane and out-of-plane. Note that a very good match is obtained, proving that the proposed 2D-EPS model correctly reproduces this mechanism, giving a correct estimation of the masonry compressive strength, as it can take into account the transverse strains thanks to the introduced enriched kinematics. Indeed, the classic 2D model assuming plane strain or plane stress conditions would fail in correctly describing the compression response of the masonry specimens (Addessi and Sacco, 2016a). The effectiveness of the adopted nonlocal formulation described in Section 10.5 is shown in Fig. 10.6, where the results obtained by using two different meshes, mesh1 and mesh2 made of 98 and 338 finite elements, respectively, are plotted. The curves with full symbols refer to the nonlocal model, and those with empty symbols to the local model. Note that the two meshes give indistinguishable results in the case of the nonlocal model, while the local formulation shows the well-known mesh dependency.

10.7.1.2 Compression
tensile test In the second test the UC is subjected to the combined effect of the compression strain in the vertical direction, kept constant during the whole history, E22 5 2 4:0 3 1024 , and the tensile strain E11 applied in the horizontal direction and varying from 0 to 30.0 3 1024, to 25.0 3 1024.

376

PART | II Modeling of unreinforced masonry 40 Nonlocal mesh1 Nonlocal mesh2 Local mesh1 Local mesh2

35

−Σ22 (MPa)

30 25 20 15 10 5 0

0

0.2

0.4

−E22

0.6

0.8

1 x10–2

FIGURE 10.6 Uniaxial compression response of the UC: 2D-EPS nonlocal model mesh1 (line with full triangles), 2D-EPS nonlocal model mesh2 (line with full circles), 2D-EPS local model mesh1 (line with empty triangles), 2D-EPS local model mesh2 (line with empty circles).

3.0

Σ11

B

C

A

0.0

E 11

(MPa)

O

–3.0 FEA PWTFA

–6.0 –0.001

0

0.001

0.002

0.003

FIGURE 10.7 Compression
tensile test: comparison of the results obtained by the 2D micromechanical model and piecewise transformation field analysis (PWTFA) procedure.

Due to the double symmetry of the geometrical scheme and loading condition, the micromechanical analysis is performed by considering only a quarter of the UC under suitable boundary conditions, adopting a regular mesh of 169 2D finite elements, based on the 2D nonlocal micromodel. Fig. 10.7 shows the UC response curve plotted in terms of the macroscopic stress component Σ11 versus the macroscopic strain E11 , where the solid curve refers to the results evaluated through the PWTFA procedure, while the diamond symbols concern the micromechanical solution obtained by performing

Homogenization and multiscale analysis Chapter | 10

(A)

Deformed UC

Damage

377

Plastic shear strain –9.98E–02 1.75E–04 1.00E–01 2.00E–01 3.00E–01 4.00E–01 5.00E–01 6.00E–01 7.00E–01 8.00E–01 9.00E–01 1.00E+00 1.10E+00

–2.21E–03 –1.99E–03 –1.76E–03 –1.54E–03 –1.32E–03 –1.09E–03 –8.70E–04 –6.47E–04 –4.24E–04 –2.01E–04 2.20E–05 2.45E–04 4.68E–04

(B) –9.98E–02 1.75E–04 1.00E–01 2.00E–01 3.00E–01 4.00E–01 5.00E–01 6.00E–01 7.00E–01 8.00E–01 9.00E–01 1.00E+00 1.10E+00

–3.12E–03 –2.62E–03 –2.13E–03 –1.64E–03 –1.14E–03 –6.47E–04 –1.53E–04 3.41E–04 8.35E–04 1.33E–03 1.82E–03 2.32E–03 2.81E–03

–9.98E–02 1.75E–04 1.00E–01 2.00E–01 3.00E–01 4.00E–01 5.00E–01 6.00E–01 7.00E–01 8.00E–01 9.00E–01 1.00E+00 1.10E+00

–1.25E–02 –8.58E–03 –4.68E–03 –7.78E–04 3.13E–03 7.03E–03 1.09E–02 1.48E–02 1.87E–02 2.26E–02 2.65E–02 3.04E–02 3.44E–02

(C)

FIGURE 10.8 Compression
tensile test: deformed configurations (first column), maps of damage (second column), and plasticity (third column) evaluated for different values of the strain E22 as indicated in Fig. 10.7.

a nonlinear FEA. The initial linear elastic branch is followed by the nonlinear phase due to the damage progression in the head joints and the subsequent damage and frictional slip of the bed joints. The unloading and reverse loading paths are characterized by an elastic and perfect plastic response, followed by the unilateral contact effect of the head joints. Note that at this point only the frictional mechanism is activated, corresponding to the ultimate strength of the UC associated to the prescribed vertical compression. The results obtained with the two models are in very satisfactory agreement. Fig. 10.8 contains the UC deformed configurations as well as the damage and shear plastic strain maps at three steps of the analysis denoted by points A, B, and C in Fig. 10.7. It emerges that damage and plastic strains are almost uniformly distributed along the longitudinal direction of the head and bed joints, which validates the main assumption on which the nonlinear homogenization procedure relies.

378

PART | II Modeling of unreinforced masonry 2.0

Σ12

B A

C

(MPa)

1.0

0.0

O

Γ12

–1.0 FEA PWTFA

–2.0 –0.002

–0.001

0

0.001

0.002

FIGURE 10.9 Compression
symmetric shear test: Comparison of the results obtained by the 2D micromechanical model and piecewise transformation field analysis (PWTFA) procedure.

10.7.1.3 Compression
symmetric shear test Then, the compression
symmetric shear test is performed combining a constant compression in the vertical direction, E22 5 2 3:0 3 1024 , and the symmetric shear strain Γ 12 varying from 0 to 20.0 3 1024 to 220.0 3 1024. Based on the symmetry of the geometrical scheme, the micromechanical analysis is performed by considering only half the UC, applying to this suitable periodicity conditions. In Fig. 10.9 the macroscopic shear stress component Σ 12 is plotted versus the macroscopic shear strain Γ 12 , where the solid curve refers to the PWTFA procedure and the diamond symbols concern the micromechanical solution. In this case, the two curves perfectly match. The UC deformed configuration together with damage and shear plastic strain maps at three steps of the analysis, that is, at points A, B, and C in Fig. 10.9, are reported in Fig. 10.10. After the linear elastic phase, the nonlinear response stage occurs due to the onset of both the damaging and plasticity mechanisms in head and bed joints (Fig. 10.10A). During the subsequent unloading and reloading paths, the evolution of the friction plasticity governs the UC response. 10.7.1.4 Flexural test The following sections are devoted to investigate the effects of the presence of the Cosserat macrostrains on the UC nonlinear response. First, only thecurvature component K2 is applied to the UC, varying this from 0 to 4.0 3 1026, to 2.0 3 1026, to 10.0 3 1026, to 0. The entire UC is now considered and discretized as reported in Section 10.7.1.1. Fig. 10.11 shows the Cosserat macroscopic couple M2 versus the curvature K2 , with the solid curve concerning the results obtained through the PWTFA procedure and the

Homogenization and multiscale analysis Chapter | 10 Deformed UC

Damage

379

Plastic shear strain

(A)

(B)

(C)

–7.29E–02 1.28E–04 7.32E–02 1.46E–01 2.19E–01 2.92E–01 3.65E–01 4.38E–01 5.11E–01 5.84E–01 6.58E–01 7.31E–01 8.04E–01

–1.35E–04 2.36E–07 1.35E–04 2.70E–04 4.05E–04 5.40E–04 6.75E–04 8.10E–04 9.45E–04 1.08E–03 1.22E–03 1.35E–03 1.49E–03

–9.56E–02 1.68E–04 9.60E–02 1.92E–01 2.88E–01 3.83E–01 4.79E–01 5.75E–01 6.71E–01 7.67E–01 8.62E–01 9.58E–01 1.05E+00

–6.70E–04 1.17E–06 6.72E–04 1.34E–03 2.01E–03 2.69E–03 3.36E–03 4.03E–03 4.70E–03 5.37E–03 6.04E–03 6.71E–03 7.38E–03

–9.98E–02 1.75E–04 1.00E–01 2.00E–01 3.00E–01 4.00E–01 5.00E–01 6.00E–01 7.00E–01 8.00E–01 9.00E–01 1.00E+00 1.10E+00

–2.26E–03 3.97E–06 2.27E–03 4.54E–03 6.80E–03 9.07E–03 1.13E–02 1.36E–02 1.59E–02 1.81E–02 2.04E–02 2.27E–02 2.49E–02

FIGURE 10.10 Compression
symmetric shear test: Deformed configurations (first column), maps of damage (second column), and plasticity (third column) evaluated for different values of the strain Γ 12 as reported in Fig. 10.9.

400.0

M2 B

(N/mm)

300.0

200.0 A

FEA PWTFA

100.0

K2 0.0 0

2E–006

4E–006

6E–006

8E–006

1E–005

FIGURE 10.11 Flexural test: Comparison of the results obtained by the 2D micromechanical model and piecewise transformation field analysis (PWTFA) procedure.

380

PART | II Modeling of unreinforced masonry Deformed UC

(A)

(B)

Damage –9.83E–02 1.72E–04 9.87E–02 1.97E–01 2.96E–01 3.94E–01 4.93E–01 5.91E–01 6.90E–01 7.88E–01 8.86E–01 9.85E–01 1.08E+00

–9.98E–02 1.75E–04 1.00E–01 2.00E–01 3.00E–01 4.00E–01 5.00E–01 6.00E–01 7.00E–01 8.00E–01 9.00E–01 1.00E+00 1.10E+00

FIGURE 10.12 Flexural test: Deformed configurations (first column) and maps of damage (second column) evaluated for different values of the strain K2 as reported in Fig. 10.11.

diamond symbols those evaluated via the micromechanical 2D model. The two curves result not in perfect agreement in this case. Indeed, the stiffness degrading process appears more severe in the curve obtained by the micromechanical analysis than that calculated by the PWTFA. This is due to the different damage evolution in the mortar joints (Addessi et al., 2010). In fact, in the finite element solution the damage distribution and, hence, the inelastic strain is not uniform in each mortar joint, while the PWTFA considers the inelastic effect constant in each subset. Fig. 10.12 illustrates the deformed shape of the UC and the damage distribution evaluated at the time steps A and B reported in Fig. 10.11, where clearly the inelastic strain distribution does not appear completely uniform in each joint. The unloading branch is linear elastic characterized by the damaged stiffness. It is worth noting that the UC response is almost linear elastic in this test, because the damage mechanisms are not very relevant and involve a small region of the UC, mainly the bed joints. This is due to the deformation state related to the application of the curvature component K2 , where relevant tensile strains driving damage evolution occur only in half UC.

Homogenization and multiscale analysis Chapter | 10

381

TABLE 10.3 Combined loading history hist1. t

E11

0 1 2 3

0 0 0 0

2.0

E22 ð 3 1024 Þ 0 23:0 23:0 23:0

f=0 f=5 f = 10 f = 20

^ Γ 12 K2 Θ K1 ð 3 1024 Þ ð 3 1026 Þ 0 0 0 0 20:0 0 f 0 220:0 0 2f 0 0 0 0 0

Σ12

(MPa)

1.0

0.0

Γ12

–1.0

–2.0 –0.002

–0.001

0

0.001

0.002

FIGURE 10.13 Shear test: Comparison of the UC response subjected to hist1 for different value of f .

10.7.1.5 Combined loading conditions Now the effect of the presence of the Cosserat curvature components when coupled with compression and symmetric shear states in the UC are investigated. The aim is to reproduce loading conditions on the UC typically occurring in a shear prestressed masonry panel. To this end, the compression
symmetric shear test presented in Section 10.7.1.3 is performed by simultaneously applying to the UC the Cosserat curvatures, K1 and K2 , according to the loading histories shown in Tables 10.3 and 10.4. First, the effect of the application of K1 is analyzed and in Fig. 10.13 the macroscopic shear stress Σ12 versus the applied shear strain Γ 12 is depicted. Four curves are reported corresponding to the cases f 5 0; 5; 10; 20. It emerges that the presence of the curvature K1 does not significantly affect the shear response of the UC, except for the case f 5 20. Indeed, the UC nonlinear response becomes more complex when higher values of the curvature K1 act on it. Then, the shear response in presence of a variation of the curvature K2 is analyzed. In Fig. 10.14, the stress Σ12 versus the shear strain Γ 12 is plotted

382

PART | II Modeling of unreinforced masonry 4.0

(MPa)

2.0

f=0 f=5 f = 10 f = 20

Σ12

0.0

Γ12

–2.0

–4.0 –0.002

–0.001

0

0.001

0.002

FIGURE 10.14 Shear test: Comparison of the UC response subjected to hist2 for different value of f .

for the cases corresponding to f 5 0; 5; 10; 20. Differently from the previous case, the presence of K2 strongly influences the UC nonlinear response even for the lower values of f . In particular, in the case f 5 0 that corresponds to the UC response under only Cauchy strains, damage localizes only in the bed joints up to the complete degradation; after that, the response is governed by the friction mechanisms. The presence of K2 makes the damaging process more severe, and this mechanism becomes prominent with respect to the friction one. In conclusion, the effect of the curvature K2 on the UC nonlinear shear response is much more significant than that of K1 . This is due to the specific texture of the analyzed UC made of two bed joints that run along the whole horizontal size. When the curvature K2 is applied on the UC a significant part of the two bed joints is subjected to a tensile state and, hence, to severe damage. In contrast, when the curvature K1 acts on the UC only a small part of the head joints is damaged.

10.7.2 Multiscale analyses In this section, the PWTFA multiscale approaches are applied to analyze the structural response of masonry beams and panels under different boundary conditions. First, the results obtained by adopting at the macrolevel the nonlocal Cauchy formulation presented in Section 10.3.1 are shown. Then, the results evaluated by introducing the Cosserat model illustrated in Section 10.3.2 at the macrolevel are discussed. In both cases, the 2D micromodel introduced in Section 10.4.3 is considered at the UC level to reduce the computational burden. Finally, the results of the two multiscale procedures are compared. The multiscale analyses are performed using four-node 2D plane stress elements, implementing at the Gauss point level the described homogenization procedure. The aim of the performed analyses is

Homogenization and multiscale analysis Chapter | 10

383

FIGURE 10.15 Vertical and horizontal masonry beams subjected to tensile loading.

to verify the capability of the presented approaches of reproducing the masonry structural behavior and overcoming the problems due to strain and damage localization, thanks to the nonlocal integral regularization adopted for the Cauchy macromodel, as well as the Cosserat formulation.

10.7.2.1 Nonlocal Cauchy multiscale model based on piecewise transformation field analysis The analysis of two masonry beams is first performed by adopting the nonlocal Cauchy model at the macrostructural level. The geometrical scheme of the two masonry beams is illustrated in Fig. 10.15. These, clamped at one edge and subjected to a tensile loading at the free edge, are characterized by same geometry and material properties, with the beam axis oriented in the vertical and horizontal direction, respectively. The geometrical data are set as L 5 860 mm, b 5 215 mm and the geometrical parameters of bricks and mortar are the following: size of the brick b 5 210 mm, h 5 52 mm and thickness of the mortar joints s 5 10 mm. The two cases are characterized by a different orientation of the UC inside the structure, which results in a completely different mechanical response of the beams, as will be shown in the following. The nonlocal radius Lc is set to equal 150 mm. Three different discretizations are considered, that is, 30, 60, and 120 finite elements are located along the beam axis, while only one finite element is placed along the beam height. A linear elastic constitutive law is assumed for the bricks, while the damage-plastic model in Section 10.5.1 is used for the mortar. The material parameters are contained in Tables 10.5 and 10.6, where a constant friction parameter is considered, while the crushing plasticity is neglected for the mortar. The value of the damage threshold normal strain is reduced to 15% only for the last element of the beam to introduce an initial defect in the structure

384

PART | II Modeling of unreinforced masonry

TABLE 10.4 Loading history hist2. t

E11

0 1 2 3

0 0 0 0

E22 Γ 12 ð 3 1024 Þ ð 3 1024 Þ 0 0 23:0 20:0 23:0 220:0 23:0 0

^ K1 Θ 0 0 0 0

0 0 0 0

K2 ð 3 1026 Þ 0 f 2f 0

TABLE 10.5 Material parameters adopted for the mortar. Em ðMPaÞ

798

νm

0.11

ε0

3.0 3 1024

γ0

1.0 3 1023

GcI ðMPaÞ

1.79 3 1023

GcII ðMPaÞ

1.26 3 1022

μi

0.75

μf

0.75

ω

0.0

TABLE 10.6 Material parameters adopted for the bricks. Eb ðMPaÞ

16,700

νb

0.15

driving the onset of the damage. To follow the softening branch of the global response curve, the analyses are performed through the arc-length technique, assuming the displacement at the beam-free edge as the evolutionary control parameter. In Figs. 10.16 and 10.17, the axial beam responses are shown in terms of axial force versus axial displacement of the free section. The initial linear stage is followed by severe softening branches in both cases, although a nonlinear hardening phase occurs for the horizontal beam. This is due to the

Homogenization and multiscale analysis Chapter | 10

385

60

mesh 60 FEs mesh 30 FEs mesh 120 FEs

50

Axial force (N)

40

30

20

10

0

0

0.1

0.2

0.3

0.4

0.5

0.6

Axial displacement (mm) FIGURE 10.16 Global response of the vertical beam for different meshes.

180

mesh 30 FEs mesh 120 FEs mesh 60 FEs

160 140

Axial force (N)

120 100 80 60 40 20 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Axial displacement (mm) FIGURE 10.17 Global response of the horizontal beam for different meshes.

0.8

386

PART | II Modeling of unreinforced masonry

damaging process starting from the weaker element and evolving along the whole beam. Then, damage remains localized in a limited zone around the weaker element during the final part of the analysis, inducing the softening branch in the equilibrium curve. The two beams clearly show significantly different behavior. In particular, the vertical beam is characterized by a more brittle response and a peak load three times lower than that of the horizontal beam. This effect is due to the different orientation of the UC in the two beams. It can be seen that the three discretizations adopted give results in very good accordance for the vertical, as well as for the horizontal beam, highlighting the mesh independence of the finite element solutions and the effectiveness of the nonlocal integral technique introduced in the framework of the PWTFA multiscale analysis.

10.7.2.2 Cosserat multiscale model based on piecewise transformation field analysis In this section, the Cosserat model is introduced at the macrolevel in the framework of the PWTFA procedure and used to study the structural response of the masonry panel shown in Fig. 10.18. The sizes of the panel are width B 5 3290 mm, height H 5 2356 mm, and thickness t 5 100 mm. Both top and bottom sides are completely restrained. The panel is subjected, initially, to a vertical compressive displacement

H

B FIGURE 10.18 Masonry panel under horizontal displacement: Geometry and boundary conditions.

Homogenization and multiscale analysis Chapter | 10

3

387

× 104

Base reaction (N)

2.5

2

1.5

Micro 2D PWTFA 20×15 PWTFA 20×15 PWTFA 15×11

1

0.5

0 0

10

20

30

40

50

60

Top displacement (mm) FIGURE 10.19 Masonry panel under horizontal displacement: Global response curve.

applied at the top side equal to 10 mm. Subsequently, a horizontal displacement monotonically increasing until the value of 60 mm is applied to the top side nodes. The geometrical parameters of bricks and mortar are the following: size of the brick b 5 210 mm, h 5 52 mm and thickness of the mortar joints s 5 10 mm. The numerical response evaluated by adopting the presented multiscale model is compared with the results obtained by the 2D micromechanical analysis, assuming a linear elastic response for the bricks and the constitutive law presented in Section 10.5.1 for the mortar. The same material parameters as in Section 10.7.2.1 are adopted, except for the friction parameter, which is assumed as constant and equal to 0.45. At the macrolevel, two different meshes are adopted with 15 3 11 and 20 3 15 finite elements, respectively, while 39,710 finite elements are used for the micromechanical discretization. The global response curves of the panel are shown in Fig. 10.19, depicting the base horizontal reaction versus the top applied horizontal displacement. The solid line refers to the micromechanical result, while the other lines refer to the multiscale analyses performed with the 15 3 11 (dashed-dotted line) and 20 3 15 (dot) meshes and restraining completely the degrees of freedom at the top and bottom sides; the dashed line refers to the 20 3 15 mesh where only the two translational degrees of freedom are restrained at the horizontal sides. Note that very good agreement is obtained in the initial elastic branch, while the curves, obtained with the multiscale model, depart a little from the micromechanical results in the

388

PART | II Modeling of unreinforced masonry

nonlinear range due to the capability of the micromechanical model of describing more accurately the nonlinear damaging and plasticity mechanisms. Furthermore, it appears that, when the rotation degree of freedom is not restrained at the horizontal sides (dashed line), the multiscale model fits better the nonlinear micromechanical response.

10.7.2.3 Comparison between the two piecewise transformation field analysis techniques Finally, to compare the performance of the two multiscale procedures, assuming the nonlocal Cauchy and Cosserat models at the macrolevel, respectively, the response of the masonry wall in Fig. 10.20 is analyzed. It was studied experimentally by Raijmakers and Vermeltfoort (1992). The dimensions of the wall are width B 5 990 mm, height H 5 1000 mm, and thickness t 5 100 mm. It is built up with 18 courses of clay bricks, the first and last of which being clamped in steel beams. The wall is subjected initially to a vertical compressive load, uniformly applied at the top side, equal to 0.30 MPa. During this phase the wall is completely restrained at the bottom side. Subsequently, the rotation of the top side is restrained and a horizontal leftward displacement monotonically increasing until the value of 4 mm is applied to these nodes. The geometrical parameters of bricks and mortar are the following: size of the

q = 0.30 MPa u

H

B FIGURE 10.20 Shearing masonry wall: Geometry and boundary conditions.

Homogenization and multiscale analysis Chapter | 10

389

TABLE 10.7 Material parameters adopted for the mortar. Em ðMPaÞ

233

νm

0.15

ε0

1.5 3 1023

γ0

4.0 3 1023

GcI ðMPaÞ

9.6 3 1024

GcII ðMPaÞ

5.7 3 1023

μi

0.4

μf

0.4

ω

0.0

TABLE 10.8 Material parameters adopted for the bricks. Eb ðMPaÞ

1850

νb

0.15

bricks b 5 210 mm, h 5 52 mm, and thickness of the mortar joints s 5 10 mm. The material mechanical parameters are given in Tables 10.7 and 10.8. Fig. 10.21 contains the global response curve of the wall, plotting the bottom overall horizontal reaction versus the top applied displacement. Three different curves are shown, referring to the experimental outcomes (line with star symbols) and to the numerical results obtained through the nonlocal Cauchy (dashed line) and Cosserat (solid line) multiscale approaches. Both the numerical curves match very well the experimental one. After the initial linear elastic branch, the nonlinear mechanisms start to occur and evolve. In the case of the Cosserat-based multiscale model, the global response curve reaches a peak load a little lower than that experimentally calculated, while the nonlocal Cauchy-based approach gives a peak load closer to the experimental test. Then, the global response curve shows a weak softening trend. Eventually, both the multiscale procedures presented are capable of efficiently describing the structural response of the studied masonry panel in all stages of the analysis, that is, linear elastic, nonlinear hardening, and nonlinear softening, satisfactorily reproducing the experimental outcomes.

390

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Base reaction (N)

50

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Experimental Cosserat PWTFA Nonlocal Cauchy PWTFA

20

10

0 0

0.5

1

1.5

2

2.5

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Top displacement (mm) FIGURE 10.21 Shearing masonry wall: Global response curves.

10.8 Conclusions Masonry is one of the most ancient construction materials. Initially, masonry constructions were built just superimposing large rough-hewn stones, characterized by good mechanical properties, without mortar. Commonly, the masonry fabric of these constructions was disordered because of the irregular shapes of the blocks. Nevertheless, even with the irregular masonry texture, builders introduced some layers characterized by more regularity to ensure higher stability of the masonry wall. In other words, old builders understood the relevance of the fabric to the mechanical behavior of masonry construction. The more the texture is ordered, the more the masonry shows good mechanical properties. Then, the geometrical arrangement of the stones in a wall plays a key role in the capacity of the construction to carry its weight and external loadings. The use of artificial bricks was introduced in the Mesopotamian age, where rocks characterized by good mechanical properties were not available. Artificial bricks have reduced strength with respect to stones, making the texture arrangement even more significant. The disposition of the bricks or blocks in masonry construction also became very important to improve the static behavior of domes and vaults, and very special textures have been developed over the years. Thus, it appears that old masonry builders were conscious that the use of regular masonry with specific texture arrangements can significantly improve the mechanical behavior of the construction. In this perspective, a very

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challenging structural problem concerns the evaluation of the safety of a masonry construction, taking into account the actual texture of the composite material. Among the different modeling procedures, multiscale approach permits us to consider the real arrangement of the bricks in the masonry structures. Two scales can be distinguished in a masonry construction: the scale of the structural element and that of the material, that is, of the bricks arrangement. In this chapter, a multiscale technique was illustrated for the analysis of masonry walls, considering the 2D Cauchy or Cosserat continuum model at the macroscale, that is, the scale of the structural element. At the material microscale, where masonry texture is distinguished, the response of the material was deduced considering the real arrangement of the bricks. To this end, a UC was introduced and 3D, 2D, and enhanced 2D Cauchy models were considered, adopting suitable nonlinear constitutive laws for brick and mortar. A homogenization technique was presented based on the TFA. Numerical applications concerning the overall behavior of the masonry material and the response of structural elements were illustrated. The results demonstrated: G

G

G

G

G

G

the ability of the proposed constitutive models for mortar and brick to satisfactorily reproduce experimental evidence; the effectiveness of the PWTFA technique in recovering the overall response of the UC; the significant effects of the Cosserat strains on the overall response of the UC and damage evolution in mortar joints; the important role of the microstructure orientation, that is, of masonry texture, in the response of the structural element; the ability of the multiscale analysis to reproduce the response of structural elements, performing comparisons with results obtained by full micromechanical analyses and experimental evidence; the satisfactory results obtained using the multiscale analysis, even when the condition of separation of the scales is not fully satisfied.

The possibility of developing multiscale analyses is becoming more common for different physical phenomena in various fields. In the last 20 years, structural analysts have become very interested in multiscale approaches. In the field of masonry construction, perhaps of monumental and historic relevance, the final objective could be to derive the response of a structure starting from knowledge of the mechanical properties of the materials used in the construction, obtained by performing laboratory tests on very small parts. This is a relevant aim, as it permits us to overcome the need for tests on masonry elements. Tests on large masonry elements are not possible as they cannot be taken from the structure and tested in a laboratory; on the other hand, in situ tests are not always reliable. Eventually, multiscale analysis may become a viable approach to the analysis of masonry structures.

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Chapter 11

Automated geometry extraction and discretization for cohesive zone-based modeling of irregular masonry T.J. Massart, K. Ehab Moustafa Kamel and H. Hernandez Department of Building, Architecture and Town Planning, Universite´ libre de Bruxelles (ULB), Bruxelles, Belgium

11.1 Introduction The intensive use of masonry-based structures throughout history has inspired many research efforts devoted to the understanding and prediction of the mechanical behavior of masonry to ensure reliable assessment of the safety of such structures. Together with experimental characterization campaigns investigating masonry and its constituents, computational models have been developed over the last few decades to simulate the mechanical response of masonry systems and structures. Many of these models focus on the behavior of periodic masonry with a regular structure. For such arrangements computational models are often classified depending on the scale at which they are formulated as suggested in Lourenc¸o (1996). Closed-form macroscopic continuum constitutive laws were developed for structural scale computations; see, for example, the contributions by Lourenc¸o et al. (1997) and Berto et al. (2002). In lower (mesoscopic) scale models, constituents properties are considered, which led to the development of dedicated corresponding laws used in simulations investigating smaller structures or structural details (Lourenc¸o and Rots, 1997; Alfano and Sacco, 2006). Such models were developed using different kinematical assumptions depending on their objectives, ranging from continuum descriptions (Pegon and Anthoine, 1997; Massart et al., 2005a) to interface modeling based on cohesive zone elements (Lofti and Shing, 1994; Lourenc¸o et al., 1997).

Numerical Modeling of Masonry and Historical Structures. DOI: https://doi.org/10.1016/B978-0-08-102439-3.00011-7 Copyright © 2019 Elsevier Ltd. All rights reserved.

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Research efforts were also devoted to building a link between the structural behavior of masonry and the mechanical properties of its constituents. These approaches made use of representative volume elements (RVEs), or for the case of periodic masonry, of unit cells (UCs) representing the smallest possible periodic RVE. The early and seminal computational contributions in this line of research were focused on the application of periodic computational homogenization to periodic masonry in the elastic regime (Anthoine, 1995). They were followed by contributions addressing the nonlinear behavior of masonry through computational homogenization, based on UC computations using higher-order damage models as in (Pegon and Anthoine, 1997). Such efforts were then exploited to investigate the cracking-induced anisotropy in periodic masonry (see Massart et al., 2004, 2005a). Using these results, computational homogenization-based concurrent multiscale techniques were defined to incorporate damage localization effects in structural computations based on RVE computations. This led to the development of localization-enhanced FE2 frameworks, developed both for inplane loaded structures (Massart et al., 2007; Mercatoris and Massart, 2009) and for out-of-plane loaded structures (Mercatoris and Massart, 2011). A RVE averaging scheme toward a macroscopic discrete formulation of equilibrium was also recently proposed in Berke et al. (2014). Focusing on masonry with nonperiodic arrangements of blocks or units, extensions of the previously mentioned approaches have been developed. An irregular masonry texture was analyzed in Milani and Lourenc¸o (2010) based on a homogenized limit analysis methodology with the assumption of infinitely resisting blocks with a Gaussian size distribution connected by cohesive-frictional interfaces. Computational tools were further used for irregular masonry with the objective of deducing the strength properties and assessing the strengthening solutions with glass fiberreinforced composites in Feo et al. (2016). This was achieved using voxelbased FE models obtained by automated image processing techniques. Similar techniques were also used in Cundari and Milani (2013) with limit analysis techniques to set up a pushover technique for irregular masonry. More recently, a procedure for the automated generation of conforming FE meshes for complex heterogeneous geometries was applied to the particular case of irregular masonry in Massart et al. (2018). The behavior of irregular masonry with thicker joints of varying thickness was also investigated in Vasconcelos and Lourenc¸o (2009). They concluded that masonry ductility and energy dissipation significantly depend on the textural arrangement of stones or blocks. Corresponding FE analyses performed in Senthivel and Lourenc¸o (2009) showed proper agreement with experimental values. These modeling contributions in the literature for irregular masonry can be classified according to the modeling strategy used for cracking and

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degradation. Image-based strategies most often make use of voxel-based FE models using continuum constitutive laws as in Feo et al. (2016), which leads to heavy computational efforts. With the exception of the contribution by Senthivel and Lourenc¸o (2009), image-based models using cohesive zones for the modeling of cracking effects are rather scarce. This may, at least in part, be caused by the difficulty of efficiently generating lumped geometries of masonry joints based on raw image-based geometries. Yet, in view of the superior efficiency of such cohesive zone models, there is an interest in building methodologies allowing automated construction of cohesive zonebased finite element models for irregular masonry based on raw geometries. Such an automated generation of FE models is the main objective of this chapter. This will be achieved by exploiting the flexible and versatile description and manipulation of irregular heterogeneous implicit geometries offered by distance fields. To this end, this chapter is organized as follows. Section 11.2 will first describe a computational technique used to generate irregular masonry RVE geometries using implicit functions. Such implicit geometry descriptions are subsequently exploited in Section 11.3 to produce equivalent cohesive zonebased models by lumping joints toward the medial axis of the blocks arrangement. The extraction of this medial axis required for this purpose will be based on operations on implicit functions. This procedure will be shown to be fully consistent with the automated procedure to produce high-quality FE meshes for heterogeneous geometries presented in Ehab Moustafa Kamel et al. (2019). The corresponding FE models will be exploited in Section 11.4 to compute failure envelopes of irregular masonry RVEs based on cohesive zone models and computational homogenization. The extension of this methodology to produce cohesive zone-based meshes for realistic geometries of irregular masonry will then be illustrated in Section 11.5 based on a structural computation. Finally, conclusions and perspectives of the proposed methodology will be given in Section 11.6.

11.2 Representative volume element geometry generation The computational description of the geometry of RVEs for their use in computational homogenization for realistic geometries is a challenging problem. Such geometry descriptions can be obtained by two different approaches. A first approach consists of defining computational generation methodologies that try to reproduce the essential features of the material morphology. Alternatively, the geometrical descriptions can be obtained based on image acquisition techniques, combined with computational tools, allowing their use in models (image treatment and segmentation). The objective of this chapter is to define an automated procedure enabling simulations on models obtained from both these approaches. Such a procedure will make extensive use of distance fields, and there is thus an

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interest in illustrating the use of distance field-based methodologies to generate computationally RVEs based on packings of inclusions.

11.2.1 Distance field-assisted packing of inclusions RVE-generation methodologies are usually tailored according to the material under investigation (e.g., tessellation methodologies for cohesive grain-based materials or polycrystals, or sequential addition methodologies for noncohesive or inclusion-based materials). In the latter category, a unified methodology was proposed in Sonon et al. (2012) to generate RVE geometries for a wide variety of materials, starting from a packing of inclusions. The approach was further developed to generate geometries for inclusion-based materials (Sonon et al., 2012); cellular materials (Sonon et al., 2015); and more recently for irregular masonry textures (Massart et al., 2018). The approach is explained here for its further use in subsequent sections to build models based on a cohesive zone approach. For the case of masonry, a random distribution of inclusions is first generated with specific features, followed by a morphing operation to obtain irregular masonry-like geometries. This methodology for generating masonry RVEs relies on a random sequential addition (RSA) process aided with distance fields. Its main features are briefly recalled here (Sonon et al., 2012; Massart et al., 2018). The problem of filling a RVE with a given volume fraction of inclusions with prescribed size distributions may be addressed using classical RSA approaches. In such approaches, a random trial position is selected in the RVE for each (sequential) addition of an inclusion in this RVE. The potential overlap of any new inclusion to be added with the previously added inclusions is then verified. In case of overlap, the selected random position is disregarded and a new trial position is selected. Such a procedure leads to a high cost of the addition process caused by an increasing number of trial position rejections when medium-to-high RVE volume fractions are reached, leading to an exponential number of overlap verifications. As presented in Sonon et al. (2012), the efficiency of RSA procedures can be dramatically improved provided distance fields describing the current state of packing in the RVE are maintained. By maintaining a map of the distances to the closest inclusions at each position in the RVE (i.e. on a regular grid of points defined on the RVE), a set of discrete positions satisfying a priori the nonoverlapping condition can be efficiently identified if a measure of the size of the inclusion to be added can be defined. Within these nonoverlapping positions, further conditions can be imposed on distances to the closest inclusions if distance maps to the second or third nearest neighbors are maintained. In the simplest implementation, the measure of the inclusion size to be added is taken as the smallest enclosing circle of radius r.

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FIGURE 11.1 Nonoverlap and nearest-neighbor distances criteria to select the random position of an inclusion to be added. The red area represents in each version of the RVE the set of positions where an inclusion of a given size can be added according to the applied criterion on distance fields. (Left) Set of acceptable new inclusion positions based on the radius of the inclusion to add according to relationship (11.1). (Center) Set of acceptable new inclusion positions according to relation (11.2) for packing efficiency. (Right) Acceptable new inclusion positions according to condition (11.3).

Then, defining DNx as the distance map to the xth nearest inclusions in a packing computed positively outside the inclusion and negatively inside, a nonoverlapping criterion for a new inclusion of size r to be added can be expressed as DN1 . r 1 d1;min

ð11:1Þ

in which d1;min is the minimal gap desired between inclusions. To increase the packing efficiency, a maximal distance d1;max to the first nearestneighbor inclusion can also be used according to r 1 d1;min , DN1 , r 1 d1;max

ð11:2Þ

This principle to optimize the packing efficiency can be further extended to other neighboring inclusions; for instance, by requiring that the added inclusion remains close to the second nearest neighbor according to r 1 d1;min , DN1 , r 1 d1;max and DN2 , r 1 d2;max

ð11:3Þ

In order to apply the principles of periodic computational homogenization (Anthoine, 1995), the distance fields have to be evaluated such that the periodicity of the RVE is taken into account in the addition process (see Sonon et al., 2012). These principles are illustrated in Fig. 11.1 for a set of inclusions of random shape. The set of possible positions at which an additional inclusion can be placed is increasingly restricted by the use of relationships (11.1)(11.3).

11.2.2 Distance field-based morphing of a set of inclusions As an additional benefit of maintaining distance maps to nearest-neighbor inclusions at any point in the RVE, the geometrical arrangement of a set of

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neighboring inclusions can be modified by manipulations of these distance functions. The inclusions neighborhood can be used to alter their shape according to interinclusion distance rules. For instance, the distance functions DN1 and DN2 to the two nearest inclusions at any point in the RVE can be used to construct the function OV ðxÞ 5 DN1 ðxÞ 2 DN2 ðxÞ

ð11:4Þ

This function vanishes at the locus of equal distance between the two nearest inclusions. The zero-level set of this function determines a Voronoı¨like diagram made of cells, with each cell enclosing an inclusion and points closer to it than to other inclusions (see Fig. 11.2, left). When arbitraryshaped inclusions are considered, this function defines a tessellation of the space of the RVE generalizing the principle of a Voronoi diagram (see Fig. 11.2, right). A set of inclusions can be morphed into an assembly of enlarged blocks with a constant thickness w of joints between them. This new assembly can be obtained using the zero level of the function OM ðxÞ 5 DN1 ðxÞ 2 DN2 ðxÞ 1 w

ð11:5Þ

Additional geometrical alterations can be prescribed thanks to a relative weight γ between the two nearest-neighbor distances according to the zero level of the function ON ðxÞ 5 DN1 ðxÞ 2 γDN2 ðxÞ 1 w

ð11:6Þ

This morphing procedure creates a geometrical modification of each inclusion that intrinsically depends on its shape and on the configuration of its neighborhood, that is, of the shape and size of its neighboring inclusions, as well as the interdistance between them. Therefore, the initial arrangement

FIGURE 11.2 Illustration of the morphing procedure for two inclusion packings. The borders of the initial inclusions are represented in white. The colormap represents the levels associated with the function (11.4), and the black lines represent the zero-level set of this function. (Left) Laguerre-like tessellation obtained based on a packing of circular inclusions. (Right) Generalized Voronoı¨ tessellation obtained based on inclusions of arbitrary shapes.

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of inclusions added before the morphing procedure strongly constrains the obtained texture after morphing, and the precise geometry obtained cannot be controlled based on the initial shape of the inclusions only.

11.2.3 Application of morphing to masonry representative volume element generation The sequential addition and distance fields-based morphing procedures explained above can be used to generate RVEs for masonry with an irregular arrangement of blocks. An irregular masonry morphology can be obtained by generating a population of inclusions with shapes that allow reproducing the main morphological aspects of the geometrical arrangement of masonry blocks. The following characteristics are requested from the generation procedure: (1) the stacking of irregular blocks should reproduce a preferential orientation of continuous joints along the horizontal direction; and (2) the morphing procedure, accounting for the neighborhood of an inclusion during its geometrical modification, should be used to replicate the choice by the block layer of consistent neighboring block shapes in irregular masonry. To reproduce such morphological features, elongated gel-like inclusions are first used in the initial packing operation. An inclusion shape made of a rectangle combined with two half-disks is considered to allow fast evaluation of the distance to the inclusion using analytical expressions. An horizontal orientation is used for the inclusions to reproduce the corresponding preferential orientation of the continuous joints. Inclusions with a variable size are considered to produce a masonry texture with blocks of different sizes. As the smallest enclosing circle is used as a measure of the inclusion size, a vertically loose packing is obtained, that is, the elongated inclusions can be packed horizontally, but are more distant vertically as illustrated in Fig. 11.3 (left). This packing of inclusions is then morphed according to the function defined by relationship 11.6 to obtain a set of masonry blocks with a controlled mortar joint thickness. The result of the morphing procedure for packings in Fig. 11.3 (left) are represented in Fig. 11.3 (center) and (right) for two values of γ and w; respectively. Other packing algorithms or experimental information obtained from image processing techniques could be considered before applying the level set-based morphing procedure, as will be illustrated in the following sections for the extraction of cohesive zone geometries. Quasiperiodic textures produced by other packing principles could also be altered using the function (11.6). Even though the resulting geometry is not completely “morphologically” controlled, it makes it possible to reach precise control of the volume fraction of blocks, as well as of the average mortar joint thickness. The implementation presented here is 2D, but a 3D extension can be straightforwardly implemented, yet accounting for a substantial increase of the cost of distance computations.

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FIGURE 11.3 Generation of two RVEs for irregular masonry. (Left) Generation of loose packing of elongated inclusions. (Center) Extraction of block geometry with thin joints based on function (11.6) with γ 5 0:9 and w 5 0:015. (Right) Extraction of block geometry with thick joints based on function (11.6) with γ 5 0:9 and w 5 0:035.

11.3 Extraction of cohesive zone geometry To produce meshes allowing the incorporation of cohesive zone elements for efficient nonlinear simulations, several ingredients have to be combined. Among those, the extraction of the medial axis of an inclusion-based packing is a crucial step. The concept of medial axis was first introduced by Blum (1967) and is widely used in image analysis to build a compact representation of geometrical and topological features of an image. Mathematically, the medial axis of a shape represents the set of points at equal distance from at least two points of interfaces. In other words, the medial axis of an interface can be defined as the locus of centers of circles in 2D and spheres in 3D, respectively, that can be placed in such a way that a circle and sphere touches the interface on at least two points without crossing it (Blum, 1967). For a multibody geometry as illustrated in Fig. 11.4, the medial axis is composed of the inner medial axis of each inclusion (in blue) as well as the outer medial axis belonging to the matrix (in red) to characterize the whole geometry. For its application in the extraction of cohesive zones for masonry joint modeling, only a part of the medial axis is of interest (e.g., the part located outside the inclusions). Thus, this section will focus on the procedure to extract this part of the medial axis and to mesh it efficiently.

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FIGURE 11.4 (Left) RVE made of a set of gels. (Right) Representation of the medial axis of the inclusions (blue) and the matrix (red). This morphology of inclusions is selected here for clarity of the explanation, and will be replaced in the sequel by irregular blocks.

This research is available on continuous or discrete methods for the extraction of the medial axis of a set of inclusions. It essentially refers to applications in the image processing community as it is a widely used shape descriptor used in applications such as object recognition or shape-matching fields. Such literature is too wide to be exhaustively reviewed here. Among the existing extraction methods, we cite approaches based on distance fields through local maxima (Persson and Strang, 2004; Coeurjolly and Montanvert, 2007); methods using morphological erosion based on the “grassfire” (Blum, 1967); or the approaches based on Voronoı¨ tessellations (Amenta et al., 1998; Gold and Snoeyink, 2001; Costa et al., 2002; Dardenne et al., 2008). Here, the discrete extraction of the medial axis is based on the Voronoı¨ tessellation, also called the Voronoı¨ diagram. This technique is based on a subdivision of the space into a number of cells, and uses “points” as seeds and the Euclidean distance as a distance measure. The resulting segments from the tesselation correspond to all points in the plane equidistant to the two nearest seeds (Voronoi, 1907). In this case, the medial axis is represented as a subset of the interfaces of the Voronoı¨ cells defined from the contours of the initial geometry (Dardenne et al., 2008). Generating a tessellation after extracting inclusion contour nodes used as “seeds” does not allow obtaining directly the medial axis. Indeed, a set of segments of the medial axis is located both in the inclusions and in the matrix (see Fig. 11.5). A filtering step is therefore required to isolate the part of the medial axis useful to generate cohesive zone modeling the mortar joints. Also, the generation of the medial axis can be very sensitive to details of the inclusion shapes and may produce nondesired “sub-branches” stemming from nonconvex inclusion shapes, as illustrated in the example in Fig. 11.5 (right). These have to be removed to produce a medial axis suitable for representing lumped masonry joints. In our approach, the geometries are defined in an implicit manner, whether they originate from RVE-generation methodologies as presented in

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FIGURE 11.5 (Left) Rubble masonry. (Right) Corresponding Voronoı¨ tesselation (grey) from the inclusion contours (black) and extracted medial axis (red).

FIGURE 11.6 (Left) First nearest neighbor distance map. (Center) Second nearest neighbor map. (Right) “Voronoı¨” level-set function (distances are given related to the size of the sample).

Section 11.2, or from real geometries. The values of the distance fields or of the grey levels are then interpolated on a regular grid of points. In order to facilitate the extraction of the medial axis for its further discretization using the PerssonStrang analogy (Persson and Strang, 2004), the geometries have to be described by distance fields. This is seamlessly obtained for the generation methods as presented in Section 11.2. For imagebased morphologies, for each point of a regular grid, the distance to the two nearest-neighbor inclusions extracted from images can be computed using a fast marching method (Sethian, 1996). This is illustrated in Fig. 11.6 for the realistic morphology presented in Fig. 11.5 (left). As mentioned, extracting the outer medial axis requires filtering operations. First, the part of the medial axis located inside the inclusions is removed thanks to the distance field, as the interior of the inclusions corresponds by convention to negative values of the first nearest-neighbor distance. Second, due to the potential nonconvexity of the inclusions and the

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large gaps between them (the mortar joints in masonry), nondesired “subbranches” of the medial axis appear (see Fig. 11.5). In this case again, the computed distance fields are useful to suppress them using the morphing procedure. As illustrated in Fig. 11.6, and similar to the case of computationally generated RVEs, it is possible by combining the first and second neighbor distance fields, to reconstruct an implicit “Voronoı¨” level-set function, the zero iso-values of which are equidistant from the two nearest inclusions according to OV ðxÞ 5 DN1 ðxÞ 2 DN2 ðxÞ

ð11:7Þ

where x represents again the coordinates of the grid points at which the distance fields are evaluated (Sonon et al., 2015). This function modifies the shape of the initial inclusions, inflating them with resulting updated inclusions “touching” each other side by side. However, it does not allow extracting an initial discretization of the medial axis. The morphing operation of inclusions defined by relation (11.5) in Section 11.2 is used as depicted in Figs. 11.6 and 11.7 to inflate the contours of the inclusions and obtain fine controlled gaps. This is achieved by using an offset of the “Voronoı¨” levelset function with a value depending on the grid resolution. OM ðxÞ 5 DN1 ðxÞ 2 DN2 ðxÞ 1 wðresolutionÞ

ð11:8Þ

The morphed inclusion contours are next extracted and used as “seed” points for the Voronoı¨ tesselation before filtering the inclusion part. The procedure is illustrated in Fig. 11.7, showing that the morphing procedure helps removing all “sub-branches” to obtain a suitable medial axis.

FIGURE 11.7 (Left) Rubble masonry blocks (black). Inflated inclusions produced by the distance field-based morphing procedure (blue contours), clean medial axis (red). (Right) Clean medial axis produced based on the contours of the morphed RVE (red) and inclusions contours (blue dashed lines).

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FIGURE 11.8 Mesh generated with (left) no cohesive zones, (center) cohesive zones and inclusions contours, and (right) cohesive zones only.

The extraction is organized in such a way that the potential periodicity of the microstructure is preserved for further use in periodic computational homogenization (see Fig. 11.8). The different segments of the medial axis are doubled (or tripled) based on their inclusion in the different Voronoı¨ cells. They are subsequently rearranged and sorted so that the interface elements can be seamlessly generated. Once the discretized medial axis is extracted, the resulting data (medial axis and initial inclusions contours) can be used for the generation of bulk elements. At this stage, a choice of modeling strategy has to be made. Essentially, the choice of the constraint in this step depends on the objective pursued. If only the medial axis mesh is frozen, the resulting bulk mesh will only conform with this medial axis. For masonry, this matches a case in which the blocks are extended to lump mortar joints toward cohesive zones. This is the option taken in this contribution for computational efficiency. Alternatively, we could decide to constrain only the inclusion (stone) contours to obtain meshes similar to the ones used in Massart et al. (2018); or even to constrain both the initial inclusion contours and the medial axis to account explicitly for the thickness of mortar joints, while keeping the option to represent their cracking by cohesive zones placed on the medial axis. Fig. 11.8 illustrates the three possible choices. To this end, the mesh generation procedure developed in Ehab Moustafa Kamel et al. (2019) and used in Massart et al. (2018) and Wintiba et al. (2017) for heterogeneous structures is exploited as illustrated in Fig 11.9. It consists of the following steps: 1. 2. 3. 4. 5.

Medial axis extraction Generation of a size map for target FE sizes Generation of an initial node distribution Meshing of the contours of the inclusions and/or the medial axis Constrained delaunay triangulation (CDT) based on the contour and/or medial axis discretization 6. Triangular bulk elements optimization 7. Creation of the interface elements based on the medial axis discretization

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FIGURE 11.9 Generation of (top left) element size map hðxÞ. (Top right) quadtree node distribution according to hðxÞ: (Bottom left) medial axis mesh. (Bottom right) optimized bulk mesh.

The map of targeted element sizes allows controlling locally the element sizes inside the RVE based on predefined geometrical parameters such as the curvature of the boundaries, or the narrowness between the inclusions. It also allows using a threshold limiting a spatially strong increase of the element sizes. The initial generation of nodes is achieved by a recursive division of the RVE based on a quadtree principle (Finkel et al., 1974). By linking this initial node distribution to the size map, it is possible to seamlessly enforce the periodicity of the mesh and to refine the mesh only in the zones where it is required. The contour meshing allows obtaining a discretization in which the size of the contour segments is consistent with the size interpolated from the size function stored at the grid points. This contour meshing starts with an initial discretization obtained by a marching square for the inclusions contours and by the Voronoı¨ tessellation for the medial axis. A PerssonStrang truss-like optimization step (Persson and Strang, 2004) is then used for this contour/ medial axis meshing step. This consists of interpreting the contour/medial axis mesh generated as a truss system. An internal force field is prescribed

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on these “truss” elements, driving nodes displacements toward positions complying with the element size map defined as target element sizes. During this step, the nodes are only allowed to move tangentially to the contours and the medial axis. Their displacement normal to the contours/medial axis is prevented, thereby generating a reaction force in the associated truss equilibrium. This is made possible by the computation of the direction normal to the contours at any position based on the gradient of the distance fields. For the inclusion contours, the procedure is similar to the one used in Massart et al. (2018). However, for the medial axis, the procedure is slightly different. The medial axis is split into segments shared by the two same inclusions, and the gradient is computed for each segment by their corresponding “Voronoı¨” level-set function in order to prevent any normal displacement of medial axis nodes. A dynamic nodes addition/suppression procedure is also used for this process. This optimization procedure allows fast convergence toward a contours/medial axis discretization that complies with the element size map. Once this contour/medial axis mesh is obtained and optimized, an initial triangular bulk mesh is generated by CDT. The resulting bulk mesh is finally optimized using the PerssonStrang analogy (Persson and Strang, 2004) applied to the triangular mesh, as developed for heterogeneous materials in Ehab Moustafa Kamel et al. (2019) and used in Massart et al. (2018) and Wintiba et al. (2017). This approach furnishes high-quality triangular meshes as illustrated in Fig. 11.8 for the three possible modeling strategies mentioned. Finally, the last operation consists of doubling the nodes of the medial axis segments to allow defining cohesive zones for damaging simulations to be performed.

11.4 Representative volume element computations for irregular masonry In this section, the methodology to generate cohesive zone-based models for irregular masonry is illustrated on computationally generated RVEs. A failure locus is determined computationally based on a RVE using computational homogenization with periodic boundary conditions. At the local level, a stacking of stiff blocks is considered separated by soft damaging interfaces to represent the joints. The block behavior is assumed elastic, while the mortar phase is modeled using cohesive zone elements with quasibrittle behavior, located at a priori known interface positions derived in the previous section. The behavior of the soft interface is modeled by an initially elastic interface element with kn and kt , the normal and tangential elastic stiffnesses, respectively. A MohrCoulomb type strength criterion (Berke et al., 2014) is used with a tension cutoff and a linear compression cap, as shown in

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Fig. 11.10. Parameters ft and fc control the tensile mode I strength of the soft interface and its compressive strength, respectively; c is the cohesion parameter; ϕ the friction angle; and θ is the angle defining the linear compression cap. A scalar damage model with an exponential evolution law is considered. The tractionseparation law links the traction vector t across the interface to the relative displacement vector d by t 5 ð1 2 DÞH:d

ð11:9Þ

where D is the scalar damage variable taking values between zero (undamaged material) and one (complete failure) and H is the elastic stiffness tensor depending on kn and kt . Note that Eq. (11.9) implies that no stiffness recovery is taken into account upon crack closure. The damage evolution law of the cohesive zone interface is given by

 ft ft ft ft D512 exp 2 κ2 ð11:10Þ for κ . κ kn Gf kn kn with Gf being the mode I fracture energy. The damage criterion has to take into account tensioncompression asymmetry. The damage-driving parame ter κ is therefore taken as κ 5 max deq ’t with 8 ft ft kt > > tan ϕdn 1 :dt : > > c c kn > < ð11:11Þ deq 5 max dn > > ft ft kt 1 > > 2 dn 1 :dt : > : c fc kn tan θ where dn and dt are the normal and tangential relative displacements, respectively. The compressive fracture energy implicitly results from the tensile fracture energy Gf and the relative values of fc and ft . Using the previously developed tools for the generation and discretization of RVEs, the effect of irregularity is now illustrated by considering the macroscopic failure envelope obtained for an RVE generated by the methodology presented in Section 11.2. The RVE has dimensions 500 3 500 mm2 , and is subjected to proportional macroscopic loading, combining vertical

FIGURE 11.10 Damage criterion used for cohesive zones representing mortar joints, consisting of a MohrCoulomb criterion with a tensile cutoff and a linear compressive cap.

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compression and shear to evaluate its maximum load-carrying capacity. The macroscopic loading is applied using classical periodic homogenization. A periodic fluctuation of the displacement field at the boundary of the RVE is enforced by linear dependency constraints (see Anthoine, 1995; Massart et al., 2005a). This results in automatic enforcement of the scale transition conditions, that is, ð ð ð 1 1 1 Σ5 σ dV; E5 ε dV; Σ:δE 5 σ:δε dV; ð11:12Þ V V V V V V The behavior of mortar is modeled using the damaging interface with the material parameters provided in Table 11.1. The blocks are assumed elastic with a Young’s modulus of 20 GPa and a Poisson coefficient of 0.2. The simulations are performed under macroscopic stress control using an advanced path-following technique to trace the softening branch of the macroscopic response of the RVE (Geers, 1999). Such advanced path-following techniques can be further complemented by techniques specifically tailored for the multiscale setting (see Massart et al., 2005b). The obtained failure envelope is given in Fig. 11.11. As expected, an increasing peak load is obtained when the shearing component of the loading is decreased with respect to the amount of vertical compression. The interfacial damage distribution corresponding to three loading directions are depicted in Fig. 11.11. For each loading direction matching a letter in Fig. 11.11, the damage distributions are represented at the peak load and in the homogenized softening regime.

TABLE 11.1 Material properties of the cohesive zones assumed for mortar joints in irregular RVE computations. kn (N/mm3 ) kt (N/mm3 ) ft (N/mm2 ) Gf (N/mm) c (N/mm2 ) ϕ ( ) fc (N/mm2 ) θ ( ) 100

65

0.2

0.1

0.3

40

12

20

FIGURE 11.11 Failure envelope corresponding to the RVE represented in Fig. 11.12, with material properties of the constituents given in Table 11.1. The RVE is subjected to stresscontrolled proportional loading. The envelope denotes the maximal load-carrying capacity of the material along each loading direction.

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  Loading direction A matches the direction Σxy ; Σyy 5 ð1; 2 1Þ in the stress space. For this loading direction, the early state of damage at peak load shows a predominant failure of joints with an orientation close to a diagonal direction (see Fig. 11.12A, left). This result is similar to the computations performed on regular running bond masonry UCs (Massart et al., 2007), in which head joints are damaged first, before reaching the homogenized softening regime. However, the irregularity of the mesostructured implies that only a part of the joints close to the vertical direction are damaged. The complete development of the failure mechanism requires the continuation of the simulation far in the softening range of the homogenized behavior. Fig. 11.12A (right) shows that further loading develops damage in

FIGURE 11.12 Damage distributions obtained for a proportionally loaded RVE. Each line denotes the damage distributions for a given loading direction, denoted by the corresponding letter in Fig. 11.11. (Left) Damage distribution at peak load; (right) damage distribution at the end of the computation.

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a part of the almost-horizontal joints to form a full staircase-shaped crack. The fully developed damage mechanism depicted in Fig. 11.12A (right) shows an inclined staircase periodic crack. This final damage pattern is heterogeneous among the joints (i.e., not all the staircase paths are cracked), unlike in a periodic UC computation performed on a regular periodic running bond masonry RVE. This is due to the irregularity of blocks that slightly perturbates the pure staircase failure. Decreasing the   proportion of shear with loading direction B corresponding to Σxy ; Σyy 5 ð0:3; 2 1Þ, an initial damage distribution similar to the one obtained for loading direction A is found at the peak load. The higher proportion of vertical compression induces an even more extended damage state in the vertically oriented crack, but for a much higher load level than for loading direction A. Subsequently, in the overall softening response of the RVE, a part of the staircase crack forms (see Fig. 11.12B, right). However, the full staircase crack does not form. Instead, Fig. 11.12B (right) shows distributed cracking predominantly oriented along the vertical direction. Large segments of horizontally oriented joints remain undamaged at the end of the homogenized softening response, while vertically and diagonally oriented joints are predominantly damaged. This effect is even more pronounced when the contribution   of shear is further reduced with a loading direction defined by Σxy ; Σyy 5 ð0:1; 2 1Þ (loading direction C), and the damage configuration at the peak becomes more distributed than for cases with more shearing. Fig. 11.12C (left) shows that vertically oriented joints are damaged, but without vertical continuity of cracks. In the softening range, damage progresses toward inclined joints (see Fig. 11.12C, right). The final damage pattern is even more distributed than for loading direction B, with horizontal joints damage much more widespread in the RVE. This means that the higher proportion of vertical compression in the loading induces compressive failure of horizontal joints. In spite of using different constitutive settings, these results are globally consistent with the failure modes obtained using gradient damage models in Massart et al. (2018). The slight changes in the shape of the failure locus are related to the higher sensitivity of the MohrCoulomb criterion to shear with respect to the modified von Mises damage criterion used in Massart et al. (2018). Yet, such slight discrepancies are more than compensated for by the much lower computing cost associated with the cohesive zones compared to gradient damage modeling.

11.5 Discretization of segmented realistic masonry geometries To further illustrate the methodology, it is now applied to larger-scale structures, testing the possibility to apply the proposed approach to realistic geometries involving larger gaps between blocks.

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To focus the application on the generation of cohesive zone models on complex geometries rather than on the image segmentation procedure itself, a computationally generated geometry is considered. Spence et al. (2008) generated geometries of masonry samples based on random fields. One of these geometries is used here and repeated by a vertical axial symmetry to obtain the geometry represented in Fig. 11.13. This image is then segmented and its medial axis extracted using the tools described in Section 11.3, that is, this segmented image is treated using the morphing approach to extract the medial axis of the set of blocks. The resulting segmented geometry is illustrated in Fig. 11.14 together with the corresponding medial axis. Following the segmentation and medial axis extraction, the level-set meshing strategy inspired by the works of Persson and Strang (2004) and briefly described in Section 11.3 is applied to obtain a mesh of the tessellation. This mesh is obtained as a set of independent inclusions with a conforming discretization of their common boundaries, yet with distinct nodes. This allows the automated introduction of interface elements along the shared segments of the medial axis. The resulting mesh is represented in Fig. 11.15. It consists of a mesh of 20,650 bulk triangular quadratic elements. The addition of cohesive zone elements is performed, leading to the incorporation of 4306 interfaces. The full model results in a set of 24,956 elements with 50,508 nodes. To illustrate further the advanced computations that can be conducted on complex realistic geometries using this automated FE model generation, the obtained wall is used in a test similar to the deep beam test proposed by

FIGURE 11.13 Irregular masonry wall obtained by exploiting computationally generated masonry samples in Spence et al. (2008). The wall is obtained by vertical axial symmetry, thereby obtaining a symmetric wall.

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FIGURE 11.14 Irregular masonry wall segmentation and distance field-based treatment to extract the medial axis of the set of blocks (depicted in red lines).

FIGURE 11.15 Irregular masonry wall segmentation discretization obtained by the distance field-based meshing methodology inspired by Persson and Strang (2004). The yellow lines represent the nodes describing the cohesive zone elements.

Page (1978) for small running bond masonry walls, and used in Lourenc¸o (1996) to assess plastic interface laws for masonry mortar joints. The corresponding problem statement is described in Fig. 11.16. The wall is assumed to have dimensions 3400 3 2100 mm2 . It is supported along its bottom boundary over one quarter of its length on its left and right ends. A centered compressive load is applied on the top boundary over half its length using a stiff component. Some weaker interfaces are considered in the mesh in order to introduce some lack of symmetry in the damage propagation.

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FIGURE 11.16 Deep beam test configuration applied to the irregular masonry wall. Lower mechanical properties are attributed to the red interfaces in order to avoid a fully symmetric response of the specimen.

TABLE 11.2 Material properties of the cohesive zones assumed for mortar joints in irregular masonry wall computations. kn (N/mm3 )

kt (N/mm3 ) ft (N/mm2 ) Gf (N/mm) c (N/mm2 ) ϕ ( ) fc (N/mm2 ) θ ( )

2

1

0.03

0.05

0.06

21

9

20

A load of up to 190 kN is applied on the structure using an adaptive loading step to ease convergence of the NewtonRaphson iterations. No attempt was made to trace the complete loaddisplacement curve including its softening part, as the goal is to demonstrate the ability to conduct analyses on complex geometries, and as it is expected that the structural behavior is strongly brittle with potential strong snap-through and/or snap-back effects (Lourenc¸o, 1996). The material properties of blocks remain unchanged with respect to the RVE application above. The properties of the interface elements are modified to comply with their order of magnitude used for rubble masonry in Senthivel and Lourenc¸o (2009). The elastic stiffness of the interface is therefore assumed with lower values, as for the strength properties (see Table 11.2). The interface elements depicted in red in Fig. 11.16 are assumed with lower strength properties decreased by a ratio 3 with respect to the values mentioned in Table 11.2.

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The structural response of the system is illustrated in Fig. 11.17. For three specific increasing loading levels, denoted by markers in Fig. 11.17, a map of the interfacial damage state is depicted in Fig. 11.18 on the structure in its magnified deformed shape (displacements are magnified by a factor 10). The damage propagation is clearly initiated at the free part of the bottom boundary with two damage “bands” oriented upward toward the free (nonloaded) part of the top boundary (see Fig. 11.18, top). This is consistent with the small-scale deep beam simulations performed in Lourenc¸o (1996). The damage levels then increase substantially in this damage band that has a tendency to widen toward the wall lateral boundaries in Fig. 11.18 (center). Finally, an almost continuous path of damaged interfaces (with damage above 0.875) is obtained in Fig. 11.18 (bottom) that corresponds to a load level close to the ultimate load-bearing capacity. This example demonstrates the versatility of the automated cohesive zone-based FE model generation that can be applied to complex geometries extracted from experimental data. This paves the way for building an integrated methodology that would couple image analysis with advanced cohesive zone models dedicated to masonry behavior as developed in Lourenc¸o and Rots (1997).

FIGURE 11.17 Deep beamstructural response of the test.

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FIGURE 11.18 Successive interfacial damage map for the three configurations depicted by the square markers in Fig. 11.17. The deformed shape of the structure is amplified by a magnifying factor of 10 (which explains the kinks present on the top boundary of the wall in the last image of Fig. 11.18).

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11.6 Concluding remarks The approach presented in this chapter can pave the way toward automated FE mesh generation for nonlinear analysis of irregular masonry structures based on cohesive zone constitutive models. The main assumption of the methodology relies on identification of the cohesive zone geometry based on the medial axis in the mortar phase. This medial axis can be extracted by exploiting combinations of distance fields constructed for packings of blocks. More precisely, a Voronoı¨-like function, built from the distance between the two nearest-neighbor blocks at any point in the structure, is used with an offset to allow neat extraction of the relevant part of the medial axis for cohesive zone modeling. This methodology then allows a seamless transition toward a meshing methodology recently developed to discretize complex heterogeneous geometries. The global outcome of this procedure is the automated production of FE meshes with an interface-based modeling of mortar joints, ready for use in nonlinear structural computations. The proposed approach was illustrated by discretizing irregular masonry RVEs produced thanks to a generation methodology using the same distance fields. A larger-scale computation with complex block shapes was also shown to assess the robustness of the methodology. Future steps of development essentially include the incorporation of the methodology in simulation loops involving image segmentation and advanced cohesive zone modeling.

References Alfano, G., Sacco, E., 2006. Combining interface damage and friction in a cohesive-zone model. Int. J. Numer. Methods Eng. 68 (5), 542582. Amenta, N., Bern, M.W., Eppstein, D., 1998. The crust and the beta-skeleton: combinatorial curve reconstruction. Graph. Models Image Process. 60 (2), 125135. Anthoine, A., 1995. Derivation of the in-plane elastic characteristics of masonry through homogenization theory. Int. J. Solids Struct. 32 (2), 137163. Berke, P.Z., Peerlings, R.H.J., Massart, T.J., Geers, M.G.D., 2014. A homogenization-based quasi-discrete method for the fracture of heterogeneous materials. Comput. Mech. 53, 909923. Berto, L., Saetta, A., Scotta, R., Vitaliani, R., 2002. An orthotropic damage model for masonry structures. Int. J. Numer. Methods Eng. 55 (2), 127157. Blum, H., 1967. A transformation for extracting new descriptors of shape. Models for the Perception of Speech and Visual Form. MIT Press, pp. 362380. Coeurjolly, D., Montanvert, A., 2007. Optimal separable algorithms to compute the reverse euclidean distance transformation and discrete medial axis in arbitrary dimension. IEEE. Trans. Pattern. Anal. Mach. Intell. 29 (3), 437448. Cundari, G.A., Milani, G., 2013. Homogenized and heterogeneous limit analysis model for pushover analysis of ancient masonry walls with irregular texture. Int. J. Architect. Herit. 7, 303338.

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Costa, L.D.F., Fabbri, R., Estrozi, L.F., 2002. On Voronoı¨ diagrams and medial axes. J. Math. Imaging Vis. 17 (1), 2740. Dardenne, J., Valette, S., Siauve, N., Prost, R., 2008. Medial axis approximation with constrained centroidal voronoi diagrams on discrete data. Comput. Graph. Int. 299306. Ehab Moustafa Kamel, K., Sonon, B., Massart, T.J., 2019. An integrated approach for the conformal discretization of complex inclusion-based microstructures, In Press, https://doi.org/ 10.1007/s00466-019-01693-4. Feo, L., Luciano, R., Misseri, G., Rovero, L., 2016. Irregular stone masonries: analysis and strengthening with glass fibre reinforced composites. Compos. B 92, 8493. Finkel, R., Bentley, J.L., 1974. Quad trees: a data structure for retrieval on composite keys. Acta Inform. 4 (1), 19. Geers, M.G.D., 1999. Enhanced solution control for physically and geometrically nonlinear problems. Part I—the subplane control approach. Int. J. Numer. Methods Eng. 46, 177204. Gold, C., Snoeyink, J., 2001. A one-step crust and skeleton extraction algorithm. Algorithmica 30 (2), 144163. Lofti, H., Shing, P., 1994. Interface model applied to fracture of masonry structures. J. Struct. Eng. 120 (1), 6380. Lourenc¸o, P.B., 1996. PhD dissertation Computational Strategies for Masonry Structures. Delft University of Technology, Delft, The Netherlands. Lourenc¸o, P.B., Rots, J.G., 1997. Multisurface interface model for analysis of masonry structures. J. Eng. Mech. 123 (7), 660668. Lourenc¸o, P.B., De Borst, R., Rots, J.G., 1997. A plane stress softening plasticity model for orthotropic materials. Int. J. Numer. Methods Eng. 40 (21), 40334057. Massart, T.J., Peerlings, R.H.J., Geers, M.G.D., 2004. Mesoscopic modeling of failure and damage-induced anisotropy in brick masonry. Eur. J. Mech. A/Solids 23 (5), 719735. Massart, T.J., Peerlings, R.H.J., Geers, M.G.D., Gottcheiner, S., 2005a. Mesoscopic modeling of failure in brick masonry accounting for three-dimensional effects. Eng. Fract. Mech. 72 (8), 12381253. Massart, T.J., Peerlings, R.H.J., Geers, M.G.D., 2005b. A dissipation-based control method for the multi-scale modeling of quasi-brittle materials. C.R. Mecanique 333, 521527. Massart, T.J., Peerlings, R.H.J., Geers, M.G.D., 2007. An enhanced multi-scale approach for masonry wall computations with localization of damage. Int. J. Numer. Methods Eng. 69 (5), 10221059. Massart, T.J., Sonon, B., Ehab Moustafa Kamel, K., et al., 2018. Level set-based generation of representative volume elements for the damage analysis of irregular masonry. Meccanica . Available from: https://doi.org/10.1007/s11012-017-0695-0. Mercatoris, B.C.N., Massart, T.J., 2009. Assessment of periodic homogenization-based multiscale computational schemes for quasi-brittle structural failure. Int. J. Multisc. Comput. Eng. 7 (2), 153170. Mercatoris, B.C.N., Massart, T.J., 2011. A coupled two-scale computational scheme for the failure of periodic quasi-brittle thin planar shells and its application to masonry. Int. J. Numer. Methods Eng. 85, 11771206. Milani, G., Lourenc¸o, P.B., 2010. A simplified homogenized limit analysis model for randomly assembled blocks out-of-plane loaded. Comput. Struct. 88, 690717. Page, A.W., 1978. Finite element model for masonry. ASCE J. Struc. Div. 104 (8), 12671285. Pegon, P., Anthoine, A., 1997. Numerical strategies for solving continuum damage problems with softening: application to the homogenization of masonry. Comput. Struct. 64 (14), 623642.

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Persson, P.O., Strang, G., 2004. A simple mesh generator in MATLAB. SIAM Rev. 46 (2), 329345. Senthivel, R., Lourenc¸o, P.B., 2009. Finite element modeling of deformation characteristics of historical stone masonry shear walls. Eng. Struct. 31, 19301943. Sethian, J.A., 1996. A fast marching level set method for monotonically advancing fronts. Proc. Natl. Acad. Sci. 93 (4), 15911595. Spence, S.M.J., Gioffre, M., Grigoriu, M.D., 2008. Probabilistic models and simulation of irregular masonry walls. J. Eng. Mech. 134 (9), 750762. Sonon, B., Francois, B., Massart, T.J., 2012. A unified level set based methodology for fast generation of complex microstructural multi-phase RVEs. Comput. Methods Appl. Mech. Eng. 223224, 103122. Sonon, B., Francois, B., Massart, T.J., 2015. An advanced approach for the generation of complex cellular material representative volume elements using distance fields and level sets. Comput. Mech. 56 (2), 221242. Vasconcelos, G., Lourenc¸o, P.B., 2009. In-plane experimental behavior of stone masonry walls under cyclic loading. J. Struct. Eng. 135 (10), 12691277. Voronoi, G., 1907. Nouvelles applications des parame`tres continus a` la the´orie des formes quadratiques. J. Reine Angew. Math. 133, 97178. Wintiba, B., Sonon, B., Ehab Moustafa Kamel, K., Massart, T.J., 2017. An automated procedure for the generation and conformal discretization of 3D woven composites RVEs. Compos. Struct. 180, 955971.

Chapter 12

Homogenization limit analysis G. Milani1 and A. Taliercio2 1

Department of Architecture, Built Environment and Construction Engineering (ABC), Politecnico di Milano, Milan, Italy, 2Department of Civil and Environmental Engineering (DICA), Politecnico di Milano, Milan, Italy

12.1 Introduction Masonry structures are comprised of units (such as bricks and/or stones), more or less regularly spaced, and usually bonded with mortar. Predicting the global (or macroscopic, or effective) mechanical properties of masonry according to the mechanical and geometrical properties of units and mortar is a goal that many authors have tried to achieve. Two approaches are mainly used in the literature for the description of the mechanical behavior of masonry, usually known as macro- and micromodeling. Macromodeling (see, e.g., Lourenc¸o, 1996, 1997; Lourenc¸o et al., 1997; Berto et al., 2002; Ushaksaraei and Pietruszczak, 2002; Pela` et al., 2011, 2013, just to quote a few) does not make any distinction between bricks and joints, and “smears” the effects of mortar through the formulation of a fictitious homogeneous material. The advantage of macromodeling is linked to the ability of analyzing entire buildings using large-size finite elements (FEs), totally disregarding the actual layout of the units. Unfortunately, it usually requires that many mechanical parameters available be available: they can be obtained by best fitting data provided by costly experimental campaigns performed on full-scale masonry specimens, which require cumbersome devices. On the other hand, specimens sufficiently large to be representative of the global behavior of masonry are virtually impossible to extract and submit to laboratory tests, especially in the case of historic buildings. In addition, the analysis of a different masonry material or brick pattern would require a new calibration of the model parameters and hence new experimentations. The alternative micromodeling approach (see, e.g., Lotfi and Shing, 1994; Lourenc¸o and Rots, 1997; de Felice and Giannini, 2001; Sutcliffe et al., 2001; Gilbert et al., 2006; Milani, 2008; Shieh-Beygi and

Numerical Modeling of Masonry and Historical Structures. DOI: https://doi.org/10.1016/B978-0-08-102439-3.00012-9 Copyright © 2019 Elsevier Ltd. All rights reserved.

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Pietruszczak, 2008; de Felice, 2011; Macorini and Izzuddin, 2011; Portioli et al., 2013; Minga et al., 2018a,b) is a distinct representation of mortar joints and bricks. The calibration of the mechanical parameters is easier and less expensive, because only laboratory tests on brick and mortar small samples have to be performed. In order to limit the number of degrees of freedom (DOF) in structural analyses, joints are usually reduced to zerothickness interfaces, but still the numerical effort remains proportional to the number of units in the structure: accordingly, this approach is feasible for small structural elements (e.g., panels or single piers). Homogenization (see, e.g., Pande et al., 1989; Pietruszczak and Niu, 1992; Anthoine, 1995, 1997; de Buhan and de Felice, 1997; Luciano and Sacco, 1997; Pegon and Anthoine, 1997; Luciano and Sacco, 1998; Zucchini and Lourenc¸o, 2002; Sab, 2003; Cluni and Gusella, 2004; Cecchi et al., 2005; Milani et al., 2006a,b,c; Cecchi et al., 2007; Mistler et al., 2007; Sab et al., 2007; Zucchini and Lourenc¸o, 2007; Cecchi and Milani, 2008; Dallot et al., 2008; Kawa et al., 2008; Milani, 2009; Sacco, 2009; Casolo and Milani, 2010; Milani and Tralli, 2011; Milani, 2011a,b,c; Taliercio, 2014; Milani, 2015; Milani and Taliercio, 2015; Stefanou et al., 2015; Milani and Taliercio, 2016) is an interesting compromise between micro- and macromodeling, because it allows a structure to be roughly discretized, but at the same time accounts for the mechanical behavior at the mesoscale at each Gauss point accurately. The practical advantage of homogenization stands is therefore the fact that only the mechanical parameters of the constituent materials (brick and mortar) are required to estimate the average behavior of masonry to be used in structural analyses. Additionally, in large-scale computations FE meshes unrelated to the brick size can be used. From a macroscopic point of view, if units are arranged according to a regular pattern, masonry is an orthotropic medium. The macroscopic behavior of masonry beyond the elasticity limit was mathematically described by Pietruszczak and Niu (1992), assuming bricks to be elastic-brittle and mortar to be elastoplastic and hardening. Their approach allowed macroscopic failure surfaces for different orientations of the principal stresses to the bed joints to be determined. Damage effects in the constituents were taken into account (e.g., by Luciano and Sacco, 1997; Zucchini and Lourenc¸o, 2007; Shieh-Beygi and Pietruszczak, 2008) to describe the brittle postpeak behavior experimentally observed in tests on masonry specimens. More recently, Sacco (2009) predicted the macroscopic behavior of 2D brickwork in the nonlinear range by assuming damage and friction effects to develop only in the mortar joints, and applying classical homogenization techniques for periodic media to any represenvative volume element (RVE). The major limitation of homogenization is related to nonlinear FE computations, because a continuous interaction between meso- and macroscale is needed beyond the linear range. This issue involves a huge

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computational effort, since the field problem has to be solved numerically at each loading step and at all Gauss points. Apart from the prediction of the incremental inelastic response of largescale masonry structures, from an engineering point of view it is interesting to get direct information on the behavior of masonry at failure (Gilbert et al., 2006; Milani, 2008, 2009, 2011a,b,c). The aim is to provide designers with reliable and efficient tools for fast estimates of the load-carrying capacity and the active failure mechanisms. In this framework, limit analysis combined with homogenization theory is an interesting technique that straightforwardly predicts the ultimate behavior of entire structures. This approach requires only a reduced number of material parameters to be known, avoids units and joints to be separately modeled, and allows analyses at the meso- and macroscales to be independently performed. In other words, at a first step homogenized failure surfaces for masonry can be estimated at the mesoscale, without nesting the mesoscale into the FE code used at the macroscale. This is a remarkable advantage that has recently allowed a specific research stream to be developed, to derive advanced models for the evaluation of macroscopic strength domains for masonry walls under in- and out-of-plane loads; see, for example, de Buhan and de Felice (1997), Luciano and Sacco (1997), Sab (2003), Milani et al. (2006a,b), Cecchi et al. (2007), Sab et al. (2007), Cecchi and Milani (2008), Milani (2009, 2011a,b,c), Milani and Tralli (2012), Milani and Taliercio (2015), Stefanou et al. (2015), and Milani and Taliercio (2016). The second step is the implementation of these domains at the structural scale, to perform FE limit analyses on entire buildings (see, e.g., Milani et al., 2006b; Milani, 2015): limit load multipliers, failure mechanisms, and stress distributions at collapse, at least at critical sections, can be obtained. Focusing on the mesoscale, assuming mortar and bricks to be rigid-perfectly plastic with associated flow rule, and within the basic assumptions of homogenization theory for periodic media, macroscopic strength domains for masonry can be estimated using the classic upper and lower bound theorems of limit analysis applied to an elementary cell (Suquet, 1983, 1987). In particular, the lower bound approach requires statically and plastically periodic microstress fields to be considered, and allows lower bound estimates of the actual homogenized failure domain to be obtained by means of the constrained maximization of the macroscopic stresses. Dually, the upper bound approach requires kinematically admissible, that is, strain rate-periodic, velocity fields to be dealt with, and allows the upper bound of the actual homogenized failure domain to be obtained using the constrained minimization of the total internal power dissipation. In both approaches, the mechanical problem is matched by (non)linear mathematical programming problem, where the total number of optimization unknowns is extremely reduced.

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This chapter is aimed at critically reviewing some available recent and effective models, with a comparison of their numerical performances. In particular, four different strategies for the evaluation of the homogenized strength domain of running or header bond masonry under in-plane and outof-plane loads are discussed and critically compared. Two of these models give lower bounds on the macroscopic strength domain of periodic masonry, and two give upper bounds. The first lower bound model (Milani et al., 2006a,c) subdivides the elementary cell into a few rectangular subdomains, in which the microstress field is expanded using polynomial expressions. In the second lower bound procedure (Milani, 2011a,c), joints are reduced to interfaces and bricks are subdivided into a few constant stress triangular (CST) elements: closed-form estimates of the homogenized strength domain can be determined. The third procedure (Cecchi et al., 2007; Cecchi and Milani, 2008) is a “compatible identification” approach, with joints reduced to interfaces and bricks assumed to be infinitely resistant. The velocity field is assumed to be a linear combination of elementary deformation modes applied to the elementary cell. The last model is a kinematically admissible procedure based on the so-called Method of Cells (MoC; see Milani and Taliercio 2015, 2016), where the elementary cell is subdivided into six rectangular subcells with prescribed polynomial strain rate-periodic velocity fields. The first and latter approaches have the advantage that the finite thickness of the joints is explicitly taken into account. In the second approach, although joints are reduced to interfaces with frictional behavior, failure inside bricks can be accounted for. The third approach is the most straightforward, but is reliable only in the case of thin joints and strong blocks. A critical comparison of the pros and cons of all models is discussed, with reference to some examples of engineering interest.

12.2 Fundamentals of homogenization for periodic media When heterogeneous media are dealt with, it is customary to replace the real medium Ω by a “homogenized” one and define its global (or macroscopic) properties through the analysis of a RVE. The RVE is the smallest part of the real medium that contains all the information required to completely characterize its average mechanical behavior. If the medium is periodic a single “unit cell” (Y) can be used as RVE. Ym and Yb will denote the parts of the cell occupied by mortar and brick, respectively. Masonry is a composite material, usually made of units bonded with mortar joints; in several instances, units and mortar are periodically arranged. Due to its periodicity, an entire masonry wall Ω, assumed to be a 2D medium, can be seen as the spatial repetition of unit cells (see Fig. 12.1). If a running- or header-bond bond pattern is considered, a possible choice for the unit cell is a rectangle, as shown in Fig. 12.1.

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FIGURE 12.1 Typical running bond assemblage of bricks and mortar and possible unit cell.

For periodic arrangements of units and mortar, homogenization techniques can be used both in the elastic and inelastic range, taking into account the microstructure only at a cell level. This leads to a significant simplification in the numerical models adopted for studying entire walls, especially for the inelastic case. The basic idea of any homogenization procedure consists in defining macroscopic stresses and strains (denoted by E and Σ, respectively) that represent the corresponding microscopic quantities σ and ε averaged over the cell: ð 1 E 5 hεi 5 εðuÞdY A Y ð ð12:1Þ 1 Σ 5 hσi 5 σdY A Y

where A is the area of the elementary (2D) cell, u is the microscopic displacement field, and h i is the averaging operator. Suitable periodicity conditions are imposed on σ and u, that is:  u 5 Ey 1 uper uper on @Y ð12:2Þ σn anti-periodic on @Y where uper is the periodic part of u and @Y is the boundary of Y. Assume both materials to be rigid-perfectly plastic and to obey an associated flow rule. Let Sm, Sb, and Shom denote the strength domains of mortar, units, and homogenized material, respectively. It was proved by Suquet (1983) that the static definition of Shom in the space of the macroscopic stresses reads: 8 9 8 ð P 1 > > > > > > σdY ðaÞ 5 hσi 5 > > > > > > > > > A > > > > > > Y

> ½½σnint 5 0 on Sσ ðcÞ > > > > > > > > > > > > σn anti-periodic on @Y ðdÞ > > > > > > > : ; : m m b b σðyÞAS ’yAY ; σðyÞAS ’yAY ðeÞ where ½½σ is the jump in microstresses across any discontinuity surface Sσ and nint is the normal to Sσ at any point. Condition (12.3b) imposes the

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microequilibrium, condition (12.3d) derives from periodicity, and condition (12.3e) represents the yield criteria for the components (brick and mortar). Any point of the homogenized failure surface can be determined by prescribing a direction in the homogenized stress space by means of a unit vector nΣ and solving the following constrained maximization problem: ð 8 1 > λn 5 σdY > Σ > > Y > > > Y > > < σn anti-periodic on @Y ð12:4Þ Find maxfλg:j > divσ 5 0 > > ( > > > Sb if yAY b > > i > : σðyÞAS ðyÞ 5 Sm if yAY m where λnΣ is a macroscopic stress on the boundary of Shom belonging to a straight line, oriented as nΣ . If masonry is assumed to be in a state of plane  T stress in the plane (x1,x2), nΣ 5 α β γ is a unit vector in the macroscopic stress space (Σ11, Σ22, Σ12); see Fig. 12.2A. A dual kinematic definition of Shom , also due to Suquet (1983), can be derived through its support function πhom ðDÞ as follows: 8 8 9 Σ:D #8πhom ðDÞ ’DAR6 > > > 9 > > > > > ð > > > > < = > > > s > > > 1 > > hom < < π ðDÞ 5 inf PðvÞ:jD 5 = v  ndS > hom v : ; Y ð12:5Þ S 5 Σ:j @Y > > > ð ð > > > > > > > > > > > > PðvÞ 5 πðdÞdY 1 πð½½v; nv ÞdS > > > > > : > ; : Y

S

where v 5 Dy 1 vper is the microscopic velocity field, and vper is its periodic part; d and D are the microscopic and macroscopic strain rates, respectively;

FIGURE 12.2 (A) In-plane homogenization problem. Meaning of the multiplier λ in the homogenized stress space (Σ11 5 Σxx 5 α, Σ22 5 Σyy 5 β and Σ12 5 Σxy 5 γ). (B) Out-of-plane homogenization problem. Meaning of the multiplier λ in the optimization problem and angles ψ and ϑ.

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S is any discontinuity surface for v in Y, and nv is the normal to S; s denotes the symmetric part of the dyadic product of two vectors; πðdÞ 5 maxσ fσ: d; σASðyÞg; πð½½v; nv Þ 5 πð½½v s nv Þ.

12.3 Polynomial expansion of the microstress field (PES) 12.3.1 Homogenized yield surface under in-plane loads The first micromechanical model presented in this chapter for the limit analysis of masonry walls under in- and out-of-plane loads was proposed by Milani et al. (2006a). The model requires a subdivision of the unit cell into 36 subdomains, as shown in Fig. 12.3, in which polynomial microstress fields are defined. Equilibrium inside each subdomain and at the interface between contiguous subdomains is prescribed, together with antiperiodicity conditions for the microscopic stress vector along the boundary of the unit cell. In each subdomain Y k , any stress component σðkÞ ij is expressed as a polynomial of degree m and can be written as follows: T σðkÞ ij 5 XðyÞSij

 where XðyÞ 5 1

y1

y2

y21

y1 y2

y22

ð12:6Þ

yAY k h

 . . . , and Sij 5 Sð1Þ ij

Sð2Þ ij

Sð3Þ ij

ð5Þ ð6Þ ~ Sð4Þ ij Sij Sij . . . is an array of N 5 ðm 1 1Þðm 1 2Þ=2 entries, representing the unknown stress parameters. Prescribing equilibrium with zero body forces within every subdomain, continuity of the stress vector at any interface, and antiperiodicity of σn, strongly reduces the total number of independent stress parameters.

FIGURE 12.3 Subdivision of the unit cell. Left: subdivision and geometrical characteristics of one-fourth of the cell. Right: subdivision of the entire cell into 36 subdomains.

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In detail, equilibrium within each subdomain reads σðkÞ ij;j ðy1 ; y2 Þ 5 0, i 5 1; 2, ’ðy1 ; y2 ÞAY k . Since σðkÞ is a polynomial expression of degree (m), ij any linear combination of its derivatives (divσðkÞ ) is a polynomial of degree (m 2 1). This gives 2 N linear independent equations in the stress coefficients, where N 5 ððm21Þ2 =2Þ 1 ð3ðm 2 1Þ=2Þ 1 1 5 ðmðm 1 1Þ=2Þ. Continuity of the stress vector across any internal interface between two contiguous subdomains ðY k ; Y r Þ sharing a common interface of normal n reads ðrÞ k r σðkÞ ij nj 2 σij nj 5 0, i 5 1; 2; ’Y ; Y (see Fig. 12.4A). The stress components are polynomial expressions of degree m along the interface: hence, other 2N 0 additional equations, with N 0 5 m 1 1, are obtained (see Fig. 12.4B). Antiperiodicity of σn on @Y gives 2N 0 additional equations per each pair of external faces ðm; nÞ (Fig. 12.5C), where the outward unit normal vectors (n1, n2) are opposite. Finally, some automatically performed elementary assemblage operations on the local variables allow the stress vector within every subdomain to be expressed as follows:

FIGURE 12.4 Contiguous subdomains. (A) Geometry and reference system for each subdomains and interfaces between adjacent subdomains. (B) Example of equilibrium conditions along a horizontal interface for the normal stress and quadratic expansion of the microstress field.

FIGURE 12.5 Micromechanical model proposed for transverse loads. (A) Subdivision of the unit cell into layers along the thickness. (B) Subdivision of each layer into subdomains. (C) Enforcement of equilibrium and periodicity conditions.

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~ ðkÞ ðyÞS~ σ~ ðkÞ 5 X

k 5 1; . . .; kmax

431

ð12:7Þ

where σ~ is the array of the in-plane stresses within the k-th subdomain; S~ is the array gathering the Nun independent unknown stress parameters; ~ ðkÞ ðyÞ is a 3 3 Nun matrix that contains only geometrical coefficients; its eleX ments are polynomial forms in the microscopic coordinate y. The approximated stress field defined in Eq. (12.7) is statically admissible, so that the constrained maximization problem (12.4) that defines the macroscopic yield surface in plane stress conditions point by point can be further specialized as follows: 8 maxfλg 8 > > max ð > > X > > 1 4k > > ~ > ~ ðkÞ ðyÞSdY > λnΣ 5 X ðaÞ > > > < > Y < k51 Y ð12:8Þ such that > > ðbÞ yAY i > > > > > > > > ~ ðkÞ ðyÞS~ > σ~ 5 X ðcÞ > > > : : k ~ σðyÞAS k 5 1; . . .; 4kmax ðdÞ ðkÞ

where Sk stands for the failure domain of the component (unit or mortar) belonging to the k-th subdomain. The optimization problem given by Eq. (12.8) is generally nonlinear as a consequence of the (possible) nonlinearity of the yield surfaces of the components. In addition, condition (12.8d) has to be checked at every point of the domain Y. Nevertheless, a continuous check is avoided using classic collocation, that is, imposing plastic admissibility only where the stress is higher. This procedure provides a rigorous lower bound only for polynomial models of order 0 and 1; in all other cases, collocation consists in enforcing, in every subdomain, the admissibility condition on a regular grid of r 3 q “nodal points.” Adopting a regular grid, the optimization problem takes the following discretized form: 8 maxfλg 8 > ð > > 1 X ~ ðkÞ ~ > > > > > X ðyÞSdY > > > λnΣ 5 Y > > > > k < > Y < ð12:9Þ yj  nodal point such that > > ðkÞ j > > j ~ ðy ÞS~ > > σ~ 5 X > > > > > > j i > > ~ σ AS j 5 1; :::; rq > > : : k 5 1; :::; 4kmax Standard linearization of the yield surfaces of brick and mortar allows the problem to be solved by linear programming. Sections of the in-plane failure surface of masonry can be represented at different angles, ϑ, between a bed joint and the macroscopic horizontal stress (Σ 11 ). By keeping ϑ fixed and varying the angle ψ 5 tan21Σ 22 /Σ11 , where

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Σ 22 is the macroscopic vertical stress, any section is drawn. Accordingly, the director cosines of vector nΣ can be expressed as: 1 nΣ;1 5 ðcos ψð1 1 cosð2ϑÞÞ 1 sin ψð1 2 cosð2ϑÞÞÞ 2 1 nΣ;2 5 ðcos ψð1 2 cosð2ϑÞÞ 1 sin ψð1 1 cosð2ϑÞÞÞ 2 1 nΣ;3 5 ðcos ψ 2 sin ψÞsinð2ϑÞ 2

ð12:10Þ

The numerical failure surfaces can be therefore obtained by solving the optimization problem  T given by Eq. (12.4), where the direction of the “load” Σ 11 Σ22 Σ 12 depends on the orientation ϑ of the principal stresses to the joints.

12.3.2 Extension to transverse loads In order to account for loads acting transversely to a wall, within the framework of the previous discretization into subdomains and polynomial expansions of the stress field, the unit cell is subdivided into a fixed number of layers along its thickness, as shown in Fig. 12.5A. In other words, the out-ofplane model requires a subdivision of the wall into nL layers of equal thickness ΔL 5 t=nL (Fig. 12.5A). According to classical limit analysis for thin plates, the out-of-plane components σi3 (i 5 1,2,3) of the microstress tensor σ vanish, so that each layer is subjected only to in-plane stresses σij with i,j 5 1,2. These stresses are assumed to be uniform along the thickness of each layer, that is, in each layer σij 5 σij ðy1 ; y2 Þ is a polynomial expansion in the in-plane geometric variables (see Milani et al., 2006a,c and Fig. 12.5B). Under the same hypotheses made for in-plane loads, any point belonging to the boundary of the homogenized flexural strength domain can be evaluated by solving the following (non)linear optimization problem: 8 maxfλg > > 8 Ð > > > N 5 k;iL σ~ ðk;iL Þ dYdy3 ðaÞ > > > > > > Ð > > > ðk;i Þ > > M 5 k;iL y3 σ~ L dYdy3 ðbÞ > > > > > > >     > > > > > M11 M12 cosðψÞ cosðϑÞ sinðϑÞ > > > >M5 > 5λ ðcÞ > < > > sinðϑÞ sinðψÞ cosðϑÞ M12 M22 < > such that ψ 5 ½0; 2π θ 5 ½0; π=2 ðdÞ > > > > > > ðk;iL Þ > ðk;i Þ > > L ~ ~ > > 5X ðyÞS ðeÞ σ~ > > > > > > > ðk;i Þ ðk;i Þ > L L > > σ~ AS ðfÞ > > > > > > > > > > k 5 1; . . .; number of sub-domains; > > > > : : iL 5 1; . . .; number of layers ðgÞ ð12:11Þ

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where λ is the load multiplier along any radial path in the space of the bending and twisting moments (M11, M22, M12; see Fig. 12.2B); ψ and ϑ are spherical coordinates in the space (M11, M22, M12), given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 M 2 , tan ψ 5 M =M ; Sðk;iL Þ denotes the (nonlinear) tan ϑ 5 M12 = M11 22 11 22 strength domain of the constituent material (mortar or brick) corresponding ~ to the kth subdomain and ith L layer; S collects all the unknown polynomial coefficients (of all subdomains, of all layers). It is worth noting that: G

G

G

For the sake of simplicity, membrane actions are assumed to be constant and independent from the load multiplier. Hence, in estimating the loadbearing capacity of transversely loaded walls, in-plane actions are only assumed to modify the flexural strength domain of masonry. This assumption is rigorous for laboratory wallettes, where a fixed in-plane vertical compressive load is always applied before any out-ofplane actions, which is then increased to failure. It also acceptable in the analysis of real buildings, which always withstand vertical, in-plane dead loads. Condition ðf Þ should be checked at every point of the domain Y, but this is impossible for polynomial expansions of degree higher than 1. The approach used is thus based on collocation, that is, admissibility is checked on a regular grid of “nodal points.” Similar to the in-plane case, the nonlinearity in the terms σ~ ðk;iL Þ ASðk;iL Þ , due to the (possible) nonlinearity of the strength domains of the components, can be avoided by a classical piecewise linearization of the domains.

12.4 Equilibrated model with joints reduced to interfaces and constant stress triangular discretization of the bricks 12.4.1 Homogenized yield surface under in-plane loads The second homogenization model presented is also based on a static lower bound approach, in which the unit cell is roughly discretized into FEs, as depicted in Fig. 12.6. Unlike the previous model, joints are reduced to interfaces of vanishing thickness, and units are discretized using a coarse mesh of three-node planestress elements (CST), as schematically sketched in Fig. 12.6. The choice of meshing 1/4 of the brick through at least three triangular elements is due to the need of capturing the presence of shear stresses in the bed joint under horizontal stretching (element 2 in Fig. 12.6). The interfaces within the bricks allow, in principle, failure of the units to be captured.

434

PART | II Modeling of unreinforced masonry

FIGURE 12.6 Lower bound approach with CST discretization of the bricks. Subdivision of the REV into 24 CST triangular elements (and 1/4 into six elements) and antiperiodicity of the ðnÞ ðnÞ ðnÞ ðnÞ 5 σðnÞ microstress field (σðnÞ 12 ). xx 5 σ11 , σyy 5 σ 22 , τ

Here, 24 CST elements are used for the discretization of the unit cell (Fig. 12.6). The superscript (n) denotes any stress component belonging to the n-th element. Assuming planestress conditions, the nonvanishing components of the Cauchy stress tensor σðnÞ within the n-th element are σðnÞ 11 (horðnÞ ðnÞ izontal stress), σ22 (vertical stress), and σ12 (shear stress). The total number of unknowns is 73, that is, 72 stress components (three per element), and the load multiplier λ. Equilibrium within each element (divσ 5 0) is a priori fulfilled, because CSTs are used. On the contrary, two equality constraints must be imposed for each internal interface, to ensure continuity of the normal and tangential components of the stress vector across the interface between contiguous elements. Antiperiodicity constraints for the stress vector are written for the pairs of triangles 16, 10 60 , 712, 70 120 , 170 , 390 , 4100 , and 6120 . For instance, for the pairs 16 and 170 , the following equality constraints must be prescribed: ) ð6Þ σð1Þ 1i 5 σ1i i 5 1; 2 ð12:12Þ ð70 Þ σð1Þ 5 σ 2i 2i Assuming the strength of both the interfaces and the triangular elements to be limited, the in-plane homogenization problem can be rewritten as follows:

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435

maxfλg

8 24 X > > > σðiÞ > 11 Ai > > i51 > > λα 5 > > > 2ab > > > > > > > 24 X > > > > σðiÞ 22 Ai > > > i51 > > λβ 5 > > 2ab > > > > > < 24 X subject to σðiÞ > 12 Ai > > > i51 > > > λγ 5 2ab > > > > > > > > > AIeq X 5 bIeq > > > > > ap > > Aap eq X 5 beq > > > > > ðiÞ ðiÞ ðiÞ > i > > > fE ðσ11 ; σ22 ; σ12 Þ # 0 ’i 5 1; . . .; 24 > > : i ðiÞ ðiÞ fI ðσI ; τ I Þ # 0 ’i 5 1; . . .; 32

ð12:13Þ

The symbols used in Eq. (12.13) have the following meaning: G

G G

G

G

G

α, β, and γ indicate the director cosines of the unit vector nΣ (see Fig. 12.2,A) in the space of the homogenized membrane stresses. The solution of the optimization problem allows a point on the homogenized failure surface, with coordinates Σ11 5 λα, Σ22 5 λβ, and Σ12 5 λγ, to be determined. Ai is the area of the i-th element (ab/8 or ab/16). X is an array of 73 entries, which collects all the optimization unknowns (elements stress components and collapse multiplier). AIeq X 5 bIeq is a set of linear equations corresponding to equilibrium constraints on all interfaces. Since 32 interfaces are present in the discretized unit cell and two equality constraints have to be fulfilled at each interface, AIeq is a 64 3 73 matrix and bIeq is an array of 64 zero entries. ap Aap eq X 5 beq collects antiperiodicity conditions, and it is therefore a set of ap 16 equations. Thus, Aap eq is a 16 3 73 matrix, and beq is an array of 16 zero entries. ðiÞ ðiÞ fEi ðσðiÞ 11 ; σ 22 ; σ12 Þ # 0, i 5 1,. . .,24, is a set of nonlinear inequalities defining the strength domain of the i-th element. Within a linear programming scheme, the yield surface is linearized a priori: linearization is usually performed so as to get a safe approximation of the strength domain, to ensure that a strict lower bound estimate of the collapse load is obtained

436

G

G

PART | II Modeling of unreinforced masonry

using a static approach. This can be easily obtained using a Delaunay tessellation. ðiÞ i fIi ðσðiÞ I ; τ I Þ # 0, i 5 1,. . ., 32 are inequalities that play the role of fE # 0 for the interfaces. Two typologies of interfaces are present in the model, namely interfaces within bricks and interfaces corresponding to mortar joints. ðiÞ σðiÞ I and τ I indicate the normal and shear stress acting on interface i, respectively.

Eq. (12.13) is a standard linear programming problem that allows the collapse loads of in-plane loaded masonry structures to be estimated using an FE approach.

12.4.2 Extension to transverse loads The generalization of the model to out-of-plane actions (under the KirchhoffLove hypothesis for thin plates) is performed in the same way followed for the polynomial expansion of the stress field in Par. 12.3.2, namely with a subdivision of the wall thickness t into nL layers of equal thickness ΔL 5 t=nL . Within these assumptions, similar to the in-plane case any point of the failure surface in the bending-twisting moment space can be estimated by solving the following linear programming problem: maxfλg

0 1 8 nL 24 > nL 24 X X > X X t 1Δ L > @ A σði;jÞ Ai > 2jΔ Δ > L L ΔL σði;jÞ 11 > 11 Ai > 2 > i51 j51 > i51 j51 > > ; N11 5 5Σ11 t50 λα5 > > 2ab 2ab > > > > > 0 1 > > > nL 24 nL 24 X X > X X t 1Δ > L ði;jÞ > ði;jÞ > 2jΔL A σ22 ΔL @ Ai Δ σ22 Ai > L > 2 > i51 j51 > j51 i51 > > ; N22 5 5Σ22 t 50 > λβ 5 > > 2ab 2ab > > < 0 1 subject to nL 24 nL 24 X X > X X t 1Δ L > ði;jÞ @ A σði;jÞ Ai > 2jΔ Δ > L L ΔL σ12 Ai 12 > 2 > > i51 j51 j51 i51 > > > ; N12 5 5Σ 12 t 50 λγ 5 > > 2ab 2ab > > > > > > > AI X5bI > > eq eq > > > ap ap > > A X5b eq > > eq > > > i;j ði;jÞ ði;jÞ ði;jÞ > fE ðσ11 ;σ22 ;σ12 Þ#0 ’i51;...;24; ’j51;...;nL > > > > : i;j ði;jÞ ði;jÞ fI ðσI ;τ I Þ#0 ’i51;...;32; ’j51;...;nL ð12:14Þ

where all the symbols have already been defined in the previous section.

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Compared to the in-plane case, the following key issues are worth noting: G

G

G

G

G

G

λ is the load multiplier along any radial path in the ðM11 ; M22 ; M12 Þ space; α, β, and γ are the director cosines of the unit vector nΣ in the ðM11 ; M22 ; M12 Þ space (see Fig. 12.2B). X is an array that collects all the unknown stresses in all the FEs of all layers. Therefore, X is an array of 3 3 24 3 nL entries. AIeq X 5 bIeq collects equilibrium constraints at the interfaces of all layers. Since no shear stresses are transferred between contiguous layers, for each layer these constraints are the same as in Eq. (12.13). AIeq is a 64nL 3 (72nL 1 1) matrix, and bIeq is an array of 64nL zero entries. ap Similar remarks apply to the set of equations Aap eq X 5 beq , which collects antiperiodicity and equilibrium conditions for all layers. Aap eq is now a is an array of 16n zero entries. 16nL 3 (72nL 1 1) matrix, and bap L eq Unlike the in-plane case, three additional equality constraints have to be imposed to ensure that the homogenized membrane forces ðN11 ; N22 ; N12 Þ vanish.

Similar to the procedure followed with the previous lower bound model, membrane actions are kept constant and independent from the load multiplier.

12.5 Compatible identification model with joints reduced to interfaces and infinitely strong units 12.5.1 Homogenized yield surface under in-plane loads The third model briefly recalled here is based on a kinematic approach, in which bricks are supposed to be infinitely resistant and joints are reduced to interfaces with cohesive frictional behavior. At the microscale, a full description of the model can be given considering a RVE consisting of a central brick and six adjacent bricks. In Fig. 12.7 a RVE is subjected to three elementary in-plane homogenized strains, namely horizontal normal strain (A), shear (B), and vertical normal strain (C). Note that when the RVE is subjected to horizontal in-plane stretching, head and bed joints contribute both to the ultimate strength, whereas in

FIGURE 12.7 Deformation modes considered in the compatible identification model.

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PART | II Modeling of unreinforced masonry

vertical stretching only the bed joints experience a nonvanishing jump in displacement. The real continuum is replaced by a standard 2D Cauchy continuum, identified by its midplane S of normal e3. The homogenization procedure is a so-called “compatible identification,” which is based on the equality of the internal power dissipated in the 3D discrete system and in the equivalent 2D continuum. The velocity at any point ξ of one of the bricks (A) is expressed in terms A of velocity (vC ) of the brick centroid, CA , and rotation rate of the brick (ΦA ). When ξ lays on any interface between two contiguous bricks A and B, its velocity can be expressed in terms of kinematic unknowns of both bricks as follows: A

vA ðξÞ 5 vC 1 MðΦA Þðξ 2 CA Þ B

vB ðξÞ 5 vC 1 MðΦB Þðξ 2 CB Þ

ð12:15Þ

The jump in velocity ½½vðξÞ at ξ is therefore given by: A

B

½½vðξÞ 5 vA ðξÞ 2 vB ðξÞ 5 vC 2 vC 1 MðΦA Þðξ 2 CA Þ 2 MðΦB Þðξ 2 CB Þ ð12:16Þ The power dissipated at the interface is: ð ð A A B B π 5 ½t ðξÞUv ðξÞ 1 t ðξÞUv ðξÞdS 5 tA ðξÞU½vðξÞdS I

ð12:17Þ

I

 T is the stress vector at ξ, and where tA ðξÞ 5 τ 13 ðξÞ τ 23 ðξÞ σ33 ðξÞ A B t ðξÞ 5 2 t ðξÞ. The velocity of a point P 5 ðxP1 ; xP2 ; xP3 Þ in the equivalent continuum corresponds to the velocity wðxÞ 5 ðw1 ; w2 Þ 5 ðw1 ; w2 Þ of the point x 5 ðxP1 ; xP2 ; 0Þ laying in the midplane of the wall. _ where The power dissipated in the equivalent continuum is π 5 NT E,  T E_ 5 E_ 11 E_ 12 1 E_ 21 E_ 22 is the array of the in-plane strain rates, and  T is an array collecting the homogenized membrane N 5 N11 N12 N22 forces. In the so-called compatible identification, the power dissipated in the heterogeneous assemblage of blocks and interfaces is assumed to be equal to that dissipated in the equivalent model. To this end, fields wðxÞ corresponding to possible actual failure mechanisms are a priori chosen as combinations of elementary deformation modes of the unit cell. From a practical point of view, a field wðxÞ corresponding to each elementary deformation mode is obtained by alternatively taking one of the macroscopic strainrate components equal to unity and by setting all the other components equal to zero: wðxÞ is then given a simple polynomial expression. Once that wðxÞ is known, the rotation rates and velocities of each

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439

brick belonging to the RVE in the heterogeneous model are determined, by assuming as point x the centroid of the brick under examination.

12.5.2 Extension to transverse loads: ReissnerMindlin model In the ReissnerMindlin (RM) plate model the angular velocity (ω) of any segment perpendicular to the midplane is independent of the transverse velocity, w3. The Compatible Identification approach can be extended to masonry walls obeying the RM model and subjected both to in-plane and transverse loads, by applying suitable homogenized curvature and transverse shear rates, as shown in Figs. 12.8 and 12.9. Fig. 12.8A shows the effect on the brickwork of a homogeneous curvature rate χ_ 11 5 ω1 ;1 all the other generalized strains being set to zero. In this case, both head and bed joints are involved in the dissipation induced by this deformation mode. Fig. 12.8B shows the effect of a homogeneous curvature rate χ_ 22 5 ω2 ;2 . It is interesting to note that only the bed joints exhibit a jump in velocity between adjacent bricks. Similarly, in Figs. 12.8C and D the cases in which only ω1,26¼0, and ω2,16¼0 are nonvanishing are examined: combining these two cases, the deformation mode under homogeneous twisting curvature rate χ_ 12 5 ω1 ;2 1 ω2 ;1 can be obtained. In the first case, no bending moment

FIGURE 12.8 Elementary homogeneous deformations applied to the representative volume element. (A) ω1;1 5 χ_ 11 ; (B) ω2;2 5 χ_ 22 ; (C) ω2;1 ; (D) ω1;2 .

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PART | II Modeling of unreinforced masonry

FIGURE 12.9 Shear deformation rates. (A) γ_ 13 ; (B) γ_ 23 .

exists in the head joints, whereas bed joints experience twisting. Conversely, in the second case, torsion is present in the head joints and the bed joints experience bending. Finally, Fig. 12.9 refers to the evaluation of the behavior of masonry under transverse shear strain rates, γ_ i3 5 w3;i 1 ωi , i 5 1,2. In particular, Fig. 12.9A shows the effects of the γ_ 13 component, whereas Fig. 12.9B shows the effects of the γ_ 23 component. Here, a numerical procedure for obtaining macroscopic homogenized failure surfaces for running bond masonry is presented. The procedure is developed under the hypotheses of RM plate theory, assuming bricks to be infinitely resistant and joints to be reduced to rigid-perfectly plastic interfaces with an associated flow rule. As the problem is dealt with in the framework of linear programming, for each interface I of area AI a piecewise linear approximation of the failure surface ϕ 5 ϕðσÞ is adopted. The surface  is defined by nlin planes of equation ðaIi ÞT σ 5 cIi 1 # i # nlin , where σ 5 σ33 σ13 σ23 , σ33 being the normal stress on the interface and σ13, σ23 the transverse shear stresses along two perpendicular directions, aIi is a 3 3 1 vector of the coefficients of the i-th linearization plane, and cIi is constant term of the equation of the i-th linearization plane. Since the jump in velocity at the interfaces is assumed to vary linearly in the discrete model (see Eq. 12.16), for each interface 3 3 nlin independent plastic multiplier rates are assumed as optimization variables. At each interface I, the following equality constraints between plastic I multiplier rate fields λ_ i ðξ 1 ; ξ 2 Þ and jump in velocity ½w ðξ1 ; ξ 2 Þ are imposed: nlin   X @ϕ I wðξ 1 ; ξ 2 Þ 5 λ_ i ðξ 1 ; ξ2 Þ @σ i51

ð12:18Þ

where (ξ 1,ξ2) is a local reference frame laying on the interface plane, so that  T is ξ 3 is orthogonal to the interface; ½wðξ 1 ; ξ2 Þ 5 Δw33 Δw13 Δw23 the jump in velocity at the I-th interface; any component Δwj3 corresponds

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I to the jump along the direction j; w is linear in (ξ1,ξ2); λ_ i ðξ1 ; ξ2 Þ is the i-th plastic multiplier rate field of the interface I, associated with the i-th lineariI zation plane of the failure surface; also λ_ i is linear in (ξ 1,ξ2). Eq. (12.18) is nothing but the specialization, for the interface I, of the _ _ ij is the plastic strain well-known normality rule ε_ ij 5 λð@φ=@σ ij Þ, where ε _ rate, λ is the plastic multiplier, and φ is the failure surface, which coincides with the plastic potential in the case of associated plasticity. In order to satisfy Eq. (12.18) at any point of any interface I, nine equality constraints have to be imposed, corresponding to Eq. (12.18) evaluated at three different points Pk 5 ðξ P1 k ; ξ P2 k Þ of the interface I. Explicitly: nlin   X @ϕ I wðξP1 k ; ξP2 k Þ 5 λ_ i ðξP1 k ; ξP2 k Þ @σ i51

k 5 1; 2; 3:

ð12:19Þ

The power dissipated at the I-th interface, defined as the dot product of the interface tractions for the jump in velocity, is evaluated using the following equation:  T ð ðX nlin nlin 3 X @ϕ 1X I I πIint 5 ½wT σ dAI 5 σ dAI 5 cIi λ_ i ðξ1 ; ξ2 Þ λ_ i ðξP1 k ; ξ P2 k ÞAI : @σ 3 i51 i51 k51 AI

AI

ð12:20Þ The external power can be written as πext 5 ðΣT0 1 λΣT1 ÞD, where Σ0 is the array of the dead loads; λ is the multiplier of the live loads; Σ1 is the array gathering the reference values of the live loads (which defines the optimization direction in the space of the macroscopic stresses); and D is the array of the generalized macroscopic strain rates. D collects in-plane deformation rates (E_ 11 E_ 12 E_ 22 ), curvature rates (χ_ 11 χ_ 12 χ_ 22 ), and transverse shear strain rates ( γ_ 13 γ_ 23 ); see Figs. 12.712.9. Introducing the classic normalization condition of the failure mechanism ΣT1 D 5 1, the external power becomes linear in D and λ and can be written as πext 5 ΣT0 D 1 λ. The core of the Compatible Identification approach is to assign prescribed microscopic strain-rate periodic velocity fields to the unit cell (represented in Figs. 12.712.9). As a consequence, for each interface I a linear relationship can be written between D and the jump in velocity as follows: ½wðξ1 ; ξ2 Þ 5 GI ðξ1 ; ξ2 ÞD;

ð12:21Þ

where GI ðξ1 ; ξ2 Þ is a 3 3 10 matrix that depends only on the geometry of the interface under consideration. By assembling the equality constraints (12.18)(12.21), and using the kinematic formulation of classic limit analysis, the following constrained

442

PART | II Modeling of unreinforced masonry

minimization problem is finally obtained to evaluate a point of the failure surface: 8 nI > X > > > πIint 2 ΣT0 D λ 5 min > > I ^ > x 5 ½D;λ ðP Þ k i > I51 < T ð12:22Þ Σ1 D 5 1 > > > nlin X > @ϕ > I > > Pk AI λ_ i ðξP1 k ; ξP2 k Þ GI ðPk ÞD 5 ½wðPk Þ 5 > : @σ i51 where nI is the total number of interfaces considered and x^ is the array of all the optimization unknowns. Similar to the homogenization models previously presented in this chapter, problem (12.22) can be easily handled numerically using either one of the well-known simplex and interior point methods, due to the very limited number of optimization unknowns involved. In fact, x^ collects only 3 3 nlin 3 nI plastic multiplier rates and the macroscopic kinematic variables D. Problem (12.22) is written in general form, as it covers both in-and outof-plane loads, and allows interaction failure surfaces of masonry to be estimated through a kinematic approach. Denoting by Σ 5 ðN11 ; N12 ; N22 ; M11 ; M12 ; M22 ; T13 ; T23 Þ an array gather^ ing all the generalized stresses, and by ΦðΣÞ the macroscopic failure poly^ for any couple of variables Σi tope for masonry, a 2D representation of Φ and Σj can be obtained by fixing a direction defined by a unit vector nΣ in the 8D Σ space, so that nΣ ðiÞ 5 cos ψ and nΣ ðjÞ 5 sin ψ (with tan ψ 5 Σj =Σ i ), and solving the following optimization problem: 8 nI > X > > > πIint 2 ΣT0 D minfλg 5 > > > > I51 < T ð12:23Þ nΣ D 5 1 nΣ ðiÞ 5 cosψ nΣ ðjÞ 5 sin ψ > > > nlin X > @ϕ > I > > GI ðPk ÞD 5 ½wðPk Þ 5 λ_ i ðξP1 k ; ξP2 k Þ > : @σ i51 where λ denotes the load multiplier for a prescribed radial load path in the ^ (Σi,Σj; see also Fig. 12.2); i and j denote the axes of projection of Φ.

12.6 Method of Cells: a method of cells-type approach 12.6.1 Homogenized yield surface under in-plane loads The so-called MoC was originally proposed by Aboudi (1991) for unidirectional composites reinforced by a regular pattern of long, parallel fibers. The MoC has been recently extended to masonry by Taliercio (2014, 2016) to

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443

evaluate in closed form the macroscopic elastic and creep coefficients, and by Milani and Taliercio (2015, 2016) to estimate macroscopic strength domains. The method, applied to running- or header-bond, consists of subdividing the unit cell into rectangular subcells, as shown in Fig. 12.10A. In each subcell, the velocity field is approximated using two sets of strain

FIGURE 12.10 (A) RVE adopted in the MoC-type approach and subdivision into subcells. (B and C) Strain-periodic kinematically admissible velocity field under horizontal or vertical macroscopic stresses (A) and shear (B).

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PART | II Modeling of unreinforced masonry

rate-periodic, piecewise-differentiable velocity fields: one for normal deformation modes and one for a shear-type deformation mode. ðiÞ Denting by uðiÞ 1 and u2 the horizontal and vertical velocity fields in the i-th cell, under macroscopic vertical and horizontal normal strains (Fig. 12.10,B) the velocities inside each subcell are given by: x1 nð1Þ x2 u 5 2 2V1 bb 2 hb 0 1 b b ðU2 2 U1 Þ@x1 2 A 2 nð2Þ u1 5 U1 1 u2nð2Þ 5 u2nð1Þ bm 0 1 0 1 h h b b ðU1 ð1 1 2αb Þ 2 U2 Þ@ 2 x2 A ðV1 2 V2 Þ@x2 2 A nð1Þ 2 2 u 2 nð1Þ unð3Þ u2nð3Þ 5 2 V1 1 1 5 u1 2 hm 2hm 0 1 h b ðU1 ð1 1 2αb Þ 2 U2 Þ@ 2 x2 A 2 nð1Þ unð4Þ u2nð4Þ 5 u2nð3Þ 1 5 u1 1 2hm 1 0 1 0 10 b 1 b h b b m b b 2 x1 A@x2 2 A ðU1 2 U2 Þ@x1 2 A ðU1 ð1 1 2αb Þ 2 U2 Þ@ 2 2 2 nð5Þ u1 5 U1 2 2 bm hm bm 0 1 hb @hb 1 hm 2 x2 A x2 2 2 2 u2nð5Þ 5 2 V2 2 V1 hm hm 11 0 1 0 10 0 U 2 U h h 2 A@ b b @U1 1 1 B x2 2 AC ðV1 2 V2 Þ@x2 2 A B 2αb 2 C 2 B C x1 Cunð6Þ 5 2 V1 1 u1nð6Þ 5 2 B U1 2 C 2 hm hm bb B B C @ A unð1Þ 1 5 2U1

ð12:24Þ

The fields in Eq. (12.24) depend on four DOFs, U1 , U2 , V1 , and V2 , with a clear physical meaning as shown in Fig 12.10B. The reference system (x1,x2) and the meaning of the geometrical parameters hb, bb, etc., is provided in Fig. 12.10A. αb 5 bm =bb is the ratio of the bed joint thickness to the brick height. It is interesting to note that the velocity fields inside each subcell are either linear (cells 1, 3, 4) or bilinear (cells 2, 5, 6).

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When a shear deformation mode is applied to the RVE, the velocity fields in the subcells are expressed as: x2 utð1Þ 1 5 2U3 hb

x1 2 tð2Þ tð1Þ utð1Þ 2 5 0 u1 5 u1

0

utð2Þ 2 5 V3

1

utð3Þ 1 5 U3 1

U4 2 U3 @ hb x3 2 A u2tð3Þ 5 2 V4 hm 2

tð3Þ utð4Þ 1 5 u1

tð3Þ utð4Þ 2 5 2 u2 0

tð3Þ utð5Þ 1 5 u1

5 2 V3 utð5Þ 2

bm

x2 2

hb 2

hm

10 1 0 1 @x1 2 bb 1 bm A@x2 2 hb A 2 hm @x1 2 bb A 2 2 2

ð12:25Þ

bm hm 0

1

x1 @x2 2 tð3Þ utð6Þ 1 5 u1

bb 2

utð6Þ 2 5 V3

hb A 2

bm hm

tðiÞ In Eq. (12.25) the symbols utðiÞ 1 and u2 denote the horizontal and vertical velocity fields in the i-th subcell under macroscopic shear. The meanings of the three independent DOFs, U3 , U4 , and V3 , are shown in Fig. 12.10C (V4 5 V3 =2). The velocity field over the RVE under any macroscopic strain can be nðiÞ tðiÞ expressed as the sum of Eqs. (12.24) and (12.25), that is, uðiÞ 1 5 u1 1 u1 ðiÞ nðiÞ tðiÞ and u2 5 u2 1 u2 where the superscript (i) indicates the (i)-th subcell. At each point of any subcell, the associated flow rule corresponds to three equality constraints. Denoting by ε_ ðiÞ pl the plastic strain rate in the (i)-th subcell, the flow rule can be written as

"

ε_ ðiÞ pl

@uðiÞ 1 5 @x1

@uðiÞ 2 @x2

# b;m @uðiÞ @u2ðiÞ ðiÞ @S 1 1 ; 5 λ_ @x2 @x1 @σ

ðiÞ

where λ_ ($0) is the plastic multiplier and Sb;m is the (non) linear yield surface of either bricks (b) or mortar (m). As outlined in the preceding paragraphs, the yield surfaces of bricks and mortar are usually linearized by m planes, so that each strength criterion is defined by a set of linear inequalities of the form Sb;m  Ain σ # bin . As ε_ ðiÞ pl varies at most (bi-)linearly within each subcell, plastic admissibility is checked only at three of the corners. Hence, nine linear equality constraints _ ðiÞ 5 0, where U per subcell are introduced in matrix form as Aeq U 1 Aeq λ UðiÞ

λðiÞ

is an array collecting the 7 DOFs describing any microscopic velocity field

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h _ ðiÞ 5 λ _ ðiÞT (i.e., U 5 {U1, U2, U3, U4, V1, V2, V3}T), λ A

_ ðiÞT λ B

_ ðiÞT λ C

iT is an

_ ðiÞ at array of 3 m entries, collecting the rates of the plastic multipliers λ J eq of the corners of the rectangular subcell (J 5 A, B, C), and AUðiÞ , Aeq λðiÞ

three are a 9 3 7 and a 9 3 3 m matrix, respectively. The plastic admissibility conditions are then assembled cell by cell, leading to the following system of equality constraints:

eq _ Aeq ð12:26Þ U U 1 Aλ λ 5 0 h iT h iT eq T T _5 λ _ ð6ÞT , and Aeq _ ð1ÞT . . . λ . . . Aeq where Aeq , λ U 5 AUð1Þ λ Uð6Þ is a (6  9) 3 (6  3 m) block matrix, which can be expressed as: eq eq eq Aeq λ 5 Aλð1Þ "Aλð2Þ "?"Aλð6Þ ;

ð12:27Þ

the symbol " denotes direct sum. Let B and C be a couple of corners at the opposite ends of one of the diagonals of the (i)-th rectangular subcell. The internal power dissipated within the subcell can be written as: πðiÞ in 5

ðiÞ  ΩðiÞ  ðiÞT _ ðiÞ _ ðiÞ 5 Ω 0 bin λB 1 bðiÞT λ 13m in C 2 2

bðiÞT in

 ðiÞ λ_ ; bðiÞT in

ð12:28Þ

where 01 3 m is an array of m zero entries and ΩðiÞ is the area of the (i)-th subcell. The power dissipated inside the whole RVE is obtained as the sum of the contributions of the single subcells: πin 5

6 X ΩðiÞ  i51

2

01 3 m

bðiÞT in

 ðiÞ λ_ : bðiÞT in

ð12:29Þ

The “external load” applied to the RVE is the macroscopic stress, corresponding to a point of the homogenized failure surface. The array of the  T macroscopic stress components can be expressed as Σ 5 λ α β γ , where λ is a load multiplier and α, β, γ are, as usual, the director cosines defining the direction of Σ in the space of the homogenized in-plane stresses. Accordingly, the power of the external loads can be written as:   πex 5 λ α β γ D ð12:30Þ In limit analysis a normalization condition is needed because the shape of the failure mode is identified, but its amplitude is undetermined:   α β γ D51 ð12:31Þ

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In the framework of the upper bound theorem of limit analysis, any point of the homogenized failure surface is determined by using following constrained minimization problem: Find min πin  8 α β γ D51 > > > > > > _ > Aeq U 1 Aeq > λ λ50 < U ð s subject to 1 > D5 v  ndS > > A > @Y > > > > : λ_ $ 0

ðaÞ ðbÞ

ð12:32Þ

ðcÞ ðdÞ

where Eq. (12.32a) is the normalization condition (12.31); Eq. (12.32b) is the set of equations representing the admissibility of the plastic flow, Eq. (12.26); and Eq. (12.32c) links the homogenized strain rate with the local velocity field.

12.6.2 Extension to transverse loads Under transverse loads, the wall is supposed to behave as a KirchhoffLove plate. Hence, only the transverse velocity field, w(x1,x2), has to considered. Assuming the wall to undergo bending moments acting about the head joints (Mxx) and/or about the bed joints (Myy), it is possible to define a C1-type transverse velocity field defined by four parameters (or DOF, W1. . .W4) as follows: wnð12:1Þ 5

4W1 x21 4W2 x22 1 ; 2 bb h2b

ð1Þ wð2Þ n 5 wn 1 4

ð1Þ wð3Þ n 5 wn 1

ð1Þ wð4Þ n 5 wn 1

W3 ð2 b2b 1x1 Þ2 ðbb 2 bm Þbm

ð12:33aÞ ð12:33bÞ

W4 ðhb 22x2 Þ2 W3 ðbb 1 bm 1 4x1 Þðhb 22x2 Þ2 ðhb 1 3hm 2 2x1 Þ 2 4ðbb 2 bm Þh3m 8h2m ð12:33cÞ W4 ðhb 22x2 Þ2 W3 ðbb 1 bm 2 4x1 Þðhb 22x2 Þ2 ðhb 1 3hm 2 2x2 Þ 2 2 4ðbb 2 bm Þh3m 8hm ð12:33dÞ

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PART | II Modeling of unreinforced masonry

ð1Þ wð5Þ n 5 wn 1

 W4 ðhb 22x2 Þ2 2ðbb 2 2x1 Þ 1 W3 1 2 bb 2 bm 8h2m

2  ðbb 1 bb ðbm 2 4x1 Þ 1 ðbm 22x1 Þ2 Þðhb 12hm 22x2 Þ2 ðhm 2 hb 1 2x2 Þ 4ðbb 2 bm Þbm h3m ð12:33eÞ

W4 ðhb 22x2 Þ2 W3 bb bm 2 4x21 ðhb 22x2 Þ2 ðhb 1 3hm 2 2x2 Þ ð6Þ ð1Þ wn 5 wn 1 2 4ðbb 2 bm Þbm h3m 8h2m ð12:33fÞ 1

For the sake of clarity, Fig. 12.11 shows a wall deformed according to the Eqs. (12.33af) assuming W1 5 W2 5 W4 5 0. Zoomed details of the deformed wall are also shown in subfigures (B) and (C), to highlight twisting in the bed joints. When the RVE is subjected to torsion (M12 5 M21), a C0-type piecewise differentiable velocity field defined only by two parameters W5 and W6 is assumed: wtð12:1Þ 5 ð1Þ wð2Þ t 5 wt 1

4W5 x1 x2 bb hb

2W6 ð2

1 x1 Þx1 x2 bm hb bb 2

ð12:34aÞ ð12:34bÞ

FIGURE 12.11 (A) Deformed masonry wall subjected to bending curvature about the head joints and (B) details of a deformed REV. (C) Details of mortar joints deformation. Brick deformation is neglected.

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ð1Þ wð3Þ t 5 wt 2 W6

ððbb 1 bm Þhb 1 4ðhb 1hm Þx1 Þðhb 2 2x2 Þ 4ðbb 1 bm Þhb hm

ð12:34cÞ

ð1Þ wð4Þ t 5 wt 1 W6

ððbb 1 bm Þhb 2 4ðhb 1hm Þx1 Þðhb 2 2x2 Þ 4ðbb 1 bm Þhb hm

ð12:34dÞ

hb ðbm hb 2 bb ðhb 1 2hm ÞÞðbb 1 bm 2 2x1 Þ 4bm ðbb 1 bm Þhb hm ð12:34eÞ bb ðbb hb 2 2hb x1 Þ 1 bm ð 2 bm hb 1 2ðhb 1 2hm Þx1 Þ 1 2W6 x2 4bm ðbb 1 bm Þhb hm

wðt 5Þ 5 wðt 1Þ 1 W6

ð1Þ wð6Þ t 5 wt 2 W6

ðbb hb 2 bm ðhb 1 2hm ÞÞx1 ð2 bm ðbb 1 bm Þhb hm

hb 2

1 x2 Þ

ð12:34fÞ

Fig. 12.12A and B show a wall deformed under torsion and neglecting brick deformation (W5 5 0). Some details of the deformed shape are shown in Fig. 12.12C and D to show how joints are subjected to twisting. The procedure to estimate a point belonging to the out-of-plane homogenized failure surface is identical to that used for the in-plane case, and is omitted for the sake of brevity. It is only worth noting that, after a

FIGURE 12 (A and B) Deformed masonry wall subjected to twisting. (C) Detail of two deformed RVEs. (D) Detail of joints deformed shape. Brick deformation is neglected.

PART | II Modeling of unreinforced masonry

450

suitable assemblage of the constraints, the constrained minimization problem can be written as follows: 8 > > > > > > > >
> > > > eq eq _ > > > AW W 1 Aλ λ 5 0 < ð s 1 min Π in subject to _ v  n dS Xx3 5 > > > > > > j j Γ > > > > Γ > > > > > > : :_ λ$0

ðaÞ ðbÞ ðcÞ

ð12:35Þ

ðdÞ

where Eq. (12.35a) is the normalization condition, Eq. (12.35b) is the set of equations representing the admissibility of the plastic flow, and Eq. (12.35c) relates the homogenized strain rate with the local velocity field. Note that the independent variables entering the optimization problem _ the (12.35) are the three components of the macroscopic curvature rate X, _ plastic multipliers λ, and the six DOFs defining the microscopic velocity field. Via the normalization condition, and equating the internal power dissipation to the power of the external loads, it can be easily shown that the collapse multiplier λ is equal to min Π in.

12.7 Homogenized strength domains: in-plane loaded masonry Some case studies are considered in this section in order to evaluate the capabilities and the limitations of the four limit analysis homogenization strategies considered in this chapter. First, the convergence of the PES model, as the degree of the polynomials increases, is discussed for running bond masonry made of common Italian clay bricks (250 3 120 3 55 mm3), with 10 mm-thick joints. The mechanical properties are listed in Table 12.1. Convergence of the model to the actual solution was demonstrated in Milani et al. (2006a), to which readers are referred to for further details. TABLE 12.1 Plane-stress MohrCoulomb mechanical properties adopted for joints to test the PES model (bricks are assumed to be infinitely resistant). Friction angle, Φ 

36

ft 5

Cohesion, c 0.1 MPa

2c cosðΦÞ 1 1 sinðΦÞ

fc 5

2c cosðΦÞ 1 2 sinðΦÞ

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FIGURE 12.13 PES model, convergence of the different polynomial expansions masonry material of Table 12.1 and direction of the principal axes parallel to that of material axes (ϑ 5 0 , Σh 5 Σ11, Σv 5 Σ22).

Obviously convergence occurs from the safe side, as the PES model is based on a static approach. In Fig. 12.13A comparison with a standard elastoplastic FE solution is provided in the tensiontension region, assuming one of the principal stresses, Σh, to be parallel to the bed joints. It can be noted that both the P3 and P4 models match the FE results fairly well, whereas for P0 orthotropy at failure is completely lost. Due to the boundary conditions to be imposed over the RVE, the results for P1 coincide with those given by P0. As the P3 model is sufficiently accurate, from here onward it will be used for comparison with the other models discussed in the chapter. For all models, the mechanical properties adopted are those summarized in Table 12.1. When joints are reduced to interfaces, that is, for the lower bound CST discretization and the Compatible Identification approach, a MohrCoulomb failure criterion with tension (ft ) and compression (fc ) cutoff is adopted for the interfaces. The values assumed for ft and fc are reported in Table 12.1, and were obtained from the friction angle Φ and the cohesion c adopted in the PES and MoC models (i.e., with thick joints in plane-stress conditions). Fig. 12.14 shows the homogenized failure surfaces obtained with the four models, at three orientations ϑ of the bed joints to one of the macroscopic principal stresses (0 degree, 22.5 degree, and 45 degree). In particular, subfigure (A) refers to the PES (P3) approach; (B) to the CST equilibrated model and the compatible identification model (which provide the same results); and (C) to the MoC-type approach. Note that, as the models to which Fig. 12.14B refer give in principle a lower and an upper bound to the real macroscopic strength domain, for brickwork consisting of rigid units and infinitely thin joints the exact solution is obtained.

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PART | II Modeling of unreinforced masonry

FIGURE 12.14 Comparison among the different homogenization models proposed in the tensiontension region for the masonry material of Table 12.1. (A) PES, P3 approach; (B) equilibrated CST discretization and Compatible Identification model; (C) MoC model (Σxx 5 Σ11, Σyy 5 Σ22).

Fig. 12.14 points out the considerable dependence of the homogenized yield surfaces from the joint thickness (compare, for instance, subfigures (C) and (B)). Thick joints give smoother and nonlinear yield surfaces. Conversely, if joints are reduced to interfaces, multilinear yield surfaces are obtained, as predicted by de Buhan and de Felice (1997). The maximum horizontal strength is obtained at ϑ 5 0 degree, but is much lower in the case of thick joints (0.26 MPa) than in the case of infinitely thin joints (0.31 MPa), the percentage difference being of the order of 15%. When joints are reduced to interfaces in the MoC model, the theoretical predictions match those given by the compatible identification (or CST equilibrated model; see Fig. 12.16A and B): this is a proof of the good predictive capabilities of the MoC.

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As the MoC approach and the PES are the only two models capable of taking the finite joint thickness into account, and as they give, respectively, upper and lower bounds to the real macroscopic strength domain of masonry, it is interesting to assess the convergence of both models to the exact solution. Referring to the same example as before, the results shown in Fig. 12.15 are obtained. It can be noted that P2 is rather inaccurate, whereas P3, as already pointed out, provides acceptable results from an engineering standpoint (also in terms of numerical efficiency). The actual macroscopic yield surface is somewhere between the results given by P4 and the MoC, so that the exact strength domain can be quite strictly bounded. The highest discrepancy occurs at ϑ 5 0 degree and ψ 5 arctg(Σ22/Σ11) 5 0 degree, the difference between the strength predicted by the MoC and PES-P4 being of about 5%—a result that is fully satisfactory for practical purposes. Replacing joints by interfaces, the homogenized strength under horizontal stretching increases to about 0.32 MPa, and the kinematic Compatible Identification approach provides everywhere results that are superimposable to those given by the static CST approach: this means that the exact solution is captured by both models (see Fig. 12.16A and B). In Fig. 12.16C, the collapse deformation mode of the RVE predicted by the MoC is also represented at ϑ 5 0 degree and ψ 5 0 degree. Similar to the predictions of all the alternative homogenization models, it can be noted that head joints are subjected to simple tension, whereas bed joints undergo pure shear. Cross-joints, conversely, exhibit a mixed failure mode, but it does not affect the ultimate homogenized strength significantly, due to their negligible size.

12.8 Homogenized strength domains: out-of-plane loaded masonry In this section, the performances of the different models in the case of transversally loaded masonry are discussed. First of all, the PES and CST models are applied to the prediction of the ultimate uniaxial bending moment of wallettes loaded in four-point bending, with a bending moment acting about an axis that forms an angle ϑ with respect to the bed joints. The results are shown in Fig. 12.17 and compared with experimental data (Gazzola et al., 1985) and numerical results (Lourenc¸o, 1997) with a macroscopic elasto-plastic orthotropic model: subfigure (A) refers to the PES model and subfigure (B) to the CST model. On the vertical axis, the flexural strength, computed as the ultimate bending moment divided by t2/6, is reported: a fictitious linear elastic triangular distribution of the stress across the thickness is assumed. The input data adopted for the constituent materials are slightly different between PES and CST (as can be seen at ϑ 5 0 degree, where the flexural strength of masonry is that of the bed joint). As a matter of fact, a slight

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PART | II Modeling of unreinforced masonry

FIGURE 12.15 Comparison between polynomial lower bound approximation and MoC in the tensiontension region, mechanical properties of Table 12.1 (Σxx 5 Σ11, Σyy 5 Σ22).

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FIGURE 12.16 Comparison between two different homogenization models proposed in the tensiontension region for a masonry element with the properties listed in Table 12.1 and joints reduced to interfaces. (A) Equilibrated CST discretization and Compatible Identification model (Σxx 5 Σ11, Σyy 5 Σ22); (B) MoC model; (C) MoC deformed shape for horizontal stretching.

update of the input data was implemented for the CST model, due to a more refined analysis of the experimental literature available. Interested readers are referred to Milani et al. (2006c) and Milani and Tralli (2011) for further details. Nevertheless, three main issues are worth noting: (1) the rather good experimental data fitting; (2) the capability to reproduce the anisotropic behavior under out-of-plane loads correctly; and (3) the slight overestimation of the strength with the CST model, which is a consequence of the reduction of joints to interfaces.

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PART | II Modeling of unreinforced masonry

FIGURE 12.17 Comparison among experimental results by Gazzola et al. (1985), plasticity model by Lourenc¸o (1997) and PES (A) or CST (B) in uniaxial bending, at different orientations ϑ of the bed joints to bending moment axis.

Consider now the homogenized biaxial flexural behavior. Assume joints to be interfaces obeying a classic MohrCoulomb failure criterion. If the wall is sufficiently thin, the Compatible Identification model reduces to a KirchhoffLove plate model, for which an analytical solution due to Sab (2003) is available in the absence of twisting moments. Provided that the number of layers into which the wall thickness is divided is sufficiently high, the safe approximations given by the CST and PES models match quite well the analytically predicted orthotropic multilinear failure surface. It is found that a subdivision into 10 layers is suitable for both the PES (Fig. 12.18) and the CST (Fig. 12.19) model. This applies also in presence of a twisting moment: in Fig. 12.19B the results obtained with the CST model are

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FIGURE 12.18 Yield surfaces in the (Mxx, Myy) 5 (M11, M22) plane obtained using the PES model with joints reduced to interfaces (number of layers nL 5 10 or 100), and the Compatible Identification model (which coincides with Sab model in case of Kirchhoff-Love plates).

FIGURE 12.19 Yield surfaces obtained using the CST and compatible identification model by increasing the number of layers across the wall thickness. (A) Biaxial bending (M11, M22); (B) horizontal bending and twisting (M11, M12).

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PART | II Modeling of unreinforced masonry

compared with those obtained by the Compatible Identification approach in the horizontal bending-twisting moment plane (M11, M12), for different numbers of layers across the wall thickness. The results presented so far were obtained in the absence of vertical membrane compressive loads (N22 5 0), a situation which occurs only in laboratory tests on small-size walls. In this case, no perceivable difference is observed between results provided by the models with joints obeying a MohrCoulomb failure criterion with and without compression cap, as the compressive strength of masonry has little effects under pure bending. For practical purposes, it is more interesting to predict the flexural strength of masonry in presence of vertical compressive loads. Let us consider again a running bond masonry built with common Italian clay bricks and joints reduced to interfaces obeying a MohrCoulomb failure criterion with tension cutoff and linear compressive cap; its mechanical properties are listed in Table 12.2. Homogenized out-of-plane yield surfaces are evaluated at fixed, increasing values of the vertical load, in order to have an insight into the effects of a prestress on the overall flexural strength. The results are summarized in Fig. 12.20, where sections of the macroscopic yield surface are shown under biaxial bending (Fig. 12.20A) and horizontal bending and twisting (Fig. 12.20B), at increasing values of the vertical membrane stress N22. It is possible to notice that vertical membrane loads affect not only the horizontal bending moment, but also the vertical one, as bed joints are activated also when masonry is subjected to vertical bending moments. From Fig. 12.20 it can be observed that at high membrane stresses (around 0.70.8 times the compressive strength of the joints), a drop in the out-of-plane load bearing capacity of the wall occurs. In agreement with experimentation, it is possible to spot out an optimal compressive load at

TABLE 12.2 Mechanical properties adopted for the out-of-plane numerical simulations in the presence of vertical precompression (standard Italian clay bricks). Mortar joints reduced to interfaces (MohrCoulomb failure criterion with tension cutoff and linearized compressive cap) Cohesion (MPa)

c

1.4 ft

Tensile strength (MPa)

ft

0.10

Compressive strength (MPa)

fc

4.0

Friction angle (degree)

Φ

37

Shape of the linearized compressive cap (degree)

Φ2

30

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FIGURE 12.20 Yield surfaces obtained at increasing values of the vertical membrane stress, Nyy 5 N22 using the CST. (A) Sections in the plane (M11, M22) 5 (Mxx, Myy). (B) Sections in the plane (M11, M12) 5 (Mxx, Mxy).

which the highest out-of-plane strength is attained. Beyond this optimum value, the out-of-plane strength starts to decrease until membrane crushing occurs. Obviously, a classic MohrCoulomb failure criterion is incapable of reproducing this important phenomenon, since failure in simple compression is not possible. In contrast, a model with limited compressive strength is able to better capture the behavior of masonry under combined in- and out-ofplane actions. An interesting issue to discuss is the influence of the joint thickness on the out-of-plane homogenized failure surface. As already pointed out for the in-plane case, reducing joints to interfaces results in a slightly overestimated

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PART | II Modeling of unreinforced masonry

FIGURE 12.21 (A) Plane stress failure criterion adopted for thick mortar joints. (B) Interface failure criterion adopted for joints reduced to interfaces.

macroscopic strength. The MoC is a quite straightforward approach that allows the role played by the joint thickness to also be quantitatively evaluated for transversely loaded walls. In a further numerical example, the MoC is therefore utilized both assuming joints to be infinitely thin, or 10 mmthick. A running bond wall built with common Italian clay bricks is considered, and the results obtained reducing joints to interfaces are compared with those derived accounting for the actual thickness. The possibility of failure in the units is discarded. Two different failure surfaces are adopted for thick and thin joints, as shown in Fig. 12.21. Fig. 12.21A shows the plane stress multisurface failure criterion used for thick joints. The strength domain is obtained as the convex envelope of a MohrCoulomb failure criterion in plane strain conditions (characterized by a cohesion c and a friction angle Φ); a Rankine failure criterion in tension (characterized by a tensile strength ft); and a linearized compression cap (characterized by three parameters, Φ2, ρ, and fc). The meaning of the symbols is explained in Fig. 12.21A. The second failure criterion applies to joints reduced to interfaces. In this case reference is made to the multilinear failure surface depicted in Fig. 12.21B, which is characterized by MohrCoulomb failure criterion, supplemented by a linear cap in compression (identified by two mechanical parameters, Φ2 and fc, defining the shape of the compression cap and the uniaxial compressive strength, respectively), and a tension cutoff (at a tensile strength equal to ft). As can be noted from Fig. 12.21, the interface failure criterion is the exact counterpart of the 2D one used for thick joints, with a slight difference on the tension cutoff coming from the theoretical definition of the Rankine criterion. The mechanical properties adopted for mortar are summarized in Table 12.3. Again, bricks are assumed to be infinitely resistant: this is a reasonable assumption for clay brick masonry subjected to transverse loads, as failure is usually a consequence of the limited tensile and shear strength of the joints.

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TABLE 12.3 Mechanical properties adopted for the out-of-plane numerical simulations (MoC model) without vertical precompression. Mortar joints reduced to interfaces or with thickness equal to 10 mm (MohrCoulomb failure criterion with tension cutoff and linearized compressive cap) Cohesion (MPa)

c

1.0 ft

Tensile strength (MPa)

ft

0.28

Compressive strength (MPa)

fc

10.0

Friction angle (degree)

Φ

36

Shape of the linearized compressive cap (degree)

Φ2

10

ρ

0.5

It is interesting to point out that the compressive strength assumed for mortar is sufficiently high to limit mortar crushing in the compressed fibers under bending. As a consequence, the homogenized out-of-plane strength basically depends only on the tensile and shear strength of mortar, so that the role of the joint thickness can be better understood. In Fig. 12.22, three sections of the failure surface obtained with the MoC are represented, both for thin and thick joints. Subfigures (A)(C) depict sections in the planes (M11, M22), (M11, M12), and (M22, M12), respectively. A 3D representation of the whole failure surface in the (M11, M22, M12) space is finally provided in Fig. 12.23: subfigure (A) refers to thin joints, whereas subfigure (B) refers to thick joints. As can be noted, the failure surface obtained by the MoC closely matches that found by the Compatible Identification approach, as well as with the CST and PES lower bounds. The ultimate bending moment about the bed joints can be easily estimated by hand calculation as M22 5 ftt2/2. Both in the presence of thin and thick joints, the prediction given by the MoC is in agreement with the theoretical value. The same remarks apply to the ultimate twisting moment, which is simply given by M12 5 ct2/4. In the presence of joints reduced to interfaces, the analytical prediction is perfectly matched, whereas when the joint thickness is taken into account, a slight underestimation is observed. This can be explained remembering that the state of stress in thick bed and cross-joints is more complex than that in an interface. The role played by the actual thickness of the joint is similar to that observed by Milani and Taliercio (2015) for the in-plane case and recalled in the previous section. Assuming comparable failure surfaces for thick and thin joints, a nonnegligible reduction in the out-of-plane homogenized

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PART | II Modeling of unreinforced masonry

FIGURE 12.22 Sections of the failure surface obtained with the MoC model proposed in the planes of the macroscopic moments. (A) Plane (M11, M22) 5 (Mxx, Myy); (B) plane (M11, M12) 5 (Mxx, Mxy); (C) plane (M22, M12) 5 (Myy, Mxy).

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FIGURE 12.23 3D representation of the out-of-plane homogenized failure surfaces obtained by the MoC: (A) joints reduced to interfaces; (B) joint of finite thickness.

strength is observed, especially in bending about the head joints (similar to what occurs for in-plane loads). As already pointed out for a twisting moment, this discrepancy is obviously a consequence of the complex state of stress in the bed joints under bending moment M11, which cannot be effectively captured reducing joints to interfaces. For some combinations of twisting (M12) and bending moments about the bed joints (M22), an extra strength is obtained taking the joint thickness into account (see Fig. 12.22C). This is likely to be a consequence of the different failure criteria adopted for thin and thick joints. Finally, note the apparent nonlinearity of the homogenized failure surface in the presence of thick joints (Fig. 12.23A) and the reduction to a multilinear yield surface when joints are reduced to interfaces (Fig. 12.23B).

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12.9 Conclusions Four simple homogenization models were proposed to derive homogenized strength domains for in- and out-of-plane loaded periodic brickwork. The first two procedures give lower bounds, whereas the last two approaches give upper bounds to the actual macroscopic strength domain. In the first model, the elementary cell is subdivided into a few rectangular subdomains, where the microstress field is expanded using polynomial expressions. Four expansions were investigated in detail (P0, P2, P3, P4). P3 and P4 proved good convergence either to the actual solution in the case of joints reduced to interfaces, or to alternative upper bound approaches and FEM. Also, the second one yields a lower bound, where joints are reduced to interfaces and bricks are subdivided into a few CST elements. The third procedure is a kinematic compatible identification, which yields an upper bound to the macroscopic strength domain, where joints have been reduced to interfaces and bricks have been assumed to be infinitely resistant. The last model, MoC, is again a kinematic procedure where the elementary cell has been subdivided into six rectangular cells with preassigned polynomial periodic velocity fields. The first and latter models allow thick joint brickwork to be analyzed. A detailed comparison of the results provided by all models was given, both for under in-plane and out-of-plane loads, focusing in particular on the role played by the joint thickness. All the proposed models give predictions that match available experimental results fairly well. Also, the numerical results obtained by the refined FE models can be reproduced at a much lower computational cost. In conclusion, the proposed models can be conveniently used to predict the loadbearing capacity of masonry structures. The choice of either one of the models depends on the joint thickness and the required degree of approximation.

References Aboudi, J., 1991. Mechanics of Composite Materials—A Unified Micromechanical Approach. Studies in Applied Mechanics, vol. 29. Elsevier, Amsterdam. Anthoine, A., 1995. Derivation of the in-plane elastic characteristics of masonry through homogenization theory. Int. J. Solids Struct. 32 (2), 137163. Anthoine, A., 1997. Homogenization of periodic masonry: plane stress, generalized plane strain or 3D modelling? Comm. Numer. Methods Eng. 13 (5), 319326. Berto, L., Saetta, A., Scotta, R., Vitaliani, R., 2002. An orthotropic damage model for masonry structures. Int. J. Numer. Methods Eng. 55, 127157. Cecchi, A., Milani, G., 2008. A kinematic FE limit analysis model for thick English bond masonry walls. Int. J. Solids Struct. 45 (5), 13021331. Casolo, S., Milani, G., 2010. A simplified homogenization-discrete element model for the nonlinear static analysis of masonry walls out-of-plane loaded. Eng. Struct. 32 (8), 23522366. Cecchi, A., Milani, G., Tralli, A., 2005. Validation of analytical multiparameter homogenization models for out-of-plane loaded masonry walls by means of the finite element method. J. Eng. Mech. ASCE 131 (2), 185198.

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Cecchi, A., Milani, G., Tralli, A., 2007. A Reissner-Mindlin limit analysis model for out-ofplane loaded running bond masonry walls. Int. J. Solids Struct. 44 (5), 14381460. Cluni, F., Gusella, V., 2004. Homogenization of nonperiodic masonry structures. Int. J. Solids Struct. 41 (7), 19111923. Dallot, J., Sab, K., Godet, O., 2008. Experimental validation of a homogenized plate model for the yield design of masonry wall. C. R. Me´canique 336, 487492. de Buhan, P., de Felice, G., 1997. A homogenisation approach to the ultimate strength of brick masonry. J. Mech. Phys. Solids 45 (7), 10851104. de Felice, G., Giannini, R., 2001. Out-of-plane seismic resistance of masonry walls. J. Earthq. Eng. 5 (2), 253271. de Felice, G., 2011. Out-of-plane seismic capacity of masonry depending on wall section morphology. Int. J. Arch. Heritage 5 (4), 466482. Gazzola, E.A., Drysdale, R.G., Essawy, A.S., 1985. Bending of concrete masonry walls at different angles to the bed joints. In: Proc. 3th North. Amer. Mas. Conf., Arlington, TX, USA, paper 27. Gilbert, M., Casapulla, C., Ahmed, H.M., 2006. Limit analysis of masonry block structures with nonassociative frictional joint using linear programming. Comput. Struct. 84, 873887. Kawa, M., Pietruszczak, S., Shieh-Beygi, B., 2008. Limit states for brick masonry based on homogenization approach. Int. J. Solids Struct. 45, 9981016. Lotfi, H.R., Shing, P.B., 1994. Interface model applied to fracture of masonry structures. J. Struct. Eng. ASCE 120 (1), 6380. Lourenc¸o, P.B., 1996. Computational strategies for masonry structures. PhD thesis, TU Delft, The Netherlands. Lourenc¸o, P.B., 1997. An anisotropic macromodel for masonry plates and shells: implementation and validation. Report 03.21.1.3.07, University of Delft, Delft, Holland and University of Minho, Guimara˜es, Portugal. Lourenc¸o, P.B., Rots, J., 1997. A multisurface interface model for the analysis of masonry structures. ASCE J. Eng. Mech. 123 (7), 660668. Luciano, R., Sacco, E., 1997. Homogenisation technique and damage model for old masonry material. Int. J. Solids Struct. 34 (24), 31913208. Lourenc¸o, P.B., de Borst, R., Rots, J.G., 1997. A plane stress softening plasticity model for orthotropic materials. Int. J. Numer. Methods Eng. 40, 40334057. Luciano, R., Sacco, E., 1998. Damage of masonry panels reinforced by FRP sheets. Int. J. Solids Struct. 35 (15), 17231741. Macorini, L., Izzuddin, B.A., 2011. A nonlinear interface element for 3D mesoscale analysis of brick-masonry structures. Int. J. Numer. Methods Eng. 85, 15841608. Milani, G., 2008. 3D upper bound limit analysis of multileaf masonry walls. Int. J. Mech. Sci. 50 (4), 817836. Milani, G., 2009. Homogenized limit analysis of FRP-reinforced masonry walls out-of-plane loaded. Comput. Mech. 43, 617639. Milani, G., 2011a. Simple lower bound limit analysis homogenization model for in- and out-ofplane loaded masonry walls. Constr. Build. Mater. 25, 44264443. Milani, G., 2011b. Kinematic FE limit analysis homogenization model for masonry walls reinforced with continuous FRP grids. Int. J. Solids Struct. 482012, 326345. Milani, G., 2011c. Simple homogenization model for the nonlinear analysis of in-plane loaded masonry walls. Comput. Struct. 89 (17-18), 15861601. Milani, G., 2015. Upper bound sequential linear programming mesh adaptation scheme for collapse analysis of masonry vaults. Adv. Eng. Softw. 79, 91110.

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Milani, G., Tralli, A., 2011. Simple SQP approach for out-of-plane loaded homogenized brickwork panels accounting for softening. Comp. Struct. 89 (1-2), 201215. Milani, G., Taliercio, A., 2015. In-plane failure surfaces for masonry with joints of finite thickness estimated by a method of cells-type approach. Comput. Struct. 150, 3451. Milani, G., Taliercio, A., 2016. Limit analysis of transversally loaded masonry walls using an innovative macroscopic strength criterion. Int. J. Solids Struct. 81, 274293. Milani, G., Lourenc¸o, P.B., Tralli, A., 2006a. Homogenization approach for the limit analysis of out-of-plane loaded masonry walls. J. Struct. Eng. ASCE 132 (10), 16501663. Milani, G., Lourenc¸o, P.B., Tralli, A., 2006b. Homogenised limit analysis of masonry walls. Part I: failure surfaces. Comput. Struct. 84 (3-4), 166180. Milani, G., Lourenc¸o, P.B., Tralli, A., 2006c. Homogenised limit analysis of masonry walls. Part II: structural examples. Comput. Struct. 84 (3-4), 181195. Minga, E., Macorini, L., Izzuddin, B.A., 2018a. Enhanced mesoscale partitioned modeling of heterogeneous masonry structures. Int. J. Numer. Methods Eng. 113 (13), 19501971. Minga, E., Macorini, L., Izzuddin, B.A., 2018b. A 3D mesoscale damage-plasticity approach for masonry structures under cyclic loading. Meccanica 53 (7), 15911611. Mistler, M., Anthoine, A., Butenweg, C., 2007. In-plane and out-of-plane homogenisation of masonry. Comput. Struct. 85, 13211330. Pande, G.N., Liang, J.X., Middleton, J., 1989. Equivalent elastic moduli for brick masonry. Comput. Geotech. 8, 243265. Pegon, P., Anthoine, A., 1997. Numerical strategies for solving continuum damage problems with softening: application to the homogenisation of masonry. Comput. Struct. 64 (1-4), 623642. Pela`, L., Cervera, M., Roca, P., 2011. Continuum damage model for orthotropic materials: application to masonry. Comp. Methods Appl. Mech. Eng. 200, 917930. Pela`, L., Cervera, M., Roca, P., 2013. An orthotropic damage model for the analysis of masonry structures. Constr. Build. Mater 41, 957967. Pietruszczak, S., Niu, X., 1992. A mathematical description of macroscopic behavior of brick masonry. Int. J. Solids Struct. 29 (5), 531546. Portioli, F., Casapulla, C., Cascini, L., D’Aniello, M., Landolfo, R., 2013. Limit analysis by linear programming of 3D masonry structures with associative friction laws and torsion interaction effects. Arch. Appl. Mech. 83, 14151438. Sab, K., 2003. Yield design of thin periodic plates by a homogenization technique and an application to masonry walls. C. R. Me´canique 331, 641646. Sab, K., Dallot, J., Cecchi, A., 2007. Determination of the overall yield strength domain of outof-plane loaded brick masonry. Int. J. Multiscale Comput. Eng. 5 (2), 8392. Sacco, E., 2009. A nonlinear homogenization procedure for periodic masonry. Eur. J. Mech. A/ Solids 28 (2), 209222. Shieh-Beygi, B., Pietruszczak, S., 2008. Numerical analysis of structural masonry: mesoscale approach. Comput. Struct. 86 (21-22), 19581973. Stefanou, I., Sab, K., Heck, J.-V., 2015. Three dimensional homogenization of masonry structures with building blocks of finite strength: a closed form strength domain. Int. J. Solids Struct. 54, 258270. Suquet, P., 1983. Analyse limite et homoge´ne´isation. C.R. Acad. Sci. Se´rie IIB Me´canique 296, 13551358. Suquet, P., 1987. Elements of homogenization for inelastic solid mechanics. In: SanchezPalencia, E., Zaoui, A. (Eds.), Homogenization Techniques for Composite Media. Springer, New York.

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Sutcliffe, D.J., Yu, H.S., Page, A.W., 2001. Lower bound limit analysis of unreinforced masonry shear walls. Comput. Struct. 79, 12951312. Taliercio, A., 2014. Closed-form expressions for the macroscopic in-plane elastic and creep coefficients of brick masonry. Int. J. Solids Struct. 51 (17), 29492963. Taliercio, A., 2016. Closed-form expressions for the macroscopic flexural rigidity coefficients of periodic brickwork. Mech. Res. Commun. 72, 2432. Ushaksaraei, R., Pietruszczak, S., 2002. Failure criterion for structural masonry based on critical plane approach. ASCE J. Eng. Mech. 128 (7), 769779. Zucchini, A., Lourenc¸o, P.B., 2002. A micromechanical model for the homogenization of masonry. Int. J. Solids Struct. 39 (12), 32333255. Zucchini, A., Lourenc¸o, P.B., 2007. Mechanics of masonry in compression: results from a homogenization approach. Comput. Struct. 85, 193204.

Chapter 13

Discrete element modeling V. Sarhosis1, J.V. Lemos2 and K. Bagi3 1

School of Civil Engineering, University of Leeds, Leeds, United Kingdom, 2National Laboratory for Civil Engineering, Lisbon, Portugal, 3Budapest University of Technology and Economics, Budapest, Hungary

13.1 Introduction Masonry structures are composed of masonry units (such as bricks and/or stones) regularly or irregularly spaced together that are bonded with or without mortar. Despite their simplicity of construction, masonry is variable in nature, anisotropic and its mechanical behavior is characterized by high nonlinearity. The composite material (e.g., masonry units and mortar) under loads could be subjected to a complex stress state behavior that could produce compressive failure of the units and mortar, tensile failure of units and joints, and/or shear failure at the mortar joints. Factors influence the degradation and damage in masonry are due to service loading (e.g., traffic load in a masonry arch), soil settlement due to subsistence, wind loading, effects of natural hazards such ground motion due to earthquakes, and effects of flooding. The need to predict the in-service behavior and ultimate load-bearing capacity of masonry structures led researchers to approach the problem of modeling masonry from different perspectives and develop several numerical methods and computational tools characterized by their different levels of complexity. The selection of the most appropriate method to use depends on, among other factors, the structure under analysis, the level of accuracy and simplicity desired, the knowledge of the input properties in the model and the experimental data available, the amount of financial resources, and time requirements and the experience of the modeler (Lourenc¸o, 2002). Ideally, the numerical approach to be used to simulate the nonlinear behavior of masonry should be able to allow for the opening, closing and shear sliding at the mortar joints taking into account any resulting changes in strength and stiffness; and capture realistically the onset of the formation of the first

Numerical Modeling of Masonry and Historical Structures. DOI: https://doi.org/10.1016/B978-0-08-102439-3.00013-0 Copyright © 2019 Elsevier Ltd. All rights reserved.

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cracking and the subsequent propagation of cracking throughout the structure up to collapse. At present, the approaches used to simulate the mechanical behavior of parts or the entire masonry structure tend to focus on the individual masonry units and the mortar (i.e., micromodels), or on the composite material (i.e., macromodels). With the discrete element method (DEM), material can be modeled as an assemblage of distinct blocks or particles interacting along their boundaries. The DEM is presented in the UDEC (Universal Distinct Element Code) and 3DEC software, developed for commercial use by Itasca Ltd. for either static or dynamic analysis of 2D and 3D structures. The DEM only applies to a numerical approach if: 1. It consists of finite-sized discrete bodies (“discrete elements”) that can move (and perhaps deform) independently of each other; 2. It allows finite displacements and rotations of the discrete elements; and 3. New contacts and loss of existing contacts between the elements are automatically recognized and updated as the calculation progresses. The formulation of the method was proposed initially by Cundall (1971) for the study of jointed rock, modeled as an assemblage of rigid blocks. Later, this approach was extended to other fields of engineering requiring detailed study of the contact between blocks or particles such as soil and other granular materials (Ghaboussi and Barbosa, 1990). More recently the approach was applied successfully to model historic masonry structures in which the collapse modes were typically governed by mechanisms in which the deformability of the blocks plays little or no role at all. This chapter aims to provide an overview of the DEM for modeling masonry and outline a series of applications for modeling masonry and historic masonry constructions.

13.2 Overview of discrete element method codes for masonry modeling and key features of Universal Distinct Element Code and 3DEC This section aims at giving an overview and classification of the most important discrete element codes suitable for modeling masonry. These codes differ from each other in many aspects—element shapes, contact recognition and contact modeling, time integration technique, etc. It is challenging to select the most appropriate technique, and the aim of this section is to help with this.

13.2.1 The elements Many discrete element codes make use of polyhedral elements (e.g., 3DEC, DDA, versions of YADE, and the contact dynamics method). In these

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models, such elements can directly correspond to the masonry voussoirs of the real structure. The elements in most of these codes cannot break into parts, so they are suitable for problems where the failure of the voussoirs is not relevant. There are a few versions with breakable polyhedral elements: the Elias model (Elias, 2014) or Munjiza’s finite element method (FEM)/ DEM (Hazay and Munjiza, 2016) belong to this class. Other codes (like PFC or EDEM) apply circular/spherical elements from which the user can compile rigid or deformable collections of particles to form the masonry blocks; this possibility is advantageous when the aim is to simulate realistic fracture of the voussoirs. The elements in DEM may be rigid or deformable. Rigid elements (in UDEC/3DEC, or in many contact dynamics versions) have translational and rotational degrees of freedom of their reference point. The equations of motion express the relations between translational or rotational accelerations of the reference points, and the forces or moments reduced to them from those forces that act on the element. Alternatively, elements can be made deformable either by subdividing them into simple finite elements whose nodes have translational degrees of freedom (like in UDEC/3DEC or in contact dynamics versions), or by assigning a simplified—mostly a uniform—strain field to the element (like in DDA) when the degrees of freedom of an element are the reference point translations, rotations, plus the strain characteristics. Subdivided discrete elements can usually give a much better approximation of the strain field than a uniform strain in the whole element, though the analysis of the deformation patterns inside the masonry blocks is often unnecessary in practical problems. No wonder rigid and FEM-subdivided elements are more common in masonry analysis, while simplified overall strain functions are applied as often. Regarding this latter approach, note that complex deformation patterns in elements with complicated shapes may be achieved by the superposition of several mode shapes for the whole body (see Williams and Mustoe, 1987). In order to represent the complexity of the deformation pattern to the required accuracy, any number of modes may be added. Thus, the approach is efficient for bodies of complicated shapes that deform in a simple manner, because only a few low modes need to be taken into account. A main disadvantage of the approach is that due to the applied superposition, material nonlinearity is very difficult to model.

13.2.2 The contacts Contact behavior representation in a DEM code consists of two main components: (1) contact recognition and (2) mechanical model based on the geometrical properties of the contact.

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13.2.2.1 Contact recognition Contact occurs if any point of an element gets into the interior of another element. This situation should be recognized as efficiently as possible: the computational time for contact recognition basically influences the calculation time of the simulated procedure. If every element is checked in detail with every other element for contact, the necessary number of computational operations would be proportional to the square of the total number of discrete elements. In the different contact detection algorithms special techniques are applied to avoid this huge number of unnecessary calculations, to reduce the CPU requirements. These techniques eliminate those couples that are far from each other and are not in contact, so that only those pairs that are close to each other should be analyzed in detail. According to bodybased search techniques, each element is considered separately: a window is assigned to the element under question, and only those other elements are checked for contact by calculations inside this window. In the case of spacebased search techniques, several (strongly overlapping) windows are specified inside the complete domain of the analysis, and these windows are analyzed one by one: only those pairs of elements inside the same window are checked by exact calculations. Modern DEM codes apply more developed versions of these techniques. Contact detection algorithms can be classified according to the CPU time which they use and this will also depend on the size of the problem to be analysed. Classical contact detection algorithms (e.g., the body-based search technique) are hyperlinear contact detection algorithms, that is, the necessary CPU time increases faster than the number of discrete elements: most DEM codes belong to this class. In linear contact detection algorithms, the CPU time is a linear function of the number of discrete elements. Such an algorithm is implemented in FEM/DEM (Munjiza et al., 1995), and is called the Munjiza
NBS contact detection algorithm. Independently of the technique according to which the majority of the element pairs is excluded, a remaining set of the pairs must be tested for contact in detail. This requires very time-consuming calculations, and there exist several techniques for the task. The most important methods are the common plane method (Cundall, 1988; Hart et al., 1988) and its modified versions and different direct approaches such as the closest point method (Beyabanak et al., 2008) or referential plane method (Liu and Lemos, 2001). The rounding scheme (Ahn and Song, 2012) simplifies the contact detection calculations by using inscribed spheres and cylinders at the convex corners and edges of polyhedra; similar rounding is applied in the 2D UDEC code. An excellent overview is given on the different contact detection techniques by Zhang et al. (2015). The common plane method is probably the most popular today. Its main advantages are its computational efficiency and the fact that the common plane may change rapidly as the two elements move and

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rotate with respect to each other, but it will not show an abrupt change at the modification in contact type (e.g., when the vertex-to-face is changed to the edge-to-edge contact type).

13.2.2.2 Mechanical model of the contacts Contact means an interpenetration between two elements. Some DEM codes (e.g., original DDA version Shi, 1988, or contact dynamics) consider this interpenetration nonphysical, and with the help of penalty functions, algorithms are used to prevent any intersection of the two contacting bodies (“hard-contact approach”). Most DEM codes, on the other hand, use the “soft-contact approach”: interpenetration causes contact forces according to the actual contact stiffness, and the arising contact forces are calculated from the depth or other characteristics of the interpenetration (e.g., UDEC/3DEC, the Elias model, FEM/DEM of Munjiza). A mechanical model has to be involved that specifies the calculation of contact forces based on the appropriate geometrical and deformational characteristics of the contact. The most widely used approach is to model the contact as a collection of separate points where the concentrated contact forces are transmitted (e.g., UDEC and 3DEC). In any model, the plane of the contact and the point where the force is transmitted has to be known before calculating the forces. The applied contact recognition algorithm usually provides them, but this is not true for all models. For example, in the Elias model (Elias, 2014) the closed 3D intersection line of the surfaces of the two polyhedra is applied, and a best-fit plane is determined for this polygon, which serves as the plane in which the friction can act and sliding can occur. The centroid of the intersecting volume is the point where the contact force acts. Munjiza’s FEM/ DEM applies a potential field-based penalty function method, which is also different from the most usual points contact model. The dependence of the normal and tangential force components on the accumulated or the incremental relative translation of the two elements (or on the overlapped volume) shows a huge variety in the literature. The simplest and most widely applied constitutive relations are the linear dependence on the relative translation increments, perhaps together with a friction limit for the tangential component, and a tension cut-off for the normal force. In many codes, like in YADE and UDEC/3DEC, the user can build in his or her own contact models, perhaps based on experimental results or special theoretical considerations. 13.2.3 Calculation of the displacements Recent DEM codes used today in practice or in research are all based on the time integration of the equations of motion of the system. Quasistatic DEM techniques based on the stiffness matrix of the system like Kishino (1988) or

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Bagi (1993) are not applied today in masonry modeling. It is worth mentioning that the displacement method with stiffness matrix seems to come in use in masonry modeling in a simplified manner: problems where the contact topology does not change can very efficiently be analyzed with such a numerical simulation technique (see Baraldi and Cecchi, 2017). That model does not keep track of the occurrence of new contacts, so it cannot be considered as a discrete element method. However, it can be computationally very efficient in situations when the occurrence of new contacts can be excluded. Focusing from now on the different timestepping methods, the approximate motion of the system, starting from a known initial state, through a series of small but finite time intervals, in such a way that the equation: MðtÞUaðtÞ 5 fðt; uðtÞ; vðtÞÞ

ð13:1Þ

would (at least approximately) be satisfied at the discrete time points t1, t2, . . ., ti, . . ., that is, the endpoints of the time intervals. Here, u and v are the generalized displacements and velocities, f is the generalized force vector in which all effects are reduced to the reference points or nodes, and M is the generalized inertia matrix. Some of the DEM codes apply explicit time integration to solve this task, which means that when considering a time interval, those u and v values belonging to the endpoint ti11 are determined in such a way that the equations of motion are compiled at time point ti and the values at ti11 are predicted from the already determined approximated u and v values belonging to previous time instants only. Consequently, the explicit techniques do not check whether the equations of motion are satisfied at the endpoint of the actual interval, that is, at ti11; the predicted values are accepted without checking the equations of motion. They then serve as the starting values for the forthcoming time interval. UDEC/3DEC, FEM/DEM, PFC EDEM, etc., use explicit time integration, namely the central difference method, which is an easy-to-code technique but also very favorable from the point of view of accuracy. Rougier et al. (2004) give a comparison of different explicit techniques applied in DEM. Other DEM codes use implicit time integration techniques. In this case, the u and v values belonging to the endpoint ti11 are calculated in such a way that the equations of motion would be satisfied at the endpoint of the time interval. This is done with the help of a gradually improving iteration scheme: the approximated values of u and v at ti11 are checked and modified again and again until a sufficiently exact match is reached; these values are then used as the starting data for the next timestep. DDA applies such an approach (Newmark’s beta method), and the iterative solver in contact dynamics can also be considered as an implicit technique. The explicit methods are, in general, much easier to implement, and require significantly less computer time for the analysis of a timestep. On

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(B)

u

1

hi +1

u 1

hi

ui+1

ui +2 1

1

ui

ti

hi

hi +1

ui +1

ui +2

ui ti +1

ti+2

t

ti

ti +1

ti +2

t

FIGURE 13.1 The explicit (A) and the implicit (B) time integration techniques.

the other hand, the predicted results can quickly deviate from the exact solution, and numerical instabilities can occur. To avoid this, the applied length of the timesteps should be limited, and several numerical “tricks” are used to ensure the stability of the calculations. The implicit methods are more reliable from a numerical point of view, and longer timesteps can be applied; on the other hand, they are computationally more expensive. Most commercial codes use explicit techniques due to their simplicity. Fig. 13.1 shows the difference between the explicit and implicit time integration techniques.

13.2.4 Loads Loads in DEM models can be of several types. Concentrated or distributed forces may act directly on the elements; chosen points/gridpoints can have prescribed (e.g., zero) velocities; moving walls can surround the structure; and servo-mechanisms can be coded in order to produce a triaxial test, etc. Periodic boundaries are sometimes used in research codes, though their relevance to masonry modeling is rather low.

13.3 Key features of Universal Distinct Element Code and 3DEC for modeling masonry Within the DEM, masonry could be represented as an assembly of rigid or deformable blocks that can take any arbitrary shape. Joints are viewed as the surfaces where mechanical interaction between blocks takes place, governed by appropriate constitutive laws. The motion of the blocks is simulated throughout a series of small but finite timesteps, numerically integrating the Newtonian equations of motion.

13.3.1 Representation of the block In UDEC and 3DEC, blocks can take any arbitrary polyhedral shape, concave or even hollow. They can behave perfectly rigidly or as deformable (Fig. 13.2). In order to describe the behavior of a rigid element, a reference point is chosen. Theoretically, the reference point could be at any arbitrary location, even outside the element itself, but the calculations are easier if the

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FIGURE 13.2 Rigid and deformable polyhedral block in 3DEC (Lemos, 1998).

FIGURE 13.3 Voronoi-cell of an internal and of a boundary node in a deformable element.

reference point coincides with the center of gravity of the element. Hence, in UDEC/3DEC the reference point is at the center of gravity. The degrees of freedom of a rigid element are the components of its translation vector and of the rotation vector of the block about the reference point. Elements are made deformable in UDEC/3DEC by subdividing them into simplexes (triangles in 2D and tetrahedral in 3D). These tetrahedra have a uniform strain or their translation field is linear. Alternatively higher-order tetrahedra, based on quadratic displacement interpolations functions, can also be chosen, though existing experiences are not convincing on the advantages of higher-order simplexes for masonry analysis (perhaps they may be necessary if the user is very much interested in the details of the stress fields inside the voussoirs). For uniform-strain simplexes, the degrees of freedom are the translation components of the FEM nodes; rotations are not considered independently. The symmetric part of the translation gradient (i.e., the strain) characterizes the deformation of the simplex and the skew-symmetric part describes the rigid-body-like rotation of the simplex. In the equations of motion of such a node, the mass is defined as that of the Voronoi-cell belonging to the node. Fig. 13.2 illustrates that the Voronoicell of a node is the set of those points of the element that are closer to the given node than to any other node. Fig. 13.3 shows two Voronoi-cells

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(blue domains); one for a node in the interior of the element and another one belonging to a node on the boundary.

13.3.2 Identification of neighbors To check all possible pairs, the search time increases quadratically with the number of the blocks. To avoid it, before a pair of blocks can be checked for contact using exact geometrical calculations by the computer program, candidate pairs are identified first. In this first step, an envelope space is assigned to every block as the smallest 2D box with sides parallel to the coordinate axes that can contain the block. Those pairs of blocks are then tested for contact in detail whose envelope spaces intersect.

13.3.3 Contacts After two blocks have been recognized as neighbors, they are tested for contact. The contact detection algorithm also provides a unit normal vector, which defines the plane along which sliding can occur. This unit normal should change direction in a continuous fashion as the two blocks move relative to one another. 3DEC applies a scheme based on a “common plane between the two blocks.” The contact detection analysis consists of the following two parts: G

G

Determine a “common plane,” that is, roughly saying, bisects the space between the two blocks; Testing both blocks separately for contact with the common plane.

The common plane is defined as the resulting plane provided by the optimization problem “maximize the gap between the common plane and the closest vertex” or, equivalently, “minimize the overlap between the common plane and the vertex with the greatest overlap.” The algorithm applies a gradual translation and rotation of the common plane in order to maximize the gap (or minimize the overlap). Contact exists if the overlap is positive, or equivalently, if the gap is negative between the two blocks. The normal vector of the common plane is the contact normal. When a face of a rigid block is in contact with the common plane, it is automatically discretized into subcontacts by triangulating the face. Subcontacts are created with the help of the nodes being located on the block face. The area “owned” by each subcontact is, in general, equal to one-third of the area of the surrounding triangles around the node (this calculation is adjusted when the subcontact is close to one or more edges on the opposing block). If the other side of the interface is also a face, then identical conditions apply: subcontacts are created, and relative displacements, and hence forces, are calculated.

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The 2D code UDEC employs a different method to define the contact plane and normal. The corners of a polygonal block are assumed to be rounded, that is, smoothed by a circular arc tangent to the adjacent edges. For a vertex
vertex interaction, the contact normal is simply given by the line joining the centers of the two arcs, and the contact plane is perpendicular to this line at the point of contact of the arcs. For a vertex
edge interaction, the contact plane is taken as parallel to the edge, with the point of contact placed at the intersection of the edge and perpendicular line going through the arc’s center. This ensures continuity of the contact normal in transitions between the two types of contact, for example, if the blocks rotate relative to each other. The basis of the mechanical calculations is the relative velocity of the subcontact under question. This is defined as the velocity of the analyzed node minus the velocity of the corresponding point of the opposite face on the other block. This latter velocity can be calculated with the help of a linear interpolation of the three nodes on the surface of the other block surrounding that opposite point. Then the relative translation vector belonging to the subcontact is calculated from the relative velocity and from the length of the timestep. This relative translation is multiplied with the actual normal and shear stiffness of the contact in order to receive the uniform distributed normal and shear forces belonging to the subcontact. The resultant along the subcontact area is assigned to the analyzed node, and the opposite of the resultant is shared among the three nodes surrounding the coincident point on the opposite face. The same is done for all nodes on the analyzed face of the first block. Then the other block is analyzed in a similar manner: nodes along its contacting face are considered, and another set of subcontacts is produced where the subcontact forces are calculated from the corresponding relative displacements. Consequently, when two blocks come together, the contact logic described above is equivalent to two sets of subcontacts in parallel, each carrying subcontact forces. The subcontact forces received in the two steps are summed and halved, then in order to receive the overall interface behavior as the average of that of both sets (see Fig. 13.4).

(A)

(B)

Kn

Contact points

Ks Fs Lc

Fn

ΔUs ΔUn

Length of contact

FIGURE 13.4 Face-to-face contact type and corresponding subcontacts where springs are assigned in both orthogonal directions. (A) Block-to-block contact; (B) interactions at contact.

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13.3.4 Constitutive models for blocks Constitutive relations for the blocks subdivided into deformable finite element simplexes specify how to calculate the stress tensor, if the strain tensor (and perhaps other properties required by the user, e.g., history data) is known. There are several optional constitutive relations offered by UDEC and 3DEC: G

G

The simplest type is the null element, an empty domain with zero material density and zero stiffness, playing no mechanical role. This type of “material” can be used, for instance, to simulate voids and holes that will later be filled up with material. The elements can be isotropic, linearly elastic, with infinite resistance to stresses (no plastic or fracture failure limit). Such elements are characterized with the Young’s modulus (E) and the Poisson coefficient (μ), or, alternatively, with the bulk modulus (K, the ratio between isotropic stress and strain) and the shear modulus (G). The two pairs of quantities can easily be calculated from each other: K5

G

E ; 3ð1 2 μÞ

G5

E 2ð1 1 μÞ

ð13:2Þ

Failure conditions can also be assigned to the elements. The Mohr
Coulomb model, the Prager
Drucker model and many others are built-in options, but the user can also prepare his or her own failure criteria if the existing options are not sufficient for the problem under consideration. These failure criteria set a limit to the stresses in the deformed simplexes, and describe how the simplex should behave if a failure criterion is met. Inside the same element, some simplexes may be in plastic or damaged state while others are still elastic, but an element always consists of the same set of simplexes as initially: the FEM subdivision is not densified.

13.3.5 Constitutive models for contacts The mechanical behavior of contacts in UDEC is modeled with the help of contact stiffness defined in the normal and shear directions, relating subcontact stresses with relative displacements characterizing the subcontact (Fig. 13.5). In the elastic range (when contact sliding and separation does not occur) the behavior is governed by the joint normal and shear stiffness (kn and ks): ΔF n 5 2 kn ΔU n Ac

ð13:3Þ

ΔF s 5 2 ks ΔU s Ac

ð13:4Þ

where ΔF n ; ΔF s is the normal and the shear force increment (resultant for the subcontact); kn ; ks is the joint normal and the joint shear stiffness;

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σ

σ tan φ + c kn

ks

un

us

–(σ tan φ + c) FIGURE 13.5 Constitutive law describing the behavior of joints under normal and shear loads.

ΔU n ; ΔU s is the normal and the shear displacement increments belonging to the subcontact; and Ac is the subcontact area. The maximum shear force allowed is given by: s Fmax 5 F n tanðϕÞ

ð13:5Þ

where ϕ is the angle of friction.

13.3.6 Equations of motion 13.3.6.1 Rigid blocks The equations of translational motion for a single block can be expressed as: x€ i 1 α_xi 5

Fi 1g m

ð13:6Þ

where x€ i is the acceleration of the centroid of the block at time ti; x_ i is the velocity of the centroid of the block at time ti; α is the viscous (mass-proportional) damping constant; Fi is the sum of forces acting on the block (contact 1 applied external forces, except gravitational forces) at time ti; m is the mass of the block; and g is the gravity acceleration vector. The rotational motion of an undamped rigid body can be efficiently described if the principal axes of inertia of the body are considered. However, blocks in UDEC are oriented typically in random directions compared to the global coordinate axes of the system. Because velocities are small, the rotational equations can be simplified: accurate representation of the inertia tensor is not essential. Therefore, in 3DEC, only an approximate moment of inertia is calculated based on the average distance from the centroid of vertices of the block. This allows the rotational equations to be referred to as the global axes: _ i 1 αωi 5 ω

Mi I

ð13:7Þ

where ω_ i is the angular acceleration about the coordinate axes attached to the reference point, at time ti; ωi is the angular velocity about the principal

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axes at time ti; α is the viscous (mass-proportional) damping constant; Mi is the total torque (reduced to the reference point) at time ti; and I is the approximate moment of inertia. Time integration of the equations of motion is done with the central finite difference scheme. The velocities and angular velocities are calculated as follows: h
  i  Fi ðtÞ x_ i11=2 5 1 2 α Δt 1 1 g UΔt U 1 11αΔt U_ x i21=2 2 m     2 Δt Mi ðt Þ 1 ωi11=2 5 1 2 α ð13:8Þ ωi21=2 1 Δt  2 I 1 1 α Δt 2 where indices i 2 1/2 and i 1 1/2 refer to values taken at ti 2 Δt/2 and ti 1 Δt/2, respectively. The increments of translation and rotation are given by:   Δt Δxi11 5 x_ i11=2 t 1 Δt ð13:9Þ 2   Δt Δt ð13:10Þ Δθi11 5 ωi11=2 t 1 2 The position of the block centroid is updated as: xi11 5 xi ðtÞ 1 Δxi11

ð13:11Þ

The new location of the vertices is calculated with the help of the displacement of the centroid plus the rotation calculated earlier.

13.3.6.2 Deformable elements The equations of motion for the nodal points of the triangular or tetrahedral subdivision do not contain rotational degrees of freedom. The equations of translational motion for a single node i can be expressed as: x€ i 1 α_x i 5

Fi m

ð13:12Þ

where x€ i is the acceleration of the node at time ti; x_ i is the velocity of the node at time ti; α is the viscous (mass-proportional) damping constant; Fi is the sum of forces at time ti, reduced the node (compiled of (1) body forces acting on the Voronoi-cell around the node, including self weight; (2) resultant of stresses acting in the simplexes the Voronoi-cell cuts through; and (3) contact forces if the node happens to be on the boundary of a discrete element and forms a contact with another element); m the mass of the Voronoicell around the node. The reduced force Fi is approximated to act exactly on the node: the eccentricity of its different contributions is neglected. (Indeed, the increasing density of the simplex subdivision decreases the error made by this approximation.)

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The nodal positions at ti11 are calculated in the same way as the translational increments of the reference points of rigid elements:     Δt Fi ðtÞ 1 ð13:13Þ x_ i11=2 5 1 2 α _x i21=2 1 Δt  2 m 1 1 α Δt 2   Δt Δxi11 5 x_ i11=2 t 1 Δt ð13:14Þ 2 xi11 5 xi ðtÞ 1 Δxi11

ð13:15Þ

13.3.7 Mechanical damping In addition to the representation of the physical phenomenon itself, damping is applied in UDEC/3DEC to decrease oscillations originating from the explicit nature of the time integration technique and to facilitate reaching a force equilibrium state as quickly as possible. Two forms of damping are applied for the solution of quasistatic problems. The first is called adaptive global damping, in which viscous damping forces acting on the nodes of the deformable blocks or on the rigid block centroids are used, but the viscosity constant is continuously adjusted in such a way that the power dissipated by damping is a given proportion of the rate of change of kinetic energy in the system. In the second form of damping different damping force and moment components are applied on every degree of freedom. Every component is proportional to the magnitude of the unbalanced force or moment. For this scheme, referred to as local damping, the direction of the damping force is always opposite to the actual translational or rotational velocity. For dynamic analysis, Rayleigh damping is available. The massproportional component is applied in the equations of motion, in the same manner as global viscous damping. The stiffness-proportional component is applied in the form of viscous contact forces, which physically correspond to contact dashpots affecting the relative block movements. For deformable blocks, this component also generates viscous stresses in the internal block elements. At the moment UDEC/3DEC does not offer a more general built-in contact damping options. The user can, however, develop and code any arbitrary individual constitutive relation for the contacts, so contact damping can be implemented this way with extra effort if necessary.

13.3.8 Numerical stability The central difference method is only conditionally stable. To avoid numerical instabilities, a limiting timestep is defined and the user is allowed only to decrease it.

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In the case of rigid elements, the limiting timestep is calculated by analogy to a simple degree of freedom linear elastic system as   Mmin 0:5 Δtb 5 frac2 ð13:16Þ Kmax where Mmin is the mass of the smallest block in the system; Kmax is the maximum contact stiffness; and frac is a user-defined value that approximately accounts for the fact that a block may be in contact with several neighboring blocks, and that the contacts or the elements can be in an inelastic state, for example, sliding, or just broken. (A typical value for frac is 0.1, a rather low value, to ensure safe approximation of the admissible timestep length.) Suitably chosen damping contributes to the numerical stability of the simulation. Adaptive global damping is usually efficient if the system strongly oscillates around the equilibrium state, while local damping is more advantageous when some parts of the system are already close to equilibrium while other others are just collapsing or strongly oscillating; or if the loads quickly change.

13.3.9 Structural elements UDEC/3DEC offers three possibilities to model the effect of cables or beams drilled into or attached to a masonry structure: (1) local strengthening of discontinuities, (2) cable reinforcements, and (3) beams. The three possibilities are briefly covered in this section; for more details the interested reader should consult the manual.

13.3.9.1 Local reinforcement This version simulates only the local effect of fiber or cable reinforcement passing through a joint between two masonry blocks. It can be thought of as a small-sized linear axial and shear spring, perhaps both having rupture limits. The direction of the reinforcement is not necessarily perpendicular to the joint (Fig. 13.6).

Axial spring Joint Shear spring

FIGURE 13.6 Representation of local reinforcement.

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Cable nodal point

Axial spring Shear spring

FIGURE 13.7 Representation of global reinforcement.

Double arrows: rotational degrees of freedom Single arrows: translational degrrees of freedom FIGURE 13.8 Representation of beam element.

13.3.9.2 Cable (“global”) reinforcement This type of strengthening takes into account not only the local effect of the reinforcement passing through a joint, but also the lengthy restraint to masonry voussoirs that deform (elastically or inelastically) as the blocks move and deform (Fig. 13.7). The cable element consists of a series of longitudinal segments, with nodal points at the segment ends. The mass of each segment is assumed to be located at the nodal points. The segments can axially elongate, but their bending resistance is neglected. Shear occurs in the imagined grout between the reinforcement and the masonry, also in axial direction. A tensile yield limit force can be assigned to the cable, and a maximal shear force between the cable and the masonry can also be defined. This type of structural element can be used for deformable blocks only. 13.3.9.3 Beam element A beam element is a straight segment of prismatic shape, connecting two points on the surface of the model (Fig. 13.8). (A long curved beam can be

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modeled as a collection of consecutive beams.) By default, isotropic, linearly elastic material behavior is assumed, and a limiting axial strength can be specified. The beam element has 12 degrees of freedom: three translational and three rotational components at each end. Corresponding to each degree of freedom, there is a force or moment that acts between the node and the neighborhood the beam end is attached to. Beam elements interact with blocks by means of point contacts established between a beam node and any block face it touches. The contact forces are a function of the relative displacements between the beam node and the block face and obey a Mohr
Coulomb behavior. The beam nodes then move in response to unbalanced forces and moments in accordance with the equations of motion, and are treated numerically in a manner similar to rigid blocks. Beam elements are also only for deformable block models (Sarhosis et al., 2013). For application of a modeling masonry wall panel containing reinforcement, readers are referred to Sarhosis et al. (2013).

13.3.10 FISH function FISH is a built-in programming language in UDEC and 3DEC. Using FISH, the user can define variables or functions and execute calculations and thus automatize the generation of model geometry, visualize simulation results in a personalized manner, code servo-control in loading procedures, etc. FISH is very helpful for the experienced UDEC/3DEC user with specific needs, but beginners may find it unnecessary or too difficult. However, automatized preparation of complex model geometries or sophisticated representation of the results are more straightforward using FISH. The UDEC and 3DEC manuals offer a complete programming guide to FISH, with several simple examples to help those inexperienced in computer programming.

13.4 Applications of modeling masonry and historical structures using Universal Distinct Element Code and 3DEC 13.4.1 Tall masonry structures and historic towers The vulnerability of towers and other tall masonry structures to horizontal loads, namely seismic action, has motivated substantial research on numerical tools to assess their safety. The drum columns of classical architecture, actually one of the first applications of DEM in this field, will be treated in the following section. The method has also been applied to other types of tall structures, for which the overturning modes of failure are common. For dynamic actions, the numerical method of analysis is required to model the rocking mode with uplift examined in the classical work of Housner (1963).

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Large rotations of the blocks have to be taken into account, with the structural geometry updated throughout the analysis. DEM models are capable of reproducing the type of behavior observed in lab experiments, which is characterized by great variability, as seen in the comparisons with free-rocking and shaking table tests of single rock blocks (Pen˜a et al., 2007). These authors also showed that the spring-dashpot contact model of DEM may be calibrated to match the response of the analytical contact models based on the notion of restitution coefficient, as in Housner’s solution. The effects of earthquakes on freestanding structures, such as columns or statues, are often clearly observed and recorded. Whether they collapsed or not, or the magnitude of permanent movements provides valuable data on the intensity of the seismic action at that location, which can be explored and understood using numerical modeling. Oliveira et al. (2002) employed 3DEC models to reproduce the observed large displacements and rotations displayed by several simple tall structures (such as a tower, a chimney, and a statue), during the 1998 Azores earthquake. Lemos et al. (2015) analyzed the observed rotations of a stone obelisk displaced during the 2011 Lorca earthquake with a rigid block model. This study stressed the importance of measuring the natural frequencies in situ in order to calibrate the joint stiffness parameters, as the structure deformability had a significant effect on the block movements. The dynamics of church spires were studied by DeJong and Vibert (2012), who compared the numerical results with a physical model experiment. More elaborate models have also been developed. For example, the minaret models presented by Cakti et al. (2015), who examined the damage and failure modes produced by various types of recorded and synthetic seismic records (Fig. 13.9). These structures often have metal clamps to connect the stones in the circumferential direction. These were represented in the numerical model by axial elements allowed to fail when the tensile force exceeded the strength. The seismic vulnerability of historic towers was investigated by Sarhosis et al. (2017), who compared the performance of simplified analytical tools and numerical models, considering both FEM and UDEC models (Fig. 13.10). Extensive simulations were performed for a range of tower slenderness and input accelerations. The simplified tools allow large numbers of simulations to be performed rapidly, but the numerical models are more suitable for indicating the likely failure mechanisms. The preferable type of discretization to be used in the UDEC models for this type of analysis is also discussed.

13.4.2 Ancient columns and colonnades DEM has been successfully used to study the mechanical behavior of ancient columns and colonnades subjected to static and dynamic analysis for almost

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FIGURE 13.9 Model of a minaret and view of internal structure (Cakti et al., 2015).

three decades. The first study of the seismic behavior of ancient multidrum columns was addressed using the commercial 2D software UDEC by Papastamatiou and Psycharis (1993). Later, several researchers adopted both 2D and 3D formulations based on DEM to investigate the stabilities of columns and colonnades structures of architectural heritage. The 3D commercial code 3DEC was also used to compare the shaking table tests of a marble column tested at the Earthquake Engineering Laboratory of NTUA in Greece (Papantonopoulos et al., 2002). The results indicated that 3DEM was able to capture quite well the main features of the response of the column. One year later, the ability of the 3DEC software to represent damage in drums and retrofitted columns containing shear links between them was investigated by Psycharis et al. (2003). A typical model showing columns with damaged drums subjected to dynamic load is shown in Fig. 13.11. In addition, Stefanou et al. (2011) demonstrated that 3DEC was not only able to capture shear sliding and rocking, but also wobbling of between the drums of the column. It was also highlighted that wobbling between drums can

488

PART | II Modeling of unreinforced masonry A1

B1

A2

B2

A3

B3

A4

B4

FIGURE 13.10 Failure mechanisms of historic towers observed from UDEC (Sarhosis et al., 2017).

significantly affect the stability and deformation capacity of the column. Recently, Sarhosis et al. (2016a,b) developed both 2D and 3D numerical models using the DEM software UDEC and 3DEC to investigate the static and dynamic stability of the two-storey colonnade of the Forum in Pompeii, Italy. The structure under investigation was a three-span, two-series system colonnade consisting of multidrum columns positioned one over the other. The peculiarity of the structure is that the lower-level columns support a series of both solid and segmental beams forming a flat arch. Sliding between drums due to deterioration and stresses (Fig. 13.12) in stone

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FIGURE 13.11 Models of columns with damaged drum under seismic loading (Psycharis et al., 2003).

FIGURE 13.12 Stress analysis in deformable blocks with fine meshes (Sarhosis et al., 2016a).

blocks of the colonnade were investigated. In addition, the seismic reliability of multidrum columns was studied by Psycharis et al. (2013) with the use of synthetic ground motions obtained from a stochastic analysis using Monte Carlo simulations. An accurate 3DEC model was developed by Nayeri (2012) to study the temple in Evora, Portugal. The shapes of the drums of the column and architraves were estimated using laser scanning.

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Dimitri et al. (2011) investigated the effect of damping and material properties on the behavior of multidrum columns subjected to harmonic loading. Other researchers developed in-house specialized software based on DEM to study the seismic behavior of multidrum columns and colonnades. Komodromos et al. (2008) developed software based on DEM and studied different columns subjected to earthquake load. The same software was also used later to study the influence of the frequency content and amplitude of the ground motions on the seismic response of columns and colonnades with epistyles (Papaloizou and Komodromos, 2009). Recently, Pulatsu et al. (2017) used software based on DEM developed by Bretas et al. (2014) to study the nonlinear static behavior of freestanding multidrum ancient columns. Capacity curves and corresponding failure mechanisms of each of the studied models were obtained. The influence of different parameters, namely drums, geometrical properties, and imperfections at columns, was also assessed to observe their influence on the response of drum assemblies.

13.4.3 Masonry walls (dry stack or masonry units joined with mortar) Damage in the form of cracking in the in-plane and out-of-plane behavior of masonry walls was predicted using models based on DEM. UDEC models with deformable blocks have also been developed (e.g., by Sincraian and Azevedo, 1998; Schlegel and Rautenstrauch, 2004). Bui et al. (2017) studied the in-plane and out-of-plane behavior of different geometry wall panels with dry joints (Fig. 13.13). Within DEM, the bricks were modeled as continuum rigid elements while the joints were modeled by line interface elements represented by the Mohr
Coulomb law. DEM models were capable of representing the crack development and load-carrying capacity of masonry structures constructed with dry joints with sufficient accuracy. Moreover, a collection of experimentally verified material parameters were given to other researchers and engineers to solve similar problems. A methodology for the

FIGURE 13.13 Comparison of experimental against numerical failure modes of dry joint masonry wall panel (Bui et al., 2017).

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FIGURE 13.14 Failure mode of panel under vertical load (Sarhosis and Sheng, 2014).

calibration of the material properties to be inputted into the DEM models was developed by Sarhosis and Sheng (2014). Using a UDEC model of a brick panel containing an opening (Fig. 13.14), the least squares difference between the experimental and the numerical results with different material properties was evaluated. An optimization procedure and set of parameters to minimize the objective function (i.e., absolute least squares difference between experimental and numerical results) were obtained. In addition, the ability of the DEM to represent different shape walls was investigated by several researchers. For example, Roberti and Spina (2001) used a UDEC model with irregular polygonal blocks to study the behavior of ancient Sardinian “Nuraghe” structures. De Felice (2011) undertook both static and dynamic analysis of a three-leaf cross-section of a traditional wall with UDEC (Fig. 13.15). The results highlighted the importance of the connection between the outer wall leaves in maintaining the integrity of the section. Lemos et al. (2011) studied the out-of-plane behavior of masonry walls with 3DEC models using static pushover analysis. Regular and irregular patterns were considered, namely Voronoi polygons and irregular course geometries, and their effects on the failure loads were examined.

13.4.4 Arches and vaults The dynamic behavior of circular freestanding arches (see Fig. 13.16) of partially ruined structures were studied using the 2D DEM software UDEC by De Lorenzis et al. (2007). Numerical results were validated against those obtained from analytical studies. Later, Dimitri et al. (2011) investigated the behavior of arches supported by buttresses. Different patterns of block arrangements were used to represent the support walls and buttresses. DEMs were also used to evaluate the structural capacity of stone masonry arch bridges. Lemos (1998) developed a 3D model using 3DEC of a masonry arch bridge that included the arch barrel and spandrel walls to study the load distribution along the spandrel. Sarhosis et al. (2014) investigated the case of skew bridges and studied the effect of arch shape and angle of skew on the load-carrying capacity. An extension of this research can be found in

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FIGURE 13.15 Analysis of failure modes of three-leaf wall sections (De Felice, 2011).

FIGURE 13.16 Deformation of arch subject to a base dynamic impulse (De Lorenzis et al., 2007).

Forg´acs et al. (2017) where different skew arches and the failure mode and limiting thickness of the arch barrel to carry the dead load was investigated (Fig. 13.17). Toth et al. (2009) made use of the 2D software UDEC to investigate the soil-to-arch barrel interaction.

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FIGURE 13.17 Failure mode of a helicoidal skew arch (angle of skew 45 degree) (Forg´acs et al., 2017).

(A)

(B)

Stress

Scale: 8.69294e–08 Minimum Prin, 5.7108E + 05 5.0000E + 05 2.5000E + 05 0.0000E + 05 –2.5000E + 05 –5.0000E + 05 –7.5000E + 05 –1.0000E + 06 –1.2500E + 06 –1.5000E + 06 –1.7500E + 06 –2.0000E + 06 –2.2500E + 06 –2.5000E + 06 –2.7500E + 06 –2.9333E + 06

Stress

Scale: 1.68139e–08 Minimum Prin, 7.0013E + 05 5.0000E + 05 0.0000E + 05 –5.0000E + 05 –1.0000E + 06 –1.5000E + 06 –2.0000E + 06 –2.5000E + 06 –3.0000E + 06 –3.5000E + 06 –4.0000E + 06 –4.5000E + 06 –5.0000E + 06 –5.5000E + 06 –6.0000E + 06 –6.5000E + 06 –7.0000E + 06 –7.0830E + 06

FIGURE 13.18 Failure modes and stress contours obtained from the numerical model. (A) Span 3 m, rise:span 1:4, rings 2; (B) span 3 m, rise:span 1:4, rings 5. From Kassotakis, N., ´ Sarhosis, V., Forgacs, T., Bagi, K., 2017. Discrete element modelling of multi-ring brickwork masonry arches. In: 13th Canadian Masonry Symposium. Halifax, Canada: Canada Masonry Design Centre.

In addition, Kassotakis et al. (2017) presented a 3DEC model to study the load-carrying capacity of multiring arches. In this way, the ultimate loadcarrying capacity, failure mode, and stress distribution of different multiring masonry arches were investigated (Fig. 13.18). The suitability of the model to predict hinge development and ring separation indicates its suitability for the assessment of such structures. Spherical domes (Fig. 13.19) and masonry cross-vaults were investigated by Lengyel and Bagi (2015) and Simon and Bagi (2016). In both cases, the voussoirs allowed to slide along each other and separate from their neighbors in any directions. A material parameter sensitivity study was undertaken and their influence on the structural stability of domes was investigated.

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FIGURE 13.19 Collapse mechanism for spherical dome (colors indicate displacement magnitude (Simon and Bagi, 2016).

13.4.5 Modeling large and complex masonry structures Structures with large sizes or complex geometries always pose a challenge to analysts. The most straightforward approach with DEM models is to represent each masonry unit by a block, whether rigid or deformable. However, this may not be practical or even possible. First, the generation of the model is time-consuming, and many details of the construction have to be inferred. Also, runtimes may be forbidding, particularly in dynamic analysis, given the small timesteps required by explicit algorithms. Furthermore, there may not be sufficient field data, for example, to characterize each of the various types of joints or interfaces. Some degree of simplification is thus always required. A simplified model that represents well the key components and their interactions is often capable of providing realistic results. Furthermore, this type of model is easier to verify and validate and allows many more parametric studies. Block sizes larger than the real ones often have to be employed in numerical representations, which simply means that a numerical block is in effect a macroblock assembling various real blocks. The key requirement is that the model must represent the possible collapse mechanisms. Even if the blocks are larger than the reality, if there are enough of them and joints to provide the failure paths, then the collapse modes and loads may be properly estimated. This type of simplification was used by C¸aktı et al. (2016) in the 3DEC study of a shake table test of a scale model of a mosque (Fig. 13.20). The complexity of the dynamic response of such structures under intense shaking often means we cannot expect to match numerically the exact displacement histories at specific points. However, if the patterns of behavior and the peak levels of response are correctly reproduced, the numerical model can provide an effective simulation tool that can be employed to study the prototype structure.

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FIGURE 13.20 Sliding in the numerical (see arrow) and experimental models under 160% Montenegro earthquake on left and right, respectively (C ¸ aktı et al., 2016).

FIGURE 13.21 Failure of stone masonry model: (left) collapse mode in static pushover analysis (0.65 g); (right) peak displacement in dynamic analysis with PGA of 1.05 g (Lemos and Campos, 2017).

For traditional stone masonry constructions, the irregular shape of units presents another difficulty for modelers. Numerical representations are commonly based on regular block shapes. A benchmark study conducted in the frameworks of the 9th International Masonry Congress involved prediction and postdiction analyses for a 1:1 model of a traditional stone masonry house tested on the shaking table (Mendes et al., 2017). A simplified 3DEC model of the structure was capable of providing the observed collapse mechanism (Lemos and Campos, 2017). Both pushover and time domain analysis were performed, giving qualitatively the same type of failure pattern (Fig. 13.21), but with the pushover collapse taking place for a lower acceleration. The

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DEM model did not try to replicate the exact blocks, but only the typical block size and structural patterns using regular brick-shaped blocks. The calibration of the joint stiffness in the postdiction analysis, mostly based on the measured natural frequencies and modes, allowed a good match of the experimental displacement response during the sequence of tests with increasing input levels. In 2D, it is easier to create complex block geometries. De Felice (2011) studied a three-leaf cross-section of a traditional wall with UDEC, simulating closely the observed irregular block geometries, namely in the very detailed representation of the inner leaf of rubble masonry. Static and dynamic analyses were performed, and the results highlighted the importance of the connection between the outer wall leaves in maintaining the integrity of the section. Bui et al. (2017) performed a series of comparisons of 3DEC models with experimental tests on brick structures with dry joints, including various types of walls and panels, three-wall arrangements, and a full-house model (Fig. 13.22). Different static loading conditions were applied in these tests, in-plane and out-of-plane loading, namely involving imposed base rotations. In these models, it was possible to replicate the real block size in the numerical models. These were able to represent quite satisfactorily the crack development and failure modes observed in the tests. Masonry dams are a quite different type of structure, built mostly in the first half of the 20th century, and require safety assessment tools, given the risk they pose to communities downstream. Bretas et al. (2014) employed a DEM model to study the failure mode of a gravity dam induced by cracking through the structure. The inner core of the structure was represented by randomly generated Voronoi polygons, allowing multiple failure paths to take place. Three different Voronoi patterns are compared in Fig. 13.23, which gave different failure surfaces, but all in the same region and for friction angles within a relatively close range. The other important potential failure mode involves sliding along the dam
rock interface, which can be addressed with simpler DEM models.

13.5 Concluding remarks The expanding group of DEMs exhibits significant variety in base assumptions and numerical implementation features. In this chapter, the UDEC and 3DEC codes were examined, placed in the diverse context of DEM codes for masonry analysis, as representative of a central line of formulation, following Cundall’s original development. Application of DEM to masonry has gradually increased, grounded in the method’s ability to address the mechanics of discontinuous media, already proven in other engineering fields. In recent years, a set of validation studies comparing DEM models with field

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FIGURE 13.22 Comparison of observed experimental failure mechanisms (Restrepo-Velez et al., 2010) with those predicted by the DEM for the masonry construction S42 (B2 mechanism).

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FIGURE 13.23 Safety analysis of masonry gravity dams (Bretas et al., 2014).

and lab data, namely shake table tests, has been published, further establishing their fitness for the structural analysis and safety assessment of masonry. The review of applications reported shows the diverse uses of DEM, from very simplified models with an idealized block structure, to detailed representations in which every physical block is replicated accurately. The latter approach is only feasible in particular cases. A key challenge for researchers is to develop robust methodologies to address complex geometries (e.g., as found in historical masonry) to achieve efficient computational tools without losing realism. Correct simulation of the constitutive behavior of joints and block material is essential, and has to be closely supported by experimental evidence. But as has been discussed in this chapter, the strength of DEM models lies in their ability to reproduce the observed complexity of behavior using only a limited set of elementary constitutive assumptions and mechanisms.

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106. Bagi, K., 1993. A quasi-static numerical model for micro-level analysis of granular assemblies. Mech. Mater. 16 (1
2), 101
110. Baraldi, D., Cecchi, A., 2017. A full 3D rigid block model for the collapse behaviour of masonry walls. Eur. J. Mech. A/Solids . Available from: https://doi.org/10.1016/j.euromechsol.2017.01.012. Beyabanak, S., Mikola, R., Hatami, K., 2008. Three-dimensional discontinuous deformation analysis (3D DDA) using a new contact resolution algorithm. Comput. Geotech. 35, 346
356. Bretas, E.M., Lemos, J.V., Lourenc¸o, P.B., 2014. A DEM based tool for the safety analysis of masonry gravity dams. Eng. Struct. 59, 248
260. Bui, T.T., Limam, A., Sarhosis, V., Hjiaj, M., 2017. Discrete element modelling of the in-plane and out-of-plane behaviour of dry-joint masonry wall constructions. Eng. Struct. 136, 277
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Cakti, E., Oliveira, C.S., Lemos, J.V., Saygili, O., Go¨rk, S., Zengin, E., 2015. Ongoing research on earthquake behavior of historical minarets in Istanbul. In: Psycharis, I.N., et al., (Eds.), Seismic Assessment, Behavior and Retrofit of Heritage Buildings and Monuments, Computational Methods in Applied Sciences, vol. 37. Springer. ¨ ., Lemos, J.V., Oliveira, C.S., 2016. Discrete element modeling of a scaled C¸aktı, E., Saygılı, O masonry structure and its validation. Eng. Struct. 126, 224
236. Available from: https://doi. org/10.1016/j.engstruct.2016.07.044. Cundall, P.A., 1971. A computer model for simulating progressive, large-scale movements in blocky rock systems. Proceedings of the International Symposium on Rock Mechanics. Nancy 129
136. Cundall, P.A., 1988. Formulation of a three-dimensional distinct element model
Part I. A scheme to detect and represent contacts in a system composed of many polyhedral blocks. Int. J. Rock Mech. Min. Geomech. Abstr. 25 (3), 107
116. De Felice, G., 2011. Out-of-plane seismic capacity of masonry depending on wall section morphology. Int. J. Architect. Herit. 5, 466
482. DeJong, M.J., Vibert, C., 2012. Seismic response of stone masonry spires: computational and experimental modeling. Eng. Struct. 40, 566
574. De Lorenzis, L., DeJong, M., Ochsendorf, J., 2007. Failure of masonry arches under impulse base motion. Earthq. Eng. Struct. Dyn. 36, 2119
2236. Dimitri, R., De Lorenzis, L., Zavarise, G., 2011. Numerical study on the dynamic behavior of masonry columns and arches on buttresses with the discrete element method. Eng. Struct. 33, 3172
3188. Elias, J., 2014. Simulation of railway ballast using crushable polyhedral particles. Powder Technol. 264, 458
465. Forg´acs, T., Sarhosis, V., Bagi, K., 2017. Minimum thickness of semi-circular skewed masonry arches. Eng. Struct. 140 (1), 317
336. Hart, R., Cundall, P.A., Lemos, J., 1988. Formulation of a three-dimensional distinct element model
Part II. Mechanical calculations for motion and interaction of a system composed of many polyhedral blocks. Int. J. Rock Mech. Min. Sci 25 (3), 117
125. Hazay, M., Munjiza, A., 2016. Introduction to the combined finite-discrete element method. In: Sarhosis, et al., (Eds.), Computational Modeling of Masonry Structures Using the Discrete Element Method. IGI Global, Hershey, PA. Housner, G.W., 1963. The behavior of inverted pendulum structures during earthquakes. Bull. Seismol. Soc. Am. 53 (1), 403
417. Kassotakis, N., Sarhosis, V., Forg´acs, T., Bagi, K., 2017. Discrete element modelling of multiring brickwork masonry arches. 13th Canadian Masonry Symposium. Canada Masonry Design Centre, Halifax, Canada. Kishino, Y., 1988. Disc model analysis of granular media. In: Satake, M., Jenkins, J.T. (Eds.), Micromechanics of Granular Materials. Elsevier, pp. 143
152. Komodromos, P., Papaloizou, L., Polycarpou, P., 2008. Simulation of the response of ancient columns under harmonic and earthquake excitations. Eng. Struct. 30 (8), 2154
2164. Lemos, J.V., 1998. Discrete element modelling of the seismic behaviour of stone masonry arches. In: Middleton, J., Pande, G.N., Kralj, B. (Eds.), Computer Methods in Structural Masonry 2 4. E&FN Spon, London, pp. 220
227. Lemos, J.V., Campos, C.A., 2017. Simulation of shake table tests on out-of-plane masonry buildings. Part (V): discrete element approach. Int. J. Architect. Herit. 11 (1), 117
124. Lemos, J.V., Costa, A.C., Bretas, E.M., 2011. Assessment of the seismic capacity of stone masonry walls with block models. In: Papadrakakis, M., Fragiadakis, M., Lagaros, N.D. (Eds.), Computational Methods in Earthquake Engineering. Springer, pp. 221
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Lemos, J.V., Oliveira, C.S., Navarro, M., 2015. 3D nonlinear behavior of an obelisk subjected to the Lorca May 11, 2011 strong motion record. Eng. Fail. Anal. 58, 212
228. Lengyel, G., Bagi, K., 2015. Numerical analysis of the mechanical role of the ribs in groin vaults. Comput. Struct. 158, 42
60. ISSN: 0045-7949. Liu, X.L., Lemos, J.V., 2001. Procedure for contact detection in discrete element analysis. Adv. Eng. Soft. 32 (5), 409
415. Mendes, N., Costa, A.A., Lourenc¸o, P.B., Bento, R., Beyer, K., de Felice, G., et al., 2017. Methods and approaches for blind test predictions of out-of-plane behavior of masonry walls: a numerical comparative study. Int. J. Architect. Herit. 11 (1), 59
71. Munjiza, A., Owen, D.R.J., Bicanic, N., 1995. A combined finite discrete element method in transient dynamics of fracturing solids. Eng. Comput. 12 (2), 145
174. Oliveira, C.S., Lemos, J.V., Sincraian, G.E., 2002. Modelling large displacements of structures damaged by earthquake motions. Eur. Earthq. Eng. 3, 56
71. Papaloizou, L., Komodromos, P., 2009. Planar investigation of the seismic response of the ancient columns and colonnades with epistyles using a custom made software. Soil Dyn. Earthq. Eng. 29 (1), 1437
1454. Papantonopoulos, C., Psycharis, I.N., Papastamatiou, D.Y., Lemos, J.V., Mouzakis, H.P., 2002. Numerical prediction of the earthquake response of classical columns using the distinct element method. Earthq. Eng. Struct. Dyn. 1 (31), 1699
1717. Papastamatiou, D., Psycharis, I., 1993. Seismic response of classical monuments
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601. Pen˜a, F., Prieto, F., Lourenc¸o, P.B., Costa, A.C., Lemos, J.V., 2007. On the dynamics of rocking motion of single rigid-block structures. Earthq. Eng. Struct. Dyn. 36 (15), 2383
2399. Psycharis, I.N., Fragiadakis, M., Stefanou, I., 2013. Seismic reliability assessment of classical columns subjected to near-fault ground motions. Earthq. Eng. Struct. Dyn. 42, 2061
2079. Psycharis, I.N., Lemos, J.V., Papastamatiou, D.Y., Zambas, C., Papantonopoulos, C., 2003. Numerical study of the seismic behaviour of a part of the Parthenon Pronaos. Earthq. Eng. Struct. Dyn. 32 (13), 2063
2084. Pulatsu, B., Sarhosis, V., Bretas, E., Nikitas, N., Lourenc¸o, P.B., 2017. Non-linear static behaviour of ancient free-standing stone columns. Struct. Build. 170, 406
418. Roberti, G.M., Spina, O., 2001. Numerical analysis of the Sardinian ‘Nuraghe’. In: Hughes, T. G., Pande, G.N. (Eds.), Computer Methods in Structural Masonry
5. Computers & Geotechnics Ltd, Swansea, pp. 190
197. Rougier, E., Munjiza, A., John, N.W.M., 2004. Numerical comparison of some explicit time integration schemes used in DEM, FEM/DEM and molecular dynamics. Int. J. Numer. Methods Eng. 61, 856
879. Sarhosis, V., Asteris, P., Wang, T., Hu, W., Han, Y., 2016a. On the stability of colonnade structural systems under static and dynamic loading conditions. Bull. Earthq. Eng. 14 (4), 1131
1152. Sarhosis, V., Asteris, P.G., Mohebkhah, A., Xiao, J., Wang, T., 2016b. Three dimensional modelling of ancient colonnade structural systems subjected to harmonic and seismic loading. Struct. Eng. Mech. 60 (4), 633
653. Sarhosis, V., Garrity, S.W., Sheng, Y., 2013. Computational modelling of low bond strength brickwork wall/beam panels with retro-fitted reinforcement. 12th Canadian Masonry Symposium. International Masonry Society, Vancouver, Canada. Sarhosis, V., Sheng, Y., 2014. Identification of material parameters for low bond strength masonry. Eng. Struct. 60, 100
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Sarhosis, V., Milani, G., Formisano, A., Fabbrocino, F., 2017. Evaluation of different approaches for the estimation of the seismic vulnerability of masonry towers. Bull. Earthq. Eng . Available from: https://doi.org/10.1007/s10518-017-0258-8. Sarhosis, V., Oliveira, D.V., Lemos, J.V., Lourenc¸o, P.B., 2014. The effect of skew angle on the mechanical behaviour of masonry arches. Mech. Res. Commun. 61, 53
59. Schlegel, R., Rautenstrauch, K., 2004. Failure analyses of masonry shear walls. In: Konietzky, H. (Ed.), Numerical Modelling of Discrete Materials in Geotechnical Engineering, Civil Engineering and Earth Sciences. Taylor & Francis, London, pp. 15
20. Shi, G.-H., 1988. Discontinuous deformation analysis
a new model for the statics and dynamics of block systems. PhD thesis, University of California Berkeley, USA. Simon, J., Bagi, K., 2016. DEM analysis of the minimum thickness of oval masonry domes. Int. J. Architect. Herit. 10 (4), 457
475. Sincraian, G.E., Azevedo, J.J., 1998. Numerical simulation of the seismic behaviour of stone masonry structures using the discrete element method. In: Bisch P., Labbe´ P., Pecker A. (Eds.) Proc. 11th European Conf. on Earthquake Eng., Balkema, Rotterdam. Stefanou, I., Psycharis, I.N., Georgopoulos, I.-O., 2011. Dynamic response of reinforced masonry columns in classical monuments. Constr. Build. Mater. 25, 4325
4337. Toth, A.R., Orban, Z., Bagi, K., 2009. Discrete element modelling of a stone masonry arch. Mech. Res. Commun. 36 (4), 469
480. Williams, J.R., Mustoe, G.G.W., 1987. Modal methods for the analysis of discrete systems. Comput. Geotech. 4, 1
19. Zhang, H., Chen, G., Zheng, L., Han, Z., Zhang, Y., Wu, Y., et al., 2015. Detection of contacts between three-dimensional polyhedral blocks for discontinuous deformation analysis. Int. J. Rock Mech. Min. Sci. 78, 57
73.

Further reading Baraldi, D., Reccia, E., Cecchi, A., 2017. In plane loaded masonry walls: DEM and FEM/DEM models. A critical review. Meccanica . Available from: https://doi.org/10.1007/s11012-0170704-3. Godio, M., Stefanou, I., Sab, K., 2017. Effects of the dilatancy of joints and of the size of the building blocks on the mechanical behaviour of masonry structures. Meccanica . Available from: https://doi.org/10.1007/s11012-017-0688-z. Lemos, J.V., 2017. Contact representation in rigid block models of masonry. Int. J. Masonry Res. Innov. 2 (4), 321
334. Munjiza, A., Andrews, K.R.F., 2000. Penalty function method for combined finite-discrete element systems comprising large number of separate bodies. Int. J. Numer. Methods Eng. 49 (11), 1377
1396. Shi, G.-H., 1992. Discontinuous deformation analysis: a new numerical model for the statics and dynamics of deformable block structures. Eng. Comput. 9 (4), 157
168. Shi, G.-H., 2001. Three dimensional discontinuous deformation analysis. In: Procs. ICADD-4, 6
8 June 2001, Glasgow, ed. N. Bicanic, pp. 1
21.

Chapter 14

Descrete macroelement modeling S. Caddemi, I. Calio`, F. Cannizzaro, B. Panto` and D. Rapicavoli Department of Civil Engineering and Architecture, University of Catania, Catania, Italy

14.1 Introduction Simulation of the nonlinear behavior of masonry structures subjected to earthquake excitations or extreme loadings is a complex computational issue for which many numerical strategies characterized by different levels of accuracy and efficiency have been proposed. Masonry is one of most ancient construction materials and today represents a large part of existing and new structures. However, the word “masonry” has to be intended as a composite material obtained by the assemblage of individual units and mortars whose property is different from the property of its components (Hilsdorf, 1969). As a consequence, the word masonry itself refers to a great variability of masonry materials characterized by different constituents, geometrical layouts, and construction techniques. This huge variability makes it difficult to define reliable numerical models and general constitutive laws suitable for all masonry structures (Lourenc¸o et al., 1998; Asteris et al., 2014). Masonry material provides its mechanical contribution also in mixed-masonry structures, like confined masonry and infilled frame structures; in these latter cases, reliable numerical simulations also require nonlinear modeling of the interaction of the different structural members contributing to the global bearing capacity of the structural system (Asteris et al., 2011; Calio` and Panto`, 2014). Many significant examples of applications of the nonlinear FEM to historical masonry buildings and churches are reported in the literature; some of these studies consider masonry as a homogenized continuum at the

Numerical Modeling of Masonry and Historical Structures. DOI: https://doi.org/10.1016/B978-0-08-102439-3.00014-2 Copyright © 2019 Elsevier Ltd. All rights reserved.

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macroscale (Mele et al., 2003; Betti and Vignoli, 2008, 2011; Araujo et al., 2012; Lourenc¸o et al., 2012; Barbieri et al., 2013; Milani and Valente, 2015), other refined FE approaches are based on detailed simulations of units and mortar as micromodeling (Lofti and Shing, 1994; Anthoine, 1997; Gambarotta and Lagomarsino, 1997; Lourenc¸o and Rots, 1997; Berto et al., 2002; Macorini and Izzuddin, 2011). Much effort is made today in the link between the micro- and macro-modeling approaches using homogenization techniques that allow the use of continuum based approaches as the nonlinear FEM simulation as well as macromodeling simplified strategies. Nonlinear FEM approaches require the adoption of sophisticated constitutive laws, huge computational costs as well as advanced skills in the model implementation and in the interpretation of the numerical results. However, practitioners need simple and efficient numerical tools, whose complexity and computational demand are appropriate for practical engineering purposes. For these reasons, in recent decades, many researchers have proposed new efficient numerical methodologies for predicting the nonlinear seismic behavior of Un Reinforced Masonry (URM) structures (Brenchich et al., 1998; Magenes and Della Fontana, 1998; Kappos et al., 2002; Calio` et al., 2005; Chen et al., 2008; Marques and Lourenc¸o, 2011; Lagomarsino et al., 2013; Raka et al., 2015). Marques and Lourenc¸o (2011) report a comparison between different simplified approaches currently used in academic research and engineering practice. A common limitation of the existing simplified numerical strategies for URM structures, currently used by many practitioners, is the basic assumption of in-plane behavior of masonry walls, making these approaches less suitable for Historical Masonry Structures (HMSs), in which the out-of-plane behavior strongly influences the seismic response. An original alternative efficient approach is represented by the “rigid body spring model”, specifically formulated with the aim of approximating the macroscopic behavior of masonry walls with reduced degrees of freedom. Some valuable applications of this approach are related to historical masonry buildings (Casolo and Pen˜a, 2007; Casolo and Sanjust, 2009; Valente and Milani, 2016). Among the simplified methods, the Equivalent Frame Model (EFM) approach represents the most commonly adopted strategy and was implemented in several academic as well as commercial software environments. Several numerical and experimental validations have been already reported in the literature and different formulations have been proposed in the last three decades. In the following, a more detailed description of the EFM approach and its recent evolution is given, highlighting the advantages and the few drawbacks of this simplified strategy. Subsequently, an alternative macromodeling strategy for the simulation of the nonlinear behavior of URM structures is presented. The approach is based on the concept of

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macroelement discretization (Calio` et al., 2012a) and was conceived with the aim of capturing the nonlinear behavior of an entire structure through an assemblage of discrete macroelements characterized by different levels of complexity, according to the role played in the global model. The basic element was first developed for the simulation of the in-plane response of masonry walls (Calio` et al., 2005) and has been validated by several numerical and experimental tests (Marques and Lourenc¸o, 2011; Calio` et al., 2012a; Panto` et al., 2015). The basic plane element can be represented through a simple mechanical scheme comprised by an articulated quadrilateral with four rigid edges and four hinged vertices connected by two diagonal nonlinear links. The interaction between the macroelements is ruled by nonlinear zero-thickness interfaces. This novel approach has also been successfully applied for infilled frame structures (Calio` et al., 2008; Caddemi et al., 2013; Calio` and Panto`, 2014; Marques and Lourenc¸o, 2014). In this latter case, the infills are modeled by the macroelements, while the reinforced concrete frames are modeled by concentrated-plasticity beam columns. The basic 2D macroelement has been solely utilized for the simulation of the nonlinear response of masonry walls in their own plane. To overcome this significant restriction—common to several simplified approaches—a third dimension, together with the relevant needed additional degrees of freedom, were introduced in a 3D macroelement (Panto`, 2007; Caddemi et al., 2014; Panto` et al., 2017a). The kinematics of the enriched 3D macroelement is governed by seven Lagrangian parameters only and allows an efficient simulation of both the in-plane and the out-of-plane response of masonry walls. One of the advantages of the proposed macroelement strategy is related to the strongly reduced computational cost, if compared to the traditional nonlinear Finite Element Modeling (FEM). However, another benefit relies on the adopted mechanical calibration strategy that, being based on a straightforward fiber discretization, allows the use of simple uniaxial constitutive laws and leads to an easy interpretation of the numerical results. Based on the above issues, the proposed discrete macroelement method can be considered not only a reliable numerical tool for academic research but also an efficient practice-oriented approach for the nonlinear simulation of masonry buildings. However, many masonry monumental constructions are characterized by the presence of structural elements with curved geometry, such as arches, vaults, and domes, which require an efficient reliable simulation. For this reason, a further enrichment of the proposed 3D macroelement, with a more general shell macro element, was subsequently introduced (Calio´ et al., 2010; Cannizzaro, 2010; Caddemi et al., 2015; Cannizzaro et al., 2018). The latter shell macroelement was conceived as an extension of the spatial element and currently represents the first macroelement proposed for curved

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masonry structures. Its nucleus is comprised by an irregular articulated quadrilateral, still characterized by four rigid layer edges, whose orientation and size are now related to the shape of the element and to the thickness of the modeled masonry portion. This more general macroelement strategy is mainly used for the numerical simulation of the seismic behavior of HMSs, masonry arch bridges and has been implemented into the software code HiStrA (Historical Structures Analysis) (Calio` et al., 2015), which simplifies the modeling of historical structures by means of several wizard generation tools, capable of managing complex curved geometries through to a powerful parametric input. In this chapter, a comprehensive review of the proposed discrete macroelement strategy is discussed. The different proposed macroelements and their capability to be applied for the structural assessment of masonry structures are discussed with reference to some relevant cases. Numerical and experimental validations are reported with reference to some benchmarks already investigated in the literature. The low computational cost and the easiness in the interpretation of the results make this method particularly suitable for the engineering community, as well as for academic research on the seismic assessment of cultural heritage buildings.

14.2 The equivalent frame models A widely used model for the global analysis of masonry buildings—assuming an in-plane response of masonry walls—is the so-called EFM. This approach can be regarded as a macroelement strategy based on the assumption that the out-of-plane response of the masonry walls is prevented and the global behavior of the structure is ruled by the in-plane reactions of the masonry walls that can horizontally interact through diaphragms. Following the macroelement approach, each wall is idealized in several macroportion, or structural components, to be represented by a suitable equivalent mechanical model. In the EFM, it is generally assumed that in-plane damage can occur on piers and spandrels while the other masonry portions are not subjected to damage. Piers and spandrels are identified as the masonry portions between horizontally and vertically aligned openings, respectively. In this macroelement approach, the masonry portions susceptible to damage are represented by equivalent nonlinear beams whose nonlinear behavior is calibrated for describing the nonlinear response of the corresponding masonry panel. In Fig. 14.1 an example of the subdivision of a typical masonry wall in piers, spandrels, and joint regions is shown; in the same figure, a geometrical scheme of the corresponding equivalent frame is also represented. Under this hypothesis the masonry portions connecting piers and spandrels are considered damage prevented and are regarded as rigid links connecting

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FIGURE 14.1 Equivalent frame modeling strategy. (A) Masonry wall geometrical layout; (B) identification of piers, spandrels and regions assumed as rigid; (C) frame model superimposed to the wall geometry; (D) equivalent frame model.

the equivalent beams representating piers and spandrels. This practically oriented approach leads to the definition of an equivalent frame for each plane masonry wall; the spatial connection of the plane frames by means of rigid or deformable diaphragms allows to obtain a spatial frame model representative of the global behavior of the overall building. The first equivalent beam-based model can be attributed to Tomazevic (1978), which introduced the so-called POR method. In this pioneering version of the EFM, each wall was idealized as a shear-type frame in which only the columns, representing the masonry piers, were assumed as susceptible to damage while both spandrels and connecting regions were assumed as rigid zones exempt from damage. Initially, only the shear capacity of the masonry piers had been considered, according to simplified elastic perfectly plastic constitutive laws. This very simple EFM strategy, although too approximate, had the advantage to recognize the need to perform nonlinear analyses for masonry buildings. The POR method was than improved by considering shear and flexural collapse mechanisms for the masonry

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piers, leading to the POR FLEX method (Braga and Dolce, 1982; Dolce, 1991). After the introduction of the POR method, some research highlighted the need to enrich the model by introducing the possibility of identifying the presence of damage in other structural components. Taking inspiration from a seismic damage scenario on URM buildings, some research groups observed that the in-plane damage in masonry walls is mainly concentrated on piers and spandrels, while the masonry portion connecting these regions is rarely subjected to significant damage. In view of this consideration, the shear-type model has been abandoned and new EFMs proposed. Magenes and Calvi (1996) introduced the SAM (simplified analysis of masonry) method, which was then further modified by Magenes and Della Fontana (1998). In the SAM method, each plane wall is represented by an equivalent frame where columns and beams represent piers and spandrels respectively. Rigid offsets describe the joint panels in which damage cannot occur. Many equivalent beam-based models (Kappos et al., 2002; Roca et al., 2005; Penelis, 2006; Belmouden and Lestuzzi, 2009) are based on the idealization of the structure as an assemblage of piers as columns and spandrels as beam elements, connected by rigid links. The main differences between these models rely on rules adopted for the definition of the equivalent frame and on the constitutive laws adopted for the description of the nonlinear behavior of piers and spandrels. A widely used EFM is implemented in the Tremuri software (Lagomarsino et al., 2013), which has been validated experimentally and numerically, and implements an original and versatile algorithm for the pushover analysis, suitable for assessing the nonlinear evolution of the lateral response of 3D masonry buildings—including the deterioration of the base shear for increasing lateral displacements after the attainment of peak strength. Recently some equivalent beam-based models proposed the use of distributed plasticity beam elements (Addessi et al., 2015; Raka et al., 2015) leading to a better description of the nonlinear flexural behavior associated with a fiber discretization of the masonry. Equivalent frame approaches represent a sleek and fast solution to assess masonry buildings, whose main advantages are listed here: 1. The needed degrees of freedom to model an entire building is limited, thus allowing to perform of nonlinear analyses with a reasonable computational burden when compared with FE approaches. 2. The main in-plane failure mechanisms of a masonry panel can be considered by means of ad-hoc constitutive laws. 3. A large part of masonry structures respects the hypothesis which this methodology relies on. 4. The implementation of such approaches in general-purpose software environments is possible.

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On the other hand, these approaches present some drawbacks that limit their employability. In particular: 1. The definition of the equivalent frame is not always straightforward, especially when the distribution of the openings on the masonry walls is irregular. 2. The geometric inconsistency between a plane masonry portion and the beam makes it difficult to simulate the interaction between reinforced concrete or steel frame structures and adjacent masonry walls. This is the case with confined masonry or infilled frame structures. 3. As in many macroelement approaches, the out-of-plane response is not considered. Many works investigated and validated the EFM, highlighting the difference between the approaches already proposed in the literature as well as the advantages and limits of applicability, explored in the recent works of Marques and Lourenc¸o (2011), Raka et al. (2015), Quagliarini et al. (2017), Siano et al. (2018).

14.3 A discrete macroelement strategy Starting from pioneering work in 2004 (Calio` et al., 2004), a research group at the University of Catania proposed a new macroelement method defined according to a unique approach within the framework of a discrete element formulation strategy. Such an approach is based on the subdivision of the structure under consideration in several macroportions; after homogenization of the mechanical properties of the components (mortar and units), each macroportion is regarded as an equivalent continuum whose mechanical properties can be assumed as isotropic or orthotropic depending on the masonry texture. The following step is the discretization by means of a mesh of macroelements chosen according to the macroportion that has to be modeled. Fig. 14.2

FIGURE 14.2 Subdivision of a dome in macroportion to be represented by macroelements.

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shows a qualitative subdivision of a dome by means of several macroportions that, according to a macroelement strategy, will be represented by shell macroelements. In this approach, each flexible macroelement interacts with the adjacent elements through nonlinear distributed zero-thickness interfaces. The nonlinear behavior of the structure is captured through an assemblage of macroelements, characterized by different levels of complexity, according to the role played by the global model. The degrees of freedom needed to describe the macroelements’ kinematics are those strictly related to the rigid body motion plus a single degree of freedom governing the element deformability. The following subsections contain a brief description of the different macroelements introduced so far.

14.4 The basic 2D macroelement The basic 2D macroelement is a plane quadrangular element endowed by four degrees of freedom (Fig. 14.3A). The 2D macroelement, first proposed in 2004 (Calio` et al., 2004), was conceived for the simulation of the nonlinear response of masonry walls in their own plane (Fig. 14.3B). The element can be regarded as an articulated quadrilateral of rigid beams connected by four hinges, leading to a kinematics governed by only four degrees of freedom. Zero-thickness interfaces govern the interaction with the adjacent elements, while the element deformability is conveniently ruled by a single, diagonal nonlinear link. The kinematics of the mechanical scheme, after a proper calibration procedure of the nonlinear links, is capable of simulating the main in-plane collapse failure modes of a masonry panel: flexural failure, diagonal shear failure, and sliding shear failure (Calio` et al., 2012a). Despite its simplicity, the assemblage of these elements allows the simulation of the global nonlinear response of masonry buildings

FIGURE 14.3 The 2D macroelement and its mechanical scheme. (A) Initial undeformed configuration; (B) deformed configuration.

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FIGURE 14.4 Typical macroelement discretization of an infilled frame in presence of a central door opening.

(also in the presence of openings) allowing a geometrically consistent simulation of the masonry walls in their own plane. Each macroelement exhibits three degrees of freedom associated with the in-plane rigid body motion, plus the additional degree of freedom, needed for the description of the in-plane shear deformability. The deformations of the interfaces are related to the relative motion between corresponding panels; therefore, no further Lagrangian parameter has to be introduced to describe their kinematics. The adopted model has the advantage of interacting with the adjacent elements along the entire perimeter, thus allowing the possibility of using different mesh discretizations, as highlighted in the following paragraphs. The numerical approach has been validated by several studies (Marques and Lourenc¸o, 2011) and it has been implemented in the software 3DMacro (Calio` et al., 2012b) currently used for research and practical applications. The geometric consistency of the elements also allows an efficient simulation of infilled frame structures; Fig. 14.4 reports an example of infilled frame model by means of a hybrid approach in which the beams are modeled as frame elements and the infill is modeled by means of mesh of plane macroelements.

14.4.1 The 3D macroelement The 2D macroelement allows the simulation of a masonry wall in its own plane but ignores the out-of-plane response. To overcome this significant restriction, a third dimension, and the relevant needed additional degrees of freedom were introduced in a 3D macroelement (Panto`, 2007; Caddemi et al., 2014; Panto` et al., 2017a). Fig. 14.5 reports the 3D macroelement (Panto`, 2007; Caddemi et al., 2014; Panto` et al., 2017a) obtained as the extension to the space of the plane element described in the previous paragraph. The kinematics of the spatial macroelement is governed by seven degrees of freedom, able to describe the in- and out-of-plane rigid body motions of the quadrilateral and the in-plane shear deformability. The interaction of the spatial macroelement with

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FIGURE 14.5 3D macroelement. (A) Simplified mechanical scheme; (B) a typical fiber discretization of the element.

the adjacent elements or the external supports is ruled by 3D interfaces. Each 3D interface possesses m rows of n orthogonal (i.e., perpendicular to the planes of the interface) nonlinear links. Consequently, each interface is discretized, similarly to what is done in classical fiber models, in m 3 n subareas (Fig. 14.5B). The 3D interfaces are endowed with additional shearsliding links (Fig. 14.5A), required to control the in-plane and out-of-plane sliding mechanisms and the torsion around the axis perpendicular to the plane of the interface. The number of nonlinear links adopted in the 3D interfaces is selected according to the desired level of accuracy of the nonlinear response. A detailed description of the mechanical calibration of the spatial macroelement and its numerical and experimental validation is reported in Panto` et al. (2017a). This model has also been applied for the simulation of infilled frame structures, accounting for the in- and out-of-plane behavior of the infills (Panto` et al., 2018).

14.4.2 The shell macroelement for modeling curved geometry The 3D macroelement (Panto` et al., 2017a) allows the simulation of the inplane and out-of-plane behavior of plane masonry walls. However, historical structures are often characterized by a curved geometry whose role in the global and local response cannot be ignored. Aiming at modeling curved geometry, a more general shell macroelement for modeling arches, vaults, domes, and masonry arch bridges has been introduced. The shell macroelement is characterized by four rigid layer edges whose orientation and dimension is now associated to the shape of the element and to the thickness of the portion of structure to be modeled (Fig. 14.6). The in-plane shear deformability is still governed by a single degree of freedom related to a diagonal spring placed along one of the diagonals of the quadrilateral. The plane interfaces rule the interaction with adjacent elements or external supports. However, due to the irregular geometry, these interfaces are in general skewed with respect

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FIGURE 14.6 Shell macroelement. (A) The orthogonal links of the interfaces; (B) the longitudinal and the diagonal links.

FIGURE 14.7 (A) Curved portion of masonry structures; (B) its flat discrete element representation.

to the average plane of the element. Curved surfaces are therefore modeled under the assumption that the behavior of a continuously curved surfaces can be adequately represented by flat macroelements. Each quadrilateral is geometrically defined by the coordinates of its vertices, the four normal vectors to the surface and the thicknesses at these points (Fig. 14.7). The most significant features of the improved shell element are: 1. interfaces no longer orthogonal to the plane of the element, thus allowing to follow the curved geometry of the structure; 2. thickness can be linearly variable at each interface; 3. shape of the element can be represented by a generic quadrangular element. Despite the complications, due to the curved geometry, the model keeps the original simplicity and computational cost. Its kinematics is still ruled by seven degrees of freedom (six rigid body motion degrees of freedom and one associated with the in-plane shear deformability). The irregular geometry implies that each link corresponds to a prismatic fiber, whose cross-sectional area varies with a parabolic trend (Fig. 14.8).

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FIGURE 14.8 Fiber discretization of the shell macroelement.

There are three nonlinear sliding links in each interface (Fig. 14.6B): one along the axis of the interface (in-plane sliding link) and two orthogonal to the axis and still lying on the plane of the interface (out-of-plane sliding links). The calibration strategy follows the same philosophy of the spatial regular model. Since those links have to simulate the occurrence of sliding along the bed joints, their nonlinear behavior is closely affected by friction phenomena and the yielding domain accounts for the influence of the normal force acting on the interface. In the subdivision of an arbitrary shell into flat elements, both triangular and quadrilateral elements should generally be used (see Fig. 14.2). The triangular elements are assumed to be rigid in their own plane and are therefore characterized by six degrees of freedom only. A detailed description of the mechanical characterization of this nontrivial shell discrete element is outside the purpose of the present chapter.

14.5 Mechanical characterization strategy of the proposed macroelement approach According to the proposed strategy, each macroelement must be representative of the corresponding finite portion of masonry wall, cut out by plane sections located at the edges of the element. The formulation follows a phenomenological description of the mechanical behavior of a masonry portion in which the zero-thickness interfaces rule the membrane-flexural response and the shear-sliding behavior of adjacent elements, while the inplane shear element deformability is related to the angular distortion of the articulated quadrilateral. The mechanical characterization of the zerothickness interfaces here is performed following a straightforward fiber calibration procedure, while the shear element deformability is calibrated through a mechanical equivalence with the reference geometric-consistent continuous model. The interface nonlinear links can be distinguished as orthogonal links and shear-sliding links. In the following paragraphs, the main steps needed for the calibration procedure are described with reference to each group of nonlinear links.

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14.5.1 Calibration of the nonlinear links orthogonal to the interfaces The orthogonal nonlinear links incorporate the mechanical properties of the represented element assuming masonry as an orthotropic homogeneous medium. Each orthogonal link encompasses the nonlinear behavior of the corresponding fiber along a given material direction (Fig. 14.5B). With a regular 3D macroelement, each link is calibrated, assuming that the uniform masonry strip is a homogeneous inelastic material, and can also consider cyclic behavior governed by fracture energy values for the tensile and compressive response, Gt and Gc, respectively, and follows different postelastic branch laws (Ch´acara et al., 2018). For clarity, reference is made to a single orthotropic panel under monotonic loadings (Fig. 14.9). In this case, the flexural behavior of the masonry panel is characterized by different mechanical properties along the two fundamental directions. Eh and Ev are the Young’s moduli of the homogenized orthotropic masonry medium; σch, σth, and σcv, σtv are the corresponding compressive and tensile maximum stresses, Gch, Gth, and Gcv, Gtv are the fracture energies in compression and tension, as shown in Fig. 14.9A. Consistently with the adopted fiber calibration strategy, the flexural stiffness calibration of the panel is simply obtained by assigning to each link the axial stiffness of the corresponding masonry strip. Each masonry strip is identified by its influence area, and the half-dimension of the panel in the direction perpendicular to the interface (Fig. 14.5B). The initial stiffness K, the compressive and tensile yielding strengths, fc and ft, and the corresponding ultimate displacements, uc and ut (under the simplified hypothesis of a (A)

σ tv Gcv

(B)

Gtv Ev

u

σ cv

kh , f th , f ch , uth ,uch

λh

σ th

H

Gch

B 2

Gth Eh

σ ch

u

H 2

λv k , f , f , u , u v tv cv cv tv

B FIGURE 14.9 Mechanical characterization of an orthotropic masonry panel: (A) constitutive laws; (B) calibration of the orthogonal links (Panto` et al., 2017a).

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PART | II Modeling of unreinforced masonry

TABLE 14.1 Mechanical calibration of the orthogonal links for a rectangular panel. Direction

K

fc

ft

uc

ut

Horizontal

Kh 5 2 Eh λBh λs

fch 5 σch λh λs

fth 5 σth λh λs

uch 5 2 Gσchch

uth 5 2 Gσthth

Vertical

Kv 5 2 Ev λHv λs

fcv 5 σcv λv λs

ftv 5 σtv λv λs

ucv 5 2 Gσcvcv

ut2 5 2 Gσtvtv

rectangular shape of the panel and linear softening) of the links relative to the horizontal and vertical interfaces are reported in Table 14.1 as a function of the mechanical and geometrical properties of the masonry panel. In Table 14.1 B and H are the length and the height of the panel, λh and λv are the in-plane distances between the springs along the interfaces arranged according to the two fundamental directions, and λs is the out-ofplane distance between the rows of links, as shown in Fig. 14.9B.

14.5.2 Calibration of the nonlinear links along the interfaces The nonlinear links, lying along the interface and denoted as shear-sliding links, govern the torsional and shear-sliding behavior along the interfaces. In the discretization shown here, one single link is considered for the in-plane model (Fig. 14.3) while three nonlinear links are considered for the spatial models (Figs. 14.5 and 14.6), this being the minimum required to obtain the possible masonry failure modes (Panto` et al., 2017a). A single in-plane shear-sliding spring, governing the in-plane sliding of the element along the interface is calibrated according to a rigid-plastic Mohr Coulomb law. The out-of-plane shear deformability is ruled by two parallel links, which take care of the out-of-plane sliding behavior and the torsional elastic and inelastic response of connected, adjacent panels. The two out-of-plane shearsliding nonlinear links are required to control the out-of-plane sliding mechanisms as well as the torsion around the axis perpendicular to the plane of the interface. With the goal of maintaining a simple fiber calibration approach, the out-of-plane shear deformability of each link connecting two adjacent panels is calibrated according to their influence volumes. Referring to two identical adjacent macroelements, with thickness s, width B and height H, shear modulus G, cohesion c, and friction coefficient μs, the calibration procedure is summarized providing the main parameters that govern the mechanical behavior of the sliding links (Table 14.2). Once the elastic shear out-of-plane stiffness has been assigned, according to the formulas reported in Table 14.2, the relative distance d between

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TABLE 14.2 Mechanical calibration of the shear-sliding links for a rectangular panel. Direction

ks

In-plane

N

Out-of-plane

ks 5

d

1 GBs 2 H

d 5 2s

fsy qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 1 s s4 3 2 0; 21 B 1 2 12B4


 fsy 5 c 1 μs N A
 fsy 5 12 c 1 μs N A

the two out-of-plane sliding links has to be set according to an equivalence with the corresponding elastic continuum in terms of torsional behavior (Panto` et al., 2017a). Aiming at obtaining a suitable torsional elastic calibration, although maintaining a simplified calibration strategy, the distance d between the two links is simply obtained considering that the torsional elastic stiffness of the corresponding geometrical consistent continuous model is equivalent to that associated to the discrete system. The yielding strength of each link is associated with the current contact area A of the interface and to the current axial force N associated to the orthogonal links of the interface.

14.5.3 Calibration of the diagonal link The diagonal shear failure (collapse of the panel) is related to a single degree of freedom; this allows to associate the shear nonlinear response to a single diagonal nonlinear link. Many different yielding criteria, strongly dependent on the compressive stresses in the wall, can be adopted to account for the shear capacity. In the elastic range, the diagonal shear link is calibrated by imposing an energy equivalence between the articulated quadrilateral, ruled by the diagonal spring, and a continuous reference elastic model. The yielding forces are associated with the limits of tensile or compressive stresses in the reference continuous model, while the postelastic behavior is ruled by a suitable constitutive law. The Mohr Coulomb law or the Turnsek Cacovic (1970) law can generally be adopted for the calibration of the diagonal link, although any constitutive law can also be considered.

14.6 Experimental and numerical validation of the proposed macroelement strategy In this section, the capability of the proposed discrete macroelement approach to simulate the nonlinear response of masonry structures is

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PART | II Modeling of unreinforced masonry

investigated. The method is validated by comparing the numerical results with those obtained by other numerical approaches or by experiments already available in the literature.

14.6.1 The 2D macroelement The first validation is relative to the case of a single 2D macroelement, which was investigated by a comparison of the proposed approach (Panto` et al., 2015) and an EFM, combined with a fiber section model recently proposed in Raka et al. (2015). The panel is restrained at its base and at its top. Initially the panel is subjected to a force-controlled application of a vertical load, then a displacement-controlled analysis, with an increasing horizontal displacement at the top of the panel, is applied. The panel is characterized by the thickness t 5 0.6 m, the width w 5 3 m, and the height h 5 2 m. The adopted mechanical properties are reported in Table 14.3; for this first example, the shear failure is considered inhibited. The results are compared with those obtained by an equivalent frame approach on a direct fiber section analysis (Raka et al., 2015). Several analyses were performed for different levels of the axial load; in Fig. 14.10A the ultimate bending moment of the base section is reported versus the considered axial load. The capability of the model to describe the axial-flexural response of a masonry wall section is assessed by comparing the M N dominium of the base section with data obtained following the closed-form expression reported in the Italian building code (NTC, 2008). A second example of a single panel is reported in Fig. 14.10B. The ultimate load obtained with the equivalent frame fiber model, as proposed by Raka et al. (2015), and the proposed macroelement are compared when either only the flexural or only shear mechanisms are considered. The two numerical models provide very close results in terms of ultimate loads, and they are consistent with the values suggested by the Italian code (NTC, 2008). The plane macroelement can also be adopted for modeling infilled frame structures. In the latter case a hybrid approach is applied: the surrounding frame is modeled using lumped plasticity beam column elements while the nonlinear response of the infill is modeled by means of the plane TABLE 14.3 Mechanical properties adopted for the masonry. G (MPa)

E (MPa)

ρ (kN/m3)

σc (MPa)

σt (MPa)

fv0 (MPa)

230

870

19

1.0

0.1

0.4

Descrete macroelement modeling Chapter | 14 3DMacro model

(A) 800

Fiber approach

700

Closed-form

600 M (KNm)

519

500 400 300 200 100 0 0

400

800

1200

1600

2000

N (KN)

(B) 700

Base shear (kN)

600 500 400 300 Shear failure fiber model Flexure failure fiber model Shear failure 3DMacro model Flexure failure 3DMacro model

200 100 0 0

100

200

300

400 500 N (kN)

600

700

800

900

FIGURE 14.10 Interaction diagrams: (A) M N; (B) V N (Panto` et al., 2015).

macroelement, already described in the previous section. The frame element interacts with the masonry panels by means of nonlinear-links distribution along discrete interfaces. Each interface is constituted by n transversal nonlinear links and a single longitudinal nonlinear link. The flexural interaction between the panel and an adjacent beam is governed by the four degrees of freedom of the beam associated to its two ends and by the n internal degrees of freedom associated to the links of the interface. For a more accurate evaluation of the nonlinear behavior of the frame element, it has been assumed that plastic hinges can occur in each sub-beam element between two nonlinear transversal links. This latter assumption provides a reliable frame element model as it can embed plastic hinges at different positions and is consistent with the adopted level of discretization for the infill interface. An experimental validation of the 2D macroelement, implemented in the software 3DMacro (Calio` et al., 2012b) has been provided by Marques and Lourenc¸o (2014) with reference to a three-dimensional building prototype.

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PART | II Modeling of unreinforced masonry

FIGURE 14.11 Experimental validation of the 2D macroelement for a prototype building (Marques and Lourenc¸o, 2014): (A) numerical model, (B) comparison in terms of capacity curve; (C) damage scenario at collapse of the south wall.

The experimental campaign was carried out at Centro peruano japones de Investigaciones Sismicas y MItigacion de Desastres (CISMID) research center in Peru (Zavala et al., 2004) on a two story building with an irregular plan, representative of a typical existing residential houses in Peru (Fig. 14.11A C). The tests were performed under quasistatic cyclic loads, applied through two actuators located at the two slabs, used to induce a constant load pattern to the structure proportional to the building height. In Fig. 14.11B, the comparison between numerical pushover curve (dotted curve) and the experimental results is reported, while Fig. 14.11C shows the damage scenario of the “south” wall at the last step of the analysis, a detailed comparison is reported in Marques and Lourenc¸o (2014).

14.6.2 The 3D macroelement An extensive numerical validation of the model on single walls with and without openings was recently carried out by Panto` et al. (2017a) considering masonry panels loaded out-of-plane with different geometries and boundary conditions. Applications of the model at a mesoscale for the out-of-plane behaviors of masonry prototypes can be found in Cannizzaro and Lourenc¸o (2017). The numerical applications here reported refer to a real scale simulation of a prototype building representative of a structural typology popular in Portugal between the end of 19th century and the beginning of the 20th century. This typology, known as “Gaioleiro” buildings, corresponds to tall structures, usually with six stories, in which walls are made of rubble masonry and lime mortar, and the horizontal diaphragms are timber floors and roofs. A four-story building with a timber roof and blind wall was investigated in the work conducted by Mendes and Lourenc¸o (2009) and Mendes (2012). Such a building was built in 1:3 scale and subsequently tested on a shaking table at Laboratorio Nacional de Engenharia Civil (LNEC), Mendes

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TABLE 14.4 Mechanical properties adopted in the numerical model. E (MPa)

σt (MPa)

Gt (N/mm)

σ c (MPa)

Gc (N/mm)

G (MPa)

γt (%)

γ u (%)

1000

0.1

0.05

100

1.6

417

0.6

1.5

et al. (2014). Details on the geometry can be found in Mendes (2012). The prototype was studied by means of an advanced FE model implemented in the software DIANA, conducting static and dynamic nonlinear analyses. The 3D macroelement method has been applied by using the software HiStrA (Calio` et al., 2015) in which the proposed macroelement strategy was implemented. The mechanical properties here assumed are reported in Table 14.4 and were determined consistently with the data proposed by Mendes (2012). ˇ coviˇc (1970) yielding criterion was assumed for the A Turnˇsek and Caˇ diagonal shear behavior characterized by a perfectly postelastic behavior until a transition drift γ t, assumed equal to 0.6%, with a subsequent linear softening branch till the achievement of a limit drift γ u equal to 1.5%. The numerical model consists of 704 elements (corresponding to an average mesh size equal to 1.1 m) with a total amount of degrees of freedom equal to 5568 (the FE model is characterized by 75,880 degrees of freedom; Mendes, 2012). The structure was initially loaded with the self weight and then subjected to horizontal mass proportional load distributions along the two main directions of the building, namely parallel and perpendicular to the fac¸ade. The target displacements for the two analyses were set according to the ultimate displacements achieved in FE model. The results reported in Fig. 14.12 show the two capacity curves obtained from the numerical simulations. In this figure, the horizontal top displacement at a monitored node versus the base shear coefficient are reported along the horizontal and vertical axes, respectively. The monitored node corresponds to the middle point of the top wall loaded in the out-of-plane direction, whereas the base shear coefficient was computed as the base shear along the load direction normalized by the self weight. As expected, the direction parallel to the fac¸ade is weaker than the perpendicular one (peak base shear coefficient equal to 0.11 vs 0.40). Despite this, it presents a much more ductile behavior (ultimate displacement equal to 200 mm vs 40 mm). In Fig. 14.13, the deformed configurations associated with the peak load and ultimate displacement are plotted with their corresponding damage patterns for the analysis in the weakest direction parallel to the main fac¸ade. Fig. 14.13A illustrates the damage pattern associated with the peak load which is mainly characterized by the failure of spandrels in the first two

522

PART | II Modeling of unreinforced masonry (A) 0.12 Base shear coefficient

0.10 0.08 0.06 0.04 DIANA

0.02

HiStrA 0.00 0

50

100

150

200

–0.02

Top displacement (mm)

Base shear coefficient

(B)

0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0 –0.05

DIANA HiStrA

10

20

30

40

50

Top displacement (mm) FIGURE 14.12 Numerical validation of the 3D macroelement through a comparison with FE results for a Gaioleiro prototype buildings (Caddemi et al., 2018). Capacity curves along the directions (A) parallel and (B) orthogonal to the fac¸ade.

FIGURE 14.13 Damage pattern at (A) the peak load and (B) at collapse for the analyses along the direction parallel to the fac¸ade (Caddemi et al., 2018).

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523

stories. The damage pattern associated with the ultimate displacement is depicted in Fig. 14.13B. In this case, it is observed that the spandrels of the upper stories present significant damage. In addition, this damage pattern was also characterized by rocking at the base of the piers. The comparison with the FE model shows good agreement of damage patterns. In the direction parallel to the fac¸ade, the damage concentrates progressively in the spandrels, from the lower to the upper stories, leading to a final damage pattern in which the overall collapse mechanism involves all the stories.

14.6.3 The shell macroelement The proposed macroelement approach was implemented in the HiStrA software, specifically devoted to historical structure analyses. The software is able to model typical masonry monumental structures with the aid of a graphical user interface that facilitates the input of the geometry and of the mechanical properties of the materials of the structure through the processing of a CAD drawing and the help of several generations of wizard tools. In Panto` et al. (2016), with the aim to provide a numerical validation for a full scale structure. The approach has been applied to an historical basilica church, characterized by the presence of arches on masonry walls and masonry columns. A similar application was reported in Panto` et al. (2017b). In this section, the capability of the shell macroelement to simulate the behavior of typical spatial curved masonry element structures is investigated. The applications discussed in the following section aim at validating the model through comparison with experimental and numerical methods. The case reported is relative to a brick masonry spherical dome with a central hole tested by Foraboschi (2006). The dome was subjected to an incremental vertical load along the edge of the central hole. Details on the experimental layout and on the mechanical properties can be found in Foraboschi (2006). The numerical model implemented to simulate the experimental tests consists of 544 quadrangular elements (17 along meridians and 32 along parallels), which correspond to a total number of degrees of freedom equal to 3808. Regarding the membrane fiber discretization, a maximum distance of the orthogonal nonlinear links equal to 5 cm along the parallels and 1.5 cm through the thickness of the dome were set, respectively. In the performed nonlinear static analysis, the model was subjected first to its self-weight, then to the external vertical load applied on the annulus of quadrilateral elements sited around the hole. The mechanical properties employed in the numerical simulations, reported in Table 14.5, were deduced by the simulations already reported in the literature (Milani et al., 2008; Milani and Tralli, 2012). The elastic properties of the masonry are represented by the Young’s modulus (E) and

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PART | II Modeling of unreinforced masonry

TABLE 14.5 Mechanical properties adopted in the numerical model. E (MPa)

σ t (MPa)

σc (MPa)

c (MPa)

μ

ν

γ (kN/m3)

850

0.07

1.9

0.12

0.37

0.25

19

(A)

(B)

(C)

60

Base reaction (kN)

50 40 30

Experiment DIANA homogeneous DIANA heterogeneous

20

DSM model QP model

10

HISTRA Milani (limit analysis)

0 0

1

2 3 Top displacement (mm)

4

5

FIGURE 14.14 Hemispherical dome (Caddemi et al., 2015): (A) failure mechanism represented in half dome; (B) damage inelastic distribution expressed in gray scale; (C) load displacement curves.

the shear Poisson’s coefficient (ν). The sliding shear failure is ruled by the cohesion (c) and the friction factor (μ). The diagonal shear behavior is considered elastic. In Fig. 14.14, the results of the nonlinear static analysis, expressed in terms of deformed shape and damage pattern at collapse, were compared to those already available in the literature. Namely, Fig. 14.14C reports the vertical top displacement as a function of the vertical load. The proposed model correctly predicts the initial stiffness and the ultimate load of the structure,

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and it is in good agreement with the available numerical results throughout all phases of the experiment. In Fig. 14.14A and B, the failure mechanism and the corresponding damage scenario at the incipient collapse condition, obtained by the numerical model implemented in HiStrA (Calio` et al., 2015), are reported. Additional details of the comparison can be found in Caddemi et al. (2015).

14.6.4 Application to masonry arch bridges A further structural typology to which the proposed approach was applied is represented by masonry arch bridges. Such structures represent a large part of the railway and road infrastructures of many countries and embeds very specific structural features to which the proposed approach was adapted, such as, the curved geometry and the 3D structural response. In order to reduce the needed effort for the implementation of the numerical model of a multiarch masonry bridge, a parametric input tool was developed considering both the complex geometry (e.g., the presence of backfill layers or the presence of tapered piers) and the automatic generation of load combinations, considering the presence of a roving vehicle load (Caddemi et al., 2019). A comparison on the results obtained on a five arches railway bridge over Esino Torrent (Italy) is here briefly summarized. The results obtained with the proposed approach were validated in the nonlinear field with those obtained with a classic nonlinear FEM approach (FEA Ltd., 2018). The adopted mechanical properties are summarized in Table 14.6, differentiated according to the structural components groups, considering the elastic modulus E, the shear modulus G, and the specific weight w. Tensile ft and compressive fc strengths of the masonry were related to a linear softening behavior governed by the corresponding fracture energies Gft and Gfc. The shear diagonal behavior is associated with a Mohr Coulomb domain characterized by a shear strength τ 0 and a friction coefficient μ 5 0.5. The two numerical models were subjected to a pushdown nonlinear analysis corresponding to a nonsymmetric vehicle load arrangement (see Fig. 14.15). The proposed approach drastically limits the required degrees of freedom (12,080 vs 349,362 in the FE model). Line loads were applied to simulate the presence of vehicles, and their intensity was magnified until the failure of the bridge. A comparison of the two models in terms of capacity curves, by monitoring the displacement of the top of the second arch, is shown in Fig. 14.16A. While the corresponding damage patterns are shown in Fig. 14.16B,C. A strong agreement between the two models is encountered considering the displacements of each of the five arches. The observed damage patterns of the two numerical models are similar as well, with significant vertical cracks on the first two piers and a spread damage on the arches.

TABLE 14.6 Mechanical properties adopted for the masonry bridge. Elements

fm (MPa)

τ 0 (MPa)

E (MPa)

G (MPa)

ft (MPa)

Gft (MPa)

Gfc (MPa)

w (kN/m3)

Abutment, pier, spandrel wall

5.8

0.4

2060

860

0.12

0.02

100

22

Masonry arches

2.6

0.3

1200

500

0.12

0.02

100

18

Backing, fill material, ballast

1.1

0.05

700

290

0.05

N

100

19

Descrete macroelement modeling Chapter | 14

527

FIGURE 14.15 Railway bridge (Caddemi et al., 2019): (A) layout of the applied loads and numerical models according to (B) proposed and (C) the FE approaches.

FIGURE 14.16 Comparison in terms of (A) capacity curves and damage pattern at collapse according to (B) proposed and (C) the FE approaches.

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PART | II Modeling of unreinforced masonry

14.7 Summary and conclusions In this chapter, a numerical strategy focused on simulating the nonlinear behavior of masonry structures is presented. The proposed numerical model, which belongs to the framework of the simplified models, is based on a simple mechanical scheme that consists of a hinged quadrilateral, endowed with a diagonal link to govern the in-plane diagonal shear behavior, and contouring interfaces that rule the interaction with contiguous elements. The proposed approach appears to be a fair compromise between oversimplified models (e.g. EFMs) and accurate models based on cumbersome strategies, which require an expert interpretation of the results. The basic model, originally conceived for the simulation of masonry walls loaded in their own plane, was repeatedly upgraded, progressively increasing the structural typologies that the proposed strategy is able to model. Within the scope of the numerical simulation of ordinary buildings with box behavior (the out-of-plane behavior is considered inhibited), interaction with frames contouring a masonry panel was enabled, thus allowing the numerical simulation of both URM and infilled masonry structures. With the goal of accurate numerical modeling of HMS, two further upgrades were considered. First, the out-of-plane degrees of freedom were enabled to assess the out-of-plane behavior of masonry walls. Then, a further improvement allowed simulating masonry structures with a curved geometry. Finally, by ruling the interaction between structural elements in correlation with their intersections, full nonlinear simulations of large historical masonry constructions were performed. The progressive improvements were obtained by simply extending the calibration procedure of the links according to the different peculiarities of the model at the various stages of complexity. However, the philosophy of the model was kept the same for all contemplated advances of the model; that is, the calibration is always straightforward and based on the same concepts. Some simple validations of the model were presented consistently with each of the described stage. The results show that the proposed strategy appears to be reliable in all the considered cases and that it represents an original approach to the nonlinear assessment of ordinary masonry buildings, historical and monumental structures.

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Araujo, A., Lourenc¸o, P.B., Oliveira, D., Leite, J., 2012. Seismic assessment of St James Church by means of pushover analysis before and after the New Zealand earthquake. Open Civil Eng. J. 6, 160 172. Available from: https://doi.org/10.2174/1874149501206010160. Asteris, P.G., Antoniou, S.T., Sophianopoulos, D., Chrysostomou, C.Z., 2011. Mathematical macromodeling of infilled frames: state of the art. J. Struct. Eng. 137, 1508 1517. Available from: https://doi.org/10.1061/(ASCE)ST.1943-541X.0000384. Asteris, P.G., Chronopoulos, M.P., Chrysostomou, C.Z., Varum, H., Plevris, V., Kyriakides, N., et al., 2014. Seismic vulnerability assessment of historical masonry structural systems. Eng. Struct. 62-63, 118 134. Available from: https://doi.org/10.1016/j. engstruct.2014.01.031. Barbieri, G., Biolzi, L., Bocciarelli, M., Fregonese, L., Frigeri, A., 2013. Assessing the seismic vulnerability of a historical building. Eng. Struct. 57, 523 535. Available from: https://doi. org/10.1016/j.engstruct.2013.09.045. Belmouden, Y., Lestuzzi, P., 2009. An equivalent frame model for seismic analysis of masonry and reinforced concrete buildings. Constr. Build. Mater. 23 (1), 40 53. Berto, L., Saetta, A., Scotta, R., Vitaliani, R., 2002. Orthotropic damage model for masonry structures. Int. J. Numer. Methods Eng. 55, 127 157. Available from: https://doi.org/ 10.1002/nme.495. Betti, M., Vignoli, A., 2011. Numerical assessment of the static and seismic behaviour of the basilica of Santa Maria all’Impruneta (Italy). Constr. Build. Mater. 25, 4308 4324. Available from: https://doi.org/10.1016/j.conbuildmat.2010.12.028. Betti, M., Vignoli, A., 2008. Assessment of seismic resistance of a basilica-type church under earthquake loading: modelling and analysis. Adv. Eng. Soft. 39, 258 283. Available from: https://doi.org/10.1016/j.advengsoft.2007.01.004. Braga, F., Dolce, M., 1982. ‘Un metodo per l’analisi di edifici multipiano in muratura antisismici. In: Proc. of the 6th I.B.MA.C International Brick Masonry Conference, Rome. Brenchich, G., Gambarotta, L., Lagomarsino, S., 1998. A macro-element approach to the threedimensional seismic analysis of masonry buildings. In: Proceedings of 11th European Conference on Earthquake Engineering. A. A. Balkema, Paris, Rotterdam, p. 602. Caddemi, S., Calio`, I., Cannizzaro, F., Panto`, B., 2013. A new computational strategy for the seismic assessment of infilled frame structures. In: Topping, B.H.V., Iv´anyi, P. (Eds.), Proceedings of the Fourteenth International Conference on Civil, Structural and Environmental Engineering Computing. Civil-Comp Press, Stirlingshire, Paper 77. Caddemi, S., Calio`, I., Cannizzaro, F., Panto`, B., 2014. The seismic assessment of historical masonry structures. In: Topping, B.H.V., Iv´anyi, P. (Eds.), Proceedings of the Twelfth International Conference on Computational Structures Technology. Civil-Comp Press, Stirlingshire, Paper 78. Caddemi, S., Calio`, I., Cannizzaro, F., Occhipinti, G., Panto`, B., 2015. A parsimonious discrete model for the seismic assessment of monumental structures. In: Kruis, J., Tsompanakis, Y., Topping, B.H.V. (Eds.), Proceedings of the Fifteenth International Conference on Civil, Structural and Environmental Engineering Computing, Civil-Comp Press, Stirlingshire, Paper 82. Caddemi, S., Calio`, I., Cannizzaro, F., Chacara, C., D’Urso, D., Liseni, S., et al., 2018. An original discrete macro-element method for the analysis of historical structures. In: Proceedings of the 16th European Conference on Earthquake Engineering, Thessaloniki, Greece, 18 21 June 2018.

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Caddemi, S., Calio`, I., Cannizzaro, C., D’Urso, D., Occhipinti, G., Panto`, B., et al., 2019. 3D discrete macro-modelling approach for masonry arch bridges. In: IABSE Symposium 2019, Guimara˜es (Portugal), 27 29 March 2019. Calio`, I., Panto`, B., 2014. A macro-element modelling approach of infilled frame structures. Comput. Struct. 143, 91 107. Available from: https://doi.org/10.1016/j.compstruc.2014.07.008. Calio`, I., Marletta, M., Panto`, B., 2004. Un semplice macro-elemento per la valutazione della vulnerabilita` sismica di edifici in muratura. In: atti dell’XI congresso nazionale l’Ingegneria Sismica in Italia, Genova 2004 (in Italian). Calio`, I., Marletta, M., Panto`, B., 2005. A simplified model for the evaluation of the seismic behaviour of masonry buildings. In: Topping, B.H.V. (Ed.), Proceedings of the Tenth International Conference on Civil, Structural and Environmental Engineering Computing, Civil-Comp Press, Stirlingshire, 195. Calio`, I., Cannizzaro, F., D’Amore, E., Marletta, M., Panto`, B., 2008. A new discrete-element approach for the assessment of the seismic resistance of composite reinforced concretemasonry buildings. In: AIP Conference Proceedings, 1020 (PART 1), 24 27 June 2008, Reggio Calabria, pp. 832 839. Calio´, I., Cannizzaro, F., Marletta, M., 2010. A discrete element for modeling masonry vaults. Adv. Mater. Res. 133 134, 447 452. Available from: https://doi.org/10.4028/www.scientific.net/AMR.133-134.447. Calio`, I., Marletta, M., Panto`, B., 2012a. A new discrete element model for the evaluation of the seismic behaviour of unreinforced masonry buildings. Eng. Struct. 40, 327 338. Available from: https://doi.org/10.1016/j.engstruct.2012.02.039. Calio`, I., Cannizzaro, F., Marletta, M., Panto`, B., 2012b. 3DMacro: A 3D Computer Program for the Seismic Assessment of Masonry Buildings. Gruppo Sismica s.r.l, Catania. Calio`, I., Cannizzaro, F., Panto`, B., Rapicavoli, D., 2015. HiStrA (historical structure analysis). In: HISTRA s.r.l (Catania, Italy). Release 17.2.3; April 2015. Available from: ,http://www. grupposismica.it.. Cannizzaro, F., 2010. The Seismic Behaviour of Historical Buildings: A Macro-Element Approach (Ph.D. thesis). Structural Engineering, University of Catania (in Italian). Cannizzaro, F., Lourenc¸o, P.B., 2017. Simulation of shake table tests on out-of-plane masonry buildings. Part (VI): discrete element approach. Int. J. Architect. Herit. 11, 125 142. Cannizzaro, F., Panto`, B., Caddemi, S., Calio`, I., 2018. A Discrete Macro-Element Method (DMEM) for the nonlinear structural assessment of masonry arches. Eng. Struct. 168, 243 256. Available from: https://doi.org/10.1016/j.engstruct.2018.04.006. Casolo, S., Pen˜a, F., 2007. Rigid element model for in-plane dynamics of masonry walls considering hysteretic behaviour and damage. Earthq. Eng. Struct. Dyn. 36, 1029 1048. Available from: https://doi.org/10.1002/eqe.670. Casolo, S., Sanjust, C.A., 2009. Seismic analysis and strengthening design of a masonry monument by a rigid body spring model: the “Maniace Castle” of Syracuse. Eng. Struct. 31, 1447 1459. Available from: https://doi.org/10.1016/j.engstruct.2009.02.030. Ch´acara, C., Cannizzaro, F., Panto`, B., Calio`, I., Lourenc¸o, P.B., 2018. Assessment of the dynamic response of unreinforced masonry structures using a macro-element modeling approach. Earthq. Eng. Struct. Dyn. 47 (12), 2426 2446. Chen, S.Y., Moon, F.L., Yi, T., 2008. A macroelement for the nonlinear analysis of in-plane unreinforced masonry piers. Eng. Struct. 30 (8), 2242 2252. Available from: https://doi.org/ 10.1016/j.engstruct.2007.12.001.

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Dolce, M., 1991. Schematizzazione e modellazione degli edifici in muratura soggetti ad azioni sismiche. L’Industria delle costruzioni 25 (242), 44 57. in Italian. FEA Ltd., 2018. LUSAS Theory Manuals, Lusas Version 16.0. FEA Ltd. Foraboschi, P., 2006. Masonry structures externally reinforced with FRP strips: tests at the collapse. In: Proceedings of I Convegno Nazionale Sperimentazioni su Materiali e Strutture (Venice) (in Italian). Gambarotta, L., Lagomarsino, S., 1997. Damage models for the seismic response of brick masonry shear walls. Part II: the continuum model and its applications. Earthq. Eng. Struct. Dyn. 26, 441 462. Available from: https://doi.org/10.1002/(SICI)1096-9845(199704)26:4 , 423::AIDEQE650 . 3.0.CO;2-#. Hilsdorf, H.K., 1969. Investigation into the failure mechanism of brick masonry loaded in axial compression. Designing, Engineering and Constructing With Masonry Products. Gulf Publishing Company, pp. 34 41. Kappos, A.J., Penelis, G.G., Drakopoulos, C.G., 2002. Evaluation of simplified models for lateral load analysis of unreinforced masonry buildings. J. Struct. Eng. 128, 890 897. Available from: https://doi.org/10.1061/(ASCE)0733-9445(2002)128:7(890). Lagomarsino, S., Penna, A., Galasco, A., Cattari, S., 2013. TREMURI program: an equivalent frame model for the nonlinear seismic analysis of masonry buildings. Eng. Struct. 56, 1787 1799. Available from: https://doi.org/10.1016/j.engstruct.2013.08.002. Lofti, H.R., Shing, P.B., 1994. Interface model applied to fracture of masonry structures. J. Struct. Eng. 120, 63 80. Available from: https://doi.org/10.1061/(ASCE)0733-9445(1994)120:1(63). Lourenc¸o, P.B., Rots, J.G., 1997. A multi-surface interface model for the analysis of masonry structures. J. Eng. Mech. 123, 660 668. Available from: https://doi.org/10.1061/(ASCE) 0733-9399(1997)123:7(660). Lourenc¸o, P.B., Rots, J.G., Blaauwendraad, J., 1998. Continuum model for masonry: parameter estimation and validation. J. Struct. Eng. 124, 642 652. Available from: https://doi.org/ 10.1061/(ASCE)0733-9445(1998)124:6(642). Lourenc¸o, P.B., Nuno Mendes, A.T., Ramos, L.F., 2012. Seismic performance of the St. George of the Latins church: lessons learned from studying masonry ruins. Eng. Struct. 40, 501 518. Available from: https://doi.org/10.1016/j.engstruct.2012.03.003. Macorini, L., Izzuddin, B.A., 2011. A non-linear interface element for 3D mesoscale analysis of brick-masonry structures. Int. J. Numer. Methods Eng. 85, 1584 1608. Available from: https://doi.org/10.1002/nme.3046. Magenes, G., Calvi, G.M., 1996. Prospettive per la calibrazione di metodi semplificati per l’analisi sismica di pareti murarie. Atti del Convegno Nazionale La meccanica delle murature tra teoria e progetto, Ed. Pitagora Bologna, 18-20 September 1996. Messina 503 512. Magenes, G., Della Fontana, A., 1998. Simplified nonlinear seismic analysis of masonry buildings. Br. Mason. Soc. Proc. 8, 190 195. Marques, R., Lourenc¸o, P.B., 2011. Possibilities and comparison of structural component models for the seismic assessment of modern unreinforced masonry buildings. Comput. Struct. 89, 2079 2091. Available from: https://doi.org/10.1016/j.compstruc.2011.05.021. Marques, R., Lourenc¸o, P.B., 2014. Unreinforced and confined masonry buildings in seismic regions: validation of macro-element models and cost analysis. Eng. Struct. 64, 52 67. Available from: https://doi.org/10.1016/j.engstruct.2014.01.014. Mele, E., De Luca, A., Giordano, A., 2003. Modelling and analysis of a basilica under earthquake loading. J. Cult. Herit. 4, 355 367. Available from: https://doi.org/10.1016/j.culher.2003.03.002.

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Mendes, N., 2012. Seismic Assessment of Ancient Masonry Buildings: Shaking Table Tests and Numerical Analysis (Ph.D. thesis). Civil Engineering, University of Minho. Mendes, N., Lourenc¸o, P.B., 2009. Seismic assessment of masonry “Gaioleiro” buildings in Lisbon, Portugal. J. Earthq. Eng. 14, 80 101. Mendes, N., Lourenc¸o, P.B., Campos-Costa, A., 2014. Shaking table testing of an existing masonry building: assessment and improvement of the seismic performance. Earthq. Eng. Struct. Dyn. 43 (2), 247 266. Milani, G., Tralli, A., 2012. A simple meso-macro model based on SQP for the non-linear analysis of masonry double curvature structures. Int. J. Solids Struct. 46, 808 834. Available from: https://doi.org/10.1016/j.ijsolstr.2011.12.001. Milani, G., Valente, M., 2015. Failure analysis of seven masonry churches severely damaged during the 2012 Emilia-Romagna (Italy) earthquake: non-linear dynamic analyses vs conventional static approaches. Eng. Fail. Anal. 54, 13 56. Available from: https://doi.org/ 10.1016/j.engfailanal.2015.03.016. Milani, E., Milani, G., Tralli, A., 2008. Limit analysis of masonry vaults by means of curved shell finite elements and homogenization. Int. J. Solids Struct. 45, 5258 5288. Available from: https://doi.org/10.1016/j.ijsolstr.2008.05.019. NTC, 2008. Decreto Ministeriale. Norme tecniche per le costruzioni. Ministry of Infrastructures and Transportations. G.U. S.O. n.30 on 4/2/2008; 2008 (in Italian). Panto`, B., 2007. The Seismic Modeling of Masonry Structure, an Innovative Macro-Element Approach (PhD Thesis). Structural Engineering, University of Catania, Catania (in Italian). Panto`, B., Raka, E., Cannizzaro, F., Camata, G., Caddemi, S., Spacone, E., et al., 2015. Numerical macro-modeling of unreinforced masonry structures: a critical appraisal. In: Topping, B.H.V., Iv´anyi, P. (Eds.), Proceedings of the Fifteenth International Conference on Civil, Structural and Environmental Engineering Computing. Civil-Comp Press, Stirlingshire. Panto`, B., Cannizzaro, F., Caddemi, S., Calio`, I., 2016. 3D macro-element modelling approach for seismic assessment of historical masonry churches. Adv. Eng. Soft. 97, 40 59. Available from: https://doi.org/10.1016/j.advengsoft.2016.02.009. Panto`, B., Cannizzaro, F., Calio`, I., Lourenc¸o, P.B., 2017a. Numerical and experimental validation of a 3D macro-model element method for the in-plane and out-of-plane behaviour of unreinforced masonry walls. Int. J. Architect. Herit. 11 (7), 946 964. Available from: https://doi.org/10.1080/15583058.2017.1325539. Panto`, B., Giresini, L., Sassu, M., Calio`, I., 2017b. Non-linear modeling of masonry churches through a discrete macro-element approach. Earthq. Struct. 12, 223 236. Available from: https://doi.org/10.12989/eas.2017.12.2.223. Panto`, B., Calio`, I., Lourenc¸o, P.B., 2018. A 3D discrete macro-element for modelling the outof-plane behaviour of infilled frame structures. Eng. Struct. 175, 371 385. Available from: https://doi.org/10.1016/j.engstruct.2018.08.022. Penelis, G.G., 2006. An efficient approach for pushover analysis of unreinforced masonry (URM) structures. J. Earthq. Eng. 10 (03), 359 379. Quagliarini, E., Maracchini, G., Clementi, F., 2017. Uses and limits of the Equivalent Frame Model on existing unreinforced masonry buildings for assessing their seismic risk: a review. J. Build. Eng. 10, 166 182. Available from: https://doi.org/10.1016/j.jobe.2017.03.004. Raka, E., Spacone, E., Sepe, V., Camata, G., 2015. Advanced frame element for seismic analysis of masonry structures: model formulation and validation. Earthq. Eng. Struct. Dyn. 44, 2489 2506. Available from: https://doi.org/10.1002/eqe.2594.

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Roca, P., Molins, C., Mar´ı, A.R., 2005. Strength capacity of masonry wall structures by the equivalent frame method. J. Struct. Eng. 131 (10), 1601 1610. Siano, R., Roca, P., Camata, G., Pela`, L., Sepe, V., Spacone, E., et al., 2018. Numerical investigation of non-linear equivalent-frame models for regular masonry walls. Eng. Struct. 173, 512 529. Tomazevic, M., 1978. The Computer Program POR: Institute for Testing and Research in Materials and Structures. ZRMK, Ljubljana. ˇ coviˇc, F., 1970. Some experimental results on the strength of brick masonry Turnˇsek, V., Caˇ walls. In: Proceedings of the 2nd International Brick & Block Masonry Conference, Stokeon-Trent, pp. 149 156. Valente, M., Milani, G., 2016. Seismic assessment of historical masonry towers by means of simplified approaches and standard FEM. Constr. Build. Mater. 108, 74 104. Available from: https://doi.org/10.1016/j.conbuildmat.2016.01.025. Zavala, C., Honma, C., Gibu, P., Gallardo, J., Huaco, G., 2004. Full scale on line test on two story masonry building using handmade bricks. In: Proceedings of the 13th World Conference on Earthquake Engineering, Vancouver, p. 2885.

Further reading Lourenc¸o, P.B., 2002. Computations on historic masonry structures. Prog. Struct. Eng. Mater. 4, 301 319. Available from: https://doi.org/10.1002/pse.120.

Chapter 15

Fiber reinforced polymer strengthened masonry: delamination, experimental and numerical issues Roberto Capozucca1, Ernesto Grande2 and Gabriele Milani3 1 3

Polytechnic University of Marche, Ancona, Italy, 2University Guglielmo Marconi, Rome, Italy, Politecnico di Milano, Milano, Italy

15.1 Introduction The most critical phenomenon influencing the effectiveness of interventions based on the use of fiber-reinforced polymer (FRP) systems is the interaction between the reinforcing system and substrate. Thus this characterization is essential to understand the involved phenomena and, consequently, to predict and eventually improve, the performance of strengthening systems. The characterization of the bond behavior of FRP strengthening systems is mainly based on experimental tests and numerical/theoretical studies, both available in the literature and still the aim of current research. Regarding the experimental characterization of the local bond behavior, most of the recent literature concerns shear lap bond tests on concrete and masonry specimens externally strengthened with FRP applied in the form of strips or sheets. These studies point out the centrality of the bond behavior in the global performance of strengthened specimens. They also underline the role of different factors, such as the type of the strengthening system, the characteristics of the material composing the substrate, the modalities of application of the strengthening system, and the influence of the mortar joints in the case of masonry substrates, on the debonding mechanism. For instance, the tests performed by Aiello and Sciolti (2003) emphasized the influence of the material properties composing the substrate on the ultimate load of strengthened specimens. Indeed, it was observed that specimens made by “Leccese stone” showed values of bond strength significantly greater than those for tuff specimens. On the other hand, this study Numerical Modeling of Masonry and Historical Structures. DOI: https://doi.org/10.1016/B978-0-08-102439-3.00015-4 Copyright © 2019 Elsevier Ltd. All rights reserved.

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underlined that the strain distribution of the reinforcement is influenced by the kind of stone only for loads approaching the failure value. In Aiello and Sciolti (2006) and in Grande et al. (2011) the effect of the bond length on the type of FRP failure mechanism was emphasized. In both studies a failure mechanism characterized by the detachment of a thicker stone layer in the case of specimens with short values of the bond length and a mechanism where the delamination of the sheets involved the detachment of a thin layer of stone in the case of specimens with high bond length values emerged. The tests carried out by Briccoli Bati et al. (2006) concerning clay bricks strengthened by FRPs underlined the important role of the setup configuration for the experimental characterization of the local bond behavior. This aspect was recently investigated in the context of the RILEM Technical Committee 223-MSC “Masonry Strengthening With Composite Materials” (TC 223-MSC), with the aim of developing a standardized and reliable procedure to study the debonding mechanism of masonry elements strengthened by composite materials. The study particularly focused on the identification of parameters useful for harmonizing laboratory experimental procedures (Valluzzi et al., 2012). Regarding the numerical/theoretical models carried out for the study of the bond behavior of FRPs externally applied on structural substrates, some authors proposed different numerical/theoretical models mainly based on the experimental outcomes of shear lap bond tests. Indeed, the main assumption characterizing most of these models is to assume damage due to the debonding phenomenon only affecting the thin layer of the material placed between the FRP strengthening and the substrate. This allowed to carry out finite element (FE) models based on the use of zero-thickness interface elements (Grande et al., 2008, 2013; Fedele and Milani, 2012; Ghiassi et al., 2013). On the other hand, other authors have proposed numerical modeling approaches based on the assumption of perfect adhesion between FRPs and the substrate (e.g., Fedele and Milani, 2010, 2011; Basilio et al., 2014). Indeed, the authors underlined the main advantage in not considering the parameters of the masonry/FRP interface since the debonding mechanism is directly modeled as a damage phenomenon affecting the material composing the substrate. At the same time, the same authors also underlined that such an approach could be hard to tackle, because of the complex damage plasticity models needed to properly describe crack propagation into the substrate. This chapter mainly aims at analyzing the main features of the bond behavior of FRPs externally applied on structural substrates from both experimental tests and numerical models carried out by different authors.

15.2 Experimental evidence Investigations suggest that the common failure mode of FRP-to-masonry joints is the delamination under shear occurring at a plane located a few

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millimeters from the surface of the masonry (Aiello and Sciolti, 2008; Capozucca, 2010; Valluzzi et al., 2012). In this section of the chapter, experimental results on the bond between FRP strips and historic brickwork are presented. The experimental results allow us to obtain strain versus anchorage length diagrams and shear stress versus slip relationships to evaluate the energy fracture value. In agreement with the structure and the contents of the majority of the studies available in the current literature, in this section we discuss both a theoretical elastic model, developed to describe the bond mechanism between FRP and masonry, and a description of the experimental evidence of some tests.

15.2.1 Theoretical bond analysis The ultimate load of FRP strips externally bonded to masonry substrates is strongly affected by the fracture energy, Gf (Tsai et al., 1995, 1998). The following presents a theoretical bond analysis with an appraisal model to predict the capacity of external bond strengthening by FRP strips on brickwork (Da Silva et al., 2009a). Fig. 15.1A shows a schematization of an FRP strip-to-brickwork bonded joint subjected to tensile load P1. The FRP strip, adherent 1, is subjected to axial and shear strains; clay brick, adherent 2, is equivalent to a rigid element while a porosity clay layer filled with polymer adhesive is an ideal L

(A)

1 x

FRP

P1

t1 tm

m t2

Brick 2

(B)

u1,bottom = u1,a

dx

u1,top

σ1(x) + dσ1(x) 1 m

y

FRP

σ1(x)

τ1 = 0

u1,bottom

τa(x) τa(x)

2

Brick

τ2 ≅ 0

FIGURE 15.1 (A) Specimen of FRP strip-to-brickwork bonded joint under pull test with (B) displacement and shear stress distribution.

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intermediate element “m” subjected to a constant shear stress. The intermediate layer “m” may be characterized by mechanical parameters, Em and Gm, respectively, Young’s modulus and shear modulus, and geometric dimensions bm, tm, respectively, width and thickness. The two adherents are made of elastic material. Fig. 15.1B shows an infinitesimal portion of the FRP strip-to-brickwork bonded joint that also shows the shear stress distribution τ 1(x,y) and displacement function u1(x,y) through the thicknesses t1 and tm. The shear stress denoted τ a is constant along the whole thickness of intermediate element tm. For a generic section, the value of the force resulting from the internal normal stresses along dy, considering an unitary depth of the adherent 1, can be written as: ð t1 P1 ðxÞ 5 σ1 Udy ð15:1Þ 0

The equilibrium of adherent 1 along the section dx is given by: ð t1 ð t1 σ1 Ub1 Udy 1 τ a ðxÞUb1 Udx 2 ðσ1 1 dσ1 ÞUb1 Udy 5 0 0

ð15:2Þ

0

From which: τ a ðxÞUdx 2

ð t1

dσ1 Udy 5 0

ð15:3Þ

0

Considering Eq. (15.1) and differentiating by x, we obtain: ð t1 dP1 ðxÞ 5 dσ1 Udy

ð15:4Þ

0

Introducing Eq. (15.4) in Eq. (15.3), the following differential equation can be obtained: dP1 2 τ a ðxÞ 5 0 dx

ð15:5Þ

Furthermore, shear stress along the adherent 1 is given by: τ 1 ðx; yÞ 5

τ a ðxÞ Uy t1

ð15:6Þ

τa Uy G1 Ut1

ð15:7Þ

with the shear strain equal to: γ 1 ðx; yÞ 5

Assuming negligible shear strain γ 1 ðx; yÞD0 in the adherent 1, along the whole thickness t1, the following displacement function is obtained: ðy u1 ðx; yÞ 5 u1;top ðxÞ 2 γ 1 ðx; yÞUdy ð15:8Þ 0

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which can be simplified as: u1 ðx; yÞDu1 ðxÞDu1; top ðxÞDu1; bottom ðxÞ From Eq. (15.1), it is: P1 ðxÞ 5

ð t1

E1 U

0

du1 ðx; yÞ du1 ðxÞ UdyDE1 U Ut1 dx dx

ð15:9Þ

ð15:10Þ

Considering Eq. (15.5), it can be written: E1 Ut1 U

d 2 u1 ðxÞ 2 τ a ðxÞ 5 0 dx2

ð15:11Þ

Since shear strain along the intermediate element m can be written as: γ m ðxÞ 5

u1 ðxÞ tm

ð15:12Þ

and shear stress is equal to: τ a ðxÞ 5 Gm U

u1 ðxÞ tm

ð15:13Þ

Eq. (15.11) can be written as: E1 Ut1 Utm d2 Uτ a ðxÞ U 2 τ a ðxÞ 5 0 dx2 Gm

ð15:14Þ

Finally, introducing a dimensional coefficient β 2 equal to: β2 5

Gm E1 Ut1 Utm

ð15:15Þ

Eq. (15.14) can be written as: d2 Uτ a ðxÞ 2 β 2 Uτ a ðxÞ 5 0 dx2

ð15:16Þ

The solution of Eq. (15.16) is given by the following function: τ a ðxÞ 5

P1 coshðβxÞ UβU sinhðβLÞ b1

ð15:17Þ

The strain at the top of the adherent 1 may be expressed as: ε1;top D

du1;a du1 ðxÞ 5 dx dx

ð15:18Þ

Considering Eq. (15.13), is it possible to write: ε1;top 5

tm P1 2 sinhðβxÞ U Uβ U sinhðβLÞ Gm b1

ð15:19Þ

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PART | III Modeling of fiber-reinforced polymer
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The maximum value of strain at the edge of the joint is: ε1;top ðx 5 LÞ 5

tm P1 2 U Uβ G m b1

ð15:20Þ

The elastic stage of deformation of the bonded joints ends when the shear stress reaches the local shear strength τ max at the slip of umax, for x 5 L. Considering L as the maximum value of bond length, it is possible to write: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P1 P1 P1 Gm τ max ðx 5 LÞ 5 ð15:21Þ UβUctghðβLÞD Uβ 5 U b1 b1 b1 E1 Ut1 Utm Assuming the interfacial fracture energy value, Gf, as: Gf 5

1 tm 1 tm 2 1 tm P21 Gm P21 Uτ max 5 U Uτ max 5 U U 2U 5 Uτ max U Gm 2 2 Gm 2 Gm b1 E1 Ut1 Utm 2Ub21 UE1 Ut1 ð15:22Þ

The value of the load capacity can be written as follows: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P1 5 b1 U 2UE1 Ut1 UGf

ð15:23Þ

where Gf is the interfacial fracture energy (i.e., the total external energy supply per unit of area) required to create and propagate we fully break a crack along the FRP strip-to-brickwork masonry without considering the mortar joints in the masonry.

15.2.2 Experimental tests on fiber-reinforced polymer strip-to-brickwork joints Regarding the experimental outcomes generally observed during shear lap bond tests, in this chapter we analyze the bond behavior of glass FRP (GFRP) strip (Capozucca and Ricci, 2016), carbon FRP (CFRP), and steel FRP (SFRP) (Capozucca, 2010) applied on masonry substrates. Experimental values of the fracture energy for each type of FRP strip to brickwork are examined in detail.

15.2.2.1 Glass fiber
reinforced polymer strip-to-brickwork joints The behavior of GFRP strip-to-brickwork joints was tested in Capozucca and Ricci (2016) by pull
push tests. In the investigation, historical masonry wallets built with handmade solid clay bricks and weak mortar with different values of thickness of bed joints were examined. The composite materials used for tests were FIDGLASS UNIDIR 300 HS 73. To know the characteristics of GFRP, specimens were subjected to tensile tests (ASTM D 3039—UNI EN ISO 527-1). Table 15.1 shows the main results of the tensile tests on the GFRP specimens.

Fiber reinforced polymer strengthened masonry Chapter | 15

543

TABLE 15.1 Experimental data of tensile tests on GFRP specimens. FIDGLASS UNIDIR 300 HS 73

a

Specimen

Pmax (N)

Af (mm2)

σft (N/ mm2)

Eft,exp. (GPa)

Type of rupturea

G1

6365

16.82

1229.524




AGM

G2

7169

18.45

1317.056

64.03

AGM

AGM 5 angled, gage, middle—ASTM D 3039.

TABLE 15.2 Experimental mechanical parameters of brickwork. Compressive strength of bricks, fb (N/mm2)

Compressive strength of brickwork, fwb (N/mm2)

Average compressive strength of mortar, fm (N/mm2)

Bending tensile strength of mortar, fm,t (kN/mm2)

15.4
28.0

8.22

12.0

3.4

FIGURE 15.2 Wallettes with different thickness of mortar bed joints: W1 with 4 mm, W2 with 8 mm, and W3 with 12 mm.

Experimental preliminary tests on wallettes and materials, brick and mortar, adopted in the specimens were also carried out by the authors and the main results are given in Table 15.2. The wallettes prepared for the investigation were 12 historic clay solidbrick specimens (Fig. 15.2): four historic specimens made for each thickness of mortar bed joints: 4, 8, and 12 mm; nine wallettes made by historic clay bricks reinforced by GFRP strips; and the others were subjected to compressive tests. In Table 15.2 the mechanical parameters of the materials, mortar and bricks, together with the compression strength of the wallettes, are summarized.

544

PART | III Modeling of fiber-reinforced polymer
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The GFRP strip specimens were characterized by a width of 50 mm, applied on one of the two surfaces of the wallets. Before applying the GFRP strips, the masonry surface was cleaned, smoothed, and covered by a bicomponent primer (type MBRACE PRIMER) along the zone to apply the strip; epoxy resin (KIMITECH EP-IN) was adopted to glue the GFRP strips. The epoxy resin presented an average tensile strength equal to fresB30 N/mm2 and Young’s modulus Eres 5 1760 N/mm2. The length of adhesion of GFRP strip was equal to 280 mm for specimens with 4 mm of mortar layers; 292 mm for the ones with 8 mm of mortar layers; and 294mm for the ones with 12 mm layers. Six strain gauges were located on the strip with different intervals (Fig. 15.3.A). The setup of tests is shown in (Fig. 15.3.B): the specimens were fixed by an anchorage system made by steelplates and clamps inside a steel frame; the load P was transferred to the GFRP strip that was connected to the load cell by a system of metallic plates. The instruments used in the tests were able to measure continuously both vertical load with load cell and strains on GFRP strips (Fig. 15.3). In Table 15.3 the values of the failure loads obtained from the pull
push tests on the historic brickwork specimens showed typical mechanisms of failure (delamination or delamination and GFRP rupture). The average failure load value was 8850 N for specimens with 4 mm of mortar layers, 8280 N for specimens with 8 mm of mortar layers, and 8470 N for specimens with 12 mm of mortar layers. For this reason, it is possible to affirm that, since the values for the failure loads were very similar, there was no evident difference between the three kinds of specimens. The delamination of composite materials was recorded in many specimens, although sometimes accompanied with rupture of bricks on the edge; delamination of GFRP strips due to a detachment of a surface layer of brick with successive compressive failure of brick on the loaded edge (Fig. 15.4). The mechanism of failure was influenced by the characteristics of strength, porosity, clay composition, and mode of execution of historic bricks, which are different although taken from the same building. Small parts of the tested specimens were observed by an electronic microscope observing the thickness of the intermediate layer equal to about 1.2 mm. During the pull
push shear tests of every specimen, strain values were recorded along the bonded portion of strips at different values of load P until failure. The diagrams of strain values of GFRP-to-historic brickwork bonded joints are shown in Fig. 15.5 for one type of wallettes with different thickness of mortar. By observing the experimental diagrams of strain along the bond length of GFRP strip, for both kinds of specimens, it is possible to understand the starting point of the delamination process. The detachment of GFRP strips from the support starts at the point where the diagram changes slope and continues until the complete detachment of the GFRP strip or its rupture.

Fiber reinforced polymer strengthened masonry Chapter | 15

545

FIGURE 15.3 (A) Historic wallets with GFRP strips and strain gauges and (B) setup of pull
push test.

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PART | III Modeling of fiber-reinforced polymer
strengthened masonry

TABLE 15.3 Experimental results of GFRP strip-to-historic brickwork with different thickness of mortar layer. Wallet

Thickness of mortar layer, h (mm)

Load, Pu (N)

Strength, σf,max (N/mm2)

Mechanism of failurea

W1A

4.00

10735

1789

D 1 GFR

W1B

4.00

9632

1605

D

W1C

4.00

6179

1030

D 1 GFR

W2A

8.00

8107

1351

D 1 GFR

W2B

8.00

8875

1479

D 1 GFR

W2C

8.00

7854

1309

D

W3A

12.00

6395

1066

D 1 GFR

W3B

12.00

9485

1581

D

W3C

12.00

9532

1589

D 1 GFR

a

D 5 delamination; D 1 GFR 5 delamination 1 glass fiber rupture.

FIGURE 15.4 (A and B) View of GFRP strips after test with typical delamination.

Maximum strains recorded during the tests assumed a value equal to about ε  7.5 3 1023 without relevant difference for different thickness of mortar joints. Fig. 15.6 shows the experimental interfacial shear stress, τ, versus slip, δ, diagrams for the tested specimens with thickness of mortar 4 mm of historic brickwork wallets. The diagrams are not linear with maximum values of shear stress τ f  1.6
1.8 N/mm2 related to slip δf  0.23
0.32 mm.

Fiber reinforced polymer strengthened masonry Chapter | 15 W1A

Strain (10–6)

Strain (10–6) 9000 P1 = 2552 N 8000 P2 = 4554 N 7000 P3 = 6247 N 6000 P4 = 8261 N 5000 P5 = 10,126 N 4000 P6 = 10,733 N 3000 2000 1000 0 0 200 250

9000 8000 7000 6000 5000 4000 3000 2000 1000 0 0

50

100 150 xi (mm)

547

W2A P1 = 3528 N P2 = 6748 N P3 = 7152 N P4 = 7188 N P5 = 7643 N P6 = 8107 N

50

100 150 xi (mm)

200

250

W3A

Strain (10–6) 9000 8000 7000

P = 2526 N P = 5754 N P = 6504 N

6000 5000 4000

P = 7130 N P = 7115 N P = 5754 N

3000 2000 1000 0

0

50

150 100 xi (mm)

200

250

FIGURE 15.5 Experimental strain values recorded for specimens made with historic bricks and thickness of mortar layers equal to 4 mm-W1A, 8 mm-W2A, and 12 mm-W3A.

W1A

τ (MPa) 1.8

τf

1.6

P = 10,157 N

1.4

1.6

1.2

1.4 1.2

1

0.8 0.6

0.6 0.4 0.2 0

P = 7687 N

1

0.8

–0.2

W1B

τ (MPa) 2.0 τf 1.8

0

0.1

0.2

0.3

δf 0.4

δ (mm)

0.5

0.6

0.4 0.2 0 –0.2

0

0.1

0.2 δf

0.3

0.4

0.5

δ (mm)

FIGURE 15.6 Experimental interface diagram shear stress, τ, versus slip, δ, for tested historic brickwork wallettes W1A
W1B with width of mortar joints equal to 4 mm.

In Table 15.4, the ultimate load values both experimental, Pu, and theoretical, P1, slip values δf, maximum value of shear stress values, τ f, and, finally, fracture energy Gf are shown for both types of wallets subjected to the pull
push shear tests. The used theoretical procedure considers axial strains in FRP strips during the tests as suggested by Ferracuti et al. (2007) and Valluzzi et al. (2012).

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PART | III Modeling of fiber-reinforced polymer
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TABLE 15.4 Experimental results by pull
push tests on historic wallets with GFRP strips. Brickwork wallets

Exp. ultimate load, Pu (N)

Fracture energy, Gf (N/mm)

Average values of Gf (N/mm)

W1A

10157

0.383

0.332

W1B

7687

0.354

W1C

5341

0.258

W2A

6658

0.451

W2B

7517

0.537

W2C

7249

0.329

W3A

5734

0.411

W3B

9474

W3C

8744

Slip, δf (mm)

Shear stress, τ f (N/mm2)

Theoretical ultimate load, P1 (N)

0.337

1.548

4011

0.231

1.733

3856

0.404

1.311

3292

0.260

1.786

4352

0.244

1.777

4749

0.212

1.470

3717

0.185

1.248

4155

0.501

0.234

1.736

4587

0.488

0.251

1.618

4527

0.439

0.467

The value of P1 was determined based on the experimental results by using Eq. (15.23); it is compared with the experimental values corresponding to the failure, that is, the delamination. The values of fracture energy Gf are relatively low varying, between Gf 5 0.26
0.54 N/mm, respectively, for specimens made by historic clay bricks. The influence due to the width of bed mortar joints is limited on the fracture energy and consequently on the theoretical value of load P1.

15.2.3 Carbon fiber
reinforced polymer/steel-reinforced polymer strip-to-historic brick joints A further set of experimental tests presented in this chapter concerns pull
push tests of specimens composed of solid historic clay bricks externally strengthened by strips of CFRP and SFRP (Capozucca, 2010). The SFRP is an innovative technology for structural retrofitting, developed as complementary techniques for FRP. In Table 15.5 the main characteristics of the composite materials used to prepare the specimens for pull
push shear tests are shown. The specimens were prepared with FRP strips of width 50 mm applied on two bed surfaces of clay bricks. The length of adhesion of FRP/SRP (steel-reinforced polymer) strip-to-historic bricks was equal to 250 mm; five strain gauges were located on composite material at intervals.

Fiber reinforced polymer strengthened masonry Chapter | 15

549

TABLE 15.5 Experimental data of CFRP/SRP used in shear tests. Composite material

Tensile strength, ffk (N/mm2)

Young’s modulus, Ef (N/mm2)

Ultimate strain, εf (%)

Thickness, tf (mm)

Density (g/m2)

CFRP

3500

240000

1.45

0.170

200

SRP

2479

118000

2.1

0.097




1. Fidcarbon UNI 200 HT 240

2. Fidsteel 3 3 2
4
12
500 hardwire

During the pull
push tests the specimens’ strains on the strips were recorded until the ultimate loads; some diagrams of strain values are shown in Figs. 15.7 and 15.8, respectively, for CFRP and SRP-to-historic brick bonded joints considering that the origin of axis x located on the compressed edge of each specimen. In Table 15.6 the experimental failure loads of the pull
push tests on the specimens and the observed failure mechanisms are reported. As can be seen, the delamination of strips was recorded in many specimens, although accompanied by rupture of bricks on the edge. Indeed, the main failure modes were delamination of CFRP/SFRP strips due to detachment of a surface layer of brick and delamination of FRP strips with successive compressive failure of brick on the loaded edge. In Table 15.6, the Gf, fracture energy, and Pu, ultimate load values for pull
push of a double-lap strip-to-historic brick bonded joint, are shown. The value of P1 was obtained by (Eq. 15.23) and is compared with the experimental value due to delamination. The authors noted that the observed mechanisms of failure were obviously influenced by the characteristics of strength, porosity, clay composition, and mode of execution of historic bricks, which are different, although taken from the same building.

15.2.4 Remarks and considerations Preservation of architectural monumental buildings and common masonry buildings of historic centers is one of the most important challenges in civil engineering with many aspects such as complexity of the geometry of structures, variability of materials used, and the loading history of buildings to be taken into consideration. In Italy, this objective has increased for existing

550

PART | III Modeling of fiber-reinforced polymer
strengthened masonry ε (10–6) 9000 8000

P = 750 N P = 4250 N P = 6750 N P = 10,500 N P = 13,250 N

7000 6000 5000 4000 3000 2000 1000 0 0

50

100

150 x (mm) Specimen C1

ε (10–6) 4000

200

250

P = 750 N P = 4250 N P = 5500 N P = 6750 N P = 8500 N

3500 3000 2500 2000 1500 1000 500 0 0

50

100

150

200

250

x (mm) Specimen C2 FIGURE 15.7 Experimental diagrams strain, ε, versus length of CFRP-to-historic brick bonded joints.

constructions due to seismic hazards on country. FRPs on masonry walls allow us to increase both the strength and collapse displacements of buildings. Externally bonded FRP strips and sheets, as known, are used for strengthening shear masonry walls, increasing the tensile capacity to support combined compression and shear action that occur during an earthquake. A strengthened shear wall has to be assessed by both local and global failure modes, which can occur in combination: the cracking of masonry in tension, its crushing in compression, its shear sliding, the failure of the fiberreinforced composite, and, finally, the delamination of FRP from masonry. Many tests indicate that a dangerous mechanism of brittle failure is due to delamination, especially if the FRP strips are glued to historic clay bricks with a weak clay surface (Aiello and Sciolti, 2003; Briccoli Bati et al., 2006; Capozucca, 2010). The experimental tests described in the previous section confirmed that composite materials bonded to historic brick joints lose their capacity due to the brittle delamination failure.

Fiber reinforced polymer strengthened masonry Chapter | 15 5000

ε (10–6)

551

P = 765 N P = 5500 N P = 7750 N P = 10,000 N

4000 3000 2000 1000 0 0

50

ε (10–6)

6000

150 x (mm) Specimen S1

100

200

250

P = 250 N P = 5000 N P = 7500 N P = 12,500 N P = 10,000 N

5000 4000 3000 2000 1000 0 50

0

x (mm)

150

200

250

Specimen S2

ε (10–6)

5000

100

P = 250 N P = 5500 N P = 7000 N P = 8000 N P = 12,000 N P = 13,250N

4000 3000 2000 1000 0 0

50

100

150

200

250

x (mm) Specimen S3 FIGURE 15.8 Experimental diagrams strain, ε, versus length of SFRP-to-historic brick bonded joints.

In the rehabilitation of masonry structures, externally bonded GFRP strips may be more convenient to use than CFRP or SRP. In this case, experimental tests show that failure is the mechanism of delamination; the average value of slip is equal to about δf 5 0.26 mm for specimens with historic clay bricks. The corresponding values of maximum shear stress assumed average values equal to τ fB1.58 N/mm2 for specimens with historic bricks.

TABLE 15.6 Experimental results from pull
push shear tests. Specimens

Failure mode

Exp. load (N)

Shear stress, τ f (N/mm2)

Exp. interfacial slip (mm) δf

δ1

Exp. interfacial law

Theoretical load, P1 (N)

Exp. energy fracture, Gf (N/mm)

C1

BR/FD

13000

2.80

0.80




Almost linear

15,000

1.120

C2

BR

8500

2.00

0.60

0.15

Nonlinear







C3

BR

8500

1.00

0.35

Almost linear







S1

BR/SD

10000

2.25

0.20




Linear

3594

0.225

S2

BR/SD

12500

2.55

0.25




Linear

4292

0.320

S3

SD

12000

1.75

0.60

0.20

Nonlinear

5468

0.525

AF, Failure trough adhesive; BR, brick fracture; FD/SD delamination; FR/SR, FRP/SRP rupture.

Fiber reinforced polymer strengthened masonry Chapter | 15

553

The values of fracture energy Gf evaluated by tests on historic wallets with different thickness of mortar joints is relatively low varying, between Gf 5 0.258
0.537 N/mm for specimens made by historic clay bricks, confirming the experimental results present in the literature (Aiello and Sciolti, 2008; Capozucca, 2010). The influence due to the width of the bed mortar joints appears rather limited on the fracture energy Gf and consequently also on the theoretical value of applied load.

15.3 Numerical models This section aims to give an overview of the numerical models available in the current literature developed with the main goal of simulating the bond behavior of FRP strengthening systems externally applied on masonry elements. By examining the literature, it can be observed that, while some of the models specifically focus the attention on the simulation of delamination phenomenon (they are indeed carried out by considering case studies generally consisting of single masonry units or unit-mortar assemblages involved in shear lap bond tests), other models aim to simulate the influence of delamination phenomenon on the global resistant mechanism of structural systems (panels, arches, vaults, facades, etc.) strengthened by FRPs (Fig. 15.9). Despite the different purposes of these models, it is clear that the efficacy of both model types strictly depends on their ability to correctly reproduce the local interaction mechanism between the strengthening system and the masonry substrate.

Single units and unitmortar assemblages

Masonry elements and structures

Delamination phenomenon

Delamination influence

FIGURE 15.9 Schematization of tests on single units and structures for studying the delamination phenomenon and its influence of the performance of strengthened structures.

554

PART | III Modeling of fiber-reinforced polymer
strengthened masonry

For this reason, numerical models are specifically considered concerning the simulation of the local bond behavior of FRP strengthening systems externally applied on masonry substrates.

15.3.1 Interface-based models A relevant part of the literature concerns modeling approaches based on the use of zero-thickness interface elements reproducing a thin layer between the reinforcement and the masonry substrate where the debonding mechanism can occur. For instance, in Ceroni et al. (2014) a study aimed at simulating experimental bond tests of bricks strengthened by FRPs by considering twodimensional (2D) and three-dimensional (3D) FE models carried out throughout various commercial computer codes was presented. Here the authors specifically introduced in the models zero-thickness interface elements interposed between the reinforcement and the masonry substrate in which activation depends on the relative displacement of the reinforcement with respect to the masonry substrate. Moreover, while in some of the models it was considered a nonlinearity exclusively concentrated in the interface elements, other analyses were developed by also introducing a nonlinear behavior of the support. The comparison among the different FE models proposed by the authors showed similar results in terms of overall load
displacement curves. Moreover, it was observed that in all the accounted cases the maximum tensile and compressive stresses attained inside the masonry substrate resulted significant smaller than its corresponding strength. This latter evidence, besides explaining the similarity of the obtained results, also confirmed the appropriate assumption of considering a linear-elastic behavior for both the masonry and strengthening. By examining the literature, it emerges that the assumption of considering the concentration of the nonlinearity of the whole model solely at an interface layer is common in the majority of the interface-based FE models for the study of the bond behavior of concrete and masonry elements strengthened by FRPs. This assumption allows us to substantially reduce the computational effort and, in agreement with the experimental evidence, to consider an interface layer that simulates the thin layer of masonry debonded from the support (Fig. 15.10). The results obtained by using this modeling strategy generally underline for both single units and unit-mortar assemblages good agreement with the experimental outcomes. Moreover, the interface-based models allow a better understanding of some of the important aspects concerning the interaction mechanism between the reinforcement and the masonry substrate. In Grande et al. (2011) the bond behavior of brick specimens strengthened by FRPs characterized by different values of bond length of the reinforcement was experimentally and numerically investigated. Here, the results

Fiber reinforced polymer strengthened masonry Chapter | 15

555

FIGURE 15.10 Interface-based FE model approach (Grande et al., 2011).

FRP strengthening FRP/brick interface Brick

from the numerical analyses underlined that the bond length not only influences the global response, but also the stress distribution in the support. Indeed, while for low values of bond length a peak value of minimum principal stresses located near the unloaded end of FRP (i.e., on the opposite side of the application of the external load) was observed, the increase of the bond length underlined peaks of minimum principal stresses at the center of the first part of the bonded strip. This was particularly useful for explaining the different failure modes experimentally observed by the authors. Indeed, while in the case of specimens characterized by lower values of bond length had observed failure modes characterized by the removal of a considerable part of the brick material at the unloaded end of the reinforcement, the specimens with a greater value of bond length generally showed the decohesion of a thin and uniform layer of the brick material. Nevertheless, as shown by Ceroni et al. (2014), failure modes characterized by considerable damage of the support material were observed in the case of masonry substrates composed of weak and porous masonry units, such as tuff. In this case, the removal of a considerable part of the masonry material also emerged in the case of high values of bond length. As a consequence, the analysis of the bond behavior for this kind of masonry substrates necessarily requires to also account for the nonlinear behavior of the substrate. Other research carried out using interface-based modeling approaches for studying the bond behavior of FRPs also concerns the influence of mortar joints. In Grande et al. (2013), in order to account for the influence of the mortar joints composing the support, interface elements were only introduced between the bricks and the reinforcement, while any connection was provided between the reinforcement and the mortar joints. The results presented by the authors underlined that, when the zone of maximum stress concentration approaches the mortar joint, a singularity appears in the stress field. This leads to a residual concentration of stresses in the substrate due to the normal stresses transferred by the detached portion of the reinforcement between two contiguous bricks (bridge effect) (Fig. 15.11). Similar evidence was emphasized by Grande and Imbimbo (2016) where a simple approach based on a discrete FE. model composed of axial and

PART | III Modeling of fiber-reinforced polymer
strengthened masonry

556

Single brick—CFRP (Valluzzi et al. 2012)

9000

Single brick—CFRP (Valluzzi et al. 2012)

2.5

(C) 7000

(B)

Force (N)

6000 5000 4000 FE model Experimental LOWER and UPPER bounds

3000 (A)

2000

Interface shear stress (MPa)

8000 2 (A)

1.5

1

D = 0.023 mm F = 2154 N

0.5

1000 0 0

0.2

0.6 0.8 1 Displacement (mm)

1.2

(B)

1.5

1

D = 0.27 mm F = 7116 N

0.5

0

20

40

80 100 120 60 Bonded length (mm)

140

160

20

40

60 80 100 120 Bonded length (mm)

140

160

Single brick—CFRP (Valluzzi et al. 2012)

2.5 Interface shear stress (MPa)

2

0

0 0

1.4

Single brick—CFRP (Valluzzi et al. 2012)

2.5 Interface shear stress (MPa)

0.4

2

(C)

1.5

S  s

: τ otherwise where τ 0, A, α, β, sf, τ f are the unknown parameters. Fig. 19.7 shows the shear stress
slip laws and the normal stress
slip curves obtained in Caggegi et al. (2017) from the calibration process for the specimens tested by the group of Cracow University of Technology. The comparison between the experimental and the calibrated curves shows a good agreement. Nevertheless, it is important to underline that the shape of the derived shear stress
slip laws is particularly influenced by the type of failure mechanism emerged from the tests: the tensile failure of the reinforcement instead of the debonding. Then, the approach carried out by the authors led to laws representing the interaction between the reinforcement and the matrix when this type of phenomenon occurs. Recently, in Grande et al. (2017, 2018) a simple theoretical study devoted to investigate the influence of the upper mortar layer on the bond behavior of FRCM-strengthening systems applied on structural supports was carried out. Here the authors presented a theoretical modeling approach based on the solution of the following system of differential equations obtained with reference to the scheme shown in Fig. 19.8 by introducing equilibrium considerations involving separately an infinitesimal portion of the strengthening component (first equation) and an infinitesimal portion of the upper mortar layer component (second equation): dσf tf bf 2 ðτ e ðse Þ 1 τ i ðsi ÞÞbf dx 5 0 ð19:24Þ dσec bf tce 1 τ e ðse Þbf dx 5 0 where σf and σec are the normal stresses in the reinforcement and in the upper mortar, respectively; tf and tce are the thicknesses of the reinforcement and

696

PART | IV Modeling of textile-reinforced mortar-strengthened masonry

FIGURE 19.7 Experimental and calibrated laws (Caggegi et al., 2017). (A) Shear stress
slip laws; (B) normal stress
slip curves. Figures reproduced from Caggegi, C., Carozzi, F.G., De Santis, S., Fabbrocino, F., Focacci, F., Hojdys, L., et al., 2017. Experimental analysis on tensile and bond properties of PBO and aramid fabric reinforced cementitious matrix for strengthening masonry structures. Compos B 127, 175
195.

the upper mortar, respectively; τ i and τ e are the shear stresses at lower and upper interfaces, respectively; and bf is the width of the reinforcement. In order to obtain an explicit solution of the above system of differential equations, the authors introduced some simplifications concerning (1) the behavior of the support and lower mortar layer, both assumed rigid; (2) the behavior of the lower and upper mortar/reinforcement interfaces, both schematized as zero-thickness elements shear deformability only; (3) the behavior of the upper mortar layer and the reinforcement, both assumed axial deformable only.

σc

σc + dσc τe

σp

τe

e

uc se

Upper mortar

τi

si

Upper interface Strengthening

P

Lower interface Lower mortar Support

FIGURE 19.8 Schematization of the accounted model (Grande et al., 2018).

σp + dσp

698

PART | IV Modeling of textile-reinforced mortar-strengthened masonry

τi(τe)

σc

τf

ft

εc

S1

S i(S e)

FIGURE 19.9 Accounted constitutive laws for the upper mortar layer (left) and the mortar/ reinforcement interfaces (right) (Grande et al., 2018).

Moreover, the authors introduced simple nonlinear constitutive laws for the interfaces and the upper mortar layer (Fig. 19.9), in order to reproduce the following mechanisms experimentally observed: G

G

G

G

undamaged stage DP0: both the interfaces and the upper mortar layer were assumed undamaged; damage pattern DP1: damage only involving the interfaces (debonding mechanism); damage pattern DP2: damage only involving the upper mortar (cracking phenomenon); damage pattern DP3: damage involving both interfaces and upper mortar (debonding/cracking mechanism).

Regarding the undamaged stage, which is a common preliminary phase to all considered damage patterns, the system of Eq. (19.24) assumes the following form: 8 d 2 si > i e > > > dx2 2 K1 ðs 1 s Þ 5 0 > < G G τf 0 1 K2 5 e G5 with K1 5 2 i 2 e > tf Ef tc E c s1 > > @d s 2 d s A 1 K2 se 5 0 > > : dx2 dx2 ð19:25Þ where Ef and Ec are the elastic moduli of the reinforcement and of the upper mortar layer, G is the shear stiffness of the interface, τ f is the maximum allowable shear stress, and s1 is the elastic slip threshold. The explicit solution of the system (19.25) was derived by the authors by introducing boundary conditions concerning: (1) the unloading state of the upper mortar at both the end sections (respectively x 5 0 and x 5 L); (2) the unloading condition of the strengthening at x 5 L; and (3) the attainment of the slip value s1 for the lower interface at the loaded section (i.e., by supposing that the limit condition corresponding to the attainment of the shear strength at the lower interface precedes the attainment of the tensile strength of the upper mortar).

Theoretical and FE models Chapter | 19

699

The authors also underlined that in the case of absence of the upper mortar layer, the system (19.25) reduces to the following equation: d 2 si 2 K1 si 5 0 dx2

ð19:26Þ

which also governs the problem of FRP strengthening systems. Analyzing the solution of both the system (19.25) and Eq. (19.26), the authors pointed out that, although the presence of the upper mortar does not significantly affect the shape of shear stresses distribution at the lower interface, it leads to an increase of the length of the transferring zone (i.e., the effective bond length Leff). This evidence is very important since the effective bond length is a design parameter at the basis of the evaluation of the bond strength of reinforcing systems. Formulas concerning this parameter are available in standard codes for the case of FRPs, while any indication specifically concerning FRCMs is available. Regarding the damage patterns, the authors examined subsequent steps, each corresponding to a specific damage level. In other words, each step is characterized by a different number of regions of the strengthening system along the bond length characterized by a specific damage or undamaged status. The behavior of each portion is governed by a system of two differential equations, similar to (19.25), depending on the value of the maximum slip attained by the interfaces, which identifies the status of the interface (i.e., the pre or postpeak stage). Then, in order to obtain the solution in terms of whole behavior of the reinforcing system, additional boundary conditions concerning the type of damage (i.e., the attainment of a specific value of the slip at the interfaces or the attainment of the tensile strength of the mortar) and in terms of continuity of the slip functions were introduced by the authors. The first two damage patterns selected by the authors were introduced for analyzing separately the influence of the damage of the interfaces (DP1) and the upper mortar (DP2) on the global response of the FRCM system. Regarding the damage pattern DP1, the authors identified three subsequent steps corresponding to different limit conditions (Fig. 19.10): G

G

G

the first step corresponds to the attainment of the shear strength at the lower interface at the loaded end (x 5 L); the second step corresponds to the attainment of the shear strength at the upper interface; and the third step is a generic subsequent step.

The results emerged from this damage pattern clearly underline that increasing the applied load after the occurrence of the debonding between the reinforcement and the upper interface does not lead to further increases of the peak value of normal stresses of the upper mortar. This means that, whether the debonding of the reinforcement respect to the upper interface

PART | IV Modeling of textile-reinforced mortar-strengthened masonry

Slip (mm)

Lower interface Upper interface

0.2 0.1 0

–0.1

0

1000 500 Bond length, L (mm)

Step 1

Shear stress (MPa)

Step 1 0.3

1.5 Lower interface Upper interface

1 0.5 0

–0.5

0

Step 2

0.2 0

–0.2

0

1000 500 Bond length, L (mm)

1.5 1 0.5 0

–0.5

0

Step 3

0

–0.5

0

500 1000 Bond length, L (mm)

0

0

1 0.5 0

–0.5 -0.5

0

500 1000 Bond length, L (mm)

1000

8 6 4 2 0

1000 500 Bond length, L (mm)

1.5

500 Bond length, L (mm) Step 2

0

1000 500 Bond length, L (mm) Step 3

Normal stress (MPa)

0.5

2

Step 3

Shear stress (MPa)

Slip (mm)

1

Upper mortar layer

–2

1000 500 Bond length, L (mm)

Normal stress (MPa)

0.4

4

Step 2

Shear stress (MPa)

Slip (mm)

0.6

Step 1

Normal stress (MPa)

700

8 6 4 2 0

0

500 Bond length, L (mm)

1000

FIGURE 19.10 Results derived by accounting the damage pattern DP1: (left) slips at interface level; (center) shear stress of interfaces; (right) normal stress of upper mortar (Grande et al., 2018).

occurs before the cracking of the upper mortar, the failure mechanism is characterized by the slippage of the reinforcement respect to both the interfaces without arising cracking on the external mortar surface. Regarding the damage pattern DP2 (see Fig. 19.11), the authors observed that after the occurrence of the first crack at the upper mortar, only the peak value of slips at the lower interface continues to increase while the peak value of slips at the upper interface does not significantly increase. In this case, it is evident a failure mechanism of the strengthening system characterized by both the debonding at the lower interface and the cracking of the upper mortar: this failure mechanism is widely experimentally observed. Finally, with the damage pattern DP3 the authors introduced the possibility of damage of the interfaces and the upper mortar layer in order to obtain the numerical response of two case studies derived from the literature (D’Antino et al., 2015). For both these cases, the following steps emerged from the numerical analyses: G

G

Step 1: corresponding to the attainment of the tensile strength of the mortar (first crack); Step 2: corresponding to the attainment of the shear strength at the lower interface (debonding);

Theoretical and FE models Chapter | 19

0.1 0.05 0

–0.05

0

500 1000 Bond length, L (mm)

Step 1 1

Shear stress (MPa)

Slip (mm)

Lower interface Upper interface

0.5

–0.5 0

0 –0.2 0

500 1000 Bond length, L (mm)

1 0

–1

0

0 –0.2 0

1000 500 Bond length, L (mm)

500 Bond length, L (mm)

1000

2 1 0

0

500 Bond length, L (mm)

1000

2 1 0

0

500 Bond length, L (mm)

1000

Step 3

3

–1

500 Bond length, L (mm)

3

–1

Normal stress (MPa)

0.2

0 –1 0

Step 3

Shear stress (MPa)

Slip (mm)

0.4

1

Step 2

2

Step 3 0.6

2

Normal stress (MPa)

0.2

500 1000 Bond length, L (mm)

Upper mortar layer

Step 2

Shear stress (MPa)

Slip (mm)

0.4

Lower interface Upper interface

0

Step 2 0.6

Step 1 3

Normal stress (MPa)

Step 1 0.15

701

1000

3 2 1 0

–1

0

500 Bond length, L (mm)

1000

FIGURE 19.11 Results derived by accounting the damage pattern DP2: (left) slips at interface level; (center) shear stress of interfaces; (right) normal stress of upper mortar (Grande et al., 2018). G

G

Step 3: corresponding to the attainment of the tensile strength of the mortar (second crack); Step 4: corresponding to the attainment of the tensile strength of the mortar (third crack).

From these steps then emerged a failure mechanism of specimens characterized by the debonding of the reinforcement at the lower interface and the progressive cracking of the upper mortar layer: this failure mechanism resulted in agreement with the experimental evidence. Indeed, considering the global response in terms of applied force versus global slip it was observed a good agreement with the experimental curves (Fig. 19.12). In the last part of the paper, the authors discussed about a further feature of FRCM systems: the grid configurations of the reinforcement and its effect in terms of the interaction between the upper and the lower mortar layer, which just occurs throughout the mesh of the grid. Indeed, the authors simulated this effect by introducing an additional interface elements which directly connects the upper mortar layer to the lower mortar layer by opportunely modifying the second equation of the system (25). The results obtained by introducing this modification underlined

PART | IV Modeling of textile-reinforced mortar-strengthened masonry

702

Bond length Lb = 450 mm Width b = 60 mm

Bond length Lb = 450 mm Width b = 80 mm

10 Applied force (kN)

Applied force (kN)

10 8 6 4 2

8 6 4 Numerical

2

0 0

2

4 6 8 Global slip (mm)

10

12

Experimental envelope

0 0

2

4 6 8 Global slip (mm)

10

12

FIGURE 19.12 Global response in terms of applied force versus the global slip: comparison between the experimental and the numerical response (Grande et al., 2018).

that the presence of the additional interface simulating the interaction between the two mortar layers leads to an increase of slips at the reinforcement/mortar upper interface and a consequent reduction of normal stresses in the upper mortar. In other words, the interaction between the two mortar layers increases the probability of failure mechanisms characterized by the slippage of the reinforcement from both the mortar layers. Other studies available in the current literature are instead mainly based on the use of numerical modeling strategies. In this case, similarly to the models proposed for FRPs, the authors propose models based on the use of zero-thickness interface elements interposed between the reinforcement and the mortar matrix for numerically simulating the interaction mechanism between these components of the strengthening system. In Carozzi et al. (2014), an experimental and numerical study concerning the bond behavior of FRCM strengthening systems externally applied on masonry bricks and bricks/mortar pillars were presented. In particular, the authors performed both shear-lap tests and, moreover, pull-off tests involving a single yarn embedded in the mortar. Considering the results emerged from the latter type of tests, the authors derived a simple constitutive law in terms of shear stress
slip to use for developing the numerical analyses (Fig. 19.13). In particular, the authors proposed two approach for numerically study the bond behavior of the accounted specimens. The first one consists of an analytical-numerical approach which specifically accounts for the interaction between the reinforcement and the mortar. Here the authors assumed a scheme composed of two in-series nonlinear springs for schematizing the unbonded and bonded zone of the reinforcement respectively. Consequently, the total deformation of the system was obtained by the sum of the axial deformation of the unbounded grid and the contribution of the pure sliding of the grid within the surrounding mortar. An elastic perfectly plastic behavior was assumed for the interface between grid and mortar, where an idealized stepped stress
slip behavior was introduced on

Theoretical and FE models Chapter | 19

f (N/mm)

5.0

703

(Carozzi et al., 2014)

Experimental 2.5

0.0

Numerical

3.0 Slip (mm)

6.0

FIGURE 19.13 Experimental and numerical curves derived from pull-off tests (Carozzi et al., 2014). Figure reproduced from Carozzi, F.G., Milani, G., Poggi, C., 2014. Mechanical properties and numerical modeling of Fabric Reinforced Cementitious Matrix (FRCM) systems for strengthening of masonry structures. Compos. Struct. 107, 711
725.

the basis of the results emerged from the pull-off tests. An elastic
perfectly plastic behavior with infinite ductility was considered for the FRP yarn, while the deformation of both brick and masonry pillar was neglected. On the basis of these assumptions, the authors derived simple formulas for the evaluation of the global stiffness of the model (KT) and the peak load (Fu1) associated to either the slipping of the grid from the mortar or the tensile failure of yarns:   Ef n y A f ΔLu 1 ΔLb 1 ny kt ð19:27Þ KT 5 ðLu 2 Lb Þ Lu Lu 1 Lb Fu1 5 ny cI Lb

ð19:28Þ

where Lu is the length of the unbonded region of the grid; Lb is the length of the bonded region of the grid; Ef is the Young’s modulus of the reinforcement; ny is the number of yarns composing the grid; Af is the cross-section area of the reinforcement; kt is the tangential elastic stiffness of yarn
mortar interface law carried out on the basis of results of pull-of tests; ΔLu is the part of the overall displacement due to the unbounded part; ΔLb is the part of the overall displacement of the bonded part. The second modeling approach proposed by the authors was a full 3DFEM nonlinear approach consisting of an extension of the procedure originally adopted in Milani and Lourenc¸o (2012) and in Milani and Tralli (2012). In particularly, in Carozzi et al. (2014) the FE model proposed by the authors is composed of a discretization characterized by 3D rigid and infinitely resistant eight-node elements coupled with quadrilateral nonlinear interfaces just interposed between the rigid elements. This allows to consider the deformation of the model both in the linear and nonlinear range lumped on the interface elements and, in addition, significantly reducing the

704

PART | IV Modeling of textile-reinforced mortar-strengthened masonry

variables. In particular, the authors introduced three translational and three rotational nonlinear springs for each interface, and used a Sequential Quadratic Programming scheme—SQP to deal with softening by approximating the actual stress
strain behavior of the interfaces by means of a stepping function. Moreover, the relationship between continuum strains and the interface displacement was directly deduced by means of well-established averaging procedures, as pointed out in Kawai (1978). In order to properly take into account the grid configuration of the reinforcement and the nonlinear interfacial behavior between a single yearn and surrounding mortar material, the authors modeled the single yarn by means of rigid infinitely resistant truss elements, each characterized by six kinematic variables: three centroid displacements and three rotations around centroid. In addition, the authors introduced translational axial springs to connect contiguous trusses. The axial behavior of the yarn was assumed elastic perfectly plastic by considering the ultimate strength of yarns. On the other hand, the tangential stress
slip relationship between the yarn and surrounding mortar was still assumed on the basis of the evidence emerged from pull-off tests. The comparison between the experimental and the numerical results obtained by using the two approaches underlined a good agreement in terms of the peak load, initial undamaged state stiffness and postpeak behavior, then emphasizing the reliability of the proposed numerical approaches. In particular, the authors underlined the ability of the numerical simulations in detecting the debonding phenomenon of the grid within mortar. At the same time, the authors also pointed out the occurrence of experimental unexpected failure modes, such as the tensile failure of yarns, due to drawbacks emerged from tests and then did not capture from numerical models. Indeed, the authors also simulate an accidental experimental imperfection due to a rotation of the loaded pad because of different tensile stresses acting on the yarns. This allowed the authors to numerically account for the increase of the tensile stress in one of the lateral yarns and then its premature rupture like emerged from tests. In D’Antino (2014) the stress-transfer mechanism in FRCM
concrete joints was investigated by means of a 3D numerical analysis carried out using the software Abaqus (2010). In particular, the data obtained from the application of the fracture mechanics approach to PBO FRCM
concrete joints with and without the external layer of matrix were directly used by the authors to calibrate the shear stress
slip laws introduced in the numerical models. Indeed, the matrix
fiber interfaces were modeled by the authors by means of a master
slave contact interaction based on a cohesive damage law where nonlinear shear stress
slip laws were employed for defining the relationship between the master and slave surfaces. In detail, a linear elastic branch modeled by using a surface-based cohesive behavior, followed by a nonlinear branch modeled by introducing a damage variable that simulates

Theoretical and FE models Chapter | 19

705

the interface degradation were considered. The authors particularly emphasized the assumption of considering the same bond behavior for the internal and external matrix layers. In the model, both the PBO fibers and the matrix layers were assumed homogeneous isotropic linear
elastic materials. The contact between the matrix and the lateral surfaces of the fiber bundle was not considered in the model and, in order to specify the contact law, the authors directly specified the normal behavior and the shear stress
slip relationship between the master and slave surfaces. In particular, the normal behavior was specified by using a hard contact pressure overclosure relationship: when the surfaces are in contact, the contact pressure can be transmitted between them; differently, the contact pressure becomes zero when the surfaces separate. Regarding the shear stress
slip law, the authors directly introduced the laws derived from the experimental tests (Fig. 19.14). Nevertheless, the authors pointed out that the cohesive behavior does not interact with the hard contact pressure. Finally, both the fiber bundle and the matrix were modeled by using eight-node brick elements with a linear behavior by introducing different mesh sizes to verify the influence of the model discretization. The results carried out in this study were compared with a significant number of experimental outcomes: they showed a good agreement (Fig. 19.14). Both studies examined above underline the importance of a correct calibration of the law characterizing the shear behavior of the interface elements used for simulating the interaction between the reinforcement and the mortar: in the above studies these laws were directly derived from experimental measurements. Indeed, as underlined in Grande et al. (2017), the different phenomena characterizing the resistant mechanism of FRCM systems may particularly influence the shear-stress transfer mechanism between the reinforcement and the upper mortar layer. In particular, when the configuration of FRCMs is characterized by a double mortar layer, the cracking phenomenon of the upper

(D’Antino, 2014)

0.4

0.0

0.8

Slip (mm)

1.6

4.5

Applied load (kN)

Shear stress (MPa)

0.8

DS_330_43_S_5 (D’Antino, 2014) Experimental

2.5 Numerical

0.0

2.0

4.0

Slip (mm)

6.0

FIGURE 19.14 Shear stress
slip law and applied load
slip curve. Figures reproduced from D’Antino, T., 2014. Bond behavior in fiber-reinforced polymer composites and fiber-reinforced cementitious matrix composites. PhD thesis, Universita` degli Studi di Padova, Italy.

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PART | IV Modeling of textile-reinforced mortar-strengthened masonry

mortar layer is generally observed. The occurrence of this phenomenon influences the shear stress transfer mechanism between the reinforcement and the upper mortar layer which results different from the one characterizing the interface between the reinforcement and the lower mortar layer. In this case, it is then evident that the common modeling strategy based on the use of interface elements interposed between the reinforcement and the mortar layers, where the nonlinearity of the whole specimen is concentrated by neglecting possible damages of the mortar, could results unsuitable for obtaining a good prediction of the experimental response of FRCMs. Recently, Grande and Milani (2018) proposed a simple approach devoted to opportunely calibrate the shear stress
slip constitutive law of the reinforcement/mortar interface layer for implicitly accounting for the influence of the damage of the mortar on the resistant mechanism of FRCM strengthening systems. The approach proposed by these authors was finalized for the use of the common interface modeling strategy without introducing nonlinear constitutive laws for the mortar also in case of resistant mechanisms of FRCMs involving the damage of the mortar. The approach was derived by the authors on the basis of interesting results emerged from numerical analyses referring to some case studies derived from the literature. The authors paid particular attention just to the role of the progressive damage of the upper mortar layer on the local shear stress-transfer mechanism between the reinforcement and the upper mortar layer. In particular, considering a simple but effective Spring Model (Fig. 19.15), where each component of the strengthening system, that is, mortar, reinforcement, support and reinforcement/mortar interface, is modeled throughout springs with linear or nonlinear behavior, the authors obtained results in good 1 Lower mortar 3

2 Reinforcement 3 Upper mortar

K

N 5

4 Lower interface 5 Upper interface

2

J

M 4

I

1

L

Upper mortar Upper interface Strengthening

P

Lower interface Lower mortar Rigid support

FIGURE 19.15 Schematization of the Spring Model proposed in Grande and Milani (2018).

Theoretical and FE models Chapter | 19

707

agreement with the experimental ones when a nonlinear behavior of both the interfaces and the upper mortar layer were introduced in the Spring Model. Indeed, it was observed that the progressive cracking of the upper mortar layer influence in a considerable manner the global response of the examined specimens, and also the shear stress-transfer mechanism between the reinforcement and the upper mortar layer. Differently, the analyses performed with a linear behavior of the upper mortar layer, that is, excluding the possibility of damage of the upper mortar, underlined an increase of the numerically obtained peak load together with higher values of tensile stresses arising in the upper mortar layer and appearing early even in the prepeak stage. This led to a significant overestimation of the effective global resistance of the examined case studies. The authors also performed numerical analyses by reducing the bond strength of the upper interface only, and still considering a linear
elastic behavior of the mortar (consistent with the occurrence of debonding on upper interface before cracking of the upper mortar). In this case, they underlined a decrease of the peak load and, at the same time, that the peak normal stresses in the upper mortar no longer increased after the occurrence of the slipping phenomenon at the upper interface. Just considering the outcomes emerged from this study, the authors proposed the simple three-step approach where: G

G

G

STEP 1—considering a straightforward interface model with a linear
elastic behavior of the mortar layer, a preliminary analysis is performed in order to deduce the peak value of the shear stress acting on the upper interface which corresponds to the attainment of the tensile strength in the upper mortar layer, that is, τ e with σc 5 ft. STEP 2—the obtained value of the shear stress is compared with the value of the bond strength of the interface (τ b ). If τ e (σc 5 ft) . τ b , that is, the debonding at the upper interface precedes cracking in the upper mortar, no modifications are introduced in the constitutive law of the upper interface. STEP 3—when τ e (σc 5 ft) , τ b , the constitutive law of the upper interface is modified by introducing an elastic-fragile behavior where a reduced value of the shear strength at the upper interface just equal to τ e (σc 5 ft) is assumed.

The authors assessed the reliability of the proposed approach with reference to case studies derived from the literature (D’Antino et al., 2015), consisting of single shear lap bond tests of concrete blocks strengthened by a bidirectional unbalanced PBO fiber net with two mortar layers, characterized by two different values of both the bond length (L 5 330 mm and L 5 450 mm) and the width of the reinforcement (bp 5 60 mm and bp 5 80 mm). In particular, the authors performed numerical analyses by using both the Spring Model proposed by the authors themselves and a 2D

PART | IV Modeling of textile-reinforced mortar-strengthened masonry

708

detailed FE model (2D FE Model; see Fig. 19.16) built within the commercial computer code DIANA 9. In both models it was considered a linear behavior of the mortar, the reinforcement and the support. On the contrary, a nonlinear shear stress
slip behavior was introduced for both the interfaces, where the constitutive law of the upper interface was opportunely modified according to the approach proposed by the authors. The results derived from these numerical analyses are presented in Fig. 19.17 in terms of applied global force
displacement curves. The same plots also report the curves derived from the experimental tests. From the plots the ability of both the Spring Model and the 2D FE Model to capture Upper interface Upper mortar

Reinforcement Upper interface

Lower mortar

Support

Y Z

X

7 6 5 4 3 2 1 0

Bond length: 330 mm Width: 60 mm

Spring-model 2D FE-model

10 Applied force (kN)

Applied force (kN)

FIGURE 19.16 2D FE Model accounted in the study of Grande and Milani (2018).

Bond length: 330 mm Width: 80 mm

8 6 4

Spring-model

2

2D FE-model

0 0

2

4

6

8

0

2

6 4

Spring-model

2

2D FE-model

0 2

4

6

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8

8 6 4

Spring-model Bond length: 450 mm

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8

2D FE-model

Width: 80 mm

0 0

6

10

Bond length: 450 mm Width: 60 mm

8

Applied force (kN)

Applied force (kN)

10

4

Displacement (mm)

Displacement (mm)

0

1

2

3

4

5

Displacement (mm)

FIGURE 19.17 Results of numerical analyses presented in Grande et al. (2018): comparison of Spring Model prediction and full 2D FE Model.

Theoretical and FE models Chapter | 19

709

the global response of specimens when the upper interface constitutive law is modified according to the approach proposed in Grande and Milani (2018) is evident, that is, implicitly accounting for the damage of the mortar. At the end of the study, the authors emphasized that both the modification of the constitutive law of the upper interface and the introduction of a linear behavior of the mortar, that is, neglecting its progressive damage, led to different local behavior in terms of distribution of normal stresses in the mortar layer and shear stresses at the upper interface. Nevertheless, they also pointed out that these modifications did not affect the distribution of shear stresses at the lower interface and, moreover, they avoided further increases of shear-stress peak values at the upper interface: two phenomena governing the failure mechanism of specimens.

19.3 Considerations and suggestions The complexity of the bond behavior of FRCM systems applied on structural supports clearly emerges from the experimental evidence reported in the literature. Indeed, they underline the occurrence of mechanisms, generally not observed in the case of FRPs, which could particularly affect the shear stress transfer mechanism at the level of reinforcement-matrix-support and, consequently, the global performance of the FRCM system. Moreover, it also emerges that these mechanisms are influenced by several factors concerning the characteristics of materials (matrix, reinforcement, support) and also the configuration of the strengthening system (unidirectional or bidirectional). Indeed, the presence of the mortar cover over the reinforcement creates an additional reinforcement/matrix interface layer. These complex experimental outcomes clearly reflect the need for development of accurate theoretical/numerical modeling approaches able to account for all the possible failure mechanisms, and the interaction among them. Indeed, some of the modeling approaches available in the literature are generally based on approximations that have the advantage of reducing the computational efforts, but at the same time only allow capturing specific mechanisms. For instance, the use of 2D models generally neglects possible interactions among yarns, which can have particular relevance in the case of specific grid configurations. Other models do not consider either the damage in the mortar composing the matrix or the damage affecting the support: this only occurs in the case of materials comprising the matrix and the support characterized by high-strength properties. These simplified modeling approaches can be suitable for specific FRCM system configurations, where the discarded phenomena do not occur. In the context of the actual state of the art, where the available experimental tests underline a significant variability in terms of occurrence of the phenomena characterizing the resistant mechanism of FRCMs, together with the absence of conclusive studies concerning this topic, particular attention

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PART | IV Modeling of textile-reinforced mortar-strengthened masonry

has to be devoted to development of theoretical, analytical, and numerical modeling approaches. Indeed, although it is evident that the experimental outcomes represent a fundamental way to assess their reliability, the main feature of these models is their ability to predict and identify the most probable mechanisms characterizing the bond behavior of FRCMs. This allows the use of modeling approaches as design tools for the optimization of FRCM strengthening systems, in light of their potentialities and limitations. However, full characterization of constitutive laws and material parameters are necessary for achieving this objective. Most of the current experimental tests have been focused on shear debonding tests rather than constitutive modeling of the bond performance. These tests, although interesting for understanding the weak interfaces in the whole composite system (FRCM-strengthened masonry), are not suitable for constitutive modeling and proposal of bond
slip laws. The bond
slip laws extracted from these tests are usually obtained with several simplifying assumptions that can lead to erroneous predictions in numerical simulations under complex loading mechanisms. Fiber pull-out tests seem to be a suitable testing methodology for constitutive modeling of the bond response in these composites, but have received only limited attention (Dalalbashi et al., 2018). The choice of modeling strategy should take into account the complexity of the actions and expected failure mode and evaluation of the reliability of the adapted constitutive laws and input parameters. In most modeling strategies, an elastic response is assumed for both the brick and mortar and nonlinearities are concentrated at the fiber-to-mortar or textile reinforced mortar (TRM)-to-substrate interfaces. The TRM-to-substrate interface is usually ignored in the simulations as (1) the experimental results show that in most cases debonding at the TRM-to-substrate does not occur; (2) this failure mode should generally be avoided at the design and application stage by accurate surface preparation; and (3) measurement of the bond behavior at this interface in debonding tests is a complicated task and thus usually accompanied by reverse fitting or simplifying assumptions. The elastic response of the bricks is reasonable in bond-response simulations as failure of brick has not been observed in experimental tests. However, at structural scale simulations this assumption does not hold as explained in more detail in the following chapters. As for mortar, mortar cracking and splitting are very common in these tests and thus an elastic assumption seems to be inappropriate unless the effect of these failure modes are indirectly considered in the simulations.

References Abaqus, 2010. Abacus 6.10 Extended Functionality Online Documentation (generated September 29, 2010). Banholzer, B., 2004. Bond behavior of multi-filament yarn embedded in a cementitious matrix. PhD thesis, RETH Aachen University, United Kingdom.

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Caggegi, C., Carozzi, F.G., De Santis, S., Fabbrocino, F., Focacci, F., Hojdys, L., et al., 2017. Experimental analysis on tensile and bond properties of PBO and aramid fabric reinforced cementitious matrix for strengthening masonry structures. Compos. B 127, 175
195. Carbone, I., 2010. Delaminazione di compositi a matrice cementizia su supporti murari. PhD thesis, Universita` degli studi ROMA TRE. Carbone, I., De Felice, G., 2008. Bond performance of fiber-reinforced grout on brickwork specimens. In: SAHC 2008 2 6th International Conference on Structural Analysis of Historical Constructions Bath, United Kingdom, July 02
04, 2008. Carbone, I., De Felice, G., 2009. Debonding of C-FRCM composite on masonry support. In: Proceedings of Protection of Historical Buildings, PROHITECH 09, Rome, Italy. Carloni, C., Focacci, F., 2016. FRP-masonry interfacial debonding: an energy balance approach to determine the influence of the mortar joints. Eur. J. Mech. A/Solids 55, 122e33. Carozzi, F.G., Milani, G., Poggi, C., 2014. Mechanical properties and numerical modeling of Fabric Reinforced Cementitious Matrix (FRCM) systems for strengthening of masonry structures. Compos. Struct. 107, 711
725. D’Antino, T., 2014. Bond behavior in fiber-reinforced polymer composites and fiber-reinforced cementitious matrix composites. PhD thesis, Universita` degli Studi di Padova, Italy. D’Ambrisi, A., Feo, L., Focacci, F., 2012. Bond-slip relations for PBO-FRCM materials externally bonded to concrete. Compos. B 43, 2938
2949. D’Ambrisi, A., Feo, L., Focacci, F., 2013. Experimental and analytical investigation on bond between carbon-FRCM materials and masonry. Compos. B: Eng. 46, 15
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fiber bond behavior in PBO FRCM composites: a fracture mechanics approach. Eng. Fract. Mech. 117, 94
111. D’Antino, T., Sneed, L.H., Carloni, C., Pellegrino, C., 2015. Influence of the substrate characteristics on the bond behavior of PBO FRCM-concrete joints. Constr. Build. Mater. 101, 838
850. Dalalbashi, A., Ghiassi, B., Oliveira, D.V., Freitas, A., 2018. Effect of test setup on the fiber-tomortar pull-out response in TRM composites: experimental and analytical modeling. Compos. B: Eng. 143, 250
268. Faella, C., Martinelli, E., Paciello, S., Perri, F., 2009. Composite materials for masonry structures: the adhesion issue. In: Proceedings of Mechanics of Masonry Structures Strengthened with Composite Materials, Venice, Italy. Ferracuti, B., Savoia, M., Mazzotto, C., 2007. Interface law for FRP-concrete delamination. Compos. Struct. 80, 523
531. Grande, E., Imbimbo, M., Sacco, E., 2017. Local bond behavior of FRCM strengthening systems: some considerations about modeling and response. In: Proceedings of Mechanics of Masonry Structures Strengthened with Composite Materials MURICO5, June 28
30, Bologna, Italy. Grande, E., Milani, G., 2018. Interface modeling approach for the study of the bond behavior of FRCM strengthening systems. Compos. B: Eng. 141, 221
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251. Hartig, J., Ha¨ußler-Combe, U., Schicktanz, K., 2008. Influence of bond properties on the tensile behavior of textile reinforced concrete. Cem. Concr. Compos. 30, 898
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Milani, G., Lourenc¸o, P.B., 2012. 3D non-linear behavior of masonry arch bridges. Comput. Struct. 110
111, 133
150. Milani, G., Tralli, A., 2012. A simple meso-macro model based on SQP for the non-linear analysis of masonry double curvature structures. Int. J. Solids Struct. 49 (5), 808
834. Pellegrino, C., Tinazzi, D., Modena, C., 2008. An experimental study on bond behavior between concrete and FRP reinforcement. J. Compos. Constr. ASCE 12 (2), 180
189. Razavizadeh, R., Ghiassi, B., Oliveira, D.V., 2014. Bond behavior of SRG-strengthened masonry units: testing and numerical modeling. Constr. Build. Mater. 64, 387
397.

Chapter 20

Advanced finite element modeling of textile-reinforced mortar strengthened masonry Elisa Bertolesi1, Gabriele Milani2 and Bahman Ghiassi3 1

ICITECH, Universitat Polite`cnica de Valencia, Valencia, Spain, 2Department of Architecture, Built Environment and Construction Engineering (ABC), Politecnico di Milano, Milan, Italy, 3 Centre for Structural Engineering and Informatics, Faculty of Engineering, University of Nottingham, Nottingham, United Kingdom

20.1 Introduction The high seismic vulnerability of existing masonry is basically a consequence of low masonry tensile strength. The previous earthquakes such as those that occurred in Italy in 2009 (L’Aquila), 2012 (Emilia-Romagna), 2016 (Center Italy), and 2017 (Ischia) stressed once again that historical buildings, essentially made by masonry in the majority of cases, are unable to withstand horizontal loads and thus are highly vulnerable to seismic actions. There are some precautions implementable on existing buildings to prevent premature failures, such as external reinforcement with steel plates or surface coating with concrete reinforced with welded steel meshes. These are all well known in the scientific community and are classified as “conventional,” because of the use of standard concepts of strengthening realized with classical civil engineering materials. Unfortunately, they have proven to be complex, time consuming, and add considerable mass to the structure, which may increase the inertia forces induced by an earthquake excitation. About 20 years ago, after the devastating 1997
98 earthquake in Italy (Umbria
Marche earthquake), the use of fiber-reinforced polymer (FRP) strips as reinforcements instead of conventional methods appeared immediately groundbreaking. In Assisi, the collapsed groin vaults in Basilica Superiore were rebuilt almost immediately as they were before, but strengthened with abundant FRP strips at the extrados. Their limited invasiveness,

Numerical Modeling of Masonry and Historical Structures. DOI: https://doi.org/10.1016/B978-0-08-102439-3.00020-8 Copyright © 2019 Elsevier Ltd. All rights reserved.

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speed of execution, and good performance at failure (Triantafillou and Fardis, 1997; Triantafillou, 1998a,b; Corradi et al., 2002; Valluzzi et al., 2002) immediately appeared to be attractive to supersede any existing alternative strengthening strategy, especially for masonry curved structures (Bertolesi et al., 2016a, 2018a). However, during the last 15 years, the specialized scientific community became progressively more skeptical on the use and abuse of FRP strengthening, because it entails several drawbacks, such as low vapor permeability, poor behavior at elevated temperatures, incompatibility of resins with different substrate materials, relatively high cost of epoxy resins, no reversibility of the installation, and excessive stiffness when applied to historical masonry (Ghiassi et al., 2013, 2014, 2015, 2016a; Maljaee et al., 2016). The use of inorganic matrices has since become a popular alternative to circumvent such problems (Triantafillou and Papanicolaou, 2005; Triantafillou, 2011; D’Ambrisi et al., 2013; Carozzi et al., 2014; Razavizadeh et al. 2014; Wang et al., 2017). As a matter of fact, it is notorious that cement-based materials have low tensile strength compatible with existing masonry and indeed plaster reinforced with steel bars is a wellknown conventional strengthening technique for masonry, which however is not considered too much effective, because of the excessive thickness required for the plaster, sometimes affected by diffused detachment. Recently, innovative and thinner types of reinforcements have been introduced, such as short fibers (fiber-reinforced concrete) and continuous fibers in a fabric form [textile-reinforced mortar (TRM)]. Fabric-reinforced cementitious matrix (FRCM) composites represent a particular type of TRM where a dry-fiber fabric is applied to a structure through a cementitious mortar enriched with short fibers (Ghiassi et al., 2016b; Carozzi et al., 2018; Dalalbashi et al., 2018a,b; Bertolesi et al., 2018b,c). The mechanical properties of FRCM depend on the bond between the fibers and the matrix and may vary if the yarns of the fabrics are preimpregnated with resin. From a theoretical point of view, FRP is easier to characterize, because its typical failure mode is a brittle debonding from the substrate. In contrast, FRCM usually exhibits more complex failure modes. Typically, the stress
strain behavior of a FRCM coupon is trilinear, with a first phase that increases linearly according to the Young’s modulus of the mortar, a second phase where the cracks in the mortar start to grow, and a last phase in which the mortar is fully cracked and the curve assumes the same slope of the stiffness of the fabric (Bertolesi et al., 2014). As is understandable, since the technology is quite new, there is still a lack of advanced and reliable numerical models for the prediction of the behavior or FRCM masonry-reinforced specimens or structural elements beyond elasticity. The present chapter is aimed at providing a comprehensive state-of-theart on the use of homogenization approaches for reliable prediction of FRCM-reinforced masonry behavior in the nonlinear range. Both cases of in-

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and out-of-plane loads are discussed. In particular, two alternative approaches for an effective prediction of FRCM-reinforced masonry behavior are reviewed, namely a detailed 3D heterogeneous and a homogenization approach. In the first model, bricks and mortar joints are meshed separately typically using 3D elements, whereas FRCM is discretized by means of trusses (fiber grid) and 3D elements for the cementitious matrix. For the 3D elements, a concrete damage plasticity (CDP) model with softening and damage in tension and compression is used. In the homogenization approach, masonry is substituted by an equivalent nonlinear orthotropic material with softening, with mechanical properties deduced using a suitable homogenization model applied at the level of the unit cell (mesoscale). A unit cell is a representative volume element (RVE) which generates an entire wall by repetition. There are many different approaches available in the literature that have proven to be sufficiently reliable in the determination of macroscopic elastic and inelastic properties for regular masonry, but here only one of them is considered (Bertolesi et al., 2016b; Milani and Bertolesi, 2017a), mainly because the objective of the chapter is to discuss numerically the FRCM reinforcement rather than the masonry. In such a simplified model, the elementary cell is discretized by means of few triangular elastic elements (bricks) and holonomic interfaces (joints) where all the nonlinearity is lumped. The obtained homogenization model is characterized by either two or three unknowns under biaxial and shear stress states, respectively. The structural implementation is obtained applying the reinforcement to the already homogenized material. Brickwork is modeled with rigid quadrilateral elements and homogenized holonomic interfaces; FRCM is discretized using equivalent trusses with limited tensile strength and fragile behavior, connecting adjoining rigid elements. Equivalent mechanical properties of the trusses can be eventually tuned accounting for FRCM debonding or rupture of the fibers. The pros and cons of the two numerical procedures are discussed, mainly evaluating two critical features, that is, their reliability in fitting experimental force
displacement curves with crack patterns and the needed computational effort. First, a preliminary section reviewing the most important linear and nonlinear mechanical properties to adopt for FRCM reinforcement is given. An important issue is indeed represented by the complex modes of rupture that FRCM composites can exhibit, which are more than those expected, for example, for a reinforcing system made by FRP. In a FRCM strengthening system, indeed (1) an axial rupture of the yarns, (2) debonding from the cementitious matrix, and (3) detachment of the yarn 1 matrix system from the support can occur (Ghiassi et al., 2016a,b). Such discussion is based on a comprehensive experimental investigation including various activities, starting from the characterization of the (G) FRP grid and two types of mortars (a cementitious and a lime-based mortar), push
pull tests on double-lap reinforcements applied to single bricks and

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pillars, and pull-out tests on single yarns immersed in a mortar block, crucial information to have about the interface behavior between grid and matrix. Analytical/numerical indications are also provided, with the aim of putting at disposal a validated tool for the design and assessment of these reinforcement systems. Such collection of experimental evidence, combined with a simplified analytical and numerical approach appears particularly attractive for design purposes as preliminary tool, since it provides a quick estimate of the nonlinear behavior of the single reinforced specimens subjected to standard pull
push tests, and the most significant parameters (loads, eccentricities, boundary conditions, mechanical properties of the constituent materials, etc.) to be further used complex 3D-FEM (finite element model) simulations, as presented in the subsequent sections.

20.2 Experimental investigation on fabric-reinforced cementitious matrix applied to masonry In this section, a concise overview of a set of experimental results on mechanical properties of FRCM strengthening systems is provided (Carozzi et al., 2014). In particular, experimental information on the constituent materials mechanical properties (FRP grid, cementitious matrix, brick, and mortar) are briefly recalled, followed by the typical behavior of reinforced single bricks subjected to standard push
pull tests.

20.2.1 Constituent material mechanical characterization Table 20.1 shows the results of tensile tests obtained in Carozzi et al. (2014) and performed according to EN ISO 10618/2005 on a single roving of glass fiber
reinforced plastic (GFRP) impregnated with Styrene Butadiene TABLE 20.1 Characterization of a typical GFRP grid for FRCM reinforcement. Tensile tests

Average failure load

CoV

Unit of measure (UoM)

(kN)

(%)

Single roving, warp direction

1.11

2.7

Singe roving, weft direction

1.03

1.7

Grid strip, width 5 cm, three rovings, warp direction

3.36

1.3

Grid strip, width 5 cm, four rovings, weft direction

4.24

2.8

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TABLE 20.2 Characterization of compressive and flexural strength. Test type

Average compression strength

Average flexural strength

Symbol

fc

fft

Unit of measure (UoM)

MPa

MPa

Cementitious mortar

27.13

8.38

2.75

1.03

Lime-based mortar

Rubber in the warp and weft directions. Five specimens were tested in each direction, and the same characterization was also performed on a grid strip of 5 cm width, again in both warp and weft directions (five specimens in each direction). The experimental results are summarized in Table 20.1. GFRP, Glass fiber
reinforced plastic

20.2.2 Cementitious and lime-based mortar Two types of mortars, namely a cementitious and a lime-based mortar, were considered (which constitute upper and lower bounds for the applications) in this study. In both cases, the compressive and flexural strength were experimentally determined for six specimens, according to EN 1015-11, and the average results are summarized in Table 20.2. As can be observed and as was expected, cementitious mortar exhibits much higher strength (one order of magnitude), meaning that lime mortars can be used only in particular applications, such as for the seismic strengthening of historical masonry where there is better compatibility with the support material.

20.2.3 Mechanical characterization of bricks Two typologies of clay bricks were considered, namely a low-strength “historical block” and a “standard modern” brick with good mechanical properties. Geometric dimensions were those of common Italian clay bricks, that is, 250 3 55 3 120 mm3 (length 3 height 3 thickness). Compressive strength, elastic modulus, and indirect tensile strength according to relevant standards were determined and the results are summarized in Table 20.3.

20.2.4 Single-brick reinforcements, push
pull tests In this experimental tests, push
pull, double-lap joint tests were carried out with a GFRP grid bonded to the two sides of a single brick, as depicted in

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TABLE 20.3 Mechanical properties of the bricks. Test type

Elastic modulus

Compressive strength

Tensile strength

Unit of measure

GPa

MPa

MPa

Standard

EN 14580

EN 772-1

EN 12390-6

Historical bricks




22.32 (12)

1.81 (3)

Modern bricks

12.33 (4)

68.87 (12)

6.24 (3)

Note: Numbers in parentheses show the number of samples.

FIGURE 20.1 Experimental setup for debonding tests on single bricks, rig in the testing machine.

Fig. 20.1. The experimental results are summarized in Table 20.4. Different bond lengths (5, 10, and 15 cm) were considered to investigate the effect of these parameters on the debonding strength. An MTS testing machine with the ultimate capacity of 250 kN was used and a special test rig was designed and realized to perform the double shear lap debonding test (see Fig. 20.1).

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TABLE 20.4 Layout of the experimental tests. Brick type

Grid type

Matrix type

Unit of measure

Bond length

Reinf. width

cm

cm

# tests

Modern brick

GFRP grid

Cem. mortar

5 2 10 2 15

5

Five for each type

Historical brick

GFRP grid

Lime mortar

10 2 15

5

Three for each type

20.2.5 Modern bricks, cementitious mortar, and glass fiber
reinforced plastic From push
pull, double-lap tests reported in Carozzi et al. (2014), it was possible to observe how the failure load increases—as expected—with the bond length, but not proportionally. Indeed, in the shorter reinforcements (5 and 10 cm), a slippage of the grid inside the matrix at loads smaller than the tensile failure of the GFRP grid was observed. In contrast, for longer reinforcements a tensile failure of the GFRP grid without slippage occurred. No perceivable debonding was observed for an anchorage length equal to 15 cm. The experimental load
displacement curves obtained experimentally for the different bond lengths are shown in Fig. 20.2 [(A) 50 mm; (B) 100 mm; (C) 150 mm]. For a 5 cm anchorage length a large scatter of experimental strength is observed, which is probably a consequence of the very short anchorage that promotes premature failure in the presence of geometric defects of the specimens.

20.2.6 Historical bricks, lime mortar, and glass fiber
reinforced plastic The historical clay bricks were reinforced with the lime-based mortar and GFRP, with the aim of reproducing a similar situation that can be found in historical buildings where cementitious matrices are not accepted by some national commissions for architectural and landscape heritage conservation. The failure mode, for 50 and 100 mm bond lengths, was a full detachment of the matrix 1 fiber grid package from the substrate and for 150 mm a tensile failure of the mortar layer. Also, in this case, it seems that the minimum value of bond length to adopt in practical applications is 15 cm, but the corresponding failure load was equal to 1.73 kN, which is significantly lower (45%) than that found for the reinforcing system with cementitious mortar.

Anchorage length: 50 mm (A) 1.2 1

λ (kN)

0.8 0.6

Test #1 Test #2 Test #3 Test #4 Test #5 Numerical simulation Simplified analytical model

0.4 0.2 0

0

2

4

6

8

10

δ (mm) Anchorage length: 100 mm

(B)

2.5

λ (kN)

2

1.5 Test #1 Test #2 Test #3 Test #4 Test #5 Numerical simulation Simplified analytical model

1

0.5

0

0

2

4

(C)

6 δ (mm)

8

10

12

Anchorage length: 150 mm

4.5 4 3.5

λ (kN)

3 2.5

Test #1 Test #2 Test #3 Test #4 Test #5 Numerical simulation FEM 1D Simplified analytical model

2 1.5 1 0.5 0

0

1

2

3

4 5 δ (mm)

6

7

8

FIGURE 20.2 Push
pull, double-lap shear tests on single modern bricks. Force
displacement experimentally obtained curves. Comparison with numerical predictions: (A) 50 mm bond length; (B) 100 mm bond length; and (C) 150 mm bond length.

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5 Test #1

4.5

Test #2

cI

4

Test #3 Numerical simulation

f (N/mm)

3.5 3

cII

2.5 2 1.5 1 0.5 0

Δb1

kt 0

1

2

Δb2 3

4

5

6

δ (mm)

FIGURE 20.3 Left: specimen to determine the mortar
GFRP interface behavior. Right: experimental tangential stress
slip behavior of the GFRP
mortar interface.

20.2.7 Glass fiber
reinforced plastic
mortar interface bonding behavior The ad hoc pull-out test shown in Fig. 20.3 was arranged with the aim of investigating the interface behavior between the GFRP grid and the cementitious mortar. The experimental setup relied on a single yarn (coming from the standard GFRP grid) immersed into a mortar parallel piped with dimensions 10 3 2 3 1 cm3. Test results, depicted in Fig. 20.3, show an elastic phase followed by a sudden decrease of the strength (at the same mutual displacement between mortar and yarn) immediately followed by a slippage of the grid at a constant strength.

20.2.8 Predictive simple analytical/numerical model A simplified analytical/numerical assessment of experimental results is summarized here. In the model, it is assumed that all the deformability and failure mechanisms occur in the reinforcing system (constituted by the grid and the mortar). Three main hypotheses are made: (1) a mechanism comprised of two nonlinear springs disposed in series and constituted by the unbounded and bonded grid. The total deformation of the system is therefore due to the sum of the axial deformation of the unbounded GFRP grid and the contribution of the pure sliding of the grid within the surrounding mortar. For the sake of

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simplicity, it is assumed that the yarn exhibit axial deformation only in the unbounded region. In the experiments discussed in the previous section, this length is Lu 5 150 mm. The number of yarns is indicated with the symbol ny. (2) The grid-to-mortar interface is elastic
perfectly plastic, exhibiting an idealized stepped stress
slip behavior as in Fig. 20.3. In the same figure, kt indicates the elastic tangential stiffness of the yarn
mortar interface, whereas cI and cII the peak and residual strength. The bonded region is indicated with the symbol Lb. (3) The FRP yarn is elastic
perfectly plastic with infinite ductility, having elastic stiffness equal to kn 5 EFRP AFRP and ultimate strength equal to Fu 5 fu-FRP AFRP , where EFRP 5 70,000 MPa is the GFRP Young modulus, fu-FRP 5 1200 MPa is the GFRP ultimate strength, and AFRP is the transversal area of the single yarn. The transversal area of the single yarn may be evaluated experimentally, once fu-FRP is known, from direct tensile tests performed on a single yarn and conducted within the present research. Using the aforementioned hypotheses, it can be easily shown that the stiffness of the mechanical system constituted by the unbounded and bonded part is
 EFRP ny AFRP ΔLu 1 ΔLb KT 5 ðLu 1 Lb Þ 1 ny k t ð20:1Þ Lu Lu 1 Lb where ΔLu is the overall displacement due to the unbounded part and ΔLb is the overall displacement of the bonded part. The peak load is evaluated as follows (if grid is yielded): Fu1 5 ny cI Lb

ð20:2Þ

The displacement at the elastic limit is Δse 5 Fu1

Lu 1 Lb KT

ð20:3Þ

The displacement when the tangential force at the interface between yarn and mortar shifts from peak to residual value is Δsu1 5 ΔLu1 1 Δb1

ð20:4Þ

where ΔLu1 5 Fu1 Lu =ðEFRP ny AFRP Þ and Δb1 , defined in Fig. 20.3, corresponds to a slip where the interface strength drops from the peak to the residual load. Similarly, the residual strength of the overall system is evaluated as Fu2 5 ny cII Lb

ð20:5Þ

The displacement value reached due to a drop of the tangential force from the residual value to zero is: Δsu2 5 ΔLu2 1 Δb2

ð20:6Þ

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where ΔLu2 5 Fu2 Lu =ðEFRP ny AFRP Þ and Δb2 (see Fig. 20.3) correspond to a slip where the interface strength drops from the residual load to zero. Conversely, if the failure is associated with an axial stress in a yarn reaching the ultimate value, fu-FRP , then the failure load is: Fu 5 ny AFRP fu-FRP

ð20:7Þ

The application of the analytical/numerical approach to the push
pull tests on single bricks is summarized in the force
displacement curves reported in Fig. 20.2 from (A) to (C) for anchorage lengths equal 50, 100, and 150 mm, respectively. The analytical/numerical curve is the red dashed one (indicated with the label “simplified analytical model”). As can be observed, the predictions are in agreement with the experiments in terms of both deformability and strength.

20.3 Model I: Full 3D heterogeneous approach Any sufficiently robust commercial software, such as Abaqus can be suitably utilized for a full 3D heterogeneous discretization of a masonry panel reinforced with FRCM. While such a procedure is without a doubt the most realistic for a detailed numerical investigation, it requires huge computational effort when nonlinear material properties and refined 3D meshes are used. As is well known, in a standard heterogeneous approach, joints, blocks, and the strengthening material are meshed separately. A CDP or a smeared crack model, both very popular and diffused in commercial codes, can be used for blocks and mortar. The CDP model—present in the Abaqus standard materials gallery—is well documented and provides a suitable description of the nonlinear behavior of brittle or quasibrittle materials such as concrete, mortar, and brick. For illustrative purposes, the general uniaxial constitutive relationships required in CDP are shown in Fig. 20.4A and B, where the different inelastic behaviors in tension and compression are worth noting. Damage is isotropic and ruled by two scalar variables mutually independent (dt and dc), which govern the degradation of the elastic stiffness during the loading
unloading processes. The multidimensional strength criterion is characterized by either a Drucker
Prager (DP) or a modified (modification parameter equal to Kc) Mohr
Coulomb (MC) failure surface, where the plastic part of the deformation is eventually nonassociated and ruled by a dilation angle ψ. To avoid numerical instabilities on both DP cone and MC pyramid vertices, surfaces are usually regularized by a parameter e called “eccentricity.” In the heterogeneous simulations discussed in this chapter, the numerical parameters ψ, e, and Kc are assumed equal to 10 degrees, 0.1, and 2/3, respectively, in agreement with the suggested values in the literature (Van Der Pluijm, 1997; Vermeltfoort, 2002; Vermeltfoort et al., 2007; Mohamad et al., 2017). Typical meshes used in a full 3D heterogeneous discretization are shown in Fig. 20.5.

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(B)

σt σt0

σC σCu σC0

E0

E0 (1–dc)E0 (1–dt)E0

εplt

εt

εelt

εplc

εc

εelc

(C) – σ'1

(D–P model) (M–C model)

–σ'3

–σ'2

FIGURE 20.4 Constitutive behaviors in (A) tension; Drucker
Prager failure surface implemented into Abaqus.

(B)

compression;

and

(C)

FIGURE 20.5 Typical meshes used in full 3D heterogeneous analyses: (A) in-plane loaded masonry (diagonal compression tests) and (B) out-of-plane loaded masonry (single way bending).

As a rule, the FRCM-strengthened system is discretized by a continuous bidirectional grid of fibers constituted by trusses embedded into a cementitious matrix meshed with 3D eight-noded brick FEs, as depicted in Fig. 20.4. Trusses representing the fiber grid are assumed elasto-brittle in tension and unable to withstand compressive stresses. In order to reproduce exactly the same geometry of the real grid, the mesh must be extremely refined. It is

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worth mentioning that a perfect bond between the fibers and the matrix is assumed, both to slightly simplify computations and because experimental evidences showed that the interaction between cementitious matrix and strengthening grid plays a negligible role at the structural level. Moreover, the premature debonding of the FRCM system from the substrate is usually disregarded simply because that such a failure mode should generally be avoided. Indeed, the strengthening system usually maintains its integrity during the entire loading history if it is properly designed. Even if, as reported (e.g., in Prota et al., 2006), some cracks are visible in the outer strengthening layers, the collapse of structural elements is characterized by other failure modes, which indirectly confirm the high compatibility of the FRCM-tomasonry substrate.

20.4 Model II: Two-step holonomic homogenization model In the simplified homogenization model proposed here, two separate steps are considered. In the first step, masonry is considered unreinforced and homogenized at the mesoscale. The application of the strengthening material occurs on the already homogenized masonry, which is discretized at a structural level (macroscale) using rigid elements (Bertolesi et al., 2016c) and homogenized nonlinear interfaces, within a rigid body and spring model (RBSM). FRCM is modeled in step 2 using equivalent trusses with no compressive resistance and with limited tensile strength and fragile behavior. The mechanical properties of the trusses are suitably tuned to take into account the experimental push
pull double-lap experimental results. Each truss is assumed perfectly bonded at the extremes to the adjoining rigid elements.

20.4.1 Step 1: Mesoscopic homogenization of the unreinforced masonry In the first step (homogenization of the unreinforced masonry), the unit cell is discretized with 24 elastic constant stress triangles (bricks) and holonomic interfaces (mortar) where all nonlinearities are lumped. Imposing on the boundary symmetry and antisymmetry, only 1/4 of the RVE is considered, as depicted in Fig. 20.6A. Furthermore, it is worth noting that the stress state of elements 4, 5, and 6 is equal to that of elements 3, 2, and 1, respectively, a feature that simplifies further the numerical model. Holonomic constitutive relationships between microstresses (normal σ and tangential τ) and total jumps of displacements adopted are multilinear and the homogenization problem path independent, because the interfaces are holonomic and triangular elements elastic. A coupling between normal and shear stresses ruled by a frictional-cohesive law (MC criterion) is imposed on interfaces in order to realistically reproduce the actual behavior

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FIGURE 20.6 Micromechanical model utilized in step 1 (homogenization of unreinforced masonry): (A) geometry of 1/4 of the elementary cell; (B) horizontal stretching; (C) vertical stretching; and (D) macroscopic shear.

of mortar joints. Full details of the homogenization model here briefly reviewer can be found in Milani and Bertolesi (2017a,b). The following symbols are used: Ux½n (Uy½n ) denotes node n horizontal (vertical) displacements; U x (U y ) the horizontal (vertical) displacement applied at the cell boundary in a strain-driven biaxial (or shear) homogenization problem; Eb is the brick Young modulus; ν b is the Poisson’s ratio; and Gb 5 Eb =½2ð1 1 ν b Þ is the shear modulus. After the imposition of equilibrium on interfaces, compatibility equations and through suitable rearrangement, the results are particularly appealing, because the biaxial homogenization problem (i.e., in absence of macroscopic shear; see Fig. 20.6B and C) can be solved finding the solution of the following nonlinear system of two equations: 2 3 2 2 2 2H 4 1 2 νb I L 1 2 ν b II 5 Curve I: η 5 U y 1 Ux 2 ξ 2 L f ðξÞ 2 f ðξÞ νbL Eb n 2H Eb t 2 3 ð20:8Þ 2 L 4 1 2 ν b II 5 Uy 2 η 2 2 Hfn ðηÞ Curve II: ξ 5 U x 1 2ν b H Eb

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I;II where ξ 5 U x 2 Ux½9 , η 5 Uy½5 1 Uy½6 and fn;t are the normal (n) or shear (t) interface (vertical I, horizontal II) laws, and ξ and η can be determined graphically finding the point of intersection of curves I, II provided in Eq. (20.8). The solution strategy can be either graphical or numerical. The knowledge of ξ and η enables utilizing equilibrium and compatibility equations, a direct evaluation of all static and kinematic internal variables. Considering a pure shear problem (see Fig. 20.6D) the boundary uniform t displacements to apply are U x 5 ð2H 1 eh ÞExy on horizontal edges and t U y 5 ðL 1 2ev ÞEyx . In such a deformation case, it has been shown that independent variables are three, namely ξ t , ηt , and κ (with ξt 5 Ux½5 1 Ux½6 , ηt 5 Uy½3 , and κ 5 τ ð1Þ ev =Gb ) and the following three-equation nonlinear system is deduced: 2 3 t t U 2 η H 4H κ t y f II ðηt 2 κÞ 1 2 Gb 5 Gb 1 2ftI ðηt Þ 2 ξt 5 2U x 2 4 2 Gb L n L L=2 2 3 t t L 2U 2 ξ κ t t ð20:9Þ 42f II ðξ t 2 U Þ 2 x Gb 1 2 Gb 5 ηt 5 U y 2 x t 2Gb L H i ev h II t t 2ft ðξ 2 U x Þ 2 ftI ðηt Þ κ5 Gb

In several papers (e.g., Bertolesi and Milani, 2016a,b; Bertolesi et al., 2016d; Milani and Bertolesi, 2017a,b,c) an ad hoc iterative scheme with κ 5 0 in the first loop is adopted, so that the previous set of equations can be presented with a simple graphical solution, very similar to that used for the biaxial stress state. A detailed description of the solution strategy that can be adopted when a macroscopic tangential deformation is applied to the unit cell is reported in Milani and Bertolesi (2017a), where the reader is again referred.

20.4.2 Step 2: Structural implementation of the homogenization model The second step is performed at a structural level and relies on the implementation of the homogenized stress
strain relationships determined in the first step into a rigid element approach—also known as RBSM where masonry continuum is discretized by quadrilateral rigid elements interconnected by shear and normal nonlinear homogenized springs for in-plane actions and flexural-torsional springs (eventually modeled with assemblages of rigid links and nonlinear trusses) for out-of-plane actions, as depicted in Fig. 20.7 (Kawai, 1978; Casolo and Milani, 2010, 2013; Casolo et al., 2013; Bertolesi et al., 2016c, 2017). Such a straightforward procedure has the

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FIGURE 20.7 Nonlinear springs and rigid elements assemblage in the macroscopic homogenized RBSM model: (A and B) in-plane behavior and (C) out-of-plane Kirchhoff
Love behavior.

undoubtable advantage that meso- and macroscale are fully decoupled, that is, homogenized stress
strain nonlinear relationships of the springs connecting rigid elements are evaluated in step 1, without the need of solving new boundary value problems at each load step in each Gauss point. The disadvantage of RBSM is the intrinsic mesh dependence of the results in case of global softening. This limitation cannot be circumvented, but it has been recently shown that the deviation from the mesh-independent solution, for the majority of practical cases, is not large when reasonably refined meshes are utilized (Kawai, 1978; Milani and Bertolesi, 2017a). The equivalent force
displacement nonlinear relationships to apply for the single trusses/ springs characterizing the in- and out-of-plane behavior of the discrete model are classically obtained by properly scaling the homogenized stress
strain relationships obtained at the mesoscale. Typically, scaling parameters are obtained by a simple energy identification between the discrete and the

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continuum model, in agreement with Kawai (1978) indications, elementary deformations by elementary deformations. Analytical relationships of the homogenized continuum/discrete macroscopic model can be found by Bertolesi et al. (2016b) for the in-plane and Silva et al. (2017a,b) for the outof-plane case, respectively. Rectangular rigid elements can be used in case of regular geometries, but no theoretical complications occur when irregular quadrilateral rigid elements are adopted, with comparable performance of unstructured and structured meshes, as shown by Milani and Bertolesi (2017a). When dealing with the FRCM reinforcement, a grid of equivalent trusses exhibiting an elasto-fragile behavior is adopted, with very low compressive strength to account for the loss of equilibrium of the system in compression due to internal buckling. To some extent, a similar procedure was also utilized before for FRP strips (see, for example, Grande et al., 2008), which relies on the substitution of the reinforcement with nonlinear truss elements perfectly bonded to the support. The step of the trusses in the homogenized FEM is not related to the real step grid of FRCM and that equivalent areas for the trusses are used. While such an approach could appear at first glance quite rough, it allows obtaining reliable results at a fraction of the time needed by a detailed FE heterogeneous discretization, where the actual step grid is modeled. To suitably consider both the possible debonding of the FRCM from the support and the rupture of the grid for axial loads, an idealized bilinear stress
strain curve is generally imposed to trusses, as in Fig. 20.8. FRCM trusses peak stress is typically assumed equal to the minimum between tensile strength of the glass grid associated to axial rupture and the debonding pull-out force divided by the area of the trusses. In the absence of any code of practice indicating standard values for FRCM debonding stress to assume, indications by existing experimental/numerical data can be considered. However, in the majority of the cases, it has been 1500 f t glass = 1276 (MPa)

σ (MPa)

1000

500 E = 20 (GPa) 0

0

0.1

0.2 ∈ (–)

0.3

0.4

FIGURE 20.8 Equivalent stress
strain relationship used for trusses at a structural level representing the FRCM reinforcement.

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shown that FRCM composite delamination on entire walls is unlikely and failure is instead driven by the tensile rupture of the wires, as also confirmed by some recent experimental investigations. As a consequence, mechanical properties can be estimated using direct tensile tests on FRCM coupons. The experimental results showed a trilinear behavior mainly characterized by three distinct phases. The initial elastic phase is affected by the presence of the cementitious matrix. When the tensile strength of the mortar materials is reached, several cracks start to open, creating a progressive decrease of the elastic stiffness, until the ultimate stress is reached. Since the matrix is completely cracked, tensile actions transfer to the grid, which starts to be subjected to not negligible levels of stresses. As indicated by Bertolesi et al. (2014), the peak tensile stress of FRCM coupons is quite similar to the tensile strength of the fibers. Taking into consideration such results and in order to slightly simplify computations, the bilinear stress
strain relationship shown in Fig. 20.8 (or similar, changing numerical values depending on the typology or reinforcement used) can be used at a structural level.

20.5 In-plane loaded masonry, experimental benchmarking In this section, an in-plane benchmark of the numerical approaches previously reviewed is discussed. A detailed analysis can be found in Bertolesi et al. (2016c), where the reader is referred for further details. The experimental campaign used as benchmark relies in some reinforced tuff wallets subjected to standard diagonal compression and tested by Prota et al. (2006), Lignola et al. (2009) at the University of Naples “Federico II” (Italy). A cement-based strengthening characterized by continuous fibers arranged in a bidirectional grid was used. The laboratory investigation included four diagonal compression tests on unreinforced tuff panels and a total of eight tests conducted taking into account different configurations of the FRCM strengthening. Here, for the sake of conciseness, both numerical models (heterogeneous and homogenized) are benchmarked only on walls reinforced with a single-grid FRCM disposed on one side. The wallets have dimensions 1030 3 1030 3 250 mm3, made with blocks of dimensions 370 3 120 3 250 mm3, laid in a single leaf running bond pattern. Mortar joints thickness is equal to 15 mm. The strengthening system is comprised of a bidirectional AR glass-coated open grid, embedded into a FRP-modified cementitious matrix for the application on masonries. The overall thickness of the system is 10 mm for each ply of strengthening applied. To properly monitor displacements, four linear displacement variable transducers were placed along the two diagonals over a length of 400 mm in order to collect vertical and horizontal experimental displacements. The nonlinear monoaxial stress
strain relationships used for masonry inelastic elements (bricks, mortar, and homogenized masonry) in both the heterogeneous (3D elements) and homogenized (springs) numerical approaches

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are depicted in Fig. 20.9. It is worth mentioning that they are kept generally in agreement with the experimental data available. In particular, as far as the heterogeneous approach is concerned, the values of the compressive peak strength are assumed to be in agreement with Prota et al. (2006), while the tensile strength is tuned according to literature indications (Van Der Pluijm, 1997; Lourenc¸o, 1998; Nazir and Dhanasekar, 2013; Acito et al., 2016). Since no information is available for tuff in tension, a safe value equal to fcm/15 was adopted. Concerning the FRCM strengthening, some direct laboratory tests were performed to assess the mechanical properties of the cementitious matrix, while the data provided by the manufacturer are used with respect to the fiber grid. In particular, a tensile strength equal to 1276 MPa is assumed. Compressive strength is neglected, as is usually done for this kind of strengthening. The nonlinear homogenized stress
strain response deduced applying the homogenization approach previously reviewed is shown in Fig. 20.9B. The nonlinear homogenized stress
strain relationships are obtained stretching the elementary cell horizontally (left) and vertically (vertically) for the determination of Σxx and Σyy ; respectively, and assuming a homogeneous shear

FIGURE 20.9 Nonlinear behavior of inelastic elements used for masonry in the (A) heterogeneous (A) and (B) homogenized approaches.

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deformation state (right) incremented up to failure to evaluate Σxy . Elastic and inelastic mechanical properties for the constituent materials are those assumed in the heterogeneous approach (see Fig. 20.9A), with the exception of the inelastic behavior of bricks, which are assumed elastic in the homogenization approach. The compressive behavior—as usually done in the literature—is numerically taken into account using a macroscopic approach. As a matter of fact, masonry crushing is not easily reproducible with 2D simplified procedures, because it is usually characterized by mortar and brick failure with considerable 3D effects. The experimental stress
strain curves obtained during the tests are depicted in Fig. 20.10A and B (subfigure (A) refers to the unstrengthened panels, whereas subfigure (B) to the strengthened ones). The shear stress τ is calculated as τ 5 0:707V=An , where V is the experimental applied vertical load and An is the uncracked section area, while the shear strain γ is taken equal to γ 5 εv 1 εh , having indicated with εv , εh the vertical and horizontal experimental strains indirectly estimated by transducers installed on diagonals. As can be noted, in the unreinforced panels, failure is mainly characterized by tensile cracks arising on both mortar bed and head joints. That mechanism is peculiar, with the formation of stepped cracks that propagate following the least resistance line. The application of FRCM strengthening increases considerably the peak load and the ductility (see Fig. 20.10B) and partially tends to inhibit the formation of a mechanism involving sliding along joints, showing a more uniform distribution of cracks on the entire panel, mainly concentrated along the line of action of the splitting load. As far as the experiments without reinforcement are concerned, global stress
strain curves (average tangential stress vs average diagonal strain) obtained numerically with both models are depicted in Fig. 20.11 (subfigures (A) and (B), respectively, refer to heterogeneous and the homogenized approach) and compared with experimental results.

FIGURE 20.10 Experimental results used as benchmark for the in-plane validation: (A) unreinforced tuff panels and (B) FRCM-strengthened walls.

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FIGURE 20.11 Comparison between the global stress
strain curves obtained experimentally and provided using the heterogeneous (A) and homogenization (B) approaches for unreinforced panels. (C) Damage in tension obtained at collapse with the heterogeneous approach. (D) Shear and axial springs damage obtained with the homogenization approach at collapse.

As can be noted in both cases, a reasonable fitting of the initial elastic stiffness is obtained, with rather good estimates of the peak tangential stress, postpeak behavior and ultimate ductility. The less stiff response of the homogenized model can be explained by the utilization of a regular discretization, which is intrinsically unable to well capture the real blocks interlocking and favors slippage on preferential planes of weakness where shear springs are laid. In Fig. 20.11C and D damage in tension obtained at collapse with the heterogeneous approach and shear and axial springs damage at collapse obtained with the homogenization approach are depicted, respectively. As can be noted, the agreement with experimental evidences is generally acceptable, with collapse modes found numerically mainly characterized by a brittle failure concentrated near the vertical symmetry axis, soon after the peak tangential stress is reached. Failure is governed by a progressive damage spreading laterally along the line of action of the external load. The same simulations are replicated for the one-side-strengthened panels. The global stress
strain curves obtained in this case (both with the homogenized and heterogeneous model) are depicted in Fig. 20.12A, where

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FIGURE 20.12 FRCM-reinforced panel. (A) Comparison between numerical and experimental shear stress
shear strain curves. (B and C) Tensile damage parameter in the heterogeneous approach on the unreinforced (B) and reinforced (C) sides. (D and E) Damage on springs in the homogenization model [(D) tension; (E) compression], peak point.

experimental data are also represented. As clearly shown in Fig. 20.12, and as already pointed out by Prota et al. (2006) and Bertolesi et al. (2016d), FRCM presence has the beneficial effect of increasing both strength and ductility of the wall, significantly changing the failure mechanism. While experimental results exhibit a quite wide scatter (around 40% on the peak strength), the agreement obtained with both numerical models can be considered satisfactory, slightly in favor of the homogenized approach, which seems to fit better the experimental lower bound. In Fig. 20.12B and C, tensile damages at peak in the heterogeneous approach (subfigure (B) refers to the unstrengthened side, whereas subfigure (C) to the strengthened one) are depicted. As can be noted, larger damaged areas in both blocks and mortar are present on the strengthened side, whereas the unstrengthened surface appears less damaged, with cracks propagating mainly along mortar joints.

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Such an outcome appears in agreement with the intuitive mechanical behavior of FRCM, which allows transferring tractions from masonry to the strengthening and justifies the much larger strength of the FRCM-reinforced panel. The large ductility is also a consequence of the beneficial effect of FRCM, since the energy dissipation is considerable in the strengthened case. Inelastic strains at peak on springs in tension and compression in the homogenization model are depicted in Fig. 20.12D and E, respectively. Comparison of these results with the damage pattern observed in the heterogeneous model is not straightforward, because the homogenized approach is bidimensional and it is not possible to distinguish between masonryreinforced and unreinforced surfaces. In addition, there is not a distinction between joints and blocks, since their mechanical properties are smeared into homogenized springs. However, the general behavior of the panel appears in good agreement with heterogeneous predictions, with a clear formation of a vertical compressed strut and tensile cracks spreading laterally along the horizontal direction, a consequence of the low masonry tensile strength. Obviously a micromodeling approach is capable of showing in much more detail the damage evolution, starting on mortar joints and then diffusing inside bricks, but the computational effort is exponentially higher. It is also interesting to point out that FRCM tends to inhibit the formation of brittle failures such as cracks zigzagging on mortar joints, typically observed on unreinforced walls, diffusing inelastic dissipation more homogeneously. Therefore it preserves the global integrity of the panels at relatively high levels of external loads, at the same time allowing a more ductile behavior. As a matter of fact, the presence of a continuous strengthening layer, differently from the application of single FRP strips, has the quite important advantage of redistributing uniformly stresses on the entire surface. The role played by FRCM is indeed to enforce the overall contribution of the panel in carrying tensions and compressions, avoiding damage localization on the weakest elements, as occurs for unreinforced walls.

20.6 Out-of-plane loaded masonry In this section, a benchmark for masonry loaded out-of-plane (vertical bending) is provided. The experimental campaign considered as reference was carried out at the University of Miami (Babaeidarabad et al., 2014a,b) with the aim of assessing the efficiency of FRCM composites for masonry subjected to flexural loads. The experiments comprised a total of 12 full-scale rectangular panels of dimensions 1422 3 1220 mm2 (width 3 height) loaded uniformly out-of-plane up to failure and simply supported at both horizontal edges. One half was reinforced using a cement-based strengthening technique, whereas the rest was tested without reinforcement. Two different types of blocks were used, namely concrete units and clay bricks disposed in running bond, so that the thickness of the two sets were slightly different, 92

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and 102 mm, respectively. In both cases, 10 mm thick joints with type M mortar were employed. The reinforced masonry panels were strengthened on the tensile surface with a bidirectional balanced carbon fiber grid embedded into a 10 mm thick cementitious matrix. The experimental setup was designed to apply a horizontal uniformly distributed pressure throughout some air bags placed between the samples and a reaction wall. Details on the experimental setup are available in Babaeidarabad et al. (2014a,b), where the reader is referred for further details on the experimentation carried out. The results, in terms of bending moment versus midheight deflection, are depicted in Fig. 20.13 for unreinforced and FRCM-strengthened clay panels. As clearly visible, the experimental response of the as-built walls is characterized by an initial elastic phase followed by a brittle collapse. All the asbuilt panels exhibited a collapse mechanism characterized by the formation of a horizontal crack close to the midheight bed joint. Due to their brittle behavior, the experimentation was carried out applying the external pressure in a single cycle of loading, whereas for the strengthened walls the external pressure was applied with a series of loading and unloading cycles. In this latter case, the failure of the reinforced specimens occurred with the formation of midheight horizontal cracks that developed in the reinforcing cementitious layer. The opening of the cracks was also accompanied by fabric slippage until the collapse of the walls. When dealing with the detailed 3D micromodeling heterogeneous approach, the meshes shown in Figs. 20.14 and 20.15 are used for the clay brick and the concrete block walls, respectively. Where possible, computations are simplified using double-symmetry conditions (see Figs. 20.14A and 20.15A). The resulting FEMs are still quite demanding, resulting in a total of

FIGURE 20.13 Out-of-plane loaded panels: (A) unreinforced clay brick panels and (B) FRCM-strengthened walls.

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FIGURE 20.14 Out-of-plain loaded clay brick walls (unreinforced): (A) Heterogeneous FE mesh used for 1/4 wall and (B) deformed shape at collapse and tensile damage map (magnified 400 times) of the entire wall.

FIGURE 20.15 Out-of-plain loaded hollow concrete block walls (unreinforced): (A) heterogeneous FE mesh used for 1/4 wall and (B) deformed shape at collapse and tensile damage map (magnified 400 times) of the entire wall.

48,466 and 34,586 FEs, respectively, for clay and concrete supports. The final meshes are further adjusted in order to allow the presence of two FEs along the thickness of both head and bed joints. Due to the expected efficiency of the proposed homogenization approach, whole panels can be easily considered without making any symmetry consideration. The resulting discretization is quite light, in this case composed by a total of 504 rigid quadrilateral elements interconnected by homogenized springs exhibiting inelastic behavior with softening for both the supports analyzed. A satisfactory agreement is found when comparing the numerical predictions and the experimental results obtained analyzing all the unreinforced

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panels using both the numerical strategies, see for instance global force
displacement curves depicted in Fig. 20.16A and the failure mechanism found in the heterogeneous model (see Fig. 20.15B). The homogenization model is able to properly capture the elastic branches as well as the maximum bending moments. Slightly different results are achieved with the sophisticated tridimensional heterogeneous approach, but the formation of a flexural hinge in correspondence of the two central bed joints proves the correctness of the approach. Similar behavior is observed for the concrete unreinforced panels. When dealing with FRCM-reinforced panels, in agreement with the procedure utilized for in-plane actions, the cement-based material is again replaced by equivalent two-noded truss elements, whose mechanical properties are calibrated according to some experimental data available from University of Miami reports. More in detail, it should be pointed out that bilinear stress
strain curves are adopted for equivalent trusses, in agreement with results of tensile tests performed on FRCM coupons. As can be noted, FRCM-reinforced panels exhibit a failure mechanism similar to the unreinforced ones and characterized by the formation of a central cylindrical hinge, but with damage spreading laterally and involving neighboring joints (Fig. 20.17). As far as the FRCM composite is concerned, tensile damages are detected in the equivalent truss elements located in a portion of the wall spreading downward (and symmetrically upward) from the midheight section toward the farthest damaged horizontal joints, as clearly indicated in Fig. 20.17B. Such an outcome suggests that slipping phenomena are occurring between fibers and the cementitious matrix. No tensile rupture of the equivalent trusses is found numerically. The same damage spreading is visible in the homogenization approach. Globally, both numerical strategies can describe the force
displacement behavior of the masonry panels analyzed (see Fig. 20.16B, also on unloading branches).

FIGURE 20.16 Out-of-plane loaded panels; comparison between experimental and numerical data: (A) unreinforced walls and (B) FRCM-reinforced walls.

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FIGURE 20.17 Tensile damage maps at collapse of the concrete masonry panel obtained using: (A) the homogenization model and (B) the heterogeneous micromodeling approach.

20.7 Conclusion This chapter discussed two different numerical procedures to use for analysis of masonry walls reinforced with FRCM, under in- and out-of-plane loads. The first numerical procedure is straightforward, because it is a full 3DFE heterogeneous discretization of the FRCM-strengthened masonry wall. Any commercial code equipped either with a smeared crack or a CDP material model can be used to perform the analyses. In such a procedure, joints and bricks can be discretized by means of eight-noded quadrilateral elements exhibiting damage and softening in both tension and compression, considering distinct mechanical properties for mortar and units. FRCM system can be then modeled in a similar way, that is, using a damaging and softening material representing the cementitious matrix and with an internal grid (the fabric strengthening) constituted by elasto-fragile trusses. It was shown that the hypothesis of a perfect bond between matrix and grid is sufficiently reliable and improves the stability of the numerical models assumed. The second approach is a homogenization model, where the heterogeneous assemblage of mortar and bricks is substituted at a structural level—through a suitably simplified solution of the homogenization problem in the unit cell—with a homogenized material exhibiting orthotropic behavior and softening in both tension and compression. An effective solution of the homogenization problem can be obtained in different ways; one of them has been discussed in the chapter, which relies in subdividing the unit cell into a discretization with few elastic constant stress elements for bricks and holonomic interfaces with softening for joints. At a structural level, to further speed up the computations, the homogenized masonry material can be discretized using rigid quadrilateral elements linked by shear and normal springs with mechanical

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properties deduced from the aforementioned homogenization model. It is then convenient to discretize FRCM using equivalent elasto-fragile trusses perfectly bonded to rigid body centroids. Both models have been benchmarked in the chapter using existing experimental tests available on wallets loaded in- and out-of-plane. From simulation results, it was found that both models fit satisfactory well with both the global force
displacement curves and the crack patterns found experimentally, with a slight advantage in favor of the heterogeneous approach. This notwithstanding, the numerical efficiency of the homogenized approach is much higher and makes it particularly suited for analysis of entire structures and large-scale walls. All situations where any heterogeneous approach would be prematurely subjected to halting, despite parallelization with large workstations can be helpful in the reduction of the still-toolong processing times needed.

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154. Milani, G., Bertolesi, E., 2017a. Quasi-analytical homogenization approach for the non-linear analysis of in-plane loaded masonry panels. Constr. Build. Mater. 146. Available from: https://doi.org/10.1016/j.conbuildmat.2017.04.008. Milani, G., Bertolesi, E., 2017b. Simple quasi-analytical holonomic homogenization model for the non-linear analysis of in-plane loaded masonry panels: Part 1, Meso-scale. In: AIP Conference Proceedings, vol. 1863. Rhodes, Greece ,https://doi.org/10.1063/1.4992621.. Milani, G., Bertolesi, E., 2017c. Simple quasi-analytical holonomic homogenization model for the non-linear analysis of in-plane loaded masonry panels: Part 2, Structural implementation and validation. In: AIP Conference Proceedings, vol. 1863. Rhodes, Greece ,https://doi. org/10.1063/1.4992622.. Mohamad, G., Fonseca, F.S., Vermeltfoort, A.T., Martens, D.R.W., Lourenc¸o, P.B., 2017. Strength, behavior, and failure mode of hollow concrete masonry constructed with mortars of different strengths. Constr. Build. Mater. . Available from: https://doi.org/10.1016/j. conbuildmat.2016.12.112. Nazir, S., Dhanasekar, M., 2013. Modelling the failure of thin layered mortar joints in masonry. Eng. Struct. . Available from: https://doi.org/10.1016/j.engstruct.2012.12.017. Prota, A., Marcari, G., Fabbrocino, G., Manfredi, G., Aldea, C., 2006. Experimental in-plane behavior of tuff masonry strengthened with cementitious matrix-grid composites. J. Compos. Constr. 10, 223
233. Razavizadeh, A., Ghiassi, B., Oliveira, D.V., 2014. Bond behavior of SRG-strengthened masonry: testing and numerical modeling. Constr. Build. Mater. 16, 387
397. Silva, L.C., Lourenc¸o, P.B., Milani, G., 2017a. Nonlinear discrete homogenized model for outof-plane loaded masonry walls. J. Struct. Eng. (United States) 143 (9). Available from: https://doi.org/10.1061/(ASCE)ST.1943-541X.0001831. Silva, L.C., Lourenc¸o, P.B., Milani, G., 2017b. Rigid block and spring homogenized model (HRBSM) for masonry subjected to impact and blast loading. Int. J. Impact Eng. 109. Available from: https://doi.org/10.1016/j.ijimpeng.2017.05.012.

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Triantafillou, T.C., 1998a. Composites: a new possibility for the shear strengthening of concrete, masonry and wood. Compos. Sci. Technol. . Available from: https://doi.org/10.1016/S02663538(98)00017-7. Triantafillou, T.C., 1998b. Strengthening of masonry structures using epoxy-bonded FRP laminates. J. Compos. Constr. . Available from: https://doi.org/10.1061/(ASCE)1090-0268(1998) 2:2(96). Triantafillou, T.C., 2011. A new generation of composite materials as alternative to fiber reinforced polymers for strengthening and seismic retrofitting of structures. In: Nicolais, L., Meo, M., Milella, E. (Eds.), Composite Materials. A Vision for the Future. Springer-Verlag Limited, London, UK. Triantafillou, T.C., Fardis, M.N., 1997. Strengthening of historic masonry structures with composite materials. Mater. Struct. . Available from: https://doi.org/10.1007/BF02524777. Triantafillou, T.C., Papanicolaou, C.G., 2005. Textile reinforced mortars (TRM) versus fibre reinforced polymers (FRP) as strengthening materials of concrete structures. In: Seventh International Symposium of the Fiber-Reinforced Polymer Reinforcement for Reinforced Concrete Structures (FRPRCS), Missouri, USA ,https://doi.org/10.14359/14827.. Valluzzi, M.R., Tinazzi, D., Modena, C., 2002. Shear behavior of masonry panels strengthened by FRP laminates. Const. Build. Mater. . Available from: https://doi.org/10.1016/S09500618(02)00043-0. Van Der Pluijm, R., 1997. Non-linear behaviour of masonry under tension. Heron 42 (1), 42. Vermeltfoort, A.T., 2002. Deformation of the brick mortar interface in compression and the use of ESPI. Heron 47, 185
209. Vermeltfoort, A.T., Martens, D.R.W., van Zijl, G.P.A.G., 2007. Brick
mortar interface effects on masonry under compression. Can. J. Civil Eng. 34, 1475
1485. Available from: https:// doi.org/10.1139/L07-067. Wang, X., Ghiassi, B., Oliveira, D.V., Lam, C.C., 2017. Modeling the nonlinear behavior of TRM-strengthened masonry structural components. Eng. Struct. 134, 11
24.

Chapter 21

Macromodeling approach for pushover analysis of textilereinforced mortar-strengthened masonry Daniel V. Oliveira1, Bahman Ghiassi2, Reza Allahvirdizadeh1, Xuan Wang3, Gemma Mininno4 and Rui A. Silva1 1

University of Minho, Guimara˜es, Portugal, 2University of Nottingham, Nottingham, United Kingdom, 3University of Macau, Zhuhai, P.R. China, 4Polytechnic University of Bari, Bari, Italy

21.1 Introduction A significant part of the world’s buildings, from world heritage monuments to vernacular constructions, are made of structural masonry, which over centuries has proven to exhibit acceptable performance under gravity loads. However, the low tensile strength of materials and weak connections have made them considerably vulnerable with respect to ground motion excitations. This weakness has been widely observed in earthquakes, which caused significant damage and collapse of many buildings, economic losses, and even the loss of life. Hence, the development of efficient strengthening solutions has received much attention in the few past decades, in an effort to enhance the strength/ductility of existing structures, but also to make them affordable and compatible. With this aim, a variety of techniques have been tested, among which composite-based solutions, such as fiber-reinforced polymers, have gained special interest. They can efficiently promote loadbearing capacities and ductility with a negligible increase in mass; however, their poor high-temperature resistance, lack of vapor permeability, low reversibility, brittle failure, and incompatibility with masonry substrates are some of their crucial drawbacks (Valluzzi et al., 2014). Thus conventional organic matrices (e.g., epoxy) and strips have been progressively substituted by inorganic matrices (typically cement- or lime-based ones) and fiber grids to overcome the aforementioned issues. This technique is usually known as fiber-reinforced cementitious matrix or textile-reinforced mortar (TRM). Numerical Modeling of Masonry and Historical Structures. DOI: https://doi.org/10.1016/B978-0-08-102439-3.00021-X Copyright © 2019 Elsevier Ltd. All rights reserved.

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In recent years, many experimental studies have been conducted to characterize the material properties, bond characteristics, and influence of TRM on the response of different structural components (Papanicolaou et al., 2007; Borri et al., 2009; Babaeidarabad et al., 2013; Bernat et al., 2013; D’Ambrisi et al., 2013a,b; Larrinaga et al., 2013; Carozzi et al., 2014; De Felice et al., 2014; Larrinaga et al., 2014; Razavizadeh et al., 2014; Ascione et al., 2015; Carozzi and Poggi, 2015; Garofano et al., 2016; Ghiassi et al., 2016; Ramaglia et al., 2016; Caggegi et al., 2017; Dalalbashi et al., 2018a,b). However, researchers still face several unknowns like the long-term behavior of the bond between the strengthening system and substrate or even between the mortar and mesh. On the other hand, it is evident that their widespread use will not be possible without enabling engineers to study them numerically for assessment purposes, as only very limited numerical studies have been carried out to date. This chapter aims to point out key aspects of the modeling of TRM systems—from material characteristics and bond properties to structural elements. In addition, the influence of applying them on masonry subassemblies is evaluated by investigating both in-plane and out-of-plane responses. With this aim, two types of masonry substrates, namely brick and rammed earth, are considered.

21.2 Macromodeling strategies for textile-reinforced mortar-strengthened masonry Generally, composite materials (such as masonry and TRM systems) are comprised of materials with different characteristics that are bonded to each other. Hence, their modeling requires not only consideration of material properties, but also its dependency on the definition of their interfaces (e.g., mortar-to-mesh and TRM-to-substrate). Considering all aforementioned aspects leads to a detailed modeling approach, which is micromodeling. In other words, the micromodeling strategy for TRM-strengthened masonry consists of modeling units, mortar joints of masonry, matrix and fiber grids, as well as their interfaces. It is evident that following this approach requires a large variety of parameters (elastic/nonlinear and in different directions) to be known, which consequently results in computationally expensive numerical models and less than stellar results. Larrinaga et al. (2014) used individual truss elements for meshes and curved shell elements for mortars to model TRM specimens subjected to uniaxial tensile loads. Conventional bond-slip behavior obtained from previously conducted single- and double-lap shear (Caggiano and Martinelli, 2012; D’Ambrisi et al., 2013a,b; Grande et al., 2013; Carozzi et al., 2014; Razavizadeh et al., 2014) tests was assigned to interfaces. Comparing the numerical outcomes with corresponding experimental responses, minor differences were found between such models with those that consider the

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interface as rigid (Larrinaga et al., 2014). The same conclusion was obtained from conducting direct shear tests on steel reinforced grout (SRG) specimens, which mostly failed by debonding rather than slippage or failure in substrate (Borri et al., 2009). On the other hand, experiments conducted on strengthened masonry components have shown that debonding between strengthening and substrate mostly occurs at the end of the tests or was not observed. Thus considering a perfect bond between TRMs and masonry would be a realistic assumption and may result in accurate numerical predictions (Basili et al., 2016; Garofano et al., 2016; Ramaglia et al., 2016). It is worthwhile to note that such micromodels precisely work in small-scale specimens, but their application to structural/component modeling considerably increases the computational costs. For this reason, the use of continuum elements has received a great attention in the literature (Garofano et al., 2016). This modeling strategy is known as macromodeling; see a schematic comparison with micromodeling for TRM-strengthened masonry components in Fig. 21.1. As can be seen, the macromodeling approach models units and mortar joints with a single material and also follows the same approach for mortar and mesh in TRM. This chapter aims to assess the influence of strengthening masonry subassemblies using TRM (mortar and fiber grid). Hence, in addition to the substrate (masonry), a relatively homogenization approach should be followed for the strengthening. It may be carried out by proposing analytical expressions or empirical ones, depending on the state of the available experimental data. Detailed information regarding macromodeling of masonry can be found in Lourenc¸o (1996), but to the knowledge of the authors no significant studies (especially analytical ones) have been conducted to propose such MicroModeling approach Head joint

MacroModeling approach Composite masonry

Bed joint

Mortar-to-mesh

Textile/mesh

TRM composite

FIGURE 21.1 Comparing micro- and macromodeling strategies for masonry and TRM.

PART | IV Modeling of textile-reinforced mortar-strengthened masonry

Stress/force

748

Crak development

Un-craked

Strain/deformation Mesh grid

Experimentally obtained behaviour of the composite

Composite behavior in FE modeling

FIGURE 21.2 Schematic view of the macromodeling strategy of TRM strengthening.

analytical stress strain relationships (yield criterion) for TRM. For this reason, using average outcomes of the uniaxial tensile/compressive tests on composites seems to be the most rational approach. Additionally, some other approaches like using crack control (Aveston Cooper Kelly theory) may also be employed. Based on this theory, the matrix is held within the strengthening only due to the presence of the fibers and a critical state of shear stresses that causes them to slide or for the composite to be cracked. Employing such approaches leads to theoretical trilinear stress strain behaviors (Larrinaga et al., 2013). As is evident, previously conducted experimental outcomes could be employed to assign a composite behavior to the whole strengthening instead of modeling mortar and mesh grid individually. These approaches are schematically illustrated in Fig. 21.2. Additionally, one of the key parameters regarding the nonlinear modeling of TRM relates to the fracture model in compression and most importantly in tension. Most of the studies surveyed in the literature employed exponential softening for tension and parabolic softening for compression with classical smeared crack formulation using a rotated or combined rotating-fixed crack model (Ghiassi et al., 2013; Giamundo et al., 2014; Larrinaga et al., 2014; Bernat-Maso et al., 2015; Basili et al., 2016; Garofano et al., 2016; Wang et al., 2017). However, it should be mentioned that most of the studies adopted the rotating crack model. This strategy is employed in the following case studies, where detailed information is provided.

21.3 Analysis of a masonry wall In this section, the efficiency of TRM strengthening on enhancing the shear in-plane and out-of-plane responses of masonry walls is investigated using an advanced nonlinear finite element (FE) model (Mininno, 2016; Mininno et al., 2017).

21.3.1 Description of the case study Aiming at studying the seismic response of masonry, two models representing the in-plane and out-of-plane behaviors are considered. Their geometry

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×

(A)

×

(B) FIGURE 21.3 Geometry of considered masonry walls: (A) in-plane model and (B) out-ofplane model.

is shown in Fig. 21.3. The in-plane model represents a fac¸ade wall containing an opening and lintel, where the load from a hypothetical timber slab or roof is assigned as a linear distributed load at the top. Consequently, the outof-plane model is similar to the former case, but two transversal walls have been added for stability.

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FIGURE 21.4 (A) Composite Rankine Hill yield criterion (Lourenc¸o, 1996) and (B) eightnode quadrilateral shell element (DIANA FEA BV, 2017).

21.3.2 Modeling considerations A macromodeling strategy is adopted for both the plain and strengthened models, using the nonlinear FE code DIANA (2017). Hence, the hypothesis of quasibrittle material with the capability of taking into account the cracking phenomenon is employed by considering parabolic hardening and exponential softening behaviors, respectively, for compression and tension, with different yield criteria for each. The aforementioned yield criterion is based on a plane stress anisotropic yield law that considers a Hill-type criterion in compression and a Rankine type criterion in tension; see Fig. 21.4A (Lourenc¸o, 1996). Quadrilateral eight-node curved shell elements are used (Fig. 21.4B). An elastic behavior is assigned to the lintel above the opening, with the same elastic parameters of the masonry. A similar approach is followed for the strengthened in-plane model; however, the TRM strengthening layer is simulated by implementing an additional set of eight-node curved shell elements for the matrix in which the textile is embedded as an equivalent grid with no possibility of slippage (see Fig. 21.5A). On the other hand, these elements in the strengthened out-of-plane model are substituted by eight-node curved layered shell

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FIGURE 21.5 (A) Embedded grid reinforcement in curved shell elements and (B) layered curved shell elements adopted for the out-of-plane model (DIANA FEA BV, 2017).

elements, where each layer may have its own material properties and is numerically integrated separately (see Fig. 21.5B). Regarding the modeling of the matrix, a total strain crack model was assigned, assuming for the tensile behavior a softening law based on the fracture energy approach (nonlinear softening with plateau function from the Japan Society of Civil Engineers (JSCE) as shown in Fig. 21.6A). On the other side, the behavior in compression is simulated by a parabolic function. Moreover, a brittle linear elastic behavior is used for the tensile behavior of the fiber, for which a fragile failure is expected after the peak strength is reached. In this study, three types of fiber grid reinforcements have been adopted: polyparaphenylene benzobisoxazole (PBO), basalt, and glass (see Fig. 21.6B). Finally, the interface between masonry substrate and the applied strengthening on both the in-plane and out-of-plane models is considered as a perfect bond (no relative displacement between substrate and strengthening). It is achieved by adopting quadrilateral 8 1 8 node elements, which couple the displacements of the masonry walls with the reinforced mortar. It is evident that such an approach may not lead to a detailed understanding of TRM strengthening behavior, but it can be effectively used for the aim of investigating the influence of the strengthening on enhancing the lateral response of walls.

21.3.3 In-plane behavior of the unstrengthened and textilereinforced mortar-strengthened models As for the boundary conditions of the in-plane model, all base nodes are fixed. Moreover, the whole wall is subjected to its own self-weight together

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FIGURE 21.6 (A) Nonlinear behavior of mortar (DIANA, 2017) and (B) uniaxial tensile behavior of reinforcement.

with the top distributed load. In the following, the seismic response of the wall is investigated using pushover analysis. It was previously shown that outcomes of pushover analysis are reliable enough in terms of average values, but some scatter on the predicted damage should be expected (Allahvirdizadeh and Gholipour, 2017). Initially, the in-plane performance of the wall subjected to horizontal loading is investigated for two different levels of precompression states, namely when it is subjected only to its own weight (SW) and when it has a distributed top load in addition to its self-weight (SW 1 DL). As can be seen, a higher level of precompression leads to a higher lateral load capacity

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(49% increment) and also ultimate lateral displacement. Hence, further investigations were conducted by considering the SW 1 DL load condition. In the following, the influence of adding a TRM composite strengthening layer with different types of reinforcement (PBO, basalt, and glass) at both sides of the wall is investigated (Fig. 21.7B). As is evident, adopting TRM

FIGURE 21.7 In-plane capacity curves: (A) influence of gravity loads and (B) TRMstrengthened wall.

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FIGURE 21.8 Contour map of the principal compressive stress: (A) plain wall at load factor α 5 1.69 and (B) strengthened wall with PBO reinforcement at load factor α 5 6.11.

composites considerably increases both strength and displacement capacities. In particular, the lateral load factor is enhanced by 160%, 207%, and 262%, respectively, for glass, basalt, and PBO reinforcements. A more detailed investigation of the structural performance may be achieved by plotting the contour map of principal compressive stress (Fig. 21.8) of both plain and strengthened walls at a lateral load equal to their peak capacity. The formation of a diagonal compressive strut is evident on both walls. The extreme values (for both cases) are concentrated at the toe of the piers and also at the corner of the opening (lintel). The adopted strengthening solution does not change the lateral load path of the wall, but the compression stresses in the strengthened model are higher than that of the plain one.

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The results seem to indicate that the failure of the in-plane model is characterized by a rocking motion. A relatively similar conclusion can be derived from the contour map of the principal tensile strains, where the largest deformations occur at the base of the piers, exactly on the opposite side where the plastic hinges are supposed to form. These regions are the locations where cracks will appear. Due to the sake of brevity, the contour map of principal tensile strains is not reported here, but these aforementioned regions are shown by circles in Fig. 21.8. Furthermore, some cracks also may be distinguished at the corners of the masonry above the opening, which shows the contribution of this part against lateral loads.

21.3.4 Out-of-plane behavior of the unstrengthened and textilereinforced mortar-strengthened models An approach similar to that followed for the in-plane model was used to investigate the influence of TRM strengthening on the out-of-plane response of masonry. Regarding the boundary conditions, the nodes at the base are fixed with respect to translation along all degrees of freedom. Additionally, an identical top distributed load is assigned to the top of the wall. The efficiency of strengthening is investigated considering two different layouts: the case in which TRM strengthening was applied only on the outer side of the wall and the case in which strengthening was adopted on both sides (inner and outer). Moreover, the results are only presented for the PBO reinforcement, due to its better performance shown on the in-plane response. In the following, pushover analysis was conducted by monotonically increasing a lateral load proportional to the mass of the subassembly. The obtained out-of-plane capacity of the plain masonry wall in comparison to the strengthened models is illustrated in Fig. 21.9A. The corresponding results of the strengthened models are reported up to the last converged step; hence, the discussion will be made on that step. The obtained pushover curves show efficiency of TRM strengthening on promoting the out-of-plane capacity (both strength and displacement). For instance, 30% and 38% enhancement can be distinguished regarding the ultimate lateral load capacity for the cases in which strengthening was applied on one and two sides of the wall, respectively. The deformed shape of the plain model and corresponding contour map is shown in Fig. 21.9B. It is worth noting that a similar same shape was observed for the strengthened model. As is clear, the transversal walls restraint rotations of the extreme part of the fac¸ade, while the midspan area is prone to rotate. Hence, the possible out-of-plane failure mechanism of the masonry wall would be as illustrated in Fig. 21.9C. This statement should be verified using detailed studies, that is, investigating principal compressive stresses and also principal tensile strains.

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Load factor, α

1.5 1.2 0.9

Reference node

0.6 0.3 0.0 0.0

0.5

1.0 1.5 Displacement (mn)

2.0

0.5

(A)

(B)

(C)

FIGURE 21.9 Out-of-plane model: (A) influence of TRM strengthening on the capacity; (B) deformation contour map of the unstrengthened model at its peak capacity; and (C) failure mechanism.

The contour map of the principal tensile strains of the plain out-of-plane model at a lateral load level equal to its peak capacity is presented in Fig. 21.10A and B. The results are plotted in the extreme integration layers and for both inner and outer sides of the walls. In fact, in the inner part of the fac¸ade, the maximum strains (crack initiation zone) are mainly located in the midspan that tends to rotate as schematically illustrated in Fig. 21.9C.

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FIGURE 21.10 Out-of-plane model: (A) principal tensile strains in the plain model (inner side); (B) principal tensile strains in the plain model (outer side); (C) tensile stresses in the strengthening (layout on one side, x local axis); and (D) tensile stresses in the strengthening (layout on one side, y local axis).

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Moreover, a crack pattern is observed in the connection between the fac¸ade and transversal walls. The latter is more clear in the inside view of the presented principal tensile strains (see Fig. 21.10A). Additionally, a high tensile strain concentration is evident on the base of the fac¸ade, particularly in the inner side. Hence, the out-of-plane failure of the wall seems to be a combination of horizontal bending and overturning. In summary, an efficient strengthening solution should prevent or postpone the aforementioned failure mechanism to enhance the out-of-plane response of the masonry wall. Finally, the efficiency of applied strengthening is assessed by investigating the principal tensile stresses in the reinforcement for the model at its peak capacity. It is worth noting that similar outcomes were obtained for both strengthening layouts. Hence, the results of the layout only on the one side are presented in Fig. 21.10C and D. These results are in terms of stresses along two planar local axes of the reinforcement. By comparing these outcomes with the previously reported response of the plain model, it is concluded that the applied strengthening works exactly in the regions where the plain wall tends to fail.

21.4 Analysis of a masonry fac¸ade In this section, the efficiency of applying TRM as a strengthening solution for improving the in-plane shear behavior of masonry walls is numerically investigated. The results are presented for a real masonry fac¸ade located in the historical center of Macau. The outcomes of this section are based on Wang (2015) and Wang et al. (2017).

21.4.1 Description of the case study The historic center of Macau is the oldest, most complete, and consolidated array of European architectural legacy standing intact on Chinese territory, mostly constructed between the 18th and 19th centuries. This case study is located in Patio da Felicidade street, where door gates and arcade, representing a typical Lingnan style of architecture for shop houses in south China are built. The buildings in Patio da Felicidade are mainly constructed using clay brick with mortar joints, with a fac¸ade thickness of about 0.2 m; see a general view and typical dimensions in Fig. 21.11. The following sections report the modeling procedure of these facades, their in-plane performance, and the influence of TRM strengthening.

21.4.2 Modeling considerations Numerical models should include both plain masonry and strengthening. Regarding the masonry, an identical approach discussed in Section 21.3.2 is followed here. The applicability of this modeling technique is validated by

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FIGURE 21.11 Buildings in Patio da Felicidade: (A) general view and (B) typical sketch and dimensions.

the experimental and numerical work from Milani et al. (2006). Furthermore, since no significant damage was observed in the interface between substrate and strengthening, it is assumed to be as perfect. Hence, this section only contains the modeling of the TRM strengthening solution. However, few studies have been previously conducted to examine the behavior of masonry shear walls strengthened with TRM composites. The mechanical behavior of TRM is typically distinguished into three phases. In the first phase, the load is carried primarily by the matrix until cracking initiates. In the second phase, the matrix undergoes a multicracking process resulting in transfer of stresses from the matrix to the mesh, with some debonding at the mesh matrix interface. At the third phase, the composite system behaves almost linearly until failure occurs due to the progressive rupture of the grid (Mininno, 2016). Thus the employed modeling technique is initially validated by reproducing the tensile and bond experimental results available in literature. It is worth noting that the models are constructed following the concepts of macromodeling techniques using the same FEs denoted in Section 21.3.2 and the analyses were performed again using DIANA software. The tensile behavior is validated by comparing numerical predictions with those results obtained from uniaxial tensile experiments on TRM composites conducted by Carozzi and Poggi (2015). Regarding the modeling of the mortar, a total strain crack model with two different sets of softening functions based on fracture energy, namely exponential softening and nonlinear softening with a plateau (according to JSCE), are adopted for tensile

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Load (kN)

6

4

2

0 0

1

2

3

4

5

6

Displacement (mm)

FIGURE 21.12 Validation of different tensile behaviors of TRM.

behavior. Moreover, a parabolic (traditionally used for concrete) function is assigned to the mortar in compression accompanied by a linear elastic behavior with fragile failure for the mesh. The numerical results and experimental envelop are shown in Fig. 21.12. The JSCE model presents a smooth increase of stiffness in the early second phase, but both models accurately reproduce the test results (both the peak load and ultimate displacement) and can be used for further investigation. It should be mentioned that no meaningful difference in the first and third phases of the response was observed between the models with rough and refined mesh sizes, although decreasing element size may lead to a smoother transition from the second phase to the third one.

21.4.3 Behavior of the unstrengthened and textile-reinforced mortar-strengthened models Site investigations showed that nine similar building walls form one block fac¸ade (see Fig. 21.11A), but initially, a single wall was considered to evaluate its performance, estimate enhancements promoted by the strengthening solution, and find the most proper layout for the strengthening application (see Fig. 21.11B). In addition to the self-weight of the structure, a distributed load coming from the slab and roof (both timber) are assigned at the corresponding level. Regarding the boundary conditions, all nodes at the base are fixed. An incremental gravity analysis is carried out to control the self-weight safety factor (Fig. 21.13A). As can be concluded, the safety factor under gravity actions is about 4.5, which states that the fac¸ade is safe enough when subjected to vertical loads. Moreover, principal tensile strains as an indicator of cracking are

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5

Safety factor

4

SF ~ 4.5 3 2 1 0 0.0

0.2 0.4 0.6 Vertical displacement (mm)

0.8

(A)

(B)

(C)

FIGURE 21.13 Single fac¸ade under vertical loads: (A) capacity curve; (B) principal tensile strains at the self-weight load level; and (C) principal tensile strains at the safety factor load level.

presented in Fig. 21.13B, both under self-weight and at the safety factor load level. The results show that cracks mainly occur in the middle of the top spandrel. Later, a pushover analysis was conducted to obtain the capacity of the unreinforced single fac¸ade and also to assess the efficiency of TRM as strengthening solution (Fig. 21.14A). Three different textile materials, that is, steel (SRG), glass, and PBO, were taken into account. The PBO and glass grids are assumed bidirectional, while steel fibers are unidirectional as is the case for commercial products. The PBO and glass are uniformly applied through all of the fac¸ade, while SRG is applied in the horizontal direction

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Load factor, α

0.6

0.4

Unreinforced facade Glass reinforced PBO reinforced Steel reinforced

0.2

0.0 0

3

6 9 Lateral displacement (mm)

12

15

(A)

(B)

(C)

FIGURE 21.14 Pushover of the single fac¸ade: (A) capacity curves; (B) principal tensile strains of the unreinforced single fac¸ade at its peak capacity; and (C) principal tensile strains of steel reinforced fac¸ade at its peak capacity.

over the spandrel walls and in the vertical direction along the piers. It is also worth mentioning that the strengthening is applied on both sides of the wall. As is evident, the nonlinear behavior of the plain structure starts at a very early stage (around a lateral displacement of 0.5 mm) and the maximum load coefficient reached is of about 0.26. This coefficient value is quite low, although in reality the fac¸ade is connected to the adjacent buildings composing a longer fac¸ade. The result shows that the SRG solution has the highest effectiveness in improving the performance of the fac¸ade (reaching a load coefficient of

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0.63), resulting in a 142% increase with respect to the plain fac¸ade. The principal tensile strains of the unreinforced fac¸ade at a lateral load equal to its peak capacity (Fig. 21.14B) show that the main cracks may locate at upper and lower corners of the spandrels and also at toes of the piers. It is worth noting that cracks have been reported in the existing building at the same places. Hence, the failure mode of the plain fac¸ade has two phases. First, the separation of the two wall piers from spandrels occurs, followed by independent rocking motion of two piers. In addition, the principal tensile strains of the SRG-reinforced fac¸ade at a lateral load equal to its peak capacity are presented in Fig. 21.14C. It can be seen that the largest cracks are again located at the upper/lower corners of spandrels and at the base. The SRGstrengthened fac¸ade has higher absolute values of principal tensile strain and less strain concentration; hence it is believed that it can dissipate more energy. It should be noted that the failure mode of the TRM-strengthened fac¸ade is the same as that of the plain fac¸ade. In the following, the effect of different strengthening schemes on the performance of the fac¸ade is investigated. Two strengthening schemes denoted as “Steel-H” and “Steel-V” are considered. As previously discussed, the failure of fac¸ade consists of two phases, that is, separation of piers from spandrels and then their overturning. The “Steel-H” strengthening scheme (SRG strengthening of spandrels only) increments the shear resistance at first phase and piers are strongly connected to each other. Meanwhile, the strengthening type “Steel-V” (SRG strengthening of piers only) aims to increase the shear resistance at the second phase as two wall piers are strongly connected to the base. The schemes and corresponding capacity curves are presented in Fig. 21.15A. It can be seen that the “Steel-H” case has higher stiffness (similar to full strengthened fac¸ade) but it reduces after a critical point. This is due to the fact that beyond a critical point the failure mode converses from the first phase to the second one. On the other side, the “Steel-V” fac¸ade has lower stiffness since spandrel walls without horizontal strengthening face earlier damage. Additionally, the principal tensile strains of both cases at a lateral load equal to their peak capacity are shown in Fig. 21.15B and C. It is clear that the “Steel-H” scheme could effectively limit cracks (separation) in corners of spandrels, but the “Steel-V” scheme prevented rocking motion from happening, which consequently leads to higher ductility. As previously mentioned, the investigated single fac¸ade is a part of a fac¸ade block containing nine of the studied one. In the following, the influence of TRM strengthening on in-plane seismic response of this block is evaluated. Due to the large dimension of the whole fac¸ade, only the horizontal SRG strengthening scheme (Steel-H) is taken into account. This layout may save considerable strengthening material and can acceptably enhance the mechanical performance. The obtained capacity of the block fac¸ade (both plain and strengthened) in comparison to the single fac¸ade is shown in Fig. 21.16A.

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FIGURE 21.15 Single fac¸ade strengthened with different SRG schemes: (A) capacity curves; (B) principal tensile strains of Steel-H strengthening scheme at its peak capacity; and (C) principal tensile strains of Steel-V strengthening scheme at its peak capacity.

Due to the supporting function of every single wall to adjacent ones, the block fac¸ade presents a higher strength. On the other hand, the results show that the applied strengthening effectively enhanced the strength (about 36%) and significantly improved the ductility. On the other hand, the principal tensile strains of block fac¸ade (both plain and strengthened) at a lateral load equal to their peak capacities are shown in Fig. 21.16B and C. Only severe strains appear at the extreme side of the pushed fac¸ade. It can be seen that largest cracks in the plain model are located at the corners of the spandrels, which is accompanied by overturning of extreme piers. On the other hand, cracking of spandrels is prevented in the strengthened model and a much higher system ductility is obtained.

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1.2

Load factor, α

1.0 Plain single facade Plain block facade Strengthened block facade

0.8 0.6

(B) 0.4 0.2 0.0 0

1

2 3 4 Lateral displacement (mm) (A)

5

6

(C)

FIGURE 21.16 Block fac¸ade: (A) capacity curves; (B) principal tensile strains of the plain fac¸ade at its peak capacity; and (C) principal tensile strain of Steel-H strengthening scheme at its peak capacity.

21.5 Analysis of a rammed earth subassembly In this section, the seismic performance (both in-plane and out-of-plane) of a rammed earth subassembly is numerically investigated. Rammed earth construction technique is one of the most widely spread masonry structural system around the world and one of the most traditional ways of using raw earth as a building material. In this technique, moist earth (with proper particle size distribution) is placed inside parallel panels and rammed. This section also evaluates the applicability of applying low-cost TRM as a possible strengthening solution of rammed earth walls. The concept of this solution consists of using compatible, affordable, and readily available materials in order to generalize its usage. Outcomes of this section are based on Allahvirdizadeh (2017) and Allahvirdizadeh et al. (2019).

21.5.1 Description of the case study Two types of structural geometries are considered, namely one with an I-shape (in-plane behavior) and the other with a U-shape (out-of-plane behavior). In order to be consistent with the practice, outcomes of previously surveyed rammed earth buildings in the Alentejo region (southern Portugal) were analyzed (Correia, 2007). It was observed that the wall thickness, in nearly all surveyed cases, was about 0.5 m; hence, the same dimension was adopted in both models. Furthermore, the average height and length of the walls were 2.19 m (with a standard deviation of 0.31 m) and 3.52 m (with a standard deviation of 1.49 m), respectively. Summarizing, the adopted subassemblies are as shown in Fig. 21.17.

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FIGURE 21.17 Geometry of the rammed earth models: (A) in-plane and (B) out-of-plane.

21.5.2 Modeling considerations It was previously observed that rammed earth material exhibits a very marked nonlinear behavior, with a very weak tensile strength. Moreover, it was noticed that the conventional parabolic constitutive law in compression, which is widely used for masonry, may not be proper for this material.

Macromodeling approach for pushover analysis Chapter | 21 1.4

Multilinear compressive behavior

σ

1.2

Stress (MPa)

767

ft I Gf

1.0

ft

0.8 0.6 0.4

I

Gf h

0.2 0.0 0.00

0.01

0.02

0.03

0.04

ε

0.05

Strain (mm/mm) (A)

(B)

FIGURE 21.18 Adopted stress strain relationships for rammed earth: (A) multilinear relationship in compression and (B) exponential relationship in tension.

However, a multilinear relationship seems to provide an acceptable agreement with experimental outcomes (Miccoli et al., 2015; Librici, 2016). Thus the current numerical investigation adopted a similar approach by using a calibrated stress strain relationship obtained from averaging results of the compression tests on cylindrical rammed earth specimens conducted by Silva et al. (2014) and detailed in Fig. 21.18A. It should be noted that the postpeak softening is continued by the same slope of the experimental outcomes due to lack of experimental data. Additionally, an exponential softening relationship is assumed for the tensile behavior (see Fig. 21.18B). As for the strengthening solution in the current study, different meshes available in the market are capable of being integrated into this strengthening system. Among them, glass (denoted as G2) and nylon (called as G8) meshes previously characterized by Oliveira et al. (2017) were preselected to be studied in the numerical modeling of the strengthened rammed earth models. Here, it was decided to adopt the tensile behavior of TRM specimens, which considers both the mortar and the mesh together as a composite material. It is clear that this approach prevents simulating the slippage of the fibers within the matrix. Summarizing, a trilinear average stress strain curve was adopted to represent the behavior of TRM (Fig. 21.19A and B). As can be seen from the experimental curves, a brittle failure is expected for both meshes immediately after the peak load. Furthermore, the G2 mesh has a linear behavior up to the fragile strength drop, whereas the G8 has much less strength with a clear hardening region. Additionally, the definition of the full behavior of the TRM system requires defining its behavior under compression. For this reason, the contribution of the mesh was disregarded and the behavior was assumed as being that of the mortar. This is achieved by averaging and smoothing experimental outcomes conducted on cylindrical specimens (Oliveira et al., 2017). Among different tested mortars, the one with 33% (wt.%) of earth, 67% of

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Experimental Numerical (G2 mesh)

Stress (MPa)

1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016

Strain (mm/mm)

(A) 0.5

1.6 Experimental Numerical (G8 mesh)

1.4

Stress (MPa)

Stress (MPa)

0.4 0.3 0.2

1.2 1.0 0.8 0.6 0.4

0.1

Experimental Average

0.2 0.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

0.0 0.0000

0.0005

0.0010

0.0015

0.0020

Strain (mm/mm)

Strain (mm/mm)

(B)

(C)

0.0025

0.0030

FIGURE 21.19 Adopted trilinear tensile stress strain curves of TRM composite specimens: (A) G2 mesh; (B) G8 mesh; and (C) adopted compressive stress strain curve of the TRM composite.

sand, and a water-to-solids (W/S) ratio of 0.17 (denoted as EM2) showed good performance, and thus it was selected for this numerical investigation. The reported compressive stress strain curves of the cylindrical specimens and the one used in numerical modeling are shown in Fig. 21.19C. Later, a trend-based descending branch (like Fig. 21.18A) was added to the postpeak part of the behavior to take into account the stress degradation of the TRM composite in compression. Again, nonlinear FE models are built using DIANA software. By comparing rammed earth walls with other types of masonry construction, it becomes evident that the thickness of components is considerable in comparison to the other dimensions. In this regard, two different modeling approaches, namely using shell and solid elements were taken into account. The employed shell elements are similar to those mentioned in Section 21.3.2, while in case of solid models, 20 nodes isoparametric brick elements were employed. It is worth mentioning that shell models were prepared considering the midsection planes of each wall. This simplification may lead to some errors. For instance, the connection between walls and the length of the wing cantilevers are not properly modeled. Furthermore, the overlapping thicknesses of the walls lead to a wrong consideration of the self-weight value

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and mass distribution, and thus of the inertial forces. These statements will be discussed in detail in the following section. Finally, the adopted strengthening is modeled using eight-node quadrilateral curved shell elements. The mesh size of these elements was identical to the plain rammed earth and both meshes are connected by means of rigid interface elements.

21.5.3 In-plane behavior of the unstrengthened and textilereinforced mortar-strengthened models In the following sections, conventional pushover analysis with a massproportional lateral load pattern is performed to investigate not only the seismic performance of considered subassemblies, but also the influence of TRM strengthening on their response. The obtained pushover curves and experienced lateral displacements of the unstrengthened in-plane model for both shell and solid elements are presented in Fig. 21.20. The experienced lateral displacement is controlled on three different nodes, located at top of the left wing, middle span, and right wing. As is evident, in all cases the right wing controls the behavior and the lateral displacement in the shell models are greater than that of the solid ones. Furthermore, a minor increase in peak capacity is observed from the shell to the solid models. The damage initiation point of the models is also highlighted, which corresponds to the onset of the crack’s opening. As can be seen, this state occurs at very low lateral load level, evidencing the great influence of the nonlinear behavior of the rammed earth on the structural behavior. On the other hand, by considering the left wing and the midweb nodes as response control, an apparent unloading occurs in the postpeak behavior. This situation can be explained by the possible detachment between the right wing and the web wall. Such detachment increases displacements on the right wing, whereas the left wing and the web unload. Hence, the sway of the right wing cannot be interpreted as ductility of the model. The principal compressive stresses and principal tensile strains of both in-plane models are presented in Fig. 21.21 to assess the load path and to understand possible failure modes. As is evident, applied lateral load is transferred using compressive struts in the web wall to the foundation. Moreover, as it was expected, most damage is concentrated in the connection between the right wing and the web. On the other hand, the difference between the solid and shell models is evident, namely with respect to the undesirable distribution of damage in the web of the shell model near the right wing. Hence, the following investigations are conducted only on solid models. In order to assess the influence of TRM strengthening on the in-plane seismic behavior of the solid model, an initial pushover analysis is performed to select the most suitable textile (G2 and G8). Since both materials cost

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Load factor, α

1.4 1.2 1.0 0.8 0.6

Right wing (RW) Left wing (LW) Web (W) Damage initiation Peak capacity

0.4 0.2 0.0 0.0

0.5

1.0

1.5

2.0

2.5

2.0

2.5

Lateral displacement (mm) (A)

1.8 1.6

Load factor, α

1.4 1.2 1.0 0.8 0.6

Right wing (RW) Left wing (LW) Web (W) Damage initiation Peak capacity

0.4 0.2 0.0 0.0

0.5

1.0

1.5

Lateral displacement (mm) (B)

FIGURE 21.20 Pushover curves of the plain in-plane models: (A) shell model and (B) solid model.

about the same, the one with best structural contribution would be selected. As can be seen from the obtained capacity as shown in Fig. 21.22A, the strengthening with the G8 mesh results in a 13% and 8% increase in displacement and load capacity, respectively, while for the G2 composite the increase is of 56% and 21%, respectively. In conclusion, the subsequent investigation is conducted using the G2 solution. Comparing the response of the strengthened in-plane model with the plain wall shows that the lateral stiffness of the wall is slightly increased. Due to the possible detachment of the right wing from the web, the control node on the middle section of the web was considered to evaluate the influence of the adopted strengthening technique. With respect to this control node, a 56.5% and 21.3% increase in the lateral displacement and load

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FIGURE 21.21 Principal compressive stress at the peak capacity of in-plane model: (A) shell model; (B) solid model—principal tensile strains at the peak capacity of in-plane model; (C) shell model; and (D) solid model.

capacities were observed, respectively. In addition to the capacity, the damage initiation point is also highlighted; no marked difference was detected between this point in the strengthened model with that of the plain one. This is probably due to a very local damage that occurs at this point. Moreover, the principal tensile strains of the strengthened model at its peak capacity and that of the plain model are shown in Fig. 21.22B and C. As can be seen, the detachment between the right wing and the web at the peak capacity of unstrengthened model is completely prevented by the TRM. This is also true for the developed diagonal shear crack on the web. Only some damage in the toe of the left wing was observed, evidencing the tendency of the wall to overturn. It can also be seen that the strengthened model experiences smaller strains in this region in comparison with the unstrengthened model. This situation can be explained by the improved integrity of the wall due to the application of the strengthening. On the other hand, the principal tensile strains at the peak capacity of the strengthened model show detachment of the right wing; however, a portion of the web follows the wing. From a kinematic point of the view, this added portion means a greater load is required to cause the right wing to detach from the wall and overturn.

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Load factor, α

2.0

1.5 RW – strenghened (G2) LW – strenghened (G2) W – strengtened (G2) RW – plain LW – plain W – plain RW – strengthened (G8)

1.0

0.5

Peak capacit y a Damage initiation

0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Lateral displacement (mm)

(A)

(B)

(C)

(D)

FIGURE 21.22 Strengthened in-plane model: (A) pushover curves; (B) principal tensile strains of strengthened model at a lateral load equal to peak capacity of plain model; (C) principal tensile strains of strengthened model at a lateral load equal to peak capacity of strengthened model; and (D) principal tensile strains of TRM strengthening at a lateral load equal to peak capacity of strengthened model.

Moreover, a diagonal shear crack was observed in the web, whose development is much more extensive than that evidenced without strengthening. This diagonal shear crack illustrates the efficiency of the adopted strengthening solution in improving and exploring the in-plane seismic behavior of rammed earth walls. Finally, the damage state of the TRM strengthening is also investigated (see Fig. 21.22D). It is evident that the strengthening efficiently contributes to stress transfer, particularly in the connection of wing to the web, diagonal of the web, and at toes. These regions are those likely to fail; hence, the adopted strengthening works exactly for the desired purpose.

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21.5.4 Out-of-plane behavior of the unstrengthened and textilereinforced mortar-strengthened models In this section, an identical approach is followed to assess the out-of-plane seismic performance of a rammed earth subassembly and evaluate the efficiency of the proposed TRM strengthening solution. It should be noted that due to the asymmetric geometry of the out-of-plane model, the wall is pushed in two directions, that is, toward wing walls (denoted as positive Y) and outside of them (denoted as negative Y). The obtained pushover curves of both plain and strengthened out-of-plane models pushed in both aforementioned directions are presented in Fig. 21.23A and B.

1.6

Load facor, α

1.4 1.2 1.0 0.8 0.6 Strengthened model Plain model Damage initiation Peak capacity

0.4 0.2 0.0

0

1

2

3

4

5

6

Lateral displacemet (mm)

Load facor, α

(A) 2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0

Strengthened model Plain model Damage initiation Peak capacity

0

2

4

6

8

10

12

14

Lateral displacement (mm)

(B) FIGURE 21.23 Pushover curves of the plain out-of-plane model versus strengthened one: (A) pushed in negative direction and (B) pushed in positive direction.

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As can be seen, the strengthening increases the prepeak stiffness of the model by controlling the cracking process. Moreover, the lateral displacement and the load capacities were increased in the case of being pushed in the negative direction by 45% and 29%, respectively. These enhancements for the case of pushing in the positive direction are 131% and 31%,

FIGURE 21.24 Principal tensile strains of the models pushed in negative direction: (A) plain model at its peak capacity; (B) strengthened model at peak capacity of plain wall; (C) strengthened model at its peak capacity; and (D) TRM strengthening at peak capacity of strengthened model.

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respectively, for the lateral displacement and load capacity. Pushing in the negative direction would be critical since the wing walls increase the flexural resistance of the web wall when pushed toward wings. Therefore further investigations focus only on the push in the negative direction. It was observed that the applied strengthening solution has no influence on the onset of damage (as for the in-plane model), although the strengthened model presents a smoother postpeak behavior when compared to the unstrengthened one. The contour map of the principal tensile strains of the plain and strengthened out-of-plane models pushed in the negative direction is shown in Fig. 21.24. It is evident that the web wall tends to detach from the wings in the plain model, especially in the upper sections. On the other hand, the midspan of the web seems to rotate while the whole web is likely to overturn. A considerable reduction in the principal tensile strains level was observed for the strengthened model at a lateral load equal to the peak capacity of the unstrengthened one. The detachment of the web from the wing walls and bending of the web’s midsection were avoided. Furthermore, the tensile strains at the base of the wall were decreased. Hence, the employed strengthening solution enables the wall to redistribute the stresses and decreases its tendency to overturn. The contour of the principal tensile strains at the peak capacity of the strengthened model shows that the failure mechanism is similar to that of the unstrengthened model, while a larger midsection of the web is bending. This larger section means that a higher lateral load is required to initiate the collapse mechanism. Finally, the performance of the TRM strengthening was assessed at the peak capacity of the strengthened model. High tensile strains are evident in the wall connections, which seems to illustrate the prevention of the web detachment.

21.6 Concluding remarks It has been shown in a variety of experimental studies that the TRM technique acceptably enhances the seismic performance of masonry components, however limited studies exist aimed to numerically investigate their response. In order to achieve such objective, enough knowledge on the behavior of the individual materials and their interfaces is required to define accurate behavior laws. Although some experimental studies have addressed the behavior of different mortars or textiles, there is still a lack of knowledge about their interfaces, particularly the bond-slip behavior. The applicability of the macromodeling technique in the modeling of TRM strengthening is discussed within this chapter. As the option of modeling the embedded mesh within a mortar layer makes models to be computationally expensive and, in the case of large models, hardly possible to be analyzed. In this regard, two approaches, namely considering a perfect bond

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between mesh and mortar or adopting results coming from uniaxial tensile tests on composites are taken into account. It should be noticed that different failure modes regarding the bond strength at mortar mesh or TRM substrate interfaces have been observed, thus considering both interfaces as perfect bonded may induce some error, which requires a complex model approach and detailed experimental information to be assessed. In spite of possible miscalculations, it is believed that the approach employed here can leads to a valuable understanding of the global structural response. The pushover analyses discussed on the three case studies seems to indicate that the applied TRM strengthening solution does not change the failure mechanisms observed for the plain models. However, failure was considerably postponed due to the presence of strengthening. In other words, the strengthened models exhibited evident greater strength and ductility and showed a much smoother postpeak behavior, so the structural safety of models was preserved for much higher lateral load levels.

References Allahvirdizadeh, R., 2017. Modeling of the Seismic Behaviour of TRM-Strengthened Rammed Earth Walls (Master thesis). University of Minho, Portugal. Allahvirdizadeh, R., Gholipour, Y., 2017. Reliability evaluation of predicted structural performances using nonlinear static analysis. Bull. Earthq. Eng. 15 (5), 2129 2148. Allahvirdizadeh, R., Oliveira, D.V., Silva, R.A., 2019. Numerical modeling of the seismic outof-plane response of a plain and TRM-strengthened rammed earth subassembly. Eng. Struct. (accepted for publication). Ascione, L., de Felice, G., De Santis, S., 2015. A qualification method for externally bonded Fibre Reinforced Cementitious Matrix (TRM) strengthening systems. Compos. B: Eng. 78, 497 506. Babaeidarabad, S., De Caso, F., Nanni, A., 2013. Out-of-plane behavior of URM walls strengthened with fabric-reinforced cementitious matrix composite. J. Compos. Constr. 18 (4), 04013057-1-11. Basili, M., Marcari, G., Vestroni, F., 2016. Nonlinear analysis of masonry panels strengthened with textile reinforced mortar. Eng. Struct. 113, 245 258. Bernat, E., Gil, L., Roca, P., Escrig, C., 2013. Experimental and analytical study on TRM strengthened brickwork walls under eccentric compressive loading. Constr. Build. Mater. 44, 35 47. Bernat-Maso, E., Gil, L., Roca, P., 2015. Numerical analysis of the load-bearing capacity of brick masonry walls strengthened with textile reinforced mortar and subjected to eccentric compressive loading. Eng. Struct. 91, 96 111. Borri, A., Casadei, P., Castori, G., Hammond, J., 2009. Strengthening of brick masonry arches with externally bonded steel reinforced composites. J. Compos. Constr. 13 (6), 468 475. Caggiano, A., Martinelli, E., 2012. A unified formulation for simulating the bond behavior of fibres in cementitious materials. Mater. Des. 42, 204 213. Caggegi, C., Carozzi, F.G., De Santis, S., Fabbrocino, F., Focacci, F., Hojdys, L., et al., 2017. Experimental analysis on tensile and bond properties of PBO and aramid fabric reinforced cementitious matrix for strengthening masonry structures. Compos. B: Eng. 127 (15), 175 195.

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Carozzi, F.G., Milani, G., Poggi, C., 2014. Mechanical properties and numerical modeling of fabric reinforced cementitious matrix (FRCM) systems for strengthening of masonry structures. Compos. Struct. 107, 711 725. Carozzi, F.G., Poggi, C., 2015. Mechanical properties and debonding strength of Fabric Reinforced Cementitious Matrix (FRCM) systems for masonry strengthening. Compos. B: Eng. 70 (1), 215 230. Correia, M.R., 2007. Rammed earth in Alentejo. Argumentum, Lisbon, Portugal. Dalalbashi, A., Ghiassi, B., Oliveira, D.V., Freitas, A., 2018a. Effect of test setup on the fiberto-mortar pull-out response in TRM composites: experimental and analytical modelling. Compos. B: Eng. 143, 250 268. Dalalbashi, A., Ghiassi, B., Oliveira, D.V., Freitas, A., 2018b. Fiber-to-mortar bond behavior in TRM composites: effect of embedded length and fiber configuration. Compos. B: Eng. 152, 43 57. D’Ambrisi, A., Feo, L., Focacci, F., 2013a. Experimental and analytical investigation on bond between carbon-FRCM materials and masonry. Compos. B: Eng. 44 (1), 524 532. D’Ambrisi, A., Feo, L., Focacci, F., 2013b. Experimental analysis on bond between PBO-FRCM strengthening materials and concrete. Compos. B: Eng. 44 (1), 524 532. De Felice, G., De Santis, S., Garmendia, L., Ghiassi, B., Larrinaga, P., Lourenc¸o, P.B., et al., 2014. Mortar-based systems for externally bonded strengthening of masonry. Mater. Struct. 47 (12), 2021 2037. DIANA FEA BV, 2017. Displacement Method Analyzer, Release 10.1. DIANA FEA BV, The Netherlands. Garofano, A., Ceroni, F., Pecce, M., 2016. Modelling of the in-plane behaviour of masonry walls strengthened with polymeric grids embedded in cementitious mortar layers. Compos. B: Eng. 85, 243 258. Ghiassi, B., Oliveira, D.V., Lourenc¸o, P.B., Marcari, G., 2013. Numerical study of the role of mortar joints in the bond behavior of FRP-strengthened masonry. Compos. B: Eng. 46, 21 30. Ghiassi, B., Oliveira, D.V., Marques, V., Soares, E., Maljaee, H., 2016. Multi-level characterization of steel reinforced mortars for strengthening of masonry structures. Mater. Des. 110 (15), 903 913. Giamundo, V., Sarhosis, V., Lignola, G.P., Sheng, Y., Manfredi, G., 2014. Evaluation of different computational modelling strategies for the analysis of low strength masonry structures. Eng. Struct. 73, 160 169. Grande, E., Imbimbo, M., Sacco, E., 2013. Modeling and numerical analysis of the bond behavior of masonry elements strengthened with SRP/SRG. Compos. B: Eng. 55, 128 138. Larrinaga, P., Chastre, C., San-Jose´, J.T., Garmendia, L., 2013. Non-linear analytical model of composites based on basalt textile reinforced mortar under uniaxial tension. Compos. B: Eng. 55, 518 527. Larrinaga, P., Chastre, C., Biscaia, H.C., San-Jose´, J.T., 2014. Experimental and numerical modeling of basalt textile reinforced mortar behavior under uniaxial tensile stress. Mater. Des. 55, 66 74. Librici, C., 2016. Modeling of the Seismic Performance of a Rammed Earth Building (Master thesis). University of Minho, Portugal. Lourenc¸o, P.B., 1996. Computational Strategies for Masonry Structures (Ph.D. thesis). Delft University of Technology, The Netherlands. Miccoli, L., Oliveira, D.V., Silva, R.A., Mu¨ller, U., Schueremans, L., 2015. Static behaviour of rammed earth: experimental testing and finite element modeling. Mater. Struct. 48 (10), 3443 3456.

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PART | IV Modeling of textile-reinforced mortar-strengthened masonry

Milani, G., Rotunno, T., Sacco, E., Tralli, A., 2006. Failure load of FRP strengthened masonry walls: experimental results and numerical models. Struct. Durab. Health Monit. (SDHM) 2 (1), 29 50. Mininno, G., 2016. Modelling of the Behaviour of TRM-Strengthened Masonry Walls (Master thesis). University of Minho, Portugal. Mininno, G., Ghiassi, B., Oliveira, D.V., 2017. Modelling of in-plane and out-of-plane performance of TRM-strengthened masonry walls. Key Eng. Mater. 747, 60 68. Oliveira, D.V., Silva, R.A., Barroso, C., Lourenco, P.B., 2017. Characterization of a compatible low cost strengthening solution based on the TRM technique for rammed earth. Key Eng. Mater. 747, 150 157. Papanicolaou, C.G., Triantafillou, T.C., Papathanasiou, M., Karlos, K., 2007. Textile reinforced mortar (TRM) versus FRP as strengthening material of URM walls: out-of-plane cyclic loading. Mater. Struct. 41, 143 157. Ramaglia, G., Lignola, G.P., Balsamo, A., Prota, A., Manfredi, G., 2016. Seismic strengthening of masonry vaults with abutments using textile-reinforced mortar. J. Compos. Constr. 21 (2), 04016079-1-16. Razavizadeh, A., Ghiassi, B., Oliveira, D.V., 2014. Bond behaviour of SRG-strengthened masonry units: testing and numerical modeling. Constr. Build. Mater. 64, 387 397. Silva, R.A., Oliveira, D.V., Schueremans, L., Lourenco, P.B., Miranda, T., 2014. Modelling the structural behaviour of rammed earth components. In: 12th International Conference on Computational Structures Technology, Naples, Italy. Valluzzi, M.R., Modena, C., De Felice, G., 2014. Current practice and open issues in strengthening historical buildings with composites. Mater. Struct. 47 (12), 1971 1985. Wang, X., 2015. Numerical Analysis of the In-plane Behaviour of FRCM-Strengthened Masonry walls (Master thesis). University of Minho, Portugal. Wang, X., Ghiassi, B., Oliveira, D.V., Lam, C.C., 2017. Modelling of the nonlinear behaviour of masonry walls strengthened with textile reinforced mortars. Eng. Struct. 134, 11 24.

Index Note: Page numbers followed by “f” and “t” refer to figures and tables, respectively.

A Abaqus 6.14 code, 182 183 Abaqus program, 169 170, 723 Ad hoc pull-out test, 721 Adaptive global damping, 482 Adaptive limit analysis, 589 Adaptive multiscale approaches, 26, 30 Advanced equivalent beam-based macroelement, 33 Advanced finite element modeling experimental investigation on FRCM, 716 723 cementitious and lime-based mortar, 717 constituent material mechanical characterization, 716 717 glass fiber reinforced plastic mortar interface bonding behavior, 721 historical bricks, lime mortar, and glass fiber reinforced plastic, 719 720 mechanical characterization of bricks, 717 modern bricks, cementitious mortar, and glass fiber reinforced plastic, 719 predictive simple analytical/numerical model, 721 723 single-brick reinforcements, 717 718 full 3D heterogeneous approach, 723 725 in-plane loaded masonry, experimental benchmarking, 730 735 out-of-plane loaded masonry, 735 738 two-step holonomic homogenization model, 725 730 Advanced path-following technique, 412 Amatrice and Norcia earthquake (2016), 55 56 Amplification effects, 630 631 Analytical-numerical approach, 702 703 Ancient columns and colonnades, 486 490 Anisotropic Rankine Hill model, 632, 633f ANSYS 8.1 software, 73

ANSYS code, 589 Arc-length method, 635 638, 643 Arches, 491 493 Aveston Cooper Kelly theory, 747 748 Azores earthquake (1998), 486

B Bastard mortars, 241, 249 BBMs. See Block-based models (BBMs) Beam element, 484 485, 484f FE, 32 33 multiscale approach, 354 Bending moment capacity, 649, 652 654 Bilinear law, 558 Bilinear b-slip constitutive law, 596 Bilinearization, 649 650, 650f Blind predictions, 148 154 Block-based models (BBMs), 4, 12 20. See also Geometry-based models (GBMs) block-based limit analysis approaches, 15f, 19 20 contact-based approaches, 14f, 17 18 interface element-based approaches, 14f, 15 17 textured continuum-based approaches, 15f, 18 19 XFEM, 15f, 20 Blocks block-based limit analysis approaches, 15f, 19 20 constitutive models for, 479 Body-based search techniques, 472 Bond bond slip laws, 710 effectiveness factor, 691 surface, 691

779

780

Index

Bond behavior of FRCM systems, 685 686, 702. See also Fiber-reinforced plastic strengthened masonry dynamic analysis considerations and suggestions, 709 710 review of available models, 686 709 theoretical and analytical/numerical models, 686 Box-like behavior, 211 213 Brick(s) brick mortar lamina properties, 297 298 constitutive law, 369 370 mechanical characterization, 717 mechanical properties, 718t tensile strength of, 116 Brittle damage model, 296 297 Brittle linear elastic behavior, 751 Brittleness anisotropy, 6

C Cable reinforcement. See Global reinforcement CAD. See Computer-aided design (CAD) Calibration strategy, 513, 690 Carbon fiber reinforced polymer (CFRP), 542, 634, 660 Carbon fiber reinforced polymer/steelreinforced polymer (CFRP/SRP) brick mortar assemblage strengthened with, 557f experimental data used in shear tests, 549t single brick unit strengthened with, 556f strip-to-historic brick joints, 548 549 Carbon-FRCM materials, 691 Cast3M software, 317, 337 338 Castle of St. George discussion of results, 180 181 FE model and eigenfrequency analysis, 172 174 nonlinear dynamic analyses, 174 180 structure description, 171 172 Cauchy model, 357 358 CCLM. See Continuum constitutive laws model (CCLM) CDF 1. See Inverse cumulative distribution function (CDF 1) CDP model. See Concrete damage plasticity model (CDP model) CDT. See Constrained Delaunay triangulation (CDT) Cement-based mortars, 241 242

Cementitious mortar, 717, 719 Central difference method, 474 CFRP. See Carbon fiber reinforced polymer (CFRP) CFs. See Confidence factors (CFs) Classic Cauchy model, 357 Classic Mohr Coulomb failure criterion, 458 459 Classic mutation operator, 599 Classical 1D macroelements, 264 Classical Voigt notation, 317, 337 Clay bricks, 717 Closed-form macroscopic continuum constitutive laws, 397 CMs. See Continuum models (CMs) COD. See Crack opening displacement (COD) Cohesion of mortar joints, 115 Cohesive bilinear law for interface behavior, 687 Cohesive boundaries, 275 277 Cohesive simulations, 570, 574t Cohesive zone-based modeling, 397 discretization of segmented realistic masonry geometries, 414 419 extraction of cohesive zone geometry, 404 410 material properties, 412t RVE computations for irregular masonry, 410 414 geometry generation, 399 403 Cohesive-friction constitutive model, 353 354 Collapse loads, 604 605 Collapse mechanisms, 150, 604 in predictions of Brick House, 154 in predictions of Stone House, 150, 151f, 152f Coloumb’s frictional law, 17 Combined cracking shearing crushing model, 95 96 Common plane method, 472 473, 477 Compatible identification model. See also Finite element heterogeneous models deformation modes in, 437f elementary homogeneous deformations, 439f extension to transverse loads, 439 442 homogenized yield surface under in-plane loads, 437 439 with joints reduced to interfaces and infinitely strong units, 437 442 shear deformation rates, 440f

Index “Compatible identification” approach, 426 Complete quadratic combination (CQC), 64 Composite interface model. See Combined cracking shearing crushing model Composite materials, 746 Compression symmetric shear test, 378 Compression tensile test, 375 377 Compressive behavior of masonry unit, 245 247 of mortar, 251 Compressive crushing, 662 Compressive fracture, 241 242 Compressive strength (fc), 241 242, 664 Computational efficiency, 631 Computational models, 397 Computer-aided design (CAD), 9 Concrete damage plasticity model (CDP model), 167 171, 571 573, 714 715, 723 Confidence factors (CFs), 202 Confining rings, 212 Constant stress triangular elements (CST elements), 426 Constitutive behavior law, 337 Constitutive equation, 663 665 Constitutive laws, 239 Constitutive models for blocks, 479 for contacts, 479 480 Constrained Delaunay triangulation (CDT), 408 Construction vulnerability, 201 Contacts, 471 473, 477 478 constitutive models for, 479 480 contact-based approaches, 14f, 17 18 detection algorithms, 472, 477 mechanical model, 473 recognition, 472 473 surface, 690 Continuum constitutive laws model (CCLM), 61 Continuum models (CMs), 4, 12, 20 30 direct approaches, 21 24, 21f homogenization procedures and multiscale approaches, 21f, 24 30 Contour meshing, 409 410 Conventional parabolic constitutive law, 766 767 Convergence checks, 671 Cosserat macromodel, 2D micromodel for, 363 364 Cosserat model, 358 359, 382 383

781

Cosserat multiscale model based on piecewise, 386 388 Cosserat-based multiscale model, 353 354 Coulomb yield function, 367 Coupling between homogenization techniques and damage theory accounting for creep of masonry components, 309 327 accounting for damaged brick mortar interface, 297 309 CQC. See Complete quadratic combination (CQC) Crack density, 299 Crack opening displacement (COD), 299 Cracking, 3 Creep of masonry components, 309 327 creep model for microcracked mortar, 312 314 FE homogenization of microcracked viscoelastic masonry periodic cell, 315 317 microcracked mortar, 314 315 objective and hypothesis, 311 312 time-dependent crack density and application, 317 326 CST elements. See Constant stress triangular elements (CST elements) CTR model, 323 327 Cultural heritage, 55 56 Curvature ductility, 650 index capacity, 650 651 Cyclic loading, 263, 267

D Damage assessment of monumental buildings, 55 and friction phenomena, 352 353 function, 667, 671 limit function, 369 Damage mechanics, 268 cyclic interface model, 16 Damage patterns (DP), 520, 521f, 698 700 Damage-driving parameter, 411 Damaged brick mortar interface, accounting for, 297 309 effective properties of brick mortar lamina, 297 298 of microcracked material, 299 301 estimation of representative law of microcrack evolution, 303 309 results, 307 309

782

Index

Damaged brick mortar interface, accounting for (Continued) simulation of confined medium-sized masonry panel, 304 306 interface constitutive law, 301 302 DBA. See Displacement-based approach (DBA) Debonding, 660, 692 modeling damage by, 669 670 Decohesion mechanisms, 638, 639f Deep beam, 615 622, 615f, 617f Deep beam test, 415 416, 417f structural response of system, 418f Deformable elements, 471, 481 482 Degrees of freedom (DOF), 423 424, 476 Delamination, 586 589 of composite materials, 544, 548 DEM. See Discrete element method (DEM) Diagonal link calibration, 516 DIANA software, 768 769 DIANA code, 589, 601 602, 617 DIANA FEA 10.2, 202 DIC method. See Digital image correlation method (DIC method) Digital image correlation method (DIC method), 328 Dilatation angle, 169 170 Direct continuum approaches, 22 Discontinuous deformation analysis, 17 Discrete element method (DEM), 17, 142 143, 143f, 145, 242, 470 applications of modeling masonry and historical structures, 485 496 ancient columns and colonnades, 486 490 arches and vaults, 491 493 masonry walls, 490 491 modeling large and complex masonry structures, 494 496 tall masonry structures and historic towers, 485 486 codes for masonry modeling and key features of UDEC and 3DEC, 470 475 calculation of displacements, 473 475 contacts, 471 473 elements, 470 471 loads, 475 key features of UDEC and 3DEC for modeling masonry, 475 485 constitutive models for blocks, 479

constitutive models for contacts, 479 480 contacts, 477 478 equations of motion, 480 482 FISH function, 485 identification of neighbors, 477 mechanical damping, 482 numerical stability, 482 483 representation of block, 475 477 structural elements, 483 485 model of Stone House, 156 Discrete macroelement strategy, 509 Discretization of segmented realistic masonry geometries, 414 419 Displacement calculation of, 473 475 vector, 358 Displacement-based approach (DBA), 145 Dissipation, 587 588 Distance field-assisted packing of inclusions, 400 401 Distance field-based morphing of set of inclusions, 401 403 Distinct element method. See Discrete element method (DEM) DOF. See Degrees of freedom (DOF) Double-planked timber floors, interventions on buildings with, 223 224 DP. See Damage patterns (DP) Drucker Prager (DP) cone, 723 failure surface, 571 573 plasticity model, 20 shear-compression law, 205 strength criterion, 169 170 Ductility index, 243 Dynamic shake table studies, 662

E Earthquakes, 141, 713, 745 EBA. See Energy-based approach (EBA) Eccentricity, 169 170 EDDTD. See Energy density dissipated by tensile damage (EDDTD) Effective bond length, 690 EFM. See Equivalent frame model (EFM) Eigenfrequency analysis, 165 of Castle of St. George, 172 174 of Palazzo Te, 191 193 of Sant’Andrea church, 182 186 Elastic anisotropy, 6

Index Elastic perfectly plastic behavior, 702 703 Elastic-brittle behavior, 638 Elasto-fragile behavior, 729 730 Elementary NURBS patches, 590 591 Elias model, 473 Emilia earthquake (2012), 55 56 Energy density dissipated by tensile damage (EDDTD), 167 Energy-based approach (EBA), 145 Enhanced multiscale model, 352 353 Enriched plane state micromodel (EPS micromodel), 355 Equations of motion, 480 482 deformable elements, 481 482 rigid blocks, 480 481 Equilibrated model with joints extension to transverse loads, 436 437 homogenized yield surface under in-plane loads, 433 436 reduced to interfaces and constant stress triangular discretization, 433 437 Equilibrium equations, 358 Equivalent beam-based approaches. See Equivalent frame model (EFM) Equivalent frame model (EFM), 31 33, 202 205, 504 509, 507f assumptions, 205 206 Eshelby-based homogenization scheme, 309 311 Euler Bernoulli hypothesis, 663 664 Eurocode 6, 241 Explicit time integration, 474, 475f Exponential law, 369 Extended finite element method (XFEM), 15f, 20 External steel plates, 585

F Fabric reinforced cementitious matrix (FRCM), 685, 714 FRCM-reinforced panels, 738 mechanical properties, 714 prediction of FRCM-reinforced masonry behavior, 714 715 Faced masonry, 638 Failure conditions, 479 mechanisms identification of macroelements, 77 78 progression, 103 FBA. See Force-based approach (FBA)

783

FE. See Finite element (FE) FEA. See Finite element analysis (FEA) FEM. See Finite element method (FEM) Fiber pull-out tests, 710 Fiber-reinforced cementitious matrix, 745 Fiber-reinforced plastic strengthened masonry dynamic analysis, 659 input ground motions, 672t model specifications, 663 671 beam model, 663 665 modeling damage by debonding, 669 670 modeling damage by masonry crushing, 665 668, 667f numerical code and procedures, 670 671 numerical results, 671 678 Fiber-reinforced polymer (FRP), 206, 629, 659, 685, 713 714 composites, 631 632 debonding, 662 “FRP in-ex” case, 677 FRP-strengthened masonry, 585 586 genetic algorithm mesh adjustment, 598 600 nonlinear modeling, 607 622 numerical simulations in limit analysis, 600 606 NURBS model, 589 598 FRP-to-masonry bond response, 632 FRP FRP interface, 593 594 FRP masonry interface, 593, 598 systems, 537 bond behavior, 537 538 experimental evidence, 538 553 experimental tests on FRP strip-tobrickwork joints, 542 548 numerical models, 553 580 theoretical bond analysis, 539 542 Fiber-to-mortar interface, 710 Field-based penalty function method, 473 Finite element (FE), 241 242, 263 264, 398, 423 approach, 57 continuum model assumptions, 205 homogenization of microcracked viscoelastic masonry periodic cell, 315 317 limit analysis, 24 model, 172 173, 319 320, 505, 538 of Castle of St. George, 172 174 of Palazzo Te, 191 193

784

Index

Finite element (FE) (Continued) of Sant’Andrea church, 182 186, 183f, 184f nonlinear homogenization reference local and global behaviors of nonlinear mortarless masonry, 335 338 reference material properties of constituents, 333 335 Finite element analysis (FEA), 92, 163 164, 355 interface element nonlinear behavior, 95 96 model, 110 118 comparison of results, 120 124 deterministic values in model, 112 mesh density study, 111 112 random variables, 112 118 variation in experimental results, 119 120 modeling approach, 94 95 macromodeling approach, 94 95 micromodeling approach, 95 Finite element heterogeneous models, 560 580, 561f. See also Compatible identification model brick and mortar uniaxial behavior, 563f curved pillars, 567 580 flat case, 560 567 mechanical properties of brick and mortar, 562t mortar and brick compressive strengths, 564t Finite element method (FEM), 142 145, 143f, 156, 202 205, 470 471, 648 649. See also Discrete element method (DEM) FISH function, 485 Flexural test, 378 380 Flexural-rocking failure, 141 142 Flooring block slabs, interventions on buildings with, 221 222 Floors, strengthening interventions on, 216 219 Force-based approach (FBA), 145 Force-based beam FE, 32 33 Fracture energy (Gf), 241 242, 692 693 of brick, 116 Fracture mechanics approach, 704 705 FRCM. See Fabric reinforced cementitious matrix (FRCM) Friction effect, 367 FRP. See Fiber-reinforced polymer (FRP)

Full 3D heterogeneous approach, 723 725 Full 3D-FEM nonlinear approach, 703 704 Funicular model, 35 36

G G2 mesh, 767 G8 mesh, 767 GA. See Genetic algorithm (GA) “Gaioleiro” buildings, 519 520 GA NURBS based limit analysis, 600 Gauss point (GP), 275 276 Gauss quadrature method, 595 Gaussian weight function, 358 GBMs. See Geometry-based models (GBMs) General-purpose mortar, 241 Generalized Maxwell model (GM model), 319 320 Genetic algorithm (GA), 588 mesh adjustment, 598 600 Geometric nonlinearity, 10 Geometry-based models (GBMs), 4, 13, 34 38. See also Block-based models (BBMs) kinematic theorem-based approaches, 37 38 static theorem-based approaches, 35 36 Glass fiber reinforced plastic (GFRP), 716 717, 719 720 GFRP mortar interface bonding behavior, 721 grid for FRCM, 716t strips, 542, 660 661 Glass fiber reinforced polymer strip-tobrickwork joints, 542 548, 546t experimental data of tensile tests on, 543t experimental interface diagram shear stress vs. slip, 547f experimental mechanical parameters of brickwork, 543t historic wallettes, 545f, 548t wallettes with different thickness of mortar bed joints, 543f Glass grids, 761 762 Global axes, 480 481 Global reinforcement, 484, 484f GM model. See Generalized Maxwell model (GM model) GP. See Gauss point (GP) Graphical catenary-based methods, 144 Gravitational load, 649 Griffith’s theory, 309 311

Index

H Haiti earthquake (2010), 141 Hard bricks, 243t Hard contact pressure, 704 705 Hard-contact approach, 473 Hashin Shtrikman bounds (HS bounds), 338 339 Hemispherical dome, 605, 606f, 607f HEMu. See Homogeneous equivalent undamaged material (HEMu) Heyman’s model, 11 Hill-type criterion, 267 Hill-type yield criterion, 632 Hill Mandel equation, 370 Historic towers, 485 486 Historical bricks, 719 720 Historical masonry structures (HMSs), 504 Historical masonry structures, 263 Historical structures analysis (HiStrA), 505 506, 519 520 HMSs. See Historical masonry structures (HMSs) Homogeneous equivalent undamaged material (HEMu), 297 298 Homogenization, 424 procedures, 12, 24 30, 311 312, 351 353 adaptive multiscale approaches, 30 priori homogenization approaches, 26 28 step-by-step multiscale approaches, 28 30 theorems, 268 269 Homogenized limit analysis of periodic masonry, 425. See also Two-step holonomic homogenization model compatible identification model, 437 442 equilibrated model with joints, 433 437 fundamentals, 426 429 MoC, 442 450 polynomial expansion of the microstress field (PES), 429 433 running bond assemblage of bricks and mortar and unit cell, 427f Homogenized strength domains, 450 463 HS bounds. See Hashin Shtrikman bounds (HS bounds) Hyperlinear contact detection algorithms, 472

I IM.. See Intensity measure (IM) Implicit time integration techniques, 474, 475f In-house specialized software, 490

785

In-plane behavior, 141 142 of unstrengthened and TRM-strengthened models, 751 755, 769 772 failure modes, 280 281 loaded masonry, 450 453 experimental benchmarking, 730 735 loading validation of unreinforced masonry panels, 284 288 shear spring, 277 278 Incompatible mode elements, 569 570 Incremental nonlinear FE approach, 589 Incremental-iterative analysis approaches, 10 11 Inorganic matrices, 714 Intensity measure (IM), 58 Interface constitutive law, 301 302 Interface elements, 638 approaches, 14f, 15 17 nonlinear behavior, 95 96 Interface nonlinear links, 513 514 Interface-based models, 554 560, 555f. See also Finite element heterogeneous models bond behavior of curved masonry pillars, 559f brick mortar assemblage strengthened with CFRP strip, 557f single brick unit strengthened with CFRP strip, 556f Interfacial debonding of FRCM concrete systems, 692 Interfacial fracture energy, 689 690 Interlocking mortarless block, 327 328 Internal dissipated power, 595 Internal dissipation, 593 Intervention techniques, 202, 206 207, 227 229 Inverse cumulative distribution function (CDF 1), 113 114 Inverse identification procedure, 687 Inverted catenary principle, 144 Iron girders, interventions on buildings with, 221 222 Irregular masonry generation of two RVEs for, 404f RVEs computations for, 410 414 texture, 398 Isoparametric approach, 592 Isotropic elastic material model, 642 Isotropic element, 479

786

Index

J

M

Japan Society of Civil Engineers (JSCE), 751, 759 760

MaC bricks. See Magnesia Carbon bricks (MaC bricks) Macroblock model (MBM), 61 Macroelement approach, 57, 156 157, 271 272 experimental and numerical validation, 517 524 application to masonry arch bridges, 524 shell macroelement, 522 523 2D macroelement, 517 519 3D macroelement, 519 522 mechanical characterization strategy, 513 516 diagonal link calibration, 516 nonlinear link calibration, 514 516 numerical examples, 279 291 Macroelement models (MMs), 4, 12 13, 30 34, 32f equivalent beam-based approaches, 31 33 spring-based approaches, 33 34 Macromodeling/macromodels, 142 143, 205, 242, 357 359, 423, 631. See also Micromodeling/micromodels approach, 94 95, 163 164 Cauchy model, 357 358 Cosserat model, 358 359 strategies for TRM-strengthened masonry, 746 748, 748f Macroscale models, 265 strategies, 267 272 Macroscopic 2D damage model, 268 Macroscopic nonlinear behavior of masonry components, 265 267 in-plane failure modes of URM components, 265f Macroscopic strain vector, 357 MADY code, 670 671 Magnesia Carbon bricks (MaC bricks), 329 Masonry, 91, 142 143, 239, 295, 351, 426, 503. See also Irregular masonry application to masonry arch bridges, 524 components, 263 264 constitutive law, 364 370 brick constitutive law, 369 370 mortar constitutive law, 364 368 constructions, 163 164 crushing, 18 modeling damage by, 665 668, 667f curved structures, 646 dams, 496 fac¸ade analysis, 758 764

K Kelvin Voigt model (KV model), 312 313 Kinematic approach, 144 Kinematic limit analysis, 592 598 Kinematic theorem, 11 kinematic theorem-based approaches, 37 38 Kirchhoff Love homogenized model, 354 Knot vector, 589 590 Knowledge level (KL), 202 Kolmogorov Smirnov test, 113 114

L L’Aquila earthquake (2009), 55 56 Laplace Carson transform (LC transform), 309 311 LCC. See Linear comparison composite (LCC) Leave-one-out cross-validation, 116 Leccese stone, 537 538 Lightweight mortar, 241 Lime-based mortars, 241 242, 717, 719 720 Limit analysis, 596 597 approach, 354 limit analysis-based solutions, 11 12 numerical simulations in, 600 606 theory, 646 Limiting timestep, 482 Linear comparison composite (LCC), 328 Linear contact detection algorithms, 472 Linear elastic behavior, 241 242 Linear modal analyses, 213 214 Linearization procedures, 328 Linearly elastic element, 479 LNEC. See National Laboratory for Civil Engineering (LNEC) Load-bearing system, 9 Load-carrying capacity, 694 Load displacement curves, 604 605 of unaged strengthened walls, 643 644 Loads, 475 Lobatto integration scheme, 643 Local damping, 482 Local reference system, 595 Local reinforcement, 483, 483f LP problem, 593, 598

Index behavior of unstrengthened and TRM-strengthened models, 760 764 description of case study, 758 modeling considerations, 758 760 interventions on masonry vaults, 227 229 masonry masonry interface, 593 594 material, 503 mechanics, 6 7 monumental constructions, 505 506 pier beam FE, 32 33 structural elements, 587 588 wedge elements, 609 615 Masonry material mechanical properties, 260t. See also Nonlinear seismic analysis of masonry structures input parameters for numerical simulations, 259 masonry properties, 254 259 compressive behavior, 255 257 elastic properties, 255 shear behavior, 259 tensile behavior, 257 259 modeling at different levels, 241 242 mortar properties, 248 251 ASTM C270 mortar types, 249t compressive behavior, 251 elastic properties, 250 251 leveling mortar, 251 recommended mortar properties in NZSEE, 250t tensile behavior, 251 mortar brick interface properties, 251 254 type of units and materials, 240 241 unit properties, 243 248 ancient masonry unit types, 246t compressive behavior, 245 247 elastic properties, 244 245 tensile behavior, 247 248 Masonry structures, 423, 469. See also Nonlinear seismic analysis of masonry structures analysis approaches, 9 12 incremental-iterative approaches, 10 11 limit analysis-based solutions, 11 12 BBMs, 13 20 continuum models, 20 30 GBMs, 34 38 incremental effect of interventions, 213 231 on buildings with double-planked timber floors, 223 224

787

on buildings with iron girders and flooring block slabs, 221 222 on buildings with r.c. slabs, 224 227 on buildings with traditional timber slabs, 219 221 incremental effects of addition of rigid slabs, 229 231 on masonry vaults, 227 229 strengthening interventions on floors and roofs, 216 219 mechanical and geometrical challenges, 5 9 geometrical challenges, 9 masonry experimental characterization, 7 8, 8f masonry mechanics, 6 7 structural details, 9 MMs, 30 34 modeling large and complex, 494 496 modeling of components and related interventions, 206 213 interventions to improve connections among components, 211 213 interventions to increase slab stiffness, 210 211 interventions to increase vault stability, 211 interventions to increase wall strength, 207 210 properties of existing masonry types and coefficients, 208t modeling strategies, 12 13 monumental and ordinary, 4f numerical strategies for, 5f strategies for modeling masonry buildings, 203 206 equivalent frame model assumptions, 205 206 finite element continuum model assumptions, 205 Masonry walls, 490 491 analysis, 748 758, 749f description of case study, 748 749 modeling considerations, 750 751 multiscale analysis of, 356 TRM-strengthened models in-plane behavior of unstrengthened and, 751 755 out-of-plane behavior of unstrengthened and, 755 758 Mass, 476 477 Master slave contact interaction, 704 705

788

Index

Material cohesive law, 693 694 Matrix fiber interfaces, 704 705 Maxwell model (M model), 312 313 MBM. See Macroblock model (MBM) Mc4 Loc software, 78 Mechanical damping, 482 Mechanical nonlinearity, 10 Medial axis, 404 Mesh density study, 111 112 Mesh generation procedure, 408 409 Mesh sensitivity tests, 671 Mesomodels, 631 Mesoscopic homogenization of unreinforced masonry, 725 727 Metal ties, 212 Method of Cells (MoC), 426, 442 450 extension to transverse loads, 447 450 homogenized yield surface under in-plane loads, 442 447 Microcracked material, effective properties of, 299 301 Microcracked mortar, 314 315 creep model for, 312 314 Micromechanical methodology coupling between homogenization techniques and damage theory, 297 327 nonlinear homogenization methods for masonry, 327 344 Micromodeling, detailed, 95, 142 143, 205, 242 Micromodeling/micromodels, 359 364, 631. See also Macromodeling/macromodels approach, 95, 163 164, 423 424 representation, 587 2D micromodel, 363 for Cosserat macromodel, 363 364 2D enriched plane state micromodel, 361 363 3D micromodel, 359 361 Micropolar Cosserat continuum models, 296 Midpoint method, 92 93 Minaret models, 486, 487f Mindlin Reissner homogenized model, 354 “Minimum intervention” principle, 56 MM model. See Modified Maxwell model (MM model) MMs. See Macroelement models (MMs) MoC. See Method of Cells (MoC) Model error (MEm), 113 statistics, 114, 118 for tensile fracture energy of brick, 116 of mortar joints, 112

Modeling strategies, 213 214 FEM, 94 96 FRPs-strengthened masonry, 630 632 macroscale, 267 272 for masonry structures, 12 13 numerical, 702 Modern bricks, 719 Modified Maxwell model (MM model), 309 311 parameters, 314 315 Modulus of elasticity (E), 241 242 Mohr Coulomb (MC) failure surface, 723 friction law, 276 law, 516 model, 479 shear-compression law, 205 type strength criterion, 410 411 Monte Carlo simulation methods, 92 93 Monumental buildings, 55 Monumental masonry structures, 3 Mortar constitutive law, 364 368 Mortar joints, 295, 560 cohesion of, 115 shear fracture energy of, 115 116 tensile fracture energy of, 112 115 tensile strength of, 112 Mortar brick interface properties, 251 254 elastic properties, 252 shear behavior, 253 254 tensile behavior, 252 253 Mortarless joint behavior characterization, 329 332 Multiscale analysis, 382 389 comparison between two PWTFA, 388 389 Cosserat multiscale model based on PWTFA, 386 388 of masonry walls, 356 nonlocal Cauchy multiscale model based on PWTFA, 383 386 approaches, 12, 24 30 failure analysis, 354 FE methods, 271 micromechanical-based approach, 28 numerical strategy, 19 procedures, 351 352 Munjiza NBS contact detection algorithm, 472 3Muri software, 78 79 Mutations, 599

Index

N National Laboratory for Civil Engineering (LNEC), 142 Natural hazards, 141 Natural selection, 599 Newmark techniques, 670 671 Newton Raphson method, 215 216, 670 671 NLKA. See Nonlinear kinematic analysis (NLKA) NLSA. See Nonlinear static analysis (NLSA) No-tension approaches, 22 (Non)linear optimization problem, 432 433 Nonassociated flow rule, 367 Nonlinear analysis methods, 629 630 Nonlinear constitutive laws, 698 Nonlinear dynamic analysis, 10, 144, 149, 156 of Castle of St. George, 174 180 of Palazzo Te, 193 196 of Sant’Andrea church, 186 189 Nonlinear FE analysis, 327 328, 768 769 Nonlinear FEA model, 92 Nonlinear FEM approaches, 503 504 Nonlinear homogenization methods for masonry, 327 344 experimental characterization of mortarless joint behavior, 329 332 procedure, 28 of refractory mortarless linings, 333 343 sustaining mean-field theories, 329 Nonlinear kinematic analysis (NLKA), 61, 78 79 Nonlinear link calibration along interfaces, 515 516 orthogonal to interfaces, 514 515 Nonlinear mean-field homogenization theories, 333 Nonlinear modeling, 607 622 circular arch with buttresses, 621f, 622f masonry wedge elements, 609 615 nonlinear numerical simulations, 615 622 Nonlinear models, 352 353 Nonlinear monoaxial stress strain relationships, 730 731 Nonlinear periodic masonry walls macromodels, 357 359 masonry constitutive law, 364 370 micromodels, 359 364 multiscale analysis of masonry walls, 356 numerical applications, 373 389 multiscale analyses, 382 389 unit cell response, 373 382 PWTFA procedure, 370 373

789

Nonlinear seismic analysis of masonry structures. See also Masonry structures macroscale modeling strategies, 267 272 macroscopic nonlinear behavior of masonry components, 265 267 in-plane failure modes of URM components, 265f numerical examples, 279 291 in-plane failure modes, 280 281 out-of-plane failure modes, 281 283 validation of macroelement representation, 283 291 3D macroelement approach, 273 279 Nonlinear static analysis (NLSA), 10, 61, 78 Nonlinearity, 270 Nonlocal Cauchy multiscale model based on PWTFA, 383 386 Nonlocal strain, 358 Nonperiodic microstructure masonry, 353 Nonsmooth contact dynamics method (NSCDs method), 18 Nonspatial analysis, 101, 103 106, 118 of URM, 93 94 Nonuniform rational b-spline model (NURBS model), 37 38, 588 598 kinematic limit analysis, 592 598 Normal stress slip curves, 695 Normalization condition, 598 NSCDs method. See Nonsmooth contact dynamics method (NSCDs method) Null element, 479 Numerical approach, 469 470 Numerical modeling/models, 203 for FRP, 553 580 finite element heterogeneous models, 560 580 interface-based models, 554 560, 555f of historical masonry structure, 163 164 for masonry structures analysis approaches, 9 12 BBMs, 13 20 continuum models, 20 30 GBMs, 34 38 mechanical and geometrical challenges, 5 9 MMs, 30 34 modeling strategies, 12 13 strategies, 702 Numerical simulations, 165 171, 620, 629 630 in limit analysis, 600 606 of masonry, 631

790

Index

Numerical stability, 482 483 NURBS model. See Nonuniform rational bspline model (NURBS model)

O One-dimension (1D) beam elements, 661 space, 687 689 Optimization problem, 431 Orthogonal links, 513 514 Orthotropic damage model, 24 Orthotropic softening continuum model, 632, 635t Orthotropy, 586 Out-of-plane behavior of masonry buildings, 141 142 of masonry walls, 661 of unstrengthened and TRM-strengthened models, 755 758, 773 775 diagonal bending spring, 279 failure modes, 266 267, 266f, 281 283 lateral loading, 93, 137 loaded masonry, 453 463, 458t, 735 738 response of masonry walls, 354

P Palazzo Te FE model and eigenfrequency analysis, 191 193 nonlinear dynamic analyses, 193 196 structure description, 190 Parabolic function, 759 760 Parameter identification process, 268 269 Participating mass ratios (PMRs), 165 Patio da Felicidade street, buildings in, 758, 759f PBA. See Performance-based assessment (PBA) PBO. See Polyparaphenylene benzobisoxazole (PBO) PCM. See Presidenza del Consiglio dei Ministri (PCM) Peak ground acceleration (PGA), 146 147, 150 152, 157, 164 Performance levels (PLs), 56 57 Performance-based assessment (PBA), 56 57 of masonry churches case study of San Clemente Abbey in Castiglione Casauria, 66 86 emblematic churches hit by seismic events, 56f

Periodic brick masonry, 6 PERPETUATE performance-based assessment procedure for churches, 57 combined use of 3D elastic model and 2D analyses, 63 66 general principles for complex architectonic assets, 58 59 identification and modeling of single macroelements, 60 61 information from 3D elastic modeling, 62 63 Persson Strang analogy, 406 truss-like optimization step, 409 410 PGA. See Peak ground acceleration (PGA) Piecewise transformation field analysis (PWTFA), 355 Cosserat multiscale model based on, 386 388 multiscale approaches, 382 383 nonlocal Cauchy multiscale model based on, 383 386 procedure, 370 373 Piers, 30, 506, 620 Plastic dissipation, 587 588, 595 Plastic hinge, 650 PLs. See Performance levels (PLs) PMRs. See Participating mass ratios (PMRs) Polynomial expansion of the microstress field (PES), 429 433 contiguous subdomains, 430f extension to transverse loads, 432 433 homogenized yield surface under in-plane loads, 429 432 micromechanical model for transverse loads, 430f subdivision of unit cell, 429f Polyparaphenylene benzobisoxazole (PBO), 687 689, 751, 761 762 POR method, 506 508 POR FLEX method, 506 508 Postdictions, 154 157, 155f Postpeak softening, 766 767 Prager Drucker model, 479 Predictive simple analytical/numerical model, 721 723 Predictor corrector algorithm, 36 Premature FRP debonding, 669 Presidenza del Consiglio dei Ministri (PCM), 56 57 Priori homogenization approaches, 25 28

Index Probabilistic models of URM comparison of spatial and nonspatial analysis, 103 106 nonspatial analysis, 101 spatial analysis, 101 103 vertical bending wall structural configuration, 96 100 Prony Dirichlet series, 309 311 Pull push tests, 542, 545f, 548t Pushover analysis, 10, 105 106, 156, 206, 761 762 of FRPs-strengthened masonry, 629 630 bond strength along FRP strips, 636f dimensions and strengthening configurations of masonry panels, 639f effect of environmental conditions on nonlinear response, 640 644 experimental and force displacement curves, 640f force displacement behavior, 644f material degradation, 642t modeling strategies, 630 632 numerical and experimental force displacement curves, 637f of strengthened masonry vaults and arches, 645 654 of strengthened masonry walls, 632 639, 634f Pushover nonlinear static analyses, 213 214 Push pull tests, 717 718 PWTFA. See Piecewise transformation field analysis (PWTFA)

Q Quadratic programming, 613, 615 Quadrilateral eight-node curved shell elements, 750 Quasiperiodic textures, 403 Quasistatic DEM techniques, 473 474

R Rammed earth subassembly analysis, 765 775 description of case study, 765 TRM-strengthened models in-plane behavior of unstrengthened and, 769 772 out-of-plane behavior of unstrengthened and, 773 775 modeling considerations, 766 769 Random field analysis, 92 93

791

Random sequential addition (RSA), 400 Random variables, 112 118 brick tensile fracture energy of, 116 118 tensile strength of, 116 mortar joint cohesion, 115 shear fracture energy of, 115 116 tensile fracture energy of, 112 115 tensile strength of, 112 stochastic analyses, 118 Rankine-type criterion, 267, 632 Rayleigh damping, 482 Rayleigh method, 670 671 RBSM. See Rigid body spring model (RBSM) Rectangular rigid elements, 727 729 Redistribution evaluation among macroelements, 74 77 Reduced order models (ROMs), 370 Reduction factors, 642 643 Refined modeling of interface behavior, 669 Refined numerical models, 663 Refractory mortarless linings, nonlinear homogenization of, 333 343 finite element nonlinear homogenization, 333 338 secant linearization schemes for assessing masonry global behavior, 338 343 Reinforced concrete coatings, 585 Reissner Mindlin model (RM model), 439 442 Representative volume element (RVE), 12, 268 269, 351 352, 365, 398, 424 426, 715 computations for irregular masonry, 410 414 damage distributions, 413f failure envelope, 412f geometry generation, 399 403 application of morphing to masonry, 403 distance field-assisted packing of inclusions, 400 401 distance field-based morphing of set of inclusions, 401 403 morphing procedure for two inclusion packings, 402f for irregular masonry, 404f Residual stresses, 253 Retrofitting scheme, 620 Reverse fitting method, 691 692 Rhino 3D model, 590 591

792

Index

Rigid blocks, 480 481 methods based on, 142 143, 143f Rigid body spring model (RBSM), 16, 27 28, 271 272, 271f, 504, 725, 727 729 Rigid elements, 471 Rigid no-tension model, 11 Rigid offsets, 506 508 Rigid slabs, incremental effects of, 229 231 Rilem Technical Committee TC 250 CSM, 694 695 RM model. See Reissner Mindlin model (RM model) Rocking failure mechanism, 265 266 ROMs. See Reduced order models (ROMs) Roofs, strengthening interventions on, 216 219 Rotational ductility, 649, 652 654 Rounding scheme, 472 473 RSA. See Random sequential addition (RSA) Running bond masonry, 269 270, 270f RVE. See Representative volume element (RVE)

S SAM. See Simplified analysis of masonry (SAM) San Clemente Abbey in Castiglione Casauria, 66 86 identification of masonry types, 73t information from 3d modeling, 73 78 failure mechanisms identification of macroelements, 77 78 redistribution evaluation among macroelements, 74 77 macroelement identification, 70 72, 71t performance-based assessment, 80 86 seismic assessment of single macroelements, 78 80 Sand, 241 Sant’Andrea church FE model and eigenfrequency analysis, 182 186, 183f, 184f nonlinear dynamic analyses, 186 189 structure description, 181 182, 183f SC scheme. See Self-consistent scheme (SC scheme) Scalar damage model, 411 Secant linearization schemes, 338 343 effective properties, 342 343 linearization step, 339 341 results, 341 343

Secant method, 215 216 Second-order computational homogenization, 29 Seismic assessment of historic masonry structures, 142 145 blind predictions, 148 154 out-of-plane response of existing masonry buildings, 142f postdictions, 154 157, 155f shake table tests, 145 148 Seismic assessment of single macroelements, 78 80 Seismic behavior, 486 490 of masonry structures, 263 Seismic response, 677 678 Seismic vulnerability assessment of historical masonry constructions, 163 164 Castle of St. George, 171 181 concrete damage plasticity model, 167 171 numerical simulations, 165 171 Palazzo Te, 190 196 Sant’Andrea church, 181 189 Self-consistent scheme (SC scheme), 309 311 Self-weight safety factor, 760 761 SEM. See Structural elements model (SEM) Sequential Quadratic Programming (SQP), 703 704 SFRP. See Steel fiber-reinforced polymer (SFRP) Shake table tests, 145 148, 660 661 Brick House, 146f behavior, 148f Stone House, 146f behavior, 147f Shear failure, 141 142 Shear fracture energy of mortar joints, 115 116 Shear lap-tests, 687 Shear stress slip laws, 555 556, 686 687, 695, 704 705 Shear-sliding links, 513 516 Shell macroelement, 522 523 for modeling curved geometry, 512 513 Simplified analysis of masonry (SAM), 506 508 Simplified micromodeling, 95, 142 143, 205, 242, 296 Single-brick reinforcements, 717 718 Single-lap shear tests, 694 695 Slab stiffness, interventions to increase, 210 211

Index Slender barrel vaults, 661 662 Soft bricks, 243t Soft-contact approach, 473 Softening anisotropic elasto-plastic continuum model, 642 Sophisticated FE approaches, 164 165 Space-based search techniques, 472 Spandrels, 30, 506 Spatial analyses of URM walls, 93 94, 101 106 in one-way horizontal bending, 106 108 description of experimental program, 108 110 finite element analysis model, 110 118 in one-way vertical bending finite element analysis modeling strategies, 94 96 probabilistic models, 96 103 in two-way bending, 124 127, 126t wall modeling simply supported on both vertical sides and bottom edge, 132 137 wall modeling simply supported on four sides, 127 132 Spring Model, 706 709, 706f Spring-based approaches, 33 34 SQP. See Sequential Quadratic Programming (SQP) Square root of sum of squares (SRSS), 64 SRGs. See Steel-reinforced grouts (SRGs) STA Data TREMURI 11.4, 202 Standard nonlocal averaging approach, 671 Static analysis, 74, 144 Static nonlinear analysis, 149 Static theorem, 11 static theorem-based approaches, 35 36 static theorem-based solutions, 36 Steel fiber-reinforced polymer (SFRP), 542 “Steel-H” strengthening scheme, 763 Steel-reinforced grouts (SRGs), 691 692 specimen, 746 747, 761 762 “Steel-V” strengthening scheme, 763 Step-by-step multiscale approaches, 26, 28 30 Step-by-step solution technique, 613 Stiff bricks, 243t Stiffness of mechanical system, 722 stiffness-proportional component, 482 Stochastic analysis method, 106 107 Stochastic FEA, 92 Strength anisotropy, 6

793

Strengthened masonry vaults and arches analysis, 645 654 Strengthened masonry wall analysis, 632 639, 634f Strengthening interventions, 210 211 on floors and roofs, 216 219 numerical simulation of, 210 Stress-transfer mechanism in FRCM concrete joints, 704 705 Stress slip response, 694 695 Stress strain diagram, 303 linearization schemes, 333 Structural analysis, 149, 587 Structural components, methods based on, 142 143, 143f Structural elements, 483 485 beam element, 484 485, 484f global reinforcement, 484, 484f local reinforcement, 483, 483f Structural elements model (SEM), 61 Structural masonry, 745 Structural-scale model, 12 Subcontacts, 477

T Tall masonry structures, 485 486 Tangential FRP/masonry interface stresses, 617 618 Tensile fracture energy of brick, 116 118 of mortar joints, 112 115 Tensile strength of brick, 116 of mortar joint, 112 Textile-reinforced mortar (TRM), 206, 229, 714, 745 macromodeling strategies for TRMstrengthened masonry, 746 748 TRM-to-substrate interface, 710 Textured continuum-based approaches, 15f, 18 19 TFA. See Transformation field analysis (TFA) Theoretical bond analysis, 539 542 Thin-layer mortar, 241 Three-dimension (3D) FE model, 63, 74, 164 165, 554 global model, 59 linear model, 65 66 macroelement, 511 512, 519 522

794

Index

Three-dimension (3D) (Continued) micromodel, 355, 359 361 micromodeling heterogeneous approach, 736 737 nonlinear dynamic analyses, 164 165 nonlinear FEA model, 96, 99t, 110 111 plasticity-damage constitutive law, 275 276 point clouds, 9 3D macroelement approach, 273 279 assumptions, 273 274 macroelement formulation, 274 279 cohesive boundaries, 275 277 constitutive law for macroelement springs, 278f in-plane shear spring, 277 278 out-of-plane diagonal bending spring, 279 3DEC software, 470 applications of modeling masonry and historical structures, 485 496 DEM codes for masonry modeling and key features, 470 485 rigid and deformable polyhedral block in, 476f Thrust line, 646, 647f, 652 Thrust network analysis (TNA), 36 Time history. See Nonlinear dynamic analysis Time integration schemes, 10 11 Time-dependent crack density, 317 318 case of compressed masonry panel, 323 326 case of periodic unit cell, 319 323 TNA. See Thrust network analysis (TNA) TNO DIANA 9 software, 95 96 Total strain crack model, 205, 207, 751 Traditional retrofitting techniques, 585 Traditional timber slabs, interventions on buildings with, 219 221 Transformation field analysis (TFA), 28, 352 353, 370 Transient nonlinear analysis. See Nonlinear dynamic analysis Translational motion equation for single block, 480 TREMURI approach, 207 210 Tremuri software, 33, 506 508 Trilinear behavior, 729 730 Triplet shear test, 110, 253 Triplet tests, 253 Triumphal arches, 661 662

TRM. See Textile-reinforced mortar (TRM) Truss elements, 638 Trusses, 724 725 Turnsek Cacovic law, 516 Two-dimension (2D) Cauchy macromodel, 370 code UDEC, 478 continuum models, 355 Cosserat continuum, 358 DIC, 330 FE models, 554, 708 709, 708f macroelement, 510 513, 517 519 masonry UC, 352 micromodel, 355, 363 for Cosserat macromodel, 363 364 2D enriched plane state micromodel, 361 363 Two-step holonomic homogenization model, 725 730 mesoscopic homogenization of unreinforced masonry, 725 727 structural implementation, 727 730 Two-steps procedure, 65 Two-way bending of unreinforced masonry, 288 291

U UB-ALMANAC (upperbound adaptive limit analysis tool), 37 38 UC. See Unit cell (UC) UDEC. See Universal Distinct Element Code (UDEC) Umbria Marche earthquake, 713 714 Undamaged stage, 698 Uniaxial compression test, 374 375 Unit cell (UC), 352, 398, 715 response, 373 382 combined loading conditions, 381 382 compression symmetric shear test, 378 compression tensile test, 375 377 flexural test, 378 380 uniaxial compression test, 374 375 Unit normal vector, 477 Unit-to-unit spatial variability, 92 Universal Distinct Element Code (UDEC), 470 applications of modeling masonry and historical structures, 485 496 DEM codes for masonry modeling and key features, 470 485

Index Unreinforced masonry (URM), 91 92, 263, 504. See also Masonry structures in-plane loading validation of panels, 284 288 spatial analyses of walls in one-way horizontal bending, 106 124 in one-way vertical bending, 93 106 in two-way bending, 124 137 two-way bending of, 288 291 Upper bound limit analysis problem, 588 URM. See Unreinforced masonry (URM)

Voronoi-cells, 476 477, 476f Voxel-based FE models, 398 399

W Water-to-solids ratio (W/S ratio), 767 768 Wedge-shaped elements, 588 589

X XFEM. See Extended finite element method (XFEM)

V VAR method, 339, 341 343 Vaults, 491 493 geometry, 588 stability, 211 Vertical bending wall structural configuration, 96 100 Viscous contact forces, 482 Voronoı¨ diagram. See Voronoı¨ tessellation Voronoı¨ tessellation, 405

795

Y Young’s modulus, 634 635

Z Zero compressive strength, 638 Zero-thickness interface elements, 702 Zooming, 600