Design of Reinforced Concrete Silo Groups 9783030136215, 3030136213

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Building Pathology and Rehabilitation

Fernando A. N. Silva Bernardo Horowitz João M. P. Q. Delgado António C. Azevedo

Design of Reinforced Concrete Silo Groups

Building Pathology and Rehabilitation Volume 10

Series Editors Vasco Peixoto de Freitas, University of Porto, Porto, Portugal Aníbal Costa, Aveiro, Portugal João M. P. Q. Delgado, University of Porto, Porto, Portugal

This book series addresses the areas of building pathologies and rehabilitation of the constructed heritage, strategies, diagnostic and design methodologies, the appropriately of existing regulations for rehabilitation, energy efficiency, adaptive rehabilitation, rehabilitation technologies and analysis of case studies. The topics of Building Pathology and Rehabilitation include but are not limited to - hygrothermal behaviour - structural pathologies (e.g. stone, wood, mortar, concrete, etc…) diagnostic techniques - costs of pathology - responsibilities, guarantees and insurance - analysis of case studies - construction code - rehabilitation technologies architecture and rehabilitation project - materials and their suitability - building performance simulation and energy efficiency - durability and service life.

More information about this series at http://www.springer.com/series/10019

Fernando A. N. Silva Bernardo Horowitz João M. P. Q. Delgado António C. Azevedo •

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Design of Reinforced Concrete Silo Groups

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Fernando A. N. Silva Department of Civil Engineering Catholic University of Pernambuco Boa Vista, Recife, Brazil

Bernardo Horowitz Department of Civil Engineering Federal University of Pernambuco Recife, Brazil

João M. P. Q. Delgado Faculty of Engineering, CONSTRUCT-LFC University of Porto Porto, Portugal

António C. Azevedo Faculty of Engineering, CONSTRUCT-LFC University of Porto Porto, Portugal

ISSN 2194-9832 ISSN 2194-9840 (electronic) Building Pathology and Rehabilitation ISBN 978-3-030-13620-8 ISBN 978-3-030-13621-5 (eBook) https://doi.org/10.1007/978-3-030-13621-5 Library of Congress Control Number: 2019931828 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Groups of reinforced concrete silos are structures commonly used in the food industry, where it is usually necessary to separate the storage of different types and sources of grain. The grouped layout of silos has numerous benefits when compared with single-cell silos in which the emphasis is on creating further space for silage, normally referred to as interstice—a space formed between the edges of the group’s cells. This economic benefit, on the other hand, raises a structural problem for the designer of this type of building, which is to assess the magnitude of bending moments and hoop forces due to the structural continuity of the walls in the interstice region of the cells. Bending moments assume extreme values exactly when the interstice is loaded and the other cells in the group are empty. In order to be able to achieve economic and safe designs, it is, therefore, essential to understand the structural behaviour of this type of structure, considering the loading imposed upon it. The purpose of this book is to present a new calculation procedure of those moments, easy to use and with satisfactory responses when compared to the three-dimensional analysis using the finite element method (FEM), which today is the state-of-the-art structural analysis of this type of construction. To develop the formulation of the proposed analysis models, a parametric study was carried out that allowed the adequate consideration of the variables involved. The book is divided into six chapters. Chapter 1 contemplates the characterization of the problem to be solved. It provides a bibliographical review on the methods of calculation of the bending moments due to the structural continuity in a group of silos available in the literature with brief comments on their foundations. Chapter 2 presents the geometry of the groups of silos analysed and a review of the aspects related to the applied loads. It is also defined as the physical parameters of the stored material and the procedures for calculating the design of horizontal pressure diagrams to be used.

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Chapter 3 is integrally dedicated to 2D analysis using the Slice Method. It includes the definition of the method, the load cases considered, the finite element meshes used in the analysis of the resulting structure and concepts related to the application of the loads to the models generated. The theoretical formulation for the imposition of constraint equations (required in the modelling with frame elements) and the procedure used to calculate the stresses resultants in the solid element modelling are also included in this chapter. In the end, a comparison of results between the analyses carried out and conclusions about the use of the exposed method is presented. Chapter 4 addresses the three-dimensional analysis of groups of reinforced concrete silos. Three types of modelling strategies are presented—solid elements, shell elements, and shell and solid elements—with their respective finite element meshes. In the modelling with shell and solid elements—here called mixed modelling—the procedure to be used in the generation of the constraint equations, necessary for the connection between the two types of mesh elements, is presented. At the end, a comparison of results between the analysed models is presented. Chapter 5 presents the interpretation of the structural behaviour and the formulation of the proposed analysis models. The parametric study developed, which enabled the establishment of the recommended calculation procedure, is also discussed. Finally, a comparison between the results obtained with the application of the proposed analysis model and those resulting from the calculation methods available in the literature and the Finite Element Method—FEM—are presented. Chapter 6 summarizes the conclusions of the study carried out and contemplates the recommendations about the analysis of groups of reinforced concrete silos. Detailed and commented description of the steps required to use the proposed model, with an example of application, are also presented. Boa Vista, Recife, Brazil Recife, Brazil Porto, Portugal Porto, Portugal

Fernando A. N. Silva Bernardo Horowitz João M. P. Q. Delgado António C. Azevedo

Contents

1 Characterization and Brief Literature Review . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Two-Dimensional Linear Elastic Analyses—Slice Method . 3.1 Modelling Using 2D Frame Elements . . . . . . . . . . . . . 3.2 Applying Nodal Loads . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Constraint Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Lagrange Multiplier Method . . . . . . . . . . . . . . . . . . . . 3.5 Modelling with Plane Strain Finite Element . . . . . . . . . 3.6 Additional Shape Functions . . . . . . . . . . . . . . . . . . . . . 3.7 Calculation of Stress Resultants . . . . . . . . . . . . . . . . . . 3.8 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 3.8.1 Deformed Geometry . . . . . . . . . . . . . . . . . . . . 3.8.2 Horizontal Normal Forces . . . . . . . . . . . . . . . . 3.8.3 Bending Moments . . . . . . . . . . . . . . . . . . . . . . 3.8.4 Bending Moments Envelope . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Three-Dimensional Analysis . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . 4.2 Boundary Conditions . . . . . . . . . . . . 4.3 Modelling with Linear Finite Element 4.3.1 Loading in the Cell . . . . . . . . 4.3.2 Loading in the Interstice . . . .

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2 Geometry and Loading . . . . . . . . . . . . . 2.1 Geometry . . . . . . . . . . . . . . . . . . . . 2.2 Loading . . . . . . . . . . . . . . . . . . . . . 2.2.1 Janssen’s Theory . . . . . . . . . 2.2.2 Calculation of Overpressures References . . . . . . . . . . . . . . . . . . . . . . .

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4.4 Modelling with Shell Elements . . . . . . . . 4.5 Modelling with Solid and Shell Elements . 4.6 Results and Discussion . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Structural Behaviour and Simplified Model Proposition . 5.1 Load Bearing Mechanism . . . . . . . . . . . . . . . . . . . . . 5.2 Proposed Simplified Model . . . . . . . . . . . . . . . . . . . . 5.3 Fitting with the Least Square Method . . . . . . . . . . . . 5.4 Parametric Study . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Procedure Proposed . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Bending Moments . . . . . . . . . . . . . . . . . . . . . 5.6.2 Horizontal Normal Forces . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Conclusions and Recommendations . . . . . . . . . . . . . . . . . . . . . . . . .

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2.1 2.2 2.3 3.1 3.2 3.3 3.4

Fig. 3.5 Fig. 3.6 Fig. 3.7 Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.

3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 4.1

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Geometry of the group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic drawing for deduction of Janssen’s formula . . . . . . Horizontal design pressure curves . . . . . . . . . . . . . . . . . . . . . . Cross section in silo groups . . . . . . . . . . . . . . . . . . . . . . . . . . Load cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modelling with 2D frame elements . . . . . . . . . . . . . . . . . . . . . Detail of the region of connection between two cells of the silo group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mesh of the slice of the silo group using plane strain finite element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Four node finite element—Q4 . . . . . . . . . . . . . . . . . . . . . . . . . Deformation of the Q4 element without additional shape function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Data for the calculation of stress resultants . . . . . . . . . . . . . . . Deformed geometry of the silo group—Load Case I . . . . . . . Deformed geometry of the silo group—Load Case II . . . . . . . Deformed geometry of the silo group—Load Case III . . . . . . Deformed geometry of the silo group—Load Case IV . . . . . . Deformed geometry of the silo group—Load Case V . . . . . . . Deformed geometry of the silo group—Load Case VI . . . . . . Bending moments—Load Cases I e II . . . . . . . . . . . . . . . . . . Bending moments—Load Cases III e IV . . . . . . . . . . . . . . . . Bending moments—Load Case V . . . . . . . . . . . . . . . . . . . . . . Bending moments—Load Case VI . . . . . . . . . . . . . . . . . . . . . Bending moment’s envelope—Frame elements . . . . . . . . . . . . Bending moment’s envelope—Q8 element . . . . . . . . . . . . . . . Structure for three-dimensional analysis—cross sections of interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Finite element mesh with linear solid elements . . . . . . . . . . . . a Horizontal normal forces—loading in cell and b variation of the maximum normal horizontal forces. . . . . . . . . . . . . . . .

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Fig. 4.11 Fig. 4.12 Fig. 4.13 Fig. 5.1 Fig. 5.2 Fig. 5.3 Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.

5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17 5.18 5.19 5.20 5.21

a Bending moment Mz —loading in cell and b maximum bending moments Mz along the height . . . . . . . . . . . . . . . . . . a Normal horizontal forces—loading in the interstice— Section 3 and b variation of the maximum normal horizontal forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a Bending Moments Mz —loading in interstice and b variation of the maximum positive and negative bending moments . . . . Mesh with shell finite element . . . . . . . . . . . . . . . . . . . . . . . . Mesh with solid and shell element . . . . . . . . . . . . . . . . . . . . . Connection of solid and shell elements . . . . . . . . . . . . . . . . . . Bending moments Mz in section 1 and 2—loading applied in the cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bending moments Mz in section 3 and 4—loading applied in the cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bending moments Mz in section 1 and 2—loading applied in the interstice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bending moments Mz in section 3 and 4—loading applied in the interstice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structural behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deformed shape of the interconnection region—loading applied in interstice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Normal stresses—longitudinal direction (in kN/m2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proposed simplified model . . . . . . . . . . . . . . . . . . . . . . . . . . . Scheme for calculating the spring constant . . . . . . . . . . . . . . . Detail of the Timoshenko beam model . . . . . . . . . . . . . . . . . . Calculation of transversal loading . . . . . . . . . . . . . . . . . . . . . . Comparison of nodal displacements . . . . . . . . . . . . . . . . . . . . Comparison of bending moments . . . . . . . . . . . . . . . . . . . . . . Parameters for geometric index H/D = 3.0 . . . . . . . . . . . . . . . Parameters for geometric index D/t = 20 . . . . . . . . . . . . . . . . Variation of parameter c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bending moments for the geometric index D/t = 20 . . . . . . . . Bending moments for the geometric index D/t = 30 . . . . . . . . Bending moments for the geometric index D/t = 40 . . . . . . . . Bending moments for the geometric index D/t = 50 . . . . . . . . Bending moments for the geometric index D/t = 60 . . . . . . . . Bending moments in Sect. 4 of Fig. 4.1 . . . . . . . . . . . . . . . . . Bending moments in Sect. 3 of Fig. 4.1 . . . . . . . . . . . . . . . . . Horizontal normal forces—plan of symmetry . . . . . . . . . . . . . Horizontal normal forces—support . . . . . . . . . . . . . . . . . . . . .

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5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10

Nodal displacements. . . . . . . . . . . . . . . . . . . . . . Negative bending moments . . . . . . . . . . . . . . . . Positive bending moments . . . . . . . . . . . . . . . . . Parameters for H/D = 3.0 . . . . . . . . . . . . . . . . . . Parameters for D/t = 20 . . . . . . . . . . . . . . . . . . . Parameters c. . . . . . . . . . . . . . . . . . . . . . . . . . . . Negative bending moments . . . . . . . . . . . . . . . . Positive bending moments . . . . . . . . . . . . . . . . . Horizontal normal forces—plane of symmetry . . Horizontal normal forces—support . . . . . . . . . . .

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Chapter 1

Characterization and Brief Literature Review

The problem to be solved is the calculation of stresses resultants on the walls of groups of reinforced concrete silos due to interstice loading. The groups of silos considered in the work are constituted of four cylindrical cells of equal diameter and it is of special interest to investigate the magnitude of bending moments at the ends of the regions between two cells, since it is in these locations that the highest values of bending moments usually occur. Such moments are often referred to as moments due to structural continuity. Live loads are pressures exerted by the material stored inside interstice space generated by the wall intersections of the cells of the group. Literature offers many methods for calculating bending moments due to the structural continuity in groups of reinforced concrete silos in an interstice-loading situation. Albigés and Lumbroso [1] provided expressions based on the analysis of a unit height horizontal slice of the group. This slice is loaded with the resulting force of the pressure applied to the interstice, with forces due to the walls intersection of the cells of the group and with the elastic forces due to the differences in shear force on the lower and upper sections delimiting the slice. The proposed formulation also considers that the wall thickness and the length of the region common to two cells are small in relation to the average radius. Ciesielski et al. [2] presented a method in which the arch supports comprising the interstice wall undergo certain tangential displacement. The solution to the problem is expressed by six equations that, due to the angular location of the section studied, provide the values of the bending moment and normal force. The proposed method considers that the interstice wall of groups formed by three, four or five silos undergoes radial loading. Timm and Windels [3] provided formulas to calculate bending moments based on the hypothesis that the region connecting two cells prevents rotation, but permits tangential displacement of the interstice wall supports, which, according to the authors, behaves like an arch undergoing radial pressure due to the applied loading. © Springer Nature Switzerland AG 2019 F. A. N. Silva et al., Design of Reinforced Concrete Silo Groups, Building Pathology and Rehabilitation 10, https://doi.org/10.1007/978-3-030-13621-5_1

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1 Characterization and Brief Literature Review

The formulas allow the calculation of bending moments, shear forces and tangential forces per unit of length for different sections of the arch, as a function of its angular location. Safarian and Harris [4] suggest a numerical solution using computational matrix analysis of a unit height section. The part of the section corresponding to the interstice wall is modelled as an arch with very rigid elements in its supports. The loading applied in a radial pressure, distributed along the element length. Haydl [5] presented a finite element solution based on general-purpose software [6], where a unit height section of the silo group is analysed using shell elements. The region connecting two cells is modeled with three elements, applying specific boundary conditions to their nodes to take into account the connection with other silos. The solution also considers that the centre of the connecting region is situated 30° of the crown of the arch comprising the interstice wall and its thickness is four times the wall thickness. The results are expressed in direct-application formulas of values of the wall thickness/silo wall thickness ratio of 1 and 2. Gurfinkel [7] proposes formulas for calculating the bending moments that are an adaptation of formulas of Kellner [8]. As suggested by Gurfinkel [7] and Safarian and Harris [4], there is also the clamped arch model. If the hypothesis considered is perfect fixed supports and the loading is in radial direction, the bending moments calculated by this method are negligible. Prato and Godoy [9] realized the importance of the three-dimensional nature of the problem, but the modelling of the interstice region used by the authors is too simplified, bearing in mind its strong influence on the applied forces range. Balkaya et al. [10] also recognizes the importance of tri-dimensional effects on the overall behaviour of silo groups but propose a 2D model based on beam elements to represent the geometry of the overlapping region that clearly underestimates complex phenomena that occur in that region. With the exception of Prato and Godoy, all other aforementioned methods are based on two-dimensional analysis that, by their very nature, overlook the influence of the three-dimensional interaction of the different cells in the group, a decisive aspect in the structural response of the silo groups under interstice loading. Moreover, because of the different boundary conditions used in the analysis, the numerical results provided by these methods are very conflicting. This fact leaves the designer with no exact benchmark for choosing the most suitable method, an even more relevant aspect when it is known that the divergence between the top and bottom value of the bending moments obtained using these methods can be thirty times or more, according to Horowitz and Nogueira [11]. On the other hand, because of the problems in modelling and interpreting results, three-dimensional analysis using finite element method is not yet easy to apply in the daily design of this type of structure. In this context, the book herein intends to offer an alternative to calculate the bending moments in groups of reinforced concrete silos, which provides quite satisfactory results compared with the three-dimensional analysis but has the benefit of being easy to apply in the preliminary and final design stages. The basis, validation, and demonstration of the effectiveness of the procedure proposed are discussed in detail.

References

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References 1. Albigés M, Lumbroso A (1964) Silos a Cellules Principales Circulaires et Intermédiaires en as de carreau. Annales de L’Institut technique du bâtiment et des travaux publics, pp 1547–1562 2. Ciesielski R, Mitzel A, Stachurski W, Suwalski J, Zmudzinski Z (1970) Behalter, Bunker, Silos, Schornsteine, Fernsehturme und Freileitungsmaste. Verlag von Wilhelm Ernst & Sohn, Berlin 3. Timm G, Windels R (1977) Silos, Sonderdruck ans Dem Beton-Kalendan. Verlag von Wilhelm Ernst & Sohn, Berlin, Germany 4. Safarian SS, Harris EC (1985) Design and construction of silos and bunkers. Van Nostrand Reinhold Company, New York 5. Haydl HM (1987) Bending of interstice walls in circular silos. J Struct Eng ASCE 113(10):2311–2315 6. SAP 2000 (2014) User’s manual, Wilson L Edward and Habibullah Ashraf 7. Gurfinkel G (1990) Reinforced-concrete bunkers and silos. In: Gaylord H, CN Gaylord (eds) Structural engineering handbook, Sect 26. Wiley, New York 8. Kellner M (1960) Silos a Cellules de grande profondeur, Organe de la technique des travaux publics et du ciment arm, pp. 612–622 9. Prato CA, Godoy LA (1989) Bending of multi-bin RC cylindrical silos. J Struct Eng ASCE 115(12):3194–3200 10. Balkaya C, Kalkan E, Yuksel SB (2006) FE analysis and practical modelling of RC multi-bin circular silos. ACI Struct J 103(2):365–371 11. Horowitz B, Nogueira FA (1999) Stress resultants due to interstice loading in group of four cylindrical silos. ACI Struct J 96(2):307–313

Chapter 2

Geometry and Loading

2.1 Geometry The geometry of the silo groups of interest is shown in Fig. 2.1. They may have a thick slab (or conical hoppers with ring beams) at the bottom, and a slab, possibly on beams, at the top. Due to the thick slab close to the foundation, we will assume the cylindrical shell clamped at the bottom slab. The objective is to compute bending moments and hoop forces on the shell due to interstice loading, in order to be able to design the horizontal reinforcement of the walls. The group of silos showed in Fig. 2.1 is composed of four cylindrical cells of the same diameter that forms an additional space for store materials, knows as interstice. The acronym SOR means Shell Overlapping Regions, which represents the area of intersections of the walls of adjacent cells. To show the quality of the model of analysis proposed, an illustrative example with H = 30 m, D = 10 m, t = 0.2 m and p = 42.46 kN/m2 is solved in Chap. 6. Regarding the diameter and heights considered in the example, also investigated values cover a wide range of groups of existing silos used to store granular materials. In fact, groups with circular cells with diameters of 4, 6, 8, 10 and 12 m and heights of 12, 15, 18, 20, 22, 24, 25, 30 m were studied. Chapter 5 presents the results of the parametric investigation performed.

2.2 Loading The loading considered is that resulting from the action of the ensiled material on the walls and bottom of the cells. This action takes place through the following pressures: • Vertical pressure; • Lateral pressure and • Vertical friction force. © Springer Nature Switzerland AG 2019 F. A. N. Silva et al., Design of Reinforced Concrete Silo Groups, Building Pathology and Rehabilitation 10, https://doi.org/10.1007/978-3-030-13621-5_2

5

6

2 Geometry and Loading

Fig. 2.1 Geometry of the group

The responsibility for supporting the weight of the material stored inside a silo is assumed by the bottom, through the vertical force coming from the weight of the material on the existing slab, and by the walls, mobilized by the vertical force of friction generated by the contact with the same ensiled material. In addition, pressures arise perpendicular to the walls, usually referred to as lateral or horizontal pressures. The horizontal pressures are loads of special interest because of result from them the stress resultants necessary for the design of cylindrical shells that constitute the silo walls. The above-mentioned pressures are commonly referred to as static loads and there are several theories available for their estimation. In the present work, the Jansen’s theory of silo loads [1] detailed in Sect. 2.2.1—was used. In silos design, it is also necessary to consider the pressure increases due to the unloading process the ensiled material. During this process, the pressure gradient is generally very high and the final pressures to be considered in the design often exceed the calculated values for the resting material—static pressures. This increase in pressures is sometimes referred to as a dynamic effect, although the term overpressure is more commonly used in the relevant literature. Several factors influence the occurrence of overpressures in silos. Among these, the following stand out: • Arching of ensiled material; • The collapse of arches formed by ensiled material; • A sudden change in the type of flow (funnel/mass).

2.2 Loading

7

There are, currently, two ways of estimating the final pressures to use in silo designs. One of them is the direct calculation of the total pressures, using existing theories available in the literature [2–10]. Another way is to multiply static pressures by overpressure coefficients [10], which take into account the dynamic effects of loading and unloading process. In the approaches carried out throughout the present work, the calculation of the design pressures was made using the overpressure coefficients indicated by the American Concrete Institute [11]. The pressures developed inside a silo are closely related to the physical properties of the stored material, and the following properties can be highlighted as being of particular importance: • Specific weight; • The coefficient of friction wall/stored material; • Internal angle of friction of the stored material or angle of repose. To establish numbers to represent each one of the properties reported above is a very important task because the magnitude of the pressures exerted by the ensiled material on the walls and bottom of the cells is strongly dependent on these values. In design practice, it is common to use tables, available in the literature, that suggest a range of variation to be observed. However, it is important to consider that the values presented in these tables are approximate values and, therefore, should be used with care and. Whenever uncertainties remain regarding the values to be adopted the most recommended procedure is to carry out laboratory tests to obtain properties desired. In the present work, the values of the physical parameters of the stored material were the following, related to an existing structure. • Specific weight = 8.1 kN/m3 • Coefficient of friction wall/stored material = 0.35 • Internal angle of friction of the stored material or angle of repose = 34°. The value of the specific weight of the ensiled material was adopted according to practical experience and the other values were obtained in the recommendations of the American Concrete Institute [11].

2.2.1 Janssen’s Theory The theory presented by Janssen in 1895 was the first to take into account the friction of the ensiled material and the silo wall. It is based on the balance of vertical forces acting on an infinitesimal horizontal layer of ensiled material, as shown in Fig. 2.2. The theory considers that the vertical pressure in a given horizontal cross-section is constant and the relationship between the horizontal and the vertical pressure is expressed by the coefficient K constant for a given depth. By verifying the equilibrium of the vertical forces acting on that layer, it is possible to obtain:

8

2 Geometry and Loading

Fig. 2.2 Schematic drawing for deduction of Janssen’s formula

  dq dy − u  pU dy = 0 q A + γ Ady − A q + dy

(2.1)

where q is the vertical static pressure, A is the cross-sectional area, γ is the specific weight of the ensiled material, p is the horizontal static pressure, U is the crosssectional perimeter and μ is the coefficient of friction between the ensiled material and the silo wall. Simplifying Eq. (2.1), substituting p for Kq, taking into accounts that the relation (U/A) is the inverse of the hydraulic radius—R—and carrying out the pertinent separation of variables, one gets at the following expression: dy =

dq  γ − u RK q

(2.2)

Equation (2.2) can be integrated by assuming, for calculation of the constant of integration, that for y = 0 we have q = 0. Thus, we got the expression that relates the vertical pressure to the depth (see Eq. 2.3).   γ R y =  ln (2.3)  uK γ − u RK q

2.2 Loading

9

Finally, solving the Eq. (2.3) to find q, we obtain the expression of the vertical pressure of the Janssen’s Theory, as it follows:   Rγ  − u RK γ 1 − e u K

q=

(2.4)

The equivalent horizontal pressure is given by the expression p = Kq: p=

  Rγ  − u RK γ 1 − e u

(2.5)

In Janssen’s theory there is no indication about the values to be adopted for the variable K, usually referred to as the pressure coefficient. In the present work, Eq. (2.6) was used. k=

1 − sin ρ 1 + sin ρ

(2.6)

where ρ is the angle of internal friction of the material. The vertical frictional force was evaluated using Eq. (2.7). V = (γ Y − 0.8q)R

(2.7)

Equations (2.6) and (2.7) are included in the recommendations of the American Concrete Institute [11] and the parameters need to calculate the static pressures and the vertical friction force are presented in Sects. 2.2 and 2.2.1 of present work. To calculate the hydraulic radius of the circular cells, the recommended value of ACI [11] was used and for the interstice space, the procedure adopted was to consider the hydraulic radius of a square of equivalent area.

2.2.2 Calculation of Overpressures For the calculation of the overpressures, the recommendations of the American Concrete Institute [11] were followed. This code recommends that the design forces and pressures, resulting from the action of the ensiled material, must be evaluated for silos with concentric discharge using Eqs. (2.8) and (2.9) as it follows. qdes = Cd q

(2.8)

pdes = Cd q

(2.9)

10

2 Geometry and Loading

Fig. 2.3 Horizontal design pressure curves

The maximum friction force on the wall occurs simultaneously with the minimum vertical static pressure, and then, in concrete silos, we must adopt the Eq. (2.10) to calculate it. Vdes = V

(2.10)

The values of the overpressure coefficient C d to be considered in the calculation of the design pressures are those given by the American Concrete Institute [11]. The numbers presented are minimum values recommended, however smaller values may be adopted if the designer can attest that such values are satisfactory. Figure 2.3 shows the variation of horizontal design pressure along the height of the silo is shown for both cell and interstice. These values correspond to an existing structure, with 10 m in diameter and 30 m in height, that is investigated in detail in this study.

References 1. Janssen HA (1895) Versuche über Getreidedruck in Silozellen. Zeitschrift Verein Deutscher Ingenieure, pp 1045–1049 2. Caquot A, Kerisel J (1956) Traite Mecanique des Sols. Gauthier Villars Editeur

References

11

3. Pieper K, Wenzel F (1964) Druckverhaltnisse in Silozellen. Verlag von Wilhelm Ernst and Sons 4. Geniev GA (1958) Voprosi Dinamiki Siputchei (Questions of the dynamics of granular mass). Government Publication of Literature on Construction and Architecture, Moscow, URSS 5. Platanov PN, Kovtum AP (1959) Davlenie Zerna na Stenki Silosov Elevatorov, Mukomolno Elevatornaia Promyshlennost 6. Theimer OF (1970) Betrachtungen Uber Druckverhaltnisse in Silozeneb. Deutsche MullerZeitung 7. Walker D (1966) M. An approximate theory for pressures and arching in hoppers, Chem Eng Sci J, pp 975–997 8. Reimbert M, Reimbert A (1980) Pressures and overpressures in vertical and horizontal silos. In: International Conference on Design of Silos for Strength and Flow, Powder Advisory Center 9. Jenike AW (1977) Construction of concrete silos. Technical Report, Norwegian Society of Chartered Engineers 10. Safarian SS (1969) Design pressure of granular material in silos. ACI J Proc, 539–547 11. ACI Committee 313 (2011) Standard practice for design and construction of Concrete silos and stacking tubes for storing granular materials (ACI 313-11). American Concrete Institute, Detroit

Chapter 3

Two-Dimensional Linear Elastic Analyses—Slice Method

The initial procedure used in the structural analysis of reinforced concrete silo groups is to remove a horizontal slice of unit height from the group of cells, as shown in perspective in Fig. 3.1, and study it for the various expected loading cases. Such a procedure is usually referred to as Slice Method and represents one of the simplified processes available for the design of silo groups. The resulting structural model is then, subjected to a two-dimensional analysis using computational matrix analysis methods [1] or simplified manual methods [2]. In the work, numerical analyses were performed using the finite element method taking into account the following modelling approaches: • Modelling with 2D frame elements; • Modelling with plane strain elements. From these analyses, one can obtain the stress resultants—horizontal axial forces and bending moments—along all cross cross-sections of the silo group necessary

Fig. 3.1 Cross section in silo groups © Springer Nature Switzerland AG 2019 F. A. N. Silva et al., Design of Reinforced Concrete Silo Groups, Building Pathology and Rehabilitation 10, https://doi.org/10.1007/978-3-030-13621-5_3

13

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3 Two-Dimensional Linear Elastic Analyses—Slice Method

Fig. 3.2 Load cases

to design the cylindrical shells that comprise silo walls. Eleven loads cases were considered, representing all possible combinations of load cases indicated in Fig. 3.2. The applied pressures values are those indicated in Fig. 2.3 and the slice studied was loaded with the maximum pressure values for the cell and the interstice, as indicated in the figure.

3.1 Modelling Using 2D Frame Elements The mesh idealized for the analysis of the silo groups with frame 2D elements is presented in Fig. 3.3. Red lines linking nodes represent the region common to adjacent cells (Detail A in Fig. 3.3). The origin of the global coordinate system is located in the centre of the interstice space. Each silo was modelled with 72 2D frame elements each one corresponding to an arc length of five degrees. The height of the analysed slice was 1 m. The linking region between two contiguous cells was treated as a rigid region, modelled as rigid lines, interconnecting the nodes of the cell. The generation of the rigid lines was implemented by imposing special Constraint Equations—Sect. 3.3. These equations establish relations between degrees of free-

3.1 Modelling Using 2D Frame Elements

15

Fig. 3.3 Modelling with 2D frame elements

dom of the nodes of the elements that compose the rigid region, in order to allow the consideration of its stiffness. In the two-dimensional space of this analysis, three constraint equations are generated for each pair of constrained nodes. These equations define the three rigid body motions in the global Cartesian System, that is, two plane translations—ux and uy —and an out-of-plane rotation—rotz . In addition, a minimum number of boundary conditions were imposed to suppress rigid body motion of the structure. Such boundary conditions were implemented through the imposition of prescribed zero displacements to certain nodes of the model.

3.2 Applying Nodal Loads There are two usual concepts in the Theory of Finite Element Method associated with the conversion of distributed load acting on the elements in nodal load, namely: • Consistent loads; • Non-consistent loads.

16

3 Two-Dimensional Linear Elastic Analyses—Slice Method

The first concept is associated with the conversion of distributed load into nodal forces and moments. The loads, which come from this procedure, are referred to as consistent loads or equivalent nodal loads, since the resulting forces and moments in a point are equivalent to those arising from the originally applied distributed load. The other concept presented relates to the conversion of distributed applied loading in nodal force without moment. The resulting loads are generally referred to as nonconsistent loads also called as lumped loads. Such a load is preferable in straight elements representing curves whose the rotation is a degree of freedom, since the presence of spurious bending moments, which are not beneficial to the required solution quality, is eliminated [3]. In the present work, the applied pressures were converted into non-consistent nodal forces.

3.3 Constraint Equations The general form of a constraint equation is as follows: CD − Q = 0

(3.1)

where C is the matrix m × n (m = number of constraint equations; n = number of DOF in D), D is the vector of nodal displacements of the model and Q is the vector of constants. In the present study, it was necessary to generate equations to allow the consideration of the stiffness of the region that links two adjacent cells. To do this let examine Fig. 3.4, where it is represented the finite element mesh of two adjacent cells, with emphasis on the nodes—i and j—interconnected by a fictitious rigid line whose existence is to be simulated. The equations required to generate the rigid line in Fig. 3.4 are as follows:

Fig. 3.4 Detail of the region of connection between two cells of the silo group

3.3 Constraint Equations

17

uj − ui = 0

(3.2)

vj − vi − 2Lθi = 0

(3.3)

θj − θi = 0

(3.4)

where ui is the translation of node i in the x direction, vi is the translation of node i in the y direction and θ i is the rotation of node i around z-axis. In matrix notation, Eqs. (3.2)–(3.4) can be written as it follows: ⎡ ⎤ ⎡ ⎤ 0 −1 0 0 100 ⎢ ⎣ 0 −1 −2L 1 0 0 ⎦ − [D] = ⎣ 0⎥ (3.5) ⎦ 0 0 −1 0 0 1 0 in which,⎡

⎤ −1 0 0 100 T  C = ⎣ 0 −1 −2L 1 0 0 ⎦ and D = ui vi θi uj vj θj with vector Q being 0 0 −1 0 0 1 equal to zero. There are two procedures widely used to implement constraint equations to a finite element model, namely: • Lagrange Multiplier Method; • Penalty Method. The Lagrange Multiplier method is the way in which the finite element program used implements the mentioned equations [4].

3.4 Lagrange Multiplier Method The analysis of an engineering problem using the Finite Element Method results in the solution of simultaneous algebraic equations, expressed in matrix form according to the following equation: Ku = F

(3.6)

The aim is to impose to the system in Eq. (3.6) constraints equations expressed as follows: C T u = C0

(3.7)

where K is the stiffness matrix, u is the displacement vector, F is the nodal forces vector and C is the coefficients matrix.

18

3 Two-Dimensional Linear Elastic Analyses—Slice Method

The Eqs. (3.6) and (3.7) can be written in the following way: L

Kkj uj = Fk

(3.8)

Cj uj = C0

(3.9)

j=1 L

j=1

with K = 1, …, L. Taking into account that L is the dependent of freedom that one wishes to eliminate, it can be written that: uL +

L−1

Cj∗ uj = C0∗

(3.10)

j=1

in which Cj∗ =

Cj C0 and C0∗ = with k = 1, . . . ,L − 1 CL CL

(3.11)

The system represented by Eq. (3.6) is the necessary and sufficient condition for the minimization of the functional Total Potential expressed as follows:  1 = uT Ku − F T u 2

(3.12)

Therefore, the problem of imposing constraints can be presented in the following form:

1 Min uT Ku − F T u (3.13) 2 with C ∗T u = C0T . It is a problem with linear equality constraints, whose necessary optimality conditions are: • Feasibility; • Stationarity. The feasibility condition indicates that the solution of the problem formulated in Eq. (3.13) must satisfy the imposed constraints and the stationarity condition expresses the necessity of the existence of Lagrange Multipliers, as many as the constraints, such that: l  

  ∇f u∗ + λi hi u∗ = 0 i=1

(3.14)

3.4 Lagrange Multiplier Method

19

where f is the Functional, u* is the solution vector, l is the internal number of equality constraints, λ is the Lagrange multiplier and h is the Equality constraints. Defining the Lagrangian Function as:   1 L(u, λ)= uT Ku − F T u + λ C ∗T u − C0∗ 2

(3.15)

The above-mentioned associated optimality conditions can be written as follows: ∇λ L = 0

(3.16)

∇u L = 0

(3.17)

Equations (3.16) and (3.17) can be written in matrix form as:      u F K C∗ = ∗T C 0 C0∗ λ

(3.18)

Since the intention is to eliminate the degree of freedom L, the static condensation of the above system for uL and λ is performed and one can write that: ⎡ ⎤ ⎡ u⎤ ⎡ F ⎤ K K∗L C ∗ ⎥ ⎢ ⎥ ⎣ K∗L KLL 1 ⎦ ⎢ (3.19) ⎣ uL ⎦ = ⎣ FL ⎦ C∗ 1 0 λ C0∗        C∗ u F K K∗L ∗ ;C = in which K = and u = , with K *L ;F = K∗L KLL uL 1 FL representing the column L of matrix K. Eliminating the second and the last columns of the matrix in Eq. (3.19), we obtain the matrix equation that follows:   ⎤⎡ ⎤ ⎡ u Kkj − Cj∗ KkL − Ck∗ KLj − Cj∗ KLL 0 0 ⎥⎢ ⎥ ⎢ ∗ u ⎣ KLj − C KLL 0 1 ⎦⎣ L ⎦ 

C∗

j

10   ⎤ ⎡ ∗ ∗ ∗ Fk − C0 KkL − C0 FL − C0 KLL ⎦ =⎣ FL − C ∗ KLL 0

λ

(3.20)

C0∗

This way, the final system to be solved, with the consideration of the desired constraint equations, becomes:

20

3 Two-Dimensional Linear Elastic Analyses—Slice Method L−1

Kkj∗ uj = Fk∗

(3.21)

j=1

with k = 1, …, L − 1; Kkj∗ = Kkj − Cj∗ KkL − Ck∗ KLj + Ck∗ Cj∗ KLL and Fk∗ = Fk − C0∗ KkL − Ck∗ FL + Ck∗ C0∗ KLL .

3.5 Modelling with Plane Strain Finite Element The finite element mesh used to model the slice of the silo groups with plane strain element is shown in Fig. 3.5. Detail A in this figure exhibits the level of refinement adopted in the region of connection between two cells. Each silo was modelled with 256 plane strain elements. Along the thickness of the walls, the mesh has two elements. Two meshes were tested: one with four-node iso-parametric two-dimensional solid elements (Q4) and other with eight-node isoparametric two-dimensional solid elements (Q8). These elements are available in the ANSYS finite element library [4] and are referred to as Plane42 and Plane82, respectively.

Fig. 3.5 Mesh of the slice of the silo group using plane strain finite element

3.5 Modelling with Plane Strain Finite Element

21

In both hypotheses considered, the angle describing the lengths of the elements was five degrees. As well as the modelling with frame elements, prescribed displacements were imposed on certain nodes of the model to avoid rigid body motion. In the analyses developed with solid elements type Plane42, additional shape functions were used. The detail of such type of interpolation function is discussed in next.

3.6 Additional Shape Functions Figure 3.6 shows the geometry of a typical Q4 element with an indication of the global Cartesian system (x-y) and the natural coordinate system (t-s). The shape functions for the element shown in Fig. 3.6 are given below, defined in natural coordinates: 1 (1 − t)(1 − s) 4 1 Hj (t,s) = (1 − t)(1 + s) 4 1 Hk (t,s) = (1 + t)(1 + s) 4 1 HL (t,s) = (1 + t)(1 − s) 4 Hi (t,s) =

(3.22) (3.23) (3.24) (3.25)

where H i (t,s) is the Lagrangian function associated to node I, in natural coordinates. The displacement field inside the element is then interpolated as follows:

Fig. 3.6 Four node finite element—Q4

22

3 Two-Dimensional Linear Elastic Analyses—Slice Method

  1 ui (1 − t)(1 − s) + uj (1 − t)(1 + s) u= +uk (1 + t)(1 + s) + uL (1 + t)(1 − s) 4   1 vi (1 − t)(1 − s) + vj (1 − t)(1 + s) v= +vk (1 + t)(1 + s) + vL (1 + t)(1 − s) 4

(3.26) (3.27)

The deformed geometry of a Q4 type element is indicated in Fig. 3.7, with the sides deforming as straight lines. The additional shape functions modify this behaviour of the Q4 elements, allowing a parabolic deformation along its sides, which is important in problems with bending. Additional Shape Functions are obtained by adding terms to incorporate modes that describe a state of constant curvature [3] to the functions that interpolate the previously defined displacements. The displacement expansions thus characterized are expressed by:   1 ui (1 − t)(1 − s) + uj (1 − t)(1 + s) + uk (1 + t)(1 + s)     u= +uL (1 + t)(1 − s) + u1 1 − s2 + u2 1 − t 2 4   1 vi (1 − t)(1 − s) + vj (1 − t)(1 + s) + vk (1 + t)(1 + s)     v= +vL (1 + t)(1 − s) + v1 1 − s2 + v2 1 − t 2 4

(3.28) (3.29)

Since the above expansions contain six shape functions, Q4 elements with Additional Shape Function are sometimes referred as Q6 elements in finite element theory books. In the present work, the Q4 element used was the ANSYS four-node linear solid [4], with the option of adding the Additional Shape Functions.

Fig. 3.7 Deformation of the Q4 element without additional shape function

3.7 Calculation of Stress Resultants

23

3.7 Calculation of Stress Resultants The stress resultants—bending moments and normal horizontal forces—in the various sections of the silos were calculated taking into account two important properties of an analysis performed using the Finite Element Method, which are independent of the refinement of the mesh considered [5]: • At a given node, the sum of the nodal forces of all the elements sharing this node is in equilibrium with the applied external loads and • Each element is in equilibrium with respect to its own nodal forces. It follows from the two properties mentioned above that the nodal forces obtained from the finite element analysis are always in global equilibrium with the applied loads and, thus, the calculation of the stress resultants computed from these forces ensures the equilibrium in each cross section analysed. Figure 3.8 indicates the data required for the calculation of the stress resultants. In this figure, two elements of the mesh located along the wall of the silo are showed, both inclined at an angle α with respect to the horizontal. Nodes 1, 2 and 3 are also indicated with the respective nodal forces, which define the section where the stress resultants of interest will be calculated.

Fig. 3.8 Data for the calculation of stress resultants

24

3 Two-Dimensional Linear Elastic Analyses—Slice Method

3.8 Results and Discussion The comparison between results obtained in the analyses is presented. The reference number of the indicated load case corresponds to that shown in Fig. 3.2.

3.8.1 Deformed Geometry The following figures show the deformed geometry of the silo groups studied for each of the load cases analysed, considering the two types of modelling strategies already discussed. These deformed geometries allow to observe that the structural behaviour of the silo group, for all the Load Cases, are quite similar, either modelling the slice with frame elements or with plane strain elements. It can also be observed that, in the case of Load Case VI—Load on the interstice—the deformed geometry assumes a configuration that causes the emergence of high values of the stress resultants, especially the bending moments (see Sect. 3.8.3). Such situation does not reflect the actual structural behaviour of the silo groups analysed, since in the Slice Method the structure is considered plane, without any influence of the bottom and top slices that comprise the slice studied. Such a consideration disregards a very important aspect the loading-carrying mechanism of this type of structure—i.e. the three-dimensional interaction between the various cells of the group (Figs. 3.9, 3.10, 3.11, 3.12, 3.13 and 3.14).

Fig. 3.9 Deformed geometry of the silo group—Load Case I

3.8 Results and Discussion

25

Fig. 3.10 Deformed geometry of the silo group—Load Case II

Fig. 3.11 Deformed geometry of the silo group—Load Case III

3.8.2 Horizontal Normal Forces The values of the normal horizontal forces along the sections located in the walls of the silos remained constant, for load in the cell, in a value of 350 kN/m (tensile), in all modelling that were used. This value was the maximum verified for all Load Cases analysed and it always occurred in the cross sections of the loaded cells.

26

3 Two-Dimensional Linear Elastic Analyses—Slice Method

Fig. 3.12 Deformed geometry of the silo group—Load Case IV

Fig. 3.13 Deformed geometry of the silo group—Load Case V

For the load acting on the interstice, the maximum horizontal normal force was of 88 kN/m (tensile) and occurred in the wall common to the cell and interstice.

3.8.3 Bending Moments Figures 3.15, 3.16, 3.17 and 3.18 represent the bending moments in several cross sections of a silo, considering all the Load Cases analysed and the three modelling

3.8 Results and Discussion

27

Fig. 3.14 Deformed geometry of the silo group—Load Case VI

approach considered. In these figures, the average surface of the silos wall is plotted in black colour with divisions indicating the angular position of the several cross sections studied. As complements to the each figure, tables summarizing the maximum bending moment values are shown, both positive and negative. The sign convention adopted was that positive bending moments elongate the inner fibres of the silo wall and ls and negatives bending moment are those that elongate the outer fibres of the silo wall. Negative bending moments are plotted outward of the wall of the silos and positive ones inwards. As shown in the figures, the values of the bending moments vary throughout the studied sections and one can observe that there are sections with positive moments and sections with negative moments. It was observed, in addition, that the maximum values of the bending moments occurred, in all Load Cases and for all models investigated, in the cross sections immediately adjacent to the region of interconnection between two cells. As well as the deformed geometries, it is observed that the behaviour of the diagrams of bending moments is quite similar for the three modelling approaches studied. This fact indicates that the methodology for calculating stress resultants from the nodal forces—used in the modelling with plane strain elements—produced very satisfactory results, aspect that highlights the efficiency of the adopted procedure and beyond demonstrate a great practical interest in the application of the Finite Element Method in the analysis and design of reinforced concrete structures.

28

3 Two-Dimensional Linear Elastic Analyses—Slice Method

Fig. 3.15 Bending moments—Load Cases I e II

3.8 Results and Discussion

Fig. 3.16 Bending moments—Load Cases III e IV

29

30

3 Two-Dimensional Linear Elastic Analyses—Slice Method

Fig. 3.17 Bending moments—Load Case V

It is further noted that the values of the bending moments for the model with Q4 elements are practically identical to those of the Q8 elements, thus indicating the good performance of the additional shape functions employed in the Q4 elements. This fact signals to the possibility of using linear solid element with Additional Shape Functions in the three-dimensional analysis of the silo groups studied, presented in the next chapter. A behaviour divergence was observed between the modelling with frame elements and plane elements in the Load Case IV, motivated by the difference of stiffness of the two models in the region of interconnection of two cells of the group. On the other hand, it was also observed that, in spite of the divergence observed, the resulting bending moments for this Load Case are very small when compared with the other Load Cases, in all the modelling approaches studied.

3.8 Results and Discussion

31

Fig. 3.18 Bending moments—Load Case VI

One can observe, also, the high magnitude of the bending moments for the Load Case VI, compared with other Load Cases studied. Such values come from the fact that the wall of the interstice is not behaving like an arc, due to the displacements of the points of intersection between two cells. Such displacements are the responsible for the generation of the high and unreal bending moment’s values that do not reflect the true behaviour of the structure, due to the applied loading. As for nodal displacements, the values of the bending moments obtained using the Slice Method analysis for all Load Cases studied do not adequately characterize the structure response and an approach that considers tri-dimensionality is really required.

32

3 Two-Dimensional Linear Elastic Analyses—Slice Method

Fig. 3.19 Bending moment’s envelope—Frame elements

3.8.4 Bending Moments Envelope Finally, Figs. 3.19 and 3.20 represent bending moment’s envelope for one of the cells, considering the modelling with Q8 elements and for frame elements, for all Load Cases studied, with the exception of Load Case VI, due to the impossibility of visualizing the values of the bending moments for this case vis-à-vis the others.

References

33

Fig. 3.20 Bending moment’s envelope—Q8 element

References 1. Stalnaker JJ, Harris EC (1992) Bending moments in walls of grouped silos due to structural continuity. ACI Struct J 89(2):159–163 2. Filho JF, An introduction to the study of silos. Master Thesis, Engineering School of Engineer, São Paulo University (in Portuguese) 3. Cook RD (1994) Finite element modelling for stress analysis. Willey 4. Ansys (2014) User and theoretical manuals. Swanson Analysis Systems, Inc 5. Bathe KJ (1996) Finite element procedures. Prentice Hall

Chapter 4

Three-Dimensional Analysis

4.1 Introduction The study of reinforced concrete silo groups using the three-dimensional finite element analysis demands from the designer hard work to formulate of the problem. Aspects such as the characterization of the geometry, the determination of the loads and choice of the appropriate boundary conditions, as well as in the interpretation of the obtained results, are examples such difficulties. Despite these difficulties, threedimensional analysis is an important tool for understanding the structural behaviour of this type of construction. Regarding the geometric representation, the choice of a finite element mesh that adequately incorporates the characteristics of the physical model is an important issue and deserves a special attention. The research considered three different modelling approaches, which are the usual ways of modelling of this type of structure, namely: 1. Modelling with solid elements; 2. Modelling with shell elements; 3. Modelling with solid and shell elements. The Model 1 and 2 are widely adopted in design of silo groups and Model 3 is less frequent. Model 1, which uses less simplified approaches regarding the structural behaviour of the silo group, will be considered as the benchmark model for the comparisons between the other two modelling approaches. Each one of the modelling strategies studied will be presented in details and the comparisons performed are deeply discussed. In all modelling approaches, the diameter of the circular cells used was 10 metros and the height adopted for the silo group was 30 m. The finite element package used was ANSYS—Engineering Simulation & 3D Design Software.

© Springer Nature Switzerland AG 2019 F. A. N. Silva et al., Design of Reinforced Concrete Silo Groups, Building Pathology and Rehabilitation 10, https://doi.org/10.1007/978-3-030-13621-5_4

35

36

4 Three-Dimensional Analysis

Regarding the applied loading, 3D analysis of reinforced concrete silo groups demands more than to study all the possible Load Cases, but the identification of which load combination lead to the highest stress resultants values necessary to the design and detailing of the structure. In the analysis of the silo group investigated, one can identify critical situations in some cross sections along the geometry of the group such as [1]: • Maximum horizontal tensile normal force in the wall of one cell in a silo group happens when this cell is fully loaded and no increase of this force takes place when the others cells of the group are also loaded. Furthermore, this force came, usually, associated with some flexion—bending moments; • The wall common to the cell and the interstice may be subjected to compression, with maximum value occurring when only the interstice is loaded; • The maximum values of the bending moments occur when the interstice is fully loaded and the other cells are empty, in cross sections located in the wall common to the cell and the interstice. In cross sections at the ends of the wall common to two contiguous cells—cross sections 1 and 4 of Fig. 4.1, where the larger values of the bending moments appear. From the two-dimensional analyses performed previously, it was possible to conclude that in the three-dimensional analysis of reinforced concrete silo groups must consider at least two specific loading situations, namely: • Loading applied to the cells; • Loading applied to the interstice. The combination of these two load conditions will allow the calculation of the stress resultants value in the cross section indicated in Fig. 4.1. Taking into account the shape and loading symmetry of the structure, one can use only one-eighth of it to generate the models for the three-dimensional analysis.

Fig. 4.1 Structure for three-dimensional analysis—cross sections of interest

4.1 Introduction

37

Regarding to the evaluation of stress resultants, especially the bending moments, the emphasis was given to the cross sections indicated in Fig. 4.1, where the highest values occur. Cross sections 1 and 4 are of interest because they lie in regions with geometric singularity and cross sections 2 and 3 are chosen because they are located over one of the symmetry planes of the structure. The sign convention used to represent the negative and positive bending moments in the several figures presented below was the same one adopted in the bi-dimensional analysis, as it was discussed in the previous chapter. For the horizontal normal forces, the sign convention adopted to their representations are: compression forces are plot outward the silo wall and tensile forces are plotted inward. The boundary conditions used are discussed in Sect. 4.2.

4.2 Boundary Conditions The analysis considered the following boundary conditions: • In the bottom, taking into account the rigidity of the existing slab as well as the connection of the wall with the foundation, the boundary condition imposed was the clamped support; • At the top, considering the existing slab acting as a rigid diaphragm for membrane loads allied to the structure symmetry, the boundary condition imposed was simple support. The use of shape and loading symmetry conditions of the structure generates two planes of symmetry on which appropriate boundary conditions have been imposed [2].

4.3 Modelling with Linear Finite Element Finite element mesh, with linear solid elements, was used to model for threedimensional analysis of the structure is shown in Fig. 4.2. The element used was the iso-parametric tri-dimensional solid of ANSYS [3]. Eight nodal points define this element, each one having three degrees of freedom—translation in x, y and z directions. Additional shape functions were also include improving the performance of such finite element, in the same way as it was carried out in the plane analyses previously discussed. In this modelling, one-meter high elements were used and, like in the 2D modelling, a division with two elements were considered along the thickness of the walls. The mesh refinement at the intersection region between two cells was the same as that used for the 2D modelling, discussed in Chap. 2, and the angle describing the element length was a maximum of seven and a half degrees.

38

4 Three-Dimensional Analysis

Fig. 4.2 Finite element mesh with linear solid elements

The model resulting from this modelling approach has 4774 nodes and 3180 elements and, considering the boundary conditions imposed at the bottom, at the top and at symmetry planes of the structure, one has a system with about 13,000 equations. The stress resultants in cross sections of interest were calculated using the same procedure adopted for the modelling with plane finite elements. The results of the analysis performed for the loading applied in the cell and in the interstice are discussed in the following section.

4.3.1 Loading in the Cell Figure 4.3a shows the diagram of normal horizontal forces in several cross sections of the silo wall, for a given transverse slice. This is a typical diagram for any crosssection of the structure in which are represented the silo wall and the values of the horizontal axial forces in several perimeter sections—graphically represented in red colour.

4.3 Modelling with Linear Finite Element

39

Fig. 4.3 a Horizontal normal forces—loading in cell and b variation of the maximum normal horizontal forces

One can observe tensile forces in all sections, with a maximum value of 370.5 kN/m occurring in the section located at 67.5° with the horizontal. Such value was the maximum one for the whole structure. Figure 4.3b shows the variation of the maximum normal horizontal forces in each cross section of the silo, along the height of the structure. It can be observed that there is a growth in the value of the normal force until its maximum value (370.5 kN/m) which occurs in the cross section located at 28 m of depth. From this depth on, there is a decrease in its value, which reaches 96.1 kN/m in the cross section located at the bottom of the cell. Regarding the bending moments, Fig. 4.4a shows the variation of the bending moments Mz —longitudinal direction—for several perimeter sections of the silo wall. Similarly to normal horizontal forces, the bending moment diagram shown in Fig. 4.4a is typical for the other cross sections of the silo group. One can observe that there are parts with positive bending moments and parts with negative ones. The highest value occurs at the lower end of the common wall to two contiguous

40

4 Three-Dimensional Analysis

Fig. 4.4 a Bending moment Mz —loading in cell and b maximum bending moments Mz along the height

cells—cross section 4 of Fig. 4.1. In addition, Fig. 4.4b shows the behaviour of the maximum bending moments Mz along the height of the silo group. It should be observed that the behaviour of the diagram in Fig. 4.4b is similar to that presented for normal horizontal forces. In fact, there is an increase in the value of the bending moment until its maximum value (15.27 kNm/m) is reached in the cross section located, identically, to 28 m of depth. From this point on, a decrease in value of bending moments begins to occur until, in the cross section located at the bottom of the cell, its value reaches 4.37 kNm/m. In addition, it is important to note that the maximum bending moment occurred in the cross section close to the maximum value of the applied horizontal pressure—see Fig. 2.3 in Chap. 2.

4.3 Modelling with Linear Finite Element

41

Fig. 4.5 a Normal horizontal forces—loading in the interstice—Section 3 and b variation of the maximum normal horizontal forces

4.3.2 Loading in the Interstice Figure 4.5a shows the normal horizontal forces in several perimeter sections. The diagram presented in this figure corresponds to the cross section located at 28 m in depth, in which the maximum value of the normal horizontal force of –159.49 kN/m (compression) was observed, occurring in section 3 of Fig. 4.1. This compression arises because the wall common to the cell and interstice is functioning as an arch, whose supports are the regions common to two contiguous cells. As a function of the three-dimensional effect, displacements restraints of the arch supports generate compression force in that wall. This behaviour, when com-

42

4 Three-Dimensional Analysis

pared to the study presented in the previous chapter, exposes the divergence between the two analyses and shows the inefficiency of the two-dimensional approach for the loading applied in the interstice. Figure 4.5b shows variation of the maximum normal horizontal forces in Section 3 (symmetry plane) along the height of the silo group. It is possible to see that part of the cell height is in compression and part is in tension. In fact, up to a depth of 18 m the silo wall is in tension with the maximum value of 21.48 kN/m occurring in the cross section located at 12 m in depth. This behaviour shows that in this region the elastic restriction to the displacement prevails to the detriment of the applied loading. From 18 m depth on, the silo wall is in compression, reaching its maximum value of –159.49 kN/m in the cross section located at 28 m in height. Such behaviour indicates that in this region the action of the applied loading prevails. Regarding the bending moments, Fig. 4.6a shows the variation of the moment Mz in several perimeter sections, for a slice located at 17 m in depth. In this cross section, the highest values of negative and positive bending moments were observed for the whole structure, –52.25 and 31.691 kNm, respectively. These values occurred in sections 3 and 4 of Fig. 4.1. Figure 4.6b exhibits the variation of the maximum positive and negative bending moments along the height of the silo group. It should be noted, additionally, that, in contrast to the loading applied in the cell, the maximum bending moment does not occur close to the cross section where the applied horizontal pressure is maximum.

4.4 Modelling with Shell Elements The mesh resulting from the discretization of the structure with shell elements is shown in Fig. 4.7. The cells were modelled with thin shell elements developing along the average surface of the walls. The characterization of the interconnecting region between two contiguous cells was made using appropriate boundary conditions in the symmetry plane that contains such region. The ANSYS quadrilateral flat shell element was used with Additional Shape Functions. Four nodal points each one having six degrees of freedom—three translations and three rotations, define this element. The considered element combines membrane and bending behaviour in its displacement field. It should be noted in Fig. 4.7 the refinement of the mesh close to the section that delimits the intersection of the silo wall and the region of interconnection of the two cells—sections 1 and 4 of Fig. 4.1. Three elements of equal length were arranged, in order to get stress resultants, using quadratic extrapolation of the values provided for the centre of the elements, from the finite element analysis. The model resulting from this discretization has 1147 nodes and 1080 elements generating a system with 5500 equations, approximately.

4.4 Modelling with Shell Elements

43

Fig. 4.6 a Bending Moments Mz —loading in interstice and b variation of the maximum positive and negative bending moments

44

4 Three-Dimensional Analysis

Fig. 4.7 Mesh with shell finite element

For the loading applied in the cell, it was found that the diagrams of horizontal tensile forces and bending moments Mz exhibited the same behaviour as those indicated in Sect. 4.3.1, and the obtained values were 363.45 and 19.49 kNm/m, respectively. Similarly for the loading applied in the interstice, the diagrams of the normal horizontal forces and the negative and positive bending moments Mz obtained are of the same pattern as those indicated in Sect. 4.3.2 with values of –151.05, −55.11 and 27.62 kNm/m, respectively.

4.5 Modelling with Solid and Shell Elements It is a mixed model, where the silos walls were modelled with thin shell elements and the region of interconnection between two cells was modelled with three-dimensional solid elements, as shown in Fig. 4.8. In this model, it was necessary to impose constraint equations to correctly establish the connection between the two types of elements used in the model, since the shell elements have six degrees of freedom per node and the solid elements have only three. This situation occurs in sections 1 and 4 of Fig. 4.1.

4.5 Modelling with Solid and Shell Elements

45

Fig. 4.8 Mesh with solid and shell element

Figure 4.9 shows details of this connection, indicating its geometry, non-deformed and deformed shape, for the loading acting in the interstice. Figure 4.9b, c show three generic nodes of the model—i, j and k—that will be used in the generation of the constraint equations that define the rigid regions interconnecting them. This procedure is similar to that presented in the 2D modelling with frame elements discussed in the previous chapter. The expressions in Eqs. (4.1, 4.2 and 4.3) define the equations required to generate the rigid region connecting the nodes i and j:   t ◦ sin 20 θz j = 0 (4.1) ui − u j − 2   t vi − v j − cos 20◦ θz j = 0 (4.2) 2     t t sin 20◦ θx j − cos 20◦ θ y j = 0 wi − w j + (4.3) 2 2 where t is the thickness of silo wall, ui is the translation of node i in the x direction, vi is the translation of node i in the y direction, wi is the translation of node i in the z direction and θ zi is the rotation of node i with respect to z-axis.

46

4 Three-Dimensional Analysis

Fig. 4.9 Connection of solid and shell elements

Analogous procedure must be done for nodes j and k. Figure 4.9d shows the deformed shape—magnified for visualization effect—of the connecting region between shell and solid elements, where the efficacy of the constraint equations imposed can be observed. The constraint equations previously related can be automatically generated in the finite element analysis program used—ANSYS—through the CERIG command.

4.5 Modelling with Solid and Shell Elements

47

Regarding the level of refinement of the mesh, the same detail shown in Figs. 4.2 and 4.7 was adopted. The mesh shown in Fig. 4.8 has 3410 nodes, 2580 elements and a system with close to 11,000 equations. For the two load cases considered, the diagrams of the normal horizontal forces and bending moments Mz exhibited the same pattern as those indicated in Sects. 4.3.1 and 4.3.2, with the maximum values occurring in the same cross section location. For the loading applied in the cell, the maximum horizontal tensile force was 363.62 kN/m and the maximum bending moment was 14.25 kNm/m. In the case of the loading applied in the interstice, the maximum horizontal compressive force was –160.96 kN/m and the negative and positive bending moments were, respectively, –52.43 and 30.03 kNm/m.

4.6 Results and Discussion Figures with the variation of the bending moments along the height of the silo group for the sections of Fig. 4.1 and loading applied in the cell and in the interstice are shown. (Figs. 4.10, 4.11, and 4.12). One can observe that the modelling that combines elements of shell and solid elements in the mesh—called mixed modelling—is the one that exhibited the closest results to the model with only solid elements. Such behaviour can be observed in the several figures presented and, in the case of loading case of special interest for the present research—loading applied in the interstice -, the values of the bending moments in the sections of the wall common to the cell and the interstice are quite similar (see Fig. 4.13). The results thus obtained indicate the possibility of using the mixed model in finite element modelling in the parametric analyses, described in the next chapter.

48

4 Three-Dimensional Analysis

Fig. 4.10 Bending moments Mz in section 1 and 2—loading applied in the cell

4.6 Results and Discussion

Fig. 4.11 Bending moments Mz in section 3 and 4—loading applied in the cell

49

50

4 Three-Dimensional Analysis

Fig. 4.12 Bending moments Mz in section 1 and 2—loading applied in the interstice

4.6 Results and Discussion

Fig. 4.13 Bending moments Mz in section 3 and 4—loading applied in the interstice

51

52

4 Three-Dimensional Analysis

References 1. Safarian SS, Harris EC (1985) Design and construction of silos and bunkers. Van Nostrand Reinhold Company, New York 2. Cook RD (1994) Finite element modelling for stress analysis. Willey 3. Ansys (2014) User and theoretical manuals. Swanson analysis systems Inc

Chapter 5

Structural Behaviour and Simplified Model Proposition

5.1 Load Bearing Mechanism The understanding of the load-bearing mechanism in the silo group analysed in this research, due to the loading acting in the interstice, can be intuitively understood observing the three figures presented below. Figure 5.1(A-1) shows, in plant, a group of reinforced concrete silos where there is no structural continuity between the various cells, nor a top slab. There is only the provision of a material with plastic characteristics in the region of connection of two cells, to avoid the leakage of the content deposited inside the interstice. The loading coming from the stored material imposes to the silo a deformed shape quite similar to a cantilever beam clamped in the bottom—Fig. 5.1(A-2). The clamp is ensured by the stiffness of the bottom slab and by the foundation of the whole silo group. Figure 5.1(B-1) represents the plant of a group of reinforced concrete silos with concrete top slab, but still without structural continuity between the cells. Under the loading action from the material stored in the interstice, the silo group assumes a deformed configuration similar to a beam clamped in the bottom and simply supported at the top—Fig. 5.1(B-2). The simple top support is provided by the stiffness of the top concrete slab on its own plane and by the symmetry of the structure. Figure 5.1(C-1) indicates the plant of a group of silos where there is structural continuity between the various cells as well as a top slab. Due to the action of the loading in the interstice, the silo group deforms as if it was a beam on elastic foundation, clamped at the bottom and simply supported at the top. The elastic foundation represents the effect of continuity between the various cells, which manifests itself through a restrain to the displacement of the wall of the interstice. Figures 5.2 and 5.3, resulting from the three-dimensional analysis performed, corroborate the intuitive interpretation of the loading bearing mechanism of the silo group discussed above. In these figures, respectively, the deformed shape of the interconnecting region between two cells, as well as the contour plot of the normal stresses (longitudinal direction), are represented for a given cross section of the silo. © Springer Nature Switzerland AG 2019 F. A. N. Silva et al., Design of Reinforced Concrete Silo Groups, Building Pathology and Rehabilitation 10, https://doi.org/10.1007/978-3-030-13621-5_5

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54

5 Structural Behaviour and Simplified Model Proposition

Fig. 5.1 Structural behaviour

The deformed shape shown in Fig. 5.2 shows that the silo, in fact, behaves like a beam (circular cross section of diameter D and vertical axis of length H), where the effects of shear deformation have influence that needs to be considered (Timoshenko beam theory). Figure 5.3 further characterizes this behaviour, where one can observe parts in compression and parts in tension in cross section, aspect that confirm the behaviour predicted by the classical beam theory (Fig. 5.4).

5.2 Proposed Simplified Model Based on the observations of the results of the three-dimensional analyses performed, especially the aspects related to the stress resultants and the load-bearing mechanism, it is proposed to approach the analysis of reinforced concrete silos groups using the following models: 1. A Timoshenko beam on elastic foundation to model the cylinders vertically (longitudinal bending); 2. A circular arch, clamped at both ends, subjected to transverse loading and imposed displacement δ to model the interstice walls horizontally (transverse bending).

5.2 Proposed Simplified Model

Fig. 5.2 Deformed shape of the interconnection region—loading applied in interstice

55

56

5 Structural Behaviour and Simplified Model Proposition

Fig. 5.3 Normal stresses—longitudinal direction (in kN/m2 )

Fig. 5.4 Proposed simplified model

5.2 Proposed Simplified Model

57

Fig. 5.5 Scheme for calculating the spring constant

The elastic properties of the proposed Model 1 − I (bending inertia), As (shear area) e k (spring constant of elastic foundation)—are modelled by dimensionless parameters α, β and γ that affect the values of I g , Ag and k t , respectively, as indicated below: I = a Ig

(5.1)

Ag = β Ag

(5.2)

k = γ kt

(5.3)

where I g is the gross moment of inertia of the silo cross-section, Ag is the gross cross-sectional area of the silo and k t is the theoretical spring constant of elastic foundation. The dimensionless parameters α, β and γ are functions of the geometric indices of the structure (D/t and H/D). Their values will be evaluated through a parametric study, where least square fitting of displacements of the Model 1 was applied to fifteen different geometries of silo groups. The silo groups were analyzed with the finite element method using the mixed model (with shell and solid elements) already discussed in Chap. 4. The gross moments of inertia and gross cross-sectional areas are obtained from the silo geometry. The theoretical spring constant of the elastic base is calculated for the flat structure modelled with frame elements, as indicated in Fig. 5.5.

58

5 Structural Behaviour and Simplified Model Proposition

Fig. 5.6 Detail of the Timoshenko beam model

In this figure, it is schematically represented the mesh with frame elements of the silo group, in which unit displacements are applied to the nodes of the elements of the regions of interconnection between two cells. The reaction forces—F—in these nodes generate a resulting force that will be considered as the elastic foundation reaction. The numerical value of the spring constant is given by: kt =

2F cos 45◦ √ 2

(5.4)

The stiffness matrix of the beam element considered in the implementation of Model 1 is presented above. ⎡ ⎢ ⎢ ⎢ ⎣

−12E I L 2 (1+φ) 6E I L 2 (1+φ) −12E I L 2 (1+φ) 6E I L 2 (1+φ)

6E I L 2 (1+φ) E I (4+φ) L(1+φ) −6E I L 2 (1+φ) E I (2−φ) L(1+φ)

−12E I L 2 (1+φ) −6E I L 2 (1+φ) 12E I L 2 (1+φ) −6E I L 2 (1+φ)

6E I L 2 (1+φ) E I (2−φ) L(1+φ) −6E I L 2 (1+φ) E I (4+φ) L(1+φ)

⎤ ⎥ ⎥ ⎥ ⎦

(5.5)

where E is the modulus of elasticity, L is the element length, I is the moment of inertia Izz, ϕ = 12EI/GAs L2 , G = E/2(1 + ν) and ν is the coefficient of Poison. Figure 5.6 shows the discretization used for the application of the Model 1. The nodal forces applied (see Fig. 5.6) are those resulting from the load portion of the interstice that acts in the silo wall, as shown in Fig. 5.7.

5.2 Proposed Simplified Model

59

Fig. 5.7 Calculation of transversal loading

The corresponding arches—twenty-nine in total—describe an angle of 50° and were modelled with frame elements of an arch length of 5°. The pressure applied on each arch was converted to nodal force at the respective node (Fig. 5.7) and the nodal translations resulting from the use of the Model 1 are applied as prescribed displacements in the supports of the clamped arches (interstice walls) as shown in Fig. 5.4b. From the solution of Model 2 the stress resultants—bending moments and normal forces—necessary to design the structure are obtained.

5.3 Fitting with the Least Square Method The determination of the dimensionless parameters α, β and γ , for each of the 15 geometric configurations studied, was made using Nonlinear Least Squares Method, as indicated in next expression. min

α,β,γ

n 

2 u mi − u ti (α, β, γ ) i=1

(5.6)

60

5 Structural Behaviour and Simplified Model Proposition

where n is the number of nodes of the model, um is the nodal displacements vector from the three-dimensional model and ut is the nodal displacements vector from the theoretical model. The displacements um are those resulting from the three-dimensional analysis using the mixed model (model with shell and solid elements) and the displacements uti are those calculated from the proposed Model √ 1. The displacements from the three-dimensional analysis are multiplied by 2, since the designed beam model develops in a plane located at 45° from the centre of the region of interconnection between two cells (Sect. 4 of Fig. 4.1). The solution of the minimization problem formulated above was obtained using a procedure developed in Matlab [1], where the nodal displacements are calculated for Model 1 and the optimization is performed with an internal least squares routine—leastsq, which uses the Levenberg-Marquardt algorithm. The results obtained for a silo group with four cylindrical cells of 10 m in diameter and 30 m in height are presented as an illustrative example of the proposed adjustment. The data required for analysis are given below. • • • • •

Longitudinal Modulus of Elasticity—E = 21,000,000 kN/m2 Transversal Modulus of Elasticity—G = 9,130,434.8 kN/m2 Gross Cross Section Area—Ag = 6.7876 m2 Moment of Inertia of the Gross Cross Section—Ig = 88.4368 m4 Elastic Foundation Spring Constant—kt = 130,970.7 kN/m

Table 5.1 presents the nodal displacements (projected in the bending plane of the cylinder—beam) of the three-dimensional analysis with the Finite Element Method (FEM) and the displacements calculated using Model 1, with the dimensionless parameters α, β and γ adjusted by minimum squares method. Figure 5.8 (graphical representation of Table 5.1) shows that the nodal displacement curve obtained with the use of the proposed Model 1 is quite similar to that resulting from the three-dimensional analysis performed. Such behaviour attests to the efficiency of the adjustment proposed for the calculation of the nodal displacements. The values of the dimensionless parameters α, β and γ associated to the displacements of the Model 1, adjusted by the least squares method, were 0.3513, 0.1464 and 0.4978, respectively. Tables 5.2 and 5.3 show the values of the bending moment’s for the threedimensional analysis and those resulting from the use of the proposed analysis procedure—Models 1 and 2. Negative bending moments occur in Sect. 4 of Fig. 4.1 and the positives ones in Sect. 3 of the same figure. The sign convention adopted is the same as the three-dimensional analysis of Chap. 4. The values shown in these tables are shown graphically in Fig. 5.9a, b. Figure 5.9a shows the negative bending moments and Fig. 5.9b the positive bending moments. In these figures, one can observe that the bending moments of the adjusted model, to a large extent of the height of the silo, are higher than those resulting from the three-dimensional analysis. In the case of negative bending moments, this difference was 18% and it happened because there is a deformability of the arch supports that is not considered in the model designed for the calculation of the bending moments.

5.3 Fitting with the Least Square Method

61

Table 5.1 Nodal displacements H (m)

FEM (mm)

Proposed model (mm)

H (m)

FEM (mm)

Proposed model (mm)

1

0.125

0.126

16

1.353

1.348

2

0.249

0.251

17

1.354

1.348

3

0.372

0.374

18

1.341

1.354

4

0.491

0.495

19

1.313

1.305

5

0.607

0.611

20

1.269

1.263

6

0.718

0.722

21

1.210

1.205

7

0.823

0.826

22

1.136

1.132

8

0.920

0.923

23

1.045

1.044

9

1.010

1.012

24

0.938

0.940

10

1.090

1.093

25

0.814

0.820

11

1.162

1.163

26

0.674

0.685

12

1.223

1.223

27

0.517

0.535

13

1.274

1.272

28

0.344

0.370

14

1.313

1.310

29

0.159

0.192

15

1.339

1.336

–

–

–

Table 5.2 Negative bending moments H (m)

Negative bending moments (kNm/m) FEM

H (m)

Proposed model

Negative bending moments (kNm/m) FEM

Proposed model

0

–

0

16

−51.92

−62.43

1

−2.33

−5.71

17

−52.43

−62.68

2

−7.06

−11.36

18

−52.43

−62.35

3

−11.75

−17.00

19

−51.91

−61.43

4

−16.37

−22.50

20

−50.88

−59.90

5

−20.89

−27.84

21

−49.33

−57.75

6

−25.26

−32.96

22

−47.24

−55.96

7

−29.42

−37.78

23

−44.60

−51.54

8

−33.31

−42.23

24

−41.41

−47.44

9

−36.88

−46.30

25

−37.64

−42.65

10

−40.14

−49.98

26

−33.40

−37.20

11

−43.06

−53.25

27

−28.92

−31.48

12

−45.64

−56.07

28

−24.12

−24.41

13

−47.83

−58.42

29

−16.34

−17.14

14

−49.63

−60.28

30

−6.04

0

15

−51.01

−61.62

–

–

–

62

5 Structural Behaviour and Simplified Model Proposition 0

Height (m)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

E 0

1.4e-4

A 2.8e-4

4.2e-4

5.6e-4

7e-4

8.4e-4

9.8e-4

0.00112 0.00126 0.0014

Displacements - m E - Finite Element 3D

A - Adjusted Model

Fig. 5.8 Comparison of nodal displacements

Regarding to positive bending moments, the mean difference was 8%. It is important to note, however, that, despite the differences observed, the results obtained are always on the safety side and, as will be discussed in the following sections, they are quite satisfactory when compared with the simplified calculation methods available in the literature. The parametric study described in the following section will present factors to be used in determining the final design bending moments.

5.4 Parametric Study Table 5.3 Positive bending moments

63

H (m)

Positive bending moments (kNm/m) FEM

Proposed model

0

–

0

1

1.36

2

4.07

3

H (m)

Positive bending moments (kNm/m) FEM

Proposed model

16

29.76

32.45

2.97

17

30.03

32.58

5.91

18

30.02

32.41

6.75

8.84

19

29.73

31.93

4

9.40

11.69

20

29.14

31.14

5

11.98

14.47

21

28.25

30.02

6

14.48

17.13

22

27.04

28.57

7

16.86

19.64

23

25.51

26.79

8

19.08

21.95

24

23.65

24.66

9

21.13

24.07

25

21.45

22.17

10

23.00

25.98

26

18.89

19.34

11

24.68

27.68

27

15.97

16.36

12

26.15

29.14

28

12.56

12.69

13

27.41

30.37

29

7.84

8.91

14

28.44

31.33

30

0.35

0

15

29.23

32.03

–

–

–

5.4 Parametric Study The objective of the parametric study is to investigate the behaviour of the dimensionless parameters α, β and γ , regarding the variation of the silo geometry, and to propose an approximate expression for their calculation. The variation of the silo geometry will be measured by two distinct indexes, which are conventionally referred as geometric indexes of the structure, listed below: 1. Diameter/Wall Thickness Ratio—D/t; 2. Silo Height/Diameter Ratio—H/D. The implementation of the parametric study considered silo groups with diameters and heights indicated in Sect. 1 of Chap. 2. The dimensions referring to the arch describing the region of interconnection between two cells (40°), the thickness in the centre of this region (30 cm) and the wall thickness of the silos (20 cm) were kept the same used before, since such values are typical in silo groups construction. Table 5.4 presents the variation of the dimensionless parameters α, β and γ for several values of the geometric index D/t, keeping the H/D = 3.0 index constant. The data in Table 5.4 are plotted in Fig. 5.10. The behaviour observed in Fig. 5.10 is typical for other values of the studied H/D geometric index (2.5, 3.0, 3.67, 3.75, 5.0 and 7.5) and in it one can observe that the

64

5 Structural Behaviour and Simplified Model Proposition

Fig. 5.9 Comparison of bending moments Table 5.4 Parameters for H/D = 3.0

D/t

Parameters α

β

γ

20

0.4464

0.1983

0.7927

30

0.3498

0.1604

0.6490

40

0.3532

0.1505

0.5664

50

0.3513

0.1464

0.4978

60

0.3385

0.1580

0.3885

5.4 Parametric Study

65

Fig. 5.10 Parameters for geometric index H/D = 3.0

Table 5.5 Parameters for D/t = 20

H/D

Parameters α

β

γ

2.5

0.4258

0.2004

0.7764

3.0

0.4464

0.1983

0.7927

5.0

0.4585

0.1916

0.8139

7.5

0.4408

0.1881

0.8192

parameter γ is more sensitive to the variation of the geometric index D/t than the parameters α and β. In fact, for D/t = 20, the calculated spring constant is about 80%, decreasing linearly to about 39% when the D/t ratio reaches 60. Within this range, the other dimensionless parameters present behaviour with little variation in their values. The behaviour related to the geometric index D/t = 20, in Fig. 5.10, was slightly different from the others due to the influence of the deformation of the cross section of the silo, an aspect that interferes with the load bearing mechanism of the structure. Table 5.5 shows the variation of the dimensionless parameters α, β and γ for several values of the geometric index H/D, keeping the D/t = 20 index constant. Figure 5.11 shows the plotting of the numbers in Table 5.5. In a similar way to that shown previously, the present figure is characteristic for the other values of the geometric index D/t studied (30, 40, 50 and 60) and in it one can observe an approximately constant behaviour in the values of the dimensionless parameters α, β and γ . Such behaviour is discretely disturbed at the point corresponding to the index H/D = 2.5, a fact that represents the greater influence of the top and bottom slabs of the silo group, for this geometry. The results of the study allow to conclude that the dimensionless parameters α and β show little variation in relation to the two geometric indexes considered. Moreover, taking into account that Fig. 5.10 exhibits a characteristic behaviour of those parameters for the silo groups studied, it is possible to consider them as constants, in their observed mean values, respectively, 0.35 and 0.15. This consideration implies to adopt, for implementing the calculation of the nodal displacements by the proposed

66

5 Structural Behaviour and Simplified Model Proposition

Fig. 5.11 Parameters for geometric index D/t = 20

Model 1—Timoshenko Beam on Elastic Foundation—35% of the gross moment of inertia of the cross section as the flexion inertia of the elements and 15% of the gross area of the section as their shear area. The value of the shear area considered corresponds approximately to the total area of the connecting region (web) between two cells. Regarding the dimensionless parameter γ , it was observed that the geometric index D/t (see Fig. 5.10) is the one that most influences its variation. In the case of the H/D index (see Fig. 5.11), there was little interference. The figures and tables presented allow concluding that the dimensionless parameters of the proposed simplified model—α, β and γ —are more sensitive to the variation of the geometric index D/t than in H/D. Among these parameters, it was also observed that the one that experiences greater influence of the D/t index is the parameter γ. In this way, the proposed analysis model will take into account a single dimensionless parameter—γ, which will have its behaviour evaluated for several values of the D/t index. In order to assess the behaviour of this parameter, the minimization problem discussed in Sect. 5.3 was slightly reformulated as follows: n 

2 u mi − u ti (α, β, γ ) min γ

(5.7)

i=1 α=0.35 β=0.15

Table 5.6 summarizes the results obtained. The first column of Table 5.6 shows the parameter γ values that solve the minimization problem described above (optimum values), for several values of geometric

5.4 Parametric Study Table 5.6 Parameters γ

67

D/t

Value Optimum

Adopted

20

0.8255

0.84

30

0.6815

0.71

40

0.6038

0.58

50

0.4899

0.45

60

0.2895

0.32

Fig. 5.12 Variation of parameter γ

index D/t. The second column indicates the values resulting from a linear fitting performed to the optimum value. The equation of the linear fitting procedure is shown below. f (x) = −0.013x + 1.1

(5.8)

where x is the Geometric index D/t. Figure 5.12 exhibits the value of Table 5.6 together with the linear fitting. In the sequence, it is shown figures with the results of positive and negative bending moments for silo groups with geometric index D/t equal to 20, 30, 40, 50 and 60, applying the proposed analysis models. For the calculation of nodal displacements of Model 1 (Timoshenko Beam on Elastic Foundation) the following values of the dimensionless parameters were considered:

68

5 Structural Behaviour and Simplified Model Proposition

Fig. 5.13 Bending moments for the geometric index D/t = 20

• α = 0.35 • β = 0.15 • γ according to linear fitting equation describe above. The positive and negative bending moments obtained with proposed analysis models are shown in Figs. 5.13, 5.14, 5.15, 5.16 and 5.17 together with the results obtained from 3D finite element analysis. Exception made to the positive bending’s moments corresponding to the geometric index D/t = 20, the other results obtained are in safety side.

5.4 Parametric Study

69

Fig. 5.14 Bending moments for the geometric index D/t = 30

The parametric study showed that it is possible to bypass the non-consideration of the deformability of the arch’s supports used in the model for the calculation of the bending moments) by establishing a 10% reduction factor for the negative bending moments values. Applying this factor to the analyses performed, the maximum error verified regarding to the three-dimensional model was 12%, for the silo group corresponding to the geometric index D/t = 20, the minimum error was 5% for the geometric index D/t = 60 and the mean error was 8.8%. The results of the parametric study showed that the positive bending moments obtained with the proposed model exhibited smaller differences regarding to the three-dimensional analysis than the negative bending moments.

70

5 Structural Behaviour and Simplified Model Proposition

Fig. 5.15 Bending moments for the geometric index D/t = 40

Regarding to the positive bending moments, the value obtained from the proposed model can be directly adopted for design purposes, without the need for any reduction. By doing do, the maximum error committed is 10%, corresponding to the geometric index of the silo group with D/t = 50, the minimum error is 2.5% for D/t = 30 and the mean error is 7%. The only case in which the bending moments from the three-dimensional model were greater than those of the proposed model was for the silo group with geometric index D/t = 20, only for the positive bending moments. The maximum difference found in this case was 5%, which is perfectly satisfactory for design purposes.

5.5 Procedure Proposed

71

Fig. 5.16 Bending moments for the geometric index D/t = 50

5.5 Procedure Proposed The proposed procedure for the calculation of the bending moments in grouped of reinforced concrete silos consisting of four cells of equal diameter for interstice loading are as follows. 1. Calculate the lateral design pressure due to the silage [2]; 2. Calculate the cross-section area and moment of inertia of one cell in the group, to obtain Ag and Ig; 3. Calculate reference stiffness of elastic foundation as indicated in Sect. 2 of Chap. 5 to obtain kr ;

72

5 Structural Behaviour and Simplified Model Proposition

Fig. 5.17 Bending moments for the geometric index D/t = 60

4. Build the Model 1 of analysis, discretizing the silo in elements of Timoshenko beam and calculating the nodal displacements (translations), considering the inertia of each element as being equal to 35% of the gross inertia of the cross section, the shear area equal to 15% of the gross area and define the participation of the spring constant, calculated in item 3 adopting the linear adjustment described in Chap. 5; 5. Build the transversal model of a clamped arch and calculate the bending moments, as indicated in Sect. 2 of Chap. 5; 6. Apply a 10% reduction factor to negative bending moments as indicated in Sect. 4 of Chap. 5. Positive bending moments do not need correction. The resulting moment diagrams are those that will be used for the design of the structure.

5.6 Results and Discussion

73

5.6 Results and Discussion The comparison of results between the methods available in the literature and the procedure proposed in the present work is presented below.

5.6.1 Bending Moments Tables 5.7 and 5.8 summarize, respectively, the values of the negative and positive bending moments that are represented graphically in Figs. 5.18 and 5.19. In addition to the values of the bending moments, the figures also indicate the variation of the applied horizontal design pressure. The figures and tables presented are for a silo group with 10 m in diameter and 30 m in height. The geometry of the group is that indicated in Chap. 2, and the diagram of the horizontal design pressures for the interstice is the one introduced in Sect. 2.3 of Chap. 2. The calculation of the bending moments of the proposed model considered the indications listed in Sect. 5 of this chapter. It was observed that the results obtained through the application of the simplified model proposed are the ones that are closest to the results of the three-dimensional analysis, benchmark for the research developed. In fact, the behaviour of the bending moment diagram resulting from the threedimensional analysis shows that the bending moments (positive and negative) are initially small, increase until reaching their maximum value in a depth of 16 m and. From that depth on there is a decreasing until it reaches, in the bottom, values of the same order of magnitude as those occurring at the top of the structure. This behaviour is characteristic of the bending moments diagrams that come from the application of the proposed model, aspect that assure the efficiency of the formulation designed for the calculation of the bending moments. The same behaviour does not occur with the other methods for calculating the bending moments available in the literature, which results in values of bending moment diagram directly proportional to the applied design pressure value. In addition, it should be noted that the simplified methods found in the literature give very conflicting results. The fixed arch method (arch clamped at both ends), suggested by Gurfinkel [3] and Safarian and Harris [4], presented the smallest values of the bending moments (positive and negative), always contrary to the safety of the structure.

−13.09 −34.47

−19.94 −52.49

−26.92 −70.89

−33.99 −89.49

−41.14 −108.32

−47.93 −126.20

−53.30 −140.32

−3.32

−5.06

−6.83

−8.63

−10.44

−12.16

−13.52

2

3

4

5

6

7

8

0

−72.05 −189.70

−76.12 −200.42

−80.02 −210.70

−83.71 −220.41

−18.28

−19.32

−20.31

−21.24

11

12

13

14

a Finite

element method

−67.73 −178.34

−17.19

10

15

−58.38 −153.72

−63.19 −166.39

−14.81

−16.04

9

−16.94

0

−6.44

−1.63

Timm and Windels

0

Haydl

1

Gurfinkel

Method

−69.66

−66.59

−63.34

−59.95

−56.36

−52.58

−48.58

−44.35

−39.88

−34.23

−28.28

– 22.40

−16.59

−10.89

−5.35

0 −10.84

−5.44

0

Proposed

−51.01 −58.30

−49.63 −57.09

−47.83 −55.39

−45.64 −53.20

−43.06 −50.57

−40.14 −47.50

−36.88 −44.03

−33.31 −40.18

−29.42 −35.96

−25.26 −31.38

−20.89 −26.52

−16.37 −21.44

−11.75 −16.20

−7.06

−2.33

–

Ciesielski FEMa

Negative bending moments—kNm/m

0

H (m)

Table 5.7 Negative bending moments

–

30

29

28

27

26

25

24

23

22

21

20

19

18

17

16

H (m)

−87.23

–

−29.63

−29.41

−29.18

−28.93

−28.66

−28.36

−27.98

−27.35

– 26.70

−26.01

−25.13

−24.57

−23.80

–

−116.77

−115.91

−115.01

−114.01

−112.94

−111.76

−110.28

−107.80

−105.22

−102.52

−99.74

−96.83

−93.87

−90.56

−22.13 −22.98

Haydl

Gurfinkel

Method

–

−307.44

−305.19

−302.80

−300.20

−297.37

−294.26

−290.35

−283.83

−277.03

−269.93

−262.62

−254.94

−246.91

−238.43

−229.76

Timm and Windels

–

−97.16

−96.45

−95.70

−94.87

−93.98

−93.00

−91.76

−89.70

−87.55

−85.31

−83.00

−80.57

−78.03

−75.35

−72.58

Proposed

–

0

–

0

−16.34 −15.70

−24.12 −22.46

−28.92 −28.73

−33.40 −34.46

−37.64 −39.62

−41.41 −44.18

−44.60 −48.11

−47.24 −51.43

−49.33 −54.14

−50.88 −56.27

−51.91 −57.80

−52.42 −58.76

−52.43 −59.15

−51.92 −59.00

Ciesielski FEMa

Negative bending moments—kNm/m

74 5 Structural Behaviour and Simplified Model Proposition

53.27

10.25

10.72

14

a Finite

15

50.93

9.75

element method

48.44

45.85

43.10

40.22

37.15

33.92

30.50

9.23

6.83

8

13

6.14

7

26.18

12

5.27

6

21.63

17.13

8.68

4.35

5

11

3.45

4

12.69

7.48

2.55

3

8.33

4.10

8.10

1.68

2

10

0.82

1

0

Haydl

9

0

Gurfinkel

Method

113.84

108.82

103.51

97.98

92.11

85.94

79.39

72.47

65.18

55.95

46.22

36.61

27.11

17.80

8.75

0

Timm and Windels

35.16

33.61

31.97

30.26

28.45

26.55

24.52

22.39

20.13

17.28

14.28

11.31

8.38

5.50

2.70

0

Ciesielski

Positive bending moments—kNm/m

0

H (m)

Table 5.8 Positive bending moments

29.23

28.44

27.41

26.15

24.68

23.00

21.13

19.08

16.86

14.48

11.98

9.40

6.75

4.07

1.36

–

FEMa

33.34

32.65

31.67

30.42

28.91

27.16

25.17

22.97

20.56

17.94

15.16

12.26

9.27

6.20

3.11

0

Proposed

–

30

29

28

27

26

25

24

23

22

21

20

19

18

17

16

H (m)

–

14.96

14.85

14.73

14.60

14.47

14.32

14.13

13.81

13.48

13.13

12.78

12.40

12.01

11.60

11.17

Gurfinkel

Method

–

74.31

73.76

73.19

72.56

71.87

71.12

70.18

68.60

66.96

65.24

63.47

61.62

59.68

57.63

55.51

Haydl

–

158.79

157.63

166.39

155.05

153.59

151.98

149.96

146.60

143.08

139.41

135.64

131.67

127.52

123.15

118.62

Timm and Windels

–

49.05

48.69

48.31

47.89

47.44

46.94

46.32

45.28

44.20

43.06

41.90

40.67

39.39

38.04

36.64

Ciesielski

Positive bending moments—kNm/m

–

0.35

7.84

12.56

15.97

18.89

21.45

23.65

25.51

27.04

28.25

29.14

29.73

30.02

30.03

29.76

FEMa

–

0

8.98

12.84

16.43

19.70

22.66

25.26

27.51

29.41

30.96

32.17

33.05

33.60

33.82

33.73

Proposed

5.6 Results and Discussion 75

76

5 Structural Behaviour and Simplified Model Proposition

Fig. 5.18 Bending moments in Sect. 4 of Fig. 4.1

5.6.2 Horizontal Normal Forces Tables 5.9, 5.10 and Figs. 5.20, 5.21 indicate the result of the comparison performed between the values of the normal horizontal forces. The tables and figures presented contain the values of the three-dimensional analysis with the finite element method, the results obtained with the use of the proposed analysis procedure and the numbers from the application of the simplified methods available in the literature. The results presented are for the sections located in the plane of symmetry (middle of the interstice wall—Sect. 3 of Fig. 4.1 of Chap. 4) and sections located at the

−44.334

−47.160

−49.824

−52.380

−54.792

−73.949

−78.662

−83.106

−87.369

−91.392

11

12

13

14

15

−34.884

−58.186

8

−41.364

−31.374

−52.331

7

−38.214

−26.928

−44.916

6

−63.741

−22.248

−37.109

5

−68.995

−17.622

−29.393

4

10

−13.050

−21.767

3

9

−8.568

−14.291

2

0

−63..011

−60.237

−57.298

−54.234

−50.984

−47.569

−43.946

−40.117

−36.080

−30.967

−25.585

−20.265

−15.008

−9.853

−4.844

−64.120

−61.300

−58.300

−55.190

−51.880

−48.400

−44.720

−40.820

−36.710

−31.510

−26.030

−20.620

−15.270

−10.030

−4.929

0

17.020

19.510

20.960

21.480

21.160

20.110

18.470

16.630

15.100

13.920

12.610

10.770

8.300

5.280

1.820

–

45.537

48.305

49.699

49.805

48.741

46.627

43.591

39.754

35.697

32.044

28.352

23.974

18.861

13.546

6.747

0

16

–

30

29

28

27

26

25

24

23

22

21

20

19

18

17

−57.096 −59.274

–

–

−127.481 −76.228

−126.550 −75.870

−125.560 −75.276

−124.479 −74.628

−123.308 −73.926

−122.017 −73.152

−120.396 −72.180

−117.693 −70.560

−114.871 −68.868

−111.929 −67.104

−108.896 −65.286

−105.714 −63.378

−102.381 −61.390

−98.869

–

−87.892

−87.251

−86.567

−85.822

−85.015

−84.125

−83.007

−81.144

−79.198

−77.170

−75.079

−72.285

−70.587

−68.165

−65.660

−7.026

−95.256

0

−4.212

0

1

Timm and Windels

Method Proposed

Gurfinkel Haydl

Ciesielski FEM

Gurfinkel Haydl

Timm and Windels

Method

Horizontal normal forces—kN/m Section 3—plan of symmetry

H (m)

Section 3—plan of symmetry

Horizontal normal forces—kN/m

0

H (m)

Table 5.9 Horizontal normal forces—plane of symmetry

–

−89.440

−88.780

−88.090

−87.330

−86.510

−85.600

−84.470

−82.570

−80.590

−78.530

−76.400

−74.170

−71.830

−69.360

−66.810

−72.751

−53.678

−35.677

−19.204

−4.626

8.074

18.956

28.092

35.526

41.300

Proposed

–

30.040

−87.590

–

0

−164.350

−159.490 −138.980

−131.340 −115.210

−101.400 −93.134

−82.980

−66.920

−51.430

−37.320

−24.980

−14.320

−5.230

2.370

8.550

13.400

Ciesielski FEM

5.6 Results and Discussion 77

Proposed

0

−3.159

−6.426

−9.787

−13.217

−16.686

−20.196

−23.531

−26.163

−28.661

−31.023

−33.251

−35.370

−37.368

−39.285

−41.094

0

−7.507

−15.271

−23.260

−31.409

−39.654

−47.996

−55.920

−62.177

−68.112

−73.726

−79.020

−84.057

−88.805

−93.361

−97.660

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

30.380

−43.570

−41.650

−39.620

−37.500

−35.250

−32.890

–

−27.740

−24.950

−21.410

−17.690

−14,010

−10.380

−6.812

−3.349

–

3.484

6.057

7.778

8.784

9.132

8.959

8.414

7.775

7.258

6.890

6.446

5.690

4.503

2.908

1.007

0

30.203

33.289

35.132

35.813

35.442

34.134

32.012

29.188

26.240

23.791

21.379

18.336

14.559

10.696

5.315

16

–

30

29

28

27

26

25

24

23

22

21

20

19

18

17

–

–

−136.224 −57.321

−135.229 −59.030

−134.170 −56.457

−133.015 −55.971

−131.764 −55.445

−130.385 −54.864

−128.652 −54.135

−125.765 −52.920

−122.749 −51.651

−119.605 −50.328

−116.364 −48.965

−112.964 −47.534

−109.402 −46.035

−105.649 −44.555

−101.767 −42.822

Haydl

Method Ciesielski FEM

Gurfinkel

Timm and Windels

Gurfinkel

Haydl

Method

–

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Timm and Windels

Horizontal normal forces—kN/m Section 4—support

H (m)

Section 4—support

Horizontal normal forces—kN/m

0

H (m)

Table 5.10 Horizontal normal forces—support

–

−60.770

−60.320

−59.850

−59.340

−58.780

−58.160

−57.390

−56.100

−54.760

−53.350

−51.910

−50.390

−48.800

−47.130

−45.400

−102.220

−83.172

−65.298

−48.331

−32.728

−18.866

−6.724

3.755

12.638

19.969

25.794

Proposed

–

−2.842

−51.878

−92.366

–

0

−168.480

−144.890

−102.459 −122.760

−91.538

−79.804

−67.213

−55.359

−43.770

−33.572

−24.397

−16.618

−9.914

−4.435

0.043

Ciesielski FEM

78 5 Structural Behaviour and Simplified Model Proposition

5.6 Results and Discussion

79

Fig. 5.19 Bending moments in Sect. 3 of Fig. 4.1

intersection of the interstice wall and region of a connection to another cell (Sect. 4 of Fig. 4.1 of Chap. 4). It was observed that, as it happened with the bending moment’s diagrams, the literature methods available present results that do not capture the actual structure response. Indeed, in all of them the resulting normal force are compressive, and there is no part in tension, contrary to behaviour that the three-dimensional analysis shows.

80 Fig. 5.20 Horizontal normal forces—plan of symmetry

5 Structural Behaviour and Simplified Model Proposition

5.6 Results and Discussion

81

Fig. 5.21 Horizontal normal forces—support

In addition, one of the simplified methods of the literature [5] indicates null stress for the sections located in the support, a fact that does not represent the actual structural behaviour observed in the three-dimensional analysis. Although the proposed analysis procedure has not been developed for the calculation of the horizontal normal forces, the diagrams of these efforts due to its application present behaviour quite similar to that from the three-dimensional analysis. Some considerations, however, must be observed. Firstly, in the case of normal forces in the symmetry section, it was observed that the proposed model shows an increase of the tensile region of the silo wall. In fact, the region of the height of the silo where a tensile force is developed is increased from 18 m in the three-dimensional analysis to 21 m in the proposed model. In addition,

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5 Structural Behaviour and Simplified Model Proposition

the values resulting from the proposed model are always higher than those of the three-dimensional analysis. On the other hand, the magnitude of the tensile force of the proposed model in this region (50 kN/m) correspond to only 14% of the values that are usually considered in the design this type of structure—internal pressure x internal radius—which, for the silo considered in the comparison, represents 350 kN/m. Regarding to the symmetry section, it was still verified that the compressions resulting from the proposed model are smaller those from the three-dimensional analysis, however, the maximum value in the two analyses performed occurred at approximately at the same depth, with close values −160 kN/m. For the normal forces in the arches supports (Sect. 4 of Fig. 4.1), there was also an increase in the tension region of the silo (from 15 m in the three-dimensional analysis to 19 m in the proposed model). Similarly to the normal forces in the symmetry section, the values resulting from the application of the proposed analysis model are also larger. However, the order of magnitude of the tensile forces of the proposed model (35 kN/m) corresponds to only 10% of those usually considered for design purposes, already discussed previously. In the compressed region, especially in the depth of 27 m, a behaviour difference of the diagram of the proposed model was observed in relation to the three-dimensional analysis. Such divergence comes from the vertical flexion of the walls, an aspect that is not considered in the proposed analysis model. The flexing effect is more expressive in the region of connection between two cells (this fact did not occur in the normal forces in Sect. 3 of Fig. 4.1). For the design purposes, it is possible to use the horizontal normal forces resulting from the application of the proposed simplified model, taking into account, in the sections of the supports, the value corresponding to the section located at 87% of the height of the silo. From this point on, it is recommended to consider a linear variation up to zero, as indicated in Fig. 5.21. The horizontal normal forces in the symmetry section can be used in the manner that results from the application of the proposed analysis model without any restriction.

References 1. Mathworks (1992) MATLAB user’s guide. Math Works, Inc., Natick 2. ACI Committee 313 (2011) Standard practice for design and construction of concrete silos and stacking tubes for storing granular materials (ACI 313–11), American Concrete Institute, Detroit 3. Gurfinkel G (1990) Reinforced-concrete bunkers and silos. In: Gaylord H, Gaylord CN (eds) Structural engineering handbook, Sect. 26. Wiley, New York 4. Safarian SS, Harris EC (1985) Design and construction of silos and bunkers. Van Nostrand Reinhold Company, New York 5. Timm G, Windels R (1977) Silos, Sonderdruck ans Dem Beton-Kalendan. Verlag von Wilhelm Ernst & Sohn, Berlin, Germany

Chapter 6

Conclusions and Recommendations

Taking into account the analyses performed throughout the research it is possible to draw the following conclusions. (1) The results presented attest to the efficiency of the proposed formulation for the calculation of the bending moments in reinforced concrete silo groups due to the interstice loading; (2) The two-dimensional analysis methods available in the literature do not present reliable results; (3) If a three-dimensional analysis is required, the Finite Element Method is preferable. In this case, solid elements, shell or mixed modelling may be used. The use of this method involves quite complex models that demand, besides time (computational and human), a strong theoretical knowledge of the technique used; (4) The three-dimensional solid modelling is one that admits less simplifying hypotheses regarding the representation of the physical model. On the other hand, it is the analysis that requires more computational effort and more time of the designer engineer in the interpretation of the results, aspect that makes difficult its use in the daily design practice; (5) The technique of stress resultants calculation from Finite Element solution (used in solid modelling) proved to be efficient and can be used as a practical application of the finite element method in reinforced concrete design of structures; (6) The three-dimensional modelling of the structure integrally with shell elements is the simplest and least demanding computational effort and time to interpret the results from the design engineer. This modelling, although satisfactory from the point of view of the structural design, presents an excess of stiffness in the region of intersection between two contiguous cells, especially in the sections near the intersection. In addition, this modelling requires the consideration of rigid connection in the regions of connection between cells, an option not always available in the structural analysis software package used in the design offices;

© Springer Nature Switzerland AG 2019 F. A. N. Silva et al., Design of Reinforced Concrete Silo Groups, Building Pathology and Rehabilitation 10, https://doi.org/10.1007/978-3-030-13621-5_6

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6 Conclusions and Recommendations

(7) Three-dimensional analysis using mixed modelling (walls modelled with shell elements and intersecting regions modelled with solid elements) is the most cost-effective; (8) The proposed simplified model presents quite satisfactory results and can be used for initial design or even for the definitive design of the reinforced concrete silo groups. The implementation of the proposed model is very simple and the results obtained are always in the safety side, within the silo groups range considered in the research; (9) The horizontal normal forces of the proposed model can be used in the design of reinforced concrete silo groups, provided that the respecting the recommendations presented in Sect. 5.6.2 of Chap. 5 are adopted.