Punching shear of structural concrete slabs: Honoring Neil M. Hawkins ACIfib International Symposium
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81
Bulletin
Punching shear of structural concrete slabs: Honoring Neil M. Hawkins ACIfib International Symposium
ACI SP315 ACIfib symposium proceedings
Bulletin
81
Punching shear of structural concrete slabs: Honoring Neil M. Hawkins
Technical report Proceedings of a symposium held in Philadelphia, PA, USA, on 25 October 2016 Edited by Carlos E. Ospina, Denis Mitchell and Aurelio Muttoni April 2017
Approval of this bulletin Subject to priorities defined by the technical council and the presidium, the results of the fib’s work in commissions and task groups are published in a continuously numbered series of technical publications called bulletins. The following categories are used. Category
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Any publication that has not met the above requirements will be clearly identified as preliminary draft. fib Bulletin 81 is published as a technical report and is a collection of contributions to a symposium that was cosponsored by the fib and the American Concrete Institute (ACI). The authors have presented their individual views. Although these contributions have not been discussed in any of the fib’s working bodies, the subject matter is highly topical and believed to be of general interest to members of the fib. This bulletin is also published as an ACI Symposium Publication, ACI SP315.
Cover images: Sawcuts of inner slabcolumn connections failed in punching. Photographs courtesy of A. Herzog, EPFL
© Fédération internationale du béton (fib) and American Concrete Institute (ACI), 2017 Although the International Federation for Structural Concrete / Fédération internationale du béton (fib) does its best to ensure that any information given is accurate, no liability or responsibility of any kind, including liability for negligence, is accepted in this respect by the organisation, its members, servants or agents. All rights reserved. No part of this publication may be reproduced, modified, translated, stored in a retrieval system, or transmitted in any form or by any means—electronic, mechanical, photocopying, recording, or otherwise—without prior written permission from the fib. ISSN 15623610 ISBN 9782883941212 Layout by Laura Vidale on behalf of the fib secretariat. Printed by DCC Document Competence Center Siegmar Kästl e.K., Germany
ACIfib International Symposium Punching shear of structural concrete slabs
Foreword fib Bulletin 81 deals with punching of slabs, which is a relevant issue in the design of reinforced concrete. Punching is one of the most frequent reasons for failure of concrete structures, which underlines the importance of this problem. Flat slabs are used in buildings, bridges and other structures. Punching may result in brittle failure, which requires that special attention be paid to the design of new structures and the assessment of existing ones. Punching and shear capacity of structures is often evaluated using empirical methods, which does not always provide a sufficient level of safety. This bulletin is a result of an international symposium where experts from fib and ACI met and exchanged their experience. The papers included in this bulletin provide new experimental evidence and a comprehensive review of analytical and numerical methods that are used to evaluate the structural performance of slabcolumn connections. This bulletin provides a review of the performance of slabs under static, dynamic and seismic loading. The individual papers generally show the studied phenomena using experiments and analytical evaluation, which makes it possible to compare results obtained according to the European code (EC2), the ACI code and the international code, the fib Model Code for Concrete Structures (MC2010). Particular attention is paid to the socalled size effect (i.e. the nominal load carrying capacity of thicker slabs is lower than that of thin slabs). The arrangement of reinforcement is important for the residual strength after a local failure and robustness, which are instrumental to avoid the progressive collapse of the entire structure after a local failure. It is confirmed that the bottom reinforcement crossing the column is very beneficial to ensure sufficient residual strength and avoid progressive collapse. A paper dealing with the shear capacity of bridge slabs illustrates an extended strip model, which may be used for efficient assessment of bridge slabs under concentrated loads. This bulletin also presents new types of shear reinforcements and shows efficiency of distribution of classical shear reinforcing elements like headed studs. Some papers deal with retrofitting and strengthening of existing slabs. Postinstalled shear reinforcing elements may be used for strengthening of existing structures. This bulletin summarizes several phenomena that influence the performance of slabs sensitive to punching. It is a valuable summary of the stateofthe art knowledge for practicing engineers, academics and also for students. It is also important to appreciate that the opinions and experiences of American and European experts are summarised in one publication. Finally, it is necessary to thank all authors, as well as the main organizers of the symposium (A. Muttoni convener of fib WP 2.2.3, and C. E. Ospina and D. Mitchell representing ACI committee 445) for their editing of this extraordinary document. Last but not least, many thanks to Laura Vidale for the preparation of the bulletin for publication.
Jan L. Vítek Chairman of fib Commission 2, Analysis and design
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ACIfib International Symposium Punching shear of structural concrete slabs
Preface fib Bulletin 81 reports the latest information available to researchers and practitioners on the analysis, design and experimental evidence of punching shear of structural concrete slabs. It follows previous efforts by the International Federation for Structural Concrete (fib) and its predecessor the EuroInternational Committee for Concrete (CEB), through CEB Bulletin 168, Punching Shear in Reinforced Concrete (1985) and fib Bulletin 12, Punching of structural concrete slabs (2001), and an international symposium sponsored by the punching shear subcommittee of ACI Committee 445 (Shear and Torsion) and held in Kansas City, Mo., USA, in 2005.
Prof. Emeritus Neil M. Hawkins
This bulletin contains 18 papers that were presented in three sessions as part of an international symposium held in Philadelphia, Pa., USA, on October 25, 2016. The symposium was coorganized by the punching shear subcommittee of ACI 445 and by fib Working Party 2.2.3 (Punching and Shear in Slabs) with the objectives of not only disseminating information on this important design subject but also promoting harmonization among the various design theories and treatment of key aspects of punching shear design. The papers are organized in the same order they were presented in the symposium. The symposium honored Professor Emeritus Neil M. Hawkins (University of Illinois at UrbanaChampaign, USA), whose contributions through the years in the field of punching shear of structural concrete slabs have been paramount.
The papers cover key aspects related to punching shear of structural concrete slabs under different loading conditions, the study of size effect on punching capacity of slabs, the effect of slab reinforcement ratio on the response and failure mode of slabs, without and with shear reinforcement, and its implications for the design and formulation in codes of practice, an examination of different analytical tools to predict the punching shear response of slabs, the study of the postpunching response of concrete slabs, the evaluation of design provisions in modern codes based on recent experimental evidence and new punching shear theories, and an overview of the combined efforts undertaken jointly by ACI 445 and fib WP 2.2.3 to generate test result databanks for the evaluation and calibration of punching shear design recommendations in North American and international codes of practice. Sincere acknowledgments are extended to all authors, speakers, reviewers, as well as to fib and ACI staff for making the symposium a success and for their efforts to produce this longawaited bulletin. Special thanks are due to Laura Vidale for preparing the bulletin for publication. The editors of fib Bulletin 81: Carlos E. Ospina (chair of ACI 445 Punching Shear Subcommittee) Denis Mitchell (ACI 445) Aurelio Muttoni (chair of fib Working Party 2.2.3)
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ACIfib International Symposium Punching shear of structural concrete slabs
Participants in the 2016 ACIfib symposium on punching shear of structural concrete slabs. Standing, L to R: Sagaseta (Surrey, UK), Vollum (Imperial College, UK), Ramos (Nova, Portugal), Fernández Ruiz (EPFL, Switzerland), Kueres (Aachen, Germany), Genikomsou (Queen’s, Canada), Walkner (Innsbruck, Austria), Topuzi (Toronto, Canada). Seated, L to R: Lantsoght (Delft, The Netherlands, and Quito, Ecuador), Polak (Waterloo, Canada), Ospina (Houston, USA), Hawkins (U. of Illinois, USA), Muttoni (EPFL, Switzerland), Mitchell (McGill, Canada), Criswell (Colorado State, USA), Alexander (Edmonton, Canada). Not shown: Bažant (Northwestern, USA), Dam (U. of Michigan, USA), Gayed (Calgary, Canada), Hueste (Texas A&M, USA). Photo by ACI staff.
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ACIfib International Symposium Punching shear of structural concrete slabs
Contents Foreword
iii
Preface
iv
Shear and moment transfer at columnslab connections
1
Scott D. B. Alexander
Size effect on punching shear strength of RC slabs with and without shear reinforcement
23
Zdeněk P. Bažant, Abdullah Dönmez
Behavior and performance levels of reinforced concrete slabcolumn connections
35
Marvin E. Criswell, Carlos E. Ospina, Neil M. Hawkins
Size effect on punching shear strength: Differences and analogies with shear in oneway slabs
59
Miguel Fernández Ruiz, Aurelio Muttoni
Flexureinduced punching of concrete flat plates
73
Ramez B. Gayed, Chandana Peiris, Amin Ghali
3D finite element analysis of punching shear of RC flat slabs using ABAQUS
101
Aikaterini S. Genikomsou, Maria A. Polak
Effect of slab flexural reinforcement and depth on punching strength
117
Neil M. Hawkins, Carlos E. Ospina
Review of test data for interior slabcolumn connections with moment transfer
141
Yan Zhou, Mary Beth D. Hueste
Maximum punching shear capacity of footings with new punching shear reinforcement elements
167
Dominik Kueres, Marcus Ricker, Josef Hegger
Bridging the gap between oneway and twoway shear in slabs
187
Eva O. L. Lantsoght, Cor van der Veen, Ane de Boer, Scott D.B. Alexander
Punching and postpunching response of slabs
215
Denis Mitchell, William D. Cook
The Critical Shear Crack Theory for punching design: From a mechanical model to closedform design expressions
237
Aurelio Muttoni, Miguel Fernández Ruiz
Punching of flat slabs under reversed horizontal cyclic loading António Ramos, Rui Marreiros, André Almeida, Brisid Isufi, Micael Inácio
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ACIfib International Symposium Punching shear of structural concrete slabs
Structural robustness of concrete flat slab structures
273
Juan Sagaseta, Nsikak Ulaeto, Justin Russell
Influence of flexural continuity on punching resistance at edge columns
299
Luis F. S. Soares, Robert L. Vollum
Seismic retrofit of concrete slabs against punching shear: Testing and modelling
319
Dritan Topuzi, Maria Anna Polak, Sriram Narasimhan
A new method for postinstalled punching shear reinforcement
337
Rupert Walkner, Mathias Spiegl, Jürgen Feix
Punching failure of slabcolumn connections reinforced with headed shear studs
353
Thai X. Dam, James K. Wight, Gustavo J. ParraMontesinos, Alex DaCosta
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Shear and moment transfer at columnslab connections Scott D. B. Alexander COWI North America, Canada
Abstract The Strip Model describes a load path for the transfer of vertical shear between a slab and column. The model is easily adapted to design but its application to the analysis of specimens tested under combined shear and moment is less clear. This paper provides a brief description of the Strip Model, updates the model to include size effect, and shows how it can be applied to interior and edge columnslab connections transferring combinations of shear and moment.
Keywords Columns, connections, punching shear, reinforced concrete, shear strength, slabs, structural design.
1
Introduction
The Strip Model for slab punching shear, originally called the Bond Model (Alexander and Simmonds, 1992), describes an internal distribution for the transfer of vertical load between a twoway slab and column. The model may be considered an extension of the Strip Method of Design (Hillerborg, 1975). The Strip Method allows a designer to define a load distribution that rigorously satisfies equilibrium at all points in a slab and to reinforce the slab for the bending moments that are the consequence of that load distribution. The Strip Method as developed by Hillerborg does not address shear strength. The Strip Model for slab punching shear is consistent with the Strip Method for flexural design but is focused on a particular problem: the development of an internal load distribution for shear transfer at concentrated loads that does not violate either shear or flexural strength limits at any point. This internal load distribution is derived from subdividing the slab into regions dominated by slender flexural behavior (Bregions) and regions dominated by deep beam behavior (Dregions). The result is a model for shear transfer that can be verified by direct measurement (Alexander et al., 1995). The distinguishing characteristic of a B or Dregion is the predominant mechanism of moment gradient (i.e. shear transfer). In a slender beam, moment gradient is mostly the result of a varying flexural tension force acting on a more or less constant moment arm. Such behavior is called beam action. In a deep beam, moment gradient results from a constant tensile force acting on a varying moment arm. This behavior is called arching action.
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ACIfib International Symposium Punching shear of structural concrete slabs
It is appropriate to describe shear transfer by beam action in terms of an average shear stress acting on the crosssection. A reasonable design strategy for slender members is to limit the average shear stress to some critical value; however, an average shear stress does not model the behavior in a Dregion. Dregions are more correctly modeled using strutandtie. Columnslab connections exhibit the characteristics of both B and Dregions. Tests show that radial arching action is an important mechanism of shear transfer between a slab and a column, suggesting that columnslab connections should be considered Dregions. In the circumferential direction, however, columnslab connections behave more like Bregions. This paper provides a brief summary of the mechanics of the strip model and extends the model to account for size effect. It then introduces the concepts of nonproportional loading and shows how these are used to describe the transfer of shear moment at a columnslab connection.
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Strip model for concentric punching
2.1
Internal load distribution
The Strip Model divides the slab into radial strips and plate quadrants, as shown in Fig. 1. No load can reach the column without passing through one of the radial strips. Within each radial strip shear is carried to the column by arching action. This is visualized as a curved arch, with maximum slope at the face of the column as shown in Fig. 2. The quadrants of twoway slab are fundamentally slender flexural elements, which means shear transfer across the boundary between a strip and its adjacent quadrant of plate is through the twoway plate equivalent of beam action.
Direction of Reinforcement
Lines of zero shear
Radial Strips
Remote end located at position of zero shear Column
Direction of Reinforcement
Figure 1:
Shaded region indicates radial halfstrip (Fig. 3)
Geometry of Strip Model.
Consider the compression arch shown in Fig. 2. The compression force in the arch is approximately constant throughout. At the column face, the vertical component of the arch accounts for the shear transferred to the column; the horizontal component provides a flexural compression.
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ACIfib International Symposium Punching shear of structural concrete slabs
Figure 2:
Beam and arching action at columnslab connection.
Moving away from the column, the slope of the arch decreases. Vertical equilibrium of the arch requires that there be a transverse stress field. The transverse stress field is internal and is generated by the twoway plate equivalent of beam action shear acting in a direction perpendicular to the arch. Thus, the model is of the interaction between the slender quadrants of plate and the radial strips acting as deep beams. Figure 3 shows a free body diagram of half of a radial strip. The halfstrip is loaded on its side face by a combination of plate bending moment, mn, torsional moment, mt, and shear, v. The strip is supported by a vertical reaction, Ps, at the columnsupported end and bending moments, Mneg and Mpos, at the column and remote ends of the strip, respectively. r 0.5 M
pos
L q
mn n
Re mo te E nd
mn v
mt
0.5 M
Co lum nE nd
neg
0.5 P
s
Figure 3:
Forces on radial halfstrip.
The internal vertical shear at any point along the side face of a radial strip is a function of the gradients of bending and torsional moments at that point. Alexander and Simmonds (1992) and Afhami et al. (1998) examine the various components of this internal shear in some detail to justify the simplified free body diagrams of radial strips at ultimate load, shown in Fig. 4. The loading term, w, is the limiting oneway shear that can be carried by the slab. An internal radial strip, such as those shown in Fig. 1, is loaded on two faces; hence the total distributed line load on the strip is 2w. At an edge column, a spandrel strip running parallel to the free edge of the slab would be loaded on only one side and so would be subject to a line load of w.
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ACIfib International Symposium Punching shear of structural concrete slabs
l 2w
Column End
Remote End Mpos
Mneg L
Ps
(a) Interior Strip l w Mpos
Mneg L
Ps
(b) Spandrel Strip
Figure 4:
Simplified loading on radial strip.
The flexural strength of the radial strip, Ms , is the sum of the negative and positive flexural capacities, Mneg and Mpos, at the ends of the strip. The loaded length of the strip is l and the total load carried by one strip is Ps, termed the nominal capacity of a strip. Equilibrium for an internal radial strip requires that: 𝑀
𝑙 = √ 𝑤𝑠
(1)
𝑃𝑠 = 2√𝑀𝑠 𝑤
(2)
The corresponding equations for a spandrel strip are: 2𝑀𝑠
𝑙=√
𝑤
𝑃𝑠 = √2𝑀𝑠 𝑤
(3) (4)
The total load that can be delivered to the column is simply the summation of the individual strip contributions. 𝑃𝑐𝑜𝑙 = ∑ 𝑃𝑠
(5)
While the load distribution described above is certainly a simplification of reality, it is nevertheless realistic. The radial strips are parallel to the stiff directions of the reinforcing mat. Alexander et al. (1995) and Afhami et al. (1998) show that bending moment gradients perpendicular to the side faces of radial strips, estimated from strain gauge measurements, account for the measured column reaction over a considerable range of loading. Using nonlinear finite elements, Afhami et al. (1998) show that yielding of flexural reinforcement through the column results in a stepped distribution of internal shear along the side face of a radial strip, consistent with what is shown in Fig. 4. Earlier work (Alexander and Simmonds, 1992) examines a limited data set of 116 concentrically loaded interior columnslab connections from eight sources, six with simply supported edges and two with rotationally restrained edges. The flexural capacities are strictly limited to those of the strip of slab defined by the width of the column. 𝑀𝑛𝑒𝑔 = 𝜌𝑛𝑒𝑔 × 𝑓𝑦 × 𝑗 × 𝑐 × 𝑑 2
4
(6a)
ACIfib International Symposium Punching shear of structural concrete slabs
𝑀𝑝𝑜𝑠 = 𝜌𝑝𝑜𝑠 × 𝑓𝑦 × 𝑗 × 𝑐 × 𝑑2
(6b)
𝜌𝑓𝑦
𝑗 = 1 − 1.7𝑓′
(6c)
𝑀𝑠 = 𝑀𝑝𝑜𝑠 + 𝑀𝑛𝑒𝑔
(6d)
𝑐
Consistent with the assumption of slender flexural behavior, the loading term w is taken as the unit oneway shear strength of the slab. Based on ACI 31814 (ref), w is given by: 𝑤 = 𝑑 × 0.17√𝑓 ′ 𝑐 (𝑀𝑃𝑎) = 𝑑 × 2√𝑓′𝑐 (𝑝𝑠𝑖)
(7)
where d is the flexural depth of the slab and 𝑓𝑐′ is the specified concrete strength. Alexander and Simmonds (1992) report an average ratio of test load to calculated load of 1.29 with a coefficient of variation of 12.3%. On the same body of test results, the ratio of test load to the load predicted by ACI code is 1.56 with a coefficient of variation of 26.5%. To illustrate a typical set of calculations used to analyze these tests, consider a hypothetical interior columnslab punching test specimen. The slab measures 2.33 m (7 ft) square, centered on a 400 mm (15.75 in) square column. The slab has 1% top reinforcement each way based on an average flexural depth of 130 mm (5.12 in). The yield strength of the reinforcement is 400 MPa (58 ksi) and the concrete strength is 30 MPa (4350 psi). The slab is loaded on its perimeter. There is no rotational restraint at the boundary (i.e. Mpos = 0). From Equations (6a) through (6d): 𝑀𝑠 = 1% × 400𝑀𝑃𝑎 × (1 − m; (18.4 ft ∙ kips)
1%×400𝑀𝑃𝑎 1.7×30𝑀𝑃𝑎
) × 400 mm × (130 mm)2 = 24.9 kN ∙
From Equation (7), 𝑤 = 130 mm × 0.17 × √30 MPa = 121 𝑁/mm ; (8.29 kips/ft). From Equation (2), 𝑃𝑠 = 2√24.9 × 121 = 110 kN (24.7 kips). Because there are four strips framing into the column, Equation (5) gives a total load of 440 kN (98.8 kips). Note that even though the top reinforcing mat will have two values of d, one for each layer of reinforcement. The average value of d, measured to the middle of the mat, is used for all calculations. One could consider the different values of d but this is needlessly complicated. The larger value of d for flexure always pairs with the smaller value of d for the loading term, w, and vice versa.
2.2
Size effect
A shortcoming of the data set used by Alexander and Simmonds (1992) is the limited range of flexural depth. The deepest slab had a flexural depth of 200 mm (7.9 in). Fortytwo of the tests were on smallscale specimens with flexural depths less than 55 mm (2.2 in). To address this shortcoming, a subset of 257 specimens from the ACI 445 Punching Shear Databank (Ospina et al., 2011) is used. The subset has results from 38 separate sources. Specimens are simply supported (i.e. 𝑀𝑝𝑜𝑠 = 0 ), steel reinforced, with either square or circular columns. While the dataset still contains a large number of smallscale specimens (39 with flexural depth 65 mm [2.6 in] or less), it has better representation of deeper slabs (29 with flexural depths over 140 mm [5.5 in]). 5
ACIfib International Symposium Punching shear of structural concrete slabs
Figure 5 shows distributions of the ratio of test load to calculated load with respect to flexural depth. In Fig. 5(a) the calculated loads are nominal capacities based on the ACI code. The average ratio of test load to calculated load is 1.45 and the coefficient of variation is 25.4%. The results show some size effect. As flexural depth increases, the ratio of test load to calculated load decreases.
3 2.5 2 1.5 1 0.5 0 0
100
200
300
400
500
600
700
500
600
700
600
700
Flexural Depth, d (mm) (a) ACI Code
3 2.5 2 1.5 1 0.5 0 0
100
200
300
400
Flexural Depth, d (mm)
(b) Strip Model
3 2.5 2 1.5 1 0.5 0 0
100
200
300
400
500
Flexural Depth, d (mm)
(c) Strip Model with Size Effect
Figure 5:
Ratio of test load to calculated load: concentric tests.
Figure 5(b) shows the same test results relative to calculated loads based on the Strip Model with no size effect considered. Here the average ratio of test load to calculated load is 1.24 with a coefficient of variation of 16.2%. While considerably less scattered than the results using the ACI code, there is still a noticeable trend to decreasing strength with increasing flexural depth. To incorporate size effect with the Strip Model, one need only calculate the loading term, w, using an expression for oneway shear value that accounts for size. It is beyond the scope of this paper to present a detailed treatment of size effect. Instead, a simpler approach will be used. A size factor, (𝑎⁄𝑑)𝜆 , where a is a reference depth, d is the flexural depth of the slab, and 𝜆 is a fraction, will be incorporated in the loading term w. 6
ACIfib International Symposium Punching shear of structural concrete slabs
The reference depth, a, is set to 100 mm (4 in). There are a couple of reasons for this choice. First, it is convenient. Second, it is approximately equal to the depth of the thinnest slabs that could still be considered fullscale. Fujita et al. (2002) assess several methods that account for size effect in oneway shear and concluded that two separate populations were represented in their body of test results. For normal and medium strength concrete, they propose 𝜆 = 1⁄4 . For high strength concrete specimens, they propose 𝜆 = 1⁄2. Since the expression for Ps places any size factor used in the expression for w under a second root sign, it follows that twoway shear is somewhat less sensitive to size than oneway shear. A compromise value of 𝜆 = 1⁄3 is used here. Revising the expression for w results in: 3
100 𝑚𝑚
𝑤 = 𝑑 × 0.17√𝑓 ′ 𝑐 × √
𝑑
(8)
Figure 5(c) shows results ratios of test load to calculated load using the revised expression for w [Equation (8)] that accounts for size effect. The average ratio of test to calculated load for these 257 specimens is 1.24 and the coefficient of variation is 14.0%. Some comments on the Strip Model:
The Strip Model shows how a oneway shear limit combined with localized arching behavior is sufficient to explain twoway shear. A magnified "twoway" limiting shear stress is not necessary. With the loading term, w, held constant, the loaded length, l, increases with increasing flexural strength. In effect, the additional flexural capacity supports a larger "critical section." As presented above, the Strip Model has in no way been calibrated to the tests it is being used to analyze. The flexural strength of the radial strip is strictly that which can be attributed to a strip of slab with width defined by the supporting column. The loading term, w, is the oneway shear per unit width that one would use for a oneway slab. All the terms used in the Strip Model to calculate a twoway punching capacity are derived from oneway tests. Because the Strip Model makes use of only conventional flexural strengths and a oneway shear strength, it allows a unification of design treatment for one and twoway systems. The Strip Model does not describe a failure mechanism for a columnslab connection. It describes a load path that does not violate strength limits for flexure or oneway shear. With sufficient ductility in the system, the load predicted by the Strip Model is a lower bound of the capacity of the structure.
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3
Connections transferring shear and moment
3.1
Proportional and nonproportional loading
To apply the Strip Model to the more general case of an interior connection under shear and moment requires a modified approach that does not rely on the level of symmetry usually found in concentric load tests. Most concentrically loaded test specimens reported in the literature have, more or less, four axes of symmetry in the distribution of slab curvature around the column. As is observed by Afhami et al. (1998), the loading is such that the nominal capacity of each radial strip is developed. This situation is called proportional loading. While most concentric load tests in the literature have this level of symmetry, most columnslab connections in practice do not. Nonproportional, concentric loading occurs when the radial strips in one direction do not each their nominal capacity. The load is still concentric if there is no moment transferred to the column but the distribution of slab curvature around the column has only 2 axes of symmetry. The skew axes are no longer axes of symmetry. In this case enhanced radial strips will develop in one direction at the expense of strip development in the other. The limiting case, illustrated in Fig. 6, occurs when fully enhanced strips, each carrying a load 𝑃𝑠𝑠 , develop in one direction. The fully enhanced strip is called a super strip. Because radial strips do not develop in the direction perpendicular to the super strip, load transfer on the side face (c1 face) of the column is limited to the oneway shear limit. The maximum total load transferred to the column with this load distribution is given by: 𝑃𝑐𝑜𝑙 = (∑ 𝑃𝑠𝑠 ) + 2 × 𝑤 × 𝑐1 = 2 × (𝑃𝑠𝑠 + 𝑤𝑐1 )
(9)
This changed loading pattern requires some revision to the flexural capacity of the super strip. With nonproportional loading, the negative moment capacity of the strip need not be limited to the reinforcement within the strip. The radial super strip acts something like a Tbeam in negative moment, with its "stem" defined column dimension perpendicular to the strip, 𝑐2 and with a "top flange" wider than 𝑐2 .
Oneway shear loading side faces of super strip and c1 faces of column.
1.5h c2 1.5h
Interior Super Strip
Figure 6:
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Nonproportional, concentric loading at interior connection.
ACIfib International Symposium Punching shear of structural concrete slabs
Consistent with usual practice, it is assumed that top reinforcement within 1.5 times the slab thickness on either side of the column (i.e. within 𝑐2 + 3ℎ ) is effective as top reinforcement for the radial super strip. Equation (6a) becomes: 𝑀𝑠𝑠,𝑛𝑒𝑔 = 𝜌𝑠𝑠,𝑛𝑒𝑔 × 𝑓𝑦 × 𝑗𝑠𝑠 × 𝑐2 × 𝑑 2 where 𝜌𝑠𝑠,𝑛𝑒𝑔 = 𝜌𝑛𝑒𝑔 ×
(10)
(𝑐2 +3ℎ) 𝑐2
The factor 𝑗𝑠𝑠 is modified to account for flexural compression stresses framing into the side faces of the column joint. Alexander and Simmonds (2003) show that the effective width of the compression block associated with a wide band of flexural reinforcement is 𝑐1 + 𝑐2 . For a super strip framing into a square or circular column, the gives: 𝜌×𝑓𝑦
𝑗𝑠𝑠 = 1 − 2×1.7×𝑓′
(11)
𝑐
The basic equations for loaded length and the total load carried are conceptually unchanged. 𝑀
𝑙𝑠𝑠 = √ 𝑤𝑠𝑠
(12)
𝑃𝑠𝑠 = 2√𝑀𝑠𝑠 𝑤
(13)
Applying Equation (9) to the data set of 257 concentrically loaded test specimens results in an average test to calculated load of 1.24 with a coefficient of variation of 15.6%. In other words, the load path defined for nonproportional loading accounts for essentially the same load as proportional loading in the case of an interior column under concentric load. The virtue of considering this alternative load path is that it is more easily adapted to test specimen under eccentric loading.
3.2
Interior connections
The Strip Model allows the designer to assess independently the load coming into each face of a connection. Analogous to a shear failure in one of four beams framing into a column, punching occurs when one of these faces is overloaded. The net moment transferred to the connection is a consequence of the loading but not particularly relevant to the shear on any particular side of the connection. For example, in a continuous system with unequal spans, it is possible to design a connection that is not concentrically loaded yet transfers no moment to the column. In contrast, for a test on an isolated columnslab connection, moment transfer is often the only measure of the degree to which one side of the connection is more heavily loaded. The following shows how the nonproportional load distribution developed above can be adapted to test specimens that meet at least one of two criteria: (1) the slab has the same reinforcing mats top and bottom and/or (2) the loading eccentricity is low enough to avoid load reversal. Figure 7 shows a side view of an interior connection specimen under nonproportional, eccentric load. The load is treated as a combination of symmetric and antisymmetric components. Punching occurs when one side of the connection reaches its limiting load.
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ACIfib International Symposium Punching shear of structural concrete slabs
Symmetric component lss +c1 / 2
lss +c1 / 2
Antisymmetric component
lss
Figure 7:
c1
lss
Simplified shear and moment transfer at interior connection.
The overall length of loading for the concentric component of load is 2𝑙𝑠𝑠 + 𝑐1 . The antisymmetric loading components are each distributed over half of this length, or 𝑙𝑠𝑠 + 𝑐1 /2. From equilibrium, the load distribution described in Fig. 7 results in: 𝑃𝑢𝑙𝑡 2
+
where
𝑀𝑢𝑙𝑡 𝑟
≤ 𝑃𝑠𝑠 + 𝑤 × 𝑐1 = 2 × 𝑤 × 𝑟
(14)
𝑟 = 𝑙𝑠𝑠 + 𝑐1 /2
The length r is analogous to a radius of gyration for the connection. The Strip Model makes this quantity dependent on the flexural reinforcement near the connection. The equivalent quantities in the ACI code are derived from geometric properties of a fixed critical section that is independent of reinforcement. Tests by Hanson and Hanson (1968) and Hawkins et al. (1989) satisfy at least one of the two criteria listed earlier. Table 1 provides details for the 37 specimens considered here. Hanson and Hanson (1968) tested 17 columnslab connections. Of these, seven specimens were of interior columnslab connections with solid slabs (no holes near the column) tested under combined shear and moment. All specimens had the same layout of reinforcement, top and bottom. Two loading methods are represented here. Type I loading involved the application of equal upward and downward line loads to the slab on either side of the column to produce high moment with little net vertical load transfer. Type III involved a single line load on one side of the column. Hawkins et al. (1989) presents results from 30 tests of interior connections under eccentric loading. Point loads, controlled to produce a fixed eccentricity, were applied around the perimeter of the slab. Table 2 provides test results as well as predicted failure loads using both the Strip Model [Equation (14)] and the ACI code. Both analyses give reasonable results although the limited size of the data set makes any statistical comparisons less compelling.
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ACIfib International Symposium Punching shear of structural concrete slabs
Table 1:
Interior connections under shear and moment: description of specimens
Mark A1 A2 B7 C8 A12 B16 C17
f'c (MPa) 30.3 31.4 33.0 32.9 33.3 30.5 36.1
6.00AH 9.60AH 14.00AH 6.00AL 9.60AL 14.00AL 7.30BH 9.50H 14.20BH 7.30BL 9.50BL 14.20BL 6.00CH 9.60CH 14.00CH 6.00CL 14.00CL 6.00DH 14.00DH 6.00DL 14.00DL 10.20FHI 10.20FHO 14.00FH 6.00FLI 10.20FLI 10.20FLO 9.60GH2 9.60GHO.5 9.60GH3
31.4 30.7 30.4 22.8 29.0 27.1 22.2 19.9 29.6 18.2 20.1 20.6 52.5 57.3 54.9 49.7 47.8 30.1 31.8 28.7 24.3 26.0 33.9 31.3 26.0 18.1 26.6 24.7 26.3 27.1
Hanson and Hanson (1968) fy c1 c2 h (MPa) (mm) (mm) (mm) 366 152 152 76.2 377 152 152 76.2 355 305 152 76.2 412 152 305 76.2 373 152 152 76.2 341 305 152 76.2 342 152 305 76.2 Hawkins, Bao, and Yamazaki (1989) 473 305 305 152 416 305 305 152 421 305 305 152 473 305 305 152 416 305 305 152 421 305 305 152 473 305 305 114 473 305 305 114 416 305 305 114 473 305 305 114 473 305 305 114 416 305 305 114 473 305 305 152 416 305 305 152 421 305 305 152 473 305 305 152 421 305 305 152 473 305 305 152 421 305 305 152 473 305 305 152 421 305 305 152 421 305 305 152 421 305 305 152 421 305 305 152 473 305 305 152 421 305 305 152 421 305 305 152 416 406 203 152 416 203 406 152 416 457 152 152
dave (mm) 57.2 57.2 57.2 57.2 57.2 57.2 57.2
𝜌 (%) 1.64 1.64 1.64 1.64 1.64 1.64 1.64
120.7 117.5 114.3 120.7 117.5 114.3 82.6 82.6 79.4 82.6 82.6 79.4 120.7 117.5 114.3 120.7 114.3 120.7 114.3 120.7 114.3 114.3 114.3 114.3 114.3 114.3 114.3 117.5 117.5 117.5
0.60 0.96 1.40 0.60 0.96 1.40 0.73 0.95 1.42 0.73 0.95 1.42 0.60 0.96 1.40 0.60 1.40 0.60 1.40 0.60 1.40 1.02 1.02 1.40 0.60 1.02 1.02 0.96 0.96 0.96
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ACIfib International Symposium Punching shear of structural concrete slabs
Table 2:
Interior connections under shear and moment: test and calculated results Test results Mcol Pcol
Mark A1 A2 B7 C8 A12 B16 C17
(kN·m) 22.4 24.3 35.8 31.5 20.5 27.4 24.8
(kN) 5.75 4.81 4.90 5.62 26.9 34.4 31.6
6.00AH 9.60AH 14.00AH 6.00AL 9.60AL 14.00AL 7.30BH 9.50H 14.20BH 7.30BL 9.50BL 14.20BL 6.00CH 9.60CH 14.00CH 6.00CL 14.00CL 6.00DH 14.00DH 6.00DL 14.00DL 10.20FHI 10.20FHO 14.00FH 6.00FLI 10.20FLI 10.20FLO 9.60GH2 9.60GHO.5 9.60GH3
90.6 97.9 110.4 32.7 34.6 43.5 39.1 45.5 51.1 12.8 16.6 15.3 95.3 113.3 133.6 36.9 49.5 65.9 98.9 24.5 34.9 74.2 90.1 102.7 27.2 26.9 35.1 82.4 89.8 83.6
169.8 187.2 205.9 244.3 257.6 319.6 79.8 94.5 102.1 130.2 142.2 162.2 186.8 218.0 252.7 273.7 362.4 134.2 195.7 232.7 282.1 153.8 183.6 206.4 227.3 240.2 290.2 165.8 182.7 165.4
Hanson and Hanson (1968) Calculated quantities Strip Model results w Mss r Mpred Ppred T/P kN/m 63.0 64.0 65.7 65.6 66.0 63.1 68.7
(kN·m) 7.02 7.22 6.86 11.03 7.19 6.57 9.30
(mm) 410 412 475 486 406 475 444
(kN·m) 20.1 20.9 28.8 29.7 17.2 21.9 21.1
(kN) 5.2 4.1 3.9 5.3 22.5 27.6 26.9
1.11 1.16 1.24 1.06 1.19 1.25 1.17
ACI code results Mpred Ppred T/P (kN·m) 14.9 15.3 27.3 28.4 13.2 21.1 23.8
(kN) 3.8 3.0 3.7 5.1 17.4 26.5 30.3
1.50 1.59 1.31 1.11 1.55 1.30 1.04
82.4 154 77.7 149 75.0 140 31.8 238 34.7 258 32.6 240 39.2 80.1 36.9 76.5 43.3 86.5 13.7 139 16.4 140 13.4 142 105 205 106 204 100 190 47.2 350 43.4 318 77.8 158 74.9 148 29.8 283 28.8 233 66.4 138 76.3 156 73.8 148 29.0 243 23.1 206 29.6 245 66.1 133 69.9 142 67.7 134 Average COV (%)
1.10 1.26 1.47 1.03 1.00 1.33 1.00 1.23 1.18 0.94 1.01 1.14 0.91 1.07 1.33 0.78 1.14 0.85 1.32 0.82 1.21 1.12 1.18 1.39 0.94 1.17 1.19 1.25 1.28 1.23 1.17 17.0
Hawkins, Bao, and Yamazaki (1989)
12
105.5 102.5 100.1 89.8 99.6 94.4 68.9 65.2 77.4 62.3 65.5 64.6 136.3 140.0 134.4 132.6 125.5 103.1 102.3 100.9 89.5 92.5 105.6 101.5 92.5 77.2 93.6 91.9 94.8 96.2 Average COV (%)
30.7 40.4 55.4 30.3 40.3 55.0 14.6 18.5 22.7 14.4 18.5 22.1 31.0 41.1 56.9 31.0 56.6 30.6 55.5 30.6 54.5 40.7 41.2 55.5 27.4 39.8 40.8 34.7 45.5 32.1
692 780 896 734 789 915 612 686 694 633 684 737 629 694 803 636 824 697 889 703 933 816 777 892 696 870 812 817 794 807
61.2 71.5 87.6 25.9 31.5 36.3 31.8 35.8 44.0 11.8 15.6 14.3 66.8 80.9 98.5 31.9 42.4 58.7 86.1 22.9 32.6 66.7 71.2 85.1 22.9 24.0 28.3 67.4 66.1 69.6
115 137 163 193 234 267 64.9 74.3 87.9 120 134 152 131 156 186 237 311 119 170 218 264 138 145 171 192 214 234 136 135 138
1.48 1.37 1.26 1.27 1.10 1.20 1.23 1.27 1.16 1.08 1.06 1.07 1.43 1.40 1.36 1.16 1.17 1.12 1.15 1.07 1.07 1.11 1.27 1.21 1.19 1.12 1.24 1.22 1.36 1.20 1.20 9.0
ACIfib International Symposium Punching shear of structural concrete slabs
Some comments on interior connections transferring shear and moment:
3.3
On average, ratios of test load to predicted load are about the same, with the Strip model producing an average of 1.20 and the ACI code producing an average of 1.17. The Strip Model provides safe predictions for all tests, with a minimum ratio of test load to predicted load (T/P) of 1.06. The corresponding minimum for the ACI code is 0.78. This is a large enough error on the unsafe side to be of some concern. The ACI code tends to overestimate the strength of more lightly reinforced specimens. All but one of the unsafe predictions are for specimens with 0.6% flexural reinforcement. The concept of nonproportional loading is easily adapted to design. As a design tool, the Strip Model becomes a guide for the lateral distribution of reinforcement.
Edge connections
For an edge columnslab connection with no edge beam, there are three radial strips that may develop: two spandrel strips and one interior strip. By considering how these strips are loaded, one can develop an interaction diagram for shear and moment transferred to the column; however, it will be seen that for many practical design cases, developing a full interaction diagram is unnecessary. An additional factor to consider at an edge connection, that was not a concern for an interior connection, is the close proximity of the free edge of the slab. This will require an assessment of the anchorage of reinforcement perpendicular to the free edge. Anchorage of reinforcement perpendicular to the free edge — with increasing eccentricity of load, the interior radial strip becomes a super strip. The negative reinforcement of this super strip will include bars perpendicular to the free edge but outside the column. Triangularshaped wings, as shown in Fig. 8, project 1.5h from either side of the column. Any force generated by bars within the wings contributes to the transfer of moment to the column by flexure; however, the proximity of the free edge limits the available length for anchorage of reinforcement perpendicular to the free edge.
1.5 x h
c2
1.5 x h
ldh
A
Figure 8:
B
C
D
Bar development at an edge columnslab connection.
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ACIfib International Symposium Punching shear of structural concrete slabs
It is assumed here that all top bars perpendicular to the free edge will be hooked. For most circumstances, the development length for a hook, ldh, is roughly half that for a straight bar. It follows that the mechanical anchorage provided by a hook develops half the strength of the bar. So, any hooked bar within 𝑐2 + 3ℎ can develop at minimum half of its strength. How much more it can develop depends upon its embedment within the column or the triangular wings. Bars perpendicular to the free edge but within the column will usually develop their full strength. They have maximum embedment and very favorable confinement within the column (This may not be the case for a roof slab). Bars outside of the column can still develop their full force if they have sufficient embedment beyond the diagonal line that defines the wing. Consider the reinforcement layout in Fig. 8. With this layout, a maximum of seven bars will contribute to the negative moment of an interior super strip. This is determined as follows:
The bars (2) within the column will develop fully. Bar A is expected to develop fully since its embedment past the diagonal line defining the wing is greater than ldh. The length of embedment of bar B beyond the diagonal is approximately 0.75 ldh. It will develop approximately 88% of its full strength. Mechanical anchorage of the hook accounts for half of the bar strength. Straight bond accounts for 75% of the remaining half of the bar strength. Bar C has embedment of approximately 0.25 ldh. It should develop approximately 62% of its strength, made up by 50% from the mechanical anchorage of the hook and 25% of the remaining 50% by straight bond. Bar D is outside the band and not expected to develop. Summing up: 2 wings × (1 + 0.88 + 0.62) = 5 bars anchored in the triangular wings plus 2 bars within the column for a total of 7 bars.
Figure 9:
14
Conceptual interaction diagram for edge column.
ACIfib International Symposium Punching shear of structural concrete slabs
Shearmoment interaction — Figure 9 shows a conceptual interaction diagram at an edge connection with moment transfer about an axis parallel to the free edge. Four points define the interaction: (1) Proportional Loading, (2) Interior Super Strip, (3) Spandrel Super Strip, and (4) Saddle Point. 1)
Proportional Loading (PL) Proportional loading occurs when each radial strip reaches its nominal capacity as defined by Equation (2) for the interior strip and Equation (4) for the spandrel strips. The total vertical load transferred to the column is simply the summation of the three nominal strip loads. The net moment about the centroid of the column is purely a function of the interior strip. 𝑀𝑐𝑜𝑙 = 𝑀𝑛𝑒𝑔 + 𝑃𝑠 ×
2)
𝑐1
(15)
2
Interior Super Strip (ISS) With increasing moment, reinforcement outside the column acts to reinforce an interior super strip. Equation (9) is used to estimate the total vertical load transferred to the column, with due consideration of the anchorage of bars perpendicular to the free edge. The moment about the centroidal axis of the column is: 𝑀𝑐𝑜𝑙 = 𝑀𝑠𝑠,𝑛𝑒𝑔 + 𝑃𝑠𝑠 ×
3)
𝑐1 2
(16)
Spandrel Super Strip (SSS) As the moment transferred to the column decreases, the spandrel strips must carry a larger fraction of the total load. Reinforcement within a band of 𝑐1 + 1.5ℎ is assumed effective. The load on each spandrel super strip is: 𝑃𝑠𝑠 = √2𝑀𝑠𝑠 𝑤
(17)
Because there is no interior radial strip, there is no net moment transferred to the column. The total load transferred to the column is: 𝑃𝑐𝑜𝑙 = 2 × 𝑃𝑠𝑠 + 𝑤 × 𝑐1 4)
(18)
Saddle Point (SP) Under extreme lateral load, there can be uplift in the spandrel concurrent with gravity loading on the interior strip. At the columnslab connection, the interior strip is in negative bending while the spandrel strips are in positive bending. The deformed shape is a saddle point. How much uplift can develop in the spandrel strip is debatable. Here it is assumed that the full capacity of the spandrel strip under proportional loading can be developed in uplift. This might be a little optimistic but great precision is not called for here; this loading combination is more a creation of laboratory testing than field experience. The moment transferred to the column is the same as for the case of the interior super strip. The total load transferred to the column is the difference between the gravity load carried by the interior super strip and the uplift load carried by two spandrel strips at nominal capacity.
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ACIfib International Symposium Punching shear of structural concrete slabs
Comparison to test results — Results from three separate investigations are examined here. Singlecolumn tests reported by Zaghlool (1971) and by Hanson and Hanson (1968) cover a wide range of shear and moment combinations, providing some experimental justification for the interaction shown in Fig. 9. The two connections reported in Afhami et al. (1998), from a fullscale, twopanel specimen with rotational restraint to model flexural continuity, are much more representative of the edge columnslab connections in a real structure. The test program by Zaghlool included four geometrically similar specimens, all with the layout of reinforcement shown in Fig. 10(a). The column was 267 mm (10.5 in) square. The slabs were 152 mm (6 in) thick with a flexural depth measured to the middle of the reinforcing mat of 121 mm (4.75 in). All reinforcing bars were US #4 (12.7 mm diameter). The average concrete strength was 33.9 MPa (4920 psi) and the average yield strength of the reinforcement was 465 MPa (67.4 ksi). The slabs were simply supported on three sides with prescribed combinations of vertical and lateral load applied through the column.
Figure 10: Top slab reinforcement for selected edge column tests.
Figure 11 shows test results compared to the calculated interaction diagram. Table 3 lists calculated values needed to generate the interaction. For all points on the interaction, the loading term, w (Eq. 8), is equal to 110 kN/m (7.54 kips/ft). Two of the interaction points are discussed in more detail below.
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ACIfib International Symposium Punching shear of structural concrete slabs
Vertical Load Transferred to Column (kN)
300 PL SSS
250
ISS
200 150
Single Point Interaction
ZV(1)
ZV(4)
100
V(V5) 50
ZV(6)
0 SP
50
0
20
40
60
80
100
Moment about Column Centroid (kN*m)
Figure 11: Interaction for tests by Zaghlool.
Table 3:
Shearmoment interaction for tests by Zaghlool (1971) and Hanson and Hanson (1968) Interior Strip
Point
Zaghlool
Nbar
Interaction Values
Spandrel Strip
As,neg
Ms or Mss
Ps or OnePss way
(mm²)
(kN*m)
(kN)
(kN)
Nbar
As
Ms or Mss
Ps or Pss
Oneway
Pcol
Mcol
(mm²)
(kN*m)
(kN)
(kN)
(kN)
(kN*m)

252.2
38.8
29.3 212.9
74.8
PL
4.00
506
24.9
104.5

4
506
24.9
73.9
ISS
8.87
1122
54.3
154.3





SSS




29.3
7
882
44.2
98.4

226.2
0.0
SP
8.87
1122
54.3
154.3

4*
506*
24.9*
73.9*

6.6
74.8
Hanson ISS 4.85 344 6.23 39.9 9.7 59.3 9.27 and SP 4.85 344 6.23 39.9 2* 142* 2.80* 18.9* 2.1 9.27 Hanson * For spandrel strips at saddle point of interaction, bottom mat reinforcement is engaged. Spandrel strip carries uplift (negative) load.
For the condition of Proportional Loading (PL), the interior strip and the two spandrel strips are each reinforced by four bars. As a result, 𝐴𝑠,𝑛𝑒𝑔 = 506 mm2 (0.784 in2) for both of these. Based on the average flexural depth of 121 mm (4.76 in), both interior and spandrel strips have the same flexural capacity, 𝑀𝑠 = 24.9 𝑘N ∙ m (18.4 ft·kips) Equation (2) predicts a capacity of 104.5 kN (23.5 kips) for the interior strip and 73.9 kN (16.6 kips) for each of the spandrel strips. This gives a total load of 252.2 kN (56.7 kips). Equation (16) gives a total moment transferred to the column of 38.8 kN ∙ m (28.6 ft·kips). With the development of an Interior Super Strip (ISS), reinforcement perpendicular to the free edge is anchored to the column. The hook development length for these bars is estimated at 153 mm (6 in). Following the procedure outlined above and in Fig. 8, it is estimated that 8.87 bars are effective (100% of the four bars through the column, 100% of bar A, 88% of bar 17
ACIfib International Symposium Punching shear of structural concrete slabs
B, and 55% of bar C). Bar D is considered too remote to be effective. Summing up the effective bars, the total area of reinforcement for the interior super strip is 1122 mm2 (1.74 in2) and the corresponding value of Mss is 54.3 kN ∙ m (40.0 ft·kips). Equation (13) gives a value of Pss equal to 154.3 kN (34.7 kips). The spandrel column faces each carry a oneshear of 29.3 kN (6.59 kips). It is worth noting that a simplified, square interaction defined by the ISS interaction point would appear to provide a reasonable approximation of the overall interaction diagram.
Vertical Load Transferred to Column (kN)
Hanson and Hanson (1968) report one edge columnslab specimen tested at high eccentricity. The concrete strength was 31.2 MPa (4525 psi). The slab was 76 mm (3 in) thick with an average flexural depth of 57 mm (2.25\ in). The column was 76 mm (3 in) square. Slab reinforcement is shown in Fig. 10(b). All bars were US #3 (9.5 mm diameter) with a yield strength of 366 MPa (53 ksi). The column extended above and below the slab, supported by the laboratory floor at the lower end and laterally braced at the upper end. A single line load was applied to the slab.
70 60
50 40
30 20 10
0 10 0
2
4
6
8
10
12
Moment about Centroid of Column (kN*m)
Figure 12: Interaction for test by Hanson and Hanson.
Figure 12 shows the test result plotted against a simplified interaction. The relevant part of the interaction is constructed by considering two points: the Interior Super Strip (ISS) and the Saddle Point (SP). Table 3 summarizes various calculated quantities. For both points, from Equation (8), 𝑤 = 63.8 kN/m (4.37 kips/ft). The hook development length is estimated to be 120 mm (4.7 in). Following the procedure outlined previously, the maximum reinforcement that can be developed perpendicular to the free edge is estimated at 4.85 bars (2 column bars @ 100%, 2 bars 38 mm (1.5 in) from column face @ 92.5%, and 2 bars 114 mm (4.5 in) from column face @ 50%). From Equation (10): 𝑀𝑠𝑠,𝑛𝑒𝑔 = 𝐴𝑠,𝑛𝑒𝑔 × 𝑓𝑦 × 𝑗𝑠𝑠 𝑑 = 346 mm2 × 365 MPa × 0.863 × 57 mm = 6.23 kN ∙ m (4.59 ft·kips). Equation (13) gives: 𝑃𝑠𝑠 = 2√𝑀𝑠𝑠 𝑤 = 2√6.23 × 63.8 = 39.9 kN (8.97 kips).
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ACIfib International Symposium Punching shear of structural concrete slabs
From Equation (17): 𝑀𝑐𝑜𝑙 = 𝑀𝑠𝑠,𝑛𝑒𝑔 + 𝑃𝑠𝑠 ×
𝑐1 2
= 6.23 + 39.9 ×
.152 𝑚 2
= 9.27 kN ∙ m (6.84 ft·kips).
Equation (9) applied to the edge column gives: 𝑃𝑐𝑜𝑙 = (∑ 𝑃𝑠𝑠 ) + 2 × 𝑤 × 𝑐1 = 39.9 + 2 × 63.8 ×× 0.152 = 59.3 kN (13.3 kips). At the saddle point of the interaction, the interior super strip is still active. An uplifting, proportionally loaded spandrel strip replaces the oneway shear on each spandrel face of the column. For each of these spandrel strips, Ms is estimated to be 2.80 kN ∙ m (2.07 ft·kips) and Ps to be an upward 18.9 kN (4.25 kips). The net moment transferred to the column is unchanged at 9.27 kN ∙ m (6.84 ft·kips). The net load transferred to the column is 39.9 kN − 2 × 18.9 kN = 2.1 kN (0.47 kips). Afhami et al. (1998) reports test results from a fullscale, twopanel slab and column structure encompassing two edge columnslab connections (north and south) and one interior columnslab connection and measuring approximately 10 m (32.8 ft) by 4.3 m (14.1 ft) in plan. The concrete strength and steel yield strength are reported at 34.8 MPa (5050 psi) and 420 MPa (60.9 ksi), respectively. The edge columns measured c1 = 254 mm (10 in) and c2 = 305 mm (12 in). Hydraulic jacks with loaddistributing frames simulated a uniformly distributed load (16 equal point loads in each panel). Rotational restraint was provided along the long edges (east and west) of the specimen. The edge connections were designed for different fractions of the panel moment, resulting in the two reinforcing layouts shown in Figs. 10(c) and 10(d). All bars parallel to the free edge are 15M (Abar = 200 mm² or 0.31 in2). The dashed lines perpendicular to the free edge represent unhooked 10M bars (Abar = 100 mm² or 0.155 in2), added for crack control but not expected to contribute to ultimate strength. For the south connection, the solid lines perpendicular to the free edge represent hooked 15M bars. For the north connection, the solid lines perpendicular to the free edge represent hooked 10M bars. Two results are reported for the north connection. Failure of the twopanel structure first occurred by punching at the interior columnslab connection. Just before this, there were visible signs of distress at the north connection indicating that it too was near failure. The load and moment carried at the north connection when the interior connection punched is reported as the first result. After punching of the interior connection, slab supports were adjusted so that the two edge columnslab connections could be tested to failure. These retests produce the result for the south connection and the second result for the north connection. Figure 13 shows simplified interaction diagrams for both the north and south columns. The interior superstrip is the only interaction point that need be considered to assess these tests. Calculated values are summarized in Table 4. The loading term, w, from Equation (8) is equal to 110 kN/m (7.54 kips/ft).
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ACIfib International Symposium Punching shear of structural concrete slabs
Vertical Load Transferred to Column (kN)
250
200
150 North Column
South Column
South North (1)
100
North (2) 50
0
0
20
40
60
80
100
Moment about Centroid of Column (kN*m)
Figure 13: Interaction for tests by Afhami et al. Table 4:
Shearmoment interaction for Tests by Afhami et al. (1998) Point
Interior Strip Nbar
North South
Interior SS Interior SS
6 6.9
As,neg (mm²) 600 1380
Mss,neg (kN*m) 27.3 57.1
Ms,pos (kN*m) 15.3 12.2
Pss (kN) 136 173
Spandrel Strip Oneway (kN) 27.5 27.5
Interaction Values Pcol Mcol (kN) (kN*m) 191 44.5 228 79.0
The hook development lengths for 15M and 10M bars are estimated to be 190 mm (7.48 in) and 134 mm (5.28 in), respectively. For the south connection, the total number of effective 15M top bars reinforcing the interior super strip is estimated to be 6.9 (four bars through the column, 87% of the first bar outside the column, 58% of the second bar outside the column). For the north connection, all six 10M hooked bars perpendicular to the free edge are considered fully effective.
10M bars
15M bars, hooked
10M bars
10M bars, hooked
1.6
0.6
1.4
0.5
yield
0.3 0.2
Strain (%)
Strain (%)
1.2
0.4
1.0 0.8 0.6
yield
0.4
0.1
0.2 0
0
(a) South Connection
(b) North Connection
Figure 14: Measured strains in slab top reinforcement at edge column.
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ACIfib International Symposium Punching shear of structural concrete slabs
Figure 14 shows measured strains in reinforcement perpendicular to the free edge for both the north and south connections. In both cases, the measured strains are in reasonable agreement with the predicted effectiveness of the bars. Because the slab was continuous, the flexural capacity of the Interior Super Strip (ISS) includes a contribution, Ms,pos, from bottom steel. The edge columns were designed to take different fractions of panel moment and the bottom reinforcing mats were adjusted accordingly. The values of Ms,pos for the north and south panels are 15.5 kN·m (11.4 ft·kips) and 12.2 kN·m (9.00 ft·kips), respectively. Some comments on edge columnslab connections:
4
The Strip Model analysis results shown in Figs. 11 through 13 are good agreement with test results. For combinations of negative moment (tension in top steel) and gravity load transferred to a column, there is no substantive interaction between moment and shear. The ISS point on the interaction diagram is all that need be considered. This is likely not true for load cases that involve reversals of either load or moment transfer. An important design issue at the edge columnslab connection is the anchorage of reinforcement so that it will be effective in transferring moment to the column by flexure. Placing reinforcement within a band c2 + 3h is by itself not sufficient to ensure full development. Reinforcing for any load case that increases the demand for transfer of negative moment to the column will automatically increase the shear capacity of the connection.
Conclusion
The Strip Model describes a load path for the transfer of vertical load from a slab to a column. The various mechanisms of load transfer assumed to be acting within the load path are consistent with observations of tests. The model fully explains twoway behavior using capacities that can be determined from tests of oneway members. The load path described in the Strip Model is easily adapted to a variety of test configurations. The calculated capacity of the connection using the Strip Model agrees well with test results for a variety of loading conditions.
5
References
ACI Committee 318 (2014) Building Code Requirements for Reinforced Concrete (ACI 31814) and Commentary (ACI 318R14), American Concrete Association, Farmington Hills, MI, 2014, 519 pp. Afhami, S., Alexander, S. D. B. and Simmonds, S. H. (1998) Strip Model for Capacity of Slabcolumn Connections, Structural Engineering Report No. 223, Department of Civil Engineering, University of Alberta, Edmonton, 231 pp. Alexander, S. D. B. (1999) Strip Design for Punching Shear, ACI SP183, American Concrete Association, Detroit, pp 161179.
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Alexander, S. D. B. and Simmonds, S. H. (1992) "Bond Model for Concentric Punching Shear," ACI Structural Journal, Vol. 89, No. 3, pp 325334. Alexander, S.D.B. and Simmonds, S.H. (2003) “Moment transfer at interior slabcolumn connections.” ACI Structural Journal, Vol. 100, No. 2, pp. 197202. Alexander, S.D.B., Xilin Lu, and Simmonds, S.H. (1995) "Mechanism of Shear Transfer in a ColumnSlab Connection," 1995 Annual Conference of the Canadian Society for Civil Engineering, Proceedings, Ottawa, Ontario, Vol. II, pp. 207216. Fujita, M., Sato, R., Matsumoto, K., and Takaki, Y. (2002) "Size effect on shear strength of RC beams Using HSC without Shear Reinforcement," Translation from Proceedings of JSCE, Vol. 711/V56, August 2002, pp. 113128. Hanson, N.W. and Hanson, J.M. (1968) "Shear and Moment Transfer Between Concrete Slabs and Columns," Journal of the PCA Research and Development Laboratories, Vol. 10, No. 1 (January), pp. 216. Hawkins, N.M., Bao, A., and Yamazaki, J. (1989) "Moment Transfer from Concrete Slabs to Columns," ACI Structural Journal, Vol. 86, No. 63, pp 705716. Hillerborg, A. (1975) Strip Method of Design, Viewpoint Publications, Cement and Concrete Association, Wexham Springs, Slough, England. Ospina, C.E., Birkle, G., Widianto, Wang, Y., Fernando, S.R., Fernando, S., Catlin, A.C. and Pujol, S. (2011) “ACI 445 Collected Punching Shear Databank,” http://nees.org/resources/3660. Zaghlool, E.R.F. (1971) "Strength and Behaviour of Corner and Edge ColumnSlab Connections in Reinforced Concrete Flat Plates," Ph.D. Thesis, Department of Civil Engineering, University of Calgary, Calgary, Alberta.
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ACIfib International Symposium Punching shear of structural concrete slabs
Size effect on punching shear strength of RC slabs with and without shear reinforcement Zdeněk P. Bažant1 and Abdullah Dönmez2 1
: Northwestern University, Evanston, Ill., USA
2
: Istanbul Technical University, Turkey
Abstract Comparison of various design codes reveals major differences among the design provisions for punching shear, especially with respect to the size effect. This indicates the need for deeper analysis of the existing test data supplemented by realistic finite element (FE) analysis. This study presents a refined statistical analysis of the ACI445 database comprising 440 punching shear tests, and an FE analysis based on concrete microplane model M7 calibrated by test data. Computer filtering of the database is used to create data subsets in which the averages of secondary variables, such as the steel ratio and shape parameters in subsequent size intervals, are almost constant. The resulting trend of the mean punching shear strength vc clearly reveals that the slope of the diagram of log vc versus log d is milder, but not much milder, than 1/2, and that the trend does not disagree with the theoretically well justified energetic size effect law (endorsed for shear failures by ACI Committee 446). A new design equation with a size effect factor, emulating previous equations in many respects, is proposed. The equation is verified and calibrated by nonlinear leastsquare multivariate regression of the database, with weights compensating for the crowding and scarcity of data in various parts of the range. The size effect and other trends are also verified by finite element fitting of selected data series with a broader range. The size effect factor validated here can be applied to improve any design equation missing the size effect, including a plastic limit analysis equation, provided it fits the smallscale test data well (for which the size effect factor is defined as 1).
Keywords Design codes, failure, finite element analysis, optimum data fitting, punching shear, quasibrittle fracture, reinforced concrete, size effect, statistical analysis.
1
Introduction
Although concrete is not a plastic material, the shear strength of beams and slabs has traditionally been analyzed according to plastic limit analysis. In this analysis, as well as in elasticity with a strength limit, the nominal strength, vc, of geometrically similar structures, is independent of structure size d. This is the case of no size effect. But if the structural failure is due to fracture or localization of cracking damage, which is typical of concrete, vc decreases with d, which is called the size effect. The size effect is of two kinds:
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ACIfib International Symposium Punching shear of structural concrete slabs
1) statistical, due to material strength randomness, which occurs when a macrocrack initiates dynamically from one of many possible places of random strength, as described by the weakest link model and Weibull theory; and 2) energetic (i.e., deterministic), which occurs when a large crack grows in a stable manner prior to reaching the maximum load (in which case the material randomness affects only the coefficient of variation of vc and the size effect is due to increase of energy release rate with structure size). The shear failures of reinforced concrete are generally of Type 2, which is the type occurring in structures with a deep notch or a deep stressfree crack formed stably before reaching the maximum load (while the Type I size effect is that which occurs in structures that fail right at crack initiation from a smooth surface) (Hoover and Bažant, 2014). The plastic limit analysis, which underlies the concrete design codes, gives realistic results for relatively small structures used in most laboratory testing. The reason is that the size l0 of the fracture process zone is in concrete very large—about 0.5 m (compared to about 1 µm for metals or fine grained ceramics). This implies that, at maximum load, the distributed cracking cannot localize into one dominant crack prior to maximum load, and thus the size effect must be negligible for small structures. The size effect becomes strong only when the structure is sufficiently larger than l0. Consequently, extending the current code provisions to large sizes is relatively easy—it suffices to multiply the vc value according to the current code, based on limit analysis, by the proper size effect factor, which is about 1 for small structure sizes. Here it is proposed how to do it for punching shear. The ACI445C database, refined for punch failures, is used in this study; see Fig. 1ad (Ospina et al., 2011). The figure shows the nominal punching shear strength 𝑣" =
$% &' (
(1)
normalized by the mean concrete strength fc as a function of the slab depth d, in comparison .// with the curves of log 𝑣" / 𝑓" (for Eurocode log 𝑣" /𝑓" ) versus log 𝑑 according to four design codes. Note that the strength in Eq. (1) is not the design strength, since the material safety factors are not included. The database for the punch failures, repeated in all four diagrams in Fig. 1, contains the results of 440 tests reported in 60 experimental studies conducted in many laboratories throughout the world (Ospina et al., 2011). The slabs had square, rectangular, circular or octagonal simply supported boundaries. The depths, d, in the database range from 30 to 668 mm (1.18 to 26.3 in.); the mean concrete strength fc from 8 to 118 MPa (1160 to 17114 psi), and the longitudinal reinforcement ratio ρ from 0.1% to 7.3%. The data points in Fig. 1 show very high scatter. But the scatter is due not only to inevitable randomness but also, and largely, to inevitable sampling bias. The objectives of this study are to: 1) filter the inevitable sampling bias from the existing experimental database (Ospina et al., 2011) to reveal the basic size effect trend; 2) propose an improved equation for vc that includes the size effect in order to avoid excessive or insufficient safety factors (Bažant and Yu, 2006); and 3) calibrate and validate the size effect by weighted multivariate regression.
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ACIfib International Symposium Punching shear of structural concrete slabs
Figure 1:
2
Normalized strength comparisons among the provisions of a) ACI (x12 for psi), b) Eurocode (x5.25 for psi), c) fib Model Code (x12 for psi), and d) proposed (x12 for psi).
Punching shear in current design codes
For the punching load capacity, Vc, due to concrete, the current standard ACI318 specifies the formulas (ACI, 2011): V2 = λb5 d f2 V2 = λb5 d f2 + A:; f
for no shear reinforcement
(2)
with shear reinforcement
(3)
where fc is in MPa for SI (or in psi for USC units); b0 = control perimeter of loading area (or column); Asw = cross sectional area of one shear reinforcement layer around the column; fyw = yield strength of shear reinforcement; λ = 1/3 for SI units (4 for USC units) if there is no shear reinforcement, 1/4 if the shear reinforcement is provided by studs (3 for USC units), and 1/6 by stirrups (2 for USC units); sw = distance between shear reinforcement layers. The Eurocode 2 (2004) specifies the formula: 𝑉" = 𝜆𝜉(100𝜌𝑓" ).// 𝑏5 𝑑 𝑉" = 0.75𝜆𝜉(100𝜌𝑓" ).// 𝑏5 𝑑 + 𝐴LM 𝑓NM,P 1.5
( LQ
for no shear reinforcement
(4)
with shear reinforcement
(5)
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ACIfib International Symposium Punching shear of structural concrete slabs
𝜉 =1+
R55
(6)
(
where ρ = flexural reinforcement ratio, and λ = 0.18 (or 5 for USC units), which must be considered to have the dimension of (MPa)2/3 if fc is in MPa (note in Eq. 5 that, in case of shear reinforcement, λ is multiplied by 0.75). The fib Model Code 2010 (“MC2010”; fib [2013]) specifies the formula: 𝑉" = 𝑘T 𝑓" 𝑏5 𝑑 𝑉" = 𝑘T 𝑓" 𝑏5 𝑑 +
𝐴LM 𝜎LM
for no shear reinforcement
(7)
with shear reinforcement
(8)
where 𝜎LM = stress in shear reinforcement and 𝑘T = 𝑚𝑖𝑛 0.6,
. ..[\5.]^_` T(
(9)
where kψ must be considered to have the dimension of MPa if fc is in MPa; dg= maximum aggregate size in mm, and kdg= 32/(dg + 16 mm); ψ is the slab rotation defined in CSCT (Muttoni and Schwartz, 1991) for different cases; for the most practical case ψ = 1.5(rs/d)(fyd/Es) in rs = distance to zero radial moment and can be taken as 0.22 times the slab span. It should also be stated that there are 4 levels of approximations to calculate the slab rotation, ψ, and the simplest one, which is the LoA I, is used in here. The accuracy of the MC2010 provision depends, of course, on the accuracy of the rotation calculations. These three code formulas are compared (for the case of no shear reinforcement) in Fig. 1ac with the raw (unfiltered) ACI445C database, in which the measured values of vc divided by 𝑓" for ACI and MC2010, and by 𝑓" .// for Eurocode, are plotted versus d in logarithmic scales. For ACI318, the formulas give a horizontal line, i.e., there is no size effect. For Eurocode 2004, the size effect term ξ in Eq. (6) approaches a horizontal asymptotic slope for d→∞. For MC2010, there is size effect given by factor kψ in Eq. (7), which has, for d→∞, an asymptote proportional to d1, i.e., an asymptote of slope −1 in a loglog plot. Fig. 1d shows the size effect proposed here, discussed later. In plotting these size effect curves, all the parameters other than d had to be fixed, and they were fixed at values equal to the average values in the database.
3
Filtering of database to remove bias in secondary variables
Consider that the size range is subdivided into several intervals of constant width in log d (ﬁve in Fig. 2). The averages of steel ratio ρ and ratios b/d and c/b calculated separately for each size interval, should ideally be about the same for all the size intervals (Bažant and Yu, 2008). But, as seen in the rows III to V of column 1 (c.1) in Fig. 2, the averages of ρ, b/d, c/b, vary signiﬁcantly through the intervals; ρ decreases with increasing size even by an order of magnitude and b/d decreases more than 3 times. Also note in Fig. 2a, row II, that the interval averages for the entire database (shown solid diamond points) are highly scatter and do not show a clear trend.
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ACIfib International Symposium Punching shear of structural concrete slabs
Figure 2:
Column 1) entire database; columns 2–6) filtered database and mean values of longitudinal reinforcement ratio (ρ), aspect ratio (b/d), shape factor (c/b) and their variations inside the size intervals; (vc= P/(b0d 𝑓" ) vs. d plots are in log scale; P = maximum punching load; b= perimeter of the column; b0= control perimeter of the column according to ACI; d= slab effective thickness; fc= concrete compressive strength (for cylindrical specimens); c= side length (or diameter) of the column) (figure continues on next page).
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ACIfib International Symposium Punching shear of structural concrete slabs
Fig. 2 (cont.): Column 1) entire database; columns 2–6) filtered database and mean values of longitudinal reinforcement ratio (ρ), aspect ratio (b/d), shape factor (c/b) and their variations inside the size intervals; (vc= P/(b0d 𝑓" ) vs d plots are in log scale; P = maximum punching load; b= perimeter of the column; b0= control perimeter of the column according to ACI; d= slab effective thickness; fc= concrete compressive strength (for cylindrical specimens); c= side length (or diameter) of the column).
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ACIfib International Symposium Punching shear of structural concrete slabs
In previous work (Bažant and Yu, 2008), a ﬁltering program was developed to delete gradually, in an unbiased way (without human intervention), the extreme points in each size interval, so as to achieve a nearly uniform variation of the secondary parameters throughout the size intervals. The resulting interval averages, obtained with this procedure, are shown, by the solid diamond points in rows III–V of c.25. As seen from the diamond points in c.25 of row II, a clear size eﬀect trend has emerged. The ﬁltering lead to clear trends of size eﬀect on vc/ 𝑓" . As seen, the filtering (with objective 1) leads to a clear size effect trend (and so would the filtering with objective 2). The trend invalidates the size effect in Eurocode, as well as the size effect in MC2010. There is no hint of a slope steeper than −1/2, and certainly not a slope approaching −1, as implied by MC2010. But there are also some drawbacks. The main one is that the ﬁltered database may have lost too many points from the original database. This could be remedied in future studies by creating many database subsets with diﬀerent levels of secondary parameters. Another is that most of the slab sizes in the ﬁltered databases are below the sizes in practical use. These drawbacks are avoided by multivariate regression analysis, which is discussed next.
4
Proposed shear strength equation
Based on the previous experience reviewed in Ospina et al. (2011) and on the energetic size effect law, which was shown to be generally applicable to quasibrittle failures and was endorsed by ACI Committee 446 (Bažant et al., 2007), the following equation is proposed: 𝑉e = 𝑉" + 𝑉L , 𝑉" = 𝑏5 𝑑𝑣" 𝑣" = 𝑣5 𝜃, 𝑣5 = 𝜆 𝑓" (100𝜌)5./ 𝜃=
.
( 5.R
" 5.g
&
&
(11) (12)
.\(/('
𝑉L = 𝜆L 𝐴LM 𝑓NM
(10)
(
" 5./
LQ
&
(13)
Here v0 = value of vc for vanishing size d; θ = size effect factor (Bažant [1984], Bažant and Planas [1997]) unanimously endorsed for shear failures by ACI Committee 446 (Bažant et al., 2007); d0 = transitional size (empirical, 60 mm according to database regression); Vu = ultimate punching capacity (or maximum load); Vc, Vs = ultimate punching capacities due to concrete and to shear reinforcement, respectively; ρ = flexural reinforcement ratio calculated for twoway slabs as the geometric average of reinforcement ratios for each direction, i.e., 𝜌 = 𝜌h 𝜌N ; Asw = cross sectional area of one shear reinforcement layer around the column; sw = spacing between shear reinforcement layers around the column, measured in radial direction; λ = empirical constant = 2.0 MPa in SI (or 24.1 psi in USC units); λs = empirical dimensionless constant = 0.9 for both in SI and American units. The exponent values 0.2, 0.3 and 0.4 of (d/b) and (c/b) were obtained by leastsquare optimization of the fit of entire database, as described later.
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ACIfib International Symposium Punching shear of structural concrete slabs
5
Verification and calibration by weighted multivariate regression for slabs without shear reinforcement
The maximum load in punching shear is reached only after stable growth of a large crack, and the location of the crack tip line at maximum load is determined by mechanics. The volume of concrete where mechanics permit the crack tip to be located is much too small compared to the volume of structure. Thus, although the material strength and fracture energy are random, the possible crack tip locations are so restricted by mechanics that a large sizedependent volume fraction of the structure cannot be sampled by the structure (Bažant, 2002). Therefore, the size effect cannot be statistical. Rather, it must be energetic, i.e., deterministic, and must follow the size effect law of quasibrittle fracture mechanics (Bažant [1984], Bažant and Planas [1997], Bažant [2002]) as expressed by the size effect factor θ in Eq. (12). The analytical form of this factor is here supported by the fits of mean trends of the filtered subsets of database; see the solid curves Fig. 2, row II, c.26. By contrast, the size effects in Eurocode 2004 and in MC2010, seen in Fig. 1bc, would not fit the interval means in Fig. 2 row II, c.16, at all. Besides, the finite asymptotic value of the Eurocode size effect (Fig. 1b) is theoretically impossible, according to fracture mechanics. So is the asymptotic slope of 1 in Fig. 1c in the size effect of MC2010 (which is shared with the previous Swiss design code). This slope even violates thermodynamic restrictions. The results of the weighted regression of the entire database are shown in Fig. 3a, in which the database points represent the values of normalized shear strength vc divided by v0, where vc are the measured punching strength values and v0 are the values from the optimized fit by Eq. (11). This is another way to eliminate the effect of variation of secondary variables with d, but, of course, it meets success only if a realistic form of the formula, with a low coefficient of variation of errors, is found. Fig. 3b (in which the solid diamond points show the interval averages) shows that the ACI446 size effect factor θ (solid curve) matches the data trend well (Bažant et al., 2007). The transition to asymptotic slope 1/2 was for shear failures experimentally verified also by Ruiz et al. (2015), and was supported by a different type of analysis.
Figure. 3:
30
a) Illustration of the size effect fit on entire database, normalized by the secondary variables found by multivariate regression, b) sizeeffect fit for the normalized mean strengths of subdivided data into size intervals.
ACIfib International Symposium Punching shear of structural concrete slabs
A comment should be made on the invisible capacity reduction (or ‘understrength’) factors which are embedded in the design equations of the design codes, to provide additional (covert) safety. These factors scale down the design equation from the mean fit of data to the lower margin of the data cloud (Bažant and Yu, 2006). To conform to this practice, the value of vc given by Eq. (10) would have to be scaled down, in this case by a factor of about 2.1 (however, it should be noted these hidden safety factors create a problem for probabilistic structural analysis; if the mean of the test data is not known to the analyst, the probabilistic analysis is meaningless).
6
Microplane FE analysis of test series wıth different slab sizes
There nevertheless exist a few punching test series where the size was varied significantly (Fig. 4ad). The data from each of these test series have been optimally fitted by a finite element (FE) program with a realistic damage constitutive law. In this program, the constitutive law was the microplane model M7 (Bažant and Caner, 2013) (which is the latest and most realistic in a series of microplane models for concrete developed at Northwestern University). Model M7 is robust, always convergent, and has been successfully used in dynamic problems with > 3·107 unknowns. Here all the simulations are conducted with commercial software ABAQUS, in which model M7 is introduced in VUMAT as a user’s material subroutine. To avoid spurious mesh sensitivity, the crack band model is used as the localization limiter (Bažant and Oh, 1983). To avoid some loss in accuracy due to scaling of the post peak (Červenka et al., 2005), a constant element size, hc (approximately equal to double the maximum aggregate size) is used in all the present computations. Linear hexahedral or tetrahedral elements are chosen. The adjustable parameters of M7 sufficed to match the main material properties. After calibration by fitting of the available punching test data of each test series (Bažant and Cao [1987], Regan [1986], Guandalini et al. [2009], Li [2000]), model M7 was used to calculate the values of vc/v0 for various sizes. Compared to the highly scattered individual test data shown by circles in Fig. 4, the calibrated FE results, shown by the xpoints in the figure, have the advantage of no scatter. Overall, Fig. 4ad demonstrate a significant transitional size effect on the punching shear strength, agreeing with the size effect law and with finite element simulations based on microplane model M7. Although the shear reinforcement of slabs does not enhance ductility, it changes in the cracking pattern and significantly elevates the load and deformation capacity because it intersects and suppresses the conical fracture produced by the punching load from the column. There is a variety of transverse reinforcement types, such as stirrups, studs and assorted profiles, and a variety of reinforcement layouts around the column. This increases uncertainties in predicting the punching load capacity. Like in shear of beams, the contribution of concrete and the contribution of steel reinforcement, calculated under the assumption of yielding, are considered to be additive (Elgabry and Ghali, 1990) (Eq. 3). However, it is not guaranteed, and is in fact doubtful for thick slabs, that the steel would actually yield at maximum load rather than later in post peak softening. Moreover, the failure of shear reinforcement can take different forms, such as the punch failure outside the shear reinforced area, delamination, and concrete crushing. Investigation of these diverse forms and their effects on the size effect curve is beyond the scope of this study.
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ACIfib International Symposium Punching shear of structural concrete slabs
Figure 4:
Size effect analysis of tests by a) Bažant and Cao (1987), b) Regan (1986), c) Guandalini et al. (2009) and created specimen with 1.45% rein. ratio for the size effect convenience, and d) FEA of calibrated test data from Li (2000), and a sample of corresponding fracture patterns for every sets obtained from FEA (left bottoms).
Unfortunately, there are no tests clearly demonstrating the size effect for slabs with shear reinforcement. Lips et al. (2012) and Birkle (2004) tested slabs with and without shear reinforcement but the size range was too narrow compared to inevitable scatter. To reveal the size effect, the shear reinforcement ratio, which is defined as ρs = Asw/bsw, should be constant, but in these and other tests, it varied with d, and so did the aspect ratio and even the concrete strength (here Asw = shear reinforcement area in the first layer from the column face, b = column perimeter, and sw = radial distance between the shear reinforcement layers). Figure 5ab shows the predicted size effect simulations calibrated from the tests by Birkle (2004) and Lips et al. (2012) for the slabs with and without shear reinforcement (solid and empty cross points in the Figs). As shown in Fig. 5, the effect of studs is to shift the size effect curve upwards. Note also that the studs, with the ACItype layout (Fig. 5), greatly increase the punching capacity.
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ACIfib International Symposium Punching shear of structural concrete slabs
Figure 5:
7
a) Size effect regression curves of FE models (with and without studs) calibrated by tests of Birkle (2004), b) Size effect regression curves of FE models (with and without studs) calibrated by tests of Lips et al. (2012).
Conclusions and key observations
1) The data trend revealed by filtering confirms that there is a significant size effect on the nominal punching shear strength vc of concrete. This trend reveals that: a) there is no indication of a sudden slope change on the size effect curve (unlike Eurocode 2004); b) the average slope of the data trend of log vc vs. log d is milder, though not much milder, than −1/2; c) there is no hint of a slope approaching −1, not even steeper than −1/2; and d) the data are compatible with the energetic size effect law. 2) The leastsquare multivariate regression of the ACI445C database with weights countering uneven data distribution indicates that the proposed formula with the ACI446 size effect factor gives a relatively good fit. 3) Including the effects of geometry (d/b, c/b) and of the longitudinal and shear reinforcement ratios in the test data fitting broadened the support of the analytical model. Nevertheless, the extensive previous justifications of the size effect factor θ, obtained and validated here, lend further credence to the model.
8
References
ACI 318 (2011) Building Code Requirements for Structural Concrete (ACI 31811) and Commentary, ACI Committee 318, Detroit, USA, 503. Bažant, Z. P. (1984) “Size Effect in Blunt Fracture: Concrete, Rock, Metal,” Journal of Engineering Mechanics, ASCE, V. 110, No. 4, pp. 518–535. Bažant, Z.P. (2002) Scaling of Structural Strength, 2nd ed. Elsevier, London 2005. Bažant, Z. P., and Caner, F. C. (2013) “Microplane Model M7 for Plain Concrete: I. Formulation,” Journal of Engineering Mechanics, ASCE, V. 139, No. 12, pp. 1714–1723. Bažant, Z.P., and Cao, Z. (1987) “Size Effect in Punching Shear Failure of Slabs,” ACI Structural Journal, V. 84, No. 1, 1987, pp. 44–53. 33
ACIfib International Symposium Punching shear of structural concrete slabs
Bažant, Z. P., and Oh, B. H. (1983) “Crack Band Theory for Fracture of Concrete,” Materials and Structures (RILEM, Paris), V. 16, No. 3, pp. 155–177. Bažant, Z. P., and Planas, J. (1997) Fracture and Size Effect in Concrete and Other Quasibrittle Materials, CRC Press, V. 16. Bažant, Z. P., and Yu, Q. (2006) “Reliability, Brittleness and Fringe Formulas in Concrete Design Codes,” Journal of Structural Engineering, ASCE, V. 132, No. 1, pp. 3–12. Bažant, Z. P., and Yu, Q. (2008) “Minimizing Statistical Bias to Identify Size Effect From Beam Shear Database,” ACI Structural Journal, V. 105, No. 6, pp. 685–691. Bažant, Z. P., Yu, Q., Gerstle, W., Hanson, J., and Ju, J. W. (2007) “Justification of ACI446 Proposal for Updating ACI Code Provisions for Shear Design of Reinforced Concrete Beams,” ACI Structural Journal, V. 104, No. 5, pp. 601–610. Birkle, G. (2004) “Punching of Fat Slabs: The Influence of Slab Thickness and Stud Layouts,” PhD Dissertation, Faculty of Graduate Studies, University of Calgary, Alberta. Červenka, J., Bažant, Z. P., and Wierer, M. (2005) “Equivalent Localization Element for Crack Band Approach to Meshsensitivity in Microplane Model,” International Journal for Numerical Methods in Engineering, V. 62, No. 5, pp. 700–726. Elgabry, A. A., and Ghali, A. (1990) “Design of Studshear Reinforcement for Slabs”, ACI Structural Journal, V. 87, No. 3, pp. 350–361. Eurocode 2 (2004) Design of Concrete Structures Part 1: General rules and rules for building. European Committee for Standardization, Brussels, 241. Guandalini, S., Burdet, O., and Muttoni, A. (2009) “Punching Tests of Slabs with Low Reinforcement Ratios,” ACI Structural Journal, V. 106, No. 1, pp. 87–95. Hoover, C. G., and Bažant, Z. P. (2014) “Universal SizeShape Effect Law Based on Comprehensive Concrete Fracture Tests,” Journal of Engineering Mechanics, ASCE, V. 140, No. 3, pp. 473479. Li, K. K. L. (2000) “Influence of Size on Punching Shear Strength of Concrete Slabs,” M.Eng Dissertation, Department of Civil Engineering and Applied Mechanics, McGill University, Montréal. Lips, S., Ruiz, M. F., Muttoni, A. (2012) “Experimental Investigation on Punching Strength and Deformation Capacity of ShearReinforced Slabs,” ACI Structural Journal, V. 109, No. 6, pp. 889–900. fib (Fédération internationale du béton) (2013) fib Model Code for Concrete Structures 2010, Ernst & Sohn, Berlin. Muttoni, A., Schwartz, J. (1991) “Behaviour of Beams and Punching in Slabs without Shear Reinforcement,” IABSE Colloquium, V. 62, No. EPFLCONF111612, pp. 703–708. Ospina, C. E., Birkle, G., and Widianto, W. (2011) “ACI 445 Punching Shear Collected Databank,” Network for Earthquake Engineering Simulation (database) Dataset, DOI:10.4231 D3TX35618. Regan, P. E. (1986) “Symmetric Punching of Reinforced Concrete Slabs,” Magazine of Concrete Research, V. 38, No. 136, pp. 115–128. Ruiz, M. F., Muttoni, A., and Sagaseta, J. (2015) “Shear Strength of Concrete Members Without Transverse Reinforcement: A Mechanical Approach to Consistently Account for Size and Strain Effects,” Engineering Structures, V. 99, pp. 360–372.
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ACIfib International Symposium Punching shear of structural concrete slabs
Behavior and performance levels of reinforced concrete slabcolumn connections Marvin E. Criswell1, Carlos E. Ospina2, Neil M. Hawkins3 1
: Colorado State University, Fort Collins, Co., USA
2
: BergerABAM Inc., Houston, Tex., USA
3
: University of Illinois at UrbanaChampaign, Ill., USA
Abstract Information is summarized on the behavior, evaluation and design of slabcolumn connections and flat plate/flat slab systems for three levels of structural performance: strength, strength combined with ductility, and strength combined with ductility and robustness.
Keywords Concrete design codes, disproportionate collapse, ductility, flat plates, flat slabs, performance levels, punching shear, robustness, slabcolumn connections
1
Introduction
A primary structural design requirement for twoway flat plate and flat slab reinforced concrete systems is to assure satisfactory performance in shear of the slab regions adjacent to and surrounding the columns. In the socalled punching shear failure mode the column punches through the slab in the vicinity of the column perimeter. The failure surface is an inverted truncated pyramidal to conical shape defined by an inclined crack extending upward from the intersection of the column perimeter and the bottom of the slab. Punching shear requirements, along with control of flexural deflections, very often set the minimum adequate thickness of the slab, and have a large impact on the economics of the slab system. Although the punching failure issue may appear to be simple, the factors affecting the strength and performance of a slabcolumn connection are many. The punching failure mode is strongly related to the flexural behavior of the slab, especially in the region near the column. Although several models have been proposed to approximate observed behaviors, most research studies of the punching shear problem have been based on experimental tests and empirical fits to those results. Unlike the situation for a beam in shear, it is difficult to define the punching shear problem as involving a welldescribed isolated element, as it often governs the strength and ductility of the entire slab system of which it is a part, even when current design provisions are satisfied. Local failures around slabcolumn connections can also be the initiating event for excessive disproportionate damage and progressive collapse.
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ACIfib International Symposium Punching shear of structural concrete slabs
Designers can have different expectations for the performance of slabcolumn connections, depending upon the functions of the slab, the loading conditions, and the evolution of design expectations. Consistent with general trends in structural design, initial design requirements were concerned with strength only. The development of seismic design was among the factors leading to increased design attention to ductility as well as strength. The ability of a structure to survive local failures or distress without experiencing disproportionate damage is now receiving increased attentions, especially for critical and iconic structures. The purpose of this paper is to examine the primary behavior of a reinforced concrete slab and the associated slabcolumn connection regions for the three primary performance levels, along with an overview of the research work and design provisions for each of these performance levels.
1.1
Performance levels for slabcolumn connection regions
The performance levels are: (1) strength, (2) strength with ductility, and (3) robustness. Over time, these performance levels have generally been addressed in design provisions in this same order. The design objective for the strength performance level is that significant flexural distress should precede any punching event. The design is to be controlled by flexure rather than shear. This was the performance expectation first adopted in slabcolumn research and the formulation of design provisions. Test results showed that flexural and shear response are related, and that the separation of failure modes into either shear or flexure is simplistic. A punching failure usually occurs even if the slab system first experiences significant distress in flexure. The performance of the slabcolumn connection can limit the attainment of the full flexural strength of the slab system. Ductility is desired in structural systems for several reasons, including as a visual warning of distress. Ductility has become of greater significance with the advent of seismic design procedures and with our understanding of the need for larger deformations to permit redistribution of moments and attainment of the full flexural capacity of a slab. Robustness (resiliency) is the ability of a structure to survive after the occurrence of a traditional failure at one or more critical locations. At the robustness performance level, this initial failure, which might be due to an excess large local loading (including explosive loadings), a construction error, or other shortcoming, should not lead to disproportionate damage. The residual failed area and the region around it should be able to redistribute the forces involved so that progressive collapse does not occur.
1.2
ACI and ASCE technical committees addressing slab shear and slabcolumn connections
The impact that slabcolumn connection regions can have on the structural performance of a slab system has resulted in several ACI technical committees addressing different portions of the overall punching shear phenomenon, sometimes without much coordination. This has now resulted in some duplication of effort, some conflicting recommendations (Ghali et al., 2015; ACIASCE Committees 352 [2011], 421[2010], and 426 [1974]), and an overall complicated evolution of committee jurisdiction for the topic of shear in slabs.
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ACIfib International Symposium Punching shear of structural concrete slabs
Historically, the ACI design provisions for punching shear originated, along with provisions for beam shear, with the technical committee that addresses the shear performance of elements and which is now designated as ACIASCE Committee 445, Shear and Torsion (previously ACIASCE 326 and 426). The basic twoway slab shear strength provisions in the ACI 31814 Code largely reflect the strength performance level. The design provisions are intended to assure that punching shear is not the initial mode of distress for a twoway slab system. Ductility, shear reinforcement, and the interaction of flexural and shear modes are among the topics also addressed by ACIASCE 445. It has long been noted that for gravity loads, tests of slab systems and multipanel slab models have shown that punching often occurs soon after the computed flexural strength has been reached and some yielding or flexural distress has been observed. A question increasingly being asked is how much slab deflection and slabcolumn connection ductility is needed before a “socalled” secondary failure, a flexuredriven punching failure, limits the slab performance. The performance of slabcolumn connections can be considered a logical extension of the extensive work on beamcolumn connections carried out by ACI/ASCE Committee 352, Joints and Connections in Monolithic Concrete. This committee’s work on beamcolumn connections addresses joint ductility and seismic design concerns, with emphasis on the confinement reinforcement and anchorage of flexural reinforcement in the connection core, the volume common to the beam and the column. Beyond the column face, the Committee 352 provisions are heavily dependent upon the beam shear provisions. The primary performance level of interest in the Committee 352 work is that of ductility (together with strength). The portions of the connection area addressed by the Committee 352 work for slabs differs substantially from that for beamcolumns in that most of the connection area of interest is outside of the connection core, the volume common to the slab and the column. Because of its confinement by the slab adjacent to the column, the punching shear mode is effectively a connection, rather than a member, phenomenon. As ductility became a performance level more commonly expected, and a better understanding of the behavior of shear reinforcement in slabs was developed, ASCEACI Committee 421, Design of TwoWay Slabs, also became involved in shear performance and the use of shear reinforcement in slabs. The performance and economy of twoway slabs is very much influenced by slab punching shear. Overall slab thickness and thus slab concrete volume and slab dead load—both affecting overall economy—, are very often set by slab shear requirements. Effective slab shear reinforcement provides the potential to allow for thinner slabs and provide the more ductile connection area needed for the moment redistribution and flexural deflections associated with general yielding. The development of effective shear reinforcement for slabs has been an active topic for this committee (ACIASCE 421 [2008, 2011]). The postpunching failure performance of slabs, the robustness performance level, has received some attention in ACI Committees and in ACI Code 318, notably by the inclusion of the structural integrity provisions, 4.10 of ACI 31814, which requires “reinforcement and connections shall be detailed to tie the structure together effectively and to improve overall structural integrity”. The robustness performance level for slabs is currently being addressed by an ASCE/SEI technical committee and a related standards committee. The ASCE/SEI Technical Committee on Disproportionate Collapse committee report emphasizes the behavior of steel and reinforced concrete beam and column frame buildings, but does address twoway slabs among the many other systems considered. The related Standards Committee
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ACIfib International Symposium Punching shear of structural concrete slabs
has produced several internal drafts of its standard. Through this future standard and the products of the technical committees, it is reasonable to expect that future building codes will also address the robustness performance level, especially – but not exclusively – for critical and iconic structures. As described above, the several technical committees addressing the slabcolumn connection area performance in shear have related but somewhat different interests based on the performance level(s) of interest. Also, the physical size of the slab region involved in the slab shear design considerations changes with the performance level. At the strength level, the problem can be somewhat isolated to the local slabcolumn region. As ductility becomes of more interest, the region of interest expands to a greater slab region around the connection. Considerations of robustness, as affected by the slabcolumn connection performance, can involve the entire slab system. This paper discusses the ways in which the slabcolumn punching shear phenomenon differs from, and compares with, the simpler beam flexural shear problem. It then provides an overview of the research work, primary findings, and major U.S. design provisions for each of the three performance levels – strength, ductility, and robustness.
2
Comparison of beam and slab behavior in shear
The shear strength and behavior of flexural members have been extensively researched around the world, with much less convergence in the design provisions for beam shear than for beam flexural behavior. Although the usual behaviors associated with beam shear strength (one or more critical shear cracks usually forming as an extension of flexural cracks, aggregate interlock, doweling, shear transfer in the compression zone above the shear crack, the contributions of any stirrups/shear reinforcement, etc.) are reasonably well understood, the design provisions for beam shear differ significantly around the world. The addition of another dimension for twoway slab shear relative to that for oneway beams adds additional behavioral complications, even though both beam and slab shear depend on the same material properties, have the same basic shear transfer mechanisms (aggregate interlock, doweling, transfer in the compression zone, shear reinforcement, etc.) and display a size effect. Some of the major differences and complications for the slabcolumn region in shear relative to those in beams, and previously noted by ACIASCE Committee 426 (1974), include: 1. Critical shear crack location and orientation: The surface of the punching shear failure extends around the column, while the flexural crack pattern in the slab is, except for a crack around the column face, predominantly in a radial pattern, following in most cases the slab tension reinforcement layout. Thus, the critical shear crack in a slab generally does not develop when a flexural crack turns over to form an inclined shear crack. Rather, it forms from diagonal tension in a direction normal to the adjoining flexural cracks. The slab shear crack formation is thus more similar to webshear cracking in beams than the more usual flexureshear cracking. 2. Restraint of critical shear crack opening by moment acting perpendicular to the shear crack: The flexural stiffness of the slab in the radial direction provides a restraint on the overall deflection and curvature of the slab region near the column, which restrains the opening of any slab shear crack, thus helping to maintain the aggregate interlock along the 38
ACIfib International Symposium Punching shear of structural concrete slabs
shear crack. Such an action does not exist in beams. This beneficial constraint on the inclined crack opening decreases as flexural yielding spreads outwards from the column resulting in a strong dependence of punching shear resistance on the slab flexural conditions in the regions near the column. 3. Limits on the critical shear crack location: The punching shear failure location is tightly constrained to a location very near the column face, as the critical perimeter increases with distance from the column face, unlike conditions for a constant width beam. 4. Distribution of horizontal forces across the slab width: For a typical beam, one without significant alongmember applied forces (prestressing) or aggregate interlock along the shear crack, the compression force above the shear crack is set by statics as equal to the flexural tension force. In a slab, the distribution of these horizontal flexural forces can vary across the width of the slab and statics alone does not dictate the force in the compression zone above the shear crack. For interior slabcolumn connections transferring small amounts of moment, the flexural tensions on either side of the potential failure cone are essentially balanced, allowing the shear crack to more easily extend to the slab compression surface rather than assuring a compression zone above the crack. 5. Formation of inplane compression forces in a slab: Lightly reinforced flexural members expand along their length after cracking, as the middepth of the member is on the tension side of the neutral axis. In a slab system, this expansion in the higher moment areas near the column is constrained by the less stressed slab areas further away, resulting in axial compressive forces in the region near a column. A similar constraint can be provided by slab supports and boundary conditions, and inplane compressive forces develop when flexural reinforcement is not in the direction of the applied moment (ACIASCE Committee 426, 1974). The mechanisms for generating inplane compressions in a slab system which tend to increase shear strength and reduce ductility are absent or much reduced for beams, and are greatly reduced relative to those for a slab system in the isolated beamcolumn connection test. 6. Benefits of biaxial stress conditions on material strengths: The benefits of biaxial compression (and tension) on material strength exists with the twoway bending action of the slab. Although there may be significant benefits of this biaxial stress condition, especially in the slab regions adjacent to the corners of rectangular columns, it is yet unclear how significant these biaxial loading effects are in determining slab shear strength and overall performance. 7. Loading intensity varying with location around the column face: For beams of usual widthtodepth ratios, the implied assumption is that conditions, including loading, are uniform over the width of the beam. In contrast conditions in a slab can vary significantly around the shear critical section. The doublycurved “inverted bowl” shape of the deflected slab results in significantly more of the vertical load transferred at the slabcolumn connection flowing toward the corners of the column. Vertical strain gauges placed in the center region of the column face show much less strain than at column corners. Slab regions near the column corners are both more heavily loaded and likely stronger (in part, due to the biaxial effects noted above) than regions near the center of the column face. 8. Transition of slab shear behavior to beam shear behavior: Another behavior related to unequal conditions around the critical shear perimeter is due to the effects of the ratio of slab depth to column face dimension. As the slab becomes thinner relative to the column size, the portion of the slab adjacent to the center of the column face experiences 39
ACIfib International Symposium Punching shear of structural concrete slabs
conditions more similar to that of a beam, or at least to the oneway shear behavior of a slab framing into a wall. The tangential slab moments decrease less rapidly and flexural cracking develops parallel to, and away from, the column face, and thus in the direction and close to the location of the inclined crack in the slab. 9. Existence of slabcolumn connection side faces and their role in moment transfer: Exterior and many interior slabcolumn connections transfer moments from the slab to the column. Conditions on the side faces, as well as on the front face and any back face, become extremely important in the overall performance of the slabcolumn connection in combined shear and moment transfer; the sides of beams can have no such roles. Even in the absence of an edge beam, a portion of the slab entering the column side faces functions as a torsion member, which combined with the pattern of vertical shear forces along the column sides, assists the shear stresses and moment capacities along the column front face and any back face in providing moment resistance. 10. Typical slab versus beam depths – size/depth effects: Floor and roof slabs typically have a thickness of 5 to 10 inches (127 to 254 mm), less than the typical depth of beams. As a result, the magnitude of any size effect for slabs is generally less than for a typical beam, although size effects for footings and other heavily loaded thick slabs can be expected similar to those for an equally deep beam. 11. Typical slab versus beam depths – shear reinforcement: The typically smaller member depth for slabs than for beams presents construction and other practical challenges. The use of traditional stirrups involves many small and closely spaced stirrups, challenges in stirrup anchorage, intrusion of the hooks and bends of stirrups into the usual cover dimensions over slab flexural reinforcement, etc. The result is a need for shear reinforcement systems specifically developed for slabs. 12. Member versus connection area design statement: Several of the previously stated differences between beams in shear and slabs in shear support the overall observation that beam shear can almost always be viewed as a member design consideration, while slab shear (unless the result of a localized load away from a column area) involves the behavior and design of a connection area. For slab structures, the slab punching shear problem is effectively synonymous to the slabcolumn connection problem.
3
Slabcolumn connections – Strength performance level
3.1
Development of basic strength design requirements and associated studies
The studies of the performance and strength of slabs in punching shear that led to the development of the familiar design provisions in the ACI 318 Code, provisions not greatly changed since 1963, were mostly conducted to determine the shear strength of slabs which failed in shear before the slab flexural capacity was reached. Several of the earliest test series addressed the strength in shear of isolated spread footings loaded by a column, for which punching is a major design concern. A similar isolated slab plus column specimen became the standard test specimen for study of the punching shear strength of elevated slabs. The slab segment was sized to include the negative moment area around the column, with vertical supports located near the perimeter of the square, rectangular, circular, or octagonal slab and load applied via a column stub. Although these isolated slabcolumn models are much more 40
ACIfib International Symposium Punching shear of structural concrete slabs
practical for experimental tests studying slab shear, they cannot completely model conditions in the full slab system. For example, the inplane constraint and resulting inplane compression in the slabcolumn connection area by the less stressed slab regions further away cannot not fully develop in the isolated slabcolumn model. With less slab inplane compression existing in the isolated model, the results of its use should be slightly lower strengths but larger rotations and deflections in the column area relative to those in a full slab system. These trends and the roles of the two primary phenomena in continuous slabs that are not included in the isolated slab model, namely slab moment redistribution between negative and positive moments and compression membrane action, have recently been addressed by Einpaul, et al. (2016a). Descriptions and summaries of these early tests and the use of these results in reaching the ACI 318 design requirements are available elsewhere (ACIASCE Committee 426, 1974; Widianto et al., 2009). Some observations and comments, but not a complete review of these early tests and design code formulation, follow. A key ratio in both the interpretation of the punching shear test results and the formulation of design provisions is φo = Vshear/Vflex = ratio of the observed shear strength to the calculated yield line flexural strength of the slab specimen. A φo value less than one indicated a shear failure occurring before the flexural strength was reached. Similarly, a φo value greater than one indicated the flexural strength was reached before punching occurred. It should be noted that the actual slab flexural strength at general yielding can exceed the calculated flexural strength, typically computed using yield line theory, for several reasons – among these being the existence of inplane compressive forces, strain hardening, and the actual reinforcement yield strength exceeding the nominal yield strength. A key step in the formulation of the current ACI Code provisions was a statistical analysis of 37 slab and 106 footing test results reported by Moe (1961). His equation shows the slab shear strength, expressed as a shearing stress at the column face, increasing with flexural strength, but not by as much as the flexural strength increase; decreasing for larger column side, c, to slab effective depth, d, ratios; and being proportional to 𝑓"# . v = V/bd = [15 (1 – 0.075c/d) – 5.25 φo] 𝑓"#
(psi units)
(1)
To obtain from this equation a lower bound on the shear strength, given that this shear strength is to be equal or less than the flexural strength, φo can be set equal to unity and the terms within the brackets become (9.75 – 1.125c/d). In 1962, ACIASCE Committee 326 modified the Moe’s equation by setting φo = 1 and found that a conservative fit to the then available test data for tests with φo not exceeding unity was v = V/bd = 4(d/c + 1) 𝑓"# = 4 [(d+c)/d] 𝑓"#
(psi units)
(2)
Defining the critical section at d/2 from the column face, rather than at the column face, gives a critical section c + d on a side, rather than c, and the (d + c)/d term drops out with the shearing stress, v, defined on this new critical section. The result is the simple design equation of the ACI 318 Code: v = V/b0d = 4 𝑓"# psi (0.33 𝑓"# MPa). It should be noted that this simple equation is not intended to be a predictor of slab shear strength; its purpose is to state a
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ACIfib International Symposium Punching shear of structural concrete slabs
minimum lower bound for the shear strength of slabs failing primarily in shear (Alexander and Hawkins, 2005). The performance level inherent in this simple design code requirement is that of strength. The goal of the provision is to preclude a brittle punching failure before the slab develops its full flexural capacity. This requirement does not necessarily provide ductile behavior, as there is a transition between a sudden shear failure and a ductile flexural failure. This is illustrated in Fig. 1, which shows the effect of φo on the slab response. Slabcolumn connections typically fail by punching soon after the flexural strength is reached in cases where φo is not much above unity. As the flexural reinforcement ratio decreases, more deflection and ductility takes place before the final punching but that final failure looks the same as that for a sudden shear failure. The simple 4 𝑓"# psi (0.33 𝑓"# MPa) design provision is quite successful in providing a reasonable lower bound on the strength of a slab failing in punching shear before the flexural strength in the column area is reached as shown in Figure 2, which plots results from 210 relevant tests included in the ACI 445 Punching Shear Test Databank (Ospina et al., 2011 and 2012). Also shown in Figure 2 is that test specimens with low reinforcement ratios, and which were defined as flexuredriven punching failures, very often failed at strengths below the limiting stress given in the ACI Code 318. In this period of emphasizing the strength performance level, most but not all studies of punching shear utilized empirical and statistical fits to the data. A notable exception was the mechanistic model formulated and reported in 1960 by Kinnunen and Nylander, a model which influenced the design requirements in Sweden, even though this lead to a complex design procedure. Also notable is that several of the studies reported in the 1974 ACIASCE Committee 426 report viewed the punching shear phenomenon as principally a local flexural failure and presented prediction equations primarily dependent upon the flexural characteristics of the slab.
Figure 1:
42
Effect of φo on slab response (Criswell, 1974).
ACIfib International Symposium Punching shear of structural concrete slabs
Figure 2:
3.2
Slab reinforcement ratio effect on shear strength.
Consideration of other slab parameters and conditions
The tests and design code developments described above addressed concentricallyloaded interior slabcolumn connections with square columns and constructed with usual concrete and steel materials. Many other geometries, material properties, and loadings are included in reinforced concrete building construction. Other test studies have explored these conditions, most often using the simple isolated slab plus stub column test specimen. Some of these conditions for the concentricallyload interior connection case include round and rectangular columns, lightweight aggregate concrete, holes in the slab adjacent or near to the column, large column to slab thickness ratios, use of fiber reinforcement, and flexural prestressing. Requirements have been added in the ACI Code 318 (Sec. 22.6.5 of ACI 31814) reducing the basic punching shear design value when the long side of a rectangular column is more than twice the short side or when the column sides are much larger than the slab thickness. In practice, slabcolumn connections transferring moment, in addition to vertical load, are common. Edge and corner slabcolumn connections lack the symmetry of interior connections and they almost always transfer moment about one or two axes. Unequal column spacings, nonuniform slab loads, and slab participation within a laterallyloaded frame are among the conditions leading to moment transfer at interior connections. Edge and corner slabcolumn connections typically transfer less vertical load and more moment than interior column connections and are numerically very common. A reinforced concrete slab building three bays wide and eight bays long has a total of 36 columns. There are 4 corner, 14 interior, and 18 edge columns. One half of all the columns are exterior columns! The 1974 ACIASCE 426 report describes the studies then available which addressed connections transferring a combination of shear and moment, for both exterior and interior column regions. The studies typically utilized isolated columnslab models and emphasized strength, not deformations and ductility. These tests and others made recently, along with procedures to rationalize the performance of these connections, demonstrate that moment transfer at slabcolumn connections is quite complex. For moment transfer about one direction, the critical section along the front and any back face of the column is loaded in 43
ACIfib International Symposium Punching shear of structural concrete slabs
moment and there is a shear stress larger on the front face and smaller on any back face than the average shear stress on the critical section. The side faces experience a combined loading in shear, torsion, and flexure. Both the design of the edge beam, if present, the slab near the edge, and the overall slabcolumn connection area with full consideration of these loadings becomes quite complicated, and the design provisions in the ACI Code 318 have long utilized simplified and approximate methods to divide the factored moment transferred to the column between unequal moments on the front and back faces and by eccentricity of shear (Sec. 8.4.4.2), with torsion on the side faces included in the shear term, to determine the shearing stress distribution. The shear and side face torsion portion of the moment transfer is analyzed assuming a linear variation of shearing stress along the sides faces. This analysis includes the use of an expression “analogous to polar moment of inertia” of a shape comprised of the critical section with a depth equal to the effective slab depth, d, taken about its horizontal centroid. The (c1 + d)3d/6 term effectively models the horizontal shears associated with torsion on the critical section parallel to the column side face with length c1. A small (c2 + d)d3/12 term for the side(s) parallel to the front (and back face, if any) of the critical section parallel to the c2 column face exists in the full polar moment of inertia equation, but this term is omitted in the moment transfer by shear expression as it would correspond to a flexural stress varying over the depth of the front (and back) face of the critical section, and not to a contribution by shearing stresses. Part of the design of the slab regions in shear adjacent to columns, a part not often considered in punching shear studies, is the analysis giving the vertical loads and moments. Tributary area concepts are often used for column loads and slab shear design. The basic tributary area concept overstates the vertical shear forces on corner and edge columns and understates loads on first interior columns, similar to the analysis results for a multispan, equal span, uniformly loaded beam. For three equal spans of length l and uniform load w, the end reaction is 0.400wl rather than the 0.500wl value for full end fixity; for four equal spans, the end reaction is 0.393wl. For the first interior support, the reaction values are 1.10wl for three spans and 1.14wl for four spans. In the slab, the edge and corner columns provide some fixity, the corner columns are end columns along both an end and a side of the slab, and the first interior column is the first interior support in both directions. Direct design, equivalent frame and computer models more consistently consider restraint and column locations. The use of vertical loads from the usual tributary area model makes the design of the slab shear at corner and edge column more conservative and the design of the first interior column region less conservative. Section 4.3.2 of the ACIASCE Committee 352.1R11 Design Guide partially recognizes the approximate nature of tributary area methods. It states that “For uniformly loaded slabs with nearly equal spans, with no more than 20 percent difference, slab shears at the connection may be determined for loads within a tributary area bounded by panel centerlines; slab shears at first interior supports should not be taken less than 1.2 times the tributary area values unless a compatibility analysis shows lower values are appropriate.” Various methods for using shear reinforcement in slabs have been proposed and assessed over the years. The usual goal of adding shear reinforcement at the time of the 1974 ACIASCE 426 Committee report was to increase the shear strength of the slab. Properly designed shear reinforcement also adds ductility. Because the purpose of shear reinforcement can be strength and/or ductility, the use of shear reinforcement will be a topic in the next section addressing the ductility performance level. There has been a gradual and consistent increase
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in the deformation behavior and design requirements to provide ductility. Rather than there being a shift from a strength performance limit to a ductility performance limit, the later could more properly be described by the longer designation of “the strength with ductility” performance level; ductility is an added consideration, not a replacement for strength.
4
Studies of slabcolumn connections – Ductility performance level
4.1
Tests showing that provisions for strength do not necessarily provide ductility
The investigations of punching shear described in the prior section and the current ACI Code provisions for slab shear strength emphasize the result where punching failure occurs before the flexural strength of the slab in the column region is reached. Punching failures occurring at a ratio of punching load to calculated flexural capacity, φo, of 1.00 or slightly higher have been described as secondary shear failures or “flexuredriven” shear failures. From the viewpoint of ductility, they may be described as premature punching failures. As shown by Fig. 1, there is a gradual change in response, not sudden, from a near elastic response when φo is significantly less than unity to a more ductile, flexuraldominant behavior when the slab flexural reinforcement ratio and thus the flexural strength is lower. The secondary punching shear condition is not currently addressed well in the ACI Code 318 for which punching can and does occur at limited slab deflections and at a load below the current ACI Code design value. Consequently, slabs with low reinforcement ratios may not have the expected shear strength and may experience punching at vertical and/or lateral deflections much less than those generally associated with ductile behavior. Several tests of slab systems designed to meet code design requirement have demonstrated a punching failure mode at low slab deflections and sometimes at column loads below the code punching shear values. One such case is the 9panel (3 x 3) 3/4 scale test conducted at the Portland Cement Association of a flat plate system (Guralnick and LaFraugh, 1963) previously tested at ¼ scale at the University of Illinois (Hatcher et al., 1965). The flexural reinforcement ratio in the column strip was 0.73%, with enough column strip bars concentrated over a width of c + 2h (column size + 2 times the slab thickness) centered on the column to give a flexural ratio of 1.5% within that region. Failure of the system occurred by punching of one of the interior column regions when the maximum of the slab deflections for the two exterior and corner panels adjacent to the punching location were about 1.5 in (38 mm), or about 1/120 of the 180in. (4.57 m) column spacing. The overall performance of this slab system is well stated in the following paragraph from Guralnick and LaFraugh: “Because extensive yielding of the negative reinforcement near the interior columns was in evidence just prior to punching of the slab, and because the observed ultimate flexural strength exceeded the capacity computed by yieldline theory, the mode of failure was classified as a secondary shear failure. Had the ultimate slab strength not been limited by shear, it is probable that the collapse load would have been considerably higher than that observed, due to the reserve flexural strength afforded by strain hardening of reinforcement and the development of membrane action in the slab.”
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The report of the ¾scale PCA slab notes that the ¼ scale model of the same slab design tested earlier by Hatcher et al. displayed deflections, crack patterns, distribution of service load moments, mode of failure, and ultimate capacity in close agreement with the larger scale slab. Although generally similar in response, the ¼ scale model had larger scaled deflections at failure (maximum of span/61), a higher shear stress at punching (116% of 4 𝑓"# ) and reached 113% of the computed yieldline flexural capacity. This trend of better performance at a smaller scale was more recently noted by Guandalini et al. (2009) for tests of isolated slabcolumn specimens at full scale, half scale, and double scale. Their conclusions state that “the tests have confirmed that, due to size effect, the punching strength decreases with increasing slab thickness. At the same time, the deformation at failure decreases”. A test including loading to failure was conducted after the 196465 New York World’s Fair on a large span (bay size of about 33 by 28 ft, 10 by 8.53 m) 2foot (610 mm) thick waffle slab designed for heavy loads (live load = 300 lb/ft2, dead load = 220 lb/ft2) and which served as the roof over the Rathskeller Building with a public plaza above. The structural response was linearly elastic until failure by punching at well below the flexural capacity (φo = 0.52) and at a maximum slab deflection of span/365 at a load level of 94% of the failure load. The shearing stress at failure was 116% of 4 𝑓"# , although well below the simplified version of Moe’s equation (Eq. 2) which contains the φo parameter. Thus, this slab met the current implied philosophy of the ACI Code requirements – punching occurred above the code requirements, but before significant flexural yielding.
4.2
Reasons for seeking ductility of slabs in shear
The following trends and design refinements over the past several decades have led to an increased attention to the importance of having ductility in slab systems, whether the performance is governed by punching shear or flexure: 1.
2. 3. 4.
The development of seismic design with its emphasis on ductile response. For twoway slab systems, this becomes especially important when the columns plus slab system have an important role in providing horizontal resistance, as is often the case in low to midrise concrete building in low to moderate seismic locations. The necessity of significant nonlinear response, i.e. ductility, to achieve the moment redistributions associated with full development of the potential slab flexural strength. Ductility is a part of the general expectation that visible structural distress should precede structural failure. Restated, large deformations should occur to serve as a warning of imminent structural distress. Ductility is needed to provide the energy absorption which may be key to a structure being able to survive dynamic loading, whether from blast forces (accidental or terroristbased), construction accidents in which equipment or materials are dropped on the slab, or other possible unusual loading scenarios.
A requirement for ductility in a slab system is effectively equivalent to requiring the brittle punching shear failure mode not control, which is effectively equivalent to requiring ductile performance of the slabcolumn connection areas. The relevance of slabcolumn connections to seismic loadings likely explains the interest of ACIASCE Committee 352, Joints and Connections in Monolithic Structures. Their 2011 committee document contains recommendations for determining the proportions and details of slabcolumn connections appropriate for connections in structures resisting gravity and lateral forces.
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4.3
Methods of providing ductility in slab systems susceptible to punching shear failure
As shown in Figure 1, one method for achieving ductility in slabs is to restrict the slab reinforcement ratios to a small enough value so that extensive flexural yielding occurs before punching takes place. To account for the fact that the low slab reinforcement ratio could lead to punching capacities less than 4 𝑓"# , two design approaches are possible: i) increase the thickness of the slab in proportion to the anticipated shear capacity drop, or ii) increase the slab tension reinforcement ratio locally across the column within 1.5h on each column side to be able to reach the ACI 31814 basic punching capacity of 4 𝑓"# . The former approach has associated material costs and increased dead load implications in a floor/roof system for which the factored dead load very often exceeds the factored live load. The latter forms the basis of a code change provision recently proposed by Hawkins and Ospina (2016). Another, very popular alternative today is to add slab shear reinforcement in the slabcolumn connection areas. Slabs without shear reinforcement will be addressed first, with shear reinforcement discussed in a separate section. The past studies of punching shear strength have provided little information, other than empirical evidence and trends, on predicting the overall ductility of slab systems susceptible to punching shear failures. One of the few mechanical models and analyses of the punching shear failure that is one of the more accurate predictors of slab response in shear is the critical shear crack theory (CSCT) reported by Muttoni (2008). His model provides information on how the slab flexural response affects its behavior in shear. The model allows prediction of both shear strength at punching failure and the corresponding slab deformations, including those for slabs with low reinforcement ratios and thus those considered as flexural controlled. The model used by Muttoni in the CSCT represents conditions in the slab after the inclined shear crack forms in the slab area adjacent to the column. In addition to the flexural crack which usually forms quite early at or very near the column face and the predominantly radial crack pattern around the slab, a diagonal crack which defines the punching shear failure surface develops in the slab well before the punching event. Using measurements of the slab thickness in several locations, Criswell (1970) observed evidence of crack formation and opening as reflected by an increasing slab thickness, and evidence that this inclined crack occurred first near the corners of slabcolumn connections with square columns. Additional data on this critical, but hidden, internal inclined crack includes refined measurements of slab thickness changes reported by Guandalini, et al. (2009). These results indicated that the development of the critical inclined shear crack does not start before about 50 to 70% of the load at punching. Once the critical inclined crack forms, most shear transfer is by aggregate interlock along the inclined crack. The flexural stiffness of the slab controls the rotation of the slab defined by the sum of the angles of the deformed slab on either side of the column relative to the original horizontal slab surface. This rotation is related to the opening of the critical inclined crack, which reduces the effectiveness of the aggregate interlock mechanism, with this decrease depending on several factors including aggregate size. In the CSCT analysis, a function predicting the slab loadrotation response in flexure (with resistance increasing with deflection) is compared to a failure criterion relating resistance to slab rotation in the column area, decreasing resistance as more slab rotation opens the critical inclined crack more and thus reduces the aggregate interlock. The point where there two relationships cross defines the 47
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strength and slab rotation when punching occurs. The resulting prediction equation contains the usual slab geometric and material values, along with the slab flexural capacity and the maximum aggregate size. Recently studies of slab rotations have been conducted by ACI Committee 445C using the CSCT model and tests on slabcolumn connections with low flexural reinforcement ratios have been carried out to validate the CSCT model (Guandalini et al., 2009). These studies verify that a slab reinforcement ratio well below a value of about 0.008, above which punching failure occurs before significant flexural yielding, is required for the slab to achieve the ductility ratios embedded in seismic design practice. These findings are consistent with the general observation made by Hawkins and Mitchell (1979) that “if a connection is forced to develop rotations larger than those at which the flexural capacity is first reached, a punching failure occurs unless the shear stress is limited to 2 𝑓"# or shear reinforcement is provided”. At present, these characteristics of slabs with low reinforcement ratios are not recognized in the ACI Code 318.
4.4
Use of slab shear reinforcement to achieve strength or strength with ductility
Shear reinforcement in slabs can be used to increase the shear strength of the slab in the area near the column and/or to increase the ductility of the slab before a punching failure limits deflections. To date, most of the studies on shear reinforcement in slabs and applications in practice have emphasized the use of shear reinforcement to increase the shear strength. With effective shear reinforcement, a slab of a given thickness can have greater capacity in shear. Given that the two factors most often controlling the minimum thickness for a flat plate or flat slab system (and thus affecting material volumes, dead load values, and floortofloor dimensions) are service level deflections and punching shear, it is not surprising that ACIASCE Committee 421 has an interest in the behaviors and strengths of slabs in shear. One result is the committee document, 421.1R08, Guide to Shear Reinforcement for Punching in Flat Plates. Design considerations for slabcolumn connections during an earthquake that results in unbalanced moments and story drifts have led to a second design guide, ACI 421.2R10, Guide to Seismic Design of Punching Shear Reinforcement in Flat Plates. Shear reinforcement in slabs can be used to strengthen the slab at the critical section by providing a vertical force component that crosses the critical diagonal crack and strengthens the slab in the immediate vicinity of the column sufficiently that the controlling critical section is moved further away from the column face; at that location, the critical section has a larger perimeter. The smaller of the strengths of the sections near the column with the shear reinforcement and that at the potential critical perimeter just beyond the effective shear reinforcement area controls. The design and utilization of shear reinforcement in the past faced two challenges. The use of stirrups, similar to the usual shear reinforcement method for beams, entails many small size closed stirrups placed close together, which presents practical problems. Stirrups going around the outer layer of reinforcement also impact the slab depth needed to provide the required cover, unlike in a beam where the usual ¾in (19 mm) minimum cover over the small bars used for stirrups does not control cover—the usual 1½ in. (38 mm) minimum cover over the larger flexural reinforcement does. Initial tests showed that stirrups were not
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fully efficient given a design assumption that the contribution of the stirrups were additive to the strength provided by the concrete, i.e. the shear strength of the slab without shear reinforcement. Later, it became evident that the contribution of the shear reinforcement can be more realistically assessed if it is assumed additive to the shear carried by the slab with no shear reinforcement at the time of the critical inclined crack initiation, about 50 to 70% of the shear strength without shear reinforcement. This resulted in Section 22.6.6.2 of ACI 31814 requiring that the contribution of the concrete, stated by the shear stress vc, when shear reinforcement is used is limited to 2 𝑓"# psi (0.16 𝑓"# MPa) for normal weight concrete when stirrups are used and to 3 𝑓"# psi (0.24 𝑓"# MPa) when headed shear stud reinforcement, usually in the form of single headed vertical studs welded to a horizontal base rail, is used. The difference is based on evidence presented in the ACIASCE 421 committee work that shear stud reinforcement is more efficient than stirrups. The use of 2 𝑓"# psi (0.16 𝑓"# MPa) or 3 𝑓"# psi (0.24 𝑓"# MPa) instead of 4 𝑓"# psi (0.33 𝑓"# MPa) results in the interesting anomaly that the addition of a small amount of shear reinforcement can reduce the design shear strength given by the ACI 318 Code. Many experimental studies have addressed the effects of slab shear reinforcement, with many reinforcement types and distributions considered. A recent paper by Einpaul, et al. (2016b), included a review of numerous past studies, European design practices, and the results of an experimental program including tests of eleven combinations of shear reinforcement types and layouts, all for a constant slab and column geometry and a slab reinforcement ratio of 1.5%. As ductility becomes more an expectation for slab systems and the associated slabcolumn connections, shear reinforcement methods, response of slabs with shear reinforcement and the associated design procedures should to continue to be active areas of research and development.
5
Slab systems – Performance level of robustness and prevention of disproportionate damage
5.1
Susceptibility of slab systems to disproportionate damage initiated by punching failures
Past experiences with concrete flat slab and flat plate systems have demonstrated that a punching failure may, and has, initiated a more general progressive collapse of at least a portion of the structural system. With the usual pattern of slab reinforcement designed to provide the needed slab moment capacities, the integrity of the slabcolumn connection area can be effectively destroyed by a punching failure. The damaged connection has little residual capacity, a resistance primarily dependent on the top slab reinforcement within the shear cone dimension, reinforcement that can easily tear loose its cover concrete or become unbonded due to the fairly short length of the column strip negative reinforcement. For the slab system to survive, the reduction in vertical load resistance at the punched column must be successfully redistributed to the surrounding columns, along with the changes in slab and connection moments. In current terminology, punching shear failures may often lead to disproportionate damage.
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Methods to prevent or minimize disproportionate damage are now receiving more attention by structural engineers, both because of increased structural safety expectations and performance experience. Damage to buildings with the intent to do harm has demonstrated some of the hazards of present world conditions. Providing robustness, resilience, so that the structure can better survive an unusual or unexpected extreme loading event has become a performance level now more often explicitly considered in design, especially for critical and iconic structures. The general concern for preventing or minimizing disproportionate damage for structures of all types has led to the formation of an ASCE/SEI technical committee to address the topic of disproportionate damage, and the formation of an ASCE/SEI standards committee on this topic. An activity receiving considerable attention in both of these committees is the definition of the response of frame structures of many types, materials, and configurations to the loss of one, or more, major structural elements such as a lower story column, along with the formulation of methods for the structure to more successfully survive such severe initial damage due to external conditions such as accidental or maliciouslyplanned explosions, large vehicle impact, or other severe localized loading. For a concrete slab system, the consequences of a punching failure are somewhat similar to the loss of a column, although the vertical load transfer from the slab to column initially involves only one level or floor. The progressive failure in Spring 1973 of a sizable portion of a 26story reinforced concrete flat plate building in Bailey’s Crossing, Virginia, near Washington, D.C., demonstrated the mechanics of a collapse initiated by a punching failure (Leyendecker and Fattal, 1977). As the construction of this castinplace apartment building neared its completion, a punching failure occurred, probably as a result of premature removal of some formwork. The failure first propagated horizontally to adjacent connections as the immature slab was unable to redistribute the forces no longer resisted by the failed connection. With no effective vertical support remaining for that floor level, the failure propagated vertically down the structure as the now unsupported slabs fell on those below. All 26 floors then in place collapsed over an approximately 80ft (24.4 m) square area bounded by the exterior of the building and expansion joints. Part of the debris from this collapse fell onto the edge portion of a large 295 by 344 ft (90 by 105 m) flat plate posttensioned garage structure with one and much of the second level completed. Damage to the posttensioned anchorage area and the consequent loss of prestress resulted in the failure propagating horizontally across this garage structure and also vertically, dropping both levels onto the foundation slab. Several past studies have addressed the general topic of progressive collapse in slabs. For example, Mitchell and Cook (1984), Hawkins and Mitchell (1979), and Qian and Li (2016). Some provisions, including those for structural integrity reinforcement now in the ACI Code 318, are among the ways the performance limit of robustness has received attention.
5.2
Methods to improve resistance of slab systems to disproportionate damage
The strategy to improve the ability of a flat plate or slab structure to better resist disproportionate damage can be to either increase the residual strength and stiffness of the damaged connection area or provide enough effective slab shear reinforcement that the punching failure mode is precluded. After a punching failure, the configuration at the failed connection becomes one in which the slab is supported by reinforcement crossing the shear
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failure surface; the usual flexural action in the failed area is lost and the slab is effective hung from the column region. As shown in Figure 3, the bars of the top reinforcement layer placed to resist the large negative moments in the column strip at the column location are quite easily torn from the top of the slab as only the concrete cover is above these bars and the length of these top bars beyond the failure location may not be enough to fully anchor these top bars.
Figure 3:
Load transfer to column after punching shear distress – i.e. postpunching. Top bars easily tear out, possibly to end of bar. Bottom bars through the column can act as suspension bars.
Thus, these top bars typically provide only limited support accompanied by a considerable vertical separation between the slab and the failure region around the column. Bottom bars through the column area are not routinely provided because they are not required for the flexural design. Any bottom bars continuous through the column area are much better anchored and able to act efficiently as inclined tensile members than are the top bars. The larger resistance provided by the bottom bars is provided with much less vertical separation of the slab from the column region. Thus, adding bottom bars through the column cage, including extending some of the bottom bars from the slab’s midspan region, is an effective way to increase the postpunching performance. There are several other ways to increase the amount of steel effective as suspension/catenary tensile ties between the slab and the distressed connection area, some illustrated in Figs. 4 and 5. Additional bars which are near the top of the slab within the column reinforcement cage region and then either curved, Fig. 4, or bent down through the critical shear crack region are thus able to function locally as shear reinforcement. When well anchored in the bottom of the slab, these bars can provide a very efficient and stiff postpunching tie between the slab and column region. Closed stirrups around the top and bottom reinforcement can keep the top bars enclosed by the stirrups from easily tearing out and thus can increase their effectiveness in the postpunching configuration. Some experimental studies of slabcolumn connection regions in shear have included data on the residual resistance after punching through continued loading/deformation of the distressed specimen. Thus some data on postpunching resistance exists. Future tests having the primary task to define the postpunching performance of specimens with reinforcement additions may include the methods shown in Figs. 4 and 5. It is recommended that continued testing after punching become more routine for all slab shear tests examining punching, whether primary or secondary, as behavior in that region will become more important for design in the future.
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Figure 4:
Use of inclined bars for postpunching resistance.
Figure 5:
Configuration of slab with vertical stirrups, making top bars more effective as suspension supports after punching.
5.3
Analysis and design to improve slab robustness as controlled by punching shear
The ability of the slab system to survive an initial punching failure depends upon several factors, including the loading pattern and load magnitude at the time of punching failure, which affects the magnitude of the forces (vertical loads and moments) needing to be redistributed, the remaining capacity of the slabcolumn connections adjacent to the initial punching location, and the residual strength and stiffness provided by the reinforcement crossing the punching failure perimeter acting as a catenary system. If all columns have similar loading conditions, the columns surrounding the column region first experiencing punching have little reserve capacity and thus a limited ability to absorb vertical forces redistributed from the failed area. If a localized loading or condition resulted in the initial failure, the surrounding columns will more likely have the additional capacity necessary to resist the redistributed forces, including changes in the column moment loading resulting from the conditions of the damaged slab. A stiffer residual support system for the punching shear failure area will reduce the resulting separation between the slab and column and thus reduce the amount of the forces needing to be redistributed, assuming the residual distressed column area has residual strength as well as stiffness.
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Design to provide robustness can take one of several approaches: (1) a full analysis is possible using a numerically welldefined postpunching resistance for the distressed column region, a good assessment method for the adjacent slabcolumn connections which includes one or more welldefined loading scenarios, perhaps probabilistically defined, to assess the potential survival of the damaged slab; (2) specify enough well anchored bottom bars through the column cage or other bars effective after punching to support a given proportion of the expected magnitude of the shear failure load at an assumed inclined bar angle; (3) specify a nominal amount of reinforcement effective in the postpunching mode; or (4) specify that enough slab shear reinforcement over an adequate area around the column be utilized so that a drop in resistance due to a punching failure is precluded. The ACI 31814 Code contains some provisions of the third type. Section 8.7.4.2.2 requires, as structural integrity reinforcement, that at least two of the column strip bars or wires in each direction pass through the column cage and be anchored at exterior supports. The structural integrity provisions of 8.7.4.2 for prestressed concrete slabs includes a basic requirement that at least two tendons of ½in. (12.7 mm) diameter or larger strand be placed in each direction and pass through the column vertical reinforcement cage. Chapter 18, EarthquakeResistant Structures, has additional requirements, including those of 18.4.5 for twoway slabs without beams in intermediate moment frames and 18.14.5, which requires significant shear reinforcement for most slabcolumn connection regions when the slab system is not a part of the seismicforceresisting system. The fourth approach, the elimination of the punching failure mode through adequate effective shear reinforcement, is advocated by Broms, 2000. His test results include loaddeflection curves extending to about five times the specimen deflection at punching for four specimens with bottom bars and either no or some shear reinforcement, either preassembled stirrup cages or bent bars through the top of the slab and through the column in each direction, then bent 35 degrees from the horizontal starting at the column edge before extending along the bottom reinforcement level, bars capable of acting as catenary hangers, similar to the bars shown in Fig. 4. Three specimens contained bottom bars through the column area, stirrup cages, and enough bent bars through the column to provide a vertical force component at d/2 from the column equal to the shear force associated with the full flexural capacity of the slab. These three specimens displayed an elastoplastic response curve out to at least five times the deflection corresponding to punching in the other specimens without any punching failure. Broms reported that the expected extra cost, including labor, for providing the stirrup cages and catenary hanger bent bars adequate to preclude punching failure to be about 1.5% of the total cost for the slab. Future research and design efforts related to the robustness performance level can be expected to be given more emphasis and use, especially once the planned ASCE/SEI Standard on Disproportionate Damage is completed.
5.4
Flat slab system displaying significant postpunching catenary behavior
Results from the tests of two 9panel (3 x 3) flat slab structures designed to carry very large loads conducted by Criswell (1970), illustrate, often in an exaggerated way, many of the pre and postpunching behaviors noted above. These tests also showed the very large deformations possible in a slab system through suspension/catenary action after punching occurs. 53
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Criswell’s slab system had prototype dimensions including 18ft (5.5 m) column spacing, 11in. (280 mm) thick slab increased to 16in. (406 mm) in the drop panel area with column capitals 66in. (1676 mm) in diameter supported on 22in. (559 mm) round columns and was designed for a 15 psi (2160 psf) blast loading. The Grade 40 (276 MPa) slab flexural reinforcement was designed by the ACI 31865 Direct Design method. Consistent with blast design principles then in use, slab compression reinforcement equal to one half the tension reinforcement amount was provided everywhere. The 3 x 3 panel slab system prototype was designed to be the roof of a blast/fallout shelter area placed just below a public area such as a park in a project sponsored by the Office of Civil Defense (now the Federal Emergency Management Agency). The prototype structure and the two quarterscale models were constructed with a basement wall integral with the slab on all four edges. Testing was done with the walls and columns supported vertically by a thick foundation slab forming a boxlike unit which was placed in a 23ft (7 m) diameter soil tank testing fixture with sand backfill to the level of the top of the slab. Static loading was done with water pressure, and the dynamic loading utilized primacord explosives within the closed soil tank. The staticallyloaded slab experienced punching at one of the four interior columns at a total uniform loading of 26.6 psi (3830 psf, 26.4 MPa) when the maximum panel deflections around the failed connection were 2.3% of the 54in. (1372 mm) model column spacing. This load was 2.1 times the yieldline flexural capacity computed with nominal material properties, no inplane loadings, and a “conservative” negative yieldline placement through the centroid of the half column capital area. The inplane compressive force from the integral walls resulted in an increase of about 60% in the capacity of the lightly reinforced slab, with additional increases of about 10% from the actual reinforcement yield being above the nominal and through the use of a less conservative yield line placement. The punching failure occurred at a shearing stress of 5.83 𝑓"# , psi (0.48 𝑓"# , MPa) with the slab shear strength also benefitting from the quite large inplane slab compressions. The measured maximum column loads at the time of the first punching, 88, 89, 89, and 91 kips, average 115% of the tributary area loads. Punching soon occurred at the other three interior columns. Primarily through the action of the bottom slab reinforcement continuing through the large column capital area, (the drop panels spalled off and dropped), the slab system after the punching of the four interior columns was able to provide a resistance of 15 to 17 psi (103 to 117 kPa) (2160 to 2450 psf) until maximum panel slab deflections reached about 7 inches (178 mm). The resistance drop from 26.6 psi (183) to about 16 psi (110 kPa) resulted primarily from the loss of negative moment flexural action for the majority of the column strip width located within the punching failure perimeter and the reduction, then loss, of the inplane compression force from the walls as the slab areas adjacent to the wall became deep horizontal beams anchoring the tensile/suspension forces of the rest of the slab acting as a catenary. Failure occurred by the slab being torn apart in tension, see Figure 4. The vertical force supporting the slab area from each of the column capital regions at the approximately 16 psi (110 kPa) loading of the slab system in its postpunching failure condition can be provided by the bottom bars passing through the punching failure perimeter inclined (as shown in Fig. 3) at 50 degrees from the horizontal.
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Figure 6:
Suspension action, rupture zone and deflected shape of slab after static test (Criswell, 1972).
The second slab model was first loaded with a longduration blast pressure with a risetime of about 8 milliseconds and a maximum pressure of 19.7 psi (136 kPa, 2840 psf). This loading caused little damage, with the structural response largely in the elastic range and with some cracking. Strain gauges on the reinforcement indicated some localized yielding in the negative moment areas. This slab was then loaded again with a considerably larger dynamic load, this second loading having a maximum pressure of 30.1 psi (207.5 kPa) (4330 psf) reached at a rise time of 4.5 milliseconds. The slab was very heavily damaged by this larger loading. This loading resulted in punching of the four interior columns at about 30 milliseconds and rupture of the slab acting as a catenary starting at about 75 milliseconds. By about 95 milliseconds, enough air volume had moved through the slab rupture zones that the pressure inside the structure and the pressure above it equalized at about 14 psi (96.5 kPa). Figure 7 shows schematically the condition of the dynamically loaded slab at about 75 milliseconds. Note the ways in which the suspension action around the column capital punching failure locations and the portions of the slab adjacent to the wall acting as deep horizontal beams support the center portions of the slab which is acting as a tensile membrane. Except for the tilting of the four columns toward the center of the slab system because of the four slab rupture regions all being at the corner panel side of a column, the condition of the statically loaded model at the time the slab failure in tension terminated the test was very similar to that shown in Figure 7. This figure illustrates well the observation that a structural system will act to avoid failure/collapse in all the ways possible by the structure as provided, and not only in the ways the designer explicitly recognized.
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Figure 7:
6
Condition of overall large deflection/postpunching behavior in dynamically loaded slab at 75 milliseconds (Criswell, 1972).
Conclusions
This paper presents on overview of research and design related to the shear strength of reinforced concrete slabs, organized by the three levels of structural performance, strength, ductility, and robustness commonly considered. Research studies on the shear strength of slabs that lead to the shear strength provisions in the U.S. design codes were concerned with the strength performance level. With the rise of seismic design provisions and improved structural modeling abilities, the performance level of ductility, along with strength, has become more emphasized. As the analytical tools available to structural designer have further developed, along with better understanding of structural response to extreme loadings, the expectations of the general public and those involved in infrastructure design have also increased and more emphasis is being given to the robustness performance level. This level seeks to provide the structure with more ability to avoid disproportionate damage following an initial local failure. The actions and effects associated with the strength and behavior of slabs in shear extend far beyond what would be implied by a naïve statement that slab shear is like beam shear except for the added dimension. The shear strength in slabs is a basic problem within the broader topic of the shear strength of reinforced concrete members. Although the shear strength of slabs may control the ability of the slab to resist localized large loads on the slab, the usual shear strength problem is that of potential punching in the column regions of the slab system and the slab shear problem is essentially that of the slabcolumn connection. The strength and performance of the slab in connection regions is a major factor in the determination of the required slab thickness, and thus affects economy and the magnitude of the dead load of the slab system. The ductility of the slabcolumn connection region impacts 56
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the ability of the slab to more fully redistribute moments by general flexural yielding. Failed slabcolumn regions can be a source of disproportionate damage and progressive collapse if there is insufficient postpunching resistance. Several ACI and ASCE Committees address structural systems or components which are impacted by the performance of slabs in shear. The performance of a slab in shear is critical in the roles assigned to these committees. This results in several groups viewing the same basic physical phenomenon from slightly different perspectives, but also causes organizational challenges in coordinating the efforts and results of the several groups. Slab punching is a topic that deserves more attention as design procedures become more powerful and the work of the technical committees needs to become more unified.
7
References
ACIASCE Committee 326 (1962) “Shear and Diagonal Tension, Part 3 – Slabs and Footings,” ACI Journal, Vol. 59, No. 3, pp. 353395. ACI Committee 318 (2014) Building Code Requirements for Structural Concrete (ACI 31814) and Commentary (ACI 318R14), American Concrete Institute, Farmington Hills, MI. ACIASCE Committee 426 (1974) “The Shear Strength of Reinforced Concrete Members – Slabs,” ASCE Journal of the Structural Division, Vol. 100, No. ST8, August 1974, pp. 15431591. ACIASCE Committee 421 (2008) Guide to Shear Reinforcement for Slabs (AC1 421.1R08), American Concrete Institute, Farmington Hills, MI. 15 pp. ACIASCE Committee 421 (2010) Guide to Seismic Design of Punching Shear Reinforcement in Flat Plates (AC1 421.2R10), American Concrete Institute, Farmington Hills, MI. 30 pp. ACIASCE Committee 352 (2011) Guide for Design of SlabColumn Connections in Monolithic Concrete Structures (ACI 352.1R11), American Concrete Institute, Farmington Hills, MI. Alexander, S.D.B., and Hawkins, N. M. (2005) A Design Perspective on Punching Shear, ACI SP232, pp. 97108. Broms, C. (2000) “Elimination of Flat Plate Punching Failure Mode,” ACI Structural Journal, Vol. 97, No. 1, Jan.Feb. 2000, pp. 94100. Criswell, M. E. (1974) “Static and Dynamic Response of Reinforced Concrete SlabColumn Connections,” ACI Special Publication, SP42, Shear in Reinforced Concrete, American Concrete Institute, Farmington Hills, MI, No. 31, pp. 721746. Criswell, M. E. (1972) Design and Testing of a BlastResistant Reinforced Concrete Slab System, Technical Report N7210, U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS, 1970. Criswell, M. E. (1970) Strength and Behavior of Reinforced Concrete SlabColumn Connections Subjected to Static and Dynamic Loadings, Technical Report N701, U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS, 1970, 388 p. Einpaul, J., Ospina, C.E., and Muttoni, A. (2016) “Punching Shear Capacity of Continuous Slabs”, ACI Structural Journal, Vol. 113, No. 4, pp. 861 – 872.
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Einpaul, J., Brantschen, F., Ruiz, M. F., and Muttoni, A. (2016) “Performance of Punching Shear Reinforcement under Gravity Loading: Influence of Type and Detailing”, ACI Structural Journal, Vol. 113, No. 4, pp. 827 – 838. Ghali, A., Gayed, R. B., and W. Dilger. (2015) “Design of Concrete Slabs for Punching Shear: Controversial Concepts”, ACI Structural Journal, Vol 112, No. 4, pp. 505514. Guandalini, S., Burdet, O., and A. Muttoni. (2009) “Punching Tests of Slabs with Low Reinforcement Ratios”, ACI Structural Journal, Vol. 106, No. 1, pp. 87 – 95. Guralnick, S. A., and LaFraugh, R.W. (1963) “Laboratory Study of a 45foot Square Flat Plate Structure,” ACI Journal, Proceedings Vol. 60, No. 9, pp. 11071185. Hatcher, D. S., Sozen, M. A., and Siess, C., P. (1965) “Test of a Reinforced Concrete Flat Plate”, Journal of the Structural Division, ASCE, Vol. 91, No. ST5, pp. 205231. Hawkins, N. M., and Mitchell, D. (1979) “Progressive Collapse of Flat Plate Structures,” ACI Journal, Proceedings, Vol. 76, No. 7, pp. 775808. Kinnunen, S., and Nylander, H. (1960) “Punching of Concrete Slabs without Shear Reinforcement,” Royal Institute of Technology, Transactions, No. 158, Stockholm, Sweden, 1960, 110 p. Leyendecker, E. V., and S. G. Fattal. (1977) Investigation of the Skyline Plaza Collapse in Fairfax County, Virginia, NBS Building Science Series 94, National Bureau of Standards, 88 p. Magura, D. D., and Corley, W. G. (1969) Tests to Destruction of a Multipanel Waffle Slab Structure,” FullScale Testing of New York World’s Fair Structures, Publication 1721, Vol. II, Building Research Advisory Board, National Academy of Sciences, Washington, D.C., pp. 100 – 135. Mitchell, D., and Cook, W. (1984) “Preventing Progressive Collapse of Slab Structures,” Journal of Structural Engineering, ASCE, Vol. 110, No. 7, pp. 15131532. Moe, J. (1961) “Shearing Strength of Reinforced Concrete Slabs and Footings, under Concentrated Loads,” Development Department Bulletin No. D47, Portland Cement Association, Skokie, IL, 130 p. Muttoni, A. (2008) “Punching Shear Strength of Reinforced Concrete Slabs without Transverse Reinforcement,” ACI Structural Journal, Vol. 105, No. 4, pp. 440450. Ospina, C.E., Birkle, G., Widianto. (2011) Wang, Y., Fernando, S.R., Fernando, S., Catlin, A.C. and Pujol, S., “ACI 445 Collected Punching Shear Databank,” http://nees.org/resources/3660. Ospina, C. E., Birkle, G., and Widianto. (2012) “Databank of Concentric Punching Shear Tests of Twoway Concrete Slabs without Shear Reinforcement at Interior Supports,” 2012 ASCE Structures Congress, Proceedings, pp. 18141832. Ospina, C. E., and Hawkins, N. M. (2013) “Considerations to Prevent Concentric Shear Failure in Reinforced Concrete TwoWay Slabs,” ASCE/SEI Structure Magazine, Jan. 2013, pp. 1316. Qian, S., and Li, B. (2016) “Resilience of Flat Slab Structures in Different Phases of Progressive Collapse,” ACI Structural Journal, Vol. 113, No. 3, pp. 537548. Widianto, Bayrak, O., and Jirsa, J.O. (2009) “Twoway Shear Strength of SlabColumn Connections: Reexamination of ACI 318 Provisions,” ACI Structural Journal, Vol. 106, No. 2, pp. 160170.
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Size effect on punching shear strength: Differences and analogies with shear in oneway slabs Miguel Fernández Ruiz, Aurelio Muttoni École Polytechnique Fédérale de Lausanne (EPFL), Switzerland
Abstract Size effect has been theoretically and experimentally acknowledged as a phenomenon influencing the shear and punching shear strength of concrete structures, with reducing unitary shear strength for increasing member sizes. For members failing in shear, as beams or oneway slabs without transverse reinforcement subjected to uniform loading, size effect has been shown to have a variable influence, with low significance for failures governed by limit analysis (strength or yield criterion) and large influence for members failing in a brittle manner. When the response of a member failing in shear can be reasonably approximated by a linear behaviour (i.e. linear relationship between the acting shear force and the crack widths), the predictions of Linear Elastic Fracture Mechanics (LEFM) can be applied to asymptotically large specimen sizes. This phenomenon can for instance be demonstrated by the Critical Shear Crack Theory (CSCT) and leads to a dependence of the shear strength with the power 1/2 of the size of the specimen. Nevertheless, in actual structures failing in shear (such as slabs or shells), the structural response is normally characterized by some level of redundancy and capacity to redistribute internal forces in the longitudinal and transversal directions. In this case, the relationship between the acting shear force and the crack widths is not linear (with lower crack widths associated with larger shear strengths) and the influence of size effect on the shear strength is milder than that predicted by LEFM. With respect to punching (shear failures due to concentrated loads in twoway slabs), a similar behaviour is observed with respect to size effect. A low dependency can be observed when limit analysis governs whereas for brittle failures, size effect becomes significant. In this case, it can be observed that the behaviour of slabs is highly nonlinear (as for redundant members failing in shear), and the crack openings are to a large extent dependent on local and structural tensionstiffening effects. This deviates the actual behaviour from the one predicted by LEFM and modifies the influence of size effect, which becomes less significant than the predicted behaviour according to LEFM. In this paper, this phenomenon is investigated by means of the CSCT, providing a consistent frame to analyse size and strain effects accounting for realistic slab responses.
Keywords Concrete structures, punching shear strength, Critical Shear Crack Theory, size effect, strain effect
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1
Introduction
The implications of size effect on the shear strength of concrete structures have been a matter of discussion for decades. Currently, there is still not a consensus on the topic and design codes present different approaches, with no direct consideration (such as ACI 318, 2014), proposing the use of empirical factors (such as Eurocode 2, 2004) or calculating it on the basis of mechanical models (such as the fib Model Code for Concrete Structures 2010, 2013 [MC2010]). For oneway slabs, it is nevertheless generally agreed that size effect has low influence for members whose strength can be suitably described by limit analysis (controlled by a strength or yield criterion, typically small specimens and/or with high reinforcement ratios) but the influence may become significant for brittle failures (typically associated with large sizes and also with low reinforcement ratios but when bending is not governing). This can for instance be described by means of the size effect law as suggested by Bažant (1984) describing the transition between both regimes, the former controlled by limit analysis and the latter by Linear Elastic Fracture Mechanics (LEFM). With respect to punching shear design, the influence of size effect on its strength has followed less research than for oneway slabs. One of the reasons is that actually testing large specimens is complex as casting of twoway slabs is required (Guandalini et al., 2009). In most cases, particularly for empirical design expressions or when the size effect factor is introduced in an empirical manner, the influence of size effect is directly adapted from that of shear failures (Eurocode 2, 2004; Bažant and Cao, 1987). This approach is however difficult to justify, as the behaviour of beams and slabs is potentially very different with reference to the cracking behaviour in the shearcritical region. In this paper, the analogies and differences with respect to the influence of size effect in failures in shear (oneway slabs) and punching (twoway slabs) are investigated. To that aim, the Critical Shear Crack Theory (CSCT, basis of current MC2010 provisions [Muttoni and Fernández Ruiz, 2010; Muttoni et al., 2013]) will be used. This theory derives the influence of size effect directly from its basic assumptions on the shear capacity related to the opening of a critical shear crack without the need of introducing any empirical correction function. A detailed summary of its theoretical ground with respect to punching shear failures can be consulted in Muttoni (2008) and Muttoni and Fernández Ruiz (2017), whereas details for the shear strength in oneway slabs and the influence of size effect can be referred to elsewhere (Muttoni and Fernández Ruiz, 2008; Fernández Ruiz et al., 2015).
2
Size and strain effects in shear failures of oneway slabs without transverse reinforcement
2.1
Size effect in shear for a linear structural response
An investigation of the influence of size effect in shear by means of the CSCT is thoroughly presented in Fernández Ruiz et al. (2015). In that work, the shear strength is shown to be characterized by a hyperbolic failure criterion accounting for various potential sheartransfer actions, namely aggregate interlock, residual tensile strength of concrete, dowelling action and inclination of compression chord. The failure criterion is determined by integration of the stresses developed accounting for the shape of the critical shear crack and its kinematics at failure (Fernández Ruiz et al., 2015); see main assumptions in Figure 1. This 60
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failure criterion determines the amount of shear that can be transferred through cracked concrete for a given crack opening. According to the CSCT, the shear capacity is justified to be dependent on the opening of the critical shear crack (lower shear capacities for larger crack openings) and on the crack roughness (assumed correlated to the maximum aggregate size) (Muttoni and Fernández Ruiz, 2008; Fernández Ruiz et al., 2015). By considering that the crack width is correlated to a reference strain times the effective depth of the member ( w µ e × d ), the following failure criterion was proposed (Muttoni and Fernández Ruiz, 2008) (Fig. 2a): Vc
1 1 = × b × d f c 3 1 + 120 e × d d dg Vc b × d fc
= 4×
1 1 + 120
e ×d
(SI units [MPa, mm])
(Customary units [psi, in])
(1)
d dg
where b refers to the member width, fc to the concrete compressive strength measured in cylinder, d to its effective depth, ddg = dg + 16 mm (5/8”) ≤ 40 mm (1.58”) where dg refers to the maximum aggregate size. With respect to the reference strain (e), it is considered at a distance equal to 0.6·d of the compression face and at d/2 of the applied load (extended considerations can be consulted elsewhere [Muttoni and Fernández Ruiz, 2008]). It can be noted that in the previous expression, the crack width accounts both for the size and for the strain of the member, thus coupling both phenomena. For beams and oneway slabs, it can be accepted that the reference strain (e) and thus the opening of the cracks (w) are linearly dependent on the acting bending moment (Muttoni and Fernández Ruiz, 2008). This assumption is confirmed by the loss of reinforcementtoconcrete bond stresses at the delamination region (refer to Fig. 1a and to Fernández Ruiz et al. [2015]) limiting the tensionstiffening effects (Cavagnis et al., 2015). A linear elastic cracked behaviour can thus be assumed, Figure 2b. For the control section located at d/2 from the load (Figure 2b), the bending moment results M = VR (a  d / 2) so that: e=
V R ( a  d / 2) M 0.6d  c 0 .6 d  c = r × b × d × E s (d  c / 3) d  c r × b × d × E s (d  c / 3) d  c
(2)
where r refers to the reinforcement ratio, Es to the modulus of elasticity of the reinforcement and c refers to the depth of the compression zone (Muttoni and Fernández Ruiz, 2008): c = r ×d
ö E s æç 2 Ec 1+  1÷ ÷ Ec çè r × Es ø
(3)
with Ec referring to the modulus of elasticity of concrete. Introducing Eq. (2) into Eq. (1) allows calculating the shear resistance (VR). This strength can be analytically calculated as (Fernández Ruiz et al., 2015): VR b × d fc
=
 1 + 1 + 4d n 6d n
(4)
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where dn is a parameter accounting for the size but also taking into account the strain effects (Fernández Ruiz et al., 2015). It can be noted that this expression is compliant with the predictions of the sizeeffect law for shear failures and in agreement with experimental evidence (Fig. 2c). For failures governed by limit analysis (dn → 0), no size effect is observed whereas for large member sizes (dn → ∞), size effect is in accordance with that of LEFM (slope 1/2 if expressed in double logarithmic scale, Fig. 2c). It is interesting to note that according to the CSCT, both size and strain effects are coupled. For instance, for a linear increase on the size of the member (d), the term e ·d does not actually increase linearly at failure because the unitary strength reduces (refer to Eq. [1]) and thus the deformation at the failure load also reduces accordingly (Eq. [2]). This leads to the fact that although the failure criterion depends on d1, their coupled response expressed only in terms of the size (Eq. [4]) depends on the term d1/2 for asymptotically large sizes.
Figure 1:
62
Shear strength according to the CSCT: (a) shape of critical shear crack and kinematics at failure; (b) crack relative movements; (c) crack opening (w) and sliding (d) along the quasivertical branch; (d) associated interlock stresses; (e) crack opening along the quasihorizontal branch; and (f) associated residual tensile strength stresses.
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Figure 2:
2.2
Size effect in shear: (a) failure criterion, cracked flexural behaviour and failure load according to the CSCT; (b) calculation of the reference strain; and (c) sizeeffect law according to the CSCT and comparison to the database of Muttoni and Fernández Ruiz (2008).
Size effect in shear for redundant structures
As previously shown, when a linear response between the actions and the crack widths (strain in the reinforcement) can be assumed, the size effect exhibits a slope of 1/2 when expressed in double logarithmic scale. This holds true for some members, such as for instance simply supported beams usually tested in laboratory (Figure 2b), and can be confirmed by means of LEFM or the CSCT. In actual structures, however, the structural response normally deviates from the linear one. Failures in shear are usually governing for planar members such as slabs and shells (refer to Figures 3ac) as well as for continuous and redundant structures (Figures 3cf). For these structures, the response between the applied shear force and the crack widths is normally not linear, as they have the capacity to redistribute the internal forces in both the longitudinal and transversal directions. As a consequence, strains and crack widths in the critical regions are reduced and this helps mitigate the influence of size on the overall response. For structures with the potential to redistribute internal forces (which are the majority in case of shear failures), it can thus be concluded that the asymptotic slope of the size effect in double logarithmic scale does not correspond to 1/2 but must be milder. This fact is also particularly evident for twoway members failing in punching shear, whose nonlinear response and interaction with tensionstiffening effects will be discussed in the following section.
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Figure 3:
Examples of actual structures without transverse reinforcement where failures in shear may be governing: (a) retaining wall; (b) cutandcover tunnel; (c) silo; (d) foundation; (e) continuous slab bridge; and (f) deck slab of bridge.
3
Size and strain effects in punching failures
3.1
Response of twoway slabs and differences between failures in shear and in punching
With respect to punching failures, a similar approach to that for shear is followed by the CSCT. In this case, the opening of the critical shear crack is correlated to the slab rotation (y, see Fig. 4b) times the effective depth of the member: w µ y × d . This term again accounts in a coupled manner for both the size and strain of the member. According to Muttoni (2008), the failure criterion can be suitably described by the following hyperbolic law (extended information on the background of this equation can be found in Muttoni and Fernández Ruiz [2017]): Vc b0 × d f c Vc b0 × d f c
=
3/ 4
y ×d
1 + 15 =
d dg
9
y ×d
1 + 15
(SI units [MPa, mm])
(Customary units [psi, in])
(5)
d dg
where d refers to the effective depth and b0 refers to the length of the control perimeter (located at d/2 of the edge of the supported area). It can be noted that the failure criterion is similar to that of shear failures, with decreasing shear strength for increasing crack openings (or decreasing crack roughness ddg).
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With respect to the opening of the cracks, however, the behaviour is significantly different to that of oneway slabs. As previously explained, the crack openings in twoway slabs are correlated to the rotation of the slab, which exhibits a marked nonlinear behaviour. This nonlinear loadrotation response of a slab is due to the distribution of bending moments in the slab (in the radial and tangential directions [Muttoni, 2008]) as well as to the momentcurvature diagram of a reinforced concrete section (simplified as a quadrilinear response in Fig. 4a). Integration of the momentcurvature diagram for the acting moments allows the corresponding loadrotation relationship of a slab to be calculated (Muttoni, 2008) as shown in Figure 4b. The loadrotation relationship (Fig. 4b) thus comprises an uncracked regime ((1) in Fig. 4b), followed by a crack development phase with all reinforcements in the elastic regime ((2) in Fig. 4b). Thereafter, local yielding of the reinforcement occurs in the vicinity of the column ((3) in Fig. 4b), but a part of the reinforcement still remains elastic until all reinforcements yield, reaching thus the flexural capacity with the development of a mechanism ((4) in Fig. 4b). As can be observed, both the elastic regime of behaviour and the elasticplastic regime are potentially governing for the punching shear strength and a linear approximation until full yielding of the reinforcement is not realistic (or even unsafe, as load will be overestimated for a given level of rotation in the elasticplastic regime).
Figure 4:
Realistic behaviour of a slab: (a) moment (m) – curvature (1/r) diagram of a slab element with tensionstiffening effects; and (b) loadrotation relationship of a slabcolumn connection with different regimes (calculated assuming a quadrilinear momentcurvature diagram [Muttoni, 2008]).
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Figure 5:
Failures in punching shear: (a) different cases; and (b) comparison to test results for thick and thin slabs with almost identical material parameters and geometrically scaled (tests PG3 and PG10 from Guandalini et al. [2009]).
With respect to tensionstiffening in slabs, it shall be noted that it affects the curvature both in the radial and tangential directions. In the radial direction, the tensionstiffening effect can eventually be reduced due to a delamination crack as for oneway slabs. Nevertheless, in the tangential direction, tensionstiffening is much more relevant as it affects the whole slab until failure. In addition, when the slab continuity is considered (instead of slab sectors near the column region), the tensionstiffening effect outside the column region becomes more dominant and may significantly vary the loadrotation relationship (Einpaul et al., 2015).
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When calculating the punching failure (refer to Fig. 5a), the intersection between the loadrotation relationship (Fig. 4) and the failure criterion (Eq. [5]) is to be determined. This is in fact the same procedure as for shear failures, since failure is defined by the situation in which the shear force and the shear capacity are equal for a given level of rotation. It can be noted that failure in punching usually occurs for the three following regimes (see Fig. 5a): (2) when all the reinforcement remains elastic, (3) when it has partly yielded and partly elastic, and (4) when all reinforcement has yielded (failures in punching at a load level controlled by the bending strength of the slab but the deformation capacity controlled by the failure criterion [Guandalini et al., 2009]). This evidence was confirmed experimentally (Guandalini et al, 2009; Muttoni, 2008) as shown in Figure 5b, where the significance of the tensionstiffening effects can furthermore be clearly noted. The consequence of the nonlinear behaviour of slabs for punching implies that the size effect for large specimens cannot be described by LEFM as for simply supported beams without transverse reinforcement failing in shear. Thus, the dependence of the strength with respect to the size does not correspond to d1/2, but to a milder influence.
3.2
Predicted size effect in punching failures according to the CSCT
By using the CSCT as presented in Fig. 5a, one can predict the influence of size effect on punching accounting for a realistic loadrotation relationship of the slabs (considering the tensionstiffening effects). This is shown in Figure 6a where theoretical predictions are compared to the test results from Guandalini et al. (2009) (scaled specimens with d = 210÷456 mm [8.3÷18 inches]). The influence of size effect is observed to be variable, depending on the size of the specimen. For low sizes, the influence of size effect becomes very low as the behaviour is mostly controlled by limit analysis (yield criterion). For large specimen sizes, the influence of size becomes more significant. Yet, the tensionstiffening effects and the nonlinearity of the loadrotation relationship do not yield to the asymptotic slope predicted by LEFM (1/2 in double logarithmic scale) but to a milder one (approximately 1/3 in double logarithmic scale) (refer to Fig. 6a). It can be noted that the slope also depends on the amount of flexural reinforcement, as it influences the plastic deformations at failure and different regimes of the loadrotation curve may govern (refer to Fig. 4b). In Figure 6b, the punching strength predictions according to the CSCT are also plotted, but neglecting tensionstiffening effects (see Fig. 6bright). The differences compared to Figure 6a are relatively limited for thin slabs. However, the differences are particularly significant for thick slabs. For asymptotically large sizes, and if the tensionstiffening effects are neglected, the slope of the punching shear strength (expressed in a doublelog scale) tends to 1/2 in agreement with LEFM. This holds true as in this case (neglecting tensionstiffening effects), the failure load is reached in the linear part of the slab response (Fig. 6bright; a detailed demonstration of the slope 1/2 obtained for asymptotically large sizes under these conditions is presented in Appendix A of this paper). This result ensures the consistency of the CSCT and LEFM, but it shall be noted that the assumptions of a linearelastic response of the slab deviate significantly from reality (refer for instance to Fig. 5b) and this prediction is not realistic for thick slabs (too strong size effect).
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Figure 6:
4
Predicted size effect by the CSCT and comparison to test series presented in Guandalini et al. (2009) (tests PG2b and PG9 for r = 0.24%; tests PG3 and PG10 for r = 0.33%, tests PG7 and PG19 for r = 0.75% and PG1 and PG6 for r = 1.50%): (a) considering tensionstiffening; (b) without consideration of tensionstiffening effects; and (c) fib MC2010.
Considerations for design – fib MC2010
A design approach based on the CSCT was introduced in fib MC2010 (Muttoni et al., 2013). For the loadrotation relationship, the analytical expression proposed by Muttoni (2008) was simplified (based on a number of assumptions justified in Muttoni et al. [2013]), leading to the following expression: r f y æ ms ö ç ÷ y = 1.5 s d E s çè mR ÷ø
3/ 2
(6)
where rs refers to the distance from the centre of the column to the line of contraflexure of bending moments, fy to the yield strength of the reinforcement, Es to its modulus of elasticity, ms to the average bending moment acting in the support strip of the slab and mR to the average flexural strength in the support strip of the slab.
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The parabola of Eq. (6) has the advantage of approximating the various regimes of the loadrotation relationship with a single expression (see Fig. 6cright), enhancing the ease of use of the approach. With respect to the failure criterion, the coefficients of Eq. (5) were adapted to account for the scatter of test results (Muttoni et al., 2013) and in order to provide a suitable level of safety for design. For design, various LevelsofApproximation (LoA) are proposed in MC2010 to estimate the punching strength of a slabcolumn connection. These levels allow refinement of the physical parameters of the loadrotation curve (such as the distance of the line of contraflexure of bending moments or the acting bending moment in the support strip) and thus the accuracy in the prediction of the strength. The simplest approach proposed in MC2010 (LoA I) aims to verify the dimensions of the slabcolumn connection to verify whether punching is governing. This LoA assumes that all flexural reinforcement yielded (ms = mR) at punching failure, ensuring a ductile behaviour of the connection and providing a potentially safe estimate of the actual strength (as connections may be overreinforced in bending to increase their punching strength). For LoA II and higher, the value of ms is calculated as function of the acting shear force (for instance, for an inner column without moment transfer, the parameter ms in Eq. [6] is estimated as V/8 for LoA II). This approach thus allows determination of the punching strength when the reinforcement in the support strip has not fully yielded. For instance, Figure 6c plots the results of this approach for the previous cases investigated with the CSCT (Figs. 6a,b). It can be noted that the predicted size effect is similar to the one obtained with the CSCT accounting for tensionstiffening effects (Fig. 6a), with an asymptotic slope in doublelog scale of approximately 1/3 (milder again to that of LEFM) when the flexural reinforcement ratio is kept constant and the specimens are geometrically scaled. In the same Figure 6c, a dashed line indicates the condition at which all flexural reinforcement yields at punching failure (regime 4 governing). In fact, this line represents the condition of LoA I, as slabs designed in compliance with LoA I will reach their flexural capacity before failing in punching. A very strong influence of the size of the member is observed for this design condition. This strong influence is justified as, when the size of a specimen increases, the nominal punching shear strength decreases (refer for instance to Fig. 5b) and, in order to ensure full yielding of the flexural reinforcement, the amount of flexural reinforcement has to be reduced accordingly. As a consequence, for LoA I design, increasing the size of the specimen has a consequence both on the size effect (reduced nominal shear strength) and on the strain effect (as a lower flexural reinforcement ratio is associated with larger strains for the same level of acting shear force and thus to lower punching capacities). This is thus not strictly a “size effect”, as the flexural reinforcement ratio is varied instead of being constant, but a combined size and strain effect. With respect to the slope observed for the LoA I (dashed line in Fig. 6c), it can be noted that the rotation leading to full yielding of the reinforcement depends only on the slenderness and on the yield strain of the reinforcement but not on the specimen size (Eq. [6]). The influence of the specimen size is thus directly dictated by Eq. (5), leading in doublelog scale, to a dependence equal to 1/1 for asymptotically large sizes.
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5
Conclusions
This paper presents a discussion on the influence of size effect for failures in punching of twoway slabs and its differences with respect to shear failures in oneway slabs without transverse reinforcement. Its main conclusions are summarized below: 1. According to the Critical Shear Crack Theory (CSCT) it is theoretically consistent that, both for failures in punching and in shear, the significance of size effect varies for members governed by limit analysis (yield or strength criteria) to members failing in a brittle manner. For the former, the influence of size effect is low. However, the influence of size effect becomes significant for brittle failures. 2. For simply supported beams without transverse reinforcement failing in shear, and for asymptotically large sizes, the predictions of Linear Elastic Fracture Mechanics for size effect are consistent and confirmed by the CSCT, corresponding to a slope 1/2 when expressed in double logarithmic scale. This holds true when the response of the member (and its crack openings) are reasonably well approximated by a linear (cracked) behaviour. 3. For failures in shear in members with structural redundancy, which is the predominant case for slabs failing in shear, the actual response of a structure is not a linear relationship between the applied shear force and the crack openings. This yields to the fact that the asymptotic slope for the size effect does not correspond to a value 1/2 (in double logarithmic scale) but to a milder one. 4. For failures in punching shear, the response (and crack openings) of twoway slabs are highly nonlinear and conditioned by tension stiffening effects. These effects do not allow to consider the behaviour of slabs as linear elastic even for very thick slabs. The size effect thus is not suitably approximated by LEFM for asymptotically large sizes; the actual response depends on the mechanical parameters of the slab as well as its static system. 5. For punching shear failures in slabs without transverse reinforcement, the size effect is milder than that predicted by LEFM. Although different regimes can be distinguished, a trend of d1/3 is observed to yield reasonable predictions for thick slabs (contrary to the slope d1/2 according to LEFM).
6
References
ACI Committee 318 (2014) Building Code Requirements for Structural Concrete, ACI 31814, American Concrete Institute, Farmington Hills, Mich., 519 pp. Bažant, Z.P., and Cao, Z. (1987) “Size effect in punching shear failure of slabs,” ACI Structural Journal, Vol. 84, 4453. Bažant, Z.P., Kim, JK. (1984) “Size Effect in Shear Failure of Longitudinally Reinforced Beams,” ACI Journal Proceedings, Vol. 81, No. 5, pp. 456468.
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Cavagnis, F., Fernández Ruiz, M., Muttoni. (2015) “Shear failures in reinforced concrete members without transverse reinforcement: a critical analysis on the basis of test results,” Engineering Structures, Vol. 103, pp. 157173. CEN European Committee for Standardization (2004) Eurocode 2, Design of concrete structures – general rules and rules for buildings, EN 199211, Brussels, Belgium, 225 pp. Einpaul, J., Fernández Ruiz, M., Muttoni, A. (2015) “Influence of moment redistribution and compressive membrane action on punching strength of flat slabs,” Engineering Structures, Vol. 86, pp. 4357. Fernández Ruiz, M., Mirzaei, Y., Muttoni, A. (2013) “Postpunching behavior of flat slabs,” ACI Structural Journal, Vol. 110, No. 5, , p. 801812. Fernández Ruiz, M., Muttoni, A., Sagaseta, J. (2015) “Shear strength of concrete members without transverse reinforcement: a mechanical approach to consistently account for size and strain effects,” Engineering Structures, Vol. 99, , pp. 360372. fib (Fédération internationale du béton) (2013) fib Model Code for Concrete Structures 2010, Ernst & Sohn, Germany, 434 pp. Guandalini, S., Burdet, O., Muttoni, A. (2009) “Punching tests of slabs with low reinforcement ratios,” ACI Structural Journal, Vol. 106, No. 1, pp. 8795. Muttoni, A. (2008) “Punching shear strength of reinforced concrete slabs without transverse reinforcement,” ACI Structural Journal, Vol. 105, No. 4, pp. 440450. Muttoni, A., Fernández Ruiz M. (2008) “Shear strength of members without transverse reinforcement as function of critical shear crack width,” ACI Structural Journal, Vol. 105, No. 2, pp. 163172. Muttoni, A., Fernández Ruiz M. (2017) “The Critical Shear Crack Theory for punching design: from a mechanical model to closedform design expressions,” in fib Bulletin 81, Punching shear of structural concrete slabs, Fédération internationale du béton, Lausanne, Switzerland. Muttoni, A., Fernández Ruiz, M. (2010) “MC2010: The Critical Shear Crack Theory as a mechanical model for punching shear design and its application to code provisions,” in fib Bulletin 57, Shear and punching shear in RC and FRC elements, Fédération Internationale du Béton, Lausanne, Switzerland, pp. 3160. Muttoni, A., Fernández Ruiz, M., Bentz, E., Foster, S.J., Sigrist, V. (2013) “Background to the Model Code 2010 Shear Provisions  Part II Punching Shear,” Structural Concrete, Vol. 14, No. 3, pp. 195203.
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ACIfib International Symposium Punching shear of structural concrete slabs
Appendix A – Consistency between CSCT and LEFM In this appendix it is demonstrated that if a linear elastic behaviour is considered for a slab, the size effect predicted by the CSCT agrees with that of LEFM. This section thus addresses the consistency between theories, but should not be considered as a realistic prediction of the actual size effect (as the actual behaviour of a slab is greatly controlled by tensionstiffening effects and thus the loadrotation relationship is not linear). To demonstrate that the CSCT yields the same size effect as LEFM, the hyperbolic failure criterion for punching of the CSCT (Eq. [5]) must be intersected with a linear response of the slab. For this purpose, the cracked tangent stiffness during the elastic regime of the reinforcement is assumed to be a predictor of the rotations (not accounting for any tensionstiffening effect) (see Fig. 4): V = (b0 × d × t c ) ×y
(7)
where t c (with stress units) defines the slope of the normalized loadrotation curve and can be calculated on the basis of mechanical properties and sizeindependent geometrical parameters of the slab (Muttoni, 2008). Equating the failure load of the loadrotation relationship (Eq. [7]) with the failure criterion (Eq. [5]) results in (V = VRc): n=
VRc b0 × d f c
=
3/ 4 f 15 1+ × c n ×d d 0 + dg tc $g! !#!!"
=
3/ 4 1 +n × a × d
(8)
a
where the coefficient a refers to a set of properties that are independent on the size of the specimen (only dependent thus on the material properties and sizeindependent geometrical parameters). By solving this equation, one can obtain: n=
 1 + 1 + 3a × d 2a × d
(9)
that, for large specimen sizes (d → ∞) turns into: n=
3/ 4 a ×d
(10)
and, expressed in doublelogarithmic scale results: 1 æ 3/ 4 ö 1 logn = logç ÷  log(d ) 2 è a ø 2
Corresponding thus to a slope 1/2 (constant before the term log d).
72
(11)
ACIfib International Symposium Punching shear of structural concrete slabs
Flexureinduced punching of concrete flat plates Ramez B. Gayed1, Chandana Peiris2, Amin Ghali3 1
: University of Calgary and thyssenkrupp Industrial Solutions, Inc., Canada
2
: MMP Engineering Ltd., Calgary, Canada
3
: University of Calgary, Canada
Abstract Local yielding of the top flexural reinforcement above a column can induce punching failure. Slabs supported directly on columns without beams or shear capitals (flat plates) are vulnerable to flexureinduced punching failure when the amount of the flexural reinforcement above the column is not adequate. Provisions of ACI 31814 and CSA A23.314 give the value of the nominal shear strength as function of specified concrete strength, geometry of slab and column and the shear reinforcement. The present research specifies the minimum amount and layout of flexural reinforcement that takes flexureinduced punching into account. These requirements are satisfied in current code design for many practical cases. A code change is proposed to cover all cases.
Keywords Concrete, flat plates, flexure, punching shear, reinforcement, yieldline
1
Introduction
Calculation of the nominal strength of slabs at their connections with columns should take shear and flexural failures into account. Typically, the available parameters for design are: specified concrete strength, geometrical properties of slab and its supports, factored shear force, Vu, and its eccentricities (ex and ey) relevant to the centroid of the shear critical section at d/2 from the column’s periphery; where d = effective slab depth = average of distances from extreme compression fiber to the centroid of tensile reinforcement running in two orthogonal directions. Moment transferred between slab and column, Mscx or Mscy, is calculated as the product Vu times ey or ex, respectively. ACI 31814 gives equations for the nominal shear strength, Vn. The value of Vn = the eccentric shear force transferred between the column and the slab that can produce punching failure. Empirical equations derived from tests are used to calculate Vn. The value of the design shear force, Vu, may produce enough yielding of steel reinforcement to develop a local yieldline mechanism. Yieldline theory—based on principles of equilibrium and compatibility—can give the magnitude of Vflex at which the yieldline mechanism is developed. Unlike punching, flexural failure increases deflection and widens cracks. But, in the vicinity of the column, wide flexural cracks can extend deep into the slab and trigger a secondary failure by punching. This is referred to here as flexureinduced punching. In slabs having low flexural reinforcement ratio, the value of Vflex may control punching strength. In thick slabs, punching can occur at a value, VSize, less than both 73
ACIfib International Symposium Punching shear of structural concrete slabs
Vn and Vflex. A procedure in the fib Model Code 2010 (2013)—based on Muttoni (2008)— calculates shear strength for thick slabs. The present paper includes a summary of this procedure and mentions an empirical equation adopted in the Canadian Standard, CSA A23.314. A code change is recommended in which a minimum amount of flexural reinforcement would be required depending on the required value of Vn. Equations 1 and 2 are requirements in current codes:
Vu £ fshear Vn
(1)
Vu £ f flex V flex
(2)
V n = vn b o d ; V u = vu b o d
(3)
where Vn and vn = nominal shear strength, expressed as a force and a stress, respectively; Vu and vu = factored shear force and stress, respectively; bo = perimeter length of the shear critical section; Vflex = shear force that develops a flexural yieldline mechanism. Equation 1 can be used to give the value of Vn required for a specified value of Vu. The equation given by ACI 31814 for the calculation of Vn is presented further down. Vflex can be determined by yieldline analysis. Accordingly, collapse occurs at ultimate load = Vflex through a system of yield lines (a fracture pattern). The failure occurs due to yielding of the steel crossing the assumed yield lines. The mechanism forms planar slab parts that can rotate freely relative to each other, while maintaining compatible deflection of adjacent parts at common yield lines. An equilibrium equation is then used to calculate Vflex. Failure occurs with the pattern giving the lowest load. Figure 1 shows two admissible yieldline patterns at an interior panel of a flat plate of equal spans. The mechanism in Fig. 1a is for a failure of a panel, referred to here as a global failure. The mechanism in Fig. 1b corresponds to a local failure at an interior square column. For load Vflex (= qu l), uniformly distributed on a slab having isotropic reinforcement in orthogonal directions, equilibrium can be expressed as (Eq. 4 and Fig. 1a):
M o l = m + m¢ = qu l 8 = V flex 8
(4)
where Mo = total factored static moment; l = distance between center lines of columns in orthogonal directions; qu = factored load intensity; m and m¢ = absolute values of ultimate flexural strengths of the slab per unit width in two orthogonal directions at midspan and at the support, respectively. The values of m and m¢ vary almost linearly with the steel reinforcement ratios r and r¢ ; where r = As/(bd) and r¢ = As¢ (b d ), with As and As¢ are, respectively, area of nonprestressed bottom and top reinforcements per unit width, b. Note that both reinforcements are subjected to tension. Here As and As¢ are average values for the yieldline considered. The Direct Design Method of ACI 31814 partitions Mo such that m and m¢ do not induce excessive cracking. The code permits varying the ratio (m m¢) without hampering serviceability. The yieldline analysis, using the same ratio (m m¢), gives the same values of r and r¢ as the Direct Design Method.
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ACIfib International Symposium Punching shear of structural concrete slabs
l/2
l/2
Panel centerline m¢
m¢
m
l
Panel centerline l
(a)
1.13 c m¢
0.4 l
V (up)
m
(b)
Figure 1:
Radial moment = 0
Yieldline patterns. (a) Global failure of an interior panel. (b) Local failure at an interior column.
For a specified Vu, Eq. 2 gives a required value of Vflex. Then Eq. 4 can be used to calculate m and m¢ and the corresponding r and r¢ that take into account the global failure of the considered span. The pattern in Fig. 1a consists of a (positive moment) yieldline at midspan and two (negative moment) yield lines at supports (where the moment strength = m¢ at the two yield lines). When the spans are not equal, the pattern in Fig. 1a and Eq. 4 underestimates the required r and r¢ . The error is commonly less than 5 percent. Here m¢ is taken as average value of ultimate strengths at the two opposite supports. For the local mechanism in Fig. 1b, equilibrium gives Eq. 5 (derived in a separate section).
V flex = 2 p m¢ (1  2.8 c l )
(5)
where c = side length of a square column; the circular column in Fig. 1b has an area = c2. The ultimate flexural strength m¢ is assumed to be the same in all directions by providing the same value of ( As¢ d ) in two orthogonal directions. The corresponding top reinforcement ratio r¢ takes the local failure mechanism into account.
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ACIfib International Symposium Punching shear of structural concrete slabs
As loading is gradually increased in tests, a critical rotation y occurs at a flexural crack in the vicinity of the column and induces punching. An empirical equation can be used to calculate y as function of the applied load and flexural strength (Muttoni, 2008). This can give the value of the load at which flexureinduced punching occurs. When the yieldline mechanism in Fig. 1b is developed at load Vflex, the rotation y can exceed the critical value that induces punching. Here, yieldline analysis is used (Eq. 5) to give a lower bound of the required flexural strength m¢ that takes flexureinduced punching into account. The corresponding flexural reinforcement ratio r¢ gives a minimum value for design against flexureinduced punching: r¢ = a y r¢min
(6)
where ay = factor less than 1.0; r¢ = reinforcement ratio corresponding to Vflex calculated by yieldline analysis; r¢min = r¢ a y = minimum reinforcement ratio to take flexureinduced punching into account by yieldline analysis. The purpose of the magnifying factor (ay)1 and a recommended value of ay are discussed in section 8.
Vn /Vc
Flexureinduced punching has been recognized by researchers without using this terminology (e.g. Muttoni, 2008; Ospina and Hawkins, 2013; Ghali et al., 2013). Ignoring flexureinduced punching would lead to erronuous interpretation of the cause of punching failure and the effectiveness of shear reinforcement in tests.
1.0
C
O
Figure 2:
A
B
D 1.0
r¢min r¢c
Proposed criterion for Vn as function of flexural reinforcement ratio r¢min .
Figure 2 explains a design criterion for the vulnerability of slabs to flexureinduced punching. Line BC, (Vn/Vc) = 1.0, represents the nominal shear strength, Vn, permitted by code (ACI 31814 or CSA A23.314). In the absence of shear reinforcmeent, Vn = Vc; where Vc = nominal shear strength provided by concrete. Point A represents the case in which r¢min r¢c = 1.0 ; where r¢c = the minimum value of r¢ , divided by ay (Eq. 6), that must be provided to allow a nominal shear strength Vn = Vc. Slabs having r¢min < r¢c are vulnerable to flexureinduced punching. Line OA represents a proposed criterion for nominal shear strength for slabs having r¢min £ r¢c . The value of r¢c depends on the requirement for nominal strength, 76
ACIfib International Symposium Punching shear of structural concrete slabs
Vn, to resist a given factored shear force, Vu (Eq. 1). Figure 2 shows Vn for slabs with or without shear or prestressing reinforcement. Examples are presented to calculate the minimum ratio r¢min to allow the value of Vn required by Eq. 1.
2
Research significance
A design for punching is presented that takes flexureinduced punching into account. Current codes, such as ACI 318 and CSA A23.3, calculate the nominal punching strength as a function of the specified concrete strength, the geometry of the slabcolumn connection and the shear reinforcement. In addition, the present research recommends a minimum amount and extent of the top flexural reinforcement mesh above the column. Calculation of the two minima is demonstrated by examples.
3
Proposed code revisions
Minor revisions of codes to take flexureinduced punching into account are proposed. The current American Concrete Institute code for structural concrete, ACI 31814 and the Canadian Standard CSA A23.314 would be affected. Equation 1, giving the value of Vn required for a factored shear force Vu, would be applied. A current code section A would specify the currently permitted value of Vn for a specific slab and the relevant requirements. The revised section A, would require also satisfying a new section B. Section B would apply Eq. 2 to determine the amount and the layout of flexural reinforcement that would take flexureinduced punching into account. In most practical cases, the new section would not require additional flexural reinforcement beyond the current flexural strength requirements.
4
Calculation of Vn according to ACI 31814 ACI 31814 requires that:
vu £ fshear vn
(7)
where the maximum shear stress, vu is given by:
(
)
vu = [Vu (bo d )] + (g vx M sc x J cx ) y + g vy M sc y J cy x
(8)
with gvx or gvy = fraction of Mscx or Mscy transferred between slab and column by eccentricity of shear; Jcx or Jcy = d multiplied by the second moment of the perimeter about xor yaxis, respectively; (x, y) = coordinates of a point on the perimeter of the shear critical section with respect to its centroidal principal axes at which vu is calculated. The value of vn is given by Eq. 9 for slabs without shear reinforcement and by Eqs. 10 and 11 for slabs with shear reinforcement. f¢ æ a dö 4 vn º vc = least of : çç 4 ; 2 + ; 2 + s ÷÷ l c MPa b bo ø 12 è
éæ ù as d ö 4 ÷÷ l f c¢ psi ú êçç 4 ; 2 + ; 2 + b bo ø úû ëêè
(9)
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ACIfib International Symposium Punching shear of structural concrete slabs
( v (= 0.25 l
) f ¢ + v ) £ 0.67 l
[(2 l f ¢ MPa [(3 l
vn = 0.17 l f c¢ + vs £ 0.5 l f c¢ MPa n
c
s
c
vs = Vs (bo d ) = Av f yt (bo s )
)
f c¢ + vs £ 6 l f c¢ psi
)
]
f c¢ + vs £ 8 l f c¢ psi
]
üï ý Headed shear studsïþ
Stirrups
(10) (11)
where vc and vs = nominal shear strength provided by concrete and shear reinforcement, respectively; b = aspect ratio of column (≥ 1.0); as = 40, 30 or 20 for interior, edge or corner columns, respectively; l = factor reflecting the reduced mechanical properties of lightweight concrete; for simplicity of presentation, only slabs made of normalweight concrete are discussed here (i.e. l = 1); f c¢ = specified compressive strength of concrete; Av = crosssectional area of vertical legs of shear reinforcement on one peripheral line parallel to the column face; fyt = specified yield strength of shear reinforcement; s = spacing between peripheral lines of shear reinforcement. When shear reinforcement is needed, Eq. 7 must be satisfied at a shear critical section at d/2 from the outermost peripheral line of shear reinforcement with: vn º vc = 0.17 l f c¢ MPa
[2 l
f c¢ psi
]
(12)
For prestressed slabs, ACI 31814 specifies the value of Vn with a different equation, which is not addressed here.
5
Ductility
In seismic design, connections need to exhibit extensive deformations without losing their strength to support gravity loads. The flexural reinforcement in the slab adjacent to the column should reach yielding due to drift reversals without punching failure. This can be achieved in design by satisfying both Eqs. 2 and 13 (Ghali et al., 2015).
f shear Vn = a d f flex V flex
(13)
where ad = ductility factor whose value is greater than 1.0, depending on the desired level of ductility (intuitively, ad = 1.2). Substituting fflex Vflex from Eq. 13 into Eq. 2 gives: Vu £ fshear (Vn ad )
(14)
The design criteria for higher ductility satisfy the inequalities of Eqs. 2 and 14. The improved ductility results from allowing some deformation without punching when Vflex is approached. For a given value of Vu, improved ductility (Eq. 14) would require ensuring higher shear strength Vn; this is typically achieved by providing shear reinforcement. For seismic design of slabcolumn connections, ACI 421.2R10 recommends Eq. 15 that provides a minimum amount and extension (≥ 3.5d) of shear reinforcement when Vu/(fVc) exceeds 0.4 or (0.70  20 DRu); with Vc (= vc bo d) being the shear strength provided by concrete in absence of shear reinforcement (Eq. 9); DRu = design storey drift ratio of a connection = differential lateral drifts of the stories above and below the considered floor divided by the storey height. vs ³ 0.25 f c¢ MPa
78
[3
f c¢ psi
]
(15)
ACIfib International Symposium Punching shear of structural concrete slabs
6
Minimum flexural reinforcement ratio to take flexureinduced punching into account
The design presented here is for slabs with d not exceeding eight inches (203 mm). In slabs with low flexural reinforcement ratio, flexure can govern punching shear strength (Eq. 2). The value of Vflex satisfying Eq. 2—determined by yieldline analysis—can be used to calculate rmin and r¢min with their layout to avoid flexureinduced punching. The yieldline theory was developed by Johanson (1932, translation 1962). The theory is available in textbooks and technical papers (e.g. Ghali and Neville, 2016; Simmonds and Ghali, 1976; Gesund and Goli, 1979). An assumed collapse mechanism gives an upper bound of Vflex. In other words, the equilibrium equation of an assumed yield pattern can overestimate the value of Vflex. Also, flexureinduced punching can occur at a smaller force than the value determined by yieldline analysis. To compensate for the two effects, the magnification factor (αy)1 is used in Eq. 6 to give a safe design value for r¢min against flexureinduced punching. The value of αy is discussed in a separate section. Equation 5 gives the value of Vflex for a square interior column transferring concentric shear force (i.e. with ex = ey = 0). The corresponding yieldline pattern is shown in Fig. 1b. The pattern suitable when ex >> 0 and ey = 0 is considered later (Fig. 8). At an interior square column of side c subjected to concentric shear force, the value Vflex that would develop a local yieldline mechanism may be calculated by Eq. 5 (Fig. 1b). To derive Eq. 5, the square column is replaced by a circular column of the same crosssectional area; zero radial moment is assumed at a circle of radius = 0.2l; where l = distance between column centerlines in two orthogonal directions; the bottom flexural reinforcement is ignored; m¢ = ultimate flexural strength of the slab per unit width in two orthogonal directions.
[
(
)]
m¢ = r¢ f y d 2 1  0.59 r¢ f y f c¢
(16)
Equations 5 and 16 can be solved for the reinforcement ratio, r¢ = As¢ (b d ) . r¢ is the flexural reinforcement ratio corresponding to m¢ giving Vflex that satisfies Eq. 2. The minimum top flexural bars above the column, in two orthogonal directions, should cover the yieldline pattern plus development lengths. A change of ACI 31814 is recommended here to require minimum top flexural reinforcement ratio, satisfying Eq. 6. The recommended design requires specified shear and flexural reinforcements to reach or exceed the factored shear force, Vu. Calibration of the proposed code change is presented below. Calculation of the minimum reinforcement that takes flexureinduced punching into account is the same for both seismic and nonseismic design. To include the effect of prestressing, the flexural strength m¢ would be calculated by a substitution of Eq. 16. Steps for calculating r¢min Design for the punching shear strength of flat plates with d ≤ 8 in. (203 mm) according to ACI 31814 or CSA A23.314, with the proposed revisions, would require a minimum reinforcement ratio, r¢min , as top mesh above columns. r¢min can be calculated in three steps:
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ACIfib International Symposium Punching shear of structural concrete slabs
Step 1: Equation 1 gives the required nominal strength Vn for a given Vu. Verify that the design satisfies the current code requirement (adequate strength of concrete and shear reinforcement). Step 2: Equation 2 gives the requirement for Vflex provided by the reinforcement in the column vicinity. Step 3: Use Eq. 5 (or equivalent) to calculate m¢ , the absolute value of flexural strength provided by the top reinforcement above the column. Calculate the corresponding reinforcement ratio r¢ (Eq. 16 or equivalent). Use the factor ay to determine r¢min by Eq. 6. The equivalent to Eq. 5 can be an analysis respecting the principles of equilibrium and compatibility. The equivalent to Eq. 16 may take into account the presence of prestressed tendons. For a required value of Vn smaller than Vc, r¢min would be a value given by (Fig. 2): r¢min = r¢c (Vn Vc )
(17)
Equation 2 gives the requirement for Vflex; this is considered the required shear strength, Vn. Thus, r¢c can be determined first for Vflex = Vn = Vc and r¢min calculated by linear interpolation, Eq. 17.
7
Size effect
Testing of slabs having d > 8 in. (203 mm) shows that punching shear strength permitted by ACI 31814 is often not reached as the depth increases; also, punching failure can occur before the full development of a yieldline mechanism. Thus, low reinforcement ratio combined with relatively large thickness is detrimental to the punching shear capacity. Because reported punching tests of thick slabs are limited, only a few tests are compiled below to show that relatively thick slabs fail by punching at a Vnvalue less than permitted by Eq. 9. The Canadian Standard CSA A23.314 specifies an empirical reduction factor for slabs with d > 300 mm (11.8 in.):
Reduction factor for vn = 1300 [1000 + d (mm)] £ 1.0
(18)
The Critical Shear Crack Theory (CSCT) developed by Muttoni (2008) is summarized here to calculate the concentric shearing strength VSize for thick slabs. Based on tests, Eqs. 19 and 21 are formulated to calculate the punching strength VSize. The curve OAB in Fig. 3 represents the relationship between the shear force, V and y; where y = rotation of the slab at a flexural crack in the vicinity of the column.
(
)(
y = 0.33 l f y d Es V V flex
V flex
80
)
32
ì 8 for interior columns ü ï ï @ m¢ × í4 for edge columns ý ï 2 for corner columns ï î þ
(19)
(20)
ACIfib International Symposium Punching shear of structural concrete slabs
where Es and fy = modulus of elasticity and specified yield strength of flexural reinforcement. The failure criterion is expressed by: VSize bo d
æ 15 y d = 0.75 ç1 + ç dg0 + dg f c¢ è
ö ÷ ÷ ø
(21)
where dg0 = reference size = 16 mm (0.63 in.); dg = maximum aggregate size. The curve CAD in Fig.3 represents VSize versus y using Eq. 21. The intersection of the two curves in Fig. 3 gives VSize. VSize
é 4.95 l f y = ê1 + ê Es d g 0 + d g ë
(
1.5
)
æ VSize ö ç ÷ ç V flex ÷ è ø
1
ù ì0.75 b d f ¢ (MPa) o c ú ´ ïí ú ï9 bo d f c¢ (psi) û î
(22)
Shear force, V
Equation 22 is adopted in the present paper only for thick slabs (with d > 8 in. (203 mm)).
C B Eq. 19 A
VSize
Eq. 22
Eq. 21 D O
yfailure
Figure 3:
8
Joint rotation, y
Calculation of VSize using CSCT procedure (Muttoni, 2008). (1 in. = 25.4 mm)
Magnification factor (αy)1
The factor αy is used to account for two approximations involved in the use of yieldline analysis to calculate r¢min that safeguard against flexureinduced punching. An assumed yieldline pattern gives an upper bound of Vflex . Thus, the value of Vflex can be overestimated. Rarely, flexureinduced punching occurs at a load slightly less than the approximate value Vflex calculated by yieldline analysis. The value αy = 0.9 is adopted here to correct the two sources of approximation. The calibration with published test results in the following section indicates that the adopted value of αy is suitable.
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ACIfib International Symposium Punching shear of structural concrete slabs
9
Calibration of proposed design taking flexureinduced punching into account
The proposed design taking flexureinduced punching into account is calibrated in the present section using provisions of ACI 318 and published test results. Table 1 (at the end of this paper) lists data of test slabs without shear reinforcement, subjected to concentric or eccentric shear force. The ultimate strength reached in the tests, VTest is normalized with respect to Vc = bo d f c¢ 3 N and mm (Columns 8 and 9). Figure 4 plots the normalized strength versus the flexural reinforcement ratio in the tests normalized with repect to the minimum ratio recommended to take flexureinduced punching into account (r¢Test r¢min ) (columns 11 and 12 of Table 1). Points above the horizontal line CB, having (VTest/Vc) > 1.0, represent safe design permitted by the current code. The code needs revision for the cases represented by points below line CB. The proposed revision would replace line CB, as limit for Vn, by the two straight lines OA and AB. Line OA represents flexureinduced punching for slabs having d ≤ 8 in. (203 mm) and a relatively small flexural reinforcement ratio (the majority of points within OCAD). Three tests, represented by squares in Fig. 4, have d = 260, 400 and 500 mm (10.2, 15.8 and 19.7 in.). These points indicate that ACI 31814 needs a revision to take into account the size effect.
(
)
The value Vn permitted by the revised code would be safe for the cases represented by points within CAO. Four points to the right of the vertical line AD represent cases in which the revised code would slightly overestimate Vn.
VTest Vc 2.0 1.8 1.6
1.4 VTest / Vc
1.2 1.0
C
B
A
0.8 0.6
Thick slabs with d > 8 in. (203 mm)
0.4
Eccentric shear test Concentric shear test
0.2 0.0
O 0.0
D 0.5
Figure 4:
82
1.0
1.5 2.0 / ¢r' rr'¢Test Test r min min
2.5
3.0
3.5
Calibration of recommended design for r¢min . Slabs without shear reinforcement subjected to concentric or eccentric shear force.
ACIfib International Symposium Punching shear of structural concrete slabs
VTest Vc 2.2 2.0
1.8 1.6
VTest / Vc
1.4 1.2
1.0
B
A
C
0.8 0.6
Eccentric shear test Concentric shear test
0.4
0.2 0.0
O 0.0
D 0.5
1.0
1.5 2.0 r'¢Test /rr'¢ min r Test
Figure 5:
2.5
3.0
3.5
min
Calibration of recommended design for r¢min . Slabs with shear reinforcmeent, subjected to concentric or eccentric shear force.
Table 2 (at the end of this paper) lists data of test slabs with shear reinforcement, subjected to concentric or eccentric shear force. The ultimate strength reached in the tests, VTest, is normalized with respect to ( bo d f c¢ 3 , N and mm = the nominal strength permitted by ACI 31814 in absence of shear reinforcement) (Columns 8 and 9). Figure 5 plots the normalized strength versus the flexural reinforcement ratio in the tests normalized with repect to the minimum ratio recommended to take flexureinduced punching into account (r¢Test r¢min ) (columns 11 and 12 of Table 2). All points in Fig. 5 are above line BC; i.e. VTest Vc > 1.0 . This can justify the nominal strength value permitted by ACI 31814 for slabs with shear reonforcement. All points in Fig. 5 represent tests having (r¢Test r¢min ) greater than 1.0, except test V1 by Stein et al. (2007). This is because in those tests, the flexural reinforcement was designed to be sufficient to exclude flexure as a cause of failure. Thus, flexureinduced punching is avoided. All tests considered in Figs. 4 and 5 show that ACI 31814, revised as proposed, would permit safe nominal punching strength.
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ACIfib International Symposium Punching shear of structural concrete slabs
Slab corners are free to uplift Slab simply supported Axis of rotation on four edges
Slab corners are restrained against uplift Slab simply supported on four edges c
ls l1
c
Vu (up) ls l1
y
l1
c
p/8
c
x
c
y
ls
Pinned supports
a
x
c
Axis of rotation
ls
(a)
ls
Vu (up)
y
l1
x
Vu (up)
ls
(b)
(c)
Points of supports Pinned supports c
4c/p
x Vu (up)
r¢Test
c
ls l1
x y
y
Axis of rotation
lB
lA
ls = 2 rs
(d)
Figure 6:
Vu (up)
lB
l1 ls
(e)
Yieldline patterns of slab tests subjected to concentric shear force and reported by: (a) Peiris and Ghali (2012), Mokhtar (1982) and Stein et al. (2007). (b) Marzouk et al. (2010), Rizk (2010) and Criswell (1974), Li (2000), Bompa and Onet (2011) and Pisanty (2005). (c) Guandalini et al. (2009). (d) Birkle and Dilger (2008). (e) Widianto et al. (2009).
The yieldline patterns used to calculate (r¢Test r¢min ) in Tables 1 and 2 depend on the test setup. Figures 6 to 8 present the assumed yieldline patterns. Appendix A and Eq. 24 give the correspomding equations for Vflex.
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ACIfib International Symposium Punching shear of structural concrete slabs
Axis of rotation
Slab simply supported on four edges
c
c
Vu (up) x
c
ls l1
Pinned supports
c
l2
y
y
l1 ls
c
l2
x y
l1
l1
ls
(a)
(b) Axis of rotation
P/4 (up)
4B (V)
P/4 (down)
4A + 8B = 1 2B (V) 2B (V)
Vu /2 (down) c
Vu (up)
(c) 4A (V)
l1/3
c
x
c
l2
x
c
ls
l1/3
y
V (up)
y
M = P l1
l1/3
M
l1
lv
(d)
c
Vu (up)
Mflex
l1
Figure 7:
c
Mflex
Mflex
ls
Points of supports
Vu (up) x
l1/3
l1/3
l1/3
(e)
Yieldline patterns of slab tests subjected to eccentric shear force and reported by: (a) Elgabry (1991) and Stein et al. (2007). (b) Tian (2007), Tian et al. (2008) and Pan and Moehle (1989 and 1992). (c) Robertson and Johanson (2004). (d) Islam and Park (1976) and Hawkins et al. (1974 and 1975). (e) Hawkins et al. (1989). These yieldline patterns are compared with the one shown in Fig. 8.
1.6 c 1.6 c e(0.15 q) Vu ex
q Vu (up)
c
Vu ex
m
x
A
A y
Vu
m′ Top view
Figure 8:
Section AA
Yieldline pattern at an interior column transferring shear force, Vu with eccentricity ex (Gesund and Goli, 1979).
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ACIfib International Symposium Punching shear of structural concrete slabs
10
Examples
Example 1: An interior column transferring concentric shear force Design the minimum flexural reinforcement that takes into account flexureinduced punching of a flat plate at an interior column transferring a concentric factored shear force, Vu = qu l2; with l = span in two orthogonal directions between centers of columns = 22 ft (6.706 m); slab thickness, h = 8 in. (203 mm); d = 6.625 in. (168 mm); square column of side, c = 16 in. (406 mm); concrete compressive strength, f c¢ = 4000 psi (27.6 MPa); specified yield strength of steel bars, fy = 60000 psi (414 MPa). Factored uniform load, qu = 250 lb/ft2 (11.97 kN/m2). Assume that, with shear reinforcement, the shear strength Vn satisfies Eq. 1. The required strength is: Vu = qu l2 = 0.250 (22)2 = 121.0 kips (538.2 kN) Equations 1 and 2 give: Vn ≥ Vu/fshear = 121.0/0.75 = 161.3 kips (717.5 kN) Vflex ≥ Vu/fflex = 121.0/0.9 = 134.4 kips (597.8 kN) Vflex expressed in terms of the flexural reinforcement ratio r¢ is (Eq. 5):
V flex = 2 p m¢ (1  2.8 c l ) æ 2.8 ´ 16 ö 134.4 = 2 p m¢ ç1 ÷ ; m¢ = 17.76 kips (79 kN) 22 ´ 12 ø è
Solve Eq. 16 for r¢ , expressed as:
[
(
)
r¢ = f y d 2  f y2 d 4  4 0.59 f y2 d 2 f c¢ m¢
]
(
)
2 0.59 f y2 d 2 f c¢
(23)
Substitution in Eq. 23 gives: r¢ = 0.00726. To take flexureinduced punching into account, a mesh of top flexural reinforcement with average ratio (Eq. 6) r¢min = 0.00726 a y in two orthogonal directions is required. The mesh should cover circular area of diameter = 0.4 l + anchorage lengths = 8.8 ft (2.682 m) + anchorage lengths. Example 2: An interior column transferring eccentric shear force Design the minimum flexural reinforcement that takes into account flexureinduced punching of a flat plate at an interior column transferring a factored shear force, Vu = 121 kips (538.2 kN) combined with eccentricity ex or ey = 13.88 in. (353 mm) due to wind (i.e. Mscx or Mscy = 1680 kipin. (190 kNm)). Spans in two orthogonal directions between centers of columns, l = 22 ft (6.706 m); slab thickness, h = 8 in. (203 mm) and d = 6.625 in. (168 mm); square column of side, c = 16 in. (406 mm); normalweight concrete of a specified compressive strength, f c¢ = 4000 psi (27.6 MPa); specified yield strength of steel bars, fy = 60000 psi (414 MPa). Assume that shear reinforcement is provided to satisfy Eq. 1.
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ACIfib International Symposium Punching shear of structural concrete slabs
The moment (Vu times ex or ey) transferred between an interior square column and a slab— that, when combined with Vu, produces the yieldline pattern shown in Fig. 8—is calculated as (Gesund and Goli, 1979):
Vu (ex or e y ) = 8 c (m¢ + m ) 
c Vu 2
(24)
The eccentric shear force transferred between column and slab should satisfy Eq. 2. Substituting fflex = 0.9 in Eq. 2 gives: Vflex ≥ 121/0.9 = 134.4 kips (597.8 kN). Then, Eq. 24 gives: 134.4 ´13.88 = 8 ´ 16 (m¢ + m ) m¢ + m = 22.974 kips
16 ´134.4 2
Assume m¢ = 3 m = 17.23 kips (76.64 kN). Substitution of m¢ and d in Eq. 16 and application of Eq. 6 gives: r¢min = 0.0070 a y and r min = 0.00232 a y . The length of the flexural bars within the width of the yieldline pattern in Fig. 8 should be approximately (2.6 c) plus a development length from each side. The calculated minimum is a design step to take flexureinduced punching into account. It is necessary to verify that the flexural reinforcement in the final design satisfies also current code requirements. Example 3: Minimum flexural reinforcement at an edge column At an edge column, the factored shear force due to gravity and lateral forces is: Vu = 30 kips with eccentricity ex = 33 in. (133 kN with ex = 0.84 m). Design the minimum flexural reinforcement ratio r¢min taking into account flexureinduced punching. Slab thickness, h = 8 in. (203 mm) and d = 6.625 in. (168 mm); square column of side, c = 16 in. (406 mm); concrete specified compressive strength, f c¢ = 4000 psi (27.6 MPa); specified yield strength of steel bars, fy = 60000 psi (414 MPa). Step 1: Equation 1 gives: Vn ≥ Vu /fshear = 30/0.75 = 40.0 kips. Verification that the strength of concrete and shear reinforcement is adequate should be done. Step 2: Equation 2 gives: Vflex ≥ Vu / fflex = 30/0.9 = 33.3 kips. Step 3: Consider the yieldline pattern in Fig. 9b, with isotropic top and bottom reinforcement. Equilibrium gives (using virtual work, Ghali and Neville, 2016):
(V
flex
) (
)
ex c  V flex 2 = 5 m¢ + 2 m
(25)
where ex = ex + 5.22 = 38.22 in. Choose m¢ = 4 m and solve for m¢ :
(33.3´ 38.22 16)  (33.3 2) = 5.5 m¢
; m¢ = 11.44 kips
The ultimate flexural strength in any two orthogonal directions is (Eq. 16):
[
(
)]
m¢ = r¢ f y d 2 1  0.59 r¢ f y f c¢
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ACIfib International Symposium Punching shear of structural concrete slabs
Solving this equation gives (Eq. 23): r¢ = 0.0045. Equation 6 gives: r¢ = a y r¢min . The minimum flexural reinforcement is: r¢min = r¢ a y = 0.0045 0.9 = 0.0050
The length of the flexural bars should extend for a development length beyond the yieldline pattern in Fig. 9b and should be properly anchored at the free edge. Free edge
Free edge
22.625
q = 45°
ex = 33 6.089
5.224
x
O
m¢
m
c = 16 in.
m¢
O
x
Vu (up) = 30 kips
c = 16
y
y
(a)
Figure 9:
m¢
m
19.3125
(b)
Edge column of Example 3. (a) Shear critical section at d/2 from column periphery. (b) Yieldline pattern used to determine m¢ (Eq. 25). (1 kip = 4.448 kN; 1 kipin. = 0.113 kNm; 1 in. = 25.4 mm)
Example 4: Corner columnslab connection with shear reinforcement Consider a flat plate at a corner column (Fig. 10a). Design the minimum flexural reinforcement ratio, r¢min to take into account flexureinduced punching. Due to gravity load combined with wind force in the y direction, the column transfers to the slab a force Vu = 12 kips (53 kN) with eccentricities ex = 30 in. (762 mm); e y = 42.5 in. (1080 mm). Slab thickness, h = 8 in. (203 mm) and d = 6.625 in. (168 mm); square column with side, c = 16 in. (406 mm); concrete specified compressive strength, f c¢ = 4000 psi (27.6 MPa); specified yield strength of reinforcement, fy = 60000 psi (414 MPa). Assume that design of shear reinforcement satisfies Eq. 1 with the requirements of ACI 31814. Consider the yieldline pattern in Fig. 10b with isotropic top reinforcement. The angle q is chosen such that tan q = ex ey = 0.706; q = 35.22°. Equilibrium gives:
(
)
ey Vu = m¢ c sin 1 q + cos 1 q + Vu y A
ey = ex sin q + ey cos q ; ex = 0 ey = 52.02 in. ; ex = 0; yA = 11.15 in.
(
)
52.02 ´12 = 16 m¢ 0.577 1 + 0.817 1 + 12 ´11.15 ; m¢ = 10.37 kips
Solve for r¢ by Eq. 23: r¢ = 0.0041. Using Eq. 6, r¢min = r¢ a y = 0.0041 0.9 = 0.0046 .
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(26)
ACIfib International Symposium Punching shear of structural concrete slabs
ex = 30 in.
Sq. column, c = 16 in.
x
x
O
O
q = 35.22°
m¢
A
ey = 42.5
y
y Vu (up) = 12 kips
Vu (up) = 12 kips
Free edge (a)
(b)
Figure 10: Corner column of Example 4. (a) Vu and its eccentricities. (b) Assumed yieldline pattern. (1 kip = 4.448 kN; 1 kipin. = 0.113 kNm; 1 in. = 25.4 mm)
11
Summary and conclusions
Punching has occurred in few tests, having relatively low flexural reinforcement, at a load smaller than the nominal shear strength permitted by American Concrete Institute code, ACI 31814 or the Canadian Standard CSA A23.3. For slabs with d not exceeding 8 inches (203 mm), premature punching is attributed to the development of a pattern of yield lines in the area above the column. A flexural crack adjacent to the column can extend deeply in an inclined direction through the slab depth inducing punching. Flexureinduced punching can be avoided by verifying that the mesh of top flexural reinforcement above the column is not less than a specified minimum. Yieldline analysis is used to determine the minimum reinforcement ratio and how far the mesh should extend over the area above the column. The minimum amount and the extension of the flexural reinforcement above the column should be code requirements for the validity of equations of permitted shear strength. For slabs with d exceeding 8 in. (203 mm), a design procedure by Muttoni (2008) is reviewed. The proposed design, with examples, is presented using only equations from ACI 31814.
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12
References
American Concrete Institute Committee 318 (2014) “Building Code Requirements for Structural Concrete, ACI 31814 and Commentary,” American Concrete Institute, Farmington Hills, MI 48331, USA. Birkle, G. and Dilger, W.H. (2008) “Influence of Slab Thickness on Punching Shear Strength,” ACI Structural Journal, Vol. 105, No. 2, Mar.Apr., pp. 180188. Bompa, D.V. and Onet, T. (2011) “Failure Analysis of Symmetric Flat SlabColumn Connections with Shear Reinforcement,” Proceedings, fib Symposium Session: New Model Code, Prague, pp. 171174. Canadian Standard Association (2014) Design of Concrete Structures, CSA A23.314, 5060 Spectrum Way, Suite 100, Mississauga, Ontario, Canada L4W 5N6. Criswell, M.E. (1974) “Static and Dynamic Response of Reinforced Concrete SlabColumn Connections,” Shear in Concrete ACI SP42, American Concrete Institute, Farmington Hills, MI, pp. 721746. Elgabry, A.A. (1991) Shear and Moment Transfer of Concrete Flat Plates, PhD Thesis, Department of Civil Engineering, University of Calgary, 266 pp. Fédération internationale du béton (fib) (2013) fib Model Code for Concrete Structures 2010. Ernst & Sohn, Berlin, Germany, 437 pp. Gesund, H. and Goli, H.B. (1979) “Limit Analysis of Flat Slab Buildings for Lateral Loads,” Journal of the Structural Division, ASCE, Vol. 105, No. ST11, pp. 21872202. Ghali, A., Dilger, W.H. and Gayed, R.B. (2013) “Punching of Concrete Slabs: Interpretation of Test Results,” Journal of Structural Engineering, ASCE, Vol. 139, No. 6, pp. 869874. Ghali, A., Gayed, R.B. and Dilger, W.H. (2015) “Design of Concrete Slabs for Punching Shear: Controversial Concepts,” ACI Structural Journal, Vol. 112, No. 4, pp. 505514. Ghali, A. and Neville, A. (2016) Structural Analysis: A Unified Classical and Matrix Approach, 7th Edition (in print), CRC Press, Taylor and Francis Group. Guandalini, S., Burdet, O. and Muttoni, A. (2009) “Punching Tests of Slabs with Low Reinforcement Ratios”, ACI Structural Journal, Vol. 106, No. 1, pp. 8795. Hawkins, N.M., Mitchell, D. and Sheu, M.S. (1974) “Cyclic Behavior of Six Reinforced Concrete SlabColumn Specimens Transferring Moment and Shear,” Progress Report 197374 on NSF Project GI3817, Department of Civil Engineering, University of Washington, Seattle. Hawkins, N.M., Mitchell, D. and Hanna, S.N. (1975) “The Effects of Shear Reinforcement on the Reversed Cyclic Loading Behavior of Flat Plate Structures,” Canadian Journal of Civil Engineering, Vol. 2, pp. 572582. Hawkins, N.M., Bao, A. and Yamazaki, J. (1989) “Moment Transfer from Concrete Slabs to Columns,” ACI Structural Journal, Vol. 86, No. 6, pp. 705716. Islam, S. and Park, R. (1976) “Tests on SlabColumn Connections with Shear and Unbalanced Flexure,” Journal of the Structural Division, ASCE Proceedings, Vol. 102, No. ST3, pp. 549567. Johansen, K.W. (1962) (translation). YieldLine Theory, Cement and Concrete Association, London, England, pp. 181.
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ACIfib International Symposium Punching shear of structural concrete slabs
Joint ACIASCE Committee 421 (2010) Guide to Seismic Design of Punching Shear Reinforcement in Flat Plates (ACI 421.2R10), American Concrete Institute, Farmington Hills, MI, 30 pp. Li, K.K.L. (2000) Influence of Size on Punching Shear Strength of Concrete Slabs, M.Eng. Thesis, Department of Civil Engineering and Applied Mechanics, McGill University, Montreal, Quebec, 78 pp. Marzouk, H., Rizk, E., Hussein, A., and Hossin, M. (2010) “Effect of Reinforcement Ratio on Punching Capacity of RC Plates,” Proceedings of 2nd International Specialty Conference, Winnipeg, Manitoba, June. Mokhtar, A. (1982) Design of Stud Shear Reinforcement for Concrete Flat Plates, PhD Thesis, Department of Civil Engineering, University of Calgary, 140 pp. Muttoni, A. (2008) “Punching Shear Strength of Reinforced Concrete Slabs Without Transverse Reinforcement,” ACI Structural Journal, Vol. 105, No. 4, pp. 440450. Ospina, C. E. and Hawkins, N.M. (2013) “Addressing Punching Failure,” Structure Magazine, National Council of Structural Engineers Associations, Vol. 20, No. 1, pp. 1416. Pan, A. and Moehle, J.P. (1989) “Lateral Displacement Ductility of Reinforced Concrete Flat Plates,” ACI Structural Journal, Vol. 86, No. 3, pp. 250258. Pan, A. and Moehle, J.P. (1992) “An Experimental Study of SlabColumn Connections,” ACI Structural Journal, Vol. 89, No. 6, pp. 626638. Peiris, C. and Ghali, A. (2012) “Flexural Reinforcement Essential for Punching Shear Resistance of Slabs,” ACI SP287: Recent Development in Reinforced Concrete Slab Analysis, Design and Serviceability, American Concrete Institute, Farmington Hills, MI, pp. 8398. Pisanty, A. (2005) “Euro Codes and North American Codes Predictions of Punching Shear Capacity in View of Experimental Evidence,” ACI SP232, American Concrete Institute, Farmington Hills, MI, pp. 3955. Rizk, E.R.M. (2010) Structural Behaviour of Thick Concrete Plates, PhD Thesis, Faculty of Engineering and Applied Science, Memorial University of Newfoundland, July, 237 pp. Robertson, I. and Durrani, A.J. (1992) “Gravity Load Effect on Seismic Behavior of Interior SlabColumn Connections,” ACI Structural Journal, Vol. 89, No. 1, Jan.Feb., pp. 3745. Robertson, I. and Johnson, P. (2004) “Nonductile SlabColumn Connections Subjected to Cyclic Lateral Loading,” 13th World Conference on Earthquake Engineering, Proceedings, Vancouver, BC, Canada, Paper No. 143. Simmonds, S.H. and Ghali, A. (1976) “YieldLine Design of Slabs”, ASCE Journal of the Structural Division, Vol. 102, Issue 1, pp. 109123. Stein, T., Ghali, A. and Dilger, W.H. (2007) “Distinction between Punching and Flexural Failure Modes of Flat Plates,” ACI Structural Journal, Vol. 104, No. 3, pp. 357365. Tian, Y. (2007) Behaviour and Modeling of Reinforced Concrete SlabColumn Connections, PhD thesis, University of Texas at Austin, 209 pp. Tian, Y, Jirsa, J., Bayrak, O., Widianto and Argudo, J.F. (2008) “Behaviour of SlabColumn Connections of Existing FlatPlate Structures”, ACI Structural Journal, Vol. 105, No. 5, pp. 561569. Widianto, Bayrak, O. and Jirsa, J.O. (2009) “TwoWay Shear Strength of SlabColumn Connections: Reexamination of ACI 318 Provisions”, ACI Structural Journal, Vol. 106, No. 2, pp. 160170. 91
ACIfib International Symposium Punching shear of structural concrete slabs
Table 1:
Interior slabcolumn connections without shear reinforcement subjected to concentric/eccentric shear force
1
2
3
ID
h (mm)
d (mm)
Birkle and Dilger (2008) 1 160 124 7 230 190 10 300 260 Peiris and Ghali (2012) S1 150 118 Mokhtar (1982) AB1 150 115 Marzouk et al. (2010) NSC1 200 154 NSC2 200 145 NSC3 150 98 HSC1 200 154 HSC2 200 151 HSC3 200 145 HSC4 200 145 HSC5 150 125 HSC6 150 115 HSC7 150 104 Criswell (1974) S20751 165 121 S20752 165 122 S21501 165 124 S21502 165 122 S40751 165 127 S40752 165 124 S41501 165 125 S41502 165 125 D20751 165 122 D20752 165 122 D20753 165 125 D21501 165 126 D21502 165 125 D21503 165 125 D40751 165 127 D40752 165 127 D40753 165 127 D41501 165 125 D41502 165 125
92
4
5
6
7
8 9 VTest r¢Test / f c¢ c fy (kN)/ Vc rTest (mm) (MPa) (MPa) ex (kN) (%) (mm) Tests subjected to concentric shear force
10
11
12
r¢min (%)
VTest Vc
r¢Test r¢mi n
250 300 350
36.2 35.0 31.4
1.54/— 1.30/— 1.10/—
488 531 524
483/0 825/0 1046/0
372 734 1185
0.64 0.54 0.49
1.30 1.12 0.88
2.17 2.16 2.01
250
28.4
0.43/—
434
252/0
309
0.55
0.82
0.70
250
36.2
1.40/—
516
408/0
337
0.53
1.21
2.39
250 250 250 250 250 250 250 250 250 250
33 35 34 60 61 61 67 69 70 70
0.52/— 2.17/— 0.40/— 0.65/— 0.98/— 1.13/— 1.67/— 2.48/— 2.68/— 1.88/—
400 400 400 400 400 400 400 400 400 400
479/0 678/0 228/0 675/0 798/0 811/0 802/0 788/0 801/0 481/0
477 451 265 643 628 595 624 518 467 411
0.54 0.58 0.76 0.72 0.74 0.76 0.80 0.89 0.95 1.02
1.01 1.50 0.86 1.05 1.27 1.36 1.29 1.52 1.71 1.17
0.86 3.36 0.48 0.81 1.19 1.34 1.89 2.50 2.53 1.66
254 254 254 254 508 508 506 506 254 254 254 254 254 254 508 508 508 508 508
32 29 30 30 27 32 35 36 34 29 31 30 29 33 37 30 29 35 39
0.79/— 0.78/— 1.54/— 1.56/— 0.75/— 0.77/— 1.52/— 1.52/— 0.78/— 0.78/— 0.76/— 1.51/— 1.52/— 1.52/— 0.75/— 0.75/— 0.75/— 1.53/— 1.53/—
331 331 331 331 331 331 331 336 336 336 346 336 346 346 336 346 325 325 325
290/0 273/0 463/0 440/0 342/0 330/0 579/0 5800/0 543/0 649/0 352/0 577/0 560/0 597/0 423/0 418/0 410/0 727/0 753/0
342 329 342 335 559 591 624 633 357 329 352 350 340 363 654 589 579 624 659
0.92 0.87 0.87 0.88 1.17 1.29 1.34 1.34 0.92 0.86 0.84 0.85 0.82 0.87 1.33 1.18 1.23 1.36 1.43
0.85 0.83 1.35 1.31 0.61 0.56 0.93 .92 1.52 1.97 1.00 1.65 1.65 1.65 0.65 0.71 0.71 1.16 1.14
0.77 0.81 1.58 1.59 0.58 0.54 1.02 1.02 0.76 0.82 0.81 1.60 1.67 1.57 0.51 0.57 0.55 1.01 0.96
ACIfib International Symposium Punching shear of structural concrete slabs
Table 1:
Interior slabcolumn connections without shear reinforcement subjected to concentric/eccentric shear force
1
2
3
ID
h (mm)
d (mm)
Rizk (2010) NS1 150 100 NS2 200 145 NS3 250 170 HS1 250 155 NS4 300 210 HS2 300 205 HS3 350 210 HS4 350 255 HS6 350 245 NS5 400 295 HS7 400 295 Li (2000) P100 135 100 P150 190 150 P200 240 200 P300 345 300 P400 450 400 P500 550 500 Pisanty (2005) 140/1 140 112 140/2 140 112 160/1 160 133 160/2 160 133 180/1 180 151 180/2 180 151 200/1 200 171 200/2 200 171 Bompa and Onet (2011) DB05 170 145
4
5
f c¢ c (mm) (MPa)
6
7
r¢Test / rTest (%)
fy (MPa)
8 VTest (kN)/ ex (mm)
9
10
11
12
Vc (kN)
r¢min (%)
VTest Vc
r¢Test r¢mi n
250 250 250 250 250 250 250 400 400 400 400
45 50 35 70 40 65 70 76 70 40 60
0.48/— 0.54/— 0.35/— 0.35/— 0.73/— 0.73/— 0.43/— 0.56/— 1.42/— 1.58/— 1.58/—
400 400 400 400 400 400 400 400 400 400 400
219/0 491/0 438/0 574/0 882/0 1023/0 886/0 1722/0 2090/0 2234/0 2513/0
313 540 563 700 815 1003 1078 1941 1763 1729 2117
1.01 0.81 0.62 0.92 0.58 0.75 0.77 0.89 0.88 0.60 0.73
0.70 0.91 0.78 0.82 1.08 1.02 0.82 0.89 1.19 1.29 1.19
0.43 0.60 0.51 0.34 1.13 0.88 0.51 0.57 1.46 2.38 1.96
200 200 200 200 300 300
39.4 39.4 39.4 39.4 39.4 39.4
0.97/— 0.90/— 0.83/— 0.76/— 0.76/— 0.76/—
488 465 465 468 433 433
330/0 583/0 904/0 1381/0 2224/0 2681/0
251 439 670 1255 2343 3348
0.43 0.41 0.39 0.36 0.38 0.34
1.31 1.33 1.35 1.10 0.95 0.80
2.04 1.95 1.91 1.91 1.81 1.99
200 200 200 200 250 250 300 300
26.4 22.8 25.0 19.0 23.3 25.5 24.1 21.8
1.31/— 1.31/— 0.96/— 0.96/— 1.18/— 1.18/— 1.04/— 1.04/—
400 400 400 400 400 400 400 400
390/0 355/0 376/0 445/0 581/0 606/0 835/0 822/0
239 222 295 257 390 408 527 501
0.61 0.57 0.53 0.47 0.52 0.55 0.53 0.50
1.63 1.60 1.27 1.73 1.49 1.49 1.58 1.64
1.92 2.06 1.61 1.84 2.02 1.93 1.77 1.86
300
43.9
0.54/—
476
495/0
570
0.65
0.87
0.74
506 478
0.61 0.58
0.61 0.84
0.55 0.77
3378 691 837 747 712 703 739
0.46 0.41 0.51 0.47 0.43 0.41 0.44
0.64 1.48 0.53 0.55 0.77 0.77 1.03
0.64 3.30 0.44 0.48 0.68 0.72 1.54
Widianto et al. (2009) (Average reinforcement ratio is used for r¢Test ) G0.5 152 127 406 31.4 0.37/— 420 311/0 G1.0 152 127 406 28.1 0.49/— 420 401/0 Guandalini et al. (2009) PG3 500 456 520 32.4 0.33/— 520 2153/0 PG1 250 210 260 27.6 1.50/— 573 1023/0 PG2b 250 210 260 40.5 0.25/— 552 440/0 PG4 250 210 260 32.2 0.25/— 541 408/0 PG5 250 210 260 29.3 0.33/— 555 550/0 PG10 250 210 260 28.5 0.33/— 577 540/0 PG11 250 210 260 31.5 0.75/— 570 763/0
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Table 1: 1
8 VTest r¢Test / f c¢ h d c fy (kN)/ ID rTest (mm) (mm) (mm) (MPa) (MPa) ex (%) (mm) PG6 125 96 130 34.7 1.50/— 526 238/0 PG7 125 100 130 34.7 0.75/— 550 241/0 PG8 130 117 130 34.7 0.28/— 525 140/0 PG9 130 117 130 34.7 0.22/— 525 115/0 Tests subjected to eccentric shear force Robertson and Johanson (2004) 0.57 52 ND1C 114 95 254 29.6 420 /0.21 /904 0.57 76 ND4LL 114 95 254 32.3 420 /0.21 /646 0.57 96 ND5XL 114 95 254 24.1 420 /0.21 /378 0.98 66 ND6HR 114 95 254 26.3 420 /0.49 /986 0.38 52 ND7LR 114 95 254 18.8 420 /0.21 /640 Islam and Park (1976) 1.14 36 1 89 70 228 27.3 356 /0.57 /852 1.14 36 2 89 70 228 31.9 374 /0.57 /1053 1.14 36 3C 89 70 228 29.7 316 /0.57 /1000 Hawkins et al. (1974) 1.18 128 S1 152 115 305 34.8 457 /0.53 /1133 0.79 142 S2 152 115 305 23.4 459 /0.46 /619 0.51 139 S3 152 115 305 22.1 462 /0.30 /386 1.18 150 S4 152 115 305 32.3 457 /0.53 /834
94
2
Interior slabcolumn connections without shear reinforcement subjected to concentric/eccentric shear force 3
4
5
6
7
9
10
11
12
Vc (kN)
r¢min (%)
VTest Vc
r¢Test r¢mi n
170 181 227 227
0.52 0.49 0.47 0.47
1.40 1.33 0.62 0.51
2.58 1.38 0.54 0.42
59
0.66
0.89
0.86
78
0.66
0.98
0.86
94
0.52
1.01
1.09
52
0.58
1.28
1.70
60
0.45
0.87
0.85
33
0.77
1.09
1.48
30
0.81
1.19
1.41
30
0.92
1.19
1.24
90
0.67
1.43
1.77
113
0.61
1.26
1.30
144
0.64
0.97
0.80
108
0.67
1.39
1.75
ACIfib International Symposium Punching shear of structural concrete slabs
Table 1:
Interior slabcolumn connections without shear reinforcement subjected to concentric/eccentric shear force
1
2
3
ID
h (mm)
d (mm)
4
5
f c¢ c (mm) (MPa)
6
7
r¢Test / rTest (%)
fy (MPa)
0.50 /0.00
469
8 VTest (kN)/ ex (mm)
9
10
11
12
Vc (kN)
r¢min (%)
VTest Vc
r¢Test r¢mi n
123
0.85
0.86
0.59
89
0.60
1.15
1.13
48
0.64
1.30
1.06
102
0.54
1.02
1.60
50
0.64
1.07
1.35
37
0.51
1.20
1.21
87
0.41
0.99
1.53
133
0.36
0.90
1.73
93
0.62
1.62
1.77
143
0.73
1.18
0.83
137
0.83
1.37
1.16
132
0.85
1.55
1.64
232
0.72
1.05
0.84
252
0.93
1.02
1.03
235
0.92
1.36
1.52
72
0.80
1.12
0.91
68
0.76
1.39
1.24
Tian (2007) and Tian et al. (2008) L0.5
152
127
406
25.6
105 /1210
Pan and Moehle (1992) 1
122
101
274
33.3
3
122
101
274
31.4
0.68 /0.25 0.68 /0.25
472 472
102 /715 62 /1542
Pan and Moehle (1989) AP1
123
103
274
29.3
AP3
123
103
274
31.7
0.86 /0.29 0.86 /0.29
484 484
104 /549 53 /1524
Robertson and Durrani (1992) A
114
91
254
33.0
B
114
91
254
30.8
C
114
91
254
32.2
116
250
35.0
0.62 /0.43 0.62 /0.43 0.62 /0.43
501 525 525
44 /1583 86 /482 121 /225
Elgabry (1991) 1
150
1.10 /0.44
452
150 /867
Hawkins et al. (1989) 6AH
150
117
305
31.3
9.6AH
150
114
305
30.7
14AH
150
110
305
30.3
6AL
150
117
305
22.7
9.6AL
150
114
305
28.9
14AL
150
110
305
27.0
7.3BH
113
80
305
22.2
9.5BH
113
80
305
19.8
0.60 /0.25 0.96 /0.45 1.40 /0.54 0.60 /0.25 0.96 /0.46 1.40 /0.54 0.73 /0.38 0.95 /0.46
472 415 420 472 415 420 472 472
169 /567 187 /567 205 /566 244 /127 257 /128 319 /128 80 /568 94 /567
95
ACIfib International Symposium Punching shear of structural concrete slabs
Table 1: 1
8 VTest r¢Test / f c¢ h d c fy (kN)/ ID rTest (mm) (mm) (mm) (MPa) (MPa) ex (%) (mm) 1.42 102 14.2BH 113 78 305 29.5 415 /0.69 /568 0.73 130 7.3BL 113 80 305 18.1 472 /0.38 /128 0.95 142 9.5BL 113 80 305 20.0 472 /0.46 /127 1.42 162 14.2BL 113 78 305 20.5 415 /0.69 /128 0.60 186 6CH 150 117 305 52.4 472 /0.25 /570 0.96 218 9.6CH 150 114 305 57.2 415 /0.45 /569 1.40 252 14CH 150 110 305 54.7 420 /0.54 /567 0.60 273 6CL 150 117 305 49.5 472 /0.25 /127 1.40 362 14CL 150 110 305 47.7 420 /0.54 /128 0.60 134 6DH 150 117 305 30.0 472 /0.25 /567 1.40 195 14DH 150 110 305 31.7 420 /0.54 /569 0.60 232 6DL 150 117 305 28.7 472 /0.25 /128 1.40 282 14DL 150 110 305 24.3 420 /0.54 /128 1.02 153 10.2FHI 150 117 305 25.9 472 /0.53 /567 1.02 183 10.2FHO 150 117 305 25.5 472 /0.53 /568 1.40 206 14FH 150 110 305 31.2 420 /0.53 /568 0.60 227 6FLI 150 117 305 25.9 472 /0.28 /127 1.02 240 10.2FLI 150 110 305 18.1 420 /0.53 /128 1.02 290 10.2FLO 150 110 305 28.5 420 /0.53 /128 1 mm = 39.4×103 in.; 1 MPa = 145 psi; 1 kN = 1000 N = 224.8 lb.
96
2
Interior slabcolumn connections without shear reinforcement subjected to concentric/eccentric shear force 3
4
5
6
7
9
10
11
12
Vc (kN)
r¢min (%)
VTest Vc
r¢Test r¢mi n
79
1.05
1.29
1.35
126
0.88
1.03
0.83
133
0.92
1.07
1.04
130
1.07
1.25
1.33
185
0.92
1.01
0.65
186
1.11
1.17
0.86
177
1.13
1.42
1.24
342
1.03
0.80
0.58
312
1.20
1.16
1.17
140
0.71
0.96
0.84
135
0.87
1.45
1.61
261
0.80
0.89
0.75
223
0.88
1.27
1.59
128
0.65
1.19
1.56
127
0.65
1.44
1.57
134
0.86
1.54
1.62
248
0.76
0.92
0.79
192
0.77
1.25
1.32
241
0.94
1.20
1.08
ACIfib International Symposium Punching shear of structural concrete slabs
Table 2:
Interior slabcolumn connections with shear reinforcement subjected to concentric/eccentric shear force
1
2
3
ID
h (mm)
d (mm)
Stein et al. (2007) V1 150 118 V2 150 118 V3 150 118 Birkle and Dilger (2008) 2 160 124 4 160 124 8 230 190 9 230 190 11 300 260 12 300 260 Mokhtar (1982) AB2 150 115 AB3 150 115 AB4 150 115 AB5 150 115 AB6 150 115 AB7 150 115 AB8 150 115
4
5
6
7
8 9 VTest r¢Test / f c¢ c fy (kN)/ Vc rTest (MPa) (mm) (MPa) ex (kN) (%) (mm) Tests subjected to concentric shear force
10
11
12
r¢min (%)
VTest Vc
r¢Test r¢min
250 250 250
29.7 26.2 25.7
0.45/— 0.98/— 0.62/—
438 438 438
329/0 438/0 365/0
316 296 294
0.55 0.52 0.52
1.04 1.48 1.24
0.73 1.69 1.08
250 250 300 300 350 350
29 38 35 35.2 30 33.5
1.54/— 1.54/— 1.30/— 1.30/— 1.10/— 1.10/—
488 488 531 531 524 524
574/0 634/0 1050/0 1091/0 1620/0 1520/0
333 381 734 736 1158 1224
0.58 0.65 0.54 0.54 0.48 0.51
1.72 1.66 1.43 1.48 1.40 1.24
2.41 2.12 2.16 2.15 2.05 1.95
250 37.8 1.40/— 516 520/0 250 23.1 1.40/— 516 545/0 250 41.2 1.40/— 516 583/0 250 39.4 1.40/— 516 583/0 250 28.6 1.40/— 516 541/0 250 35.2 1.40/— 448 572/0 250 29.9 1.40/— 448 508/0 Tests subjected to eccentric shear force
344 269 359 351 299 332 306
0.54 0.43 0.56 0.55 0.47 0.60 0.55
1.51 2.03 1.62 1.66 1.81 1.72 1.66
2.35 2.96 2.25 2.30 2.68 2.11 2.28
83
0.34
1.69
1.32
68
0.36
2.07
2.49
73
0.36
1.92
1.72
72
0.59
2.07
1.87
139
0.57
2.17
2.14
137
0.62
2.19
2.24
217
0.56
2.08
2.49
Stein et al. (2007) VM1
150
118
250
24.2
VM2
150
118
250
24.6
VM3
150
118
250
25.5
0.45 /0.45 0.89 /0.89 0.62 /0.62
438 438 438
140 /750 140 /993 140 /921
Elgabry (1991) 2
150
116
250
33.7
3
150
116
250
39
4
150
116
250
40.8
5
150
116
250
45.6
1.10 /0.44 1.23 /0.44 1.39 /0.44 1.39 /0.44
452 452 446 446
150 /1080 300 /473 300 /500 450 /233
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ACIfib International Symposium Punching shear of structural concrete slabs
Table 2:
Interior slabcolumn connections with shear reinforcement subjected to concentric/eccentric shear force
1
2
3
4
5
6
7
ID
h (mm)
d (mm)
c (mm)
f c¢ (MPa)
r¢Test / rTest
fy (MPa)
(%)
8 VTest (kN)/ ex (mm)
9
10
11
12
Vc (kN)
r¢min (%)
VTest Vc
r¢Test r¢min
124
0.81
1.83
1.73
126
0.77
1.58
1.24
127
0.78
1.78
1.23
237
0.93
1.49
1.50
228
0.85
1.27
1.13
252
0.93
1.37
1.03
28
0.99
1.29
1.16
27
0.97
1.34
1.17
27
0.86
1.35
1.32
80
0.60
1.66
2.16
64
0.56
1.99
2.20
77
0.60
1.65
2.08
83
0.63
1.53
1.43
Hawkins et al. (1989) 14EH.49
150
110
305
25.1
9.6EH.34
150
114
305
25.5
9.6EH.48
150
114
305
25.8
14EL.49
150
110
305
26.9
9.6EL.34
150
114
305
23.4
9.6EL.56
150
114
305
28.5
1.40 /0.54 0.96 /0.45 0.96 /0.45 1.40 /0.54 0.96 /0.46 0.96 /0.46
420 415 415 420 415 415
226 /566 199 /568 226 /566 354 /127 290 /128 345 /128
Islam and Park (1976) 6CS
89
70
228
28.2
7CS
89
70
228
29.7
8CS
89
70
228
22.1
1.14 /0.57 1.14 /0.57 1.14 /0.57
290 304 293
36 /1073 36 /1165 36 /975
Hawkins et al. (1975) 1.29 133 459 /0.59 /1128 1.24 127 SS3 152 115 305 25.9 455 /0.59 /1449 1.24 127 SS4 152 115 305 27.6 455 /0.59 /1189 0.90 126 SS5 152 115 305 32.2 463 /0.59 /1198 1 mm = 39.4×103 in.; 1 MPa = 145 psi; 1 kN = 1000 N = 224.8 lb. SS1
98
152
115
305
27.6
ACIfib International Symposium Punching shear of structural concrete slabs
Appendix A — Vflex corresponding to yieldline patterns in Figures 6 and 7 Equations used to calculate the concentric or eccentric shear force, Vflex that develops each of the yieldline mechanisms of Figs. 6 and 7 are given below. The symbols of Eqs. (A.1) through (A.10) are shown in Figs. 6 and 7.
Yieldline mechanism of Fig. 6a 6b
6c
6d 6e 7a 7b 7c 7d 7e
Vflex =
Eqn.
8 m ls (l1  c )
(A.1)
é l 0.293 ls  0.172 l1  0.121 c ù 8mê s ú ls  0.293 l1  0.707 c û ë l1  c
(A.2)
é ê mê ê ls ë
(
ù ú + cú 2 cos (p 8)  c l 2 cos (p 8)  ú s 2û æ 16 rs sin (p 8) ö ÷÷ m çç è r1  (2 c p) ø l 4m s l1  c 4 ls  c
)
(
)
é 3 l (l + c ) (l1  c )(ls + c ) ù mê s 1 + + cú l1 + c ë l1  c û
c
(A.3)
(A.4) (A.5)
(l1 + c )ù é êex + 2 ú ë û
é 2 (l1 + l2 ) ì ì c (l1 + l2  ls )ü c (l1 + l2  ls ) ü ù mê íls ý + 2 íls + ý + cú 2 (l1 + l2  c ) þ 2 (l1 + l2  c ) þ úû êë (l1 + l2  2 c ) î î l2 (m + m¢) é c ù ê1 + ú ex ë l1  c û é æ cö cù 2 l2 m êex çç1  ÷÷ + lv  ú 2 úû êë è l1 ø m ls B [(l1 3)  c ] + 2 A (l1  c )
(l1 + l2  c)ù é êex + ú 2 ë û
(A.6) (A.7) (A.8) (A.9) (A.10)
99
ACIfib International Symposium Punching shear of structural concrete slabs
3D finite element analysis of punching shear of RC flat slabs using ABAQUS Aikaterini S. Genikomsou, Maria A. Polak University of Waterloo, Canada
Abstract Punching shear failure of reinforced concrete slabs has been examined by many researchers through laboratory experiments. However, the existing punching shear testing database cannot address all aspects of the punching shear stress transfer mechanism. Advanced 3D finite element analysis (FEA) can be used to supplement the existing testing background and for parametric investigations. In this way, different aspects of punching shear failure may be explored in detail, to enable understanding of the phenomena that control the response and to support drafting design code requirements. This paper describes research on calibrating constitutive and finite element models in ABAQUS to capture punching shear behavior of concrete slabs. The coupled damagedplasticity model is used for modeling the concrete. Two interior reinforced concrete slabcolumn connections previously tested under static loading are presented: one slab is without shear reinforcement (SB1) and the other slab is with shear bolts (SB4). The developed formulation is calibrated using the results for specimen SB1, where the tension stiffening response, the damage parameters and the support conditions are examined. Then, the adopted FEA and concrete model are used for the analysis of slab SB4, which was retrofitted with shear bolts. Finally, both test and numerical results are compared to the ACI 31814 provisions.
Keywords Concrete damaged plasticity model, finite element analysis, punching shear, reinforced concrete slabs, shear bolts.
1
Introduction
Punching shear can lead to brittle failure in reinforced concrete slabs. Installation of shear reinforcement around the slabcolumn connection can be the most effective way to avoid punching shear failure, because both the capacity and ductility of a slab are increased. Many tests have been done on slabs where different types of shear reinforcement were examined (Dilger and Ghali [1981], Islam and Park [1976], Corley and Hawkins [1968], Tan and Teng [2005]). For retrofit techniques, a different type of shear reinforcement (shear bolts) was tested by Adetifa and Polak (2005). The shear bolts can be installed as a retrofit method in existing slabs that were designed without shear reinforcement or with an insufficient amount of shear reinforcement.
101
ACIfib International Symposium Punching shear of structural concrete slabs
Nowadays, in modern research, punching shear failure of reinforced concrete slabs can be examined using advanced numerical tools such as nonlinear finite element analysis (FEA). FEA can be viewed as an extension of and in some cases as a replacement for testing, since important information can be provided for the predicted failure modes of the slabs in a rapid and inexpensive way. However, it is essential to first properly calibrate and validate the FEA models, where in punching shear problems of reinforced concrete slabs, the constitutive behavior of concrete and its modeling are crucial. In punching shear, many researchers have used FEA methods to simulate reinforced concrete slabs. Early investigations were conducted by performing shellelement analyses using the layered approach (Polak, 2005) and only in recent years have numerical simulations been conducted using three dimensional FEA models (Genikomsou and Polak, 2015). A remaining obstacle in punching shear simulations using FEA is the analysis of reinforced concrete slabs with punching shear reinforcement. Not many researchers have tried to analyze slabs with shear reinforcements, with the most recent research done by Genikomsou and Polak (2016). The complex behavior of concrete requires the adoption of advanced constitutive models. Many constitutive models of concrete were developed based on plasticity, damage mechanics and coupled damageplasticity. Classical plasticity models (e.g. Pramono and Willam [1989], Imran and Pantazopoulou [2001], Grassl et al. [2002]) cannot describe the stiffness degradation of concrete when it is subjected to cyclic loading. Thus, damage mechanics models were developed over the past years that took into account the stiffness degradation using the concept of effective stresses (e.g. Kachaconov [1986], Mazars [1986], Mazars and PijaudierCabot [1989], Lemaitre and Chaboche [1990]). Later, coupled damageplasticity models were investigated, where damage mechanics is used to take the concrete deterioration into account and plasticity theory models are used to model the permanent deformation and volumetric expansion (e.g. Simo and Ju [1987a, 1987b], Ju [1989], Lubliner at al. [1989], Lee and Fenves [1998]). The aim of this paper is to present a constitutive model of concrete (concrete damaged plasticity model) used in ABAQUS and then to show the calibration of this model with the appropriate damage parameters in both tension and compression. The calibration of the concrete damaged plasticity model uses a specimen that has no shear reinforcement. Then, this calibrated concrete model is used to simulate the shear reinforced slab. Interesting conclusions are drawn based on all analyses with the objective of presenting a numerical tool that can be used for efficient punching shear simulations of reinforced concrete slabs.
2
Test specimens
Two interior reinforced concrete slabcolumn connections that were previously tested by Adetifa and Polak (2005) are analyzed. Both slabs were tested under static loading through the column stub and their dimensions in plan are 1800x1800 mm (70.866x70.866 in.) with simple supports at 1500x1500 mm (59.055x59.055 in.). The square columns (150x150 mm) (5.906x5.906 in.) extend 150 mm (5.906 in.) from the top and the bottom faces of the slabs. The thickness of the slabs is 120 mm (4.724 in.) and the effective depth is equal to 90 mm (3.543 in.). The dimensions of the slabs and the loading process are presented in Fig. 1. The flexural reinforcement consists of 10M bars that are placed at distance 100 mm (3.937 in.) and 90 mm (3.543 in.) in the tension side and 200 mm (7.874 in.) in the compression side. Slab SB1 has
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ACIfib International Symposium Punching shear of structural concrete slabs
no shear reinforcement while SB4 is retrofitted with four rows of shear bolts (see Fig. 1). The shear bolts were postinstalled in slab SB4 and they consist of smooth steel bars having a forged circular head on the one end while the other end is threaded. Prior to testing, holes of 16 mm (0.630 in.) diameter were drilled around the slabcolumn connection of SB4 in order to install the 9.5 mm (0.374 in.) diameter shear bolts. The placement of the shear bolts was concentric and parallel to the perimeter of the column. In slab SB4, four rows of shear bolts were installed and each of these rows had two parallel bolts to each face of the column, resulting in eight bolts in each row in total. The first row of the shear bolts was placed at a distance of 45 mm (1.772 in.) from the face of the column, while all subsequent rows were spaced at approximately 80 mm (3.150 in.) according to Polak and Bu (2013). The material properties of the concrete and reinforcement for both slabs are shown in Table 1, while Table 2 presents the test results. Specimen SB1 failed in punching shear at a load of 253 kN (56.877 kips), while slab SB4 failed in flexure at a load of 360 kN (80.931 kips). Slab SB4 showed that its tensile flexural reinforcement yielded first at a load of 240 kN (53.954 kips). Then a general yielding of the flexural reinforcement at a load of 360 kN (80.931 kips) followed. There was no further increase of loadcarrying capacity but SB4 continued to deform. Finally, SB4 failed by punching shear (secondary failure) outside the shearreinforced zone. However, this punching shear failure happened because the test specimen, even if it had already failed in flexure (general yielding of the flexural reinforcement), was pushed further in order to punch. Figure 2 shows the test results for both slabs in terms of loaddeflection response, while Fig. 3 and Fig. 4 present the crack patterns at failure of slab SB1 and SB4, respectively. Specimen SB1, which failed in punching, did not fully form flexural yield lines. No cracks at the compression surface of slab SB1 were observed. Slab SB4 first achieved its flexural capacity and then failed in punching outside the shearreinforced area, where the flexural yield lines were fully developed. It should be noted that cracking occurred at the compression surface.
Figure 1:
Schematic drawings of the slabs (dimensions, loading). (Note: 1 mm = 0.0394 in.)
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ACIfib International Symposium Punching shear of structural concrete slabs
Table 1: Slab
Material properties of the slabs
f'c, MPa (ksi)
f't, MPa (ksi)
G f, N/mm (kips/in.)
E c, MPa (ksi)
f y, MPa (ksi)
f t, MPa (ksi)
E s, MPa (ksi)
fy,bolts, MPa (ksi)
SB1
44 2.2 0.082 (6.382) (0.319) (4.7e04)
36483 (5291.410)
455 (65.992)
620 (89.923)
200000 (29007.540)
– (–)
SB4
41 2.1 0.077 (5.947) (0.305) (4.4e04)
35217 (5107.793)
455 (65.992)
620 (89.923)
200000 (29007.540)
381 (55.259)
Table 2:
Test results
Slab
No. of rows of bolts
Failure load, kN (kips)
Failure displacement, mm (in.)
Failure mode
SB1
0
253 (56.877)
11.9 (0.469)
Punching
SB4
4
360 (80.931)
29.8 (1.173)
Flexure
400
350
Load (kN)
300 250 200
150 100
SB1
50
SB4
0
0
5
10
15
20
25
30
35
40
Deflection (mm)
Figure 2:
Loaddeflection response of the slabs (test results). (Note: 1 mm = 0.0394 in.)
Figure 3:
Crack pattern of slab SB1(tension side) at failure during the test.
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ACIfib International Symposium Punching shear of structural concrete slabs
Figure 4:
3
a) b) Crack pattern of slab SB4 at failure during the test: a) tension surface and b) compression surface.
Concrete damaged plasticity model
The concrete damaged plasticity model uses both the tensile cracking and the compressive crushing of concrete as possible failure modes. The yield function of the concrete model takes into account the effective stress space as proposed by Lubliner et al. (1989) with # modifications made by Lee and Fenves (1998). The effective stress is defined as: 𝜎 = = (%&')
𝛦* ∙ (𝜀 − 𝜀 ./ ), where 𝛦* denotes the initial modulus of elasticity, 𝜀 ./ is the equivalent plastic strain and 𝑑 is the damage variable that denotes the stiffness degradation. The development of ./ ./ the yield function is controlled by the hardening variables 𝜀1 and 𝜀2 . The softening behavior of concrete and the stiffness degradation often create convergence difficulties in ABAQUS/Standard. Thus, viscoplastic regularization according to the DuvautLions approach (Duvaut and Lions, 1976) can be also taken into account by introducing the viscous parameter (𝜇) that upgrades the plastic strain tensor. The viscoplasticity permits the stresses to be outside of the yield surface and causes the tangent stiffness of the softening concrete to be positive for small strain increments. The plastic potential function, which is employed in the model, is a nonassociated DruckerPrager hyperbolic function in which the definition of the dilation angle is needed. The dilation angle is measured in the pq plane at high confining pressure. For the visualization of cracking, the concrete damaged plasticity model assumes that the cracking starts at points where the maximum principal plastic strain is positive. The direction of the cracking is assumed to be parallel to the direction of the maximum principal plastic strain and it is viewed in the Visualization module of ABAQUS/CAE. In compression, the response of concrete is linear until the initial yield stress 𝜎1* and then it is characterized by stress hardening followed by strain softening beyond the ultimate stress 𝜎14 (Fig. 5). The uniaxial stressinelastic strain curve is converted automatically by ABAQUS into stressplastic strain curve. In tension the stressstrain curve is linear elastic until the failure stress 𝜎2* (Fig. 6). At this failure stress, the microcracking in the concrete starts. Beyond this failure stress, the microcracking is presented macroscopically with a stressstrain softening response that prompts strain localization in concrete. The uniaxial stresscracking strain curve is converted automatically by ABAQUS into stressplastic strain curve. When the concrete is unloaded on the strain softening part of the stressstrain curve, the elastic stiffness of concrete degrades. This degradation is described by two damage variables 𝑑1 (compression) and 𝑑2 (tension), which are functions of the plastic strains. 105
ACIfib International Symposium Punching shear of structural concrete slabs
Figure 5:
Uniaxial loading of concrete in compression (definition of the inelastic strain 𝜺𝒊𝒏 𝒄 ).
Figure 6:
Uniaxial loading of concrete in tension (definition of the craking strain 𝜺𝒄𝒌 𝒕 ).
Typically in ABAQUS, the reinforcement of reinforced concrete structures is modeled as rebar embedded in concrete. Bond slip and dowel action are modelled indirectly by introducing tension stiffening into the concrete damaged plasticity model in order to simulate the load transfer across the cracks through the rebar. Tension stiffening can be modeled using the fracture energycracking criterion. Hillerborg’s (1976) fracture energy criterion can be used, which defines the fracture energy as the energy required to open a unit area of crack. With this approach, the brittle behavior of concrete is characterized by a stressdisplacement response instead of a stressstrain response. The execution of the stressdisplacement response into the FEA model needs the definition of the characteristic length based on the integration point. This crack length comes from the geometry and formation of the element, which is the typical length of a line across an element for a firstorder element. The definition of the characteristic crack length is considered because the direction in which the cracking happens is not known. Thus, elements with aspect ratios close to one are preferred. The nonlinear loaddisplacement response is the objective of the analysis. The internal loads (I) acting on a node are caused by the stresses in the elements that contain this node. In order to have static equilibrium, the net force at each node should be zero. Thus, the internal forces (I) and the external forces (P) should be in equilibrium (Fig. 7). ABAQUS/Standard 106
ACIfib International Symposium Punching shear of structural concrete slabs
solves the equations of the system implicitly at each solution increment using a stable stiffnessbased solution technique, the NewtonRaphson method. The solution is found by applying the loads gradually and incrementally by having many load increments. Then, the approximate equilibrium at the end of each load increment is found, where the summation of all the incremental responses is the approximate solution of the nonlinear analysis.
Figure 7:
External (P) and internal (I) loads on a body.
Taking into account the material modeling of concrete in ABAQUS and the solution procedure that FEA software uses for such type of problems, the next section presents the calibration of the concrete and FEA model by analyzing the slab without shear reinforcement (SB1).
4
Calibration of FEA model on slab without shear bolts (SB1)
Slab SB1 (without shear reinforcement) is used for the calibration of the concrete and FEA models. One quarter of the real slabcolumn connection is modelled in ABAQUS due to symmetry. Static analysis under displacement control is conducted in ABAQUS/Standard; and viscoplastic regularization is taken into account, with the viscosity parameter taken equal to 0.00001. First, a parametric investigation based on the different options that we can use to define the tensile stresscrack displacement relationship is done for specimen SB1. During this investigation different values for the maximum compressive damage parameter are examined. Concrete is modelled with 3D 8noded hexahedral elements with reduced integration (C3D8R) and the flexural reinforcement is modelled with 3D 2noded linear truss elements (T3D2). The concrete and reinforcement and reinforcement are considered to have perfect bond through the embedded method in ABAQUS. Bondslip behavior of reinforcing bars is not taken into account in the model, since the loaddeflection response of the slabs is of primary interest and there is less concern for cracking and in general for local stresses. Thus, the smeared crack approach, used by the concrete damage plasticity model in ABAQUS, is an excellent choice. With this perfect bond, the stresses of the reinforcement are underestimated, however a common way to take the tension stiffening into account in the smeared crack models is to consider a descending branch in stressstrain curve of concrete in tension. The mesh size is taken as equal to 20 mm (0.787 in.) and simple supports are introduced along the edges of the slab at the bottom.
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Three different options are available for modeling the tension stiffening response of concrete: linear, bilinear and exponential (Fig. 8). The bilinear tension stiffening response proposed by Hillerborg (1976) is used in the numerical investigation. At a stress of 𝑓2= /3, the crack displacement is taken as equal to 0.8𝐺D /𝑓2= and then, when the stress is zero the cracking displacement is equal to 3.6𝐺D /𝑓2= . The exponential tensionstiffening curve is calculated according to Cornelissen et al. (1986), where the following equations should be used: 𝜎 𝑓2= = 𝑓 𝑤 − (𝑤 𝑤1 )𝑓(𝑤1 ) 𝑓 𝑤 = 1+
𝑤1 = 5.14
𝑐% 𝑤 𝑤1
J
𝑒𝑥𝑝 −
𝐺D 𝑓2=
(1) 𝑐N 𝑤 𝑤1
(2) (3)
where, 𝑐% and 𝑐N are material constants and can be taken as equal to 3 and 6.93 for normal concrete, respectively. The fracture energy of concrete (𝐺D ) represents the area under the tensile stresscrack displacement curve; it depends on the concrete’s strength and aggregate size and can be calculated using Eq. (4) (CEBFIP Model Code 1990). 𝐺D =𝐺D* (𝑓1Q /𝑓1Q* )R.S
(4)
According to CEBFIP Model Code 1990, 𝑓1Q* = 10 𝑀𝑃𝑎 (1.450 𝑘𝑠𝑖), 𝑓1Q is the mean value of the compressive strength associated with the characteristic compressive strength (𝑓1Z ), (𝑓1Q = 𝑓1Z + 8 𝑀𝑃𝑎 (1.160 𝑘𝑠𝑖)) and 𝐺D* is the base fracture energy that depends on the maximum aggregate size, 𝑑Q[\ . The value of the base fracture energy 𝐺D* for an aggregate size (𝑑Q[\ ) equal to 10 mm (0.394 in.) is equal to 0.026 N/mm (1.5e04 kips/in.). Fig. 9 shows the tensile damagecrack displacement relationship for all three tensionstiffening approaches. Fig. 10 shows the compressive stressstrain relationship for slab SB1, where the compressive stressinelastic strain relationship and the compressive stressplastic strain relationship are shown. In ABAQUS the compressive stressinelastic strain curve is given as input; this curve is then automatically converted to compressive stressplastic strain if compressive damage parameters are specified. Fig. 11 illustrates the compressive damage parameter (𝑑1 ) versus the plastic strain curves. Different compressive damage parameters are used based on the given equivalent plastic strains. All relationships between plastic and inelastic strains for both compression and tension are based on Fig. 5 and Fig. 6, respectively. By taking into account the relationship between inelastic and plastic strain, the damage parameters (needed as input data for the model) can be specified. Finally, a comparison between test and numerical results can calibrate the concrete model that should be used.
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Tensile stress (MPa)
2.4 exponential
2.0
linear
1.6
bilinear 1.2 0.8 0.4 0.0 0
0.05
0.1
0.15
0.2
w (mm)
Figure 8:
Tensile stresscrack width curves for slab SB1. (Note: 1MPa = 0.145 ksi; and 1 mm = 0.0394 in.)
1.0 0.9
Damage (dt)
0.8
0.7 0.6 0.5 0.4
exponential
0.3 0.2
linear
0.1
bilinear
0.0 0
0.05
0.1
0.15
0.2
w (mm)
Figure 9:
Tensile damagecrack width curves for slab SB1. (Note: 1 mm = 0.0394 in.)
50
45 40
Stress (MPa)
35 30 25 20
ε εin εpl=0.8εin εpl=0.7εin εpl=0.6εin
15
10 5 0 0
0.001
0.002
0.003
0.004
Strain (mm/mm)
Figure 10: Compressive stressstrain curves for slab SB1. (Note: 1MPa = 0.145 ksi; and 1 mm = 0.0394 in.)
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Damage (dc)
1.0 0.9
εpl=0.6εin
0.8
εpl=0.7εin
0.7
εpl=0.8εin
0.6 0.5 0.4 0.3 0.2
0.1 0.0
0
0.001
0.002
0.003
0.004
Plastic strain (mm/mm)
Figure 11: Compressive damageplastic strain curves for slab SB1. (Note: 1 mm = 0.0394 in.)
Figures 12 and 13 show the numerical results for slab SB1 in terms of loaddeflection response for the three tension softening options. The analyses in Fig. 12 consider the plastic strain 𝜀 ./ equal to 0.8𝜀 ]^ , while in Fig. 13 the plastic strain is equal to 0.7𝜀 ]^ . The relationship between plastic and inelastic strains implies the values for the given compressive damage variables. The linear tension softening response underestimates the punching shear capacity of the slab in both analyses. When the plastic strain (𝜀 ./ ) is equal to 0.8𝜀 ]^ , the bilinear tension stiffening approach overestimates the load capacity of the slab, while the exponential tension stiffening approach accurately predicts the loaddeflection response of the slab. If we consider now the numerical results in Fig. 13, where the compressive damage is increased, both the bilinear and exponential tension stiffening approaches closely capture the test loaddeflection response of the slab. Fig. 14 presents the cracking at the ultimate load when the εpl=0.7εin.
300
SB1 slab Gf=0.082 N/mm εpl=0.8εin
Load (kN)
250 200 150 100
Linear Bilinear Exponential Test
50 0
0
5
10
15
Displacement (mm)
Figure 12: Loaddeflection response of slab SB1 (εpl=0.8εin). (Note: 1kN = 0.225 kips; and 1 mm = 0.0394 in.)
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300
SB1 slab Gf=0.082 N/mm εpl=0.7εin
Load (kN)
250 200 150 100
Linear Bilinear Exponential Test
50 0
0
5
10
15
Displacement (mm)
Figure 13: Loaddeflection response of slab SB1 (εpl=0.7εin). (Note: 1kN = 0.225 kips; and 1 mm = 0.0394 in.)
a) linear
b) bilinear
c) exponential
Figure 14: Cracking at ultimate load (εpl=0.7εin).
Further investigation with different boundary conditions for slab SB1 is performed. Neoprene is simulated around the bottom of the slab as happened in the real test (neoprene pads 25 mm (0.984 in.) thick and 50 mm (1.969 in.) in width were installed along the supporting lines). Neoprene is a hyperelastic material considered to be isotropic and nonlinear that exhibits instantaneous elastic response up to large strains and, like most elastomers, it has low compressive strength compared to its shear flexibility. There are several forms of strain energy potentials available in ABAQUS to model isotropic elastomers. Among these models the MooneyRivlin model is chosen. The mechanical properties of the neoprene are determined by performing a uniaxial compressive test on a neoprene specimen. ABAQUS allows the option of defining the uniaxial compression test data that can be used for the parametric modelling identification of the material coefficients. By using the material evaluation option in ABAQUS we can obtain all needed material coefficients for specifying the MooneyRivlin form for the neoprene. The material parameters according to the ABAQUS evaluation are: the shear modulus is equal to 9.297 MPa and the Poisson’s ratio is equal to 𝜈 = 0.49. The numerical results using the neoprene to simulate the boundary conditions of the isolated specimen are in good agreement with the test results and in the initial uncracked state they better describe the real loaddeflection response of the isolated slab compared to the simple supports (Fig. 15). Fig. 16 shows the crack pattern at the ultimate
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load. Neoprene supports seem to present the cracking in more realistic way without concentrated strains in the supports as happens in the simply supported slabs. 300
SB1 slab Gf=0.082 N/mm εpl=0.7εin Bilinear response
Load (kN)
250 200
150 100 Simple supports Neoprene supports Test
50
0 0
5
10
15
Displacement (mm)
Figure 15: Loaddeflection response of slab SB1 (simple vs. neoprene supports).
Figure 16: Crack pattern of SB1 slab with neoprene supports at ultimate load.
5
FEA of slab with shear reinforcement (SB4)
The previously presented calibrated concrete damaged plasticity model on slab SB1 is used for the analysis of slab SB4. Both slabs were cast together with the same aggregate size and were cured in the same manner. These slabs had identical dimensions and boundary and loading conditions during the test. The only difference between these two slabs was the addition of shear bolts as punching shear reinforcement in specimen SB4. Therefore, the concrete model calibrated on the slab without shear reinforcement (SB1) is valid for the analysis of the slab with shear reinforcement (SB4). The shear bolts are modelled with 3D quadratic beam elements (B32) based on a previous research (Genikomsou and Polak, 2016). Herein, the response of the SB4 slab is examined using damage parameters in the concrete model and neoprene supports. The bilinear tension stiffening approach is used and the plastic strain in compression is taken equal to 0.7 times the inelastic strain. Fig. 17 shows the comparison between numerical and test results for slab SB4 in terms of loaddeflection response. Failure load and displacement in the FEA are in good agreement with the test results. Fig. 18 presents the crack pattern at the ultimate load, where the final failure is through a punching shear cone outside the shear reinforced area as happened in the test.
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400
SB4 slab Gf=0.077 N/mm εpl=0.7εin
350
Load (kN)
300 250 200 150 100 Bilinear
50
Test
0 0
5
10
15
20
25
30
35
40
45
Displacement (mm)
Figure 17: Loaddeflection response of slab SB4.
Figure 18: Crack pattern of SB4 slab at ultimate load.
6
Comparison with ACI 31814
The punching shear resistance of slab SB1 (without shear reinforcement) according to ACI 31814 is calculated based on Eq. (5), where 𝑏* is the control perimeter at distance 𝑑/2 from the face of the column, 𝑑 is the effective depth of the slab and 𝑓′1 is the compressive strength of concrete. 𝑉1 = 0.33𝑏* 𝑑 𝑓′1 (𝑘𝑁) 𝑉1 = 4𝑏* 𝑑 𝑓′1 (𝑘𝑖𝑝𝑠)
(5)
The punching shear resistance of slab SB4 (with shear reinforcement) is calculated based on Eq. (6) and Eq. (7), where the critical section is located at distance 𝑑/2 from the face of the column and at distance 𝑑/2 from the outermost peripheral line of the shear reinforcement, respectively. The governing failure mode (punching inside the shear reinforced area or punching outside the shear reinforced area) is estimated as the minimum punching shear strength given from Eq. (6) and Eq. (7). The shear bolts are considered to be headed shear stud reinforcement in Eq. (6), 𝐴kl is the area of the shear reinforcement (one row of shear bolts), 𝑓mk is the specified yield strength of the shear bolts, 𝑠 is the spacing between the
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perimeters of the shear bolts and 𝑏*42 is the control perimeter at distance 𝑑/2 from the last line of the shear bolts. 𝑉n = 0.25𝑏* 𝑑 𝑓′1 + 𝑉n = 3𝑏* 𝑑 𝑓′1 +
opq Drp ' l
𝐴kl 𝑓mk 𝑑 (𝑘𝑁) 𝑠
(6)
(𝑘𝑖𝑝𝑠) 𝑉n = 0.17𝑏*42 𝑑 𝑓′1 (𝑘𝑁)
𝑉n = 2𝑏*42 𝑑 𝑓′1 +
opq Drp ' l
(7)
(𝑘𝑖𝑝𝑠)
Table 3 presents the predicted punching shear capacity for the two slabs according to ACI 31814. The punching shear resistance of specimen SB4 is calculated based on Eq. (7) since failure outside the shear reinforced area is the governing failure mode. ACI 31814 predicts safe ultimate punching shear loads for both slabs. However, the ultimate load that ACI 31814 predicts for slab SB4 is conservative and it seems that ACI underestimates the contribution of the amount of the shear bolts for the analyzed specimens presented in this study. According to Alexander and Hawkins (2005), the ACI code provides a lower limit since it does not predict accurately the punching shear strength of variety of slabs.
Table 3: Slab
7
Failure load, kN (kips) Ultimate load, kN (kips) Test
FEA
ACI 31814
SB1
253 (56.877)
252 (56.652)
189 (42.489)
SB4
360 (80.931)
373 (83.854)
237 (53.280)
Conclusions
The modeling and analysis of the mechanical behavior of reinforced concrete slabs is very demanding due to the complex constitutive concrete model needed to simulate punching shear failure. Calibration of numerical models based on the comparison with test findings will always be necessary to verify the accuracy of FEA models. The paper presents FEA of punching shear of reinforced concrete slabs. The calibration of the concrete model, called the concrete damaged plasticity model in ABAQUS, is first presented based on the response of slab SB1 (slab without punching shear reinforcement). Then, the calibrated model is used to model and analyze slab SB4 (slab with shear bolts). The following conclusions can be drawn: 1. Damage parameters should be taken into account after calibrating the concrete model. The relationship between inelastic and plastic strains should be adopted based on the test results. In the examples presented, the plastic strain is taken as equal to 0.7 times the inelastic strain. 2. Both bilinear and exponential tension stiffening approaches can model the response of slab SB1 in a good manner compared to the test results. The linear tension stiffening
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response underestimates the ultimate load of specimen SB1, since it assumes that the tensile stress becomes zero earlier compared to the other approaches. 3. The FEA results using neoprene to simulate the boundary conditions of specimen SB1 are in good agreement with the test results and they better describe, especially in the initial uncracked state, the loaddeflection response of the slab when compared to the simple supports. The simulation of the boundary conditions using neoprene overcomes the initial stiffer response obtained from simulations by using simple supports. 4. Beam elements simulate successfully the shear bolts on specimen SB4 using the same concrete model adopted in the analysis of slab SB1. 5. Test and numerical results in terms of loaddeflection response and cracking propagation are in good agreement, showing that the calibrated concrete damaged plasticity model in ABAQUS accurately predicts the responses of both slabs. ACI 31814 predicts safe ultimate punching shear loads for both slabs. The punching shear load that ACI 31814 predicts for slab SB4 seems to be more conservative compared to the punching shear load of slab SB1. ACI 31814 does not include the effect of the flexural reinforcement ratio and the size effect in the design equations, and thus conservative results appear for thin slabs with high reinforcement ratios.
8
Acknowledgements
The authors would like to acknowledge the financial support for the presented work provided by a research grand from the Natural Sciences and Engineering Research Council (NSERC) of Canada.
9
References
ABAQUS Analysis user’s manual 6.123 (2012) Dassault Systems Simulia Corp, Providence, RI, USA. ACI Committee 318 (2014) Building Code Requirements for Structural Concrete (ACI 31814) and Commentary. American Concrete Institute, Farmington Hills, MI. Adetifa, B., and Polak, M.A. (2005) “Retrofit of interior slabcolumn connections for punching using shear bolts,” ACI Structural Journal, V. 102, No. 2, pp. 268274. Alexander, S.D.B., and Hawkins, N.M. (2005) “A design prospective on punching shear,” Punching Shear in Reinforced Concrete Slabs, ACI SP23206, pp. 97108. Comité EuroInternational du Béton. (1993) CEBFIPmodel Code 1990, Thomas Telford, London. Corley, W.G., and Hawkins, N.M. (1968) “Shearhead reinforcement for slabs.” ACI Journal, 65(10), 811824. Cornelissen, H., Hordijk, D., and Reinhardt, H. (1986) “Experimental determination of crack softening characteristics of normal weight and lightweight concrete,” Heron V.31, pp. 4556. Dilger, W. H., and Ghali, A. (1981) “Shear Reinforcement for Concrete Slabs.” J. of Struct. Division, 107(12), 24032420. Duvaut, G., and Lions, J.L. (1976) Inequalities in Mechanics and Physics, Springer, New York.
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Genikomsou, A.S., and Polak, M.A. (2015) “Finite element analysis of punching shear of concrete slabs using damaged plasticity model in ABAQUS,” Engineering Structures, V. 98, No. 4, pp. 3848. Genikomsou, A.S., and Polak, M.A. (2016) “Finite Element Analysis of Reinforced Concrete Slabs with Punching Shear Reinforcement,” Journal of Structural Engineering, ASCE, V. 142, No. 12, DOI: 10.1061/(ASCE)ST.1943541X.0001603. Grassl, P., Lundgren, K., and Gylltoft, K. (2002) “Concrete in compression: a plasticity theory with a novel hardening law,” International Journal of Solids and Structures, V. 39, No. 20, pp. 52055223. Hillerborg, A. (1985) “The theoretical basis of a method to determine the fracture energy GF of concrete,” Materials and Structures, V. 18, No. 4, pp. 291296. Imran, I., and Pantazopoulou, S.J. (2001) “Plasticity model for concrete under triaxial compression,” Journal of Engineering Mechanics, 127(3):281290. Islam, S., and Park, R. (1976) “Tests on slabcolumn connections with shear and unbalanced flexure.” J. of Struct. Division, 102(3), 549568. Ju, J.W. (1989) “On energybased coupled elastoplastic damage theories: Constitutive modeling and computational aspects,” International Journal of Solids and Structures, V. 25, No. 7, pp. 803833. Lee, J. and Fenves, G.L. (1998) “Plasticdamage model for cyclic loading of concrete structures,” ASCE Journal of Engineering Mechanics, V. 124, pp. 892900. Lemaitre, J., and Chaboche, J.L. (1990) Mechanics of solid materials, Cambridge University Press, New York. Lubliner, J., Oliver, J., Oller, S., and Oñate, E. (1989) “A plasticdamage model for concrete,” International Journal of Solids and Structures, V. 25, No. 3, pp. 299326. Mazars, J. (1986) “A model of unilateral elastic damageable material and its application to concrete,” Fracture toughness and fracture energy of concrete, F.H. Wittmann, ed., Elsevier Science Publishers, Amsterdam, The Netherlands, pp. 6171. Mazars, J., and PijaudierCabot, G. (1989) “Continuum damage theoryapplication to concrete,” Journal of Engineering Mechanics ASCE, V. 115, No. 2, pp. 345365. Polak, M.A. (2005) “Shell finite element analysis of RC plates supported on columns for punching shear and flexure,” International Journal for ComputerAided Engineering and Software, V. 22, No.4, pp. 409428. Polak, M.A., and Bu, W. (2013) “Design considerations for shear bolts in punching shear retrofit of reinforced concrete slabs,” ACI Structural Journal, V. 110, No. 1, pp. 1525. Promono, E., and Willam, K. (1989) “Fracture energybased plasticity formulation of plain concrete,” Journal of Engineering Mechanics ASCE, V. 115, No. 6, pp. 11831203. Simo, J.C., and Ju, J.W. 1987a. “Strain and stress based continuum damage models  I. Formulation,” International Journal of Solids and Structures, V. 23, No. 7, pp. 821840. Simo, J.C., and Ju, J.W. 1987b. “Strain and stressbased continuum damage modelII. Computational aspects,” International Journal of Solids and Structures, V.23, No. 7, pp. 841869. Tan, Y., and Teng, S. (2005) “Interior SlabRectangular Column Connections Under Biaxial Lateral Loadings.” Punching Shear in Reinforced Concrete Slabs, ACI SP23209, pp. 147174.
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Effect of slab flexural reinforcement and depth on punching strength Neil M. Hawkins1, Carlos E. Ospina2 1
: University of Illinois at UrbanaChampaign, Ill., USA
2
: BergerABAM Inc., Houston, Tex., USA
Abstract Recommendations are presented for proposed revisions to the punching shear provisions of ACI 31814 in order to recognize the limitations imposed on punching shear capacity by low amounts of slab flexural reinforcement and by slab depth effects. The influence of those two factors is investigated by examining relevant experimental results from tests on slabcolumn assemblages and slab systems.
Keywords Concrete design codes, depth effect, flat slabs, moment transfer, punching shear, reinforcement ratio, size effect, slabcolumn connections
1
Introduction
The nominal punching shear strength for normal weight concrete, vc = 4√f’c, in psi (0.33√f’c in MPa) specified in Eq.(a) in Table 22.6.5.2 of ACI 31814 for twoway members without shear reinforcement, has remained essentially unchanged since its introduction in the 1963 edition of ACI 318. This value, and the critical section associated with it, were developed by ACIASCE Committee 326 (1962), largely based on the work of Moe (1961) along with results from earlier investigators. Most of the analyzed results were for slab to interior column connection assemblages where, for the slab, the effective depth was limited to 4.5 in. (114 mm) or less, the negative moment flexural reinforcement ratio was 1.06 % or greater, and concrete strengths ranged from 3,000 to 4,000 psi (20.7 to 27.6 MPa). Committee 326 noted that it is not economically feasible to test slab systems to determine punching shear strengths, and that because the punching problem can be considered a localized condition involving only a portion of the slab system, it is reasonable to base the nominal punching shear strength on the results of slabinterior column assemblage tests. In the expression Moe derived, vc is a function of ϕo, which equals the shear force at punching, Vtest, divided by the shear force for flexural failure, Vflex. While it is easy to calculate Vflex for the slab of a slabinterior column assemblage based on yieldline theory, it can be difficult to calculate Vflex for practical slab systems. Further, yield line theory is generally not used for design for flexure in American practice. While Committee 326 saw ϕo as an important variable for analyzing slabcolumn data, they reasoned it was not an important variable for slab systems because the
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shear capacity of the system should exceed its flexural capacity. They derived the 4√f’c value by taking ϕo as unity and assuming that vc should approach infinity as the ratio of the column side dimension to the effective slab depth decreased. Since the 1960s American concrete design practice has evolved considerably with reinforcing steel grades of 100 (690 MPa) or more being used in slabs and concrete strengths often being greater than 4,000 psi (27.6 MPa). For typical office and residential building designs, column connections are routinely stressed to ϕvc and slab column strip negative moment flexural reinforcement ratios are 1.0% or less. Further, that reinforcement is commonly placed at a uniform spacing in the slab region around the column, even though the slab moment is a maximum at the column. This uniform distribution of steel is considered permissible due to the ability of slabs to redistribute moments over a significant width. Moe (1961) conducted two series of tests where the flexural reinforcement was concentrated in the vicinity of the interior column while the reinforcement ratio for the slab as a whole was kept constant. Moe found that such concentration did not increase the punching shear strength but did increase the flexural rigidity of the slab. For slab to interior column connections transferring shear only, the results of tests (Criswell, 1974; Muttoni, 2008; Peiris and Ghali, 2012; Tian et al., 2008, and Widianto et al., 2009) and the compilation of a comprehensive database of existing test results (Ospina et al., 2011, 2012) have shown that the punching shear strength provisions of ACI 31814 can be nonconservative in two situations: (1) where the slab flexural tension Grade 60 (414 MPa) reinforcement in the immediate vicinity of the column is less than 1%; and (2) where slabs have effective depths greater than 10 in. (250 mm). The objective of this paper is to summarize the test data that justify ACI 31814 change proposals that address those two deficiencies and report the basis of such proposed code changes.
2
Slabs with flexural tension reinforcement ratios less than 1%
2.1
General principles
For oneway action, it is easy to distinguish between a flexural and a shear failure. Shear failures seldom develop once the rotations associated with flexural yielding occur. The behavior for twoway action is different. It is difficult to distinguish visually between a “pure” punching failure and a “flexuredriven” punching failure. Under gravity loading, inclined cracking develops around the perimeter of the loaded area or column, and within the body of the slab, at a shear stress of about 2√f’c psi (0.165 √f’c MPa) even though a “pure” punching failure does not occur until a stress of about 4√f’c psi (0.33 √f’c MPa) or more. The increase in strength between inclined cracking and failure is resisted, in large measure, by aggregate interlock along the inclined crack. This situation is illustrated in Fig.1 which shows the inclined crack and relationship between crack width and slab rotation ψ.
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Figure 1:
Correlation between inclined crack opening, slab thickness, and rotation (Muttoni, 2008).
As the flexural tension reinforcement in the slab around the column decreases, the flexibility of the connection at a given shear stress level increases. Experience shows that where Grade 60 (414 MPa) reinforcement is less than about 1%, yielding of this reinforcement adjacent to the column increases rotations. Consequent crack openings permit sliding along the internal inclined crack. “Flexuredriven” punching failures may occur at shear stress levels less than 4√f’c psi (0.33√f’c MPa) (Widianto et al., 2009; Tian et al., 2008; Muttoni, 2008; Dam and Wight, 2016). Slab rotations that develop before such failures are often only marginally greater than those associated with a “pure” punching failure. The nominal shear strength vc must be applicable for moment and shear transfer as well as for shear transfer only. Shown in Fig. 2 are the stresses inferred from measured steel strains across the slab width with increasing load for a slabinterior column assembly test (Hawkins et al., 1989) with 0.96% tension reinforcement in the slab, with both shear and moment transferred to the column, and with shear reinforcement surrounding the bars passing through the column.
Figure 2:
Distribution of steel stresses across slab width (Hawkins et al., 1989). (1 ksi = 6.895MPa)
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In Fig. 2, V is the shear transferred to the column and Vflex is the shear for yielding across the full width of the slab. The slab reinforcement passing through the column began yielding at about 50% of Vflex. A punching failure occurred at about 80% of Vflex after yielding had spread to the first of the flexural bars passing outside the column. For similar connections with 0.6% slab reinforcement, without shear reinforcement, the failure mode was similar but the shear stress at failure was as low as 3.2√f’c psi (0.26 √f’c MPa). For a slabinterior column assembly, yielding of the slab reinforcement in the vicinity of the column must occur before Vflex can be developed and local yielding can result in a “flexuredriven” punching failure. In a reallife slab, where continuity effects would make the calculation of Vflex a rather difficult task, it is sensible to deal instead with the shear force associated with the flexural strength of the slab locally around the column, Vly. The issue then becomes what is the appropriate way to determine Vly of the slab for comparisons with the measured slab punching shear strength. The critical shear crack theory (CSCT) (Muttoni, 2008) analyzes conditions for sliding along the internal shear crack. Its use, in accordance with procedures of the Swiss Concrete Code, gives good agreement with available data for slabcolumn connection tests exhibiting both “pure” and “flexuredriven” punching failures and having a wide range of properties, including low reinforcement ratios and effective depths greater than 10 in. (250 mm). However, use of the CSCT is relatively complicated and its application to slab systems with a wide variety of column layouts and span lengths can be difficult. While such complications can be readily addressed through the use of computer analyses, the shear strength of the slabinterior column connection often controls slab depth and system span length choices. Therefore, a procedure that retains the simplicity of the 4√f’c psi (0.33 √f’c MPa) approach of ACI 31814, while ensuring that “flexuredriven” punching failures are avoided, is preferable. For slabinterior column connections, calculated strengths are similar to those of the CSCT for low reinforcement ratios, provided the punching shear capacity at the column is limited to the shear associated with the local yield strength of the tension reinforcement in the slab surrounding the column. For most laboratory test specimens, the shear associated with that local yield strength, Vly, is a function of the distance from the column face to the edge of the test slab. However, in practice, at interior column connections, there is often no defined slab edge and Vly can be approximated as slightly greater than 8 m (Peiris and Ghali, 2012) where m is the nominal flexural strength of the slab per unit width for the reinforcement within 1.5h of the column perimeter. The corresponding local yield strength for edge and corner columns can then be approximated by reducing the constant 8 in direct proportion to the ratio of the contact perimeter between the slab and the column for the noninterior column versus the same contact length for the same column as an interior column. The local flexural strength Vly can be approximated as: Vly = 0.2 αs m
(1)
where αs, as defined in 22.6.5.3 of ACI 31814, is 40 for interior columns, 30 for edge columns, and 20 for corner columns. Those values for edge and corner columns are for slabs extending the full length of the column side dimensions. Note Equation 1 has units of force.
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2.2
Slabinterior column connections without shear reinforcement and transferring shear only
The application of the foregoing concepts to slabinterior column connections transferring shear only are discussed in greater detail in Ospina and Hawkins (2013). Shown in Fig. 3 is the agreement between measured and computed strengths for available slab to interior column connection data when Vtest is limited to Vflex and where the connection is transferring shear only. Vflex in Fig. 3 is the theoretical yield line capacity for the test specimen. On the ordinates, kv is a correction for slab depth effect as discussed later in the paper. Shown in Table 1 are properties for all the specimens, displaying either flexuredriven or pure punching failures, where the abscissa values of Fig. 3 are less than 5.0. Test specimen properties were taken directly from the NEES databank (Ospina et al., 2011 and 2012). The 45degree line in Fig. 3 captures reasonably well the expected failure load of test slabs displaying flexuredriven punching failures, confirming that punching for these slabs can in fact occur at a stress below 4√f’c psi (0.33 √f’c MPa), as noted, among others, by Widianto et al. (2009) and Peiris and Ghali (2012). For test slabs displaying pure punching failures, Fig. 3 shows that the current ACI design provisions provide a reasonable lower bound for the selected test data. It is worth noting that not until the ACI 445 punching test databank was developed (Ospina et al., 2011), test slabs displaying flexuredriven punching failures were commonly disregarded in earlier punching test databank compilation and evaluation efforts on the assumption that these slabs do not provide an explanation for pure punching failure. Ignoring this important experimental evidence hides the fact that slabs with low amounts of flexural reinforcement can in fact punch at loads below the basic value defined in traditional design codes.
Figure 3:
Correlation of measured and calculated strengths for slabinterior column assemblages transferring shear only.
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Table 1:
Properties of slabinterior column subassemblage tests transferring shear only, with Vflex /(bod√f’c) 1 and gfMT/MR > 1
2.0
1.5
Test/Calc 1.0 Subassemblages  Shear Controlled
0.5
Subassemblages  Moment Controlled
Frames  Shear Controlled Frames  Moment Controlled
0.0 0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
M e = Vc c Figure 4(a): Moment transfer strengths per ACI 31814 for data in Appendix A. 2.5 Test/Calc Ratio per ACI 31814 evaluated as
greater of VT/Vo+MT/Mo > 1, gfMT/MR > 1 and VT/Vf+MT/Mo > 1
2.0
1.5
Test/Calc 1.0 Subassemblages  Shear Controlled
0.5
Subassemblages  Moment Controlled
Frames  Shear Controlled Frames  Moment Controlled
0.0
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
M e = Vc c Figure 4(b): Moment transfer strengths per proposed revision for data in Appendix A.
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4
Code limitation for low slab flexural reinforcement ratios
4.1
Lessons from design examples
As documented in Appendix A, the impact on design of the limitation of vu to the lesser of ϕ vc or ϕ vly for slabcolumn connections without shear reinforcement and the lesser of ϕ(vc/2+ vs) or ϕ vly for slabcolumn connections with shear reinforcement was examined for two typical structures designed by a leading structural engineering firm. One structure had 28 in. (711 mm) square columns, orthogonal spans of 30 ft (9.14 m) and a slab with an overall depth of 10in. (250 mm). The second structure had 12 in. (305 mm) square columns, orthogonal spans of 24 ft (7.32 m) and a slab with an overall depth of 8 in. (203 mm). For both structures, the design moments and shears to be transferred to the columns, the number of bars and their sizes for the column strip and the middle strip for a typical interior span, an end span with an exterior column transferring moment normal to the edge and a corner column (end span in orthogonal directions), were provided. In both designs, the column strip reinforcement at the interior column was less than 1%. For the edge and corner columns, no changes were needed to the prior design with the proposed limitation on vu. For both designs, revisions to the detailing of the reinforcement in the vicinity of the interior column were needed. For the 30 ft (9.14 m) span structure, there was a need to change the uniformly spaced slab tension reinforcement so that there was slightly more reinforcement within lines 1.5h on either side of the column. With that adjustment, however, the spacing of the remaining tension reinforcement in the column strip outside the column area was still reasonable. For the 24 ft (7.32 m) span structure, shear reinforcement was required to meet existing ACI 31814 provisions. To satisfy the proposed revision to the vu limit, again slightly reduced reinforcement spacing within lines 1.5h on either side of the column was needed. The results of these design analyses suggest that, for new construction, rather than requiring a calculation of Vly, it is more appropriate to specify a minimum amount of reinforcement within lines 1.5h on either side of the column or concentrated load area. For evaluations of the strength of existing construction, a calculation of Vly may still be needed.
4.2
Reinforcement requirements for slabs without shear reinforcement
For slabs without shear reinforcement the ρfy value required to meet the nominal punching shear strength specified in Table 22.6.5.2 of ACI 31814 can be derived as follows: For interior column connections, Vly = 8 m = 8 ρfyd2
(2)
In Eq. 2, m has been taken as ρfyd2, which is a reasonable simplification for slabs. For a punching failure,
Vc = 4λ√f’c(bod) in psi (0.33λ√f’c(bod) in MPa
(3)
For Eq.1 to not control,
ρfy ≥ λ√f’c(bo/2 d)
(4)
so that
ρfy ≥ 2λ√f’c(c1/d +1) for square columns with side length c1
(5)
and
ρfy ≥ (boλ√f’c αs)/ 80d in psi (boλ√f’c αs)/ 960d in MPa
(6)
For typical variations in c1 /d and f’c, the ρ values for Grade 60 (414 MPa) reinforcement are given in Table 2.
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Table 2:
Variation in ρ with c1/d and f’c
c1/d
f’c, psi (MPa)
2
3
4
3,000 (20.7)
0.0055
0.0073
0.0091
4,000 (27.6)
0.0063
0.0084
0.0105
5,000 (34.0)
0.0071
0.0094
0.0118
ρfy in Eq. 4 is made extendable to edge and corner columns by expressing it in terms of αs where αs is given in 22.6.5.3 of ACI 31814. Use of a ρfy value is recommended rather than a ρ value because designs using reinforcement yield strengths greater than 60ksi (414 MPa) are becoming increasingly common. Shown in Fig.5 is the correlation between ρ and Vtest for the same test data as that shown in Fig.3. The reinforcement ratio was calculated within 1.5h on either side of the column or loaded area. On the vertical axis, VACI = Vc = 4λ√f’c(bod) in psi (0.33λ√f’c(bod) in MPa) and kv is a correction factor for the slab depth effect discussed later.
2.5
2.0
1.5
Vtest V ACI k v 1.0
0.5 Pure Punching Failures
Flexuredriven Punching Failures
0.0 0.000
0.010
0.020
0.030
r Figure 5:
Correlation between Vtest and ρ for connections transferring shear only.
Figure 6 shows a figure similar to Fig. 5 except that the calculated shear capacity has been taken as the lesser of VACI = Vc = 4λ√f’c(bod) in psi (0.33λ√f’c(bod) in MPa), and that defined by Eq. 2. Figure 6 shows that the nonconservatism of the ACI 31814 design provisions for slabs with flexuredriven punching failures can be eliminated for most of the tests using the proposed recommendations.
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2.5
2.0
Vtest Vc k v
1.5
1.0
Pure Punching Failures
0.5
Flexuredriven Punching Failures
Note: Vc is the lesser of VACI and Vly 0.0 0.000
0.010
0.020
0.030
r Figure 6:
4.3
Correlation between Vtest and ρ for connections transferring shear only using proposed recommendations.
Reinforcement requirements for slabs with shear reinforcement
For slabs with shear reinforcement, ACI 31814 specifies that vn = vc/2 + vs, when stirrups are used and vn = 3vc/4 + vs when headed studs are used. For simplicity in design, it is recommended that the same ρ limit be used regardless of the type of shear reinforcement. Per Table 22.6.6.2 of ACI 31814, the maximum value for v n when stirrups are used is 6√f’c psi (0.50√f’c MPa) so that for consistency with requirements for slabs without shear reinforcement, the minimum required ρfy for slabs with shear reinforcement could be as much as 50% greater than that for slabs without shear reinforcement. Because vn is rarely provided up to the maximum 6√f’c psi (0.50√f’c MPa) limit, a 4/3 multiplication is proposed for the minimum flexural reinforcement ρfy for slabs with shear reinforcement.
5
Effect of slab depth on punching shear strength
Numerous investigators have reported a decrease in punching shear strength with increasing slab depth. See for instance Mitchell et al. (2005). A statistical study (Dönmez and Bažant, 2016) using the information in the NEES database (Ospina et al., 2011) has demonstrated the dependence of punching strength as a function of 1/√(1+d/do), where do is a constant. Selected test results from the NEES database indicate that this depth effect factor can be applied for values of d greater than 10 in. (250 mm). Corrections to the basic expressions for punching shear strength for depth effects are currently included in the Canadian, EC2, and Japanese building codes. Shown in Fig. 7 are measured punching strengths for laboratory tests where investigators made systematic changes to the slab depth of their test specimens. Reinforcement ratios varied from 0.33 to 1.15%. Details of the data for the investigations shown in Fig. 7 can be found in the NEES database. References are listed in Ospina and Hawkins (2013). Measured strengths for tests decrease with increasing slab depth and are less than the ACI 31814
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predicted strengths for effective depths greater than 10 in. (250 mm). The influence of the slab reinforcement ratio when combined with the slab depth effect is rather inconclusive for the available test data. Figure 7 includes alternative depth effect factors that produce reasonable strength predictions for the available test data. The socalled “smooth” formulation (1.4/√(1+d/10) (d in inches)) (1.4/√(1+d/250) (d in mm)) provides a continuous correction across the full range of d values. This factor is to be applied for depths greater than 10 in. (250 mm). A reasonable lower bound can be also approximated by a depth effect relationship of 2.5/d2/5 (d in inches) (9/d2/5, d in mm). A third factor of the form 3/√d (d in inches) (15/√d) (d in mm)) produces the most conservative predictions. It is recommended that the values of Table 22.6.5.2 of ACI 31814 be modified by 1.4/√(1+d/10) (d in inches) to recognize the depth effect on punching strength. It is worth noting that the single test result by Guandalini et al. (2009) shown below the proposed design curve corresponds to a slab with a very low reinforcement ratio. Calculations (not reported herein) show that this slab would have punched at a load level close to what the proposed depth effect formulation predicts, had the minimum amount of flexural reinforcement proposed herein been provided to this test slab.
10
Guandalini et al (2009), rho = 0.33%
9
Birkle (2004), rho = 1.1  1.15%
8
Li (2000), rho = 0.8  1.0% Regan (1986), rho = 1.0%
Vtest bo d
7
KTH (1972, 1980), rho = 0.6  0.8%
6
Nylander & Sundquist (1972), rho = 0.6%
ACI 31814
5
f
' c
1 .4 1+
4
d 10
3 2
d
2 .5 d 2/5
20
25
3
1 0 0
5
10
15
30
d (in)
Figure 7:
Variation in punching shear strength with slab effective depth. (1 in. = 25.4 mm)
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6
Conclusions and recommendations
Two limitations to the existing punching shear provisions of ACI 31814 should be recognized: 1. There can be flexuredriven punching shear failures resulting from yielding of the slab flexural reinforcement in the immediate vicinity of a column or concentrated load area. Those failures are similar in appearance to “pure” punching shear failures and the additional ductility resulting from a flexuredriven, as compared to a “pure” punching shear failure, can be small. 2. For punching shear failures, there is a decrease in the nominal shear strength with increasing slab depth. To address the flexuredriven punching shear issue, ρfy for the slab flexural tension reinforcement within 1.5h of the perimeter of the column or concentrated load area should be equal to or greater than (boλ√f’c αs)/80d psi ((boλ√f’c αs)/960d MPa) for slabs without shear reinforcement and 1/3 greater for slabs with shear reinforcement. To address the slab depth effect, the existing twoway nominal shear strength values of Table 22.6.5.2 of ACI 31814 should be limited to slabs with effective depths of 10 in. (250 mm) or less. For greater depths, the values should be reduced in proportion to 1.4/√(1+d/10 (d in inches) (1.4/√(1+d/250) (d in mm).
7
Notation The notation of this paper is that of ACI 31814 with the following additions: kv = Slab depth effect factor = 1.4/√(1+d/10) (d in inches) (1.4/√(1+d/250) (d in mm). m = Nominal flexural strength of slab per unit width for the reinforcement within 1.5h of the column or concentrated load area MR = Calculated moment capacity of slab MT = Measured unbalanced moment in test Vflex= Shear force for slab flexural failure Vly = Shear force for yielding of the slab flexural tension reinforcement within 1.5h of the column or concentrated load area VT = Measured shear force in test
8
References
ACIASCE Committee 326. 1962. “Shear and Diagonal Tension  Slabs,” ACI Journal, Proceedings, V.59, No. 3, pp.352396. Dam, T.X. and Wight, J.K. 2016. “FlexurallyTriggered Punching Shear Failure of Reinforced Concrete SlabColumn Connections Reinforced with Headed Shear Studs Arranged in Orthogonal and Radial Layouts,” Engineering Structures (Elsevier), 110, pp. 258268.
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Dönmez, A., and Bažant, Z.P. 2016. “Size Effect on Punching Shear Strength of Reinforced Concrete Slabs Without and With Shear Reinforcement,” Report No. 1512/936s, Civil and Environmental Engineering, Northwestern University, Evanston, IL, p. 23. Hawkins, N.M., Bao, A., and Yamazaki, J. 1989. “Moment Transfer from Concrete Slabs to Columns” ACI Structural Journal, V. 86, No. 6, pp. 705716. Mitchell, D., Cook, W.D., Dilger, W. 2005. “Effects of Size, Geometry and Material Properties on Punching Shear Resistance,” SP232 Punching Shear of Reinforced Concrete Slabs, American Concrete Institute, Farmington Hills, MI. pp.3955. Moe, J. 1961. “Shearing Strength of Reinforced Concrete Slabs and Footings Under Concentrated Loads,” Development Department Bulletin No. D47, Portland Cement Association, Skokie, IL, 130 pp. Muttoni, A. 2008. “Punching Shear Strength of Reinforced Concrete Slabs without Shear Reinforcement,” ACI Structural Journal, Vol. 105, No. 4, pp. 440450. Ospina, C.E., Birkle, G., Widianto, ., Wang, Y., Fernando, S.R., Fernando, S., Catlin, A.C. and Pujol, S. 2011. “ACI 445 Collected Punching Shear Databank,” http://nees.org/resources/3660. Ospina, C. E., Birkle, G., and Widianto, . 2012. “Databank of Concentric Punching Shear Tests of Twoway Concrete Slabs without Shear Reinforcement at Interior Supports,” 2012 ASCE Structures Congress, Proceedings, pp. 18141832. Ospina, C. E., and Hawkins, N.M. 2013. “Addressing Punching Failure,” Structure Magazine, National Council of Structural Engineers Associations, V. 20, No. 1, p. 1416. Peiris, C., and Ghali, A. 2012. “Flexural Reinforcement Essential for Punching Shear Resistance of Slabs,” SP 28706 Recent Developments in Reinforced Concrete Slab Analysis, Design and Serviceability, American Concrete Institute, Farmington Hills, MI. Stein, T., Ghali, A., and Dilger, W. 2007. “Distinction Between Punching and Flexural Failure Modes of Flat Plates,” ACI Structural Journal, V. 104, No. 3, pp. 357365. Tian, Y., Jirsa, J.O., Bayrak, O., Widianto, and Argudo, J.F. 2008. “Behavior of SlabColumn Connections of Existing FlatPlate Structures,” ACI Structural Journal, Vol. 105, No. 5, pp. 561569. Widianto, Bayrak O., and Jirsa, J.O. 2009. “TwoWay Shear Provisions of SlabColumn Connections: Reexamination of ACI 318 Provisions,” ACI Structural Journal, V. 106, No. 2, pp. 160170.
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Appendix A A.1
Overview
Summarized in this appendix are most available test data on the strength for simultaneous transfer of moment and shear to columns in flat plate concrete frames with flexural reinforcement ratios for the slab of about 1.1% or less in the vicinity of interior columns. The data are analyzed according to both the existing ACI 31814 requirements and additional limitations proposed in this paper. With two exceptions, data for only interior columns are analyzed. The two exceptions are for slabexterior column connections where moment was transferred to the column parallel to the edge. ACI 31814 recognizes that unbalanced moment transfer for such columns is very similar to that for unbalanced moment transfer to interior columns. For exterior columns (moment transferred normal to the edge), the ACI 31814 provisions effectively allow all unbalanced moment to be transferred by reinforcement within lines 1.5h on either side of the column when the shear being transferred along with that moment does not exceed 0.75ϕvc for edge, and 0.50ϕvc for corner columns. Therefore, the effect of unbalanced moment on shear strength does not have to be considered, but can be considered, if an increase in the amount of slab reinforcement is acceptable. By contrast, for interior slabcolumn connections, there is little ability to redistribute the unbalanced moment and it is nearly always necessary to consider the effect of that moment on punching shear strength. Given at the end of Appendix A are relevant references for the column to slab connections analyzed here. The analysis is for 48 subassemblies and 12 slab systems where slab top reinforcement ratios were 1.1% and less, columns were square, and there was no shear reinforcement in the slabs. Omitted from the data are results for most tests with slab reinforcement ratios higher than 1%, lightweight concrete, rectangular columns, column capitals, tests dealing with any strengthening methods, and any results for subassemblies with slab overall depths less than 4.5 in. (115 mm). Those omissions are deliberate as they are not directly relevant to understanding the validity of the code limitation discussed here. Further, when slab overall depths are less than 4.5 in. (115 mm) slight variations in reinforcement depths can significantly affect results. The analysis of the data is in two parts. Part I is for tests on subassembly specimens with a single central column. Part II is for the interior column of slab system tests. For the slab system tests the slab thickness requirement of 4.5 in. (115 mm) or more was not imposed because all but one of the tests had lesser thickness. In Part I the properties of the test specimens are provided in Table A1.1. Column 1 lists the reference where the data for the specimen listed in column 2 are found. Column 3 lists the specimen type. Columns 4 and 5 list the reported concrete strength at time of test and the measured yield strength for the slab top flexural reinforcement. Column 6 lists three quantities: (1) the reinforcement ratio for the top flexural bars in the slab based on the total width of the slab; (2) the same ratio for the top bars within lines 1.5h on either side of the column; and (3) the same ratio for the bottom bars within lines 1.5h on either side of the column. In general, the first ratio is consistent with the value provided in the reference. To derive the second ratio, an examination was made of the actual number of bars within the width c2 + 3h and the ratio calculated accordingly unless the spacing between bars was such that the distance between the line at 1.5h and the next bar was less than half the bar spacing. In that case, the number of bars was taken as one greater than the number within the width
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c2 + 3h. The reinforcement ratio for the bottom bars was calculated in the same manner as for the top bars. Column 7 lists two quantities, the first being the average effective depth for the top bars, and the second the overall depth of the slab. While the average effective depth is used for shear and flexural evaluations and the reinforcement ratios reported in column 6, a slightly different quantity, the actual depth for the slab reinforcement in the direction of moment transfer for both top and bottom bars was used for evaluations of the fraction of the unbalanced moment transferred by the slab flexural reinforcement. However, some researchers omitted exact details for bar size, spacing, and yield stress, and therefore calculations were based on the d values in column 7. Column 8 lists the column side dimension. All specimens had square columns. Column 9 lists the slab outside to outside dimension with the dimension in the direction of moment transfer listed first. Column 10 lists the type of loading to which the specimen was subjected. Table A1.2 contains results of analyses in accordance with ACI 31814 and the proposed code change. Column 1 lists the specimen name. Columns 2 through 5 list, respectively, the shear, VT, acting on the connection at failure; the unbalanced moment, MT, acting on the connection at failure; the nominal shear strength, V0 = 4√f’c*bo*d according to ACI 31814 for direct shear transfer only; and the corresponding quantity, M0 = 4√f’c*Jc/(γv*(c1/2 + d/2)), for transfer of moment only. Column 6 lists the sum of the ratios for measured to calculated shear and moment strengths. For values of that sum greater than 1.0 a punching shear failure is predicted. Listed in column 7 is the amount of unbalanced moment calculated by ACI 31814 transferred to the column by the flexural reinforcement in the slab within lines 1.5h on either side of the column. Listed in column 8 is the moment MR that can be transferred by the top and bottom flexural reinforcement in the slab within the lines 1.5h on either side of the column. The ratio in column 9 is γf MT divided by MR. For values greater than 1.0, a flexuredriven punching failure resulting from inadequate moment transfer reinforcement is predicted. Column 10 lists the theoretical nominal flexural strength for the connection for shear transfer only. That prediction is 8m where m is the average nominal moment per unit width provided by the top flexural reinforcement in the slab within lines 1.5h on either side of the column for the length of the column perimeter. Column 11 lists the sum of the ratios of the measured to calculated unbalanced moment strength plus the ratio of the measured shear to the shear 8m. If that ratio is greater than 1.0 a flexuredriven punching failure is predicted. Column 12 lists the failure mode as apparent from measured loaddeflection results. If the ductility, defined in the manner reported by Hawkins et al. (1989) exceeded 1.8, the failure mode of column 12 is reported as F/P, a flexuredriven punching failure. If the ductility is less than 1.8, the failure mode is defined as a punching failure, P. In Part II, the properties of the test frames are listed in Table A2.1 and the test results are listed in Table A2.2. The columns in Tables 2.1 and 2.2 have the same meaning as those for subassembly tests in Part I. Results for 12 interior columns that were part of test frames are reported in Part II. In all 12 frame tests, punching failures occurred first at the interior, rather than the exterior, column connections.
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A.2
ACI 31814 strength predictions and ACI 31814 strength predictions amended as proposed
In Tables A1.2 and A2.2, the highest of the three ratios (columns 6, 9, and 11 in Table A1.2 and columns 5, 8, and 10 in Table A2.2) is shown in bold. An examination of those results indicates the appropriateness or nonappropriateness of this change proposal for the moment and shear transfer situation. Comparison of the ratios for the two columns 6 and 9 in Table A1.2 and 5 and 8 in Table A2.2 indicates the same situation for the existing code provisions for moment and shear transfer. For Table A1.2, for the ACI 31814 provisions, there are 25 results controlled by the punching shear strength provision of column 6 and 23 results by the flexural reinforcement strength limitation of column 9. The mean of the results where column 6 prevails is 1.23 and individual values range from 0.79 to 2.02. The mean of the results where column 9 prevails is 1.29 and individual results range from 0.99 to 2.17. For analysis using the ACI 31814 provisions amended as proposed, there are 19 results where column 6 prevails, 18 results where column 9 prevails, and 11 results where column 11 prevails. The corresponding mean values and the range in individual values for columns 6, 9 and 11 are 1.35 (1.002.02), 1.38 (0.992.17) and 1.10 (0.951.22). More importantly, for the existing code there are seven results out of the 48 where the measured to calculated strength is less than 1.00. The lowest result is 0.79. With the proposed change only two results are less than 1.00 and the lowest result is 0.95. Adoption of the change improves the likelihood of the failure mode, as listed in column 12, agreeing with the calculated prediction. The results in Part II for slab system tests are not as definitive as the results for subassembly tests. This is probably due to the difficulty of making such tests and the marked changes in properties with small changes in reinforcement positions within the slab. In Table A2.2, for the existing code provisions, there are six results for which column 6 (shear) controls (mean 1.12, range 0.911.36) and six results for which column 9 (moment transfer reinforcement) controls (mean 1.46, range 1.171.72). When the limitation proposed here is used there are four results for which shear controls, six results for which moment transfer reinforcement controls, and two results where the flexuredriven punching failure controls. Thus, for the systems tested, the introduction of the flexuredriven criteria of the proposed code change has little effect when there is moment transfer to the columns in addition to shear. However, the result for GS1 shows the marked change in the strength prediction for a low reinforcement ratio when there is shear transfer only. The introduction of the change provision also results in a better prediction of the mode of failure. One issue apparent from the slab system tests is how to address the calculation of the flexuredriven shear capacity when there are different reinforcement ratios in orthogonal directions. This issue has not been addressed in the subassembly tests but is immediately apparent in slab system tests where there are different span lengths between columns in orthogonal directions. The best agreement with test data was obtained when the m value used to calculate VF was taken as the m value for the reinforcement in the direction of the longer span. This finding was confirmed by using yield line analysis to find how VF changed with changing orthogonal span lengths and therefore different amounts of slab flexural reinforcement in orthogonal directions.
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Table A1.1 – Properties for subassemblage tests Ref.
Spec. No.
Spec. type
f’c psi
fy ksi
ρt / ρc+3ht/ ρc+3hb %
d/h in.
1
1 3 ND1C ND4LL ND5XL ND6HR ND7LR 1C SCO HHC0.5 HHC1.0 NHC0.5 NHC1.0 CO 6AH 9.6AH 6AL 9.6AL 7.3BH 9.5BH 7.3BL 9.5BL 6CH 9.6CH 6CL 6FLI SM0.5 SM1.0 1 S2 S3 S7 S8 EL1 EL2 HHS0.5 HHS1.0 HLS0.5 HLS1.0 NHS1.0 NHHS0.5 NHHS1.0 C02 SW1 SW5 L0.5 LG0.5 LG1.0
SC SC SC SC SC SC SC SC SC SC SC SC SC SC SC SC SC SC SC SC SC SC SC SC SC SC SC SC SC SC SC SC SC SEC SEC SC SC SC SC SC SC SC SIC SC SC SC SC SC
4825 4550 4290 4680 3500 3810 2730 5130 5700 10960 10490 5330 5130 5600 4550 4450 3300 4200 3220 2880 2630 2910 7600 8300 7190 3760 5330 4840 5075 3400 3200 3840 4470 4620 3520 10730 10710 6270 6190 5250 4930 5110 4480 5080 6670 3710 4820 4000
68.4 68.4 60* 60* 60* 60* 60* 61 76.1 66.7 66.7 66.7 66.7 66.0 68.5 60.2 68.5 60.2 68.5 68.5 68.5 68.5 68.5 60.2 68.5 68.5 69.0 69.0 65.5 67.1 66.0 67.1 66.0 67.1 65.0 66.7 66.7 66.7 66.7 66.7 66.7 66.7 65.8 75.4 75.4 68 66 61
0.76/0.98/0.33 0.76/0.98/0.33 0.53/0.76/0 0.53/0.76/0 0.53/0.76/0 0.93/1.50/0 0.39/0.38/0 0.49/1.00/0.50 0.86/0.90/0.51 0.50/0.49/0.27 1.00/0.98/0.27 0.50/0.49/0.27 1.00/0.98/0.27 0.46/0.54/0.23 0.56/0.56/0.25 0.89/0.89/0.47 0.56/0.56/0.25 0.89/0.89/0.47 0.68/0.68/0.25 0.95/0.95/0.30 0.68/0.68/0.25 0.95/0.95/0.30 0.56/0.56/0.25 0.89/0.89/0.47 0.56/0.56/0.25 0.56/0.84/0.25 0.5/0.5/0.35 1.0/1.10/0.35 1.1/1.0/0.44 0.9/0.9/0.49 0.57/0.57/0.40 0.9/0.9/0.49 0.57/0.57/0.40 0.81/0.81/0.40 1.07/1.07/0.49 0.5/0.6/0.27 1.0/0.97/0.27 0.5/0.6/0.27 1.0/0.97/0.27 1.0/0.97/0.27 0.5/0.6/0.27 1.0/0.97/0.27 0.96/0.97/0.48 1.09/1.09/0.56 1.09/1.09/0.56 0.5/0.5/0 0.5/0.5/0 1.0/1.0/0
3.98/4.8 3.98/4.8 3.73/4.5 3.73/4.5 3.73/4.5 3.73/4.5 3.73/4.5 3.73/4.5 3.63/4.5 4.64/5.9 4.64/5.9 4.64/5.9 4.64/5.9 5.12/6.0 4.75/6.0 4.63/6.0 4.75/6.0 4.63/6.0 3.25/4.5 3.25/4.5 3.25/4.5 3.25/4.5 4.75/6.0 4.63/6.0 4.75/6.0 4.75/6.0 4.75/6.0 4.75/6.0 4.6/5.9 4.63/6.0 4.75/6.0 4.63/6.0 4.75/6.0 5.13/6.5 5.0/7.0 4.93/5.9 4.68/5.9 4.93/5.9 4.68/5.9 4.68/5.9 4.93/5.9 4.68/5.9 3.23/4.5 3.54/4.72 3.54/4.72 5.00/6.0 5.00/6.0 5.00/6.0
2
4 6 7
8 9
10 11 12
13 14
16 17 18
Col. size Slab size Load in. ft type 10.8 10.8 10.0 10.0 10.0 10.0 10.0 9.84 10 9.84 9.84 9.84 9.84 10.0 12 12 12 12 12 12 12 12 12 12 12 12 12 12 9.8 12 12 12 12 12 16 9.84 9.84 9.84 9.84 9.84 9.84 9.84 12 7.87 7.87 16 16 16
12x12 12x12 10x10 10x10 10x10 10x10 10x10 9.8x9.8 9.5x6.5 6.3x6.3 6.3x6.3 6.3x6.3 6.3x6.3 9.5x9.5 7x7 7x7 7x7 7x7 7x7 7x7 7x7 7x7 7x7 7x7 7x7 7x7 6x6 6x6 6x6 13 x 7 13 x 7 13 x 7 13 x 7 13 x 4 13 x 4 6x6 6x6 6x6 6x6 6x6 6x6 6x6 9x9 5x5 5x5 14 x 14 14 x 14 14 x 14
C C C C C C C C C C C C C C M M M M M M M M M M M M M M M C C C C C C M M M M M M M C C C C C C
Notes for Table A1.1: Specimen type: SC = slabinterior column; SIC = slabinterior concentrated load; SEC = slabedge column with moment transferred parallel to edge. Loading: C = cyclic; M = monotonic. G = gravity; *Grade 400 MPa bars. Actual yield stress not reported. (1,000 psi = 1 ksi = 6.895 MPa; 1 inch = 25.4 mm; 1 ft = 0.305 m)
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Table A1.2: Results for subassemblage tests Spec. VT MT Vo M0 VT/V0 + γf MT MR VF VT/VF + Failure g M /M No kips kipin. kips kipin. MT/M0 kipin. kipin F T R kips MT/M0 mode 1 26.0 557 65.4 821 1.08 334 343 0.97 81.7 1.00 P 3 16.3 865 63.5 797 1.35 519 343 1.51 81.7 1.28 F/P ND1C 13.28 347 53.7 626 0.80 208 142 1.46 31.8 0.97 F/P ND4LL 15.94 379 56.1 654 0.86 221 142 1.56 31.8 1.08 F/P ND5XL 22.75 275 48.5 566 0.97 165 142 1.16 31.8 1.21 P ND6HR 14.73 492 50.6 590 1.12 295 256 1.15 61.8 1.07 P ND7LR 11.02 227 42.8 499 0.72 136 103 1.27 24.4 0.91 F/P 1C 8.5 516 58.1 671 0.92 310 290 1.07 49.1 0.94 P SCO 16.0 543 59.7 707 1.04 326 314 1.04 68.1 1.00 F/P HHC0.5 28.1 1190 118 1393 1.09 714 329 2.17 74 1.23 F HHC1.0 28.1 1442 115 1363 1.30 865 505 1.72 148 1.25 F/P NHC0.5 28.1 889 84.3 971 1.25 533 329 1.62 74 1.29 F/P NHC1.0 28.1 1126 78.4 953 1.54 676 505 1.34 148 1.37 P CO 28.1 912 92.7 1203 1.06 547 370 1.48 71.6 1.15 P 6AH 38.1 800 85.9 1223 1.10 480 405 1.19 66.5 1.22 F/P 9.6AH 42.0 865 82.4 1153 1.25 519 604 0.86 88.2 1.23 P 6AL 54.8 289 76.1 1032 1.04 173 399 0.43 66.5 1.10 P 9.6AL 57.8 306 80.3 1133 0.99 184 602 0.30 88.2 0.93 P 7.3BH 17.9 345 44.8 575 1.00 207 190 1.09 36.4 1.10 F/P 9.5BH 21.2 402 42.4 551 1.25 330 265 1.25 48.6 1.17 F/P 7.3BL 29.2 113 41.1 514 0.94 68 184 0.37 36.0 1.03 F/P 9.5BL 31.9 147 42.5 544 1.03 88 265 0.33 48.6 0.93 P 6CH 41.9 842 110. 1588 0.90 505 411 1.23 67.7 1.15 F/P 9.6CH 48.9 1001 111. 1589 1.07 601 622 0.97 88.2 1.18 F/P 6CL 61.4 326 108. 1552 0.79 196 411 0.48 67.6 1.12 F/P 6FLI 51.0 240 78.5 1091 0.87 144 520 0.28 68.6 0.96 P SM0.5 29.0 888 92.9 1297 0.99 533 359 1.48 60.3 1.16 F/P SM1.0 29.0 1128 88.4 1235 1.24 677 601 1.13 117 1.16 P 1 33.7 1151 77.9 969 1.62 691 612 1.13 118 1.47 P S2 32.0 778 71.1 1010 1.22 467 629 0.74 94.2 1.11 P S3 31.2 475 72.6 1033 0.89 285 402 0.71 63.9 0.95 F/P S7 60.8 376 76 1075 1.15 226 633 0.21 95.4 0.99 P S8 52.8 289 85.2 1204 0.86 173 416 0.66 65.0 1.05 F/P EL1 17.0 779 65.4 1053 1.00 450 518 0.87 80.6 0.95 P EL2 18.6 1159 77.5 1588 0.97 673 681 0.99 114 0.89 P HHS0.5 45.0 1044 122. 1513 1.06 626 374 1.67 72.0 1.32 F/P HHS1.0 59.0 1173 114. 1413 1.35 704 575 1.22 111 1.36 F/P HLS0.5 59.9 393 92.2 1156 0.99 236 366 0.64 71 1.18 F/P HLS1.0 91.8 467 87.4 1061 1.49 280 557 0.50 108 1.29 P NHS1.0 36.8 1040 80.0 990 1.51 624 546 1.14 106 1.40 P NHHS0.5 36.9 865 82 1030 1.29 519 360 1.44 69.8 1.37 F/P NHHS1.0 56.3 1028 78.2 979 1.77 617 544 1.13 106 1.59 F/P C02 20.2 394 52.7 677 0.96 236 237 1.00 49.4 0.99 P SW1 24.73 608 46.0 448 1.90 365 314 1.16 75.9 1.69 P SW5 36.0 689 52.8 514 2.02 413 319 1.29 77.2 1.81 P L0.5 23.5 1094 103 1818 0.83 656 414 1.58 64.9 0.96 F/P LG0.5* 26.8 1028 117 2071 0.73 617 404 1.53 63.3 0.92 F/P LG1.0* 24.1 1360 106 1885 0.95 816 743 1.10 116.5 0.93 F/P Notes for Table A1.2: Failure mode: P =punching; F/P = flexure/punch. * After lateral loading specimens were loaded to failure under gravity load only. LG0.5 punched at 72.8 kips (1.12 VF) after some yielding. LG1.0 punched at 89.9 kips (0.77 VF) after little yielding. (1 kip = 4.448 kN; 1 kipin = 0.113 kNm)
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A.3
Design examples
The practicality of the proposed code changes was examined using the designs for two reinforced concrete structures completed by a major consulting firm. One structure had 28 by 28 inch (711 by 711mm) square columns, 30 ft (9.1 m) spans in orthogonal directions, and a 10 in. (250 mm) thick slab. The factored reaction for the typical interior column was 240 kips (1068 kN) and no moment transfer. The required top reinforcement was 25#6 bars for each direction for the column strip negative moment reinforcement. The second structure had 12 by 12 inch (305 by 305 mm) columns, 24 ft (7.3 m) spans in orthogonal directions, and an 8 in. (203 mm) thick slab. The factored reaction for the typical interior column was 135 kips (600 kN) and no moment transfer. The required top reinforcement was 16#6 bars for each direction for the column strip negative moment reinforcement along with shear reinforcement consisting of four legs of #3bars@3 in. (76 mm) on all sides of the column. For both designs the specified concrete strength was 4,000 psi (27.6 MPa) along with Grade 60 (414 MPa) reinforcement. In addition to slab dead load, there was a superimposed dead load of 30 psf (1.44 MPa) and a live load of 50 psf (2.39 MPa). The number of bars and their sizes for the column strip, and middle strip, for a typical interior span, an end span with the exterior column transferring moment normal to the edge, and a corner (end span in orthogonal directions) were specified in accordance with ACI 31814 requirements. The impact of the recommended code changes on the design of the slab reinforcement in the vicinity of a typical interior column, edge column, and corner column was examined. For both building designs, while the column strip reinforcement was nominally less than 1% for the edge and corner column designs, no changes were needed to the prior designs. The existing code requirement to concentrate column strip bars within lines 1.5h on either side of the column to transfer the fraction of the unbalanced moment not transferred by shear resulted in reinforcement ratios in that region greater than 1% and therefore there was no effect of the proposed flexuredriven punching shear requirement. For the interior column for the 30 ft (9.1 m) span, a slight concentration of the slab negative moment column strip reinforcement within the lines 1.5h on either side of the column was adequate to satisfy the proposed requirement. No additional column strip reinforcement was needed. For the interior span for the 24 ft (7.3 m) span, shear reinforcement was specified in the original design to meet existing code requirements. Even with the enhanced shear strength provided by that shear reinforcement, some concentration of top bars within lines 1.5h on either side of the column was adequate to ensure that the proposed limit was satisfied. Again, no additional column strip reinforcement was needed.
Part II: Frame data Robertson and Durrani (1992) and Durrani et al. (1995): Tests of interior column of twobay (9.5 ft (2.9 m) c.t.c. of columns); 6.5 ft (2.0 m) wide slab. Sherif and Dilger (2000): Fullscale test of frame consisting of one end frame with and edge column and one interior column with an overhang extending to the line of contraflexure. End frame 16 ft5 in. (5.0 m) c.t.c. of columns and interior overhang of 8 ft3 in. (2.5 m). Slab width of 16 ft5 in (5.0 m). Rha et al. (2014): Half scale tests of a 2 bay x 2 bay frame (one interior column, four edge columns and two corner columns). Bays of 9 ft (2.75 m) in lateral load direction and 5 ft5 in. (1.65 m) in transverse direction.
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Table A2.1: Specimen properties Ref.
Spec. No.
f’c psi
fy ksi
ρt / ρc+3ht / ρc+3hb %
d/h in.
Col. size in. x in.
Load type
10 10 10 10 10 10 9.8 9.84 9.84 9.84 9.84 9.84
C C C C C C M G G M M C
3
A 4790 72.6 0.49/0.83/0.38 3.66/4.5 B 4460 76.1 0.49/0.83/0.38 3.66/4.5 C 4670 76.1 0.49/0.83/0.38 3.66/4.5 5 DNY1 5120 54 0.55/0.74/0 3.81/4.5 DNY2 3730 54 0.55/0.74/0 3.81/4.5 DNY3 3570 54 0.55/0.74/0 3.81/4.5 15 S1 4060 64.4 1.29/1.29/0.20 5.03/5.9 19 GS1 3770 59.5 0.74/0.72/0.39 2.57/3.5 GS2 3770 59.5 0.98/1.26/0.39 2.57/3.5 LMS2 3770 59.5 0.74/1.26/0.42 2.57/3.5 LMS3 3770 59.5 0.98/1.26/1.05 2.57/3.5 LCS2 3770 59.5 0.74/1.26/0.42 2.57/3.5 Note: 1,000 psi = 1 ksi = 6.895 MPa; 1 inch = 25.4 mm; 1 ft = 0.305 m
Table A2.2: Test results and strength calculations for the interior column of the frame Spec. No.
VT kips
MT kipin
Vo kips
M0 VT/V0 + γf MT MR g M /M kipin MT/M0 kipin kipin F T R
A 10 623 55.6 639 1.15 B 19.3 366 53.4 616 0.96 C 28.8 240 27.1 630 0.91 DNY1 15.4 418 60.2 706 0.85 DNY2 19.2 296 51.4 602 0.87 DNY3 11.9 390 50.3 590 0.95 S1 89.7 3.54 72 967 1.25 GS1 34.34 0 31.36 0 1.10 GS2 35.88 0 31.36 0 1.14 LMS2 11.0 332 31.36 327 1.37 LMS3 13.8 300 31.36 327 1.36 LCS2 10.9 255 31.36 327 1.13 Note: 1 kip = 4.448 kN; 1 kipin = 0.113 kNm
374 220 144 251 178 257 
251 263 263 149 149 149 
1.49 0.84 0.55 1.68 1.19 1.72 
199.2 180 153
127.8 176 127.8
1.55 1.02 1.19
VF kips 60.2 63.0 60.4 43.3 43.3 43.3 164.8 23.4 35.1 23.4 30.9 23.4
VT/VF + Failure MT/M0 mode 1.04 0.90 0.85 0.95 0.94 0.99 0.54 1.47 1.02 1.49 1.37 1.13
F P P F P F P F/P P F/P P P
Appendix A – References Akiyama, H., and Hawkins, N.M. 1984. “Response of Flat Plate Concrete Structures to Seismic and Wind Forces”, Report SM 841, Department of Civil Engineering, University of Washington, Seattle, WA, 220 pp. Bu, W., and Polak, M.A. 2009. “Seismic Retrofit of Reinforced Concrete SlabColumn Connections Using Shear Bolts”, ACI Structural Journal, V.106, No.4 JulyAugust, pp. 514522. Durrani, A. J., Du, Y., and Luo, Y. H. 1995. “Seismic Resistance of Nonductile SlabColumn Connections in Existing FlatSlab Buildings,” ACI Structural Journal, V. 92, No. 4, pp. 479487.
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Elgabry, A., and Ghali, A. 1987. “Tests on Concrete SlabColumn Connections with StudShear Reinforcement Subjected to ShearMoment Transfer”, ACI Structural Journal, V. 84, No. 5, pp. 433442. Emam, M., Marzouk, H., Hilal, M.S. 1997. “Seismic Response of SlabColumn Connections Constructed with HighStrength Concrete,” ACI Structural Journal, V. 94, No. 2, pp. 197205. Ghali, A., Elmasri, M. Z., and Dilger, W. 1976. “Punching of Flat Plates under Static and Dynamic Horizontal Forces”, ACI Journal, V. 73, No. 10, pp. 566572. Hawkins, N.M., Bao, A., and Yamazaki, J. 1989. “Moment Transfer from Concrete Slabs to Columns,” ACI Structural Journal, V. 86, No. 6, pp. 705716. Kang, T. H., and Wallace, J.W. 2008. “Seismic Performance of Reinforced Concrete SlabColumn Connections with Thin Plate Stirrups,” ACI Structural Journal, V. 105, No. 5, pp. 617625. Marzouk, H., Emam, M., and Hilal, M. 1996. “Effect of HighStrength Concrete Columns on the Behavior of SlabColumn Connections”, ACI Structural Journal, V. 93, No. 5 pp. 545554. Marzouk, H., Emam, M., and Hilal, M.S. 1998. “Effect of High Strength Concrete Slab on the Behavior of SlabColumn Connections”, ACI Structural Journal, V. 95, No. 3, pp. 227237. Pan, A., and Moehle, J. P. 1992.“An Experimental Study of SlabColumn Connections, “ ACI Structural Journal, V. 89, No. 6, pp. 626638. Rha, C., Kang, T. HK., Shin, M., and Yoon, J. B. 2014. “Gravity and Lateral LoadCarrying Capacities of Reinforced Concrete Flat Plate Systems”, ACI Structural Journal, V. 111, No. 4, JulyAug., pp. 7532014. Robertson, I., and Durrani, A. J. 1992. “Gravity Load Effect on Seismic Behavior of Interior SlabColumn Connections,” ACI Structural Journal, V. 89, No. 1, pp. 3745. Robertson, I., and Johnson, G. 2006. “Cyclic Lateral Loading of NonDuctile SlabColumn Connections, “ACI Structural Journal, V. 103, No. 3, pp. 356364. Robertson, I., Kawai, T., Lee, J., and Enomoto, B. 2002. “Cyclic Testing of SlabColumn Connections with Shear Reinforcement,” ACI Structural Journal, V. 99, No. 5, pp. 605613. Sherif, A. G., and Dilger, W.H. 2000. “Tests of FullScale Continuous Reinforced Concrete Flat Slabs”, ACI Structural Journal, V. 97, No. 3, pp. 455467. Stark, A., Binici, B., and Bayrak, O. 2005. “Seismic Upgrade of Reinforced Concrete SlabColumn Connections Using Carbon Fiber Reinforced Polymers”, ACI Structural Journal, V. 102, No.2, pp.324333. Tian, Y., Jirsa, J.O., Bayrak, O., and Argudo, J. F. 2008. “Behavior of SlabColumn Connections of Existing FlatPlate Structures”, ACI Structural Journal, V. 105, No. 5, Sept.Oct. pp. 561569. Wey, E. H., and Durrani, A.J. 1992. “Seismic Resistance of Interior SlabColumn Connections with Shear Capitals,” ACI Structural Journal, V. 89 No. 6, pp. 682691.
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Design examples Structure 1: 28 in. square columns and 30 ft spans; d = 8.5 in; vc = 4√f’c = 253 psi Interior column: bo = (28 + 8.5)*4 = 146 in. ϕVn = ϕVc = ϕ 4√f’c bo d = 0.75*0.253*146*8.5 = 236 kips Vu = 240 – 2.6 = 237.4 kips ≈ ϕVn Slab flexural reinforcement requirements within 1.5h of column perimeter Required ρfy =(bo√f’c αs)/80d Required ρ = 146*63*40/80*8.5*60,000 = 0.0090 For uniform top bar spacing over column strip width s = (span/2)/no. of bars s = 15 *12/25 = 7.2 in. and existing ρ = 0.44/(7.2*8.5) = 0.0072 Reduce bar spacing within (c + 3h) to 5.75 in. No of bars in (c + 3h) = (28 + 3*10)/5.75 = 10.1 bars. Say 10 bars. ρ = 0.44/5.75*8.5 = 0.0090 OK Bar spacing outside (c + 3h) = (15*12  58)/(25  10) = 8.1 in. OK No extra reinforcement is needed. Only requirement is concentration of about half of required column strip reinforcement within lines 15 in. on either side of column.
Structure 2: 12 in. square columns and 24 ft spans; d = 6.5 in; vc = 4√f’c = 253 psi Interior column: bo = (12 + 6.5)*4 = 74 in. ϕVn = ϕVc = ϕ 4√f’c bo d = 0.75*0.253*74*6.5 = 91.3 kips Vu = 135 kips and therefore shear reinforcement is required Required Vs = (135 – 91.3/2)/0.75 = 119 kips For four leg #3 stirrups at 3 in. spacing on all four sides Vs = n Asfy d/s = 16*0.11*60* 6.5/3 = 229 kips OK Slab flexural reinforcement requirements within 1.5h of column perimeter Required ρfy =(bo√f’c αs)/60d Required ρ = 74*63*40/60*6.5*60,000 = 0.0080 For uniform top bar spacing over column strip width s = (span/2)/no. of bars s = 12 *12/16 = 9 in. and existing ρ = 0.44/(9*6.5) = 0.0075 Reduce bar spacing within (c + 3h) to 7.5 in. No of bars in (c + 3h) = (12 + 3*8)/7.5 = 4.8 bars. Try 5 bars. ρ =0.44/(6.5*7.5) = 0.0090 OK Bar spacing outside (c + 3h) = (12*12  36)/(16  5) = 9.8 in. < 16 in. OK No extra reinforcement is needed. Only requirement is concentration of about one third of required column strip reinforcement within lines 12 in. on either side of column.
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Review of test data for interior slabcolumn connections with moment transfer Yan Zhou, Mary Beth D. Hueste Texas A&M University, College Station, Tex., USA
Abstract The twoway shear response of slabcolumn connections has been evaluated by a significant number of experiments. These experiments provide physical tests to examine and calibrate design methods. This paper documents an updated database of slabcolumn connection tests in the literature using consistent criteria for selecting key response parameters including the limiting lateral drift capacity and gravity shear ratio. The collected test results include interior reinforced concrete (RC) and posttensioned (PT) concrete slabcolumn connections with and without shear reinforcement under combined lateral and gravity shear demands. The laboratory test data and specimen parameters are compared to current ACI design requirements and recommendations and trends are noted. As observed in previous studies, the laboratory test data indicates that the gravity shear ratio has a significant influence on the limiting lateral drift for both RC and PT slabcolumn connections without shear reinforcement. In general, the presence of shear reinforcement and prestressing improve the lateral drift capacity of slabcolumn connections. The ACI 31814 relationship to evaluate the design lateral deformation demand for slabcolumn connections is reviewed with respect to the updated data. Possible modifications to this relationship for both RC and PT slabcolumn connections are presented.
Keywords Reinforced concrete, posttensioned reinforcement, experimental database
1
concrete,
slabcolumn
connections,
shear
Introduction
Flat slabs are a common floor system for commercial and residential buildings. Past earthquake damage has shown that slabcolumn (SC) frames are not a suitable main lateralforceresisting system (LFRS) in regions of high seismic risk because of their relative flexibility and potential for brittle punching shear failures. However, SC frame systems are common LFRSs in regions of lowtomoderate seismic risk, as well as gravity systems in regions of high seismic risk where moment frames or shear walls are provided as the main LFRS. In such cases, the posttensioned (PT) and reinforced concrete (RC) SC connections must still maintain sufficient strength and ductility to resist gravity loads under the presence of inelastic deformations.
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Slabcolumn connections experience very complex behavior when subjected to lateral displacements or unbalanced gravity loads. The portion of the slab around the column must transfer a combination of shear, flexure, and torsion. Flexural and diagonal cracking of the slab are coupled with significant inplane compressive forces induced by the restraint of the surrounding unyielding portions of the slab. The first punching shear design specification in the US was introduced by the Joint Committee on Concrete and Reinforced Concrete in 1912 (Cheng, 2009); a maximum value for punching shear stress was given. In 1963, the ACI 318 Building Code first required the investigation of punching shear stress due to unbalanced moment caused by gravity, wind, or earthquake demands. The “eccentric shear stress model” used in the ACI 318 Building Code is based on the work by DiStasio and Van Buren (1960) and reviewed by ACIASCE Committee 326. Experiments have been conducted for nearly five decades to evaluate the performance of SC connections under the combined effects of gravity and lateral loading. Pan and Moehle (1989) noted that these tests indicate a relationship between the limiting lateral drift ratio (DR) and the gravity shear ratio (VR) that can be used for deformation compatibility checks. According to ACI 31814, DR is the design story drift divided by story height (Dx/hsx). VR is defined as Vug /(fVc) where Vc is the punching shear capacity of SC connections without punching shear reinforcement and without moment transfer between the slab and column, and f = 0.75 for design (ACI 31814). The factored gravity shear force, Vug, is determined using the load combination of [1.2D + 1.0L (or 0.5L) + 0.2S] as specified in ACI 31814, where D, L, and S are the dead, live, and snow loads, respectively. Note that in the design process it is not possible for VR to be equal to or greater than unity for connections without shear reinforcement, as it is required that the factored gravity shear Vu for 1.2D + 1.6L (> Vug) is always less than fVc. Additional details are provided below. ACI 31805 incorporated a simplified relationship between DR and VR, given in Eq. (1): DR = larger of [(0.035 – 0.05VR) and 0.005]
(1)
This relationship is used to evaluate the design lateral deformation demand for SC connections that are not designated as part of the seismicforceresisting system in structures assigned to Seismic Design Categories (SDCs) D, E, and F. Further, ACI 31805 requires that shear reinforcement be provided for SC connections if the design story drift exceeds DR. In such cases, the shear stress capacity provided by the shear reinforcement vs according to Eq. (7) in (psi or MPa) must satisfy vs ≥ 3.5 fc'(psi) or 0.29 fc'(MPa), where fc' is the concrete compressive strength. In addition, the shear reinforcement must extend at least 4h away from, and perpendicular to, the face of the support, drop panel, or column capital; where h is the slab thickness. These requirements are still in place in ACI 31814. Physical tests documented in the literature provide laboratory test data to examine and calibrate design methods. However, details for these experiments can sometimes be difficult to obtain, and differences in defining key response parameters may lead to some inconsistencies in the summarized test results. Design recommendations have in some cases been developed based upon a limited subset of the available laboratory test data that does not include the most recent tests. To avoid these limitations, this paper provides a review of the database of interior SC connection tests documented in the literature using consistent criteria for selecting key response parameters. The collected test results include interior RC and PT connections with and without shear reinforcement under combined lateral and gravity shear demands.
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2
Review of available research
2.1
Typical SC specimen test setup
Most experiments evaluating SC connections under gravity load with lateral displacement have tested individual SC connections or portions of a structure containing SC connections. It is critical to choose a test setup that is able to realistically model the effect of loads and deformations on the connection. Figure 1 depicts three typical methods of load application for individual PT and RC SC connections. These specimens include a column section extended above and below the slab and terminated at the assumed points of contraflexure for lateral load. The slab is typically square in plan and extended to the inflection points, which is equivalent to the midspan location of the prototype structure, in the lateral loading direction. Three methods of gravity load application have been used for these specimens (Cheng, 2009): •
application of a concentrated axial load through the column (i.e. column jacking) (Figure 1(b)),
•
application of distributed point loads to the slab around the connection (Figure 1(c)), or
•
a combination of both methods (Figure 1(a)).
In some cases an initial axial load is applied to the top of the column to induce a typical column axial stress, while VR is maintained through the application of distributed loading to the slab. In such cases, the gravity load application to the SC connection is considered to be consistent with Figure 1(c). Methods to induce unbalanced moments at the SC connection include: •
application of lateral displacements at the top of the column with the base fully fixed or restrained against translation and with slab edges roller supported (Figure 1(a)),
•
application of lateral displacements at both column ends with slab edges simply supported (Figure 1(b)), or
•
application of an upward and downward displacement couple at opposite edges of the slab with both ends of the column restrained (Figure 1(c)).
There are potential concerns with inducing gravity load using column jacking, as shown in Figure 1(b). For this case, the initial gravity shear VR in the connection region is equal to the applied column load as intended. However, challenges with this method include an unrealistic momentshear relationship in the slab. Also, additional column jacking is needed to maintain VR under lateral loading. However, unrealistic deformed shapes can develop in the connection region as compared to the intended response under combined gravity and lateral load effects (Cheng, 2009; Matzke et al., 2015).
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(a)
(b)
Δ
l/2
(c)
Δ
Lateral loads
Lateral loads
l/2
Lateral loads
h
h
Δ Δ'
h'
Gravity loads
Gravity loads Lateral loads
Gravity loads
Figure 1:
2.2
Gravity loads
Typical load simulation for individual interior SC connections.
Key parameters and modes of failure
The SC connection test data were reviewed using a consistent methodology, and key parameters were determined for each test. The actual gravity shear ratio VR is determined as Vg/ϕVc, where Vg is the reported direct shear force transferred at the critical shear perimeter, Vc is calculated in accordance with ACI 31814 Sect. 22.6.5 using the reported measured material properties and geometry, and ϕ = 1.0. The failure mode for each specimen is categorized as punching shear (P), flexure (F), or a combination of flexure and punching shear (FP) where a punching shear failure occurred at a higher drift level following yielding of the slab reinforcement. The drift ratio (DR) is defined as the ratio of column lateral displacement to the column height, or the ratio of slab displacement to slab length. For the application of unbalanced moments shown in Figure 1(c), the slab deflections on either side of the column were not equal for most tests. DR was therefore determined using Eq. (2). DR = [∆/(l/2) + ∆'/(l/2)] / 2
(2)
Figure 2 depicts the measured lateral load versus drift ratio relationships for four specimens under reversed uniaxial cyclic lateral loads. The selected specimens exhibit a wide range of behaviors for the database of SC connection tests. “DR at peak load” corresponds to the lateral drift at the maximum (peak) applied lateral load; “DR limit” corresponds to the lateral drift ratio at which failure occurs (P, FP, or F). For flexure failures (F) only, when the lateral load drops by more than 20% of the peak lateral load, the DR limit is taken as the drift at 80% of the peak lateral load, consistent with Kang et al. (2008). If the test was terminated prior to failure (P, FP, or F), the mode is noted as no failure (NF). The DR at peak load and the DR limit are not necessarily identical, as Figure 2 indicates. Figure 2 illustrates some specific examples for interpreting DR limit, as follows. • • •
144
When a P or FP failure occurred only in the positive or negative lateral load direction, the value of the DR limit corresponds to that failure point (see Figure 2(a)). When a P or FP failure occurred in both the positive and negative lateral load directions, the DR limit is taken as the average of the drift ratios corresponding to both failure points (see Figure 2(b)). For specimens that failed by flexure (F) or had no failure (NF), DR limit is the drift corresponding to 80% of the peak lateral load or the drift at which testing was terminated, whichever is smaller. The DR limit is taken as the greater drift value for both lateral
ACIfib International Symposium Punching shear of structural concrete slabs
•
load directions. Figure 2(c) shows that the ultimate lateral load is below 80% of the peak lateral load in both directions; therefore, DR limit is taken as the drift corresponding to 80% of the peak lateral load in the direction for which this parameter is largest. Figure 2(d) shows an example of a flexure failure, as well; however, the ultimate lateral load is greater than 80% of the peak lateral load in both directions. The DR limit of this specimen is taken as the larger drift ratio corresponding to the end of test.
ND1C
(a) Flexurepunching failure (FP) (Robertson and Johnson, 2004) DR at peak load: +3.00% / 3.00% DR limit: 8.00%
1C
(b)
Punching failure (P) (Robertson et al., 2002) DR at peak load: +3.50% / 3.00% DR limit: 3.25%
PS2.5
(c) Flexure failure (F) (adapted from Kang and Wallace, 2008) DR at peak load: +3.85% / 3.85% DR limit: 4.85% (80% of peak load) Figure 2:
4HS
(d) Flexure failure (F) (Robertson et al., 2002) DR at peak load: +5.00% / 5.00% DR limit: 8.00%
Examples of DR at peak load and DR limit. (1 kN = 0.2248 kips)
For specimens subjected to bidirectional lateral loads, the DR limit is reported as resultant drift ratios. In each principal vector direction, the DR limit is determined as noted above for unidirectional lateral load. To compare with tests under unidirectional lateral loads, the resultant DR limit is found as shown in Eq. (3).
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DR limit = DR2x + DR2y
(3)
where DRx and DRy are the DR limit values corresponding to the two principal lateral loading directions. The nominal twoway shear strength provided by the concrete, Vc = vc Ac, was calculated using the ACI 31814 provisions, where Ac = bo d is the area of the concrete section resisting shear transfer (in2 or mm2). For interior RC SC connections, the nominal twoway shear strength provided by the concrete vc (psi or MPa) is found using Eq. (4): 4 λ fc' (psi) vc = min
0.33 λ fc' (MPa)
4 (2+ ) λ fc' (psi) β αs d (2+ ) λ fc' (psi) bo
or
vc = min
2 β αs d 0.083(2+ ) bo
0.17(1+ ) λ fc' (MPa)
(4)
λ fc' (MPa)
where the value of fc' should not exceed 100 psi (0.69 MPa); λ is the modification factor for lightweight concrete; β is the ratio of long side to short side of the column; αs is a constant taken as 40 for interior columns, 30 for edge columns, and 20 for corner columns; d is the average distance from the extreme compression fiber to the centroid of the longitudinal tension reinforcement for both slab directions (in. or mm); and bo is the perimeter length of the critical section for twoway shear at a distance d/2 from the face of the support and in the shape of the support (in. or mm). For interior PT SC connections, vc is found using Eq. (5): vc = min
vc = min
3.5λ fc' (psi) + 0.3fpc + (1.5+
αs d )λ bo
Vp bo d
fc' (psi) + 0.3fpc +
0.29λ fc' (MPa) + 0.3fpc + 0.083(1.5+
αs d )λ bo
or
Vp bo d
(5)
Vp bo d
fc' (MPa) + 0.3fpc +
Vp bo d
where the value of fc' should not exceed 70 psi (0.48 MPa), fpc is the average compressive stress in concrete due to prestressing in both slab directions (psi or MPa) and should not be less than 125 psi (0.86 MPa) and not exceed 500 psi (3.45 MPa), Vp is the vertical component of all effective prestress forces crossing the critical section (kips or kN), and d is the average distance from extreme compression fiber to the centroid of the resultant total tension force considering both the prestressed tendons and mild flexural reinforcement for both slab directions (in. or mm) (PTI, 2006). Note that d need not be taken less than 0.8h for the calculation of Vc in prestressed members (ACI 31814; PTI, 2006). For the reported data, d is taken as 0.8h for all interior PT SC connection specimens because the calculated d values are all less than 0.8h. The term Vp is taken as zero to determine vc because the values are negligible for all the reviewed PT SC connection specimens. The vc values for slabs with fpc less than 125 psi (0.86 MPa) are determined using the expressions for nonprestressed slabs in Eq. (4).
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2.3
Interior RC SC connection specimens
Summary of laboratory test data – The reviewed database includes the available laboratory test data for interior RC SC connection specimens with and without shear reinforcement under combined gravity and lateral loads. Table 1 includes 83 interior RC SC connection specimens with no shear reinforcement. As noted in Table 1, 10 SC connections were tested under monotonic (M) lateral loading, 65 SC connections were tested under unidirectional cyclic (UC) lateral loading, and eight SC connections were subject to bidirectional cyclic (BC) lateral loading. The majority of test specimens in Table 1 are individual interior RC SC connection specimens, as depicted in Figure 1. In addition, a limited number of tests have been conducted with multiconnection specimens, multipanel specimens, and multistory frame specimens. The multiconnection specimen is a twobayonestory frame that contains two exterior and one interior connections, tested by Robertson (1990) and Du (1993). The multipanel specimen includes two types: a ninepanel, onestory frame specimen tested by Hwang (1989) and a fourpanel, onestory frame specimen tested by Rha et al. (2014). A fullscale threestory flatplate structure was tested by Fick et al. (2014). The typical slab spantothickness ratio (l1/h) was 20.034.3. However, tests conducted by Ghali et al. (1976), Cao (1993), Emam et al. (1997), and Brown (2003) had lower l1/h ratios from 12.012.7. The majority of the tests used square columns; except for several tests of SC connections with rectangular columns (Hanson and Hanson, 1968; Hwang, 1989; Farhey et al., 1993; Tan and Teng, 2005). Specimens tested by Tan and Teng (2005) had a column aspect ratio up to 5.0. Table 1:
Test data for interior slabcolumn connection specimens with no shear reinforcement
Source
ID
Hanson and Hanson (1968)
B7 C8 S1 S2 S3 S4 SM 0.5 SM 1.0 SM 1.5 1 2 3C S6 S7 S8 S1 S2 S3 S4 S5
Hawkins et al. (1974) Ghali et al. (1976) Islam and Park (1976) Symonds et al. (1976)
Morrison and Sozen (1981)
Zee and interior Moehle (1984)
Lat. load M M UC UC UC UC M M M M M UC UC UC UC UC UC UC UC UC
l 1, in. 84 84 156 156 156 156 72 72 72 120 120 120 156 156 156 72 72 72 72 72
c 1, in. 12.0 6.0 12.0 12.0 12.0 12.0 12.0 12.0 12.0 9.0 9.0 9.0 12.0 12.0 12.0 12.0 12.0 12.0 12.0 12.0
c 2, in. 6.0 12.0 12.0 12.0 12.0 12.0 12.0 12.0 12.0 9.0 9.0 9.0 12.0 12.0 12.0 12.0 12.0 12.0 12.0 12.0
h, in. 3.0 3.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 3.5 3.5 3.5 6.0 6.0 6.0 3.0 3.0 3.0 3.0 3.0
d, in. 2.25 2.25 4.50 4.63 4.75 4.50 4.75 4.75 4.75 2.75 2.75 2.75 4.50 4.63 4.75 2.36 2.44 2.28 2.28 2.36
fc', psi 4780 4760 5050 3400 3200 4690 5329 4835 5794 3960 4630 4310 3360 3840 4470 6643 5091 4917 5062 5105
V g, kips 1.1 1.3 28.8 32.0 31.2 33.7 29.0 29.0 29.0 8.0 8.0 8.0 60.2 60.8 52.4 1.3 1.3 1.3 3.2 6.4
UC
80
5.4
5.4
2.4
2.03
3800
4.3
VR
Mode
0.04 0.05 0.34 0.45 0.43 0.41 0.31 0.33 0.30 0.25 0.23 0.24 0.87 0.80 0.62 0.03 0.03 0.04 0.09 0.17
DR, % 3.80 5.80 1.84 1.07 1.23 2.05 6.50 2.70 2.00 4.44 5.00 5.19 0.83 0.47 0.64 4.83 2.84 4.29 4.29 4.65
0.29
3.94
FP
P P P P P P FP P P P P P P P F F F F FP FP
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Table 1:
Test data for interior slabcolumn connection specimens with no shear reinforcement (cont.) Source ID Lat. l 1, c 1, c 2, h, d, V g, VR DR, Mode fc', load in. in. in. in. in. kips % psi b2 BC 108 9.6 9.6 3.2 2.66 3160 7.2 0.24 4.47 P c2 BC 108 6.4 12.8 3.2 2.66 3160 7.6 0.26 4.85# P Hwang # (1989) b3 BC 108 9.6 9.6 3.2 2.66 3160 7.2 0.25 5.00 P c3 BC 108 6.4 12.8 3.2 2.66 3160 7.6 0.26 5.32 P AP1 UC 156 10.8 10.8 4.8 4.07 4250 23.3 0.37 1.54 FP Pan and AP2 BC 156 10.8 10.8 4.8 4.07 4400 23.3 0.36 1.83 FP Moehle AP3 UC 156 10.8 10.8 4.8 4.07 4600 12.0 0.18 4.76 FP (1988) AP4 BC 156 10.8 10.8 4.8 4.07 4500 12.0 0.18 3.56 FP 1 UC 114 10.0 10.0 4.5 3.60 5510 10.1 0.17 3.00 NF 2C UC 114 10.0 10.0 4.5 3.60 4790 10.0 0.18 5.00 F 3SE UC 114 10.0 10.0 4.5 3.60 6380 9.5 0.15 4.00 FP Robertson 5SO UC 114 10.0 10.0 4.5 3.60 5506 10.0 0.17 3.25# P (1990) 6LL UC 114 10.0 10.0 4.5 3.60 4670 28.0 0.52 0.85# P # 7L UC 114 10.0 10.0 4.5 3.60 4460 19.2 0.37 1.45 P 8I UC 114 10.0 10.0 4.5 3.60 5700 10.5 0.18 4.50 F CD1 UC 75 9.8 9.8 5.9 4.53 5858 67.4 0.85 1.04 P Cao (1993) CD5 UC 75 9.8 9.8 5.9 4.53 4524 45.0 0.64 1.26 P CD8 UC 75 9.8 9.8 5.9 4.57 3915 33.7 0.51 1.48 P DNY_1* UC 114 10.0 10.0 4.5 3.81 5115 12.2 0.20 4.50 F DNY_2* UC 114 10.0 10.0 4.5 3.81 3731 15.3 0.30 2.00 P Du (1993) * DNY_3 UC 114 10.0 10.0 4.5 3.81 3566 12.2 0.24 4.50 F DNY_4* UC 114 10.0 10.0 4.5 3.81 2772 12.2 0.28 4.70 FP 1 UC 106 11.8 7.9 3.1 2.35 5088 0.0 0.00 5.60 FP 2 UC 106 11.8 7.9 3.1 2.35 5088 0.0 0.00 5.10 FP Farhey et al. (1993) 3 UC 106 11.8 7.9 3.1 2.35 2181 5.6 0.26 3.80 FP 4 UC 106 11.8 4.7 3.1 2.35 2181 5.6 0.30 2.50 FP Luo et al. II* UC 114 10.0 10.0 4.5 3.81 3000 3.47 0.08 5.00 F (1994) HHHC05 UC 75 9.8 9.8 5.9 4.69 10985 28.1 0.26 5.20 F 75 9.8 9.8 5.9 4.69 10486 28.1 0.26 5.20 FP Emam et al. HHHC10 UC (1997) NHHC05 UC 75 9.8 9.8 5.9 4.69 5330 28.1 0.35 3.80 FP NHHC10 UC 75 9.8 9.8 5.9 4.69 5184 28.1 0.36 3.80 P Ali and SPA UC 165 14.0 14.0 6.0 4.17 4771 26.3 0.31 4.50 FP Alexander SPB UC 165 14.0 14.0 6.0 4.17 4945 26.3 0.31 3.50 P (2002) Robertson et 1C UC 120 10.0 10.0 4.5 3.74 5128 8.7 0.15 3.25# P al. (2002) SJB6 M 73 9.8 9.8 5.9 4.49 5192 33.7 0.45 2.30 P Brown (2003) SJB7 UC 73 9.8 9.8 5.9 4.49 4177 33.7 0.51 1.70 P YLL1 UC 177 35.4 7.1 5.9 4.80 5800 0.17 7.10 F Tan and YLH2 BC 177 35.4 7.1 5.9 4.80 5800 0.28 2.39# P Teng (2005) YLL2 BC 177 35.4 7.1 5.9 4.80 5800 0.17 2.81# P ND1C* UC 120 10.0 10.0 4.5 3.94 4293 12.4 0.22 8.00 FP * UC 120 10.0 10.0 4.5 3.94 4685 16.0 0.27 3.50# FP Robertson ND4LL * and Johnson ND5XL UC 120 10.0 10.0 4.5 3.94 3495 22.7 0.44 2.00 P (2006) ND6HR* UC 120 10.0 10.0 4.5 3.94 3814 14.7 0.27 4.00# P * ND7LR UC 120 10.0 10.0 4.5 3.94 2727 11.0 0.24 5.00 FP
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Table 1: Source
Choi et al. (2007) Kang and Wallace (2008) Tian et al. (2008) Park et al. (2012) Song et al. (2012) Kang et al. (2013)
Test data for interior slabcolumn connection specimens with no shear reinforcement (cont.) ID Lat. l 1, c 1, c 2, h, d, V g, VR DR, Mode fc', load in. in. in. in. in. kips % psi ND8BU* UC 120 10.0 10.0 4.5 3.94 5685 16.0 0.24 5.00 FP S1 UC 94 11.8 11.8 4.7 3.54 4858 18.3 0.30 3.00 P S2 UC 94 11.8 11.8 4.7 3.54 5989 33.8 0.50 3.00 P S3 UC 94 11.8 11.8 4.7 3.54 5481 19.4 0.30 3.00 P C0
UC
120
10.0
10.0
6.0
5.12
5597
28.1
0.30
1.85
P
L0.5* LG0.5* LG1.0* RCA RCB
UC UC UC UC UC
168 168 168 118 118
16.0 16.0 16.0 11.8 11.8
16.0 16.0 16.0 11.8 11.8
6.0 6.0 6.0 5.3 5.3
5.00 5.00 5.00 4.47 4.47
3710 4820 4000 3263 5613
23.5 26.8 24.1 29.7 35.7
0.23 0.23 0.23 0.45 0.41
2.00 1.25 1.25 1.50 1.60
P NF NF FP FP
RC1
UC
118
11.8
11.8
5.3
4.47
5612

0.43
1.80
P
RC
UC
118
11.8
11.8
5.3
4.72
5046
35.3
0.40 2.60#
FP
Floor 1 UC 240 18.0 18.0 7.0 6.00 3920 0.21 2.39 NF Floor 2 UC 240 18.0 18.0 7.0 6.00 4030 0.21 3.31 P Floor 3 UC 240 18.0 18.0 7.0 6.00 4070 0.21 3.08 NF LMS2C5 M 108 9.8 9.8 3.5 2.76 3770 13.8 0.40 5.38 FP Rha et al. LMS3C5 M 108 9.8 9.8 3.5 2.76 3770 13.8 0.40 0.74 FP (2014) LCS2C5 UC 108 9.8 9.8 3.5 2.76 3770 13.8 0.40 1.50 P l1 is the slab dimension in the direction of loading, c1 is the column dimension in the direction of loading, c2 is the column dimension transverse to the loading direction, h is the slab thickness. * Bottom slab reinforcement is discontinuous at interior connection. # When P or FP occurs in both the positive and negative lateral load directions, the average drift is given. F = flexural failure, P = punching shear failure, FP = flexural and punching shear failure, NF = no failure. 1 in. = 25.4 mm, 1 kip = 4.448 kN, 1 ksi = 1000 psi = 6.895 MPa Fick et al. (2014)
Table 2 summarizes 66 interior RC SC connection specimens with shear reinforcement. As noted in Table 2, two SC connections were tested under monotonic (M) lateral loading, 56 SC connections were tested under unidirectional cyclic (UC) lateral loading, and eight SC connections were subject to bidirectional cyclic (BC) lateral loading. Typical shear reinforcement includes shear stud reinforcement (SSR), closed and single stirrups (CS and SS), bentup bars (BUP), and shearheads (SH). More recently, several new types of shear reinforcement have been tested including “ductility reinforcement” (DUR), shearbands (SB), steel fiber reinforcement (SFR), and lattice reinforcement (LTR). The majority of test specimens in Table 2 are individual interior RC SC connection specimens, as depicted in Figure 1. In addition, three multiconnection specimens were tested including one by Robertson (1990) and two by Dechka (2001). The typical l1/h ratio used in tests was also 20.034.3. However, tests conducted by Cao (1993), Brown (2003), Gayed and Ghali (2006), and Broms (2007) had lower l1/h ratios from 12.315.6. All but one test program used square columns; the specimens tested by Tan and Teng (2005) had a column aspect ratio of 5.0.
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Table 2:
Test data for interior RC slabcolumn connection specimens with shear reinforcement
Source
ID
Hawkins et al. (1975)
Islam and Park (1976) Symonds et al. (1976) Robertson (1990) Dechka (2001)
SS1 SS2 SS3 SS4 SS5 4S 5S 6CS 7CS 8CS SS6 SS7
Lat. load UC UC UC UC UC M M UC UC UC UC UC
l 1, in. 156 156 156 156 156 120 120 120 120 120 156 156
4S
UC
114 10.0 10.0 4.5 3.60 6360 9.8
UC UC UC UC UC UC UC UC UC UC UC UC UC UC UC UC BC UC
S1 S2 CD3 CD4 Cao (1993) CD6 CD7 2CS Robertson et 3SL al. (2002) 4HS SJB1 SJB2 SJB3 Brown SJB4 (2003) SJB5 SJB8 SJB9 Tan and YLH2V Teng (2005) YLH1V Gayed and Ghali ISP0 (2006) 17c 17d Broms (2007) 18c 18d Kang and PS2.5 Wallace PS3.5 (2008) HS2.5
150
c 1, in. 12.0 12.0 12.0 12.0 12.0 9.0 9.0 9.0 9.0 9.0 12.0 12.0
c 2, in. 12.0 12.0 12.0 12.0 12.0 9.0 9.0 9.0 9.0 9.0 12.0 12.0
h, in. 6.0 6.0 6.0 6.0 6.0 3.5 3.5 3.5 3.5 3.5 6.0 6.0
d, in. 4.50 4.63 4.50 4.50 4.63 2.75 2.75 2.75 2.75 2.75 4.63 4.50
fc', psi 4000 3730 3750 4000 4670 4630 4430 4090 4310 3210 3510 3900
V g, kips 29.9 28.4 28.5 28.7 28.3 8.0 8.0 8.0 8.0 8.0 59.7 60.4
VR 0.40 0.38 0.39 0.38 0.34 0.23 0.23 0.24 0.24 0.27 0.82 0.81
DR, Mode Shear % reinf. 3.89 F CS 4.44 P CS 5.76 F CS 5.63 F CS 5.00 FP CS 5.37 P BUP 4.63+ F SH 4.44+ F CS 4.07+ F CS 5.56+ F CS 1.59 FP CS 2.25 FP CS
0.16 7.00
F
CS
197 197 75 75 75 75 120 120 120 73 73 73 73 73 73 73 177 177
9.8 9.8 5.9 4.65 6119 39.0 0.46 4.40 9.8 9.8 5.9 4.65 4756 35.1 0.47 6.40 9.8 9.8 5.9 4.53 5162 67.4 0.90 3.41 9.8 9.8 5.9 4.53 4974 45.0 0.61 4.81 9.8 9.8 5.9 4.33 4553 45.0 0.68 4.89 9.8 9.8 5.9 4.53 4147 33.7 0.50 5.19 10.0 10.0 4.5 3.74 4552 7.6 0.14 8.00 10.0 10.0 4.5 3.74 6296 5.6 0.09 8.00 10.0 10.0 4.5 3.74 5543 8.0 0.13 8.00 9.8 9.8 5.9 4.49 4670 33.7 0.48 4.50# 9.8 9.8 5.9 4.49 4975 33.7 0.46 4.90 9.8 9.8 5.9 4.49 4699 33.7 0.48 4.90 9.8 9.8 5.9 4.49 5758 33.7 0.43 6.40 9.8 9.8 5.9 4.49 4844 33.7 0.47 7.60 9.8 9.8 5.9 4.49 5076 33.7 0.46 5.70 9.8 9.8 5.9 4.49 4525 33.7 0.49 7.10 35.4 7.1 5.9 4.80 5800 0.28 5.60# 35.4 7.1 5.9 4.80 5800 0.28 8.14
P FP FP F F F F F F FP FP FP FP FP FP FP FP F
SSR CS SSR SSR SSR SSR CS SS SSR SSR SSR SSR SSR SSR SSR SSR SSR SSR
UC
75
10.0 10.0 6.0 4.49 3820
0.84 3.76
FP
SSR
UC UC UC UC UC UC UC
110 110 110 110 120 120 120
11.8 11.8 11.8 11.8 10.0 10.0 10.0
0.65 0.65 0.67 0.67 0.32 0.34 0.31
FP FP FP FP F FP P
DUR DUR SSR SSR SB SB SSR
11.8 11.8 11.8 11.8 10.0 10.0 10.0
7.1 7.1 7.1 7.1 6.0 6.0 6.0
5.91 5.91 5.91 5.91 5.12 5.12 5.12
5340 5340 5020 5020 5090 5090 5090
80.0 80.0 80.0 80.0 28.1 29.9 27.0
3.00 3.00 3.00 3.00 4.85 3.45 5.20
ACIfib International Symposium Punching shear of structural concrete slabs
Table 2:
Test data for interior RC slabcolumn connection specimens with shear reinforcement (cont.)
Source
ID
Lat. l1, c 1, c2, h, d, fc', Vg, VR DR, Mode Shear load in. in. in. in. in. % reinf. psi kips SU1 UC 108 12.0 12.0 4.0 3.25 8490 30.8 0.42 5.00 F SFR SU2 UC 108 12.0 12.0 4.0 3.25 6940 21.6 0.33 5.00 F SFR Cheng SB1 BC 204 16.0 16.0 6.0 4.75 5360 46.4 0.40 3.25 P SFR (2009) SB2 BC 204 16.0 16.0 6.0 4.75 4470 43.7 0.41 2.95 P SFR SB3 BC 204 16.0 16.0 6.0 4.75 6450 55.0 0.43 1.46 P SSR LRA1 UC 118 11.8 11.8 5.3 4.50 3263 29.7 0.44 7.00 F LTR LRA2 UC 118 11.8 11.8 5.3 4.50 3263 29.7 0.44 4.90 F LTR SRA UC 118 11.8 11.8 5.3 4.47 3263 29.7 0.45 4.00 F SSR SBA UC 118 11.8 11.8 5.3 4.47 3263 29.7 0.45 5.10 F SB UC 118 11.8 11.8 5.3 4.47 3263 29.7 0.45 3.00 F CS Park et al. STA (2012) LRB1 UC 118 11.8 11.8 5.3 4.50 5613 35.7 0.41 4.70 F LTR LRB2 UC 118 11.8 11.8 5.3 4.50 5613 35.7 0.41 3.60 F LTR SRB UC 118 11.8 11.8 5.3 4.47 5613 35.7 0.41 5.10 F SSR SBB UC 118 11.8 11.8 5.3 4.47 5613 35.7 0.41 6.50 F SB STB UC 118 11.8 11.8 5.3 4.47 5613 35.7 0.41 3.20 F CS SR1 UC 118 11.8 11.8 5.3 4.47 5612 0.43 3.90 F CS Song et al. SR2 UC 118 11.8 11.8 5.3 4.47 5612 0.43 5.40 F SSR (2012) SR3 UC 118 11.8 11.8 5.3 4.47 5612 0.43 6.00 F SB LRA UC 118 11.8 11.8 5.3 4.72 3915 31.3 0.40 5.10 F LTR LRB UC 118 11.8 11.8 5.3 4.72 4843 34.4 0.40 3.20# FP LTR Kang et LRC UC 118 11.8 11.8 5.3 4.72 5133 35.8 0.40 4.80 F LTR al. (2013) LRD UC 118 11.8 11.8 5.3 4.72 5916 38.1 0.40 5.10 F LTR LRE UC 118 11.8 11.8 5.3 4.72 5032 35.3 0.40 3.60# FP LTR B1 BC 204 16.0 16.0 6.0 4.75 5900 54.9 0.45 2.62 FP SSR B2 BC 204 16.0 16.0 6.0 4.75 4900 50.1 0.45 2.62 FP SSR Matzke et. al. (2015) B3 BC 204 16.0 16.0 6.0 4.75 5700 56.2 0.47 2.95 FP SSR B4 BC 204 16.0 16.0 6.0 4.75 6100 50.7 0.41 3.25 FP SSR l1 is the slab dimension in the direction of loading, c1 is the column dimension in the direction of loading, c2 is the column dimension transverse to the loading direction, h is the slab thickness. * Bottom slab reinforcement is discontinuous at interior connection. # When P or FP occurs in both the positive and negative lateral load directions, the average drift is given. + DR is based on the slab displacement at peak lateral load as the author reported. F = flexural failure, P = punching shear failure, FP = flexural and punching shear failure. 1 in. = 25.4 mm, 1 kip = 4.448 kN, 1 ksi = 1000 psi = 6.895 MPa
Relationships between limiting drift and gravity shear ratio – The relationships between the DR limit and VR for interior RC SC connections with and without shear reinforcement are shown in Figure 3. Both charts include the drift limit relationship provided in ACI 31814 (see Eq. (1)). As shown in Figure 3(a), the test data indicates that VR has a significant influence on the DR limit for RC SC connections without shear reinforcement. Most test specimens without shear reinforcement failed by punching. All FP failure points had a VR below 0.5, and most F and NF points had a VR less than 0.3. Figure 3(b) shows the DR limit versus VR for specimens with different types of shear reinforcement. No tests have been conducted for shear reinforced RC SC connections with VR ranging from 0.50.6 and 0.70.8. Shear stud reinforcement (shear stud rails) are common in twoway floor slabs in the US due to their ease in placement during
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ACIfib International Symposium Punching shear of structural concrete slabs
construction. The largest number of tests have been conducted for shear stud reinforcement, particularly in the range of VR = 0.40.5. There are a significant number of tests with steel stirrups, a traditional option that has become less popular for current construction as placement is more labor intensive. Less conventional shear reinforcement types have been tested more recently including shearbands, steel fiber reinforcement, ductility reinforcement, and lattice reinforcement. The various methods show comparable limiting drifts for the VR range tested. However, there is significant scatter in the data, even for a particular reinforcement type. This variability may be due to a number of additional variables related to testing protocol and specimen parameters not captured in the chart shown in Figure 3(b). 10
10
Punching FlexurePunching Flexure No failure ACI 31814 limit
9 8
8
Drift Ratio Limit (%)
7
Drift Ratio Limit (%)
Shear studs Steel stirrups Bent up bars Shearheads Shearbands Steel fiber reinf. Ductility reinf. Lattice reinf. ACI 31814 limit
9
6 5 4 3
7
6 5 4 3
2
2
1
1
0
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Gravity Shear Ratio (Vg /ϕVc)
(a) Connections with no shear reinforcement Figure 3:
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Gravity Shear Ratio (Vg /ϕVc)
(b) Connections with shear reinforcement
Test data for interior RC SC connections.
Evaluation of ACI 318 limiting drift relationship for RC specimens – Specimens with data points below the ACI 31814 limit were reviewed to determine whether certain unfavorable conditions were present that may explain these lower drift values. Figure 4 shows only the data points below the ACI 31814 limit, and they are identified by the corresponding research study. •
•
Tian et al. (2008) tested six specimens without continuous bottom flexural reinforcement at the column to examine the lateral drift capacity of existing flat plate structures. Three of the six specimens were subject to combined vertical and lateral loads, whereas LG0.5 and LG1.0 were only loaded vertically, without lateral loading, after the drift ratio reached 1.25%. Neither specimen failed under combined vertical and lateral loads; however, they eventually failed when loaded under gravity load only. The first floor interior connections of the threestory flatplate structure tested by Fick et al. (2014) had no failure at the maximum applied drift. Each floor had two interior connections, and punching failure occurred at one connection on the second floor. The reported data for the first floor does not correspond to a failure. The DR limit of interior RC SC connection LMS3C5 is the lowest among the three specimens tested by Rha et al. (2014). This specimen had a high concentration of bottom integrity bars with six bars spaced at 1.5 in. (38 mm) within the column width and a higher bottom reinforcement ratio (0.98%) within c2+3h width than the other two specimens (0.39%) tested making the connection less flexible, and it exhibited less ductility.
•
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10
Hawkins et al. (1974) Morrison & Sozen (1981) Pan & Moehle (1988) Robertson (1990) Tian et al. (2008) Kang & Wallace (2008) Fick et al. (2014) Rha et al. (2014) ACI 31814 limit
9
Drift Ratio Limit (%)
8 7 6 5 4
3 2 1 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Gravity Shear Ratio (Vg /ϕVc)
Figure 4:
Data for interior RC SC connections without shear reinforcement below the ACI lateral drift limit.
10
Cao (1993) Dechka (2001) Brown (2003) Song et al. (2012) Park et al. (2012) Cheng (2009) Matzke et al. (2015) ACI 31814 limit
9
Drift Ratio Limit (%)
8 7 6 5 4
Bidirectional Lateral Load
3 2 1 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Gravity Shear Ratio (Vg /ϕVc)
Figure 5:
Data for interior RC SC connections with SSR (VR = 0.40.5).
Specimens with the noted unfavorable conditions are shown with red symbols in Figure 4, while those that did not fail under the combination of lateral and gravity loads, the drift and VR reported are shown in green. There are six remaining symbols, shown in black, of the 78 total data points that are only slightly below the ACI drift limit (less than 10 percent). This may be acceptable given that for design, the VR values will include a reduction factor (f = 0.75 for shear) that will increase the value of VR, such that the allowable drift limits for checking the design story drift are reduced. However, a more conservative option, shown with a dashed line targeting 3% drift at VR = 0, could be considered. 153
ACIfib International Symposium Punching shear of structural concrete slabs
Variability of drift limits for RC specimens with shear reinforcement – Figure 3(b) shows significant scatter in the DR limit for interior RC SC connection specimens for a given VR. In particular, there are a large number of tests with SSR and VR = 0.40.5. These laboratory tests were reviewed to identify possible reasons for the observed variability. Table 3 and Figure 5 summarize additional parameters for interior RC SC connections with SSR and VR = 0.40.5. All tests are unidirectional unless noted otherwise. As indicated in Table 3, five specimens were tested with bidirectional cyclic lateral loading, leading to the lowest drift limits (below 3.5%) for the data shown in Figure 5. Four of the 13 specimens tested with unidirectional cyclic lateral loading exhibited limiting drifts greater than 6%; among them are three specimens tested by Brown (2003). Table 3:
Summary of RC SC connections with SSR within the VR range of 0.40.5
0.50 0.46 0.47 0.48 0.46 0.48 0.43 0.47 0.46
DR, % 5.19 4.40 6.40 4.50 4.90 4.90 6.40 7.60 5.70
Extension, in. 25.6 16.7 16.9 11.3 11.3 18.9 18.9 18.9 18.9
v s, psi 365 482 656 400 400 400 400 400 400
SJB9
0.49
7.10
18.9
400
SR2
0.43
5.40
13.5
478

SRA SRB
0.45 0.41
4.00 5.10
13.5 13.5
443 443
SRB had fc' (5613 psi = 38.7 MPa) and SRA had fc' (3263 psi = 22.5 MPa).
SB3
0.43
1.46
21.5
151
Bidirectional lateral loading.
Source
ID
VR
Cao (1993) Dechka (2001)
CD7 S1 S2 SJB1 SJB2 SJB3 SJB4 SJB5 SJB8
Brown (2003)
Song et al. (2012) Park et al. (2012) Cheng (2009)
Note Lightly concentrated top flexural reinf. Lightly concentrated top flexural reinf. Heavily concentrated top flexural reinf. Diagonal column orientation. Heavily concentrated top flexural reinf. with diagonal column orientation.
B1 0.45 2.62 22.0 276 B2 0.45 2.62 28.5 187 Bidirectional lateral loading. B3 0.47 2.95 27.7 402 B4 0.41 3.25 26.2 670 Extension is the extended length of the slab shear reinforcement from the column face. 1 in. = 25.4 mm, 1 kip = 4.448 kN, 1 ksi = 1000 psi = 6.895 MPa. Matzke et. al. (2015)
Comparisons between different studies are not always ideal because specimen dimensions; flexural reinforcement ratio; slab spantothickness ratio; type, amount, and geometry of shear reinforcement; and test setup are all factors that may affect the DR limit. The following observations are based on identical specimens with one parameter varied. • •
Dechka (2001) tested two multiconnection specimens and the specimen with a larger vs (S2) exhibited a higher drift capacity. Brown (2003) conducted tests to examine the influence of top flexural reinforcement concentration and column orientation as well as shear reinforcement extension. The results showed increased drift capacity as the extended length of the SSR increases and as the concentration of top flexural reinforcement increases. Brown also found that the column orientation did not have a significant effect on the strength or ductility of the connection.
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•
In addition to VR, Park et al. (2012) found that fc' could influence drift capacity. Specimen SRB with a higher fc' but similar VR, as compared to Specimen SRA, exhibited a larger DR limit. Five bidirectional lateral loading tests were conducted in the same laboratory and had similar test setups (Cheng, 2009; Matzke et al., 2015). The results indicated that whether increasing the extension of SSR, raising the vs value, or increasing both, the DR limit increases. Specimen SB3 had the lowest drift capacity among these five bidirectional tests. It also had the least vs and the shortest extended length of the SSR among the five specimens.
•
2.4
Interior PT SC connection specimens
Summary of laboratory test data – The reviewed database includes the available laboratory test data for interior PT SC connections with and without shear reinforcement under combined gravity and lateral loads. Table 4 includes 18 interior PT SC connections with no shear reinforcement. Among these connections, four were repeated lateral (RL) loading in one direction, 10 were unidirectional cyclic (UC) lateral loading, and four were bidirectional cyclic (BC) lateral loading. Table 5 summarizes eight interior PT SC connection specimens with shear reinforcement. Among these connections, six were individual connections, as depicted in Figure 1, with UC lateral loading. Kang (2004) tested a onethird scale, twostory, twobay by twobay PT SC frame on a shake table with reversed cyclic loading in one direction. Table 4: Source Trongtham and Hawkins (1977) Qaisrani (1993) Han et al. (2006)
Han et al. (2009)
Test data for interior PT slabcolumn connection specimens with no shear reinforcement ID 1
Lat. l1, c1, c2, h, d, fc', fpc1, fpc2, Vg, VR DR, Mode load in. in. in. in. in. psi psi psi kips % DB RL 156 14.0 14.0 5.5 4.4 3850 160 160 66.8 0.78 1.31 FP
3
BD
RL 156 14.0 14.0 5.5 4.4 3650 275 160 67.8 0.76 3.66
FP
4
DD
RL 156 14.0 14.0 5.5 4.4 3800 160 160 69.9 0.82 2.74
FP
5
DD
RL 156 14.0 14.0 5.5 4.4 3600 160 160 25.2 0.30 5.01
F
I1
DB* BC 147 7.9 DB* BC 147 7.9 DB* BC 147 7.9
I2 I3
7.9 3.5 2.8 4075 230 230 25.5 0.73 1.80
P
7.9 3.5 2.8 4075 230 230 23.5 0.67 2.11
P
7.9 3.5 2.8 4010 230 230 19.5 0.56 2.26
P FP
PIB30
BD
PID30
DB
UC 181 11.8 11.8 5.2 4.2 4684 175 175 18.4 0.24 5.90 UC 181 11.8 11.8 5.2 4.2 4684 175 175 18.4 0.24 5.40
PIB70
BD
UC 181 11.8 11.8 5.2 4.2 4684 175 175 41.2 0.53 2.75
FP
PIB70X BD* UC 181 11.8 11.8 5.2 4.2 4684 175 175 41.2 0.53 0.75
P
PIB50
BD
FP
UC 181 11.8 11.8 5.2 4.2 4684 175 175 29.7 0.38 3.35
FP
PIB50X BD* UC 181 11.8 11.8 5.2 4.2 4684 175 175 29.7 0.38 1.75
P
PID50 Prawatwong et al. (2012) Himawan and Teng (2014)
TL
DB
UC 181 11.8 11.8 5.2 4.2 4684 175 175 29.7 0.38 4.00
FP
PID50X DB* UC 181 11.8 11.8 5.2 4.2 4684 175 175 29.7 0.38 2.75
P
S1
BD
UC 224 19.7 9.8 4.7 3.8 5960 249 249 26.5 0.30 2.00
P
PI1
BD
UC 138 35.4 7.1 5.9 4.7 5235 247 132 36.9 0.25 2.50
P
PI2
BD
BC 138 35.4 7.1 5.9 4.7 4930 235 138 38.4 0.26 2.13
P
See Table 5 for notation and unit conversions.
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Table 5:
Test data for interior PT slabcolumn connection specimens with SSR
Source
ID
TL
Lat. l1, c1, c2, h, d, fc', fpc1, fpc2, Vg, VR DR, Mode load in. in. in. in. in. psi psi psi kips % Floor 1 BD UC 112 8.0 8.0 3.0 2.4 401 202 199 8.6 0.31 3.77 FP Kang (2004) 2 202 199 8.6 0.31 4.25 F Floor 2 BD UC 112 8.0 8.0 3.0 2.4 401 2 IPS3 BD UC 75 10.0 10.0 5.9 4.7 388 60 60 54.0 0.78 5.61 P 0 IPS5 BD UC 75 10.0 10.0 5.9 4.7 415 90 90 54.0 0.75 5.96 P Gayed and 0 IPS5R DB UC 75 10.0 10.0 5.9 4.7 412 90 90 54.0 0.76 5.06 P Ghali 0 130 130 54.0 0.71 5.45 P IPS7 BD UC 75 10.0 10.0 5.9 4.7 446 (2006) 0 160 160 54.0 0.77 6.02 P IPS9 BD UC 75 10.0 10.0 5.9 4.7 340 0 160 160 54.0 0.74 4.60 P IPS9R DB UC 75 10.0 10.0 5.9 4.7 373 0 TL denotes tendon layout. l1 is the slab dimension in the direction of loading, c1 is the column dimension in the direction of loading, c2 is the column dimension transverse to the loading direction, h is the slab thickness *Bottom slab reinforcement is discontinuous at interior connection. F = flexural failure, P = punching shear failure, FP = flexural and punching shear failure. 1 in. = 25.4 mm, 1 kip = 4.448 kN, 1 ksi = 1000 psi = 6.895 MPa
The posttensioning in the various test specimens consisted of one of three different tendon arrangements: banded in the lateral loading direction and distributed in the transverse direction (BD), distributed in the loading direction and banded in the transverse direction (DB), or distributed in both directions (DD). In recent years, the majority of PT flat slab construction uses banded tendons in one direction and distributed tendons in the second direction; as opposed to distributed tendons in both directions found in older construction (Kang et al., 2008). The compressive stress in the concrete due to the effective posttensioning force for the full specimen width (fpc) varied from 60275 psi (0.411.90 MPa) in the lateral loading direction (fpc1) and 60249 psi (0.411.72 MPa) in the transverse direction (fpc2), as noted in Tables 4 and 5. The ACI limits for fpc are not less than 125 psi (0.86 MPa) and not more than 500 psi (3.45 MPa). It is noted that the majority of the PT SC specimens had fpc values within this range. Three specimens tested by Gayed and Ghali (2006) had values below the minimum limit. All but one specimen had tendons draped in a parabolic profile and running through the column cages. Prawatwong et al. (2012) tested connections with a straight tendon profile. The specimens varied in terms of the amount of top and bottom conventional mild reinforcement. The typical slab spantothickness ratio (l1/h) was 23.347.6, and square columns were used for most specimens. The PT SC connections had higher l1/h ratios than the RC SC connection specimens, as is typical in building construction. Tests conducted by Gayed and Ghali (2006) had a lower l1/h ratio of 12.7. The column aspect ratio of connections tested by Prawatwong et al. (2012) and Himawan and Teng (2014) were 2.0 and 5.0, respectively. Relationships between limiting drift and gravity shear ratio – The relationships between the DR limit and VR for interior PT SC connections with and without shear reinforcement are shown in Figure 6. Both charts include the drift limit relationship provided in ACI 31814 (see Eq. (1)) for SC connections not designated as part of the seismicforceresisting system in structures assigned to SDCs D, E, and F, discussed earlier.
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10 9 8
Kang (2004) Gayed & Ghali (2006) ACI 31814 limit
9 8
7
Drift Ratio Limit (%)
Drift Ratio Limit (%)
10
Punching FlexurePunching Flexure ACI 31814 limit
6 5 4
3
7 6 5 4 3
2
2
1
1
0
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
Gravity Shear Ratio (Vg /ϕVc)
(a) Connections with no shear reinforcement Figure 6:
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Gravity Shear Ratio (Vg /ϕVc)
(b) Connections with shear reinforcement (all SSR)
Test data for interior PT SC connections.
As shown in Figure 6(a), the test data indicate that VR also has a significant influence on the DR limit for PT SC connections without shear reinforcement. Most PT SC connection specimens without shear reinforcement failed by punching or a combination of flexure and punching, and one connections had a flexural failure. All PT SC connections had a VR above 0.2, and all but one specimen had a VR less than 0.8. Figure 6(b) shows the DR limit versus VR for specimens with shear reinforcement. Only two studies have documented PT SC connection tests with shear reinforcement and both used SSR (Kang, 2004; Gayed and Ghali, 2006). The tests conducted by Gayed and Ghali (2006) exhibited higher DR limits with significantly larger VR values than the specimens tested by Kang (2004). Note that no tests are documented in the literature for shear reinforced PT SC connections with VR ranging from 0.350.70. Figure 7 compares DR limit versus VR for interior RC and PT SC connections without shear reinforcement. PT connections tend to exhibit a higher DR limit for a given VR, as noted by Kang and Wallace (2006) and Kang et al. (2008). However, there are significantly fewer data points for PT connections. The dashed line segments indicate a potential lateral drift limit for design that provides a lower bound to most of the available PT data. Although this limit is lower than that suggested by Kang et al. (2008) and ACI 352.1R11 (0.5% lower for a given VR), it may provide an appropriate level of conservatism for PT SC connections not designated as part of the seismicforceresisting system in structures located in SDCs D, E, and F.
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10
RC without shear reinf. PT without shear reinf. ACI 31814 limit Suggested lateral drift limit for PT
9
Drift Ratio Limit (%)
8 7 6 5 4 3 2 1 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Gravity Shear Ratio (Vg /ϕVc)
Figure 7:
Test data for interior RC and PT SC connections without shear reinforcement.
Figure 8 provides additional details for PT SC connection specimens, and each study is indicated by a unique symbol in both graphs. Figure 8(a) shows the level of the average compressive stress in the concrete due to prestressing in both slab directions fpc, which should not be less than 125 psi (0.86 MPa) and not exceed 500 psi (3.45 MPa) when calculating vc for PT SC connections according to ACI 31814. It is noted that most PT SC connection specimens have fpc values above the minimum of 125 psi (0.86 MPa), however there are no tests for SC connections with fpc values between 275500 psi (1.903.45 MPa). The relationship between the DR limit and VR for interior PT SC connections with and without shear reinforcement is shown in Figure 8(b).
700
Trongtham & Hawkins (1980) Kang (2004) Han et al. (2006) Han et al. (2009)
Trongtham & Hawkins (1980) Qaisrani (1993) Kang (2004)  SSR Gayed & Ghali (2006)  SSR Han et al. (2006) Han et al. (2009) Prawatwong et al. (2012) Himawan & Teng (2014)
9 8
500
Drift Ratio Limit (%)
fpc2 (the other direction, psi)
600
10
Qaisrani (1993) Gayed & Ghali (2006) Prawatwong et al. (2012) Himawan & Teng (2014)
400
300
200
Suggested lateral drift limit for PT
7 6
5
SSR
4 3 2
100
ACI 31814 limit
1
0
0
0
100
200
300
400
500
600
700
0
fpc1 (the testing direction, psi)
(a) fpc values compared to ACI limits Figure 8:
158
Additional details for PT SC connection tests.
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Gravity Shear Ratio (Vg /ϕVc)
(b) DR limit versus VR
0.8
0.9
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ACIfib International Symposium Punching shear of structural concrete slabs
PT SC connection tests conducted more recently have resulted in some lower DR limits, including five points below the suggested limiting lateral drift relationship. Han et al. (2009) tested six specimens to examine the influence of bottom mild reinforcement and tendon layout. Specimen PIB50X and PIB70X exhibited lower DR limits than the other four specimens. No continuous bottom mild reinforcement through the interior column was a factor that led to the decreased drift capacity of the two connections. The tendon layout could be another factor (DB corresponded to a higher drift capacity versus BD), as indicated in Table 4, for two specimens that also had discontinuous bottom reinforcement. Prawatwong et al. (2012) tested two interior rectangular PT SC connections to examine the effectiveness of drop panels. Specimen S1 was the control specimen without a drop panel. Most of the interior PT SC specimens in the database had a square column and used continuous tendons through the column in both slab directions. This specimen, however, had an aspect ratio of 2.0 with no tendon through the column in either direction. Both factors may have contributed to reducing the DR limit for this specimen. Himawan and Teng (2014) tested two interior PT SC connections with high column aspect ratio of 5.0, and concluded that the rectangular column section may have reduced the effectiveness of the prestressing in enhancing the twoway shear strength. The ACI 31814 estimation of Vc for prestressed twoway members does not consider the effect of column aspect ratio, and may overestimate Vc for interior PT SC connections with a high aspect ratio. To include the effect of the column aspect ratio, Himawan and Teng (2014) proposed Eq. (6) for estimating the twoway shear strength of interior PT SC connections, 4 β
vc (psi) = (1.5 + ) λ fc' (psi) + 0.3fpc + 4 β
Vp b od
or
vc (MPa) = 0.083(1.5 + ) λ fc' (MPa) + 0.3fpc +
Vp
(6)
b od
where the parameters are the same as defined for Eqs. (4) and (5). Using Eq. (6) for vc, the VR values for the two specimens tested by Himawan and Teng (2014) are approximately 0.35. Further, because bidirectional lateral loading lowers the ductility of SC connections (Pan and Moehle, 1989), the DR limit of specimen PI2 is lower than PI1 as expected.
2.5
Review of test data with shear reinforcement
Figure 9 provides a comparison of the data for interior RC and PT SC connections with and without shear reinforcement. The data shows the beneficial effect of shear reinforcement in providing an overall increase, on average, in the DR limit for a given VR for both RC and PT SC connections. The seismic provisions for ACI 31814 require that minimum shear reinforcement be provided for SC connections when the design story drift exceeds the limiting DR, or some other modification must be made to the design to satisfy the drift limit. When minimum shear reinforcement is used, the shear reinforcement must meet a minimum strength requirement and extend at least 4h away from, and perpendicular to, the face of the support, drop panel, or column capital; where h is the slab thickness. However, there is significant variation in the shear strength and extension length of the shear reinforcement used in test specimens.
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10 9
PT without shear reinf. PT with shear reinf. ACI 31814 limit
9
8
8
7
7
Drift Ratio Limit (%)
Drift Ratio Limit (%)
10
RC without shear reinf. RC with shear reinf. ACI 31814 limit
6 5 4 3
6 5 4
3
2
2
1
1 0
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Gravity Shear Ratio (Vg /ϕVc)
(a) Interior RC SC connections Figure 9:
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Gravity Shear Ratio (Vg /ϕVc)
(b) Interior PT SC connections
Comparison of test data with and without shear reinforcement.
It is noted that many SC connection tests were conducted prior to the inclusion of the minimum shear reinforcement requirements in ACI 318, and some tests intentionally varied shear reinforcement parameters to study the effects. However, it is useful to review the specimen shear reinforcement as compared to the current minimum requirements in the ACI 31814 seismic provisions. The graphs in Figure 10 show the extended length of the shear reinforcement lext (in. or mm) normalized to the minimum length of 4h. In addition, the shear strength provided by shear reinforcement vs (psi or MPa) is normalized to the minimum required shear strength provided by the shear reinforcement, vs,min = 3.5 fc'(psi) or 0.29 fc'(MPa). The value of vs is determined using the following relationship from ACI 31814, vs=
Av fyt b o s
(7)
where s is centertocenter spacing of shear reinforcement perpendicular to the column face (in. or mm), Av is the area of shear reinforcement within spacing s (in2 or mm2), fyt is the specified yield strength of shear reinforcement (psi or MPa), and bo is the perimeter of the critical section for twoway shear (in. or mm). As shown in Figure 10(a), only a few RC specimens have used just the minimum shear strength and extended length of shear reinforcement. Most specimens have more than the minimum shear strength (45) with the majority (32) providing less than the minimum extended length. Seven interior RC SC connections with shear reinforcement failed by punching. Three of these specimens used shear stud reinforcement, two specimens used steel fiber reinforcement, one specimen used bentup bars, and one specimen used steel stirrups (see Table 2). Many parameters can affect the potential for brittle punching shear failures in SC connections with shear reinforcement. Along with the strength and extended length of the shear reinforcement, spacing and layout of the shear reinforcement are also important in the design. In addition to the requirements in ACI 31814, additional guidance is provided in ACI 352.1R11 (ACIASCE Comm. 352, 2011).
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Figure 10(b) shows the interior PT SC connection specimen parameters with respect to the ACI 31814 minimum shear reinforcement requirements in the seismic provisions. All but one specimen eventually failed by punching (see Table 5). Tests of shear reinforced interior PT SC connections are much fewer than for interior RC SC connections with shear reinforcement. The amount of shear reinforcement in tests of PT SC connections to date exceeds the minimum requirements, and the extension of the shear reinforcement is less than prescribed by ACI 318. The tests conducted by Gayed and Ghali (2006) gave larger DR limits with higher VR values than those by Kang (2004). Gayed and Ghali (2006) provided SSR with a longer normalized extended length and more shear strength than used in the tests by Kang (2004). Also, the testing protocol, slab span, and slab thickness were significantly different.
2
vs /vs,min
vs = vs,min
1
3
Hawkins et al. (1975) Islam & Park (1976) Symonds et al. (1976) Elgabry & Ghali (1987) Robertson (1990) Cao (1993) Dechka (2001) Robertson et al. (2002) Brown (2003) Tan & Teng (2005) Gayed & Ghali (2006) Broms (2007) Kang & Wallace (2008) Cheng (2009) Song et al. (2012) Park et al. (2012) Kang et al. (2013) Matzke et al. (2015)
Kang (2004) Gayed & Ghali (2006)
2
vs /vs,min
3
extended length of shear reinforcement = 4h
1 vs = vs,min
extended length of shear reinforcement = 4h
0
0 0
1
2
lext /4h
(a) RC SC connections
3
0
1
2
3
lext /4h
(b) PT SC connections
Figure 10: Shear reinforcement parameters for RC and PT SC connection specimens.
3
Conclusions
Based on a detailed review of available tests for interior RC and PT SC connections with and without shear reinforcement under combined lateral and gravity shear demands, key findings are as follows. •
The updated laboratory test data set for interior RC and PT SC connections without shear reinforcement confirms that the limiting lateral drift tends to decrease as the gravity shear ratio increases.
•
The presence of prestressing tends to enhance the limiting lateral drift for interior PT SC connections. However, it is noted that there are significantly fewer data points for PT SC connections as compared to RC SC connections.
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•
The presence of shear reinforcement tends to increase the limiting lateral drift for interior RC and PT SC connections subject to varying levels of gravity shear. However, detailing of the shear reinforcement is important. Along with the provided strength and extended length of the shear reinforcement, the spacing and layout are critical parameters. In addition to the requirements in ACI 31814, guidance is provided in ACI 352.1R11.
•
The limiting lateral drift is highly variable among the RC SC connection tests with shear reinforcement. This variability can be due to differences in specimen dimensions; flexural reinforcement ratio; slab spantothickness ratio; type, amount, and geometry of shear reinforcement; and test setup. Additional testing parameters that may have contributed to differences in the limiting drift for RC SC connections with SSR and VR = 0.40.5 include bidirectional lateral loading, the amount of added shear strength, the extended length of the SSR, the concentration of top flexural reinforcement, and the magnitude of fc'.
•
The ACI 31814 limiting relationship between the design drift demand and the gravity shear ratio; which is used to evaluate the design lateral deformation demand for SC connections not designated as part of the seismicforceresisting system in structures assigned to SDCs D, E, and F; was reviewed with respect to the updated data. For RC SC connection specimens without shear reinforcement, the relationship is nearly a lower bound; however, a slightly lower limit is provided as a possible more conservative option for design. In addition, the ACI 31814 relationship may be increased slightly for PT SC connections, and still provide an approximate lower bound to the PT test data that may be appropriate for design.
4
Acknowledgments
The authors wish to acknowledge Professor Neil Hawkins for his pioneering and significant contributions to the laboratory testing of slabcolumn connections, including some of the first tests under combined gravity and lateral loading that are contained in this database. The authors also wish to acknowledge members of ACIASCE Committee 352 and ACI Subcommittee 318J who continue to motivate the need to review laboratory test data for SC connections in the development of design recommendations and requirements. Both committees have provided valuable input for the proposed limiting drift relationship for PT SC connections. The suggested limit is based on recent discussions by ACI Subcommittee 318J. Finally, the authors wish to acknowledge the support from the China Scholarship Council and the Zachry Department of Civil Engineering at Texas A&M University.
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5
References
ACI Committee 318 (1963) “Building Code Requirements for Reinforced Concrete (ACI 31863),” American Concrete Institute (ACI), Detroit, MI. ACI Committee 318 (2005) “Building Code Requirements for Structural Concrete (ACI 31805) and Commentary,” American Concrete Institute (ACI), Farmington Hills, MI. ACI Committee 318 (2014) “Building Code Requirements for Structural Concrete (ACI 31814) and Commentary,” American Concrete Institute (ACI), Farmington Hills, MI. ACIASCE Committee 326 (1962) “Report of ACIASCE Committee 326: Shear and Diagonal Tension: Part 3,” Journal of the American Concrete Institute (ACI), Detroit, MI. ACIASCE Committee 352 (2011) “Guide for Design of Slabcolumn Connections in Monolithic Concrete Structures (ACI 352.1R11),” American Concrete Institute (ACI), Farmington Hills, MI. Ali, M. A. and Alexander, S. D. (2002) “Behaviour of SlabColumn Connections with Partially Debonded Reinforcement Under Cyclic Lateral Loading,” Structural Engineering Report No. 243, Department of Civil Engineering, University of Alberta. Brown, S. J. (2003) “Seismic Response of Slab Column Connections,” Ph.D. dissertation, Univ. of Calgary, Alberta, 341 pp. Broms, C. E. (2007) “Flat Plates in Seismic Areas: Comparison of Shear Reinforcement Systems,” ACI Structural Journal, 104(6), 712721. Cao, H. J. (1993) “Seismic Design of SlabColumn Connections,” Masters dissertation, Univ. of Calgary, Alberta, 201 pp. Cheng, M. Y. (2009) “Punching Shear Strength and Deformation Capacity of Fiber Reinforced Concrete SlabColumn Connections Under EarthquakeType Loading,” Ph.D. dissertation, Univ. of Michigan, Ann Arbor, 347 pp. Choi, M. S.; Cho, I. J.; Han, B. S.; Ahn J. M.; and Shin, S. W. (2007) “Experimental Study of Interior SlabColumn Connections Subjected to Vertical Shear and Cyclic Moment Transfer,” Key Engineering Materials, 348, 641644. Dechka, D. C. (2001) “Response of ShearStudReinforced Continuous SlabColumn Frames to Seismic Loads,” Ph.D. dissertation, Univ. of Calgary, Canada, 472 pp. Di Stasio, J. and Van Buren, M. P. (1960) “Transfer of Bending Moment between Flat Plate Floor and Column,” ACI Journal Proceedings, 57(9), 299314. Du, Y. (1993) “Seismic Resistance of SlabColumn Connections in NonDuctile FlatPlate Buildings,” Masters dissertation, Rice University, Houston, 166 pp. Emam, M.; Marzouk, H.; and Hilal, M. S. (1997) “Seismic Response of SlabColumn Connections Constructed with HighStrength Concrete,” ACI Structural Journal, 94, 197205. Farhey, D. N.; Adin, M. A.; and Yankelevsky, D. Z. (1993) “RC Flat SlabColumn Subassemblages under Lateral Loading,” ASCE Journal of Structural Engineering, 119(6), 19031916.
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Fick, D. R.; Sozen, M. A.; and Kreger, M. E. (2014) “Cyclic Lateral Load Test and the Estimation of Elastic Drift Response of a FullScale ThreeStory FlatPlate Structure,” ACI Special Publication, 296, 114. Gayed, R. B. and Ghali, A. (2006) “Seismicresistant Joints of Interior Columns with Prestressed Slabs,” ACI Structural Journal, 103(5), 710719. Ghali, A.; Elmasri, M. Z.; and Dilger, W. (1976) “Punching of Flat Plates under Static and Dynamic Horizontal Forces,” ACI Journal Proceedings, 73(10), 566572. Hanson, N. W. and Hanson, J. M. (1968) “Shear and Moment Transfer Between Concrete Slabs and Columns,” Journal, PCA Research and Development Laboratories, 10(1), 216. Han, S. W.; Kee, S.; Kang, T. H.; Ha, S.; Wallace, J. W.; and Lee, L. (2006) “Cyclic Behaviour of Interior PostTensioned Flat Plate Connections,” Magazine of Concrete Research, 58(10), 699712. Han, S. W.; Kee, S. H.; Park, Y. M.; Ha, S. S.; and Wallace, J. W. (2009) “Effects of Bottom Reinforcement on Hysteretic Behavior of Posttensioned Flat Plate Connections,” ASCE Journal of Structural Engineering, 135(9), 10191033. Hawkins, N. M.; Mitchell, D.; and Sheu, M. S. (1974) “Cyclic Behavior of Six Reinforced Concrete SlabColumn Specimens Transferring Moment And Shear,” Progress Report 197374, Division of Structures and Mechanics, Department of Civil Engineering, Univ. of Washington. Hawkins, N. M.; Mitchell, D.; and Hanna, S. N. (1975) “The Effects of Shear Reinforcement on the Reversed Cyclic Loading Behavior of Flat Plate Structures,” Canadian Journal of Civil Engineering, 2(4), 572582. Himawan, A. and Teng, S. (2014) “Cyclic Behavior of PostTensioned SlabRectangular Column Connections,” ACI Structural Journal, 111(1), 177187. Hwang, S. (1989) “An Experimental Study of FlatPlate Structures under Vertical and Lateral Loads,” Ph.D. dissertation, Univ. of California, Berkeley, 278 pp. Islam, S. and Park, R. (1976) “Tests on SlabColumn Connections with Shear and Unbalanced Flexure,” Journal of the Structural Division, 102(3), 549568. Kang, T. H. K. (2004) “Shake Table Tests and Analytical Studies of Reinforced and PostTensioned Concrete Flat Plate Frames,” Ph.D. dissertation, Univ. of California, Los Angeles, 309 pp. Kang, T. H. and Wallace, J. W. (2006) “Punching of Reinforced and PostTensioned Concrete SlabColumn Connections,” ACI Structural Journal, 103(4), 531540. Kang, T. H. K.; Robertson, I. N.; Hawkins, N. M.; and LaFave, J. M. (2008) “Recommendations for Design of PostTensioned SlabColumn Connections Subjected to Lateral Loading,” Journal of the PostTensioning Institute, 6(1), 4559. Kang, T. H. and Wallace, J. W. (2008) “Seismic Performance of Reinforced Concrete SlabColumn Connections with Thin Plate Stirrups,” ACI Structural Journal, 105(5), 617625. Kang, S. M.; Park, H. G.; and Kim, Y. N. (2013) “LatticeReinforced SlabColumn Connections under Cyclic Lateral Loading,” ACI Structural Journal, 110(6), 929939.
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Luo, Y. H.; Durrani, A. J.; and Conte, J. P. (1994) “Equivalent Frame Analysis of Flat Plate Buildings for Seismic Loading,” ASCE Journal of Structural Engineering, 120(7), 21372155. Matzke, E. M.; Lequesne, R. D.; ParraMontesinos, G. J.; and Shield, C. K. (2015) “Behavior of Biaxially Loaded SlabColumn Connections with Shear Studs,” ACI Structural Journal, 112(3), 335346. Morrison, D. G. and Sozen, M. A. (1981) “Response of Reinforced Concrete PlateColumn Connections to Dynamic and Static Horizontal Loads, Volume 1,” Technical Report to NSF, Univ. of Illinois at UrbanaChampaign. Pan, A. and Moehle, J. P. (1988) “Reinforced Concrete Flat Plates under Lateral Loading: An Experimental Study Including Biaxial Effects,” Report to the National Science Foundation, Univ. of California, Berkeley. Pan, A. and Moehle, J. P. (1989) “Lateral Displacement Ductility of Reinforced Concrete Flat Plates,” ACI Structural Journal, 86(3), 250258. Park, H. G.; Kim, Y. N.; Song, J. G.; and Song, S. M. (2012) “Lattice Shear Reinforcement for Enhancement of SlabColumn Connections,” ASCE Journal of Structural Engineering, 138(3), 425437. PostTensioning Institute (PTI). (2006) PTI TAB.106: PostTensioning Manual  6th Edition, Phoenix, AZ, 354 pp. Prawatwong, U.; Warnitchai, P.; and Tandian, C. (2012) “Seismic Performance of Bonded PostTensioned SlabColumn Connections with and without Drop Panels,” Advances in Structural Engineering, 15(10), 16531672. Qaisrani, A. N. (1993) “Interior PostTensioned FlatPlate Connections Subjected to Vertical and Biaxial Lateral Loading,” Ph.D. dissertation, Univ. of California, Berkeley. Rha, C.; Kang, T. H.; Shin, M.; and Yoon, J. B. (2014) “Gravity and Lateral LoadCarrying Capacities of Reinforced Concrete Flat Plate Systems,” ACI Structural Journal, 111(4), 753764. Robertson, I.N. (1990) “Seismic Response of Connections in Indeterminate FlatSlab Subassemblies,” Ph.D. dissertation, Rice University, Houston, 278 pp. Robertson, I.N.; Kawai, T.; Lee, J.; and Enomoto, B. (2002) “Cyclic Testing of SlabColumn Connections with Shear Reinforcement,” ACI Structural Journal, 99(5), 605613. Robertson, I.N. and Johnson, G. P. (2004) “NonDuctile SlabColumn Connections Subjected to Cyclic Lateral Loading,” Proceedings of 13WCEE, Vancouver, Canada. Robertson, I. and Johnson G. (2006) “Cyclic Lateral Loading of Nonductile SlabColumn Connections,” ACI Structural Journal, 103(3), 356364. Song, J. K.; Kim, J.; Kim, H. B.; and Song, J. W. (2012) “Effective Punching Shear and Moment Capacity of Flat PlateColumn Connection with Shear Reinforcements for Lateral Loading,” International Journal of Concrete Structures and Materials, 6(1), 1929.
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Symonds, D. W.; Hawkins, N. M.; and Mitchell, D. (1976) “SlabColumn Connections Subjected to High Intensity Shears and Transferring Reversed Moments,” Technical Report SM 762, Department of Civil Engineering, Univ. of Washington. Tan, Y. and Teng, S. (2005) “Interior SlabRectangular Column Connections under Biaxial Lateral Loadings,” ACI Special Publication, 232, 147174. Tian, Y.; Jirsa, J. O.; Bayrak, O.; and Argudo, J. F. (2008) “Behavior of SlabColumn Connections of Existing FlatPlate Structures,” ACI Structural Journal, 105(5), 561569. Trongtham, N. and Hawkins, N. M. (1977) “Moment Transfer to Columns in Unbonded PostTensioned Prestressed Concrete Slabs,” Department of Civil Engineering, Univ. of Washington. Zee, H. L. and Moehle, J. P. (1984) “Behavior of Interior and Exterior Flat Plate Connections Subjected to Inelastic Load Reversals,” Technical Report to NSF, Univ. of California, Earthquake Engineering Research Center.
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Maximum punching shear capacity of footings with new punching shear reinforcement elements Dominik Kueres1, Marcus Ricker2, Josef Hegger1 1
: RWTH Aachen University, Germany
2
: Biberach University of Applied Sciences, Germany
Abstract Due to more compact dimensions and soilstructure interaction, footings and ground slabs achieve significantly higher punching shear capacities than flat plates. Since the inclination of shear cracks is much steeper in footings, punching shear reinforcement elements with inclined bars seem to be more efficient than vertical bars. On the basis of previous experimental investigations on footings with punching shear reinforcement, a new punching shear reinforcement element with inclined bars was developed. In a first experimental campaign, seven tests on reinforced concrete footings with the new punching shear reinforcement elements and a failure inside the shearreinforced zone were performed. Based on the results of the first test series, a second experimental campaign was conducted to investigate the maximum punching shear capacity with the new punching shear reinforcement. The main parameters investigated in this test series were the shear spandepth ratio, the ratio of column perimeter and effective depth, the concrete compressive strength, and the layout of punching shear reinforcement. The comparison of the failure loads of the specimens with the punching shear capacity without punching shear reinforcement according to ACI 31814 shows a high efficiency of the new punching shear reinforcement, especially when compared to vertically arranged reinforcement elements like stirrups and studs. The results of the second test series are presented in this paper.
Keywords Reinforced concrete footings, punching shear, shear spandepth ratio, specific column perimeter, punching shear reinforcement, maximum punching shear capacity
1
Introduction
The punching shear behavior of reinforced concrete footings without punching shear reinforcement has been investigated extensively by various researchers in the past (Talbot, 1913; Richart, 1948; Dieterle and Steinle, 1981; Dieterle and Rostásy, 1987; Hallgren et al., 1998; Hegger et al., 2006; Hegger et al., 2007b; Hegger et al., 2009; Siburg and Hegger, 2014; Kueres et al., 2015). According to these investigations, punching shear resistance mainly depends on the flexural reinforcement ratio, concrete compressive strength, and footing dimensions (e.g. effective depth, shear spandepth ratio, size effects), which is in line 167
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with previous investigations on flat plates. However, due to more compact dimensions and soilstructure interaction, footings and ground slabs achieve significantly higher punching shear capacities than flat plates (Hegger et al., 2006; Hegger et al., 2007b; Hegger et al., 2009; Siburg and Hegger, 2014; Kueres et al., 2015). Fewer experimental investigations have been conducted on reinforced concrete footings with punching shear reinforcement (Hegger et al., 2009; Siburg and Hegger, 2014). The test results indicate, that vertical punching shear reinforcement elements like stirrups (Beutel and Hegger, 2002; Fernandez Ruiz and Muttoni, 2009; Hegger et al., 2007a) and studs (Andrä, 1981; Mokthar et al., 1985; Ricker and Häusler, 2014; Ferreira et al., 2014) are less efficient in footings than in flat plates. Due to the steeper inclination of shear cracks in footings, a higher efficiency of inclined punching shear reinforcement elements (e.g. inclined shearband reinforcement (Pilakoutas and Li, 2003), lattice girders (Park et al., 2007), bentup bars (Einpaul et al., 2016)) can be assumed. A punching test on a reinforced concrete footing with bentdown bars seems to confirm this assumption (Dieterle and Rostásy, 1987). Based on the results of previous test series (Dieterle and Rostásy, 1987; Hegger et al., 2006; Hegger et al., 2007b; Hegger et al., 2009; Siburg and Hegger, 2014), a new punching shear reinforcement system with inclined bars was developed. In a first test series (Kueres et al., 2016a), the punching shear behavior of footings with the new punching shear reinforcement element and a failure inside the shearreinforced zone was investigated. Based on the results of the first test series, the maximum punching shear capacity of footings with the new punching shear reinforcement element is investigated in the present study. A series of seven punching tests on reinforced concrete footings was conducted. All test specimens were provided with the new punching shear reinforcement. The main parameters investigated in this test series were the shear spandepth ratio, the ratio of column perimeter and effective depth, the concrete compressive strength, and the layout of punching shear reinforcement.
2
New punching shear reinforcement element
The new punching shear reinforcement elements (Fig. 1 (a)) have an optimized form, which allows the shear crack widths to be efficiently controlled. Due to the inclination of the different sections, the Sshaped elements cross the shear cracks several times (Fig. 1 (b)). The rigid anchorage of the Sshaped elements by means of forged heads, as well as the effective upper anchorage, consisting of a clamped steel sheet, also contributes to the increased failure loads of the test specimens. Another advantage of the upper anchorage element is that no expensive welding is necessary. The mandrel diameter chosen for the new punching shear reinforcement is smaller than allowable according to ACI 31814 due to space reasons. It was shown by photomicrographs of the hooked bars that the reduced mandrel diameter does not lead to an increased number of microcracks. The fullscale tests of the previous test series (Kueres et al., 2016a) confirmed the assumption that the increased concrete pressure due to the small mandrel diameter does not cause premature concrete failure adjacent to the looped anchorage elements.
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Figure 1:
3
New punching shear reinforcement element.
Experimental campaign
The experimental campaign included seven tests on reinforced concrete footings. The tests were planned taking into account the results of a previous test series (Kueres et al., 2016a) on reinforced concrete footings with the new punching shear reinforcement elements (Fig. 1) and a failure inside the shearreinforced zone. While in the previous test series new punching shear reinforcement elements with diameters of either 10 mm (0.4 in.) or 12 mm (0.5 in.) were installed, in the present test series the diameter was increased to 14 mm (0.6 in.) to investigate the maximum punching shear capacity. The test parameters included the shear spandepth ratio aλ/d, the ratio of column perimeter and effective depth u0/d (specific column perimeter), the concrete compressive strength, and the layout of punching shear reinforcement.
3.1
Materials
For all test specimens, commercial ready mixed concrete was used. The concrete mixture was designed to produce a 28day target cylinder strength of fc,cyl = 25 MPa (3481 psi) and fc,cyl = 55 MPa (7977 psi), respectively. For the lower concrete strength, ordinary CEM II 42.5 R Portland cement and a watercementratio (w/c) of 0.65 to 0.73 was used, resulting in a slump of approximately 480 mm (18.8 in.). For the concrete with higher strength, ordinary CEM I 52.5 R Portland cement and a watercementratio (w/c) of 0.40 to 0.43 was used, resulting in a slump of approximately 470 mm (18.5 in.). The maximum coarse aggregate size was 16 mm (0.6 in.) for both mixtures. To prevent premature failure, ultrahigh performance concrete (UHPC) with concrete compressive strengths between fc,cyl = 111.3 MPa (16143 psi) and 122.2 MPa (17724 psi) was used for the column stubs. Additionally, the column stubs were strengthened with a steel collar made of 10 mm (0.4 in.) steel plates. For all test specimens, the flexural reinforcement consisted of highgrade steel St 900/1100 with yield strengths varying from fy = 996 MPa (145 ksi) to 1034 MPa (150 ksi), a tensile strength of approximately ft = 1177 MPa (170.7 ksi), and a Young’s modulus of approximately Es = 194300 MPa (28181 ksi). The highgrade steel was used to prevent a premature flexural failure. The new punching shear reinforcement elements were produced of
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steel B 500B, with measured yield strengths varying from fy = 553 MPa (80.2 ksi) to 569 MPa (82.5 ksi), tensile strengths in a range of ft = 631 MPa (91.5 ksi) and 646 MPa (93.7 ksi), and a Young’s modulus between Es = 199100 MPa (28877 ksi) and 199800 MPa (28979 ksi). Table 1 summarizes the properties of the materials used. Table 1: Test
d
c
b
u0/d aλ/d
fc,cyl
Æ
fy
ρl
Æw
Lay ψtest Vtest out mm mm mm mm  MPa mm MPa % mm MPa  mrad kN (in.) (in.) (in.) (in.) (psi) (in.) (ksi) (in.) (ksi) (kips) DF_N8 450 397 300 1300 3.02 1.26 23.0 20 996 0.79 14 553 I 2.2 7743 (17.7) (15.6) (11.8) (51.2) (3336) (0.8) (145) (0.6) (80.2) (1741) DF_N0N 450 396 300 1900 3.03 2.02 22.4 20 1034 0.83 14 569 II 7.2 6380 (17.7) (15.6) (11.8) (74.8) (3249) (0.8) (150) (0.6) (82.5) (1434) DF_N9 450 397 300 2700 3.02 3.02 25.1 20 996 0.81 14 553 III 13.3 5629 (17.7) (15.6) (11.8) (106) (3641) (0.8) (145) (0.6) (80.2) (1266) DF_N11 450 397 300 1900 3.02 2.02 53.8 20 996 0.83 14 553 II 10.3 9831 (17.7) (15.6) (11.8) (74.8) (7803) (0.8) (145) (0.6) (80.2) (2210) DF_N12 450 397 300 2700 3.02 3.02 51.3 20 996 0.81 14 553 III 19.0 7850 (17.7) (15.6) (11.8) (106) (7440) (0.8) (145) (0.6) (80.2) (1765) DF_N13 450 395 400 2000 4.05 2.03 22.8 20 996 0.86 14 553 II 7.6 7416 (17.7) (15.6) (15.7) (78.7) (3307) (0.8) (145) (0.6) (80.2) (1667) DF_N14 450 396 600 2200 6.06 2.02 21.7 20 996 0.86 14 553 II 7.9 8692 (17.7) (15.6) (23.6) (86.6) (3147) (0.8) (145) (0.6) (80.2) (1954) h: slab thickness; d: effective depth; c: square column dimension; b: square footing dimension; u0/d: specific column perimeter; aλ/d: shear spandepth ratio; fc,cyl: concrete compressive strength; Æ: diameter of longitudinal reinforcement; fy: yield strength of longitudinal reinforcement; ρl: longitudinal reinforcement ratio; Æw: diameter of shear reinforcement; fyw: yield strength of shear reinforcement; layout: layout of punching shear reinforcement; ψtest: rotation of specimen at failure; Vtest: ultimate failure load.
3.2
h
Details of test specimens and failure loads fyw
Test specimens
The test series consisted of seven reinforced concrete footings with side dimensions of 1300, 1900, 2000, 2200, and 2700 mm (51.2, 74.8, 78.7, 86.6, and 106.3 in.) in square and a slab thickness of 450 mm (17.7 in.). The square column stubs had side dimensions of 300, 400, and 600 mm (11.8, 15.7, and 23.6 in.) and were cast monolithically at the center of the footing. The distance between the outer compression fiber and the centroid of the tension reinforcement (effective depth) was approximately d = 400 mm (15.7 in.), resulting in shear spandepth ratios between aλ/d = 1.25 and 3.00. The specific column perimeter was in the range of u0/d = 3.00 and 6.00. The flexural reinforcement ratio varied between ρl = 0.79% and 0.86% and the diameter of the inclined bars of the new punching shear reinforcement elements was 14 mm (0.6 in.) for all specimens. The different layouts of punching shear reinforcement investigated are shown exemplarily in Fig. 2 for test specimens DF_N0N (Layout II) and DF_N9 (Layout III). Layout II consisted of eight punching shear reinforcement elements in the first row and eight elements in the second row. In the slender specimens (DF_N9 and DF_N12, aλ/d = 3.00), a third row, consisting of eight punching shear reinforcement elements, was installed. Specimen DF_N8 (Layout I) was tested with a shear spandepth ratio aλ/d = 1.25. Hence, only one row of punching shear reinforcement could be installed due to space limitations.
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Figure 2:
3.3
Layout of punching shear reinforcement for test specimens DF_N0N and DF_N9.
Test setup and measurements
The test specimens were loaded by a uniform surface load using the test setup shown in Fig. 3. The footings were tested upside down. A uniform pressure was simulated by means of 25 hydraulic jacks. All the hydraulic jacks were linked to a common manifold and applied the same load, independent of the displacement. In order to avoid any formation of membrane forces in the test specimens, polytetrafluoroethylene (PTFE)coated sliding and deformation bearings of dimensions 140 x 140 mm (5.5 x 5.5 in.) were placed between the footing and the hydraulic jacks. During testing, the vertical displacement of the test specimens was recorded at the corners of the column stub and at the footing’s corners using linear variable differential transformers (LVDTs). To investigate the development of the inner shear cracks, the increase in the slab thickness was measured at several points and the penetration of the column into the slab was monitored. Strain gages were used to measure the strains in the flexural reinforcement at six locations and at several points in the punching shear reinforcement elements. To obtain the average strain at the bar’s center of gravity, two strain gages were attached to opposite side faces of the reinforcing bars at each measuring point. The concrete strains were recorded at four locations on the compression face of the footing near the column.
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Figure 3:
3.4
Test setup.
Test procedure
The load was applied under load control in increments of 400 kN (89.9 kips). To simulate lifetime loading, the load was cycled ten times between a calculated service load Vservice and half of its value. The service load was defined as the predicted failure load of an identical footing without punching shear reinforcement according to DIN EN 199211:201101 and DIN EN 199211/NA:201304. After the load cycles, the test specimens were continuously loaded until failure took place.
4
Experimental results
4.1
Failure characteristics
All tests failed in punching of the footing. The failure loads Vtest are listed in Table 1. Before the failure occurred, increasing slab thickness, increasing strains in the punching shear reinforcement and penetration of the column stub into the slab were observed.
4.2
Cracking characteristics
After testing, the footings were sawn in half to examine the inner crack patterns (Fig. 4). All crack patterns showed finely distributed shear cracks crossing the punching shear reinforcement elements at several locations. Strain measurements confirmed that the punching shear reinforcement was activated. The more compact footings (aλ/d ≤ 2.00) with ratios of column perimeter and effective depth u0/d ≤ 4.00 (DF_N8, DF_N0N, DF_N11, DF_N13; Fig. 4 (a, b, d, f)) showed a steep failure crack, which developed between the column face and the lower bend of the punching shear reinforcement elements in the first row. In specimen DF_N14 (aλ/d = 2.00, u0/d = 6.00) a slightly flatter failure crack formed between the column face and the second row of punching shear reinforcement (Fig. 4 (g)). Because the number of reinforcement elements in the first row was kept constant for all specimens, the tangential distance between the reinforcement elements was increased for this specimen due to the large column. Since the yield strength of the punching shear reinforcement was not reached when failure occurred, failure of the shear reinforcement can be excluded and the punching failure 172
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developed between the reinforcement elements in the first row. In contrast, in the slender specimens DF_N9 and DF_N12 (aλ/d = 3.00) the failure crack developed from the third row of punching shear reinforcement, which indicates a failure outside the shearreinforced zone (Fig. 4 (c, e)). As observed in previous test series conducted on footings with punching shear reinforcement (Hegger et al., 2009; Siburg and Hegger, 2014; Kueres et al., 2016a), the inclination of the shear cracks seems not to be influenced by the shear spandepth ratio if a failure outside the shearreinforced zone can be excluded.
Figure 4:
4.3
Saw cuts of test specimens.
Loaddeflection characteristics
The previous test series (Kueres et al., 2016a) on footings with the new punching shear reinforcement elements proved a significant increase in punching shear capacity compared to identical specimens without punching shear reinforcement and with stirrups as punching shear reinforcement. Since the specimens failed inside the shearreinforced zone (Kueres et al., 2016a), higher punching shear capacities can be assumed by increasing the amount of punching shear reinforcement. In this context, Fig. 5 (a) shows the comparison of the measured loaddeflection curves for specimens DF_N8 (aλ/d = 1.25, u0/d = 3.00) and DF_N0N (aλ/d = 2.00, u0/d = 3.00) with new punching shear reinforcement elements consisting of 14 mm (0.6 in.) bars, as well as for two identical footings DF_N7 (aλ/d = 1.25, u0/d = 3.00) and DF_N4 (aλ/d = 2.00, u0/d = 3.00) with new punching shear reinforcement elements made of 10 mm (0.4 in.) bars (Kueres et al., 2016a). The concrete compressive strength of the compared specimens was in the same range and varied between fc,cyl = 20.4 MPa (2959 psi) and 23.0 MPa (3336 psi). While specimens DF_N7 and DF_N4 173
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failed inside the shearreinforced zone (Kueres et al., 2016a), specimens DF_N8 and DF_N0N were designed to reach the maximum punching shear capacity. After a stiff initial response, first bending cracks appeared, leading to reduced stiffness for the compared footings. The measured loaddeflection curves of specimens DF_N8 and DF_N0N (Æw = 14 mm (0.6 in.)) are slightly steeper compared to specimens DF_N7 and DF_N4 (Æw = 10 mm (0.4 in.)). Specimen DF_N7 failed in punching shear at 6573 kN (1478 kips) and specimen DF_N8 failed in punching shear at 7743 kN (1741 kips) resulting in a load increase of 18%. Specimen DF_N4 failed in punching shear at 5045 kN (1134 kips) and specimen DF_N0N failed in punching shear at 6380 kN (1434 kips), resulting in a load increase of 26%. The comparison of the failure loads indicates a disproportionate increase in punching shear capacity with increasing amounts of punching shear reinforcement (Av,Ø14 ≈ 2·Av,Ø10), which is in line with previous investigations on the punching shear behavior of flat plates with different types and amounts of punching shear reinforcement (Beutel, 2003; Siburg, 2014).
Figure 5:
Measured loaddeflection curves.
The measured loaddeflection curves of the footings with the new punching shear reinforcement elements consisting of 14 mm (0.6 in.) bars and a concrete compressive strength of approximately fc,cyl ≈ 23 MPa (3336 psi) are shown in Fig. 5 (b). Regardless of the specific column perimeter and the layout of the punching shear reinforcement, a stiff initial response corresponding to the uncracked stage was observed. At approximately 800 kN 174
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(180 kips), first bending cracks appeared, leading to a reduced stiffness, but still to a nearly linear trend of the loaddeflection curves. The gradient of the loaddeflection curves of the footings with a shear spandepth ratio of aλ/d = 2.00 (DF_N0N, DF_N13, DF_N14) is comparable. In contrast, the gradient of specimen DF_N8 with a shear spandepth ratio aλ/d = 1.25 is much steeper and the gradient of specimen DF_N9 with a shear spandepth ratio aλ/d = 3.00 is much flatter. This was already observed in previous test series on footings with variable shear spandepth ratio (Hegger et al., 2009; Kueres et al., 2016a). Fig. 5 (c) shows the comparison of the measured loaddeflection curves for specimens DF_N0N (aλ/d = 2.00, u0/d = 3.00) and DF_N9 (aλ/d = 3.00, u0/d = 3.00) with a concrete compressive strength of approximately fc,cyl ≈ 24 MPa (3481 psi), as well as for two identical footings DF_N11 (aλ/d = 2.00, u0/d = 3.00) and DF_N12 (aλ/d = 3.00, u0/d = 3.00) with a concrete compressive strength of approximately fc,cyl ≈ 53 MPa (7687 psi). The loaddeflection curves of specimens DF_N0N and DF_N11, as well as DF_N9 and DF_N12, respectively, are comparable. The vertical shift of the loaddeflection curves for specimens DF_N11 and DF_N12 can be explained by the higher tensile strength of the concrete used for these specimens. Specimen DF_N0N failed in punching shear at 6380 kN (1434 kips) and specimen DF_N11 failed in punching shear at 9831 kN (2210 kips) resulting in a load increase of 54%. Specimen DF_N9 failed in punching shear at 5629 kN (1266 kips) and specimen DF_N12 failed in punching shear at 7850 kN (1765 kips) resulting in a load increase of 40%. The comparison of the failure loads also indicates a high efficiency of the new punching shear reinforcement elements for higher concrete compressive strengths.
Figure 6:
Measured strains of flexural reinforcement for test specimens DF_N8 (a), DF_N0N (b), and DF_N9 (c).
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4.4
Strains in flexural reinforcement
Fig. 6 depicts the measured strains in the flexural reinforcement of test specimens DF_N8 (aλ/d = 1.25), DF_N0N (aλ/d = 2.00) and DF_N9 (aλ/d = 3.00). The strains, corresponding to the yield strength of the highgrade steel used, are indicated in the diagrams. As already observed in previous tests (Hegger et al., 2009; Kueres et al., 2016a), the highest strains were confined to the region of the column face. While most of the recorded strain values were clearly below the yield limit (Fig. 6 (a, b)), in some tests the flexural reinforcement in the vicinity of the column face yielded before punching took place (Fig. 6 (c)). However, the fact that the flexural reinforcement only locally reached the yield strength indicates that the flexural capacities were not reached when failure occurred.
4.5
Strains in punching shear reinforcement
Fig. 7 (a, c, e) shows exemplarily the measured steel strains of the punching shear reinforcement elements in the first row for test specimens DF_N0N (Fig. 7 (a)), DF_N11 (Fig. 7 (c)), and DF_N9 (Fig. 7 (e)). While specimens DF_N0N (fc,cyl = 22.4 MPa (3249 psi)) and DF_N11 (fc,cyl = 53.8 MPa (7803 psi)) were tested with a shear spandepth ratio aλ/d = 2.00 and two rows of punching shear reinforcement (Layout II), in specimen DF_N9 (fc,cyl = 25.1 MPa (3641 psi)) three rows of punching shear reinforcement (Layout III) were installed due to the larger shear spandepth ratio aλ/d = 3.00. The strains corresponding to the yield strength of the punching shear reinforcement are indicated in the diagrams. In the tests, substantial steel strains were first observed at a load level coinciding more or less with the beginning of the inner shear crack formation. This was confirmed by the measured increase in the slab thickness at approximately the same load level. Regardless of the layout of punching shear reinforcement and the concrete compressive strength of the specimens, all measuring points (S17, S26, S29) showed considerable strains when failure occurred due to punching. While the measuring points S26 and S29 showed no yielding of the punching shear reinforcement (measured steel strains between 0.8‰ and 2.2‰), in specimens DF_N0N and DF_N11 the measuring point close to the column face (S17) reached the yield strength at maximum load level (Fig. 7 (a, c)). This can be explained by the inner shear crack formation. Specimens DF_N0N and DF_N11 (Fig. 4 (b, d)) showed a steep failure crack, which developed between the column face and the lower bend of the punching shear reinforcement elements in the first row. Hence, the short inclined bars in the first row of punching shear reinforcement (measuring point S17) were crossed by the failure crack and significantly activated due to the high stresses at the column face and the good anchorage of the shear reinforcement. In contrast, in specimen DF_N9 (Fig. 4 (c)) the failure crack developed from the third row of punching shear reinforcement. Thus, the short inclined bars in the first row (measuring point S17) were not crossed by the failure crack.
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Figure 7:
Measured strains of punching shear reinforcement in different rows for test specimens DF_N0N (a, b), DF_N11 (c, d), and DF_N9 (e, f, g).
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Fig. 7 (b, d, f) depicts the measured steel strains of the second row of punching shear reinforcement for test specimens DF_N0N (Layout II, Fig. 7 (b)), DF_N11 (Layout II, Fig. 7 (d)), and DF_N9 (Layout III, Fig. 7 (f)). Regardless of the layout of punching shear reinforcement and the concrete compressive strength of the specimens, the measuring points in the second row of punching shear reinforcement (S27, S28) showed considerable strains when the specimens failed in punching. The recorded strain values were below the yield strength for all tests. For specimens DF_N0N and DF_N11 the measured steel strains in the second row of punching shear reinforcement (Fig. 7 (b, d)) reached comparable values as in the first row (Fig. 7 (a, c)). In contrast, for specimen DF_N9 the measured steel strains in the second row of punching shear reinforcement (Fig. 7 (f)) were slightly higher than in the first row (Fig. 7 (b). The measured steel strains (measured steel strains for all specimens higher than 1.0‰), as well as the crack patterns (Fig. 4), confirm that the second row was activated in all specimens. The measured steel strains in the third row of punching shear reinforcement for specimen DF_N9 (Layout III) are shown in Fig. 7 (g). The steel strains in the third row were slightly lower than in the second row (Fig. 7 (f)) and on a comparable level as in the first row (Fig. 7 (e)). The crack pattern of specimen DF_N9 (Fig. 4 (c)) showed shear cracks especially in the region of the second and third row of punching shear reinforcement, which confirms this observation.
4.6
Changes in thickness of the specimens
The changes in the thickness of the test specimens were continuously measured during the tests at several points. The results are presented exemplarily for different load levels in Fig. 8 for test specimens DF_N8 (aλ/d = 1.25), DF_N0N (aλ/d = 2.00) and DF_N9 (aλ/d = 3.00). For the specimens, the inner shear crack formation initiated at approximately 30  40% of the maximum load. Regardless of the layout of the punching shear reinforcement, the highest changes in thickness were measured close to the column face. While specimen DF_N8 (Fig. 8 (a)) showed no changes in thickness outside the first row of punching shear reinforcement, in specimen DF_N0N (Fig. 8 (b)) minor changes in thickness between the first and the second row of punching shear reinforcement were observed. This observation can be verified by the measured steel strains in the second row of punching shear reinforcement (Fig. 7 (b)), as well as the crack pattern (Fig. 4 (b)). In contrast, the measured changes in thickness of specimen DF_N9 showed very high values outside the third row of punching shear reinforcement. Especially between 80% and 90% of the maximum load a high increase in the measured slab thickness was observed, which indicates a failure outside the shearreinforced zone. Strain measurements of the punching shear reinforcement (Fig. 7 (e, f, g)), as well as the crack pattern (Fig. 4 (c)), seem to confirm this assumption. The evaluation of the changes in thickness (Fig. 8), the tensile strains in the punching shear reinforcement (Fig. 7), as well as the crack patterns (Fig. 4), indicate a clear failure on the level of the maximum punching shear capacity for all specimens with shear spandepth ratios aλ/d ≤ 2.00 and specific column perimeters u0/d ≤ 4.00. In these tests, a steep failure crack developed between the column face and the lower bend of the punching shear reinforcement elements in the first row (Fig. 4 (a, b, d, f)). In specimen DF_N14 (aλ/d = 2.00, u0/d = 6.00) a slightly flatter failure crack formed between the column face and the second row of punching shear reinforcement due to the increased tangential distance between the reinforcement elements in the first row (Fig. 4 (g)). For the slender specimens DF_N9 and DF_N12 (aλ/d = 3.00), the punching shear failure occurred outside the shearreinforced zone, which 178
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can be verified by the evaluation of the changes in thickness (Fig. 8 (c)) and the crack patterns (Fig. 4 (c, e)). Due to the fact that the strains of the punching shear reinforcement in the slender specimens (aλ/d = 3.00) reached values comparable to the more compact specimens (aλ/d ≤ 2.00), it can be assumed that the failure crack outside the shearreinforced zone formed just before a failure on the level of the maximum punching shear capacity could have occurred. Thus, the failure loads of tests DF_N9, DF_N12, and DF_N14 can be considered as a lower bound of the maximum punching shear capacity.
Figure 8:
Measured changes in thickness for different load levels and locations of the measured points for test specimens DF_N8 (a), DF_N0N (b), and DF_N9 (c).
5
Discussion of experimental results
5.1
Effect of concrete compressive strength fc,cyl
The influence of the concrete compressive strength on the punching shear capacity of flat plates was investigated by various researchers in the past (Elstner and Hognestad, 1956; Regan, 1986; Gardner, 1990; Hallgren and Kinnunen, 1996; Ramdane, 1996). As a result of these investigations, different punching shear design provisions (e.g. ACI 31814, fib Model Code 2010) accounting for this influence by the square root of the concrete compressive strength (√fc,cyl). The influence of the concrete compressive strength on the punching shear capacity of footings without punching shear reinforcement was investigated in two test series (Hegger et al., 2009; Siburg and Hegger, 2014). In this context, Fig. 9 (a) shows the normalized failure loads Vtest / √fc,cyl for specimens DF12 (Hegger et al., 2009) and DF21 (Hegger et al., 2009) (aλ/d = 1.50), as well as for specimens DF13 (Hegger et al., 2009), DF22 (Hegger et al., 2009), and DF39 (Siburg and Hegger, 2014) (aλ/d = 2.00). Regardless of the shear spandepth ratio, the tests show no trend with increasing concrete compressive strength, which confirms the approach of √fc,cyl for the presented footings.
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To verify this effect for footings with the new punching shear reinforcement elements, specimens DF_N0N and DF_N11 (aλ/d = 2.00), as well as DF_N9 and DF_N12 (aλ/d = 3.00), can be considered (Fig. 9 (a)). As already observed for footings without punching shear reinforcement, the approach of the square root of the concrete compressive strength allows for a realistic description of the punching shear capacity of footings with the new punching shear reinforcement elements.
Figure 9:
5.2
Effects of concrete compressive strength fc,cyl (a), shear spandepth ratio aλ/d (b), and specific column perimeter u0/d (c) on punching shear strength of reinforced concrete footings.
Effect of shear spandepth ratio aλ/d
In previous test series (Hegger et al., 2006; Hegger et al., 2007b; Hegger et al., 2009; Siburg and Hegger, 2014), the effect of the shear spandepth ratio aλ/d on the punching shear resistance of footings without and with stirrups as shear reinforcement was investigated. Fig. 9 (b) shows the normalized failure loads Vtest / √fc,cyl for specimens DF11 (aλ/d = 1.25), DF12 (aλ/d = 1.50), and DF13 (aλ/d = 2.00) without punching shear reinforcement and specimens DF16 (aλ/d = 1.25), DF17 (aλ/d = 1.50), and DF18 (aλ/d = 2.00) with stirrups as punching shear reinforcement (Hegger et al., 2009). The shearreinforced specimens failed at maximum load level, which was indicated by the measured steel strains far below the corresponding yield strain (Hegger et al., 2009). The tests indicate that the punching shear resistance decreases with increasing shear spandepth ratio. This influence is less pronounced for footings with stirrups as punching shear reinforcement than for footings without punching shear reinforcement. Thus, the efficiency of stirrups decreases with decreasing shear spandepth ratio. To verify this effect for the footings with the new punching shear reinforcement elements, specimens DF_N7 (aλ/d = 1.25), DF_N4 (aλ/d = 2.00), and DF_N5 (aλ/d = 2.00), with a failure inside the shearreinforced zone (Kueres et al., 2016a), as well as the presented specimens DF_N8 (aλ/d = 1.25), DF_N0N (aλ/d = 2.00), DF_N11 (aλ/d = 2.00), DF_N9 (aλ/d = 3.00), and DF_N12 (aλ/d = 3.00) can be considered (Fig. 9 (b)). All these specimens were tested with a specific column perimeter u0/d = 3.00. In contrast to the test results of the previous test series (Hegger et al., 2009), a similar correlation between punching shear resistance and shear spandepth ratio as for footings without punching shear reinforcement can be observed regardless of the failure mode. Hence, the efficiency of the new punching shear reinforcement with inclined bars seems not to be affected by the shear spandepth ratio. 180
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5.3
Effect of specific column perimeter u0/d
Previous experimental investigations on the punching shear behavior of flat plates showed a strong correlation between the punching shear resistance and the specific column perimeter u0/d (Regan, 1986; Regan, 2004). For small specific column perimeters u0/d, the punching shear resistance decreases, which is taken into account in the code provisions of for example DIN EN 199211/NA (DIN EN 199211:200101; DIN EN 199211/NA:201304; Hegger et al., 2008) or a new uniform design method for punching shear in flat plates and column bases (Kueres et al., 2016b). A verification of this effect for footings by means of the results of the previous test series (Hegger et al., 2006; Hegger et al., 2007b; Hegger et al., 2009; Siburg and Hegger, 2014) on footings without and with stirrups as punching shear reinforcement is not possible, since the tests were conducted with a uniform u0/d ratio of 2.0. Fig. 9 (c) shows the normalized failure loads Vtest / √fc,cyl for specimens DF_N3 (u0/d = 2.00), DF_N4 (u0/d = 3.00), DF_N5 (u0/d = 3.00), and DF_N6 (u0/d = 4.00), with a failure inside the shearreinforced zone (Kueres et al., 2016a), as well as for specimens DF_N0N (u0/d = 3.00), DF_N11 (u0/d = 3.00), DF_N13 (u0/d = 4.00), and DF_N14 (u0/d = 6.00), with a failure on the level of the maximum punching shear capacity. Regardless of the failure mode, the tests indicate an increasing punching shear resistance with increasing specific column perimeter.
5.4
Load increase with the new punching shear reinforcement elements
Reinforced concrete slabs with punching shear reinforcement may fail in punching inside and outside the shearreinforced zone, as well as on the level of the maximum punching shear capacity. A failure inside the shearreinforced zone may develop if the amount of punching shear reinforcement is not sufficient to limit the inner shear crack growth, resulting in a failure of the punching shear reinforcement due to yielding or pullout of anchorages. A failure outside the shearreinforced zone may take place if the length of the shearreinforced zone is too short. Once a failure inside and outside the shearreinforced zone can be excluded, the punching failure occurs on maximum load level, which is strongly influenced by the multiaxial stress state along column face and the slab rotation (Beutel and Hegger, 2002; Beutel, 2003; Fernandez Ruiz and Muttoni, 2009). In most punching shear design provisions, the check of the maximum punching shear capacity is performed by limiting the punching shear capacity to a multiple of the punching shear capacity of an identical slab without punching shear reinforcement (e.g. ACI 31814, fib Model Code 2010) as
Vmax = α × Vc
(1)
where Vmax is the maximum punching shear capacity, α is an increase factor, and Vc is the punching shear capacity without punching shear reinforcement. The increase factor depends on the efficiency of the punching shear reinforcement and should be verified experimentally for each reinforcement type. Besides the type of punching shear reinforcement, the arrangement of the reinforcement may also influence the maximum punching shear capacity (Beutel and Hegger, 2002; Beutel, 2003; Hegger et al., 2007a; Siburg, 2014). In this context, ACI 31814 defines the maximum punching shear capacity as
Vmax = 1.5 × Vc
(for stirrups)
(2)
Vmax = 2.0 × Vc
(for studs)
(3)
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The evaluation of the increase factor for footings with the new punching shear reinforcement elements is performed on the basis of ACI 31814. For the following calculations, all material and strength reduction factors in the code equations are taken as unity. Since the control of the crack width is not relevant for the evaluation of test results, the limitation of the yield strength of the punching shear reinforcement (413 MPa (60000 psi)) is neglected. For the evaluation, all tests presented in this paper are considered, which will lead to a safe estimation of the increase factor.
Table 2:
Load increase with the new punching shear reinforcement elements
u0/d
aλ/d
fc,cyl
Æw
Failure
b0
vc,ACI
Vc,ACI
Vtest,red


MPa (psi)
mm (in.)

m (in.)
MPa (psi)
kN (kips)
kN (kips)
DF_N8
3.02
1.26
23.0 (3336)
14 (0.6)
m
2.79 1.583 (109.8) (229.6)
1752 (394)
5524 (1242)
3.15
DF_N0N
3.03
2.02
22.4 (3249)
14 (0.6)
m
2.78 1.562 (109.4) (226.6)
1722 (387)
5531 (1243)
3.21
DF_N9
3.02
3.02
25.1 (3641)
14 (0.6)
o
2.79 1.653 (109.8) (239.7)
1830 (411)
5255 (1181)
2.87
DF_N11
3.02
2.02
53.8 (7803)
14 (0.6)
m
2.79 2.420 (109.8) (351.0)
2679 (602)
8508 (1912)
3.18
DF_N12
3.02
3.02
51.3 (7440)
14 (0.6)
o
2.79 2.364 (109.8) (342.9)
2616 (588)
7327 (1647)
2.80
DF_N13
4.05
2.03
22.8 (3307)
14 (0.6)
m
3.18 1.576 (125.2) (228.6)
1979 (445)
6238 (1402)
3.15
DF_N14
6.06
2.02
21.7 (3147)
14 (0.6)
m
3.98 1.537 (156.7) (222.9)
2425 (545)
6914 (1554)
2.85
Average COV
3.03 0.06
Test
Vtest,red / Vc,ACI 
u0/d: specific column perimeter; aλ/d: shear spandepth ratio; fc,cyl: concrete compressive strength; Æw: diameter of shear reinforcement; failure: failure mode (m: maximum punching shear capacity; o: failure outside shearreinforced zone); b0: control perimeter according to ACI 31814; vc,ACI: punching shear capacity without punching shear reinforcement according to ACI 31814; Vtest,red: ultimate failure load reduced by the effective soil pressure within the control perimeter (Vtest,red = Vtest(1−A0/A); Vtest,red / Vc,ACI: ratio of experimental failure load and punching shear capacity without punching shear reinforcement according to ACI 31814; COV: coefficient of variation.
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Table 2 shows the comparison of the ultimate test loads reduced by the soil pressure within the control perimeter at a distance 0.5d from the column face and the punching shear capacities of similar footings without punching shear reinforcement according to ACI 31814. Due to the good anchorage performance and the Sshaped form of the new punching shear reinforcement elements, the punching shear capacity of the specimens presented in this paper is significantly increased compared to similar footings without punching shear reinforcement. For the seven specimens, the average value of Vtest,red / Vc is 3.03 (COV: 0.06), which is twice as high as the increase factor for stirrups and 50% higher than the increase factor for studs (Eqs. (2, 3)). The evaluation proves a high efficiency of the new punching shear reinforcement elements, especially when compared to vertical punching shear reinforcement elements like stirrups and studs. The high loadincrease factor underlines the potential of the new punching shear reinforcement.
6
Comparison of predictions and experimental results
The main parameters investigated in the present test series were the shear spandepth ratio aλ/d, the ratio of column perimeter and effective depth u0/d, and the concrete compressive strength fc,cyl. Fig. 10 depicts the comparison of the ultimate failure loads reduced by the soil pressure within the control perimeter and the punching shear capacities of similar footings without punching shear reinforcement according to ACI 31814, separately for the different influences. While the influence of the concrete compressive strength (Fig. 10 (a)) is taken into account realistically by the code equations, the ratio Vtest,red / Vc shows a minor decrease with increasing shear spandepth ratios (Fig. 10 (b)) and specific column perimeters (Fig. 10 (c)), respectively. While other punching shear design provisions (fib Model Code 2010; Kueres et al., 2016b) directly account for these influences, in ACI 31814 these effects are solely considered implicitly by the definition of the control perimeter. Hence, the code equations according to ACI 31814 might underestimate the influences of shear spandepth ratio and specific column perimeter. To verify this observation, further experimental investigations on footings without and with punching shear reinforcement with variable shear spandepth ratios and specific column perimeters are necessary.
Figure 10: Comparison of predictions and experimental results (concrete compressive strength fc,cyl (a), shear spandepth ratio aλ/d (b), and specific column perimeter u0/d (c)).
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7
Conclusions
The results of the experimental investigations on reinforced concrete footings with new punching shear reinforcement elements allow the following conclusions to be drawn: 1. The new punching shear reinforcement elements with inclined bars significantly increase the punching shear capacity of reinforced concrete footings. The high efficiency of the new punching shear reinforcement elements is also evident for footings with higher concrete compressive strength. 2. Regardless of the layout of punching shear reinforcement, the measured steel strains confirm that in the present tests all rows of punching shear reinforcement were activated and thus contributed to the punching shear resistance. 3. While punching tests on reinforced concrete footings with stirrups indicate a reduced efficiency of the shear reinforcement with decreasing shear spandepth ratio aλ/d, the efficiency of the new punching shear reinforcement elements seems not to be affected by the shear spandepth ratio aλ/d. 4. Due to the good anchorage performance and the Sshaped form of the new punching shear reinforcement elements, the average load increase of the presented specimens is 3.03 compared to the punching shear capacity without punching shear reinforcement according to ACI 31814. The evaluation proves a high efficiency of the new punching shear reinforcement elements, especially when compared to vertical punching shear reinforcement elements like stirrups and studs. 5. While the influence of the concrete compressive strength is taken into account realistically by the code equations according to ACI 31814, the presented tests show slightly decreasing ratios of predicted and experimental failure load with increasing shear spandepth ratios aλ/d and specific column perimeters u0/d, respectively. To verify this observation, further experimental investigations on footings without and with punching shear reinforcement are necessary.
8
Acknowledgements
The investigations presented were supported by Deutsche Forschungsgemeinschaft (German Research Foundation, DFGGZ HE 2637/211) and the authors wish to express their sincere gratitude. The authors would also like to acknowledge the support of Halfen GmbH Langenfeld, Germany.
9
References
ACI Committee 318 (2014) Building Code Requirements for Structural Concrete (ACI 31814) and Commentary (ACI 318R14), American Concrete Institute, Farmington Hills, MI, 2014. Andrä, H.P. (1981) “Zum Tragverhalten von Flachdecken mit DübelleistenBewehrung im Auflagerbereich,” Beton und Stahlbetonbau, V. 76, No. 3, pp. 5357 (Part 1), No. 4, pp.100104 (Part 2). (in German) Beutel, R. (1981) “Durchstanzen schubbewehrter Flachdecken im Bereich von Innenstützen,” PhD thesis, RWTH Aachen University, Institute of Structural Concrete, 2003. (in German)
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Beutel, R., Hegger, J. (2002) “The effect of anchorage on the effectiveness of the shear reinforcement in the punching zone,” Cement and Concrete Composites, V. 24, No. 6, pp. 539549. Dieterle, H., Rostásy, F.S. (1987) “Tragverhalten quadratischer Einzelfundamente aus Stahlbeton,” Deutscher Ausschuss für Stahlbeton, Heft 387. (in German) Dieterle, H., Steinle, A. (1981) “Blockfundamente für Stahlbetonfertigstützen,” Deutscher Ausschuss für Stahlbeton, Heft 326. (in German) DIN EN 199211:201101, “Eurocode 2: Bemessung und Konstruktion von Stahlbeton und Spannbetontragwerken – Teil 11: Allgemeine Bemessungsregeln und Regeln für den Hochbau,” Deutsche Fassung EN 199211:2004 + AC:2010. (in German) DIN EN 199211/NA:201304, “Nationaler Anhang – National festgelegte Parameter – Eurocode 2: Bemessung und Konstruktion von Stahlbeton und Spannbetontragwerken – Teil 11: Allgemeine Bemessungsregeln und Regeln für den Hochbau,” Deutsche Fassung EN 199211/NA: 201304. (in German) Einpaul, J., Brantschen, F., Fernández Ruiz, M., and Muttoni, A. (2016) “Performance of Punching Shear Reinforcement under Gravity Loading: Influence of Type and Detailing,“ ACI Structural Journal, V. 113, No. 4, pp. 827838. Elstner, R.C, Hognestad, E. (1956) “Shearing Strength of Reinforced Concrete Slabs,” Journal of the American Concrete Institute, V. 28, No. 1, pp. 2958. Fernandez Ruiz, M., Muttoni, A. (2009) “Applications of Critical Shear Crack Theory to Punching of Reinforced Concrete Slabs with Transverse Reinforcement,” ACI Structural Journal, V. 106, No. 4, pp. 485494. Ferreira, M.P., Melo, G.S., Regan, P.E., Vollum, R.L. (2014) “Punching of Reinforced Concrete Flat Slabs with DoubleHeaded Shear Reinforcement,” ACI Structural Journal, V. 111, No. 2, pp. 363374. fib (Fédération international du béton) (2013) fib Model Code for Concrete Structures 2010, Ernst & Sohn, Berlin, Germany. Gardner, N.J. (1990) “Relationship of the Punching Shear Capacity of Reinforced Concrete Slabs with Concrete Strength,“ ACI Structural Journal, V. 87, No. 1, pp. 6671. Hallgren, M., Kinnunen, S. (1996) “Increase of punching shear capacity by using high strength concrete,” Proceedings of 4th International Symposium on Utilization of HighStrength/HighPerformance concrete, Paris, France. Hallgren, M., Kinnunnen, S., Nylander, B. (1998) “Punching Shear Tests on Column Footings,” Nordic Concrete Research, V. 21, No. 3, pp. 122. Hegger, J., Häusler, F., Ricker, M. (2008) “Critical Review of the Punching Shear Provisions According to Eurocode 2,” Beton und Stahlbetonbau, V. 103, No. 2, pp. 93–102. (in German) Hegger, J., Häusler, F., Ricker, M. (2007a) “Maximum Punching Capacity of Flat Slabs,” Beton und Stahlbetonbau, V. 102, No. 11, pp. 770777. (in German) Hegger, J., Ricker, M., Sherif, A.G. (2009) “Punching Strength of Reinforced Concrete Footings,” ACI Structural Journal, V. 106, No. 5, pp. 706716.
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Hegger, J., Ricker, M., Ulke, B., Ziegler, M. (2007b) “Investigations on the punching shear behaviour of reinforced concrete footings,” Engineering Structures, V. 29, No. 9, pp. 22332241. Hegger, J., Sherif, A.G., Ricker, M. (2006) “Experimental Investigations on Punching Behavior of Reinforced Concrete Footings,” ACI Structural Journal, V. 103, No. 4, pp. 604613. Kueres, D., Ricker, M., Hegger, J. (2016a) “A new punching shear reinforcement system for precast footings,” Proceedings of 2016 PCI Convention & National Bridge Conference, Nashville, USA. Kueres, D., Siburg, C., Herbrand, M., Classen, M., Hegger, J. (2016b) “Uniform design method for punching shear in flat slabs and footings,” Beton und Stahlbetonbau, V. 111, No. 1, pp. 919. (in German) Kueres, D., Wieneke, K., Siburg, C. (2015) “Investigations on the punching behaviour of eccentrically loaded footings,” Beton und Stahlbetonbau, V. 110, No. 9, pp. 609619. (in German) Mokthar, A.S., Ghali, A., Dilger, W. (1985) “Stud shear reinforcement for flat concrete plates,” ACI Journal, Proceedings, V. 82, No. 5, pp. 676683. Park, HG., Ahn, KS., Choi, KK., and Chung, L. (2007) “Lattice Shear Reinforcement for SlabColumn Connections,” ACI Structural Journal, V. 104, No. 3, pp. 294303. Pilakoutas, K., Li, X. (2003) “Alternative Shear Reinforcement for Reinforced Concrete Flat Slabs,” Journal of Structural Engineering, V. 129, No. 9, pp. 11641172. Ramdane, K.E. (1996) “Punching shear of high performance concrete slabs,” Proceedings of 4th International Symposium on Utilization of HighStrength/HighPerformance concrete, Paris, France. Regan, P. E. (1986) “Symmetric punching of reinforced concrete slabs,” Magazine of Concrete Research, V. 38, No. 136, pp. 115–128. Regan, P.E. (2004) “Punching of slabs under highly concentrated loads,” Structures and Buildings, V. 157, No. 2, pp. 165–171. Richart, F.E. (1948) “Reinforced Concrete Wall and Column Footings,” Journal of the American Concrete Institute, V. 20, No. 2, 1948, pp. 97127 (Part 1), No. 3, pp. 237261 (Part 2). Ricker, M., Häusler, F. (2014) “European punching design provisions for doubleheaded studs,” Structures and Buildings, V. 167, No. SB8, pp. 495506. Siburg, C. (2014) “Zur einheitlichen Bemessung gegen Durchstanzen in Flachdecken und Fundamenten,” PhD thesis, RWTH Aachen University, Institute of Structural Concrete. (in German) Siburg, C.; Hegger, J. (2014) “Experimental investigations on the punching behaviour of reinforced concrete footings with structural dimensions,” Structural Concrete, V. 15, No. 3, pp. 331339. Talbot, A.N. (1913) “Reinforced Concrete Wall Footings and Column Footings,” Publications of the Engineering Experiment Station, Bulletin No. 67.
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Bridging the gap between oneway and twoway shear in slabs Eva O. L. Lantsoght1,2, Cor van der Veen2, Ane de Boer3, Scott D.B. Alexander4 1
: Universidad San Francisco de Quito, Ecuador
2
: Delft University of Technology, The Netherlands
3
: Ministry of Infrastructure and the Environment, Utrecht, The Netherlands
4
: COWI North America, Canada
Abstract The shear capacity of slabs under concentrated loads is particularly of interest for bridge decks under concentrated live loads. Often, oneway shear will be analyzed by considering the slab as a wide beam (without taking advantage of the transverse load redistribution capacity of the slab) and twoway shear by considering the punching area around the load. Since experiments have shown that the failure mode of slabs under concentrated loads is a combination of oneway and twoway shear as well as twoway flexure, a method was sought that bridges the gap between the traditional oneway and twoway shear approaches. The proposed method is a plasticitybased method. This method is based on the Strip Model for concentric punching shear and takes the effects of the geometry into account for describing the ultimate capacity of a slab under a concentrated load. The model consists of “strips” that work with arching action (oneway shear) and slab “quadrants” that work in twoway shear. As such, the resulting Extended Strip Model is suitable for the design and assessment of elements that are in the transition zone between oneway and twoway shear.
Keywords Arching action, extended strip model, flexure; plasticity, oneway slabs, punching, shear, strip method, strip model
1
Introduction
1.1
Application to existing bridges
Over the past few years, more attention has been paid to the shear capacity of slabs without shear reinforcement (Lantsoght et al., 2013a; Reißen and Hegger, 2013; Rombach and Latte, 2009). These structures have been studied in detail because the shear capacity of a number of existing bridges does not fulfill the requirements of the recently introduced Eurocodes in terms of shear capacity as given by NENEN 199211:2005 (CEN, 2005) and live loads as given by NENEN 19912:2003 (CEN, 2003). In the Netherlands, 600 slab bridges are the subject of discussion. The shear provisions of the code, however, do not take into account beneficial effects such as transverse load redistribution (Lantsoght et al., 2015d). Since slabs 187
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have a high level of redundancy, it is possible to refine the analysis and assessment for reinforced concrete slab bridges and to take the redistribution into account. Plastic design methods are wellsuited for this purpose. Recent research on slabs under concentrated loads has resulted in a number of proposals for design, and experimental analysis has resulted in a better understanding of the behavior of slabs. The study of the shear capacity of slabs under concentrated loads at different locations of the slab is particularly of interest for bridge decks under concentrated live loads. The goal of any proposed design and assessment method should be to determine the maximum load on the bridge under study.
1.2
Plastic design methods
Plastic solutions exist for oneway shear (Braestrup, 2009; Cho, 2003; Ibell et al., 1997), twoway shear (Kuang and Morley, 1993; Salim and Sebastian, 2002) and flexure (Alexander, 1999b; Hillerborg, 1975; 1982; 1996). For Dregions, the use of strutandtie methods (He et al., 2012; Mattock, 2012; Reineck, 2010; Schlaich et al., 1987) has become part of daily design practice, which is reflected in the ACI 31814 code (ACI Committee 318, 2014). ACI 31814 is the first ACI Building Code in which strutandtie methods are part of the text of the code and no longer in the annex. Simplified plastic material models can be used together with plastic design methods, for example to determine the oneway (Windisch, 1988) or twoshear capacity (Windisch, 2000) based on the capacity of the concrete compressive zone. More complex and complete plastic material models, such as the concrete damaged plasticity model (Grassl et al., 2013), can be used in conjunction with nonlinear finite element models. These rational methods generally give good and conservative results when compared to experiments. Hillerborg’s Strip Method for flexure (Hillerborg, 1975; 1982; 1996) is commonly used in the engineering profession to design oddly shaped slabs or slabs with holes. The Advanced Strip Method is, contrary to its name, an easytouse method that allows for the design of slabs subjected to concentrated loads or supported by columns by using a cornersupported element. The Strip Method and the Advanced Strip Method are simplifications of the equilibrium method, for which the torsional moments are taken as equal to zero. As such, the method is based on the lowerbound theorem of plasticity. An upperbound method for analyzing the flexural capacity of slabs is available as well. This method is called the yield line method (Park and Gamble, 2000), and is based on the collapsed state of the slab by the formation of a mechanism. For slabs subjected to concentrated loads or supported by columns, a fan of yield lines is introduced. For beams in shear, an analysis method based on a mechanism is also available (Nielsen and Hoang, 2011). A weakness of the plasticitybased methods is that some methods require the use of an effectiveness factor on the concrete compressive strength. When a compressive strut is used, often the full concrete compressive strength cannot develop because of tensile stresses perpendicular to the strut, reinforcement crossing the strut and disturbing the stresses, and the effect of cracking. Determining the correct effectiveness factor is an element of empiricism in the currently used plasticitybased methods.
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2
Research significance
This study shows the application of the Strip Model to reinforced concrete slabs under concentrated loads failing in a combination of oneway shear, twoway shear and flexure. The resulting Extended Strip Model provides an estimate of the maximum concentrated load that can be applied to the structure without violating defined material capacities. This loading case is important in the assessment of existing bridges subjected to concentrated live loads. The presented Extended Strip Model is suitable for a loading situation that was not given much attention in the past. The validity of the model is proven by a comparison with laboratory experiments and a case study of an existing bridge tested to failure.
3
The design gap between oneway and twoway shear
Often, for the design and assessment of existing reinforced concrete bridges subjected to concentrated live loads, oneway shear will be analyzed by considering the slab as a wide beam (without taking advantage of the transverse load redistribution capacity of the slab) and twoway shear by considering the area of the load on the slab in the same way as a slabcolumn connection in buildings. For Dregions, strutandtie models can be developed, but little guidance is available on how to elaborate threedimensional strutandtie models for complex loading cases such as slab bridges subjected to concentrated live loads. Since experiments (Lantsoght et al., 2013b; Lantsoght et al., 2015b) have shown that the failure mode of slabs under concentrated loads (representing bridge decks) is a combination of oneway and twoway shear as well as flexure (see Figure 1), a method that bridges the gap between traditional oneway and twoway shear approaches was sought. In the experiments on slabs, flexural cracking was observed in the transverse and longitudinal direction. Moreover, inclined cracks on the bottom of the slab indicating shear distress were observed towards the ultimate load, as well as partial cracks showing the punching perimeter around the load. Cracks indicating torsional distress at the edges where the slab edges want to curl up were also observed for slabs loaded with a load close to the free edge. An overview of the different types of distress after one experiment (S6T4) are shown in Figure 1. Similar observations were made for slabs under a combination of loads, mimicking the typical live load models of concentrated loads and distributed loads (Lantsoght et al., 2015a). Clearly, for slabs under concentrated loads, the different failure mechanisms cannot and should not be isolated, as current design methods imply. In experiments on elements with a smaller width (Lantsoght et al., 2014), only interaction between oneway shear and flexure was observed. When the width of specimens subjected to concentrated loads was increased, a transition from oneway shear to a combination of oneway and twoway shear was observed (Lantsoght et al., 2015c). This transition zone is typically not considered in the codes, which make a strict distinction between beam shear (oneway shear) and punching shear (twoway shear). However, measurements of the force increment in structural elements failing in oneway or twoway shear showed values that were very similar (Olonisakin and Alexander, 1999). These observations show that there is a fundamental link between oneway and twoway shear. Design solutions for loading cases in this transition zone between a beam in three or fourpoint bending and a slabcolumn connection are not available.
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Figure 1:
Observed cracking after failure of slab S6 in experiment S6T4 (Lantsoght et al., 2013b): (a) combination of oneway shear (inclined cracks on the bottom face), torsional distress at the edge and flexural distress represented by longitudinal and transverse cracks; (b) oneway shear failure crack at the edge; (c): twoway shear cracks on the top face of the slab.
4
Lower bound plasticitybased methods for slabs
4.1
Hillerborg’s Strip Method for flexure
As introduced previously, Hillerborg’s Strip Method (Alexander, 1999a; Hillerborg, 1975; 1982; 1996) is a lowerbound plasticitybased solution for the flexural design of slabs. The method is particularly powerful in cases where traditional methods such as direct design cannot be applied, such as slabs with openings, or slabs not supported on all edges. The Strip Method has been under development since the late 1950s, first only for slabs supported over a full edge. Later, a cornersupported element resulted in the Advanced Strip Method (Hillerborg, 1982). While the name “Advanced Strip Method” might sound rather daunting, the method itself is as easy to apply as the regular Strip Method. The ease of application of the method was shown in a book with design examples (Hillerborg, 1996).
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The Strip Method is a lower bound plasticitybased method, which means that (Hillerborg, 1975) if there is a load Q1 for which it is possible to find a moment field that fulfills all equilibrium conditions and the moment is not higher than the yield moment at any point, then Q1 is a lowerbound value of the carrying capacity. The slab can certainly carry the load Q1. The Strip Method is a simplification of the equilibrium theory, which is based on the design principle that a moment field is first determined to fulfil the equilibrium and edge conditions, after which the strength of the slab at each point is designed for this moment field. The simplification makes it possible to fulfil the lowerbound theorem and at the same time achieve a good economical arrangement of reinforcement. The basic assumption (Alexander, 1999b) of the Strip Method is that, because of the ductility and redistribution capacity of slabs, virtually any combination of the distributed loads in the main directions qx, qy and under 45° qxy, the load that determines the torsional reinforcement may be used as the basis for the design of a slab, as long as: 1. the slab is designed at all points to resist the moments resulting from the assumed load distribution; 2. the shear and moments resulting from the assumed load distribution do not violate any boundary condition; and 3. the assumed load distribution satisfies equilibrium: qx + qy + qxy = q In the Strip Method, the torsional moments are set to zero, so that qx + qy = q. Then, strips of the slab in the x or ydirection can be treated as oneway beams and designed to carry the loads acting on them. The Strip Method gives a clear indication of where reinforcement should be placed to be of greatest benefit. Since complete moment diagrams are determined for the strips/beams, bar cutoffs can easily be determined. For unsupported edges or large openings, strong bands can be used. A strong band is a strip of slab that acts like a virtual beam, on which the other strips can rest. Therefore, the strong band must be designed to carry the applied load plus the reactions of the strips carrying off to the strong band.
4.2
Strip Model for concentric punching shear
A plasticitybased solution for concentric punching shear exists as the Bond Model (Alexander, 1990; Alexander and Simmonds, 1992), later renamed the Strip Model (Afhami, 1997; Afhami et al., 1998; Ospina et al., 2003). The model subdivides the load transfer mechanism at a slabcolumn connection into the elementary beam shear carrying mechanism. The slab is split into strips branching out of the column, which work in arching action, and quadrants limited by the boundaries of the strips, which work in twoway flexure. An overview of the layout of the strips and quadrants is shown in Figure 2a. The interface between the quadrants working in twoway flexure and the strips working in arching action is then the weakest link in the assumed load transfer mechanism. Therefore, failure is assumed to occur at the interface between the strips and quadrants, which is governed by the limiting oneway shear capacity, working in beam shear. The loading on the strip and the resulting moment diagram is shown in Figure 2b.
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Figure 2:
Overview of the Bond Model for concentric punching shear: (a) layout of the slab, divided in quadrants and strips; (b) loading situation on strip and resulting moment diagram (Alexander and Simmonds, 1992).
The model was originally called the Bond Model, as it refers to the force gradients in the reinforcement in the strips working in arching action. The force gradients are transferred by bond, and originally the model was studied by defining the bond strength of the reinforcement in unconfined splitting failure (Alexander and Simmonds, 1992). For this loading case, the maximum force gradient in the reinforcement bars perpendicular to the strip determines the maximum load. Later, the equivalence between beam action shear and a limiting nominal shear stress was used to find the maximum load. The interchangeability of the beam shear capacity and a limiting force increment in the reinforcement are the main assumptions of the
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original Bond Model. However, later the model was renamed “Strip Model”, to link the name of the model to the principle of the method in which radial strips are used, and to avoid confusion with models that describe the force transfer at the interface of a reinforcement bar and the surrounding concrete (i.e., that study only the mechanism of bond). Experiments confirmed the arching action in the strips (Alexander, 1990). Strain measurements showed that the interaction between the arch and the adjacent quadrants determined the geometry of the curved arch, which led to the development of the failure criterion at the intersection between the strips and quadrants. In the Strip Model, the total capacity is determined as the sum of the capacities of the four strips (as shown in Figure 2a). For the case of concentric punching shear, the capacity of each strip is equal. Analyzing a single strip, Figure 2b, shows that the strip has a length lstrip and is loaded over the loaded length lw so as to maximize the occurring moments. The maximum possible moments are the moment capacities of the crosssections Msag for the sagging moment and Mhog for the hogging moment. The moments can be taken over the full width for simplicity. The total flexural capacity Ms of the slab is the sum of these moment capacities, Ms = Mhog + Msag. The Strip Model thus assumes that yielding of the steel occurs at the ultimate. This assumption is valid for practical cases of building slabs and bridge slabs, but is sometimes violated in laboratory experiments that are designed to achieve a shear failure. The acting loading is 2w, obtained by summing the maximum loads on both sides of the strip, which has an interface between the strip and the quadrant on both sides. Using vertical and moment equilibrium on the strips (Figure 2b) results in the following expressions: Ms =
2wlw2 2
PAS,1 = 2 wlw
(1) (2)
with Ms the total flexural capacity, combining the hogging and sagging moment capacities; w
the maximum shear at the interface between the strip and the quadrant;
lw
the loaded length, optimized to get the largest moments, as shown in Figure 2b.
Solving Eq. (1) for the unknown loaded length lw and substituting this into Eq. (2) results in the shear capacity of a single strip, PAS,1: PAS,1 = 2 M s w
(3)
The total capacity of the studied slabcolumn connection is then the sum of the capacities of four strips: PAS = 8 M s w
(4)
Since the load w is determined by the maximum shear stress at the interface between the strips and the quadrants, the oneway shear capacity wACI from ACI 31814 (ACI Committee 318, 2014) was proposed for use with the Bond Model (Alexander and Simmonds, 1992). This expression was first formulated in the late 1950s and determines the inclined cracking load (Morrow and Viest, 1957): 193
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é kN ù wACI ê = 0.166 ´ d [ mm ] f c' [ MPa ] ú ëmû
(5)
é lbf ù wACI ê = 2.00 ´ d [in ] f c' [ psi ] ú ë in û
with fc’
the concrete cylinder compressive strength; and
d
the effective depth.
Finite element analyses (Afhami et al., 1998) showed the validity of this simple expression for the maximum load. The shear on the side faces of the strips, when approximated by a rectangle in the region near the column, had about the same value as wACI.
5
Extended strip model applications to slab bridges
5.1
Oneway slabs as used in slab bridges
The proposed method for slabs under concentrated loads is a plasticitybased method, called the Extended Strip Model. This method is based on the Strip Model for concentric punching shear, as described previously, and takes the effects of the geometry into account for describing the ultimate capacity of a slab under a concentrated load. Like the original Strip Model (Alexander and Simmonds, 1992), the Extended Strip Model consists of “strips” that work in arching action (oneway shear) and slab “quadrants” that work in twoway flexure. As such, this method is suitable for the design and assessment of elements that are in the transition zone between oneway and twoway shear, and that show significant flexural distress upon failure, such as slab bridges under concentrated loads. Slab bridges are designed so that a flexural failure (ductile failure mode) occurs before a shear failure (brittle failure mode). A full description of the theoretical background and derivations of the Extended Strip Model is provided elsewhere (Lantsoght et al., (2017)). Here, the main novelties of the Extended Strip Model as compared to the original Strip Model will be discussed. The first difference between the slab of infinite dimensions studied by the Strip Model for concentric punching shear (i.e. a slabcolumn connection as used in buildings) and a bridge deck slab, is that the bridge deck slab will have different dimensions, reinforcement ratios and effective depths in the longitudinal and transverse direction. For clarity, the longitudinal direction will be named the xdirection, and the transverse direction the ydirection (see Figure 3). The effective depth of the longitudinal reinforcement (larger value) is then dx and of the transverse reinforcement (smaller value) is dy. For bridge deck slabs, typically a larger reinforcement ratio is used in the xdirection (main load carrying direction) than in the ydirection. As a result, the capacity of the flexural reinforcement in the xdirection Ms,x will be larger than the capacity in the ydirection Ms,y. The load on the ydirection strips will be determined by dx, since the crosssection of the intersection between the strip and the quadrant has the xdirection reinforcement as bending reinforcement. Similarly, the load on the xdirection strips is determined by dy. The resulting loads are wACI,x on strips in the xdirection and wACI,y on strips in the ydirection. The resulting expressions for the loads on the strips then are, with fc’ the concrete compressive strength:
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é kN ù ' wACI , x ê ú = 0.166 ´ d y [ mm ] f c [ MPa ] m ë û
(6)
é lbf ù wACI , x ê = 2.00 ´ d y [in ] f c' [ psi ] ú ë in û é kN ù wACI ,y ê = 0.166 ´ d x [ mm ] f c' [ MPa ] ú ëmû
(7)
é lbf ù wACI ,y ê = 2.00 ´ d x [in ] f c' [ psi ] ú ë in û An overview of the considered loads on the x and ydirection strips for a bridge deck slab subjected to a concentrated load is given in Figure 3.
Figure 3:
5.2
Overview of loads on strips in x and ydirection and implications of static equilibrium.
Implications of static equilibrium
The second step to apply the Strip Model to oneway slabs under concentrated loads close to supports is to analyze the static equilibrium of the situation. The shear stress is not the same on each side of strips perpendicular to the main span direction. Failure will take place when the maximum shear stress is reached on the interface between the strip and the quadrants on the side of the strip that is more heavily loaded because of the shear diagram. To find the capacity of the strips, the implication is that the total load on the strip will not be 2wACI anymore. From the shear diagram, it is known that the averaged stresses v1 and v2 (as shown in Figure 3) caused by a load P are:
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v1 =
L  aM P L b
(8)
v2 =
aM P L b
(9)
with aM the smallest value of the distance between the center of the load and the center of the support, and the distance between the center of the load and the point of contraflexure L
the distance between the points of contraflexure, or the span length for a simply supported slab;
P
the concentrated load;
b
the slab width.
The stresses are averaged over the full width, since they are used here only to find the relative maximum shear that will occur on the side of the strip that is less heavily loaded. The ratio of the stresses equals:
aM v2 aM = L = v1 L  aM L  aM L
(10)
If the shear capacity wACI,y is reached in v1, the stress v2 equals:
v2 =
aM wACI , y L  aM
(11)
These loads are shown in Figure 3. The maximum value of the load, replacing 2wACI,y in the original Strip Model, then becomes:
qmax,y =
L wACI , y L  aM
(12)
with qmax,y the total load on the ydirection strips; wACI,y the maximum shear on the interface between the quadrant and the strip in the ydirection as given by Eq. (7).
5.3
Reduction for selfweight
The maximum capacity of the strip is the total maximum stress that can occur at the position of the interface between the strip and the quadrant for all considered loads. When studying a slab subjected to a single concentrated load, the concentrated load as well as the selfweight of the structure contribute to the shear stress on the studied section. Therefore, to find the maximum value of the concentrated load, the effect of the selfweight has to be 196
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subtracted from the total available capacity. For slabcolumn connections, this reduction will be small because the depths of the slabs are small. For slab bridges, on the other hand, much larger crosssections are used, and the effect of the selfweight becomes more important. For oneway spanning slabs, it is assumed that the selfweight only acts in the main span direction, and thus only affects the interface between the quadrants and the ydirection strips. The sectional shear at the position of the load can be transformed into a distributed load (units [kN/m] or [lbf/in] like wACI) by dividing the sectional shear by the total width of the member. For the strips in the ydirection, the total load then becomes: qmax,y =
L ( wACI , y  vDL ) L  aM
(13)
The effects on the maximum loads on the strips is shown in Figure 4. The effect of torsion is represented in Figure 4 by the factor β, which will be discussed later in this paper when considering asymmetric loading conditions, where torsion reduces the available capacity further.
5.4
Size effect
The expression for the shear capacity at the interface between the strips and quadrants, as given in Eqs. (6) and (7), has one major drawback: it gives good results for small specimens, but becomes unconservative for larger specimens (Collins and Kuchma, 1999). This problem is caused by the size effect in shear (Bazant and Kim, 1984; Kani, 1967): as the depth of members increases, their shear capacity does not increase proportionally. A recommendation (Alexander, 2016) for taking into account the size effect on the shear capacity of slabs was built into the expression for the shear capacity, resulting in the following expressions: 1
æ 100mm ö 3 é kN ù ' wACI , x ê = 0.166 ´ d mm f MPa [ ] [ ] çç ÷÷ y c ú ëmû è d [ mm] ø
(14)
1
æ 3.94in ö 3 é lbf ù ' wACI , x ê = 2.00 ´ d in f psi [ ] [ ] çç ÷÷ y c ú ë in û è d [in ] ø 1
æ 100mm ö 3 é kN ù ' wACI , y ê = 0.166 ´ d mm f MPa [ ] [ ] çç ÷÷ x c ú ëmû è d [ mm] ø
(15)
1
æ 3.94in ö 3 é lbf ù ' wACI , y ê = 2.00 ´ d in f psi [ ] [ ] çç ÷÷ x c ú ë in û è d [in ] ø
with: d
the average effective depth (average of dx and dy).
The introduction of this size effect term leads to a good correspondence with the experimental results (Lantsoght, 2016a), as will be shown later with the case study of the Ruytenschildt Bridge.
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Figure 4:
5.5
Overview of loads on strips in x and ydirection and implications of static equilibrium, including reduction of selfweight and effect of torsion.
Loads close to supports
For loads close to the support, as is the governing case for the shear capacity of slab bridges under concentrated live loads, a direct compression strut can develop between the load and the support. This strut leads to an increase in the shear capacity (Grebovic and Radovanovic, 2015). For example, in Figure 4, if the load is placed close to the support (with av ≤ 2dx), the xdirection strip between the load and the support will have an enhanced capacity as a result of direct load transfer. Here, av is the clear shear span (facetoface distance between the load and the support) and dx the effective depth to the longitudinal reinforcement. To take this effect into account, the following enhancement factor is proposed: 1£
2d x £4 av
(16)
This factor is based on the proposed enhancement of the side of the punching perimeter facing the support as proposed by Regan for slabs under concentrated loads close to supports (Regan, 1982). The effect of torsion, which will be discussed in section 6 on asymmetric loading conditions, also needs to be considered for loads close to the support.
5.6
Different longitudinal and transverse reinforcement
The original Strip Model was developed for building slabs, which have similar reinforcement layouts in both directions. For bridge slabs, on the other hand, the reinforcement in the transverse direction is typically much less than the reinforcement in the longitudinal direction. In European practice, around 20% of the longitudinal reinforcement is used for the transverse reinforcement. As a result, the moment capacities of the strips in the ydirection will be much lower than the moment capacities in the longitudinal directions. The difference in effective depth (dx for the longitudinal reinforcement and dy for the transverse reinforcement) needs to be taken into account for the determination of the moment capacities as well as for the maximum shear, as given by Eqs. (14) and (15).
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The original Strip Model introduced a method of counting the bars present in the strip and close to the strip, which is necessary for building slabs in which larger concentrations of reinforcement can be used in the column strips. For bridge slabs, however, the reinforcement is distributed evenly. A simplification of the method is thus that the reinforcement ratios ρx (for the longitudinal direction) and ρy (for the transverse direction) can be used instead of counting the present bars.
5.7
Continuity at the supports
For continuous bridges, the effect of both the hogging and sagging reinforcement needs to be taken into account when the load is placed close to one of the mid supports, as the change of the sign of the moment will influence the behavior. The quadrants between the load and the mid support will be subject to a change in moment from hogging moment over the support Msup to sagging moment at the position of the concentrated load in the span Mspan (see Figure 5). These two quadrants that are affected by the moment diagram are bordered by three strips: the ydirection strips and the xdirection strip between the load and the support. The moment capacity of these three strips thus has to be based on the hogging and sagging reinforcement. This effect is taken into account (Lantsoght, 2016b) with the following factor: lmoment =
M sup M span
£1
(17)
The moments are determined based on the loading conditions. The total moment capacity in the three strips, where the hogging moment capacity over the mid support has an effect on the total moment capacity, is then determined as: M s = M sag + lmoment M hog
(18)
with Msag the sagging moment capacity; Mhog the hogging moment capacity. This proposed method was shown to lead to good results when compared with slab shear experiments (Lantsoght, 2016b).
Figure 5:
Definition of Msup and Mspan.
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6
Considerations for asymmetric loading conditions
6.1
Edge effect
When a load is applied flush with the edge, only three strips can develop. This situation is shown in Figure 6a, in which one can see that only two quadrants are activated. The free edge itself does not contribute to the loadcarrying capacity of the strips, so that for the xdirection strips, the critical shear can only be reached on one side of the strips. For the ydirection strip, both sides of the strip have an interface with a quadrant, so that the critical shear capacity is studied on both sides. When a load is applied close to the free edge, as shown in Figure 6b, four strips can develop. In this case, the socalled “edge effect” can take place. If the length of the ydirection strip between the load and the free edge ledge is smaller than the loaded length lw of the strip, the full capacity of the strip cannot develop. The strip can then carry load only over the length ledge. In this case, the value of ledge will replace lw in the determination of the capacity of the strip.
6.2
Effect of torsion
For loads close to the support and for asymmetric loading conditions, the effect of torsion is important and results in a reduction of the available capacity of the strips. In the model, this effect is taken into account by a factor β on the applied distributed load on the strips. To derive an expression for β, a number of linear finite element models were used (Valdivieso et al., 2016), in which the ratio of the torsional moments mxy to the bending moments mx and my were studied. This ratio was linked to the geometry of the position of the load, and resulted in the following expression: b = 0.8
b =2
a b 1 a br for 0 £ £ 2.5 and 0 £ r £ dx b 2 dx b
a br > 2.5 and 0 £ br £ 1 for dx b b 2
(19)
(20)
with br
the distance from the center of the load to the free edge (ledge + lload/2);
a
the centertocenter distance between the load and the support.
The reduction factor β is then applied to the distributed load in the quadrants between the load and the free edge, as shown in Figure 7. In Figure 7, w is used for the load to show the general idea. The distribution in the x and ydirection is as shown in Figure 3.
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Figure 6: Loads close to the edge: (a) load flush with the free edge; (b) load close to the free edge
Figure 7: Effect of torsion on capacity of strips: (a) sketch of strips; (b) loading on ydirection strip between load and free edge.
6.3
Additional loading considerations
The Extended Strip Model can also be applied for slabs with loads reinforced with plain bars, where the capacity wACI,x and wACI,y from Eqs. (14) and (15) is reduced with a factor 0.7 to take into account the reduced bond between the plain bars and the concrete as compared to deformed bars. For oneway slabs not supported over their entire width but supported by discrete bearings, the compression strut that develops between the load and the support will have a smaller width for anchorage. As a result, the factor from Eq. (16) needs to be replaced with a factor resulting in a smaller increase in capacity of the strip between the load and the support. Further details about the application of the Extended Strip Model to slabs with plain bars and slab supported by bearings are given elsewhere (Lantsoght, 2016b; Lantsoght et al., (in review)). A stepbystep description on how to analyze a slab subjected to a concentrated load with the Extended Strip Model is also given elsewhere, as well as a discussion of the advantages and limitations of the presented model (Lantsoght et al., (in review)). In this paper, the application of this stepbystep method will be shown based on the case study of the Ruytenschildt Bridge, which was tested in its second span to failure.
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7
Comparison between experiments and the Extended Strip Model
In this section, the maximum load resulting from the Extended Strip Model will be compared with the maximum concentrated load obtained in the slab shear experiments carried out in the Stevin II Laboratory of Delft University of Technology. The results of the first series of experiments (Lantsoght et al., 2013b; Lantsoght et al., 2015b), on slabs S1 to S18, are used for this comparison. In these experiments, a reinforced concrete slab, representing a halfscale slab bridge, was loaded to failure by a concentrated load close to the support. The position of the load along the width was varied, as well as the distance between the load and the support and the size of the loading plate. Slabs of normal strength and high strength concrete were tested. The amount of transverse flexural reinforcement was varied. Slabs on discrete elastomeric bearings and slabs reinforced with plain bars were tested as well. For the comparison between the tested and predicted values, the mean values of the material properties are used. The comparison shows that the average ratio of the tested to predicted values, Pexp/PESM (with Pexp the maximum value of the concentrated load in the experiment and PESM the predicted capacity based on the Extended Strip Model) equals Pexp/PSM = 1.61 with a standard deviation of 0.21 and a coefficient of variation of 13.3%. This obtained coefficient of variation is an excellent result for a shear problem. Moreover, in the series of experiments that are used for this analysis, a multitude of parameters are varied, to represent the different cases of bridge slabs in practice as close as possible. The results are shown as well in Figure 8. This graphical representation shows that the results follow a line parallel to the 45° line, which means that the performance of the Extended Strip Model over the entire range of capacities is uniform. The histogram of the tested to predicted results is given in Figure 9. The results of the histogram show that the 5% lower bound (characteristic value) of Pexp/PESM equals 1.225, which is larger than 1. Therefore, the method is suitable for design.
Figure 8:
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Comparison between the maximum value of the concentrated load in the experiment Pexp and the prediction with the Extended Strip Model PESM. Conversion: 1 kN = 0.225 kip.
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Figure 9:
Histogram of tested to predicted results for slabs S1 – S18.
8
Case study: Ruytenschildt Bridge span 2
8.1
Description of bridge geometry
The Ruytenschildt Bridge is a reinforced concrete slab bridge of five spans, with a skew angle of 18°. Since the bridge had to be replaced for functional reasons, it was possible to carry out a field test on this bridge. The goal was to study load testing procedures by carrying out a load test on the Ruytenschildt Bridge, as well as to learn more about the capacity and behavior of reinforced concrete slab bridges. For the latter reason, the bridge was tested to failure after carrying out the load testing cycles. An overview of the bridge and the reinforcement is given in Figure 10. Staged demolition was used, and in the first stage a saw cut was made, so that the width of the tested part of the bridge was 7.365 m (24.16 ft). Testing of the Ruytenschildt Bridge took place in spans 1 and 2. Since not enough load was available to reach failure in span 1, this paper will focus solely on the calculation of the capacity with the Extended Strip Model for span 2, where large flexural distress was achieved together with a significant settlement of the pier at support 2 (Lantsoght, 2015). In span 2, the value of dx is 499 mm (19.65 in) (effective depth to the longitudinal reinforcement) and dy = 483 mm (19.02 in) (effective depth to the transverse reinforcement). The average value is d = 491 mm (19.33 in).
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Figure 10: Ruytenschildt Bridge: (a) overview of spans; (b) reinforcement (of one half of the slab; the bridge is symmetric). Dimensions in [cm], elevations in [m], bar diameters in [mm] and bar spacing in [cm]. Conversion: 1 cm = 0.4 in; 1 m = 3.3 ft.
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8.2
Material parameters
Material testing was carried out on cores drilled from the bridge slab as well as on samples of the plain reinforcement bars used in the bridge. The cylinder concrete compressive strength was found to be 52 MPa (7.542 ksi). Since an old Dutch type of plain reinforcement (known as QR24) was used, extra attention was paid to testing its properties. For the ϕ 12 mm bars (0.47 in diameter, roughly a #4 bar), the yield strength was found to be fy = 352 MPa (51 ksi) and the ultimate strength fu = 435 MPa (63 ksi). For the ϕ 22 mm bars (diameter of ϕ = 0.87 in, roughly a #7 bar), the yield strength was found to be fy = 309 MPa (45 ksi) and the ultimate strength fu = 360 MPa (52 ksi).
8.3
Capacity according to Extended Strip Model
In span 2, the loads are applied close to the second support. These loads are four concentrated loads with a wheel print of 400 mm × 400 mm (15.7 in × 15.7 in), a transverse spacing between the centers of the wheels of 2 m (6.56 ft) and an axle spacing of 1.2 m (3.94 ft) (i.e. the geometry of the design truck of Load Model 1 of NENEN 19912:2003 (CEN, 2003)). The edge distance is 600 mm (23.62 in). The face of the first axle is applied at a distance of 2.5dx = 1.25 m (4.10 ft). These dimensions and the strips that will be analyzed are shown in Figure 11. As can be seen in this figure, the four wheel prints are combined into one large rectangle that serves as the departure point of the four radial strips. The second support is a mid support in a continuous bridge. Therefore, both the hogging and sagging moment reinforcement need to be taken into account. The capacity according to the Extended Strip Model is determined based on the procedures and theory outlined earlier in this paper. The capacity is calculated for each strip separately.
Figure 11: Geometry of the load and resulting strips. Conversion: 1 mm = 0.04 in: 1 m = 3.3 ft.
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8.4
Strip between load and support
First, the strip between the load and the support is analyzed. The capacity of this strip, Psup, is determined as: Psup =
2d x 2 (1 + b ) M s , x wACI , x av
Since this distance is larger than 2dx, the fraction before the square root takes the value of 1. The factor β is determined as (see Eq. (20)):
b =2
br 600mm =2 = 0.163 b 7365mm
The expression for Psup then simplifies to: Psup = 2.326 M s , x wACI , x
The shear stress at the interface between the slab quadrant and the strip is determined according to Eq. (14): 1
æ 100mm ö 3 é kN ù ' wACI , x ê = 0.166 d mm f MPa [ ] [ ] çç ÷÷ y c ú ëmû è d [ mm] ø
Since here experimental results are compared, the average concrete compressive strength of 52 MPa (7.542 ksi) is used, resulting in wACI,x = 340 kN/m (23.3 kip/ft). As can be seen in Figure 10b, the top reinforcement in the xdirection consists of four layers of ϕ 22mm bars (diameter of ϕ = 0.87 in, roughly a #7 bar) with a spacing of 27 cm (10.63 in). Above the support, the value of As,x,top is thus over 1 m (3.3 ft): As , x ,top = 4p
100cm 2 (11mm ) = 5632mm2 = 8.73in 2 27cm
The bottom reinforcement in the xdirection of the span can be seen from Figure 10b as two layers of ϕ 22mm bars (diameter of ϕ = 0.87 in, roughly a #7 bar) with a spacing of 27 cm (10.63 in) and one layer of ϕ 16 mm bars (diameter of ϕ = 0.63 in, roughly a #5 bar) with a spacing of 27 cm (10.63 in). As a result, As,x,bot equals for 1 m or 3.3 ft: As , x ,bot = 2p
100cm 100cm 2 2 (11mm ) + p (8mm ) = 3560mm2 = 5.52in 2 27cm 27cm
For the capacity of the strips in the xdirection, the reinforcement right above the support and the reinforcement in the span are now taken into account. The moment capacity equals:
M s , x = M sag , x + lmoment M hog , x with the value of λmoment based on the moments that occur in span 2 at the support and in the span when the selfweight and two axles of 1995.5 kN (448.6 kip, the maximum load in the experiment) are applied: lmoment =
206
M sup M span
=
2005kNm = 0.97 2065kNm
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The values of Msup and Mspan are taken from a simple beam model, subjected to the selfweight and the two loading axles. The values of Msag,x and Mhog,x have to be calculated based on the reinforcement given above. First, the value of Msag,x is determined by using As,x,bot = 3560mm2 (5.52 in2) and fy = 309 MPa (44.8 ksi): a=
As , x ,bot f y 0.85 f c'b
=
3560mm2 ´ 309MPa = 24.9mm = 0.98in 0.85 ´ 52MPa ´1000mm
aö 28.9mm ö æ æ M sag , x = As , x ,bot f y ç d x  ÷ = 3560mm 2 ´ 309MPa ç 499mm ÷ = 535kNm = 395kft 2ø 2 ø è è
This bending moment capacity corresponds to a strip with a width of 1 m (3.3 ft). For a strip of 2.4 m (7.87 ft) width, the capacity of the actual strip becomes Msag,x = 1285 kNm (948 kft). Next, the hogging moment capacity Mhog,x is determined with As,x.top = 5632 mm2 (8.73 in2) and fy = 309 MPa (44.8 ksi): a=
As , x ,top f y 0.85 f c'b
=
5632mm 2 ´ 309MPa = 39.4mm = 1.55in 0.85 ´ 52MPa ´1000mm
aö 39.4mm ö æ æ M hog , x = As , x ,top f y ç d x  ÷ = 5632mm 2 ´ 309MPa ç 499mm ÷ = 834kNm = 615kft 2ø 2 ø è è
This bending moment capacity is found for a strip with a width of 1 m (3.3 ft). For the actual strip of 2.4 m (7.87 ft) width, the capacity equals Mhog,x = 2002 kNm (1477 kft). M s , x = M sag , x + lmoment M hog , x = 1285kNm + 0.97 ´ 2002 = 3227kNm = 2380kft
The capacity of the strip is then calculated as: Psup = 2.326M s , x wACI , x = 2.326 ´ 3227kNm ´ 340
kN = 1598kN = 359kip m
Strip between load and the far support – For the capacity of the strip in the xdirection, only Mneg,x (sagging moment capacity) is taken into account, so that the capacity is easily determined as: Px = 2.326M sag , x qACI , x = 2.326 ´1285kNm ´ 340
kN = 1008kN = 227kip m
Strip in ydirection between load and span – Next, the capacity is sought of the strip in the ydirection:
Py = 2M s , y
L ( wACI , y  vDL ) L  aM
First, the value of the bending moment capacity Ms,y is determined, taking into account the hogging and sagging reinforcement:
M s , y = M sag , y + lmoment M hog , y
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In the reinforcement drawing, Figure 10b, it can be seen that in the ydirection of the second span, the top and bottom reinforcement are the same: ϕ = 12 mm (0.47 in diameter, roughly a #4 bar) at 20 cm (7.9 in) centertocenter, or, in other words, that per 1 m (3.3 ft) of width 5 bars of ϕ 12 mm (0.47 in diameter, roughly a #4 bar) are used. The reinforcement in 1 m (3.3 ft) width equals: As , y ,bot = As , y ,top = As , y = 5p ( 6mm ) = 565mm 2 = 0.876in 2 2
The bending moment capacity is again determined with Whitney’s stress block diagram: a=
As , y f y 0.85 f c'b
=
565mm2 ´ 352MPa = 4.5mm = 0.18in 0.85 ´ 52MPa ´1000mm
aö 4.5mm ö æ æ M s , y , hog = M s , y , sag = As , y f y ç d y  ÷ = 565mm 2 ´ 352MPa ç 483mm ÷ = 95.6kNm = 70.5kft 2ø 2 ø è è
This capacity is again determined for a strip of 1 m (3.3 ft) width. For the strip width of 1.6 m (5.25 ft), the capacity is 153 kNm (113 kft). The total moment capacity in the ydirection is then: M s, y = M sag , y + lmoment M hog , y = (1 + 0.97)153kNm = 301kNm = 222kft
The value of wACI,y is: 1
wACI , y = 0.166 ´ d x
1
kN kip æ 100mm ö 4 æ 100mm ö 3 f ç = 24.1 ÷ = 0.166 ´ 499mm ´ 52MPa ´ ç ÷ = 351 m ft è d ø è 491mm ø ' c
Next, the distance between the points of contraflexure is determined, as well as the effect of the dead load. Using a beam model, the shear caused by the dead load can be found at the position of the center of the loading system, see Figure 12. A shear force of 155.3 kN (34.9 kip) is found. This translates into a stress that needs to be subtracted from the strip of vDL =
155.3kN kN kip = 21.09 = 1.45 7.365m m ft
Figure 12: Shear diagram caused by the distributed load (dead load). Units: [kN]. 1 kN = 0.225 kip.
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Figure 13: Moment diagram caused by the distributed load and concentrated loads of the tandem for the second span. Units: [kNm] 1 kNm = 0.74 kft.
The length between the points of contraflexure is found based on a beam diagram with the dead load and the concentrated axle loads as shown in Figure 13. The length between the points of contraflexure is found to be 6.22 m (20.41 ft). The distance aM to the center of the tandem is 1.45 m (4.76 ft). All information necessary to determine the capacity of the strip is now available: Py = 2M s , y
L 6.22m kN kN ö æ wACI , y  vDL ) = 2 ´ 301kNm  21.09 ( ç 351 ÷ = 509kN = 114kip L  aM 6.22m  1.45m è m mø
Strip in ydirection between load and free edge – Finally, the capacity of a strip in the ydirection between the load and the free edge is determined:
Pedge
ì L M s , y ( wACI , y  vDL ) if lw < ledge ï 2b d L  aM ï =í L ïb w  vDL ) ledge if ledge £ lw ï d L  a ( ACI , y M î
with lw =
2M s , y
bd
L ( wACI , y  vDL ) L  aM
The loaded length is then found to be: lw =
2M y , s 2 ´ 301kNm = = 2.93m = 9.61ft L 6.22m kN kN ö æ bd wACI , y  vDL ) 0.163 351 21.09 ( ç ÷ L  aM 6.22m  1.45m è m m ø
The loaded length is thus larger than the distance between the load and the free edge, ledge = 0.6 m (1.97 ft), so that the edge effect needs to be considered. Pedge = bd
L 6.22m kN kN ö æ wACI , y  vDL ) ledge = 0.163 ( ç 351  21.09 ÷ 0.6m = 42kN = 9.4kip L  aM 6.22m  1.45m è m mø
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8.5
Comparison with maximum load in test
The capacity of the four strips is now known, so that the sum of these capacities can be made to determine the maximum load that is expected in the experiment: Psup + Px + Py + Pedge = 1598kN + 1008kN + 509kN + 42kN = 3157kN = 710kip
The capacity in the experiment was 3991 kN (879 kip), or in other words, the ratio of the tested to predicted capacity is 1.26. The Extended Strip Model thus gives a good and conservative estimate for the capacity of the tested second span.
9
Summary and conclusions
For the practical application of reinforced concrete solid slab bridges subjected to concentrated live loads, a combination of oneway shear, twoway shear and twoway flexure needs to be considered. Since slabs have a high level of redundancy, plasticitybased methods are very suitable. In the current design and assessment practice, oneway shear provisions consider slabs as wide beams and twoway slabs as slabcolumn connections that are studied based on the shear stress on a punching perimeter around the load. The proposed model, the Extended Strip Model, aims to bridge the gap between these two approaches. Experiments showed that the failure mode is indeed a combination of oneway shear, twoway shear, flexure and torsion. Currently existing lower bound plasticitybased methods are Hillerborg’s Strip Method for the design of slabs in flexure, and Alexander’s Strip Model for design and assessment of the concentric punching shear capacity. The latter method served as a starting point for the development of the Extended Strip Model, suitable for bridge deck slabs under concentrated loads. The proposed Extended Strip Model contains the following elements: • The different properties in the longitudinal and transverse direction are taken into account. • The implications of the static equilibrium are taken into account. This consideration is based on the observation that for the ydirection strips, one side of the strip will achieve failure before the other side. • The result of the model is a maximum concentrated load. Therefore, the effect of the selfweight is subtracted from the capacity of the strips in the ydirection. • The size effect is added to the determination of the capacity of the interface between the strips and quadrants. • The method takes into account the enhanced capacity for loads close to supports that results from the direct compression strut that forms between the load and the support. • The effect of continuity at the support is taken into account. • For loads close to the free edge, the edge effect can occur. The edge effect occurs when the actual length of the strip is smaller than the loaded length.
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• When large torsional moments occur, the capacity is reduced based on a factor resulting from the analyses of several cases with linear elastic finite element models. When the proposed method was compared with experiments on slabs under concentrated loads, it was found that the comparison is excellent. The coefficient of variation of the tested to predicted results is 13.3%, which is good considering the large number of different parameters that were tested and analyzed. Furthermore, the Extended Strip Model was applied to predict the capacity of the span that was tested to failure on the Ruytenschildt Bridge. The occurring mode was flexural failure, and again a very good prediction was achieved with the proposed model.
10
Acknowledgements
The authors wish to express their gratitude and sincere appreciation to the Dutch Ministry of Infrastructure and the Environment (Rijkswaterstaat) for financing the experimental part of this research work. Finalizing the theoretical work on the Extended Strip Model was possible with the support of the program of Chancellor Grants of Universidad San Francisco de Quito USFQ (Ecuador). This funding is gratefully acknowledged.
11
References
ACI Committee 318 (2014) Building code requirements for structural concrete (ACI 31814) and commentary, American Concrete Institute; Farmington Hills, MI, 503 pp. Afhami, S. (1997) "Strip model for capacity of flat platecolumn connections," Ph.D. thesis, University of Alberta, 310 pp. Afhami, S., Alexander, S. D. B. and Simmonds, S. H. (1998) "Strip model for capacity of slabcolumn connections," Dept. of Civil Engineering, University of Alberta; Edmonton, 231 pp. Alexander, S., 1999a, "Strip Design for Punching Shear," ACI SP183: Design of TwoWay Slabs, pp. 161180. Alexander, S., 1999b, "Strip Method for Flexural Design of TwoWay Slabs," ACI SP183: Design of TwoWay Slabs, pp. 93117. Alexander, S. D. B. (1990) "Bond model for strength of slabcolumn joints," Ph.D. thesis, University of Alberta, 1990., 228 pp. Alexander, S. D. B. and Simmonds, S. H. (1992) "Bond Model for Concentric Punching Shear," ACI Structural Journal, V. 89, No. 3, pp. 325334. Alexander, S. D. B. (2016) "Shear and moment transfer at columnslab connections," in fib Bulletin 81, ACIfib International Symposium: Punching shear of structural concrete slabs, Fédération internationale du béton, Lausanne, Switzerland, pp 122. Bažant, Z. P. and Kim, J. K. (1984) "Size Effect in Shear Failure of Longitudinally Reinforced Beams," Journal of the American Concrete Institute, V. 81, No. 5, pp. 456468. 211
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Braestrup, M. W. (2009) "Structural concrete beam shear – still a riddle?," SP265 Thomas T.C. Hsu Symposium on Shear and Torsion in Concrete Structures, New Orleans, LA, USA, pp. 327344. CEN (2003) Eurocode 1: Actions on structures  Part 2: Traffic loads on bridges, NENEN 19912:2003, Comité Européen de Normalisation, Brussels, Belgium, 168 pp. CEN (2005) Eurocode 2: Design of Concrete Structures  Part 11 General Rules and Rules for Buildings. NENEN 199211:2005, Comité Européen de Normalisation, Brussels, Belgium, 229 pp. Collins, M. P. and Kuchma, D. (1999) "How safe are our large, lightly reinforced concrete beams, slabs, and footings?," ACI Structural Journal, V. 96, No. 4, JulAug, pp. 482490. Cho, S. H. (2003) "Shear strengthprediction by modified plasticity theory for short beams," ACI Structural Journal, V. 100, No. 1, pp. 105112. Grassl, P., Xenos, D., Nyström, U., Rempling, R. and Gylltof, K. (2013) "CDPM2: A damageplasticity approach to modelling the failure of concrete," International Journal of Solids and Structures, V. 50, No. 24, pp. 38053816. Grebovic, R. S. and Radovanovic, Z. (2015) "Shear Strength of High Strength Concrete Beams Loaded Close to the Support," Procedia Engineering, V. 117, pp. 492499. He, Z.Q., Liu, Z. and Ma, Z. J. (2012) "Investigation of LoadTransfer Mechanisms in Deep Beams and Corbels," ACI Structural Journal, V. 109, No. 4, JulAug, pp. 467476. Hillerborg, A. (1975) "Strip Method of Design," Viewpoint Publications, Cement and Concrete Association; Wexham Springs, Slough, England, 256 pp. Hillerborg, A. (1982) "The advanced strip method  a simple design tool," Magazine of Concrete Research, V. 34, No. 121, pp. 175181. Hillerborg, A. (1996) Strip method design handbook, E & FN Spon; London; 302 pp. Ibell, T. J., Morley, C. T. and Middleton, C. R. (1997) "A plasticity approach to the assessment of shear in concrete beamandslab bridges," The Structural Engineer, V. 75, No. 19, pp. 331338. Kani, G. N. J. (1967) "How Safe Are Our Large Reinforced Concrete Beams," ACI Journal Proceedings, V. 64, No. 3, pp. 128141. Kuang, J. S. and Morley, C. T. (1993) "A Plasticity Model for Punching Shear of Laterally Restrained Slabs with Compressive Membrane Action," International Journal of Mechanical Sciences, V. 35, No. 5, pp. 371385. Lantsoght, E. O. L., van der Veen, C., de Boer, A. and Walraven, J. C. (2013a) "Recommendations for the Shear Assessment of Reinforced Concrete Slab Bridges from Experiments " Structural Engineering International, V. 23, No. 4, pp. 418426. Lantsoght, E. O. L., van der Veen, C. and Walraven, J. C. (2013b) "Shear in Oneway Slabs under a Concentrated Load close to the support," ACI Structural Journal, V. 110, No. 2, pp. 275284. Lantsoght, E. O. L., van der Veen, C., De Boer, A. and Walraven, J. (2014) "Influence of Width on Shear Capacity of Reinforced Concrete Members," ACI Structural Journal, V. 111, No. 6, pp. 14411450.
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Lantsoght, E. O. L. (2015) Ruytenschildt Bridge: analysis of test results at the ultimate limit state (in Dutch), Stevin Report Nr. 25.51505, Delft University of Technology, Delft, The Netherlands, 89 pp. Lantsoght, E. O. L., van der Veen, C., de Boer, A. and Walraven, J. (2015a) "Oneway slabs subjected to combination of loads failing in shear," ACI Structural Journal, V. 112, No. 4, pp. 417426. Lantsoght, E. O. L., van der Veen, C., Walraven, J. and de Boer, A. (2015b) "Experimental investigation on shear capacity of reinforced concrete slabs with plain bars and slabs on elastomeric bearings," Engineering Structures, V. 103, pp. 114. Lantsoght, E. O. L., van der Veen, C., Walraven, J. C. and de Boer, A. (2015c) "Transition from oneway to twoway shear in slabs under concentrated loads," Magazine of Concrete Research, V. 67, No. 17, pp. 909922. Lantsoght, E. O. L. (2016a) Extended Strip Model: Effect of selfweight and size effect, Stevin Report 25.51603, Delft University of Technology, 42 pp. Lantsoght, E. O. L. (2016b) Modified Bond Model: Improvements and deeper theoretical analysis, Stevin Report nr. 25.51602, Delft University of Technology, 87 pp. Lantsoght, E. O. L., van der Veen, C., de Boer, A. and Alexander, S. D. B., (2017), "Extended Strip Model for Slabs under Concentrated Loads." ACI Structural Journal, Vol. 114, No. 2, pp. 565574. Lantsoght, E. O. L., van der Veen, C., de Boer, A. and Walraven, J., 2015d, "Transverse Load Redistribution and Effective Shear Width in Reinforced Concrete Slabs," Heron, V. 60, No. 3, pp 145180. Mattock, A. H. (2012) "Strutandtie Models for DappedEnd beams," Concrete International, V. 34, No. 2, pp. 3540. Morrow, J. and Viest, I. M. (1957) "Shear Strength of Reinforced Concrete Frame Members Without Web Reinforcement," ACI Journal Proceedings, V. 53, No. 3, pp. 833869. Nielsen, M. P. and Hoang, L. C. (2011) "Limit analysis and concrete plasticity," 3rd ed. CRC; Boca Raton, FL, 816 pp. Olonisakin, A. A. and Alexander, S. D. B. (1999) "Mechanism of shear transfer in a reinforced concrete beam," Canadian Journal of Civil Engineering, V. 26, No. 6, pp. 810817. Ospina, C. E., Alexander, S. D. B. and Cheng, J. J. R. (2003) "Punching of TwoWay Concrete Slabs with FiberReinforced Polymer Reinforcing Bars or Grids," ACI Structural Journal, V. 100, No. 5, pp. 589598. Park, R. and Gamble, W. L. (2000) "Reinforced concrete slabs," 2nd ed. Wiley; New York, 716 pp. Regan, P. E. (1982) "Shear Resistance of Concrete Slabs at Concentrated Loads close to Supports," Polytechnic of Central London, London, United Kingdom, 24 pp. Reineck, K. H. (2010) "Strutandtie models utilizing concrete tension fields," Proceedings of the Third fib International Congress, Washington DC, USA, pp. 112. Reißen, K. and Hegger, J. (2013) "Experimental investigations on the effective width for shear of single span bridge deck slabs," Beton und Stahlbetonbau, V. 108, No. 2, pp. 96103. (in German)
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Rombach, G. and Latte, S. (2009) "Shear Resistance of Bridge Decks without Transverse Reinforcement," Beton und Stahlbetonbau, V. 104, No. 10, pp. 642656. (in German) Salim, W. and Sebastian, W. M. (2002) "Plasticity model for predicting punching shear strengths of reinforced concrete slabs," ACI Structural Journal, V. 99, No. 6, pp. 827835. Schlaich, J., Schafer, K. and Jennewein, M. (1987) "Toward a Consistent Design of Structural Concrete," Journal Prestressed Concrete Institute, V. 32, No. 3, pp. 74150. Valdivieso, D., Lantsoght, E. O. L. and Sanchez, T. A. (2016) "Effect of torsion on shear capacity of slabs," SEMC 2016, Cape Town, South Africa, pp. 6. Windisch, A. (1988) "The Charactersitc Free Body Model  A means of dimensioning the special regions of reinforced concrete structures," Beton und Stahlbetonbau, V. 83, pp. 251274 (in German). Windisch, A. (2000) "Towards a Consistent Design Model for Punching Shear Capacity," International Workshop on Punching Shear Capacity of RC Slabs  Proceedings, Silfwerbrand, J. and Hassanzadeh, G., eds., Stockholm, Sweden, pp. 293301.
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Punching and postpunching response of slabs Denis Mitchell, William D. Cook McGill University, Montreal, Canada
Abstract Factors influencing the punching shear resistance of twoway slabs are presented. Factors discussed include: construction errors; effects during construction; earthquake effects; deterioration of parking garage slabs; and design with older, deficient codes of practice. Experiments on the size effect are discussed and the provisions of the CSA Standard for the Design of Concrete Structures for the treatment of the size effect are presented. The provisions of the CSA Standard requiring structural integrity reinforcement in order to provide a secondary defense mechanism capable of preventing progressive collapse are explained. The results from experimental and analytical studies on the postpunching response of twoway slabs are described. The effects of deterioration of parking garage slabs subjected to chloride contamination and the provisions of the CSA Standard on Parking Structures to improve durability are discussed. The effects of delamination due to corrosion of the reinforcement in older parking structures are discussed and experimental studies on the effects of simulated delamination are presented. The progressive collapse in 2008 of an older parking structure in StLaurent, Quebec, is described to illustrate some key factors influencing such a collapse.
Keywords Twoway slabs, punching shear, postpunching response, progressive collapse, structural integrity, deterioration, corrosion, delamination, codes
1
Introduction
This paper summarizes the factors influencing punching shear failures that can lead to progressive collapse of a structure. Parking garage structures in colder climates are particularly susceptible to premature punching shear failures due to concrete deterioration and corrosion of the reinforcing bars due to the use of deicing chemicals. Experimental results illustrating the effects of common deficiencies in older twoway slab structures are presented. The use of structural integrity reinforcement to improve the postpunching shear response of twoway slabs is discussed. An investigation of the progressive collapse of a twoway slab structure suffering from deterioration is described in some detail to illustrate some key behavioral features.
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2
Factors influencing punching shear failures
There are a large number of factors that influence the punching shear strength of twoway slab systems. Examples of some of these factors are discussed below.
2.1
Construction errors
A common problem with existing twoway slab structures is misplacement of the top reinforcement due to inadequate support of the reinforcement during concrete casting. Fig. 1(a) shows the reduction of the effective depth in the slabcolumn connection region due to misplacement of the top mat of reinforcement that can result in significant reductions in both flexural and punching shear resistances. Another defect arises when the column formwork protrudes into the slab resulting in a reduction in the effective slab depth resisting both punching shear and flexure (Aoude et al., 2013). Fiberboard forms used for circular columns are particularly susceptible to this defect if insufficient support is offered by the slab formwork at the column interface during casting (see Fig. 1(b)).
(a) Misplaced top reinforcement Figure 1:
(b) Fiberboard form protruding into bottom surface of slab
Reduction in effective depth due to construction errors.
Figure 2 shows the reduction in punching shear resistance as the effective depth, dav, is reduced (Lee et al., 1979). This figure also shows how the maximum crack width varies as the shear increases. Both punching shear specimens had a slab thickness of 150 mm (5.9 in.) and were supported by 225 mm (8.9 in.) square columns. The top reinforcement consisted of #5 bars at 100 mm (3.9 in.) in each direction. The concrete compressive strength for specimens PS1 and PS2 were 25.9 MPa (3760 psi) and 35.1 MPa (5090 psi), respectively and the yield stress of the reinforcement was 337 MPa (48.9 ksi). Specimen PS1 had a concrete cover of 20 mm (0.79 in.) resulting in an average effective depth, dav, of 114 mm (4.5 in.). Specimen PS2 simulated the misplacement of the top reinforcement, with a cover of 65 mm (2.56 in.) and dav of 69 mm (2.72 in.). Due to the misplacement, the punching shear resistance dropped from 393 kN (88.3 kips) to 241 kN (54.3 kips), a reduction of 39% (see Fig. 2(a)). Figure 2(b) shows the significant increase in crack widths due to this misplacement. Figure 3 shows the change in cracking pattern as the top reinforcing mat is misplaced, resulting from an increase in the concrete cover from 20 mm (0.79 in.) to 65 mm (2.56 in.). For the case of small concrete covers, the reinforcing bars close to the surface act as crack attractors, resulting in an orthogonal cracking pattern with many cracks of small widths. The misplacement results in a larger cover and a reduced effective depth resulting in a radial crack pattern with fewer, wider cracks. Variations in the overall thickness of the slab during construction will typically affect the top concrete cover dimension that could be an issue for durability. 216
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Figure 2:
Influence of misplacement on punching shear strength and crack width (test results from Lee et al., 1979).
(a) Specimen PS1 with 20 mm (0.75 in.) cover Figure 3:
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(b) Specimen PS2 with 65 mm (2.56 in.) cover
Influence of misplacement of top reinforcement on cracking pattern.
Failures during construction
The 1973 collapse of the Skyline Plaza apartment building in Virginia resulted in 14 deaths (Leyendecker and Fattal, 1973). The apartment building was under construction and it is believed that a punching shear failure occurred on the 23rd floor after premature removal of forms supporting this floor and after placing some concrete on the 24th floor. The slabs were constructed with lightweight concrete and the column aspect ratio (long side to short side) was 4 to 1 in the region where the collapse initiated. The collapse spread horizontally over an interior portion of the building and then spread vertically over the entire height of the building. The impact from the debris and a construction crane falling on the adjacent posttensioned parking structure resulted in progressive collapse of the parking structure as well. The 1981 collapse of the Harbour Cay Condominium structure in Cocoa Beach, Florida resulted in 11 deaths (Lew et al., 1982). The 5story structure was under construction at the time of collapse during the placement of concrete for the roof slab. Lew et al. (1982) concluded that the causes of the initial shear failure that triggered the progressive collapse 217
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were inadequate design for punching shear and the fact that the supporting chairs for the top reinforcing steel resulted in a 25 mm reduction in the expected effective depth. Before the collapse a worker reported observing a “spiderweb” shaped cracking pattern on the top surface of the slab near a column. This type of cracking pattern may be a warning sign that the top slab reinforcement might be misplaced (see Fig. 3).
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Failures during earthquakes
Figure 4(a) shows an example of the progressive collapse of a 7story flat plate structure in Mexico City due to punching shear failures that took place during the 1985 Mexico earthquake (Mitchell et al., 1986). The punching shear failures were characterized by the ripping out of the top flexural reinforcement from the slabcolumn connections, with the spread of punching shear failures extending both horizontally and vertically. Figure 4(b) shows punching shear failures of the slabcolumn connections at the roof and upper floor of a department store in Northridge, California (Mitchell et al., 1995). It is evident that no top or bottom slab bars connected the slab to the circular column. This structure suffered a progressive collapse.
(a) Seven story flat plate structure after 1985 Mexico earthquake Figure 4:
2.4
(b) Department store after 1994 Northridge earthquake
Examples of progressive collapses due to punching shear failures during earthquakes.
Failures due to deterioration
The use of deicing chemicals in colder regions of North America has had a major impact on the durability of older parking structures. The deposit of chlorides from vehicles in parking garages, many of which are heated during winter, results in chloride contaminated concrete slabs and ideal conditions for chloride penetration and corrosion of the reinforcement. The commentary to the 1987 CSA Standard S413 on Parking Structures (CSA, 1987) indicated that surveys of unprotected parking structures built in Canada before 1987 showed signs of deterioration, including delamination, five years after construction with major repairs typically needed within ten years of construction. 218
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Figure 5 shows the progressive collapse of an indoor parking garage slab that occurred in 2008 in StLaurent, Quebec. The slab had suffered severe concrete deterioration, corrosion of the reinforcing steel due to exposure to chlorides and delamination just below the top mat of reinforcement.
Figure 5:
Progressive collapse of indoor parking garage in 2008 in StLaurent, Quebec.
Figure 6 shows the loaddeflection responses for punching shear tests with different degrees of delamination (Aoude et al., 2014). The load in this figure has been normalized to the nominal ACI punching shear capacity. The delamination was simulated by placing a plastic sheet at the bottom of the top reinforcing mat during casting. The slabs were all 150 mm thick and contained 1415M (0.63 in. diameter) bars at a spacing of 164 mm (6.5 in.) in one direction and 1615M (0.625 in. dia.) bars at 144 mm (5.7 in.) in the other direction. The concrete cover was 25 mm (1 in.). The 15 M bars had a yield stress of 458 MPa (66.4 ksi). The concrete compressive strengths were 25.4 MPa (3680 psi) for specimens S15, and S15D1 and 32.4 MPa (4700 psi) for specimens S15D2 and S15D3. The 2.3 m by 2.3 m (90.6 by 90.6 in.) slabs were supported by 225 mm (8.9 in.) square stub columns. All of these slabcolumn specimens were without structural integrity reinforcement. As shown in Fig. 6 the region of simulated delamination was varied from zero to a region measuring 1.87 by 1.87 mm (74 by 74 in.). It is apparent that a very small region of delamination has no significant effect on the response. As the region of delamination increases, the punching shear capacity and stiffness of the slabcolumn connection decreases. Figure 7(a) shows specimen S15–D3 after failure. The region of simulated delamination is evident in Fig. 7(b).
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Figure 6:
Load deflection response of slabcolumn connections with simulated delamination (test results from Aoude et al., 2014).
(a) Overall view Figure 7:
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(b) View of delamination plane
Slabcolumn Specimen S15D3 after punching shear failure.
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2.5
Design to earlier codes
A key factor influencing the punching shear strength of older twoway slab structures is the code used for design. For example, in Canada the following significant changes concerning punching shear in the CSA Standard A23.3 were adopted: a) In 1959, the minimum slab thickness for a flat plate without drop panels was taken as the centertocentre span divided by 36. b) In 1973, the minimum slab thickness was increased to the clear span divided by 32.7 for 60 ksi reinforcement (414 MPa). In addition the effects of combined shear and moment transfer were included. c) In 1977, the influence of column rectangularity on the punching shear capacity was included. d) In 1984, structural integrity reinforcement at slabcolumn connections was required. e) In 1994, the minimum slab thickness was increased to the clear span divided by 30 for Grade 400 reinforcement. An addition equation for the punching shear capacity was added to account for the perimeter of the critical punching shear section. f) In 2004, a factor accounting for the size effect on the punching shear capacity was added for effective depths greater than 300 mm (12 in.). Another very significant change occurred in 1987, with the introduction of CSA Standard S413 on Parking Structures that provided requirements for improving durability. It is noted that all of these changes except for the provisions of structural integrity reinforcement and the size effect as well as the durability provisions followed developments in the ACI Code.
2.6
Size effect on punching shear
Figure 8 summarizes the results from a series of slabcolumn tests to investigate the size effect on the punching shear strength (Li, 2000; Mitchell et al., 2005). The overall thickness of the slabs varied from 135 to 550 mm (5.3 to 21.7 in.), resulting in effective depths varying from 100 to 500 mm (3.9 to 19.7 in.). Each slab was tested upside down by loading the centrally located highstrength concrete (80 MPa (8700 psi)) square column and reacting against neoprene pads on HSS sections sitting on the strong floor around the slab perimeter. Figure 8(a) shows the testing of the specimen with the largest d. All of the slabs were cast from the same batch of readymix concrete and the average concrete compressive strength of the slab concrete was 39.4 MPa (5700 psi). It is noted that the flexural reinforcement ratio was constant at 0.76% for the three thickest slabs, with ρ varying from 0.83% to 0.98% for the thinner slabs. The 2004 CSA Standard introduced a size effect factor of 1300/(1000+d), where d is in mm, for slabs with effective depths greater than 300 mm (12 in.). Figure 8(b) shows the punching shear strengths of the six slabs compared with the constant nominal shear stress of 0.33 f c¢ in MPa units, predicted using the ACI Code expression. Also shown are the predictions, including the size effect factor using the nominal resistance of the CSA Standard, which fit the trend of the experiments better than the ACI Code for effective depths greater than 300 mm.
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(a) Test of 550 mm thick slab Figure 8:
2.7
(b) Variation of shear strength with effective depth
Size effect on punching shear capacity.
Top bar cutoff in thick slabs
The 2014 ACI Commentary (ACI 318R, 2014) cautions designers about the need to extend the minimum lengths required for top bars in the column strip region for slabs with ! n / h ratios less than about 15. Premature failure may occur in older thick slabs because the punching shear cracks may not intercept the top reinforcement (Mirzaie, 2010; Fernandez Ruiz et al., 2013).
3
Structural integrity reinforcement
3.1
Brittleness of punching shear failures without structural integrity reinforcement:
Figure 9(a) illustrates the brittle punching shear failure mechanism for a slabcolumn connection that does not contain structural integrity reinforcement. After a brittle punching shear failure occurs the top reinforcement rips out of the top slab surface causing complete and sudden loss of support of the slab and often leading to progressive collapse as the slabcolumn connections become overloaded in shear and moment at adjacent columns (Hawkins and Mitchell, 1979). Figure 9(b) illustrates that properly designed and detailed structural integrity reinforcement, consisting of bottom bars that are effectively continuous through the column, enables the slab to hangoff the columns after punching shear failure has occurred. The structural integrity reinforcement provides a secondary defence mechanism and is capable of preventing progressive collapse.
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(a) Punching shear failure without structural integrity reinforcement resulting in ripping out of top reinforcement and sudden loss of slab support
(b) Postpunching resistance provided by bottom structural integrity reinforcement Figure 9:
3.2
Role of structural integrity reinforcement in providing a secondary defence system (adapted from Hawkins and Mitchell, 1979).
Structural integrity provisions of the Canadian Concrete Standard CSA A23.314
Since 1984, CSA A23.3 requires that all twoway slabs without beams contain structural integrity reinforcement (Mitchell and Cook, 1984). The resistance of the structural integrity bars after a punching shear failure is determined assuming that these bars form an angle of 30º from the horizontal, resulting in a resistance equal to the vertical component of the yield force in the structural integrity bars connecting the slab to the column. Hence, the total area of structural integrity reinforcement, considering only the bottom bars that connect the slab to the column, must satisfy the following equation:
å Asb =
2Vse fy
(1)
where å Asb is the sum of the area of bottom reinforcement connecting the slab to the column on all faces of the column, Vse is the shear transmitted to the column due to the unfactored service loads. The structural integrity reinforcement must have at least two bars or two tendons that extend through the column core or column capital region in each span direction. The structural integrity reinforcement is provided by one or more of the following: a) Bottom reinforcement extended such that it is lap spliced over a column or column capital with the bottom reinforcement in adjacent spans using a Class A tension splice; b) Additional bottom bars passing over a column or column capital such that an overlap of 2! d is provided with the bottom reinforcement in adjacent spans;
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c) At discontinuous edges, bottom reinforcement extended and bent, hooked, or otherwise anchored over the supports such that the yield stress can be developed at the face of the column or column capital; d) Continuous tendons draped over columns or column capitals, with a minimum total area of prestressing steel calculated using Eq. (1), but with fy replaced by fpy. Figure 10 illustrates the required detailing of the bottom structural integrity reinforcement (CSA 2014). For an interior slabcolumn connection the bottom bars can be simply extended into the column in both span directions as shown in Fig. 10(a) or additional bottom bars can be added, passing through the column, in the two span directions as shown in Fig. 10(b). At exterior columns two alternative details are shown in Fig. 10(c) and (d). The bottom bars terminating in the column, are anchored by development length or hooks to develop their yield stress at the column faces (see Fig. 10(c) and (d)).
(a) Interior connection with bottom flexural bars extended through column
(b) Interior connection with additional integrity bars to overlap bottom flexural bars Figure 10: Elevation and plan views of structural integrity reinforcement required in CSA A23.314 (CSA 2014), only structural integrity reinforcement shown.
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(c) Exterior connection with bottom flexural bars extended through column
(d) Exterior connection with additional integrity bars to overlap bottom flexural bars Figure 10 (cont’d): Elevation and plan views of structural integrity reinforcement required in CSA A23.314 (CSA 2014), only structural integrity reinforcement shown.
It is noted that the ACI Code (ACI 2014) requires that all bottom bars in the column strip be continuous or spliced with mechanical, fullwelded or Class B tension splices. At least two bottom bars in each direction shall pass within the region bounded by the longitudinal reinforcement in the column and shall be anchored at exterior supports.
3.3
Modelling the complete postpunching response
While Eq. (1) gives a simplified design expression for the postpunching shear resistance it is possible to predict the complete postpunching shear response. Figure 11(a) illustrates the resisting mechanism in a slabcolumn connection after a punching shear failure. As the slab deflects the top bars rip out of the top surface due to breakout failures in the cover concrete. The structural integrity bars, located in the bottom of the slab, have a much greater breakout resistance due to the greater thickness of concrete above the bars. The use of equilibrium, compatibility of deformation and the nonlinear stressstrain relationships for the reinforcing steel enables the complete postpunching behaviour to be predicted (Habibi et al., 2014) including the progressive concrete breakout failures above the top and bottom reinforcement. Fernandez Ruiz et al. (2013) also developed a mechanical model to predict the postpunching shear resistance of slabs. Figure 11(b) shows the test setup used for the postpunching shear experiments of interior slabcolumn connections. Figure 11(c) shows the shear versus deflection response of a 200 mm (7.9 in.) thick slab supported by a 225 mm (8.9 in.) square column (specimen SS). The top and bottom cover was 25 mm (1.0 in.) and the concrete compressive strength was 26 MPa (3770 MPa). The top mat consisted of 1315M bars in one
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direction and 1515M bars in the other direction. These top slab bars were more concentrated in the immediate column region resulting in a spacing of 160 mm (6.3 in.) in one direction and 115 mm (4.5 in.) in the other direction, giving reinforcement ratios of 0.75% and 1.14%, respectively. This amount of reinforcement simulated the concentration of flexural reinforcement required in this region by the CSA Standard (2014). Specimen SS had structural integrity reinforcement in the form indicated in Fig. 10(b) with 215M bars (As of 200 mm2 (0.31 in.2) per bar) passing through the column in each direction and protruding a distance of twice the development length from the column faces. All of the 15M bars had a yield strength, fy of 460 MPa (66.7 ksi)). The bottom flexural reinforcement consisted of 10M bars at a spacing of 240 mm in each direction. These bottom 10M bars had hooks at the exterior edges of the slab test specimen to simulate continuity of this reinforcement.
(a) Model of postpunching resistance
(b) Test setup
(c) Predicted postpunching response of slabcolumn specimen SS Figure 11: Modelling the postpunching behaviour of a slabcolumn connection (test results from Habibi et al., 2014).
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After punching it is assumed that the shear resistance is provided by the vertical component of force in the reinforcing bars. In the detailed model (Habibi et al., 2014), the individual postpunching responses of each layer in the top and bottom mats are predicted (see Fig. 11(c)). After a punching shear failure occurs, the upper bars in the top mat rip out of the top cover, reaching a maximum load at point A. The lower bars in the top mat reach their maximum load capacity at point B. Beyond point B only the structural integrity bars are predicted to resist shear because the two layers of top bars have ripped out of the top surface. Due to the brittle breakout of the layers of the top mat, these bars become ineffective at an early stage in the postpunching response. With increased vertical displacements and progressive damage extending outwards from the column, as shown in Fig. 11(a), the inclination of the structural integrity reinforcement increases and the stresses increase in these bars until they breakout of the damaged slab concrete or suffer a pullout failure if the anchorage of these bars is reduced to the development length (points C and D). The complete postpunching response shown in Fig. 11(c) was obtained by incrementing the slab displacements and determining the sum of the resistances provided by the four layers of reinforcement. Figure 12 shows top bars that have suffered concrete breakout and the inclined structural integrity bars (bottom bars) that are able to provide significant resistance while the slab undergoes considerable displacements. The structural integrity reinforcement offered significant resistance up to a deflection of about 165 mm (6.5 in.) (see Fig. 11(c)).
(a) Top bars ripped out and inclined structural integrity bars
(b) After slab removal showing bottom inclined structural integrity bars
Figure 12: Slabcolumn connection specimen SS after punching shear failure.
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3.4
Experimental verification
Knowing the amount and yield strength of the structural integrity reinforcement, the predicted postpunching shear resistance can be estimated from the CSA Standard expression given in Eq. (1). For slabcolumn specimen SS, described above, the structural integrity reinforcement consisted of 215M bars in each direction passing through the column, resulting in a predicted postpunching shear resistance of: VCSA = 0.5å Asb f y = 0.5(4 ´ 2 ´ 200 ´ 460) = 368 kN
(2)
The postpunching shear resistance, Vtest, obtained from the test was 397 kN and hence this prediction is conservative. It is interesting to note that Hawkins and Mitchell (1979) concluded that “the postpunching capacity should be taken as not greater than 0.5 times the yield strength of the bottom reinforcement continuous through the column”. Figure 13 compares the predicted postpunching shear resistances with test results of slabcolumn connections that contained structural integrity reinforcement. For the postpunching shear resistance the simple design expression given by Eq. (1) was used. The average of the test to predicted strengths is 1.08, with a standard deviation of 0.18 and a coefficient of variation of 17.2%. The predictions agree reasonably well with the test results.
Figure 13: Predicted versus test postpunching shear resistances. Test results from Habibi et al. 2012, Ghannoum (1998), Mirzaei (2010) and Melo and Regan (1998).
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Influence of structural integrity reinforcement on slabcolumn connections having delamination
Although new parking garage slabs designed and constructed in Canada after 1987 should perform better, having the additional protection requirements of the CSA Standard for Parking Structures (CSA, 1987, 2014) it is of interest to examine whether or not the structural integrity reinforcement can offer a secondary defense system after delamination takes place at the level of the top mat of reinforcement. Figure 14 shows the shear versus slab displacement responses of two slabcolumn connections reported by Reilly et al. (2014). Both slabcolumn connections contained structural integrity reinforcement consisting of 215M bars passing through the column in both directions. Specimen SS did not have any delamination whereas companion specimen SSD contained simulated delamination using plastic sheets over an area 1.875 m by 1.875 m (74 in. by 74 in.). The responses shown in Fig. 14 illustrate the 25% drop in the punching shear capacity and the drop in the slab initial stiffness due to the delamination. Specimen SSD has negligible postpunching resistance resulting from the top bars due to the delamination at the level of the top mat, whereas for specimen SS the postpunching contribution from the two layers of top bars is evident up to a deflection of about 75 mm (3in.). The structural integrity reinforcement offers considerable resistance in both specimens, however the additional damage to specimen SSD due to delamination restricts the ability to resist considerable loads at very large displacements.
Figure 14: Influence of delamination on the load versus deflection responses of slabcolumn connections containing structural integrity reinforcement (test results from Reilly et al., 2014).
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Figure 15 shows the postpunching conditions of specimens SS and SSD. Specimen SS (Fig. 15(a)) shows the surfacing of the punching shear failure plane while Fig. 15(b) shows the lifting of the top bars and cover concrete due to the delamination that was simulated.
(a) Specimen SS (no delamination) (b) Specimen SSD (with delamination) Figure 15: Postpunching appearance of slabcolumn specimens containing structural integrity reinforcement.
5
Investigation of a progressive collapse failure of a deteriorated flat plate structure
The progressive collapse of a twoway slab in a parking garage is described to provide an example of an investigation of a collapse and to discuss different factors affecting the performance of an existing deteriorated slab structure.
5.1
Description of collapse
Figure 16 shows the partial collapse of a flat plate structure located at the base of a 14story apartment building in StLaurent, Quebec. The collapse of the twostory indoor parking garage initiated at a slabcolumn joint due to a punching shear failure with the collapse progressing over a region consisting of 3 bays by 4 bays. The structure was constructed in 1966 and the collapse occurred in November 2008 resulting in the death of one person. The slab had a number of deficiencies as described below.
5.2
Slab thickness
No structural drawings were available and therefore the structural features and reinforcement details had to be determined on site. Examination of the failed regions together with Ground Penetrating Radar (GPR) enabled the reinforcement details to be assessed. Samples of the reinforcing bars outside of the collapsed region were tested and together with the bar markings it was clear that the specified yield strength was 60 ksi (414 MPa). Core samples inside the collapsed region and outside of repaired regions indicated an average slab thickness of 6.92 in. (176 mm) and a concrete compressive strength of 4770 psi (32.9 MPa). It was determined that the design slab thickness was 7.25 in. (184 mm) based on measurements at the free edge of the slab where it could be well controlled during construction. The 1963 ACI Code would have required a minimum thickness of 6.81 in. (173 mm). The current 2014 CSA Standard would require a minimum thickness of 7.87 in. (200 mm). 230
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Figure 16: Progressive collapse of parking garage slab in StLaurent, Quebec.
5.3
Original design
The original design satisfied the working stress design provisions of the 1963 ACI Code (ACI 1963) and the 1965 National Building Code of Canada (NBC 1965) for flexure and just satisfied the requirements for punching shear resulting from direct shear stresses. The service live load for design was 50 psf (2.4 kN/m2). It is noted that the punching shear requirements for direct shear plus moment transfer were not introduced until the 1970’s (ACI 1971, CSA 1973). Hence the punching shear stresses would not satisfy current code requirements.
5.4
Addition of asphalt topping
A one inch thick (25 mm) asphalt topping was added to the structure several years after construction, increasing the service loading by 8%.
5.5
Construction errors
Construction errors and misplacement of the top reinforcement was evident at several locations. Figure 17 shows a slabcolumn connection after punching shear failure. The effective slab thickness is reduced due to misplacement of the top reinforcement and due to the projection of the fiberboard column forms into the bottom of the slab. Due to these two effects the average effective depth, dav, for shear was 5 in. (127 mm) at this slabcolumn connection. In the collapsed region dav varied from 4.9 to 5.4 in. which is significantly less than the design value of 6 in. (152 mm). Figure 18 shows the misplacement of the top flexural reinforcement along a column line. The significant misplacement (3 in. (76 mm) cover instead of ¾ in. (19 mm)) had resulted in a deep flexural crack along this line due to the significantly reduced effective depth. The darker colored concrete in the upper portion of the slab in Fig. 18 is the face of an old deep flexural crack.
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Figure 17: Reduction of effective shear depth due to construction errors (repair concrete evident in top cover).
Figure 18: Misplacement of top reinforcing bars along column line.
5.6
Corrosion and delamination
The indoor heated parking structure was exposed to deicing salts brought in by the vehicles, producing ideal conditions for corrosion in the presence of chlorides. The initiation of collapse had likely started at a slabcolumn joint near the entrance to the garage where the chloride exposure was the highest. The top No. 4 (12.5 mm diameter) reinforcing bars had experienced significant corrosion resulting in delamination of the concrete at the level of the top reinforcing mat.
5.7
Repair of slab
There was evidence that the top concrete had been removed around some columns, with some corroded bars being replaced. The repair concrete is evident in Fig. 17.
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5.8
Lack of structural integrity reinforcement
The slab did not contain structural integrity reinforcement and hence the collapse spread horizontally from the location of the initial punching shear failure. Requirements for the provision of structural integrity reinforcement were introduced in the 1984 CSA Standard (CSA 1984). The current requirements of CSA A23.314 are described above. Figure 19 shows the postpunching condition at one of the columns. No structural integrity reinforcement was present. After the punching shear failure the top bars ripped out of the top of the slab. Because of the presence of delamination at the level of the top mat of reinforcement the top concrete cover remained attached to the top mat after failure.
Figure 19: Brittle postpunching shear response due to the lack of structural integrity reinforcement.
5.9
Durability considerations
Although the concrete cover of ¾ in. (19 mm) satisfied earlier codes it is insufficient for durability considerations when the slab is exposed to chlorides. In 1987 CSA Standard S413 for Parking Structures (CSA 1987) was introduced. The requirements of the current version of this standard (CSA S41314) for slabs exposed to chlorides can be met if the following conditions are satisfied: minimum top bar cover of 40 mm; minimum concrete compressive strength at 56 days of 50 MPa; maximum water/cement ratio of 0.40; maximum permeability index of 1000 coulombs at 91 days; and a membrane is provided.
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6
New requirements for existing multistory parking structures in Quebec
Following the collapse of the parking structure in StLaurent, the building regulatory agency (RBQ) in the province of Quebec instituted mandatory maintenance and inspection requirements for existing multistory parking structures (RBQ 2016). These requirements include: 1.
Keeping of a register – the owner is required to keep a register on the premises that contains the following information: (a) the owner’s contact information, (b) structural drawings, (c) descriptions of repairs and modifications, (d) description of any recurrent repairs to solve a given problem, (e) copies of the annual verification reports and reports on any problems, and (f) copies of the indepth verification reports for the garage.
2.
Annual verification – The owner must carry out an annual visual inspection of the garage. The observations must be recorded on the RBQ information sheet accompanied by dated photographs. The information sheet includes items for the slab inspection such as: signs of pot holes, cracking, rust stains, water infiltration, efflorescence, deteriorated concrete, and exposed reinforcement.
3.
Indepth verification by an engineer – every five years the owner must have an indepth verification of the structure. This indepth verification of the structural elements must include: a description of the inspection techniques used, the concrete characteristics, the state of corrosion, a description of defects, a description of necessary corrective work and a schedule for implementing this work, a summary in the report stating that the garage is not in a dangerous state, and photographs.
4.
Reporting of dangerous conditions – The engineer must notify the owner and the RBQ of dangerous conditions. The engineer must describe emergency measures implemented or to be implemented without delay to correct the dangerous situation. The owner must ensure that the work is completed and the engineer must issue a verification report confirming the corrective work has been completed and that the garage is no longer in a dangerous condition.
7
Conclusions
There are many factors that affect the punching shear strength of twoway slabs. Defects in existing slabs, particularly in older structures can reduce the punching resistance considerably. Older parking garages in colder climates in North America face particular challenges due to chloride contamination of the concrete, corrosion of the reinforcing steel and delamination caused by the use of deicing chemicals. The provision of structural integrity reinforcement is capable of providing a secondary defense mechanism for preventing progressive collapse of twoway slab structures.
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8
Acknowledgements
The authors gratefully acknowledge the financial support of the Natural Sciences and Engineering Research Council of Canada for funding the experimental and analytical research presented in this paper. The research reported in this paper was possible due to the experimental programs carried out by Hassan Aoude, Michael Egberts, Carla Ghannoum, Farshad Habibi, Yoon Moi Lee, Kevin Li, Erin Redl, Julie Reilly, and Li Zhang.
9
References
ACI (1963) ACI Standard Building Code requirements for Reinforced Concrete – ACI 31863. American Concrete Institute, Detroit, MI, 144 p. ACI (1971) ACI Standard Building Code requirements for Reinforced Concrete – ACI 31871. American Concrete Institute, Detroit, MI. ACI Committee 31814 (2014) Building Code Requirements for Structural Concrete and Commentary (ACI 318R14) American Concrete Institute, Farmington Hills, MI, 519 p. Aoude, H., Cook, W.D. and Mitchell, D. (2014) “Effects of Simulated Corrosion and Delamination on the Response of Twoway Slabs”, ASCE Structural Journal, 140(1), 9 p. CSA Standard A23 Series (1959) Concrete and Reinforced Concrete, Canadian Standards Association, Ottawa, ON, 44 p. CSA Standard A23.373 (1973) Code for the Design of Concrete Structures for Buildings, Canadian Standards Association, Rexdale, ON, 123 p. CSA Standard A23.377 (1977) Code for the Design of Concrete Structures for Buildings, Canadian Standards Association, Rexdale, ON, 131 p. CSA Standard A23.384 (1984) Design of Concrete Structures for Buildings, Canadian Standards Association, Rexdale, ON, 281 p. CSA Standard A23.304 (2004) Design of Concrete Structures, Canadian Standards Association, Mississauga, ON, 214 p. CSA Standard A23.314 (2014) Design of Concrete Structures, Canadian Standards Association, Mississauga, ON, 290 p. CSA Standard S41387 (1987) Parking Structures, Canadian Standards Association, Rexdale, ON. CSA Standard S41314 (2014) Parking Structures, Canadian Standards Association, Mississauga, ON, 125 p. Fernandez Ruiz, M., Mirzaei, Y. and Muttoni, A. (2013) “PostPunching Behavior of Flat Slabs”, ACI Structural Journal, 110(5), pp. 801812. Ghannoum, C. M. (1998) “Effect of highstrength concrete on the performance of slabcolumn specimens”. M. Eng. thesis, Department of Civil Engineering and Applied Mechanics, McGill University, Montreal, QC.
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Habibi, F., Redl, E., Egberts, M., Cook, W.D. and Mitchell, D. (2012) “Assessment of CSA A23.3 Structural Integrity Requirements for TwoWay Slabs”, Can. J. Civil Engineering, 39(4), pp. 351361. Hawkins, N. M., and Mitchell, D. (1979) “Progressive Collapse of Flat Plate Structures,” ACI Structural Journal, 76(7), pp.775808. Lew, H. S. Carino, N.J., and Fattal, S.G. (1982) “Cause of the Condominium Collapse in Cocoa Beach, Florida”, Concrete International: Design and Construction, 4(8), pp. 6473. Leyendecker, E.V. and Fattal, S.G. (1973) Investigation of the Skyline Plaza Collapse in Fairfax County, Virginia”, Report BSS 94, Center for Building Technology, National Bureau of Standards, Washington, DC, 57 p. Li, K. K. L. (2000) “Influence of Size on Punching Shear Strength of Concrete Slabs,” M. Eng. thesis, Department of Civil Engineering and Applied Mechanics, McGill University, Montreal, QC, 78 p. Melo, G. S. S. A., and Regan, P. E. (1998) “Postpunching resistance of connections between flat slabs and interior columns.” Magazine of Concrete Research, 50(4): 319327. Mirzaei, Y. 2010. “Postpunching behavior of reinforced concrete slabs.” Ph.D. thesis, School of Architecture, Civil and Environmental Engineering, École polytechnique fédérale de Lausanne (EPFL), Switzerland. Mitchell, D., Adams, J., DeVall, R.H., Lo, R.C., and Weichert, D. (1986) “Lessons from the 1985 Mexican Earthquake”, Canadian Journal of Civil Engineering, 13(5), pp. 535557. Mitchell, D. and Cook, W.D., (1984) “Preventing Progressive Collapse of Slab Structures”, Journal of Structural Engineering, American Society of Civil Engineers, 110(7), pp. 15131532. Mitchell, D., Cook, W.D. and Dilger, W. (2005) Effects of Size, Geometry and Material Properties on Punching Shear Resistance, Special Publication 232 on Punching Shear in Reinforced Concrete Slabs, American Concrete Institute, pp. 39 56. Mitchell, D., DeVall, R.H., Saatcioglu, M., Simpson, R., Tinawi, R. and Tremblay, R. (1995) “Damage to Concrete Structures due to the 1994 Northridge Earthquake”, Canadian Journal of Civil Engineering, 22(2), pp. 361377. NBCC (1965) National Building Code of Canada, Associate Committee on the National Building Code, National Research Council of Canada, Ottawa, ON. RBQ (2016) Parking Garages – Maintenance and Inspection, Régie du Bâtiment Québec, www.rbq.gouv.qc.ca/en/building/technicalinformation/buildingchapterfromthesafetycode/parkinggaragesmaintenanceandinspection.html. Reilly, J.L., Cook, W.D. Bastien, J. and Mitchell, D. (2014) “Effects of Delamination on the Performance of Twoway Reinforced Concrete Slabs”, ASCE J. Performance of Constructed Facilities, 28(4), 8 p.
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The Critical Shear Crack Theory for punching design: From a mechanical model to closedform design expressions Aurelio Muttoni, Miguel Fernández Ruiz École Polytechnique Fédérale de Lausanne (EPFL), Switzerland
Abstract The Critical Shear Crack Theory (CSCT) is a consistent approach used for shear design of one and twoway slabs failing in shear and punching shear respectively. The theory is based on a mechanical model that allows the amount of shear force that can be carried by cracked concrete to be determined, accounting for the opening and roughness of a critical shear crack leading to failure. The theory was first developed for punching design of slabcolumn connections without shear reinforcement. Its principles were later extended to other cases, such as slabs with shear reinforcement, fibrereinforced concrete or slabs strengthened with CFRP strips and oneway slabs without shear reinforcement. The generality, accuracy and easeofuse of this theory led to its implementation in design codes (such as the fib Model Code 2010 or the Swiss Code for concrete structures). The design expressions of the CSCT consist of a failure criterion and a loaddeformation relationship, whose intersection defines the load and the deformation capacity at punching failure. They are clear and physically understandable, and can be written in a compact manner for the design of new structures. With respect to the assessment of the maximum punching capacity, the conventional design expressions of the CSCT can also be used, although they required being solved iteratively. In order to enhance the usability of the design equations of the CSCT, particularly for the punching assessment of existing structures, this paper presents closedform design expressions developed within the framework of the CSCT. These expressions allow for direct design and assessment of the failure load. The closedform expressions keep the generality and advantages of the CSCT approach, but they allow for faster and more convenient use in practice. In this paper, the derivation of these expressions on the basis of the CSCT principles is presented, as well as its benefits and comparison to experimental results and the original design formulation.
Keywords Concrete structures, punching shear strength, Critical Shear Crack Theory, design, assessment
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1
Introduction
The Critical Shear Crack Theory (CSCT) is a theory that was originally developed for punching shear design of slabcolumn connections without transverse reinforcement. The basic principles of the CSCT were first introduced in the Draft Code Proposal (Muttoni, 1985) of the Swiss Code for Structural Concrete (SIA 162, 1993). The main theoretical ground of the theory was published some years later (Muttoni, 1989; Muttoni and Schwartz, 1991) and it eventually became the basis for punching design of slabs without transverse reinforcement in the Swiss Code SIA 162. The theory was later extended to beams and oneway slabs without transverse reinforcement (Muttoni, 2003), as well as to twoway slabs with transverse reinforcement (Fernández Ruiz and Muttoni, 2009). Based on its mechanical model, accurate and simplified design formulations were developed and thoroughly checked against test results (Muttoni, 2008). Currently, the CSCT supports the design models of the Swiss Code for Structural Concrete (SIA, 2013) (both for one and twoway slabs failing in shear) as well as the fib Model Code 2010 (fib MC2010, with reference to punching failures in twoway slabs taking further advantage of its potential to be implemented following a LevelsofApproximation approach [Muttoni et al., 2013; Muttoni and Fernández Ruiz, 2012]). The physical model grounding the theory has shown to be consistent and has allowed its applications to be extended to fibrereinforced concrete and FRPreinforced slabs (Maya Duque et al., 2012; Faria et al., 2014), different types of shear reinforcement (Lips et al., 2012; Einpaul et al., 2016a), linearly supported slabs subjected to concentrated loads (Natário et al., 2014, fire conditions (Bamonte et al., 2012), fatigue issues (Fernández Ruiz et al., 2015b), impact loading (Micallef et al., 2014), slab continuity and membrane effect (Einpaul et al., 2016c; 2015), prestressed slabs (Clément et al., 2014) and others. The basic assumption of the CSCT states that the capacity of concrete to transfer shear forces decreases as a function of the opening of a critical shear crack and also depends upon its roughness. For punching of slabcolumn connections (refer to Fig. 1a), Muttoni (2008; Muttoni and Schwartz, 1991) proposed to estimate the opening of the critical shear crack (w) as proportional to the rotation of the slab (y) times the effective depth of the member (d): w µ y × d . With reference to the roughness of the critical shear crack, it is considered to be dependent on the aggregate size (dg, referring to the crack microroughness [Fernández Ruiz et al., 2015a]). Yet, this value is to be corrected to account for the fact that the crack surface does not develop as a perfect plane but as a rough (undulated) surface (crack mesoroughness [Fernández Ruiz et al., 2015a]). The roughness of the crack is thus usually characterized as dg + 16 [mm]. It can be noted that the crack roughness allows special types of concrete to be accounted for, such as lightweight or highstrength concrete, where the crack surface can potentially develop through the aggregates and thus dg = 0. In addition to these considerations on the micro and mesoroughness, the general shape of the crack can be considered as a macroroughness, defining the kinematics of the lips of the critical shear crack (Fernández Ruiz et al., 2015a). Calculating the capacity of concrete to transfer shear forces within the CSCT (on the basis of the opening of the critical shear crack) can be performed using different approaches. The most general manner is probably that presented by Guidotti (Guidotti, 2010; Muttoni and Fernández Ruiz, 2010), where a numerical integration of the stresses is directly performed throughout the failure surface. Guidotti’s approach accounts for the interlock stresses developed across the failure surface calculated according to Walraven (1981) but takes realistic kinematics at failure into account (comprising both a rotational and translational 238
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components). In addition, also the residual tensile strength of concrete is accounted for. The results of this approach can be seen in Figure 1b, indicating the direction of the compressive forces transferred across the critical shear crack. It can be noted that the forces are concentrated on an inclined conical strut (Fig. 1b) around the supported area. This result is consistent and in agreement with the Kinnunen and Nylander model (Kinnunen and Nylander, 1960), as well as further developments of it by Broms (1990, 2016) and Hallgreen (1996).
Figure 1:
Mechanical approach of the CSCT for punching shear failures: (a) conical strut carrying shear and critical shear crack; (b) numerical integration of the stresses across the failure surface according to Guidotti (2010); (c) numerical results of Guidotti (2010) expressed in terms of the opening and roughness of the critical shear crack; and (d) simplified hyperbolic criertion and comparison to the database presented in Muttoni (2008).
According to Guidotti’s analyses (2010), a decay of the shear strength is observed for increasing openings of the critical shear crack (as the capacity of the various sheartransfer actions decreases for increasing crack openings). In fact, failures are predicted to occur in a narrow region (failure region in Figure 1c, calculated for extreme variations of the various geometrical and mechanical parameters of a flat slab). It can be noted that the CSCT is thus a strainbased approach, accounting for the opening of a crack developing through the compression strut. The approach has thus analogies with that of Kinnunen and Nylander (1960) or Broms’ Tangential Strain Theory (2016), which assume a similar stress field to carry the shear force, and assume failure to be triggered by a strain condition in the tangential direction.
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Similar approaches to that of Guidotti (2010) can also be performed in an analytical manner (refer to Fernández Ruiz et al. [2015] for oneway slabs failing in shear accounting for aggregate interlock, residual tensile strength and dowelling of the reinforcement). These approaches confirm the decay of the various sheartransfer actions for increasing crack openings (or decreasing concrete roughness) in agreement with the basic assumption of the CSCT and Guidotti’s general numerical approach. For design purposes, however, performing a full integration of the various sheartransfer actions at the failure surface (with numerical [Guidotti, 2010] or analytical [Fernández Ruiz et al., 2015a] approaches) is not convenient. Instead, accounting for the fact that all failures occur within a relatively narrow region, the sheartransfer capacity can be expressed as a function of a single failure criterion. For instance, Muttoni (2008) proposed the following hyperbolic equation to approximate the failure condition predicted by the CSCT (Fig. 1d): VRc b0 × d f c
3/ 4
= 1 + 15
y ×d
(1)
dg0 + dg
where the coefficient ¾ is valid for SI units (to be replaced by 9 for fc expressed in [psi]), d refers to the effective depth, fc to the concrete compressive strength measured in cylinder and b0 refers to the length of the control perimeter (defined at d/2 from the edge of the column). This equation has proven to lead to excellent results when compared to test data and to be robust for its application to domains other than monotonic punching of structural concrete slabcolumn connections (Maya Duque et al., 2012; Faria et al., 2014; Einpaul et al., 2016a; Bamonte et al., 2012; Micallef et al., 2014; Einpaul et al., 2016c; Einpaul et al., 2015; Clément et al., 2014) . For the calculation of the punching strength and its associated deformation capacity (characterized by the rotation ψ of the slab defined in Fig. 1a), the loadrotation relationship of the slab has to be considered. This law relates the rotation of the slab to the applied shear force and exhibits a highly nonlinear behaviour (due to crack development and to local yielding of the flexural reinforcement) and with a significant influence of the tensionstiffening effect (Fernández Ruiz and Muttoni, 2016). When the shear force (defined by the loadrotation relationship) equals the shear strength (defined by the failure criterion) for a given level of rotation, failure occurs. Figure 2a shows this approach (intersection of the failure criterion and the loadrotation relationship). Failure can occur for different regimes of behaviour of the slab (Fig. 2a) (Fernández Ruiz and Muttoni, 2016), namely: when all flexural reinforcement remains elastic, when a part of the flexural reinforcement has yielded and a part remains elastic, or when all flexural reinforcement has yielded (the strength is thus controlled by the bending strength, but the rotation capacity by the failure criterion [Guandalini et al., 2009]).
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Figure 2:
Calculation of the punching failure load with the CSCT: (a) potentially governing regimes; and (b) laws to characterize the loadrotation relationship (Muttoni, 2008).
For the evaluation of the loadrotation relationship, realistic laws should be used to accurately estimate the punching strength (Muttoni and Fernández Ruiz, 2010). However, determining realistic laws and their intersection with the failure criterion may be timeconsuming and require numerical procedures. Alternative approaches to determine a suitable loadrotation relationships based on analytical formulae have been developed in the past. For instance, Muttoni (2008) derived an analytical law accounting for concrete cracking and tensionstiffening effects (based on a quadrilinear momentcurvature diagram in bending), see Figure 2b. This law has been proved to be consistent compared to a wide range of test results (Muttoni, 2008) and to provide accurate predictions of the behaviour of slabs (Fernández Ruiz and Muttoni, 2016). Yet, for practical design, laws defined by simpler equations are even more suitable. With this respect, the previous law can be simplified on a consistent basis (Muttoni et al., 2013), leading to simpler loadrotation relationships. For design purposes, a parabolicshape (Muttoni et al., 2013) is generally accepted as convenient and sufficiently approximate (Muttoni, 2008) (refer to Fig. 2b):
r f æm ö y = 1.5 s y çç s ÷÷ d Es è mR ø
3/ 2
(2)
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where rs refers to the distance of the column axis to the line of contraflexure of bending moments, fy to the yield strength of the reinforcement, Es to its modulus of elasticity, ms to the average unitary moment for calculation of the bending reinforcement in the support strip (bs = 1.5·rs for axissymmetric conditions [fib MC2010]) and mR to the average unitary bending strength in the support strip. With respect to coefficient 1.5, it can be taken as 1.2 if refined estimates of rs and ms are accounted for (fib MC2010). Both the hyperbolic failure criterion (Eq. (1)) and the parabolic loadrotation relationship (Eq. (2)) are defined by simple expressions and can be used in a simple and efficient manner for design purposes (Muttoni et al., 2013). This can be done by means of the utilization ratio of the bending reinforcement (ratio ms/mR in Eq. (2)). The value of this ratio can be selected by the designer (1.0 corresponding to a plastic design and lower values to overreinforcement of the flexural capacity at the slabcolumn connections) without the need to actually determine other parameters as the bending reinforcement ratio (r) or the flexural strength (mR), which are normally only known after a complete bending design of the slab. Alternatively, the acting bending moment can be introduced into the loadrotation relationship (Eq. (2)) to calculate the corresponding rotation in order to verify whether, for that level of rotation, the strength of the specimen (defined by the failure criterion, Eq. (1)) is larger or not than the acting shear force (Muttoni et al., 2013). With respect to a direct calculation of the punching strength by intersecting the failure criterion (Eq. (1)) and the loadrotation relationship (Eq. (2)), this has to be performed iteratively or numerically. This calculation might be necessary for the assessment of the actual level of safety of existing structures or to optimize some design parameters. Although the iterative procedure converges quickly, it may be argued that a closedform expression is more convenient for practical purposes. In this paper, the possibility of developing closedform expressions for punching design of slabs without transverse reinforcement according to CSCT is presented. To that end, the required modifications in the equations describing the failure criterion and the loadrotation relationship are investigated. Under a number of reasonable assumptions, it is shown that the intersection between both curves can be calculated in a closedform manner. Finally, the resulting expressions are reviewed with reference to the conventional formulation of the CSCT and their validity is compared against a test dataset.
2
Closedform solutions of the CSCT
As previously stated, calculating the intersection between Eq. (1) (failure criterion) and Eq. (2) (loadrotation relationship) cannot be performed in a closedform manner. For derivation of closedform expressions that directly provide the failure load (and its associated rotation capacity), some considerations on the failure criterion and loadrotation relationship are required and will be discussed below.
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2.1
Failure criterion
The failure criterion defined by Eq. (1) corresponds to a function where the punching strength experiences an hyperbolic decay as a function of the term y ·d. Alternative analytical expressions defining the failure criterion can be selected provided that the shape and trend of the failure region (Fig. 1c) is respected. For instance, by considering a power law with asymptotes (vertical and horizontal) crossing at the origin of coordinates, the following expression can be considered:
æ d dg ö ÷÷ VRc = VRc,0 çç 25 × y × d è ø
2/3
£ VRc,0
(3)
where d dg = d g + d g 0 refers to the failure surface roughness (see Eq. (1)) and parameter VRc,0 refers to the maximum achievable punching shear strength, with a suggested value of:
VRc,0 = 0.55b0 d f c
(4)
where the coefficient 0.55 is valid for SI units (to be replaced by 6.6 when fc is expressed in [psi]). A comparison of the hyperbolic failure criterion (Eq. (1)) and the powerlaw (Eq. (3)) to the numerical results of the CSCT calculated by Guidotti (2010) as well as to 99 test results taken from the literature (Muttoni, 2008) is presented in Figure 3. This comparison shows that the power law fits the hyperbolic equation very closely and is in good agreement with the numerical and test results. For moderate to large values of the product y·d, the results are almost coincident. For low values of the term y·d, the powerlaw criterion predicts a higher slope (yet in good agreement with the numerical and test results) that is cut off by the limit of VRc,0 (consistent with the limitations introduced into the failure criterion for design purposes, for instance by fib MC2010).
Figure 3:
Comparison of CSCT hyperbolic failure criterion (Eq. (1)), and powerlaw (Eq. (3)) to: (a) numerical results from Guidotti (2010); and (b) to test results (99 specimens gathered in Muttoni, [2008], ddg).
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2.2
Loadrotation relationship
With respect to the loadrotation relationship, as previously explained, various expressions have been proposed in the past (Muttoni, 2008). For design purposes, the simplified parabolicshape law (Eq. (2)) is convenient and yields sufficient accuracy (Muttoni, 2008). Provided that a refined estimate of the values rs and ms is used for the punching design, the formula reads (value 1.2 instead of 1.5 for the coefficient of the parabola [fib MC2010]):
r f æm ö y = 1.2 s y çç s ÷÷ d Es è mR ø
2.3
3/ 2
(5)
Closedform solution, associated deformation capacity
As suggested by Muttoni (2008) for the simplified loadrotation relationship of the CSCT, one can assume that the acting bending moment at the support strip is a function of the applied shear force:
ms =
V a
(6)
where for typical inner slabcolumn connections, the value of parameter a can for instance be approximated by 8. According to this assumption, when the flexural strength is reached in the support strip, the bending mechanism of the slab develops: Vflex = a·mR (Muttoni, 2008). As a consequence, the utilization ratio of the bending reinforcement (ms/mR) can be expressed as a function of the acting shear force and the flexural strength of the slab ms/mR = V/Vflex. By using this relationship, the intersection between the failure criterion (Eq. (3), providing the shear strength VR) and the loadrotation relationship (Eq. (5), expressed in terms of the acting shear force V) can be found by equalling the acting shear force to the shear strength (V = VRc). In so doing, one can solve the system of Equations (3) and (5) in a closedform manner:
æ d dg ö ÷÷ VRc = VRc,0 çç è 25 ×y × d ø
2/3
d dg æ VRc,0 ö ç ÷ ®y = 25d çè VRc ÷ø
3/ 2
r f y æç VRc ö÷ = 1.2 s d Es çè V flex ÷ø
3/ 2
(7)
which eventually leads to: 1/ 3
VRc @ VRc,0 × V flex
æ d dg Es ö ×ç × ÷ ç 30 × r f ÷ s y ø è
£ VRc,0
(8)
This expression has the advantage that it allows for an explicit calculation of the failure load. It can be noted that this expression already accounts for the size effect on the punching strength. Apart from cases governed by the strength limit criterion (VRc,0), where no size effect is observed, the size effect resulting from Eq. (8) depends (in doublelog scale) upon the power 1/3 of the size of the member. This influence of size effect is consistent with the theoretical predictions of the CSCT and thoroughly discussed in Fernández Ruiz and Muttoni (2016) (lower size effect influence than the asymptotic value predicted by LEFM, as the behaviour and crack openings of slabs in punching are not linearelastic but influenced by development of cracking and tension stiffening effects [Fernández Ruiz and Muttoni, 2016]).
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In addition to the failure load, the associated rotation capacity at failure could be easily determined by introducing the failure load into Eq. (5). A detailed comparison between the previous test results and the closedform solution of Eq. (8) is plotted in Figure 4. The results are very accurate, with an average measuredtocalculated shear strength of 0.99 and a Coefficient of Variation of 10% (similar to those obtained for the classical design equations [Muttoni, 2008]).
Figure 4:
3
Comparison of CSCT closedform expression for the power law (Eq. (8)) with the test database presented in Muttoni (2008) with the results presented as a function of: (a) reinforcement ratio; (b) concrete strength; and (c) effective depth.
Design expression based on mechanical parameters
With respect to the formula for calculation of the punching shear strength (Eq. (8)), it can be noted that it depends upon the flexural strength of the member. This is considered very convenient for design, as it incorporates in a compact and consistent manner a number of detailed information on the static system (inner, edge or corner column and moment transfer) and bending reinforcement. Yet, solutions of this expression can be developed for specific cases showing the influence of the various mechanical parameters on the punching strength. In addition, such a format (based on mechanical parameters) allows direct comparison of the resulting expressions to other design formulae, with different background or even with empirical nature. To develop design formulas based directly upon mechanical parameters, one can elaborate for instance the term Vflex for inner columns without moment transfer (a = 8 [Muttoni, 2008]). To that end, the unitary bending strength of the slab (mR) can be approximated for instance according to a multiplicative formula of powers of its various parameters (in a development similar to that of Zsutty [1963]), so to observe their influence:
r × fy ö æ ÷÷ » 0.75 × r 0.9 × d 2 × f y 0.9 × f c 0.1 mR = r × d 2 × f y × çç1  0.59 fc ø è
(9)
Thus, for inner slabcolumn connections:
V flex = a × mR » 6 × r 0.9 × d 2 × f c × f y 0.1
0.9
(10)
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Based on Eq. (8), the punching strength then results (SI units): 1/ 3
ö VRc 0.05 1 3 æ d = 0.551 / 2 b01 2d 1 2 f c1 4 61 2 d × r 0.45 f y0.45 f c × 30 1 3 f y × çç dg × Es ÷÷ b0 × d è rs ø
(
£ 0.55 f c
(11)
)
By approximating 0.55 0.5 × 6 0.5 × 30 1 3 × f c0.033 r × f y » 0.6 (3.2 for fc and Es expressed in [psi]), Eq. (11) can be simplified by grouping similar exponents as: 12
d dg ædö æ VRc = 0.6 çç ÷÷ × çç r × Es × f c × b0 × d rs è b0 ø è
0.117
1/ 3
ö ÷÷ ø
£ 0.55 f c
(12)
where the coefficients 0.6 and 0.55 have to be replaced by 3.2 and 6.6 when fc and Es are expressed in [psi]. This formula can be considered to be composed of the following terms: • term b0/d (power 1/2) accounting for the column sizetoeffective depth ratio; • term r·Ec (power 1/3) depending on the reinforcement stiffness; • term fc (power 1/3) depending on concrete compressive strength and • term ddg/rs (power 1/3) referring to the effect of member size, member slenderness and concrete type. This term is dimensionless, as the concrete type is described as a reference dimension depending on the roughness of cracks (ddg = 32 mm (1.25”) for usual concrete) and the combined size and slenderness effects are defined by radius rs (distance between column axis and point of contraflexure in the slab). A comparison of this formula (Eq. (12)) to the dataset from Muttoni (2008) is presented in Figure 5. The results are again very accurate and with a low scatter (average measuredtocalculated strength of 0.99 with a Coefficient of Variation of 11%).
Figure 5:
246
Comparison of CSCT closedform expression based on mechanical parameters (Eq. (12)) with the test database presented in Muttoni (2008) with the results presented as a function of: (a) reinforcement ratio; (b) concrete strength; and (c) effective depth.
ACIfib International Symposium Punching shear of structural concrete slabs
It is also interesting to compare this expression for instance to the empirical formula of the current Eurocode 2 (CEN, 2004) (written without safety format and reformulated for a control section located at a distance d/2 from the column):
(
VRc 12 = 0.84 × min 2;1 + (d ref d ) b0 × d
)× æçç1 + 3pb× d ö÷÷ × (r × f )
1/ 3
è
0
ø
c
(13)
where the coefficient 0.84 has to be replaced by 23 when fc is expressed in [psi] and dref is a reference size (200 mm (7.9”)). It can be noted that the closedform CSCT expression and the empirical formula of Eurocode yield identical dependencies for the punching strength with respect to the concrete strength and flexural reinforcement ratio (power 1/3). Other phenomena are not accounted for in the Eurocode formula, as the influence of the slenderness (Einpaul et al., 2016b), the concrete type (aggregate size) (Muttoni, 2008) and modulus of elasticity of the reinforcement (associated to the strain effects) which is relevant for FRP reinforcements. In addition, it can be noted that the Eurocode 2 size effect factor is highly inconsistent, predicting no size effect influence for asymptotically large sizes (Fernández Ruiz et al., 2015a).
4
Consideration of slab continuity and membrane action
Taking advantage of the mechanical model of the CSCT, Einpaul et al. (2015, 2016c), investigated the role of slab continuity and membrane action in twoway slabs. This topic is extremely relevant for practice as tests on slabcolumn connections usually only represent a small portion of the slab and membrane action, as well as slab continuity, are thus neglected. According to these works (Einpaul et al., 2015; 2016c), there is a nonnegligible influence for flat slabs of both slab continuity (allowing for moment redistribution between hogging and sagging bending regions) and membrane action (compressive stresses developed near the column regions) on the rotation of slabcolumn connections and consequently on the punching shear strength. Both numerical (Einpaul et al., 2015) and analytical approaches (Einpaul et al., 2016c) have been proposed within the framework of the CSCT to consistently account for these phenomena.
4.1
Closedform approach to account for slab continuity within the CSCT
In order to develop closedform design expressions accounting for slab continuity, the analytical approach presented in (Einpaul et al., 2016c) can be introduced in a simple manner within the previous closedform expressions. According to Einpaul et al. (2016c), the influence of slab continuity and compressive membrane action can be accounted for by considering a reduction of the rotations of a slab sector (Eq. (2) referring to an isolated portion of a slab without slab continuity or compressive membrane action) in the following manner:
r f æm ö y = kcs × 1.2 s y çç s ÷÷ d Es è mR ø
3/ 2
(14)
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where kcs is a coefficient that reduces the slab rotation to account for slab continuity and compressive membrane action (additional implications of the compressive membrane action on the punching strength are discussed elsewhere [Clément et al., 2014]). A suitable formulation for coefficient kcs was proposed by Einpaul et al. (2016c) derived on the basis of an axissymmetric slab model. According to this work, the value of coefficient kcs has to decrease for increasing values of the ratio mcr/mR, where mcr refers to the cracking moment of the slab and mR to the flexural strength at the hogging region (bending moment at the slabcolumn connection as defined in this paper). The use of term mcr is justified because the confinement at the column region is provided by the surrounding concrete during the crack development stage (Einpaul et al., 2016c). Based on these considerations, and for practical purposes, coefficient kcs is suggested to be evaluated according to the following expression:
æ m k cs = çç 0.08 R mcr è
ö ÷÷ ø
3/ 4
£ 1.0
(15)
To determine the failure load in a closedform manner accounting for slab continuity and compressive membrane action (applicable to inner slabcolumn connections of slabs where tensile axial forces due to slab inplane restraints are negligible), Eq. (8) yields:
VRc @ VRc,0 × V flex
æ d dg Es ×ç × ç 30 × r k × f s cs y è
1/ 3
ö ÷ ÷ ø
£ VRc,0
(16)
The results of Eq. (16) are plotted in Fig. 6 and compared to those of an isolated slab offcut (with no membrane action or hoggingsagging moment redistribution) and to those of a refined prediction of the punching shear strength of a continuous slab obtained by means of a refined numerical analysis (Einpaul et al., 2016c). It can be observed that the slab continuity and membrane forces reduce the influence of the hogging bending reinforcement on the punching strength with respect to an isolated slab offcut and may significantly increase the punching strength, particularly for slabs with low flexural reinforcement. It can also be noted that the approximated formula, despite its simplicity, yields good and safe estimates of the behaviour of a continuous flat slab, close to those resulting from the refined numerical analysis. As an alternative to Eq. (16), one can note that the same result is obtained when Eq. (8) is used but with a corrected value of the flexural strength to account for slab continuity and compressive membrane action (Vflex,cs): 1/ 3
VRc @ VRc,0 × V flex,cs
æ d dg Es ö ×ç × ÷ ç 30 × r f ÷ s y ø è
£ VRc,0
(17)
where Vflex,cs results from Eqs. (15,16):
V flex,cs = a × 12.5 × mcr × mR ³ a × mR
(18)
where coefficient a was defined in Eq. (6) as the ratio between the acting shear force and the average bending moment at the support strip for a slab sector.
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Figure 6:
4.2
Comparison of calculated punching strength for an isolated slab offcut and a continuous flat slab accounting for slab continuity and compressive membrane action) for the closedform design expression and a detailed numerical analysis according to Einpaul et al. (2016c) (rsag = 0.5%; fc = 35 MPa (5’080 psi); fct = 3.2 MPa (460 psi); fy = 420 MPa (60.9 ksi); L = 7 m (23.0 ft); d = 210 mm (8.27 in); h = 250 mm (9.84 in); column size 350 mm (13.8 in)).
Influence of design parameters on punching strength of continuous slabs
In Eq. (12), the influence of the various mechanical and geometrical parameters was directly observed for the closedform expression of the CSCT. It is interesting to note that some of these parameters modify their influence on the punching strength when slab continuity and compressive membrane action are accounted for. This can be investigated by introducing into Eq. (11) the role of coefficient kcs as defined by Eq. (15). With mcr = fct·h2/6 and adopting fct = 0.3fc2/3 as defined in (CEN, 2004), Eq. (11) leads to (SI units): 12
12
1/ 3
æ d ö æhö æ d dg ö VRc = 0.559 çç ÷÷ × ç ÷ × f c0.442 r 0.225 f y0.108 çç × Es ÷÷ b0 × d è b0 ø è d ø è rs ø
£ 0.55 f c
(19)
This equation shows that the influence of the flexural reinforcement ratio on the punching strength is severely decreased, from a power 0.45 to a power 0.225 (= 0.45 – 0.9·3/4·1/3). This is in agreement with the comments on Figure 6 and with specific design guidelines for design continuous slabs when membrane action is considered (UK Highway Agency, 2002). In addition, the influence of concrete strength is slightly increased and the slab thickness h has also an influence that is nonnegligible in the case of thin slabs.
5
Conclusions
This paper presents closedform design expressions for the use of the Critical Shear Crack Theory (CSCT) for punching design of slabs with and without transverse reinforcement. Its main conclusions are listed below: 1. Closedform expressions can be derived within the framework of the CSCT. This can be performed by introducing some minor modifications into the failure criterion.
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2. The closedform solutions for simple cases provide results almost identical to an iterative calculation of the failure load based on the use of the hyperbolic CSCT failure criterion and a parabolic loadrotation relationship 3. The closedform design expressions are shown to be practical and convenient for conventional cases and also to provide a simple explanation of the role of the various parameters involved. Additionally, they are consistent with the general formulation of the CSCT, which allows a more refined analysis of the connections to be performed if necessary. 4. Closedform solutions can also be tailored in a simple manner to take into account the behaviour of actual flat slabs, accounting for compressive membrane action and slab continuity. 5. Expressions accounting for slab continuity and compressive membrane action show that the influence of the bending reinforcement ratio is much smaller than generally assumed when experimentally investigating isolated slabcolumn connections. In addition, the slab thickness also has an influence on punching shear strength, which is nonnegligible in the case of thin slabs.
6
References
Bamonte, P., Fernández Ruiz, M., Muttoni, A. (2012) “Punching shear strength of R/C slabs subjected to fire,” Proceedings of the 7th International Conference on Structures in Fire, Zurich, Switzerland, June 68, 10 p. Broms C. E. (1990) “Punching of Flat Plates  A Question of Concrete Properties in Biaxial Compression and Size Effect,” ACI Structural Journal, Vol. 87, No 3, pp. 292304. Broms C. E. (2016) “Tangential Strain Theory for Punching Failure of Flat Slabs,” American Concrete Institute, Structural Journal, Vol. 113, No 1, pp. 95104. CEN European Committee for Standardization (2004) Eurocode 2. Design of concrete structures – general rules and rules for buildings, EN 199211, Brussels, Belgium, 225p. Clément, T., Pinho Ramos, A., Fernández Ruiz, M., Muttoni, A. (2014) “Influence of prestressing on the punching strength of posttensioned slabs,” Engineering Structures, Vol. 72, pp. 5669. Einpaul, J., Brantschen, F., Fernández Ruiz, M., Muttoni, A., 2016a, “Performance of punching shear reinforcement under gravity loading: influence of type and detailing,” American Concrete Institute, Structural Journal, Vol. 113, No. 4, pp. 827838. Einpaul, J., Bujnak, J., Fernández Ruiz, M., Muttoni, A., 2016b, “Study on influence of column size and slab slenderness on punching strength,” American Concrete Institute, Structural Journal, Vol. 113, No. 1, pp. 135145. Einpaul, J., Fernández Ruiz, M., Muttoni, A. (2015) “Influence of moment redistribution and compressive membrane action on punching strength of flat slabs,” Engineering Structures, Vol. 86, pp. 4357.
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Einpaul, J., Ospina, C.E., Fernández Ruiz, M., Muttoni, A., 2016c, “Punching shear capacity of continuous slabs,” American Concrete Institute, Structural Journal, Vol. 113, No. 4, pp. 861872. Faria, D., Einpaul, J., Pinho Ramos, A., Fernández Ruiz, M., Muttoni, A. (2014) “On the efficiency of flat slabs strengthening against punching using externally bonded fibre reinforced polymers,” Construction & Building Materials, Vol. 73, pp. 366377. Fernández Ruiz M., Muttoni A. (2009) “Applications of the critical shear crack theory to punching of R/C slabs with transverse reinforcement,” ACI Structural Journal, Vol. 106 N° 4, pp. 48549400Fernández Ruiz M., Muttoni A. (2016) Size effect on punching shear strength and differences with shear in oneway slabs, ACI/fib International Punching Shear Symposium, Philadelphia, October 2016. Fernández Ruiz, M., Muttoni, A., Sagaseta, J. (2015a) “Shear strength of concrete members without transverse reinforcement: a mechanical approach to consistently account for size and strain effects,” Engineering Structures, Vol. 99, pp. 360372 Fernández Ruiz, M., Zanuy, C., Natário, F., Gallego, J.M., Albajar, L., Muttoni, A. (2015b) “Influence of fatigue loading in shear failures of reinforced concrete members without transverse reinforcement,” Journal of Advanced Concrete Technology, Japan Concrete Institute, Vol. 13, pp. 263274. fib (Fédération internationale du béton) (2013) fib Model Code for Concrete Structures 2010, Ernst & Sohn, Germany, 434 p. Guandalini S., Burdet O., Muttoni A. (2009) Punching tests of slabs with low reinforcement ratios, ACI Structural Journal, V. 106, N°1, pp. 8795. Guidotti R. (2010) “Poinçonnement des planchersdalles avec colonnes superposées fortement sollicitées,” Thèse EPFL N°4812, Lausanne, 230 p. Hallgren M. (1996) “Punching Shear Capacity of Reinforced High Strength Concrete Slabs, PhD thesis,” KTH Stockholm. Kinnunen S. and Nylander H. (1960) “Punching of Concrete Slabs Without Shear Reinforcement,” Transactions of the Royal Institute of Technology, N° 158, 112 p. Lips S., Fernández Ruiz M., Muttoni A. (2012) “Experimental Investigation on Punching Strength and Deformation Capacity of ShearReinforced Slabs,” ACI Structural Journal, Vol. 109, pp. 889900. Maya Duque L. F., Fernández Ruiz M., Muttoni A., Foster S. J. (2012) “Punching shear strength of steel fibre reinforced concrete slabs,“ Engineering Structures, Vol. 40, pp. 9394Micallef, K., Fernández Ruiz, M., Muttoni, A., Sagaseta, J. (2014) “Assessing punching shear failure in reinforced concrete flat slabs subjected to localised impact loading,” International Journal of Impact Engineering, Vol. 71, pp. 1733. Muttoni A. (2008) “Punching shear strength of reinforced concrete slabs without transverse reinforcement,” ACI Structural Journal, V. 105, N° 4, pp. 440450Muttoni, A. (1985) Punching shear – Draft code proposal, SIA 162, Working Group 5, Swiss Society of Engineers and Architects, Zürich, 15 p.
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Muttoni, A. (1989) The applicability of the theory of plasticity in the design of reinforced concrete (in German, Die Andwendbarkeit der Plastizitätstheorie in der Bemessung von Stahlbeton), Institut für Baustatik und Konstruktion, Report Nr. 176, ETH Zürich, 159 p. Muttoni, A. (2003) “Shear and punching strength of slabs without shear reinforcement,” (in German, “Schubfestigkeit und Durchstanzen von Platten ohne Querkraftbewehrung”), Beton und Stahlbetonbau, Vol. 98, pp. 7484. Muttoni, A., and Schwartz, J. (1991) Behaviour of Beams and Punching in Slabs without Shear Reinforcement, IABSE Colloquium, V. 62, Zurich, Switzerland, pp. 703708Muttoni, A., Fernández Ruiz, M. (2010) “MC2010: The Critical Shear Crack Theory as a mechanical model for punching shear design and its application to code provisions, fédération internationale du béton,” in fib Bulletin 57, Shear and punching shear in RC and FRC elements, pp. 3160. Muttoni, A., Fernández Ruiz, M. (2012) “The LevelsofApproximation approach in MC 2010: applications to punching shear provisions,” Structural Concrete, Vol. 13, No. 1, pp. 3241Muttoni, A., Fernández Ruiz, M., Bentz, E., Foster, S.J., Sigrist, V. (2013) “Background to the Model Code 2010 Shear Provisions  Part II Punching Shear,” Structural Concrete, Vol. 14, No. 3, pp. 195203Natário, F., Fernández Ruiz, M., Muttoni, A. (2014) “Shear strength of RC slabs under concentrated loads near linear supports,” Engineering StructuresVol. 76, pp. 1023. SIA (1993) Code 162 for Concrete Structures, Swiss Society of Engineers and Architects, Zürich, 86 p. SIA (2013) Code 262 for Concrete Structures, Swiss Society of Engineers and Architects, Zürich, 102 p. UK Highway Agency (2002) BD 81/02, “Use of compressive membrane action in bridge decks,” Design manual for roads and bridges, V. 3, Section 4, Part 20, United Kingdom, 20 p. Walraven J. C. (1981) “Fundamental Analysis of Aggregate Interlock,” ASCE Journal of Structural Engineering, Vol. 107, No 11, pp. 22452270. Zsutty T.C. (1963) “Ultimate Strength Behaviour Study by Regression Analysis of Beam Test Data,” ACI Journal, Vol. 60, No. 5, pp. 635654.
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Punching of flat slabs under reversed horizontal cyclic loading António Ramos, Rui Marreiros, André Almeida, Brisid Isufi, Micael Inácio Universidade NOVA de Lisboa, Portugal
Abstract Flat slab structures are a very common structural solution nowadays, due to their architectural and economic advantages. However, flat slabcolumn connections may be vulnerable to punching failure, especially in the event of an earthquake, with potentially high human and economic losses. This type of structural solution is adequately covered by design codes and recommendations in North America, due to the large amount of experimental research that has been carried out. In Europe, the situation is different: specific guidance to flat slab design under earthquake action is missing from most European codes. The ACI 31814 prescriptive approach to the gravity shear ratiodrift ratio relationship shows good agreement with experimental results. Following a similar approach and, based on a databank containing cyclic horizontally loaded tests of slabcolumn connections found in the literature, proposals are made that are applicable to Eurocode 2 and the fib Model Code 2010.
Keywords Drift, flat slabs, gravity shear ratio, horizontal cyclic loading, performance based design, punching
1
Introduction
Flat slab buildings are advantageous for their ease of construction, opportunity for reduced overall building height, cost effectiveness for common span lengths in residential, office and commercial buildings, and freedom in interior architecture. As a result, such buildings are common worldwide. Nevertheless, the actual situation for this type of structural solution differs between Europe and North America mainly in two aspects: code coverage in case of earthquake loading (cyclic loading in general) and the amount of experimental research on flat slabs under horizontal cyclic loads. Regarding code coverage, the provisions for structural design in Europe are covered by the Eurocodes, which have now superseded local codes in most European countries. Besides the Eurocodes, the Model Code has been developed as a “futureoriented” code, implementing the most advanced research results and continuously influencing design codes. Its 1990 version (CEBFIP MC 90) significantly influenced the current version of the Eurocode. The latest version is the fib Model Code 2010 (MC2010) (fib, 2013). Eurocode 2 (EC2) (CEN, 2004a) covers the design rules of flat slabs for Serviceability and Ultimate Limit States. Eurocode 8 (EC8) (CEN, 2004b), which is meant to supplement Eurocode 2 for seismic design, does not fully cover concrete buildings for which the primary lateral force resisting system contains flat slabs (paragraph 5.1.1 of EC8). The applicability of paragraph 5.1.1 of EC8 is not related to the seismicity level, i.e. the design of flat slabs as
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primary seismic elements (elements that are considered to be part of the lateral force resisting system) is not covered, even in cases of moderate seismicity. Nonetheless, the usage of flat slabs as a “primary seismic element” in buildings is uncommon in Europe. Usually, shear walls or moment frames are used to handle the lateral displacements and to provide lateral resistance and stiffness, thus turning the flat slabs into “secondary seismic elements”. According to EC8, the strength and stiffness of such secondary elements shall be neglected, but second order effects (where relevant) must be considered and their gravity load bearing ability must not be impaired under seismic displacements corresponding to the design level earthquake. However, specific guidance is missing. EC8 has no rules for modeling, analysis, design or detailing of flat slab frames under seismic actions. MC2010 also lacks specific provisions regarding the design of flat slab buildings for seismic design. It does contain, however, provisions for “integrity reinforcement” in order to avoid progressive collapse following a sudden failure of a slabcolumn connection. This type of collapse has already been observed in several events and the phenomenon is already taken into consideration in ACI 421.2R10 and ACI 352.1R11. Regarding the amount of experimental research on this subject, the literature mainly contains tests performed in universities in USA and Canada. In the case of isolated slabcolumn connections, the test setups are quite similar. Experimental work on horizontal cyclic loading of slabcolumn connections does exist in Europe, although to a lesser extent. One example is the research at the Civil Engineering Department, Faculty of Science and Technology at Universidade NOVA de Lisboa (DEC/FCT/UNL) in Portugal. The test setup developed at DEC/FCT/UNL is briefly described in this paper along with the most common ones used in previous research. It overcomes some of the most common issues of the other setups as it allows bending moment redistribution, line of inflection mobility and ensures equal vertical displacements at the borders and symmetrical shear forces. The lack of coverage by design codes and scarce experimental research create a gap between the design of flat slabs in seismic regions in Europe and the understanding of North American research findings and codes. This paper focuses on the interpretation of the available horizontal cyclic loading tests conducted worldwide for the study of punching shear strength of flat slabs while simultaneously considering three design codes: ACI 31814, Eurocode 2 and the fib Model Code 2010. The purpose is to bridge the gap between the three codes in order to make use of the experimental data gathered both in Europe and especially in North America.
2
Literature review of cyclic loading tests of isolated slab specimens without shear reinforcement
2.1
Brief description of test setups and loading
A considerable number of specimens have already been tested under cyclic loading conditions by various researchers. The majority of the tests found in the literature consist of isolated slabcolumn connections, although some tests based on slab systems or multipanel slabs can be found. Specimens discussed in this section correspond to interior slabcolumn connections without shear reinforcement. The columns of all the specimens presented herein have rectangular (or square) cross section.
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Figure 1:
Schematic representation of test setups: a) column pinned at bottom and load applied at top column end; b pinned column ends and load applied at slab ends; c) pinned slab ends and load applied at column ends; d) test setup with nonzero moments at slab specimen ends (EA= axial stiffness and EI = flexural stiffness).
Different test setups were used. The sketches in Figure 1 show the basic information regarding support conditions and position of application of cyclic loading. The test setup shown in Figure 1a was used for specimens from Tian et al. (2008), Robertson and Johnson (2006), Widianto et al. (2006), Stark et al. (2005), Robertson et al. (2002), Pan and Moehle (1989, 1992), Zee and Moehle (1984), Wey and Durrani (1992), Morrison et al. (1981, 1983), as well as Tan and Teng (2005), Kang and Wallace (2008) and Song et al. (2012). In this setup, the column is pinned at the bottom end, the slab is simply supported and free to move horizontally while the horizontal load is applied at the top column end. The setup of Figure 1b was used for specimens from Hawkins et al. (1974), Symonds et al. (1976) and Islam and Park (1976). In contrast to the first test setup, displacements are applied at slab ends while both column ends are pinned. Figure 1c represents the setups for specimens from Cao and Dilger (1993), Brown and Dilger (2003), Marzouk et al. (2001) and Emam et al. (1997). Though similar to the setup in Figure 1b, this setup has pinned support at slab ends and load applied at column ends. Specimens 1 to 4 from Farhey et al. (1993) were tested under conditions similar to those of Figure 1c, but with application of cyclic loading only at one end of the column. The first three test setups (Figure 1a to c) have been commonly accepted and used as good and simple representatives of slabcolumn connections. However, they use simplified boundary conditions, such as borders that are bending moment free and vertically fixed, which leads to a static position of the zero moment line and prevents bending moment redistribution and supported borders that can absorb vertical load. In order to overcome some of these limitations, a different test setup was developed at DEC/FCT/UNL (Figure 1d). The
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test setup allows vertical displacements at the opposite slab borders; equal magnitude shear forces, bending moments and rotations at the slab edges; mobility of the line of inflection location along the longitudinal direction and high vertical load ratios, all driven through the column. Thus for vertical loads, the shear forces and rotations at the opposite borders are zero and the vertical displacements and bending moments are the same at the opposite longitudinal borders. For horizontal actions, the vertical displacements and shear forces should be equal in magnitude, but with opposite signs. Three passive systems were used to accomplish this. To guarantee equal shear forces and vertical displacements, a passive mechanical seesawlike system was designed, as shown in Figure 2a, allowing the vertical displacements that depend exclusively on the slab’s stiffness and force balance. Regarding shear forces at the opposite longitudinal borders, and for vertical loads, the balance beams move freely, without introducing shear forces on the slab edges, as expected at the slab’s midspan. When horizontal displacements take place, shear forces are applied to prevent the slab’s rigid body rotation. The balance system ensures that those forces are equal in magnitude and opposite in direction, again, as theoretically expected. As the slab stiffness degrades, deformation due to the constant vertical load increases. This ensures that the vertical load is driven through the column only, also simulating the effect of the whole slab. In order to have positive bending moments at the slab’s midspan and equal rotations at the opposite borders, a system consisting of a horizontal double pinned steel frame suspended on the slab’s borders by two vertical fixed columns is used (Figure 2b). The frame has variable length by means of a hydraulic jack and a load cell to measure the frame’s axial load. For the vertical load, a positive moment of equal magnitude is introduced at both borders. For the horizontal action, the slab rotations are the same at the opposite borders, and the moments are equal, but with the opposite sign. The vertical load is applied through steel plates to the top slab surface at eight points to better approximate a uniformly distributed load, and is intended to be constant throughout the whole test; therefore, it must be independent from the slab’s deformation and horizontal displacement. For this to be accomplished, a closed structure to apply the vertical load was created using a system of spreader beams and steel tendons, as shown in Figure 2c. That structure follows the slab horizontal deformations. The bottom beams are supported by the column corbels, directing all the vertical load through the column. This way, applied vertical loads are directed to the inferior column, avoiding exterior conflicts. The load is applied using four hydraulic jacks connected to a load maintainer machine. Once the target vertical load is reached, it is kept constant during the test. A photograph of the setup is shown in Figure 3. A more detailed description of this test setup is included in Almeida et al. (2016). The research team at DEC/FCT/UNL already performed tests using this setup on slabs with or without shear reinforcement, using fiber reinforced concrete and high strength concrete. Within this paper, only the slabs without shear reinforcement are considered, either with normal strength concrete or with high strength concrete (Almeida et al., 2016; Inácio et al., 2015).
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a)
b)
c) Figure 2:
Elevation drawing of the test setup at DEC/FCT/UNL: a) vertical displacement and shear force compatibilization system under slab’s vertical and horizontal deformation; b) rotation and bending moment compatibilization system under slab’s vertical and horizontal deformation; c) complete test setup.
Figure 3:
Photo of the test setup at DEC/FCT/UNL.
The actual test setups have also differences between them in terms of support conditions, such as number of supports, arrangement of supports, etc. For instance, specimens from Marzouk et al. (2001) and Emam et al. (1997) are supported along the entire outer perimeter of the slab (Figure 1a), via either a line (continuous) support or multiple point supports
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distributed along the edges. Specimens from Robertson and Johnson (2006), Stark et al. (2005), Cao and Dilger (1993), Brown and Dilger (2003), Robertson et al. (2002), Farhey et al. (1993), Hawkins et al. (1974), Symonds et al. (1976), Zee and Moehle (1984), Islam and Park (1976), Morrison et al. (1981, 1983), Kang and Wallace (2008), Song et al. (2012) and Wey and Durrani (1992) have line supports or multiple point supports for the slab in one direction only (Figure 1b). In specimens from Pan and Moehle (1989, 1992) and Tan and Teng (2005), there are struts located at the corners of the slab (Figure 1c) and at the midpoints of the edges. In specimens from Tian et al. (2008) and Widianto et al. (2006) the struts are arranged along the edges but the corners are free to move (Figure 1d).
Figure 4:
Arrangement of supports of the slab: a) continuous support on all edges; b) line supports in one direction; c) struts at corners of the slab; d) struts along the edges and free corners.
In all the tests described herein, the gravity loading was static or quasistatic. Different gravity and lateral loading histories were applied to the specimens. For instance, after the application of the vertical load, specimen L0.5 from Tian et al. (2008) was subjected to lateral deformation reversals until failure. Specimens LG0.5 and LG1.0 from Tian et al. (2008) were loaded in two phases. In the first phase, they were loaded in the gravity and lateral directions and then punched in the second phase by increasing the gravity load on the already damaged specimen. The six specimens from Robertson and Johnson (2006) were subjected to a displacement test routine in which cyclic loading is applied in two phases. In the first phase, displacement reversals were applied to the undeformed state, but in the second phase the cycles were applied to a deformed state, with a minimum drift of zero in the loading history (i.e. it did not reach negative values). Drift ratios up to 10% were applied in the second phase. Widianto et al. (2006) applied lateral loading up to failure for specimen SPControl 1 and lateral loading up to a 1.5% drift ratio followed by punching due to gradually increasing gravity loads under a changed test setup for specimen SP1. In Stark et al. (2005), the specimens (C02 and C63) were subjected to reversed lateral displacements with drift ratios up to 6.0%. The specimen SJB7 from Brown and Dilger (2003) was laterally loaded in the
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direction of the diagonal to the square column (the reinforcement layout and the column were rotated 45 degrees). Specimens CD1, CD5 and CD8 from Cao and Dilger (1993) were subjected to cycles with ever increasing lateral load to failure (i.e. with no repetition of cycles of the same lateral load level). A displacement routine with two phases similar in concept to the routine applied by Robertson and Johnson (2006) (described previously) was applied for specimen 1C from Robertson et al. (2002). Drift ratios up to 8.0% were applied in the second phase. The six specimens from Marzouk et al. (2001) were laterally loaded after the application of the gravity load according to a pattern of two cycles with the same drift level followed by a cycle with lower drift. The pattern is repeated with increased drift ratios up to approximately 6.0%. Similar routine was applied to all four specimens from Emam et al. (1997). Specimens 1 to 4 from Farhey et al. (1993) were loaded to failure in a force controlled manner with reversed application of the lateral load. Specimens 1 and 3 from Pan and Moehle (1992) were subjected to reversed cyclic displacements applied in only one direction of the specimen (as all the other specimens described so far) with absolute maximum drifts of 3.2%, whereas specimens 2 and 4 were bidirectionally loaded. The same lateral displacement history was used in specimens 2 and 4, but bidirectional effects were produced by following a twodimensional path for the lateral displacement application. The specimens from Tan and Teng (2005) were tested under a similar loading history, with biaxial loading for specimens YLH2 and YLL2 and loading in one direction for YLL1. In specimens S1, S2, S3, S4 from Hawkins et al. (1974), S6 and S7 from Symonds et al. (1976), as well as 3C from Islam and Park (1976), cyclic loading effects were produced by applying reversed cycles of loading at slab ends up to failure. The interior specimens (INT) from Zee and Moehle (1984) and SC0 from Wey and Durrani (1992) were subjected to a displacement history with zones of ever increasing peak values of the drift and zones with cycles with increasing peak values followed by a cycle with lower drift. In Morrison et al. (1983), the five specimens were subjected to columnslab joint reversed rotations following a history of repetition of various levels of rotation. In the last cycle, the lateral loading was monotonically increased to failure. Almeida et al. (2016) and Inácio et al. (2015) tested the specimens by applying cycles that were repeated three times for each drift level. In the test campaigns by Kang and Wallace (2008) and Song et al. (2012), the specimens C0 and RC1 respectively served as control specimens amongst other specimens with shear reinforcement, tested under reversed cyclic loading. Specimen RC1 was subjected to a loading history with drift levels repeated three times up to a predefined value, after which the drift is continuously increased in each cycle. The tests described above are summarized in Table 1, in which the author(s), the corresponding names for each specimen given by the author(s) and the main variables in each test program are contained in columns 1, 2 and 3 respectively. The main variables were the slab flexural reinforcement ratio, gravity load (gravity shear ratio), loading conditions, type of concrete (high or normal strength and lightweight or normalweight). Fewer tests were conducted with the purpose of studying the effect of continuity of bottom reinforcement at column region and column (loaded area) dimensions (refer to Table 1).
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Table 1:
Specimens without shear reinforcement tested under cyclic loading
Author, reference and year Hawkins et al. (1974)* Islam and Park (1976)
Specimens S1, S2, S3, S4 3C
Symonds et al. (1976) Morrison et al. (1981, 1983) Zee and Moehle (1984)
S6, S7 S1, S2, S3, S4, S5 INT
Pan and Moehle (1989, 1992)
AP1, AP2, AP3, AP4
Robertson (1990) Wey and Durrani (1992)
8I, 2C, 3SE, 5SO, 6LL, 7L SC0
Cao and Dilger (1993) Farhey et al. (1993)
CD1, CD5, CD8 1, 2, 3, 4
Emam et al. (1997)
Robertson et al. (2002)
H.H.H.C.0.5, H.H.H.C.1.0, N.H.H.C.0.5, N.H.H.C.1.0 HSLW0.5C, HSLW1.0C, NSLW0.5C, NSLW1.0C, NSNW.05C, NSNW1.0C 1C
Brown and Dilger (2003)
SJB7
Stark et al. (2005) Tan and Teng (2005)
C02, C63 YLL1, YLH2, YLL2
Robertson and Johnson (2006) Widianto et al. (2006) Kang and Wallace (2008)
ND1C, ND4LL, ND5XL, ND6HR, ND7LR SPControl 1, SP1 C0
Tian et al.(2008) Song et al. (2012)
L0.5, LG0.5, LG1.0 RC1
Inácio et al. (2015)
CHSC2, CHSC3
Marzouk et al. (2001)
Almeida et al. (2016) C50, C40, C30 * information complemented by Akiyama and Hawkins (1984)
Distinction between specimens reinforcement ratio part of an experimental campaign containing shear reinforced specimens reinforcement ratio reinforcement ratio, gravity loading part of an experimental campaign containing exterior connections gravity loading, unidirectional and bidirectional loading slab overhang, stiff edge beams control specimen part of an experimental campaign containing specimens with shear capitals gravity loading gravity loading; column dimensions, loading history type of concrete: high strength – normal strength; reinforcement ratio type of concrete: high strength or normal strength; lightweight or normal weight; reinforcement ratio control specimen in an experimental campaign containing shear reinforced specimens control specimen in an experimental campaign containing shear reinforced specimens reinforcement detailing gravity loading, unidirectional and bidirectional loading continuity of reinforcement, gravity load, reinforcement loading conditions control specimen in an experimental campaign containing shear reinforced specimens loading conditions and slab reinforcement control specimen in an experimental campaign containing shear reinforced specimens type of concrete: high strength or normal strength gravity loading (shear ratio)
Some examples of tests based on slab systems or multipanel slabs can also be found in literature, although they are scarcer. Robertson (1990) studied the adequacy of current procedures, code specifications and recommendations for flat slabcolumn connections for seismic regions. Nine specimens were tested, varying the shear ratio (ratio of the acting vertical load by the punching failure vertical load without eccentricity), slab shear reinforcement, slab overhang and the existence of a stiff edge beam. Durrani et al. (1995)
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tested four twobay slabs with geometry and test setup similar to Robertson (1990). Rha et al. (2014) tested five twobytwo bay slabs with varying reinforcement ratios and loading histories. Hwang and Moehle (2000) tested a nine panel specimen under vertical and horizontal cyclic loading to assess the flat slab behavior and bending moment distribution. In a different approach, Zaharia et al. (2006) reported a full scale structure test, representative of a threestory waffle flat slab building, aiming to study the general behavior of flat slab structures under seismic action and focusing on slabcolumn edges, corner and interior connections, as well as column orientation and cross section dimensions. The test was performed in a pseudo dynamic way instead of using symmetrical cycles. Another full scale test was reported by Fick et al. (2014), in which a threestory flat slab building was tested under reversed lateral loads.
2.2
Test results
Among the available tests reported in Table 1, only the specimens without shear reinforcement tested to failure under cyclic loading are reported here. Specimens tested under eccentric horizontal monotonic loading are not included, since they have potentially higher drift capacity. In Almeida et al. (2016) two similar slabs were tested: one under eccentric horizontal loading (E50) and another one under cyclic horizontal loading (C50) with the same shear ratio. E50 punched for an interstory drift of about 1.8%, whereas C50 failed for an interstory drift of only 1.1%, with lower horizontal load. Also left out were tests in which the load was monotonically increased to failure after the completion of a certain number of cycles (for instance, some of the specimens reported in Tian et al. [2008], Widianto et al. [2006] and all the specimens reported in Morrison et al. [1983]). Tests from Tan and Teng (2005) are also excluded due to the very large ratio between the sides of the column (900 mm (35.43 in) / 180 mm (7.09 in) = 5), which is very different from the rest of the specimens and is classified as a “shear wall” in most codes.
Figure 5:
Definition of “failure” in a typical unbalanced moment – drift ratio relationship.
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Table 2:
Main properties of specimens and test results
Author and year Hawkins et al. (1974)
ID
S1 S2 S3 S4 Islam and Park (1976) 3C Symonds et al. (1976) S6 S7 Zee and Moehle (1984) INT Pan and Moehle AP1 (1989, 1992) AP2 AP3 AP4 Robertson (1990) 8I 2C 3SE 5SO 6LL 7L Wey and Durrani (1992) SC0 Cao and Dilger (1993) CD1 CD5 CD8 Farhey et al. (1993) 1 2 3 4 Emam et al. (1997) HHHC0.5 HHHC1.0 NHHC0.5 NHHC1.0 Marzouk et al. (2001) HSLW0.5C HSLW1.0C NSLW0.5C NSLW1.0C NSNW0.5C NSNW1.0C Robertson et al. (2002) 1C Brown and Dilger (2003) SJB7 Stark et al. (2005) C02 C63 Robertson and Johnson ND1C (2006) ND4LL ND5XL ND6HR ND7LR Kang and Wallace (2008) C0 Tian et al. (2008) L0.5 Song et al. (2012) RC1 Inácio et al. (2015) CHSC2 CHSC3 Almeida et al. (2016) C50 C40 C30
262
d mm (in) 114 (4.50) 117 (4.63) 121 (4.75) 114 (4.50) 51 (2.03) 114 (4.50) 117 (4.63) 121 (4.75) 104 (4.08) 104 (4.08) 104 (4.08) 104 (4.08) 90 (3.53) 90 (3.53) 90 (3.53) 90 (3.53) 90 (3.53) 90 (3.53) 70 (2.75) 107 (4.21) 107 (4.21) 107 (4.21) 65 (2.56) 65 (2.56) 65 (2.56) 65 (2.56) 119 (4.69) 119 (4.69) 119 (4.69) 119 (4.69) 119 (4.69) 119 (4.69) 119 (4.69) 119 (4.69) 119 (4.69) 119 (4.69) 95 (3.74) 114 (4.49) 82 (3.24) 82 (3.24) 100 (3.94) 100 (3.94) 100 (3.94) 100 (3.94) 100 (3.94) 118 (4.65) 127 (4.99) 122 (4.80) 118 (4.65) 118 (4.65) 118 (4.65) 119 (4.69) 118 (4.65)
ρl % 1.32 0.89 0.55 1.32 1.15 1.88 0.89 0.68 0.75 0.75 0.75 0.75 0.71 0.71 0.71 0.71 0.71 0.71 0.65 1.32 1.32 1.32 0.77 0.77 0.77 0.77 0.50 1.00 0.50 1.00 0.50 1.00 0.50 1.00 0.50 1.00 0.61 1.32 1.17 1.17 0.60 0.60 0.60 1.04 0.41 0.54 0.49 1.25 0.96 0.96 0.96 0.96 0.96
fc MPa (ksi) 34.8 (5.05) 23.4 (3.40) 22.1 (3.20) 32.3 (4.69) 29.7 (4.31) 23.2 (3.36) 26.5 (3.84) 24.8 (3.60) 33.3 (4.83) 33.3 (4.83) 31.4 (4.55) 31.4 (4.55) 39.3 (5.70) 33.0 (4.79) 44.0 (6.38) 38.0 (5.51) 32.2 (4.68) 30.8 (4.46) 39.3 (5.70) 40.4 (5.86) 31.2 (4.53) 27.0 (3.92) 29.2 (4.23) 29.2 (4.23) 13.9 (2.01) 13.9 (2.01) 75.8 (10.99) 72.3 (10.49) 36.8 (5.33) 35.4 (5.13) 70.0 (10.15) 70.0 (10.15) 35.0 (5.08) 35.0 (5.08) 35.0 (5.08) 35.0 (5.08) 35.4 (5.13) 28.8 (4.18) 30.9 (4.48) 30.9 (4.48) 29.6 (4.29) 32.3 (4.68) 24.1 (3.50) 26.3 (3.81) 18.8 (2.73) 38.6 (5.60) 25.6 (3.71) 38.7 (5.61) 120.2 (17.43) 123.6 (17.93) 52.4 (7.60) 53.1 (7.70) 66.5 (9.65)
Gravity shear ratio ACI EC2 MC2010 0.34 0.33 N/A 0.45 0.45 N/A 0.44 0.51 N/A 0.42 0.39 N/A 0.24 0.24 N/A 0.89 0.71 N/A 0.81 0.83 N/A 0.31 0.35 N/A 0.35 0.40 0.46 0.35 0.40 0.46 0.18 0.21 0.24 0.18 0.21 0.24 0.18 0.22 0.19 0.19 0.22 0.19 0.16 0.19 0.18 0.18 0.21 0.19 0.54 0.63 0.52 0.38 0.44 0.36 0.24 0.29 N/A 0.94 0.88 N/A 0.71 0.64 N/A 0.57 0.50 N/A 0.00 0.00 N/A 0.00 0.00 N/A 0.23 0.27 0.24 0.23 0.29 0.24 0.25 0.35 0.43 0.25 0.28 0.23 0.36 0.44 0.43 0.36 0.36 0.29 0.31 0.64 0.49 0.31 0.51 0.30 0.43 0.81 0.50 0.43 0.64 0.36 0.36 0.45 0.50 0.36 0.36 0.30 0.15 0.19 0.10 0.51 0.45 N/A 0.38 0.41 0.39 0.38 0.41 0.39 0.24 0.28 0.19 0.27 0.32 0.23 0.46 0.53 0.34 0.28 0.27 0.17 0.24 0.30 0.23 0.30 0.37 N/A 0.23 0.31 N/A 0.43 0.43 N/A 0.32 0.39 0.41 0.33 0.41 0.43 0.49 0.52 0.51 0.40 0.43 0.42 0.28 0.31 0.31
dr,u % 3.8 2.0 2.0 2.6 4.1 1.1 1.0 3.9 1.6 1.5 3.7 3.5 4.3 5.0 4.6 3.5 1.5 1.4 4.5 1.0 1.1 1.3 5.6 4.1 3.7 1.9 5.2 4.9 3.7 2.8 6.0 5.9 4.4 5.2 4.1 4.9 3.5 3.4 2.9 2.2 6.0 4.0 2.0 5.0 5.0 1.9 2.1 1.4 3.0 3.0 1.0 1.5 2.0
FM P P P P P P P P P P P P P F P P P P P P P P P P P P F P P P F FP F FP F FP P P P P FP FP P P FP P P P P P P P P
ACIfib International Symposium Punching shear of structural concrete slabs
The main properties and test results for the specimens are summarized in Table 2. In this table, the average effective depth of the slab is denoted by “d” and it is calculated as the average effective depth of the slab measured in the x and y directions. The ratio of longitudinal reinforcement (ρl) is calculated as the geometric mean of the reinforcement ratios in x and y directions, which in turn are calculated based on the average bar spacing over the column width plus three times the average effective depth (d) on each side. The mean concrete strength (fc) measured in cylinders with a diameter of 150 mm (6 inches) and height of 300 mm (12 inches) is reported for each test. In cases where the concrete compressive strength tests were performed using different shapes or dimensions, the transformations summarized in Reineck et al. (2003) are used to make the conversion. The other columns of Table 2 contain the gravity shear ratio calculated in accordance with ACI 31814, EC2 and MC2010 (discussed further below) and the main test results represented by the drift at failure and the failure mode (abbreviated as “FM”). Since various authors have used different criteria to define “failure”, the drifts and unbalanced moments at failure of specimens from the literature may not refer to the same state of the specimen. In this study, the specimen is considered to have failed when the unbalanced moment (and lateral load) drops to 80% of its peak value. To avoid ambiguity, the drift at failure is then defined as the maximum drift attained in the last cycle before failure. A graphic explanation of the unbalanced moment at failure (Mu) and the corresponding drift dr,u are defined in Figure 5, where a typical unbalanced momentdrift relationship is shown. Based on this definition of failure, Mu may coincide with the maximum unbalanced moment (Mmax) for some specimens. When only lateral column displacement is available, drift is calculated taking into account the dimensions of the column and the test setup. For the test setups in which the effect of unbalanced moments is applied by imposing deformations on the slab ends, drift is calculated as in Pan and Moehle (1989). In case of specimens tested under bidirectional loading, the results are reported for the direction in which the maximum unbalanced moment was attained. The last column in Table 2 shows the modes in which the specimens failed. Flexural failure is denoted by “F”, punching shear failure is denoted by “P” and intermediate failure modes are reported as flexurepunching (“FP”).
3
Gravity shear ratio – code comparison
Gravity shear ratio is an important concept that must also be clearly defined before comparing available tests results with various codes. It is usually defined as the ratio of the gravity shear load at failure to the nominal punching strength of the slab for pure shear loading (i.e. without moment transfer) and it has been shown to have a major impact on the ability of flat slabs to resist cyclic lateral drifts (Robertson and Johnson, 2006; Pan and Moehle, 1989 and 1992), since the deformation capacity of interior slabcolumn connections decreases with the increase of the shear force on the connection. However, since various codes predict different values of punching shear strength, the gravity shear ratio reported in the literature for laboratory tests is dependent on the design code that was originally considered by the researcher. In order to overcome this issue, the gravity shear ratios for the available specimens are calculated based on ACI 31814, EC2 and MC2010 and compared to each other.
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The nominal punching shear strength was taken for all considered codes as: V0 = vc b0 d
(1)
where vc is the nominal shear strength (in stress units), b0 is the length of the perimeter at the critical section, and d is the average effective depth. The value of vc in ACI 31814 for nonprestressed normalweight concrete slabs is:
ì ï 0.33 fc ï ïï æ 2ö vc = min í 0.17 ç1 + ÷ fc β è ø c ï ï æα d ö ï0.083 ç s + 2 ÷ fc ïî è b0 ø ì ï 4 fc ï ïï æ 4 ö vc = min í ç 2 + ÷ fc ï è βc ø ïæ αs d ö + 2 ÷ fc ïç ïîè b0 ø
(inSI units:MPa, mm)
(2)
(in UScustomaryunits:psi, in)
where αs equals 40, 30 and 20 for interior, edge and corner columns, respectively; b0 is the length of the control perimeter located at a distance 0.5d from each point of the column’s perimeter (but allowed to be simplified as a rectangle for rectangular columns); fc is the concrete compressive strength; and βc is the ratio of the long side to the short side of the column’s cross section. When using the current version of Eurocode 2, the nominal shear strength for nonprestressed slabs is: vc = 0.18k (100 ρl fc ) vc = 5.0k (100 ρl fc )
1/3
1/3
³ vmin (inSI units:MPa, mm)
³ vmin
(3)
(in UScustomaryunits:psi, in)
where ρl is the geometric mean of reinforcement ratios in each direction (but limited to 0.02) and k is a factor accounting for size effect, as defined by the following expression: k = 1+ k = 1+
264
200 d 7.87 d
£ 2.0
(SI units:mm)
(4) £ 2.0
(US customary units:in)
ACIfib International Symposium Punching shear of structural concrete slabs
Eurocode 2 recommends a value vmin below which the shear strength calculated using Equation (3) need not be taken for design: vmin = 0.035k
3/2
vmin = 0.4215k
3/2
fc
(SIunits:MPa)
fc
(US customary units:psi)
(5)
According to MC2010, the nominal shear strength in terms of stress is calculated as:
vc = kψ fc
(6)
The parameter kѱ in Equation (6) depends on the rotation ѱ of the slab around the support region (loaded area, column), according to the expression below: kψ =
1 1.5 + 0.9k dg ψd
kψ =
£ 0.6
1 0.125 + 1.9k dg ψd
£ 7.2
(SIunits:mm)
(US customaryunits:in, psi for fc in the subsequent calculations)
(7)
where kdg is a parameter taking into account the influence of the maximum aggregate size (dg) in the punching shear resistance (Equation (8)). k dg = k dg =
32 16 + d g
³ 0.75
1.26 0.63 + d g
(SIunits:mm)
(8) ³ 0.75 (US customaryunits:in)
For the purposes of this paper, the gravity shear ratio is defined numerically as Vg/V0, where Vg is the slab shear force due to loading in the gravity direction and V0 is defined by Equation (1). In the application of Equation (1), it should be noted that the value of b0 is not the same in all the design codes. As it was previously shown, in ACI 31814 the perimeter can be considered to be a rectangle (for rectangular columns), whereas for EC2 and MC2010 this simplification is not allowed and the perimeter should be the minimum possible. Also, the distance of the control perimeter b0 from the perimeter of the column (loaded area in general) is d/2 in ACI 31814 and MC2010 but 2d in EC2. No partial safety or strength reduction factor is included in the expressions of vc above (Equations (2) to (8)). LevelofApproximation III was used when applying the MC2010. The values of the shear ratios calculated using Equations (2) to (8) for the specimens from literature are given in Table 2. Specimens in which it was not possible to calculate the shear ratio due to lack of data are noted by “N/A” (not available) in the corresponding cell in the table. The nominal shear strength calculated using Equations (2) to (8) corresponds to normalweight concrete slabs. For specimens HSLW0.5C, HSLW1.0C, NSLW0.5C and NSLW1.0C (Marzouk et al., 2001), which were made of lightweight concrete, the ACI shear strength (Equation (2)) was multiplied by 0.85 (as required in ACI 31814 and in compliance with the 265
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considerations of the authors of the experiment). In the EC2 calculations for lightweight concrete structures, the factor “0.18” in Equation (3) for SI units and the factor “5.0” for US customary units are changed to 0.15η1 and 4.167η1 respectively. The factor η1 depends on the density of lightweight concrete and for the four lightweight specimens of Table 2, its value is 0.67. In the MC2010 calculations, dg of Equation (8) is taken zero (as required by the code) for lightweight concrete.
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Figure 6:
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Code punching predictions as a function of the reinforcement ratio (fc = 30MPa = 4.35 ksi and d = 250 mm = 9.84 in).
Because the punching capacity provision expressions from ACI318, EC2 and MC2010 have different backgrounds, it is expected that the predicted punching capacity may differ for the very same specimen. Since this predicted value is also used to determine the flat slab’s shear ratio, which is crucial to flat slab seismic design, code comparison becomes important. To assess the differences between the three codes, a parametric analysis was made in which all slab characteristics were kept constant except for the concrete compressive strength, the effective depth and the reinforcement ratio. The fictional specimen was considered to have a slenderness (L/h) of 27, a square column with a 0.35×0.35 m (13.78×13.78 in) cross section and a maximum aggregate size of 16 mm (0.63 in). For the reinforcement bars, a yield strength of 500 MPa (72.52 ksi) and a Young’s modulus of 200 GPa (29007.54 ksi) were used. The remaining parameters were varied, one at the time as follows: the effective depth was changed from 150 mm (5.91 in) to 300 mm (11.81 in), the concrete compressive strength varied from 30 MPa (4.35 ksi) to 120 MPa (17.40 ksi) and the reinforcement ratio from 0.5% to 1.5%. For the EC2 calculations, the k parameter limitation for effective depths less than 200 mm (7.87 in) was applied (Equation (4)). For the ACI 31814 calculations, the concrete compressive strength was limited to a maximum of 69 MPa (10 ksi) as required by the code. The results are presented in Figure 6 and Figure 7. The results show that the differences between code predictions for the punching capacity may be significant, which means that shear ratios calculated according to each code may vary considerably. This can be also observed from the shear ratios presented in Table 2 for the experimental specimens. For low values of the longitudinal reinforcement ratio ρl, ACI 31814 gives higher values for the predicted punching capacity compared to EC2 and MC2010 (Figure 6). For larger values of ρl, ACI 31814 predicts a more conservative punching shear
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strength compared to the other two codes (the punching capacity prediction of ACI 31814 is lower than that of EC2 and MC2010 for high values of ρl). This is due to the fact that the ACI equations are not dependent on ρl (Equation (2)).
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With the increase of concrete strength fc, the punching capacity predictions of EC2 and MC2010 monotonically increase (Figure 7). Since the ACI 31814 code imposes a limit on the value of fc to 69 MPa (10 ksi), for slabs of higher concrete strength the code predicted punching capacity remains constant. All three code punching predictions increase with the effective depth “d” (Figure 7).
4
Proposal for EC2 and MC2010 based on the ACI 31814 prescriptive approach
Although flat slabs are generally not considered as part of the lateral force resisting system in seismic zones, they have to be able to carry the gravity loading when subjected to displacements induced by the seismic action corresponding to the design level earthquake. ACI 31814, Section 18.14.5.1, presents a prescriptive approach in which, instead of calculating the induced effects in the slabcolumn connection under the design displacement caused by the seismic action, it gives an alternative way of evaluating the connection based on a relationship between the design story drift ratio and the gravity shear ratio. The line defined in ACI 31814, Section 18.14.5.1 is shown in Figure 8 along with the data corresponding to the specimens presented in Table 2. That line is a reasonable lower bound limit for this data set, corresponding to specimens without shear reinforcement and tested under horizontal cyclic loading. Similar results can be found in Hueste et al. (2007), although this study also included specimens tested under eccentric monotonic horizontal loading and slabs containing shear reinforcement. 8 ACI 31814 Ch18 Limit Test Setup A Test Setup B Test Setup C Test Setup D Multi Frame
Drift [%]
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Figure 8:
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Test data and prescribed drift ratiogravity shear ratio limit line according to ACI 31814.
Following an approach similar to that of ACI 31814, a proposal based on a relationship between the design story drift ratio and the gravity shear ratio is given for the EC2 and MC2010 codes. For EC2 and MC2010 a characteristic line corresponding to a 5% percentile is shown. Similar to the concept described in ACI 31814, no shear reinforcement is required if the dot representing a certain slabcolumn connection is under the line. Otherwise, for dots above the line, shear reinforcement needs to be provided to the connection. An optimization algorithm was used to obtain the lines presented in Figure 9, considering that for a gravity shear ratio of 1, the story drift ratio should be 0 since it can be assumed that slabs with a unity shear ratio punch before eccentric loading can be applied. 268
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In Figures 8 and 9, the test setups A to D correspond to those described in Figure 1a to d, respectively. It was not possible to apply MC2010 for any of the specimens tested using test setup B (Figure 9b) due to lack of information. It can be observed in Figures 8 and 9 that the test setup influences the interstory driftshear ratio relationship. 8
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Test data and proposed drift ratio gravity shear ratio limit lines: a) EC2; b) MC2010.
In the general form, the proposed limit lines corresponding to EC2 and MC2010 can be obtained through the following equation:
d r = a × 10
 b× SR
+c
(9)
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Constants for the drift ratio–gravity shear ratio proposed limit lines for EC2 and MC2010 EC2
5
MC2010
a
b
c
a
b
c
4.82
0.83
0.71
4.64
0.94
0.53
Conclusions
The gravity shear ratio, defined as the gravity shear load to the nominal punching strength of the slab in pure shear loading, can significantly vary when different codes are used, because different codes provide different nominal punching resistances. The limit line of the prescriptive approach presented in section 18.14.5.1 of ACI 31814 is a sensible lower bound limit for the data set of specimens without shear reinforcement presented within this paper.
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The driftgravity shear ratio limit lines proposed for specimens without shear reinforcement based on EC2 and MC2010 following an approach similar to that recommended in ACI 31814, also show good agreement with the experimental results. Nevertheless, more future research is needed about this subject since a significant number of slabs presented in the databank have low effective depths with potential large scale effects. The presented results also show considerable scatter of interstory drifts for similar shear ratios. The test setup and load protocol influence the interstory driftshear ratio relationship. Furthermore, cyclic loading tests with low and high shear ratios are scarce.
6
References
Akiyama, H., and N. M. Hawkins (1984) "Response of Flat Plate Concrete Structures to Seismic and Wind Forces (Report SM 841)," Department of Civil Engineering, University of Washington, Seattle, Washington, USA. Almeida, A. F. O., M. M. Inácio, V. J. Lúcio and A. Pinho Ramos (2016) "Punching behaviour of RC flat slabs under reversed horizontal cyclic loading," Engineering Structures, vol. 117, no. p. 204–219. ACI (American Concrete Institute) Committee 318 (2014) Building Code Requirements for Structural Concrete (ACI 31814) and Commentary on Building Code Requirements for Structural Concrete (ACI 318R14), Farmington Hills, MI, USA: American Concrete Institute. ACI (American Concrete Institute), Joint ACIASCE Committee 352 (2012) Guide for Design of SlabColumn Connections in Monolithic Concrete Structures (ACI 352.1R11) Farmington Hills, MI, USA: American Concrete Institute. ACI (American Concrete Institute), Joint ACIASCE Committee 421 (2010) Guide to Seismic Design of Punching Shear Reinforcement in Flat Plates (ACI 421.2R10), Farmington Hills, MI, USA: American Concrete Institute. Brown, S. J., and W. Dilger (2003) Seismic Response of Slab Column Connections, Calgary, Alberta, Canada: University of Calgary. Cao, H., and W. H. Dilger (1993) Seismic Design of SlabColumn Connections (Master of Science Thesis), Calgary, Alberta, Canada: University of Calgary. CEBFIP (1993) CEBFIP Model Code 1990: Design Code, Thomas Telford, London, UK. CEN (European Committee for Standardization) (2004a) EN 199211, Eurocode 2: Design of concrete structures – Part 11: General rules and rules for buildings, Brussels, Belgium. CEN (European Committee for Standardization) (2004b) EN 19981, Eurocode 8: Design of structures for earthquake resistance – Part 1: General rules, seismic actions and rules for buildings, Brussels, Belgium. Durrani, A. J., Y. Du and Y. H. Luo (1995) "Seismic Resistance of Nonductile SlabColumn Connections in Existing FlatSlab Buildings," ACI Structural Journal, vol. 92, no. 4, pp. 479487. Emam, M., H. Marzouk and M. S. Hilal (1997) "Seismic Response of SlabColumn Connections Constructed with HighStrength Concrete," ACI Structural Journal, vol. 94, no. 2, pp. 197204. 270
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Farhey, D. N., M. A. Adin and D. Z. Yankelevsky (1993) "RC Flat SlabColumn Subassemblages Under Lateral Loading," Journal of Structural Engineering, ASCE, vol. 119, no. 6, pp. 19031916. fib (Fédération internationale du béton) (2013) fib Model Code for Concrete Structures 2010, Ernst & Sohn, Berlin, Germany. Fick, D. R., M. A. Sozen and M. E. Kreger (2014) "Cyclic Lateral Load Test and the Estimation of Elastic Drift Response of a FullScale ThreeStory FlatPlate Structure," ACI Special Publication SP296, Vols. SP296, no. 13, pp. 13.113.14. Hawkins, N. M., D. Mitchell and M. S. Sheu (1974) "Cyclic Behavior of Six Reinforced Concrete SlabColumn Specimens Transferring Moment and Shear (Progress Report 197374)," Department of Civil Engineering, University of Washington, Seattle, Washington, USA. Hueste, M. B. D., J. Browning, A. Lepage and J. W. Wallace (2007) "Seismic Design Criteria for SlabColumn Connections," ACI Structural Journal, vol. 104, no. 4, pp. 448458. Hwang, S.J., and J. P. Moehle (2000) "Vertical and Lateral Load Tests of NinePanel FlatPlate Frame," ACI Structural Journal, vol. 97, no. 1, pp. 193203. Inácio, M. M. G., A. Almeida and A. Pinho Ramos (2015) "Punçoamento em Lajes de Betão de Elevada Resistência Sujeitas a Ações Horizontais Cíclicas (in Portuguese  Punching of High Strength Concrete Slabs Subjected to Cyclic Horizontal Actions). Report No.4, Project HiCon," Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Lisbon, Portugal. Islam, S., and R. Park (1976) "Tests on SlabColumn Connections with Shear and Unbalanced Flexure," Journal of Structural Division, Proceedings of the American Society of Civil Engineers, vol. 102, no. pp. 549569. Kang, T. H.K., and J. W. Wallace (2008) "Seismic Performance of Reinforced Concrete SlabColumn Connections with Thin Plate Stirrups," ACI Structural Journal, vol. 105, no. 5, pp. 617625. Marzouk, H., M. Osman and A. Hussein (2001) "Cyclic Loading of HighStrength Lightweight Concrete Slabs," ACI Structural Journal, vol. 98, no. 2, pp. 207214. Morrison, D. G., I. Hirasawa and M. A. Sozen (1983) "LateralLoad Tests of R/C SlabColumn Connections," Journal of Structural Engineering, ASCE, vol. 109, no. 11, pp. 26982714. Morrison, D., and M. A. Sozen (1981) "Response of Reinforced Concrete PlateColumn Connections to Dynamic and Static Horizontal Loads," University of Illinois at UrbanaChampaign, Urbana, Illinois, USA. Pan, A. D., and J. P. Moehle (1992) "An Experimental Study of SlabColumn Connections," ACI Structural Journal, vol. 89, no. 6, pp. 626638. Pan, A., and J. P. Moehle (1989) "Lateral Displacement Ductility of Reinforced Concrete Flat Plates," ACI Structural Journal, vol. 86, no. 3, pp. 250258. Reineck, K.H., D. A. Kuchma, K. S. Kim and S. Marx (2003) "Shear Database for Reinforced Concrete Members without Shear Reinforcement," ACI Structural Journal, vol. 100, no. 2, pp. 240249.
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Rha, C., T. H.K. Kang, M. Shin and J. B. Yoon (2014) "Gravity and Lateral LoadCarrying Capacities of Reinforced Concrete Flat Plate Systems," ACI Structural Journal, vol. 111, no. 4, pp. 753764. Robertson, I. N. (1990) Seismic response of connections in indeterminate flatslab subassemblies (PhD Thesis), Houston, Texas, USA: Rice University. Robertson, I. N., T. Kawai, J. Lee and B. Enomoto (2002) "Cyclic Testing of SlabColumn Connections with Shear Reinforcement," ACI Structural Journal, vol. 99, no. 5, pp. 605613. Robertson, I., and G. Johnson (2006) "Cyclic Lateral Loading of Nonductile SlabColumn Connections," ACI Structural Journal, vol. 103, no. 3, pp. 356364. Song, J.K., J. Kim, H.B. Song and J.W. Song (2012) "Effective Punching Shear and Moment Capacity of Flat PlateColumn Connection with Shear Reinforcements for Lateral Loading," International Journal of Concrete Structures and Materials, vol. 6, no. 1, pp. 1929. Stark, A., B. Binici and O. Bayrak (2005) "Seismic Upgrade of Reinforced Concrete SlabColumn Connections Using Carbon FiberReinforced Polymers," ACI Structural Journal, vol. 102, no. 2, pp. 324333. Symonds, D. W., D. Mitchell and N. M. Hawkins (1976) "SlabColumn Connections Subjected to High Intensity Shears and Transferring Reversed Moments (Report SM 762)," Department of Civil Engineering, University of Washington, Seattle, Washington, USA. Tan, Y., and S. Teng (2005) "Interior SlabRectangular Column Connections Under Biaxial Lateral Loadings," in SP232 Punching Shear in Reinforced Concrete Slabs,, Farmington Hills, Michigan, USA, American Concrete Institute, pp. 147174. Tian, Y., J. O. Jirsa, O. Bayrak, Widianto and J. F. Argudo (2008) "Behavior of SlabColumn Connections of Existing FlatPlate Structures," ACI Structural Journal, vol. 105, no. 5, Sept.Oct., pp. 561569. Wey, E. H., and A. J. Durrani (1992) "Seismic Response of Interior SlabColumn Connections with Shear Capitals," ACI Structural Journal, vol. 89, no. 6, pp. 682691. Widianto, Y. Tian, J. Argudo, O. Bayrak and J. O. Jirsa (2006) "Rehabilitation of Earthquake Damaged Reinforced Concrete FlatPlate SlabColumn Connections for TwoWay Shear," in Proceedings of the 8th U.S. National Conference on Earthquake Engineering, San Francisco, California, USA. Zaharia, R., F. Taucer, A. Pinto, J. Molina, V. Vidal, E. Coelho and P. Candeias (2006) "Pseudodynamic Earthquake Tests on a FullScale RC FlatSlab Building Structure," European Communities, Institute for the Protection and Security of the Citizen, European Laboratory for Structural Assessment (ELSA), Ispra, Italy. Zee, H. L., and J. P. Moehle (1984) "Behavior of Interior and Exterior Flat Plate Connections Subjected to Inelastic Load Reversals (Report UCB/EERC84/07)," Earthquake Engineering Research Center, University of California, Berkeley, California, USA.
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Structural robustness of concrete flat slab structures Juan Sagaseta1, Nsikak Ulaeto1, Justin Russell2 1
: University of Surrey, Guildford, UK
2
: University of Warwick, Coventry, UK
Abstract Current building regulations for design against progressive collapse normally use prescriptive rules and riskbased qualitative scales, which are insufficient to cover current design needs. Structural robustness of concrete flat slab structures is examined using different theoretical models to capture the dynamic behavior under accidental events. In such extreme events, the large dynamic reactions at the connections could potentially lead to punching and progressive collapse. Punching formulae based on loaddeformation response relationships such as the Critical Shear Crack Theory (CSCT) are particularly useful in dynamic situations. The DuctilityCentred Robustness Assessment developed at Imperial College London is also used in this paper to derive simple design formulae to assess punching of adjacent columns in the sudden column removal scenario, which is commonly adopted in practice. The approach can be extended to assess flat slab systems when considering membrane action in the slab and postpunching behavior in the connections. Analytical models for tensile membrane are used in combination with the CSCT to demonstrate that the tying forces required in codes of practice cannot be achieved without prior punching of the connections. It is also shown that numerical modelling of postpunching is a promising tool to review detailing provisions for integrity reinforcement.
Keywords Punching shear, structural integrity, robustness, progressive collapse, flat slabs, accidental actions
1
Introduction
Reinforced concrete flat slabs, especially the flat plate variation without drop panels and column capitals, are commonly used in infrastructure and the construction industry due to their efficient spandepth ratio and uniform soffit. The design of flat slabs at ultimate is mainly governed by the detailing of the slabcolumn connection in order to provide sufficient punching shear and deformation capacity. Punching is a brittle type of failure with almost no warning signs. The shear resistance at a column after punching can be very low unless special considerations are made in design to mitigate this effect (e.g. integrity reinforcement). Therefore, punching of a column can cause large redistribution of loads, including an increase in shear and eccentricities (moment transfer) at the adjacent columns due to the irregular residual spans. These effects worsen due to the dynamic nature of the load redistribution, leading to an amplification of the reactions and slab deflections from the global response. 273
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Local failure at one column can propagate horizontally to adjacent columns in the slab as reported by Hawkins and Mitchell (1979) or Regan (1981); some examples of horizontal propagation collapses in underground car parking structures have been reported by Fernández Ruiz et al. (2013). Horizontal propagation of failure can lead to the slab falling onto the floor below, introducing large additional demands in the columnslab connections due to dynamic effects after the impact. This event can trigger the vertical progression of the collapse, leading to a disproportionate collapse to the original cause. Examples of vertical collapses with severe economic and social consequences can be found in America (e.g. Skyline Plaza Complex in Virginia, USA, 1973), Asia (e.g. Sampoong Department Store, South Korea, 1995) and Europe (e.g. Bluche underground car parking, Switzerland, 1981). It is worth noting that there are also examples in which the vertical progressive collapse was successfully arrested by the slab, leading to only partial collapse of the structure (e.g. Pipers Row car park in Wolverhampton, UK, 1997). This highlights the need to better understand the structural response of the system subjected to local failure in order to avoid introducing rules for design against progressive collapse that are unsafe or overly conservative. The horizontal and vertical propagation of failure is highly dependent on the response of the columnslab connections in terms of their load and deformation capacity. This response is influenced by dynamic effects and the development of alternative load path mechanisms that could develop in the slab for small and large deformations, such as compressive and tensile membrane action respectively. This paper analyses these relevant factors by means of simplified formulae based on theoretically sound approaches for punching and alternative load path analysis. A key aspect considered in this work is the use of loaddeformationbased formulae for punching, which allow the use of energy balance principles to estimate the peak dynamic deformation. In this context, the Critical Shear Crack Theory (CSCT) by Muttoni (Muttoni, 2008; fib, 2013) is used and extended to dynamic cases of column removal situations, which could be used in an alternative load path analysis for design for structural robustness. For the alternative load path approach, the DuctilityCentred Robustness Assessment (DRA) developed by Izzudin et al. (Izzudin et al., 2008; Izzudin and Nethercot, 2009; Vlassis et al., 2008) was adopted. This approach, which was originally derived for the assessment of progressive collapse of multistory buildings, has been widely recognized and applied to real steelframed composite multistory buildings (Vlassis et al., 2008) and more recently to flat slab concrete buildings (Liu, 2014). This paper shows that simple design formulae can be derived to assess punching at the adjacent connections after sudden column removal in flat slab concrete buildings (Fig. 1(a)) based on the DRA and CSCT approaches. A case study is shown of a real office building that was previously analyzed by the authors (Olmati et al., 2017) using a more complex dynamic nonlinear finite element time history analysis. The role of membrane action (compressive and tensile) and postpunching response of the connection is also discussed based on analytical and numerical modelling carried out in this work, which has been validated using existing experimental data from isolated slab tests.
2
Design against progressive collapse
Structural robustness or integrity is commonly defined as the insensitivity of a structure to local failure. Understanding the causes of progressive collapse in building structures is essential for the development of design methods to assess and improve structural robustness. The UK Building Regulations (Minister of Housing and Local Government, 1970; Department of Communities and Local Government, 2010) pioneered the inclusion of
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specific requirements against disproportionate collapse in codes of practice after the Ronan Point collapse in 1968. These general regulations were refined and used to feed into materialspecific design codes; for concrete structures these were covered by BS 8110 (BSI, 1997) in the UK and subsequently passed on to some extent to Eurocode 2 (CEN, 2004) in Europe. In the US, design against progressive collapse is currently covered by General Services Administration (GSA, 2000) and Department of Defense guidelines (DoD, 2009) and to a lesser extent by ACI 318 (ACI, 2014) (section 7.13 and 13.3.8.5 for twoway slabs). An extensive review in this field by Arup (2011) showed that there are four recognized methods to design for structural robustness in general structures: (i) tying force provisions, (ii) alternative load path methods, (iii) key element design and (iv) riskbased methods. Tying force provisions are generally recommended for structures with a low risk of progressive collapse. Eurocode 2 (CEN, 2004) (section 9.10) defines a set of prescriptive detailing rules for structures that are not specifically designed to withstand accidental actions, which aims to provide a suitable “tying system” as shown in Fig. 1(c). Similarly, ACI 318 provides structural integrity requirements for detailing, which is only intended to provide “improved redundancy and ductility” (ACI, 2014). According to Eurocode 2, the objective of the tying system is to provide “alternative load paths” after local damage by means of different mechanisms (CEN, 2004). However, it has been questioned over the years whether the structure can achieve sufficient deformations to develop these mechanisms, as prescriptive rules do not explicitly check the ductility of either the connections or the system (Izzudin and Nethercot, 2009; Arup, 2011; Merola, 2009). Moreover, the tying force requirements in Eurocode 2 were originally conceived for precast panel construction under blast or extreme wind conditions, similar to that of the Ronan Point collapse, and it is questionable whether these rules are directly applicable to other types of construction, such as flat slabs under progressive collapse conditions. Flat slabs provide enhanced continuity and alternative load paths due to twoway bending and membrane actions (Qian and Li, 2013, 2016; Russell et al., 2015); however, they are prone to punching at small slab deflections. In this paper, it is shown analytically that punching would occur prior to the formation of the tensile membrane action (TMA) required by the tying system. Increasingly popular are the alternative load path methods for considering progressive collapse, which offer a quantitative and more performancebased type of analysis, although several simplifications are generally required (Izzudin et al., 2008; GSA, 2000; DoD, 2009). For example, scenarioindependent approaches may be adopted (the hazardtriggering local failure is not considered), which requires caution as key factors can be overlooked. In flat slab construction, some examples in which scenariodependent approaches might need to be considered include fire or blast situations due to the continuity and redundancy of the structure (e.g. uplift pressures in blast or restraint strains after thermal gradients under hightemperature conditions). Otherwise, sudden column removal (scenarioindependent) is a widely accepted approach, which provides a standard dynamic test of structural robustness that can cover different extreme events (GSA, 2000; DoD, 2009). There is a noticeable growing tendency in research and practice to focus on energybased approaches as described by Izzudin and Nethercot (2009), which look at the deformation energy (strain energy) done by the external forces and can be also analyzed as the flux of energy during the collapse (Szyniszewski et al., 2009).
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(a)
(b)
(c) Figure 1:
Internal column removal scenario in concrete flat slab building: (a) SDOF approximation; (b) compatibility condition between vertical displacement and slab rotation; and (c) tying approach.
A limitation of the application of the alternative load path approach in concrete flat slab design is the general lack of knowledge of the dynamic response (load and deformation capacity) of the columnslab connections. Design formulae for punching shear have traditionally focused on load capacity only and therefore they can only be used in loadcentered alternative load path methods, such as the simplified static modelling proposed by DoD (2009). The main drawback in such methods is the need to define load dynamic amplification factors (𝜆" ), which is cumbersome (Izzudin and Nethercot, 2009). A value of 𝜆" equal to 2 is normally adopted in design for concrete structures, which corresponds to a linear response without damping and therefore provides overly conservative results (GSA, 2000; DoD, 2009). Test results on concrete flat slabs by Qian and Li (2016), Russell et al. (2015) and dynamic FE analysis by Liu (2014) and Olmati et al. (2017) show consistent values of 𝜆" between 1.2 and 1.6, depending on the gravity load and extent of the damage in the slab. The proposed simplified approach presented in this paper for assessing punching of adjacent columns under column removal situations provides values of 𝜆" consistent with those described in the literature Liu, 2014; Olmati et al., 2017; Russell et al., 2015). If alternative load path analyses fail to demonstrate suitable capacity for the redistribution of loads, riskbased methods and/or the key element design method can be adopted. In the UK, for buildings above fifteenstory and/or of large occupancy (Class 3 buildings; Department of Communities and Local Government [2010]), a systematic risk assessment is 276
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required, which might include the design of specific key elements or alternative load paths. The design of key elements (removal of which will lead to the collapse) could be based on prescriptive loads, such as (Department of Communities and Local Government, 2010), or on actual loads from hazardspecific guidelines (e.g. DoD [2009]). Significant work is currently being carried out on riskbased approaches as recognized in a recent COST project (COST, 2008) and more emphasis is being made on probabilitybased approaches in which the uncertainty of basic parameters is included in the analysis. For instance, a simplified reliability analysis was proposed by Olmati et al. (2017) in which the magnitude of the gravity applied load is considered to be a stochastic variable; this is a relevant parameter affecting the dynamic response of the flat slab structure, as discussed in the following sections.
3
Response of flat slab buildings under accidental actions
Extreme events such as industrial accidents (e.g. vehicle impact, blast), unexpected local failure due to overloading or poor design, extreme weather, or fire can lead to significant local damage. Such damage can include column failure, flexural failure of the slab, punching of the slabcolumn connection and spalling of concrete. The latter can be relevant because the size of debris and impact in other areas in the structure might be significant and could trigger progressive collapse due to overloading. It is worth noting that most of these events have a strong dynamic component; even local shear failures from a nondynamic source (e.g. punching around a connection due to quasistatic loading) trigger a dynamic redistribution of loading due to the sudden loss of strength. Most codes acknowledge that the design against all possible accidental scenarios is not feasible, since extreme events are normally unforeseen events and engineering judgement should prevail. However, the lack of theoretical models looking at the local and global response makes it difficult for the designer to tackle some of these issues. 3.1
Local vs. global response
The study of local damage is normally carried out for single structural elements as shown schematically in Fig. 2(a) for localized punching of a flat slab due to closein detonation and impact. Analytical models exist to assess local perforation in such cases, which can be based on SDOF models and the CSCT for punching (e.g. Micallef et al. [2014] and Sagaseta et al. [forthcoming] for impact and blast cases respectively). Work by Micallef et al. (2014) showed that the punching capacity increases with strainrates, as shown in Fig. 2(c), due to the increase in material strength with loading rate (increased residual tensile strength and aggregate interlock action along the critical shear crack). In cases of impulsive behavior, such as those shown in Fig. 2(a), punching may occur a short time after the action, while the deformations are small compared to the peak deflection. The global response of the slab in such cases has been shown to have a negligible effect on the occurrence of punching, which is primarily governed by the large shear demand around the localized action (Micallef et al., 2014; Sagaseta et al. (forthcoming). Due to the continuity and monolithic nature of flat slab systems, the interaction between local and global response is not straightforward and the development of hybrid models (localglobal) is useful for different types of loading with different load durations.
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(a)
(b)
(c) Figure 2:
Local vs. global response: (a) punching of the slab during local impulsive behavior; (b) punching of the columnslab connection at peak dynamic response; and (c) influence of slab rotation and strainrate on punching strength according to CSCT (Muttoni, 2008) and Micallef et al. (2014).
In cases where local failure is not critical, the dynamic loads in the slab are amplified and transferred to the columnslab connections (dynamic reactions) by means of the global dynamic response, as shown schematically in Fig. 2 (b). In this case, the maximum reactions take place at the time of maximum peak deflections (i.e. maximum resistance in the flexural response). The magnitude of the gravity loads applied in the slab (dead 𝐷 + live 𝐿 loads) is relevant as it influences the response of the system, which in turn affects the response of the connections (Olmati et al., 2017; Russell et al., 2015). As discussed by Russell et al. (2015), the increase in gravity loads affects the response of the system due to (a) changes in the natural frequency due to mass changes and (b) increase in damage to the structure, which reduces the stiffness and natural frequency, leading to an increase of the peak response. Punching in both local and global situations can be examined using the CSCT (Muttoni, 2008; fib, 2013), which considers that the punching capacity reduces with increasing slab rotation 𝜓 as shown in Fig. 2(c) (strain effect on punching shear). The slab rotation 𝜓 is measured outside the critical shear crack, as shown in Fig. 1(b) and 2(c). In local dynamic punching of the slab, failure occurs when the capacity is large (i.e. low rotations), whereas in cases of punching around columnslab connections, shear failure might occur at the moment of lowest capacity (i.e. large rotations during peak deflections). The shear demand in both situations can be extremely large and therefore punching could take place prior reaching the maximum global deformations. The simplified approach presented in this paper for the columnslab connections considers the dynamic effects described above. 278
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It is worth noting that whilst the proposed approach in this paper is relevant towards assessing structural robustness, the occurrence of local punching might not necessarily imply the development of progressive collapse, which should be assessed on a systembased analysis taking into account punching and postpunching mechanisms. The nonlinear response of the slab can activate different mechanisms (Mitchell and Cook, 1984) including flexure, compressive membrane action (CMA) and tensile membrane action (TMA) as shown in Fig. 3. These mechanisms have traditionally been investigated separately and their interaction with punching is not well known. Punching at the connections is observed in regions 2 and 3 (Fig. 3), in which CMA and flexure are predominant; the snapthrough behavior in region 4 is followed by full catenary action that could potentially develop after punching. Punching shear models based on deformations (e.g. CSCT) have the potential to capture the transition between these four regions shown in Fig. 3, as discussed further in this paper.
Figure 3:
Structural response of flat slab structures (adapted from Mitchell and Cook [1984]).
4
Assessment of punching of adjacent columns after column removal
4.1
Idealization of column damage
The sudden column removal scenario as proposed in codes of practice (GSA, 2000; DoD, 2009) is probably the most common type of design situation considered in the assessment of structural robustness. This approach studies the indirect response of the building subject to an idealized theoretical structural damage with focus on the ability of the structure to find alternative load paths. In this scenario, the selected support is removed instantaneously, which provides an upper bound in terms of the dynamic response. Whilst the concept and purpose behind this approach is clear, the approach is not meant to cover all accidental actions as recognized by DoD (2009), since the nature of the load causing the failure of the column is not considered. As reported by (Arup, 2011), there is scarce information in the literature regarding whether this is a realistic or overly conservative approach. The sudden column removal neglects any residual strength and stiffness of the column after local failure, which is clearly not the case for instance when the triggering event (local failure) is punching at the 279
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support, as shown by testing on postpunching (Fernández Ruiz et al., 2013; Melo and Regan, 1998; Habibi, 2012). As shown by Keyvani and Sasani (2014), even in demolition tests in a posttensioned flat slab building in which a column was exploded intentionally to remove it completely, some residual resistance was observed at the column due to residual deformations and some reinforcement bars surviving the explosion. 4.2
Demandcapacity ratio of adjacent column after sudden column removal
The assessment of punching of adjacent columns after sudden column removal is challenging due to the uncertainty in the magnitude and role of the main parameters involved. Equation (1) shows the main factors affecting the demandcapacity ratio (𝐷𝑅) of the adjacent column after this event (𝐷𝑅 >1, meaning that punching will occur). The contribution of each parameter is described in this section. DR = DR0 (∆V λd )∆M ∆S
(1)
Parameter 𝐷𝑅( is the demandcapacity ratio of the connection being investigated for a specified loading prior to the removal of an adjacent column, ∆* is the increase ratio of static shear force at the adjacent column after the column removal, 𝜆" is the dynamic load amplification factor, ∆+ is the increase ratio of shear demand due to moment transfer increase and ∆, is the decrease ratio of punching capacity due to the increase in span length (i.e. increased slenderness). 𝐷𝑅( is influenced by the gravity load considered and the design punching capacity of the connection (𝑉." ) considering nominal sizes of rebar and number of effective shear reinforcement provided around the column. Regarding the gravity loads, in this work Eurocode 0 (CEN, 2002) is used, which gives combination of actions for accidental loading including the quasipermanent (D+0.3L) and frequent (D+0.5L) combination cases in buildings with load categories A (domestic, residential) and B (office areas). In terms of static behavior, it is widely assumed that in slabs with equal spans in both directions, the load carried out by the removed column is transferred equally to the four closest adjacent columns, leading to an increase in the total shear at the adjacent columns of 25% (i.e. ∆* = 1.25). A linear finite element analysis (LFEA) of a flat slab with infinite number of spans on either side and 𝐿0 /𝐿2 = 1 (column size to effective depth 𝑐/𝑑 between 0.5 and 2 and column size to span 𝑐/𝐿 between 0.025 and 0.1) shows that ∆* can vary between 1.33 and 1.35 (Fig. 4(a)), which is due to the unbalanced load and unloading of the second row of columns (slab continuity). The corner column adjacent to the removed one does not contribute significantly to this redistribution; the contour diagrams in Fig. 4(a) show that the deformations around this column are similar to columns two spans from the removed one. Parameter ∆* can increase significantly in buildings with irregular distribution of columns. For instance, if the span length in one direction is significantly shorter than in the other direction (e.g. 𝐿0 /𝐿2 >1.5) then the load carried out by the removed column can be transferred almost entirely to the closest columns (i.e. ∆* ≈1.5) as shown in Fig 4(a). In slabs with irregular bays with columns with different contributable areas, the removal of highly demanded columns can result in values of ∆* closer to 2, as shown by Olmati et al. (2017). Moreover, the increase in total shear in the column will be amplified due to dynamic effects by a factor 𝜆" ,which could vary depending on the gravity loads and column distribution.
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(a) Figure 4:
(b) Influence of column removal on: (a) static reaction in rectangular column layouts (contour diagrams indicate the deflections in the slab for different cases of 𝐿8 /𝐿9 ); and (b) distribution of shear forces along control perimeter at 0.5d from column face for 𝐿8 /𝐿9 = 1.5 according to shear field analysis.
Column removal results in an increase in the moment transfer in the adjacent columns, which in turn will cause a high concentration of shear forces in the control perimeter at the side of the removed column as shown in Fig. 4(b) obtained from a shear field analysis (thin lines represent the flow of principal directions of the shear stress vectors and thick lines represent the magnitude of the shear per unit length along the control perimeter at 0.5𝑑 from the columm face). In design codes, moment transfer is considered to be an increase in shear stress demand by means of increase factors based on a given shear distribution (Eurocode 2, ACI 318). Alternatively, other codes such as Model Code 2010 use a reduced length of the control perimeter, also known as “shear resisting control perimeter” by means of a reduction factor 𝑘; . Therefore, parameter ∆+ can be easily obtained using existing design formulae in the different codes (Model Code 2010, EN 199211, ACI 318). It is worth noting that due to the irregular residual spans, recommended constant values for these factors normally do not apply and formulae based on obtained eccentricities need to be used in such cases. Dynamic effects are likely to influence ∆+ although the variations observed in the shear stress demand factors are small with respect to those obtained from a static analysis with residual spans (Olmati et al., 2017). The influence of slenderness on punching has been investigated by Einpaul et al. (2016), showing that an increase of slenderness in isolated tests leads to a decrease in stiffness of the loadrotation response of the slab and hence a reduction in punching capacity due to larger crack widths. This reduction in punching capacity ∆, is more significant for slabs with shear reinforcement (Einpaul et al., 2016) and is not considered in current Eurocode 2 and ACI 318 provisions. In a regular orthogonal column distribution, the slenderness of the critical span will double after the loss of one column. Assuming a common office building in the UK with a slenderness (0.22𝐿)/𝑑 of around 5, the column removal will result in a punching capacity reduction according to the CSCT of around 20% (∆, = 0.80) for a connection with shear reinforcement and heavily reinforced in flexure (e.g. 𝜌 = 1.5%), and as low as 10% (∆, = 0.9) if the connection has no shear reinforcement. In addition, dynamic amplification of the deflections will result in a further reduction of the punching capacity due to wider cracks. As shown in the next section, this dynamic effect could be captured using the CSCT with an 281
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increased equivalent static slenderness or “pseudostatic slenderness”. This is a similar analogy, but opposite to that observed in local punching in cases of impulsive impact and blast, in which the small flexural deformations at punching could be viewed as a “reduced effective slenderness”, which is also referred to by in the literature as “reduced span” (Micallef et al., 2014; Delhomme et al., 2007). In this paper, it is shown that for quasistatic column removal, the effect of increased slenderness on the flexural response due to the increase in span lengths is captured accurately using Model Code 2010 formulae (fib, 2013). The example shown in Fig. 5 (a) and (b) corresponds to a fourstory office building described in Technical Report 64 (Concrete Society, 2007), which is commonly referred to in flat slab design in the UK. Equation (2) is proposed in this work, based on the loadrotation relationship given by (fib, 2013) (Level of Approximation LoAII), considering the increase in slenderness after the column removal at a shear force equal to 𝑉( . Figure 5(c) shows that the predictions from Eq. (2) are comparable with those obtained by Olmati et al. (2017) from a nonlinear finite element analysis (NLFEA) with quasistatic loading. Figure 5(c) also shows that the sudden change in slope in the loadrotation curve (Eq. (2)) after column removal is captured correctly by increasing the slenderness from 𝑟E( /𝑑 = 0.22𝐿( /𝑑 to 𝑟EF /𝑑 = 0.22𝐿F /𝑑 where 𝑟E is the approximate distance from the support axis to the point of zero radial bending moment (subscripts 0 and 1 denote before and after column removal respectively) and 𝐿( and 𝐿F are the length of the original and residual span respectively (𝐿( = 6 m [19.7 ft] and 𝐿F =12m [39 ft] for the case considered). 𝑟EF 𝜓 = 1.5 𝑑
𝑓0 𝐸E
𝑉 − 0.37𝑉( 𝑘𝑚.
N/O
, ∀𝑉 > 𝑉(
(2)
where according to Model Code 2010 (fib, 2013), 𝑘 = 8(2𝑏E )/(2𝑏E + 8 𝑒U,V ) and 𝑏E is the width of the support strip (normally equal to 1.5𝑟E for internal columns with regular span layout) and 𝑒U,V is the eccentricity of the resultant of the shear forces with respect to the centroid of the basic control perimeter in the direction investigated (direction of residual span). For shear forces 𝑉 lower than 𝑉( , Eq. (2) still applies by replacing terms (𝑟EF /𝑑) and (𝑉 − 0.37𝑉( ) by (𝑟E( /𝑑) and 𝑉 respectively. A more refined analysis (LoAIII [fib, 2013]) was carried out using a LFEA showing very small differences in the values adopted for 𝑟E and 𝑘; in particular, parameter 𝑘 was close to 8 using LoAII and III for the eccentricities obtained in the case study shown in Fig. 5. Evaluating Eq. (1) using plausible values in design shows that 𝐷𝑅 can easily reach values near 0.8 for frequent load combinations (D+0.5L). For example, the case study investigated by Olmati et al. (2017) (Fig. 5) for the load combination (D+0.5L) resulted in the following: 𝐷𝑅( = 0.3, ∆* = 1.7, ∆+ = 1.15 and ∆, = 0.85, which gives 𝐷𝑅 = 0.5𝜆" . Assuming a conservative dynamic factor of 2 will result on imminent punching (𝐷𝑅 = 1.0), whereas assuming more realistic values of 𝜆" (e.g. 𝜆" =1.3 from a refined dynamic analysis [Omati et al., 2017]) shows that punching is not predicted to occur. This justifies the use of dynamic analysis to assess 𝜆" and ∆, in a more rigorous and systematic manner. A dynamic nonlinear analysis is considered the most theoretically rigorous and refined method to address this problem, however this approach is highly complex and computationally demanding. A simplified approach is presented in the following section based on a nonlinear static pushover analysis and simplified dynamic response based on energy balance as proposed by Izzudin et al. (Izzudin et al., 2008; Izzudin and Nethercot, 2009; Vlassis et al., 2008). 282
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(a)
(b)
(c) Figure 5:
4.3
Nonlinear static response of column B2 after removal of column C2: (a) plan view of flat slab structure; (b) shear reinforcement at B2; and (c) loadrotation response Eq. (2) and quasistatic NLFEA (Olmati et al., 2017).
Application of the DuctilityCentred Robustness Assessment (DRA)
The DRA proposed by Izzudin et al. (Izzudin et al., 2008; Izzudin and Nethercot, 2009; Vlassis et al., 2008) is applied to flat slab structures in this work. This approach can be applied to different levels of structural idealization. For instance, Liu (2014) used DRA and CSCT in their analysis of a concrete flat slab building system by means of a macro numerical model. In this paper, the focus is on the isolated response of the columnslab connection (subsystem) until punching failure; simplified analytical equations are derived for this particular case. As reported by Izzudin and Nethercot (2009), ductilitycentered approaches such as the DRA allow consideration of three critical factors for structural robustness, namely, energy absorption capacity, redundancy and ductility supply. Moreover, this approach proposes the system pseudostatic capacity as a single measure of structural robustness. The DRA consists of three steps: (i) determination of the nonlinear static response of the system considered, (ii) dynamic assessment using a simplified approach based on energy balance and (iii) ductility assessment of the connections by means of compatibility conditions between the system and
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the subsystem. In this case, the flat slab system represents the bay of the removed column and the adjacent columns with the floors above; this system can be modelled as a single degree of freedom (SDOF) system consisting of the vertical deflection at the point of the removed column 𝑢 as shown in Fig. 1(a). The static response of the flat slab system is clearly nonlinear due to cracking of the concrete and yielding of the reinforcement in the slab in the hogging and sagging regions (Fig. 3). Loadrotation relationship (Eq.2) can be adopted in which the reactions, and hence the gravity load, are proportional to 𝜓 O/N , where 𝜓 is the slab rotation in the direction of the longest span outside the column region. The rotation of the slab varies along the span, from zero at the column to a maximum value right outside the column region, and then remains constant and gradually reduces to zero at midspan. The exact distribution of slab rotation will vary depending on the loading and amount of hogging and sagging reinforcement in the column and midspan regions respectively due to moment redistribution (Einpaul et al., 2015). The compatibility condition between the system and subsystem degrees of freedom is in this case simple to use 𝑢 ≈ 1.5𝐿( 𝜓 (Fig. 2(b)), which has been verified by the authors against experimental and numerical results of continuous slabs with different hogging and sagging reinforcement ratios (Einpaul et al., 2015). It is worth noting that the value of the constant of proportionality of this relationship is not relevant in the dynamic study. It can be concluded that the nonlinear static response of the system until punching will follow a parabolic relationship 𝑃 ∝ 𝑢O/N , as shown in Fig. 3. The linear elastic response (region 1 in Fig. 3) is neglected as commonly assumed in punching assessments in quasistatic loading; the loads causing punching normally result in cracking and crushing of the concrete during yielding of the flexural reinforcement. This assumption holds true for dynamic punching in columnslab connections governed by the global response of the structure. The second step according to the DRA involves an energy balance calculation in order to obtain the peak dynamic vertical deflection of the system (𝑢"0Z ). The total gravity load applied in the system (𝑃) after the column removal can be divided into two components: one is a static component that is resisted by the static reactions after the column removal (𝑉F ) and the other component is dynamic that is equal to the reaction of the column removed that is applied suddenly into the system. As shown by Izzudin et al. (2009), the application of a sudden load will introduce an amplification of the deflections so that the maximum dynamic response is obtained when the kinetic energy of the system is zero and therefore the external work introduced by the gravity loads equals the internal energy absorbed by the system. For instance, in the linear system shown in Fig. 6(a), if the load 𝑃E[\[ is applied gradually, the system will deflect 𝑢 (external work introduced = internal strain energy absorbed), whereas if the same load were applied suddenly, the deflection would need to be 𝑢"0Z = 𝜆U 𝑢 = 2𝑢 to balance external work and internal energy (𝜆U is the dynamic deflection amplification factor). If the system follows a parabolic relationship 𝑃 ∝ 𝑢O/N , it can be easily demonstrated that the deflection would need to be 𝑢"0Z = 𝜆U 𝑢 = 2.15𝑢. If the system follows a parabolicplastic behavior as shown in Fig. 6(b), with a limit value 𝑃]V^ , factor 𝜆U will be constant and equal to 2.15 for static gravity loads lower than 𝑃]V^,_E (e.g. load 𝑃E[\[,F ), whereas for static gravity loads higher than 𝑃]V^,_E (e.g. load 𝑃E[\[,O ) 𝜆U will increase significantly in order to achieve balance between internal strain energy and external work (shaded areas in Fig.6(b)). In Fig.6(b), 𝑃]V^,_E is the static gravity load giving a dynamic deflection 𝑢"0Z which is equal to 𝑢]V^ , where 𝑢]V^ is the value of the deflection in the static response at which 𝑃]V^ is reached. The applied load versus maximum displacement obtained from the energy balance approach 284
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is defined by Izzudin and Nethercot (2009) as the “pseudostatic response,” as it can be obtained using static analysis. Figure 6(b) shows that for a given static gravity load 𝑃E[\[ , the peak deflection can be obtained from the pseudostatic response and the corresponding amplified dynamic load (𝑃"0Z = 𝜆" 𝑃E[\[ ), can be obtained from the nonlinear static response. The pseudostatic response (𝑃E[\[ , 𝑢"0Z ) can be written in terms of the static shear force around the column and pseudostatic rotation (𝑉, 𝜓_E ), as shown in Eq. (3). Equation (3) was derived using the static response (𝑉, 𝜓) from Eq. (2) and applying the energybased principles by multiplying 𝜆U to the difference in slab rotation (𝜓 − 𝜓( ) to obtain 𝜓_E , where 𝜓( is the slab rotation near the column investigated before the column removal. Equation (3) is used subsequently in the ductility assessment in the last step of the DRA approach. 𝜓_E
𝑟EF = 1.5 𝜆U 𝑑
𝑓0 𝐸E
𝑉 − 0.37𝑉( 𝑘𝑚.
N/O
(𝜆U − 1) 𝑉( − 2𝜆U 𝑘𝑚.
N/O
, ∀𝑉 > 𝑉(
(3)
where 𝜆U is the dynamic deflection amplification factor to the slab rotation near the column, which is equal to 2.15 for 𝜓_E ≤ 𝜓]V^ , where 𝜓]V^ corresponds to the deflection 𝑢]V^ shown in Fig. 6(b). In the plastic region (𝜓_E > 𝜓]V^ ), factor 𝜆U is given by Eq. (4), which was also derived from energy balance (Fig. 6(b)): a/N
𝜆U =
𝜓_E O/N
a/N N/O
𝜓_E 𝜓]V^ − 0.4𝜓]V^
, ∀𝜓_E > 𝜓]V^
(4)
The pseudostatic response of the connection given by Eq. (3) follows a similar form to the static response except for the introduction of coefficient 𝜆U obtained from energy balance. In Eq. (3), term (𝜆U 𝑟EF /𝑑) can be interpreted as an equivalent “pseudostatic slenderness” that accounts for dynamic amplification effects. The negative term in Eq. (3) accounts for the fact that the dynamic amplification takes place from the loadrotation point (𝑉( , 𝜓( ).
(a) Figure 6:
(b) Pseudostatic response according to simplified dynamic assessment (Izzudin et al., 2008): (a) linear response and (b) parabolicplastic response proposed in this work.
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The dynamic amplification factor of the load (𝜆" ) can be obtained separately from the static and the corresponding pseudostatic response. For a linear system, this approach gives 𝜆" = 2 as expected, whereas for a linearplastic system this factor reduces significantly for loads near the plastic region. For the columnslab connections under investigation, Eq. (2) and (3) are used to calculate 𝜆" leading to Eq. (5) for loads below the plastic region 𝑉( 𝑉( 𝜆" = 0.37 𝑉 + 𝜆U 1 − 0.37 𝑉
N/O
(𝜆U − 1) 𝑉( − 2 𝑉
N/O O/N
, 𝑉( < 𝑉 < 𝑉]V^ /𝜆"
(5)
where 𝜆U is 2.15 for this region. For loads near the plastic region where 𝜆U increases significantly according to Eq. (4), the dynamic load factor is simply 𝜆" = 𝑉]V^ /𝑉, which decreases tending to 1 for large rotations, assuming that only flexural behavior is activated (i.e. without tensile membrane action). The reduction of 𝜆" for ductile systems has been recognized for some time and it has been considered by codes of practice (DoD, 2009) for some types of construction. It is worth noting that the activation of tensile membrane action for large deformations would result in a sudden increase of 𝜆" (Izzudin and Nethercot, 2009), however in flat slabs, punching of the connection would precede in cases such as those shown in the next section. The proposed Eq. (5) for punching gives values of 𝜆" between 1.67 and 1.2 depending on 𝑉( /𝑉, which is a function of the column distribution. These values are consistent with those obtained experimentally (Russell et al., 2015) or numerically (Olmati et al., 2017) for flat slabs with similar levels of gravity loads. For cases of sudden corner column removal, Qian and Li (2013) reported values of 𝜆" between 1.13 and 1.23. In order to carry out the ductility assessment in the third step of the DRA, the failure criterion in the CSCT is applied. The proposed method to assess punching shown in Fig. 7 is described as follows: • For a given static gravity load (𝑃E[\[ ), obtain the static shear before (𝑉( ) and after (𝑉F ) the column removal. • Obtain the corresponding pseudostatic rotation 𝜓de for 𝑉 = 𝑉F from Eq. (3). • For the obtained pseudostatic rotation, calculate the dynamic demand 𝑉"0Z = 𝜆" 𝑉F from Eq. (5) or using the static response, and the dynamic capacity 𝑉.,"0Z from the CSCT failure criterion (Muttoni, 2008; fib, 2013). • Check whether the demand is lower than the capacity 𝑉"0Z < 𝑉.,"0Z ; if so, the procedure indicates there should not be punching of the adjacent column under consideration.
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Figure 7:
Proposed approach for punching assessment of sudden column removal based on the CSCT and pseudostatic response from DRA.
As shown in Fig. 7, the philosophy behind the proposed approach is similar to punching for quasistatic loading due to the use of the pseudostatic relationship. The proposed method is simple and it allows both load and deformation capacity to be checked taking into account dynamic effects in a consistent manner. The dynamic factors are automatically taken into account and refined predictions of the static response can be obtained by increasing the LoA adopted. Strainrate effects could be considered in the punching capacity and demand. For the punching shear capacity, refining the failure criterion as a function of the strainrate is possible (Micallef et al., 2014) as shown in Fig.2(c). For the punching shear demand, a refined loadrotation response can be considered using material enhancement factors dependent on the strainrate (fib, 2013), as shown by Liu (2014) (FE model with shell elements) or SDOF models as shown by Micallef et al. (2014). Furthermore, the proposed approach neglects shear redistribution around the perimeter (only the radial direction along the residual span giving the maximum rotation is considered); as shown by Sagaseta et al. (2011), more refined predictions can be obtained using the CSCT if sufficient deformation capacity is provided in the direction of the maximum rotation, taking into account the loadrotation in both orthogonal directions. The simple proposed method and the different refinements described above can be written in a LoA format similar to that proposed in Model Code 2010 so that it could be easily be applied in practice at different design stages.
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4.4
Example of application of proposed method and comparison with dynamic NLFEA
The proposed method was applied to the case study of the office building shown in Fig. 5 which was designed for flexure and punching according to Eurocode 2. This structure was previously analyzed by the authors as part of a reliability study (Olmati et al., 2017) using a more refined time history dynamic NLFEA for different gravity load combinations. The span layout and main geometry are summarized in Fig. 5(a) and (b). In the dynamic NLFEA, the gravity load was introduced gradually in the system and then column C2 was removed suddenly. Figure 8 shows that the dynamic NLFEA predicted that punching would not occur in the adjacent column B2 for the frequent load combination (D+0.5L) (𝐷𝑅 =0.9), whereas punching was predicted to occur for the characteristic load combination (D+L) (𝐷𝑅 =1.1), which demonstrates that the structure is acceptable for accidental loading according to Eurocode 0 (CEN, 2002). The dynamic NLFEA in Fig. 8 shows that after the sudden column removal, a slight unloading was predicted followed by a large amplification of the load and deformations following roughly the nonlinear static response given by Eq. (2). As shown in Fig. 8, after reaching the peak deflections the structure goes into the free vibration phase around a shear force equal to the static value (𝑉F ), as expected. Figure 8 shows that the proposed approach gives similar predictions of punching to the dynamic NLFEA with 𝑉"0Z < 𝑉.,"0Z (no punching) in the frequent load combination (Fig. 8(a)) and 𝑉"0Z ≈ 𝑉.,"0Z (punching) in the characteristic load combination (Fig. 8(b)). The dynamic amplification factor of the load 𝜆" , obtained according to Eq. (5), for the frequent combination (𝑉( = 508 kN [114 kip] and 𝑉F =833 kN [187 kip]) was equal to 1.35, compared to 1.36 from the dynamic NLFEA (Olmati et al., 2017). For the frequent load combination (𝑉F ), the ratio between the maximum dynamic rotation and the static one (including rotation 𝜓( ) is 1.76, compared to 1.80 obtained from the dynamic NLFEA. The maximum vertical deflection of the system 𝑢"0Z predicted by the NLFEA had a dynamic amplification factor of 1.82 after the column removal, which is reasonably similar to the one derived in this work from energy balance (𝜆U =2.15). Some differences between the results from the dynamic NLFEA and the proposed method are expected, as the latter approach does not take into account damping. The dynamic amplification factors obtained in this work are also consistent with those obtained by Liu (2014) for a different prototype building designed according to ACI 318 (4 span building with orthogonal column grid with 𝐿2 = 𝐿0 = 6 m [20ft]); in their analysis 𝜆U obtained from their macromodel was 2.10 instead of 2.15 and 𝜆" was 1.35, whereas in this work a value of 1.25 is obtained using Eq. (5) for the column distribution considered 𝑉F /𝑉( =1.3).
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(a)
(b)
Figure 8:
Application of proposed approach to case study and comparison with dynamic NLFEA (Olmati et al., 2017): (a) frequent load combination (𝐷𝑅 =0.9); and (b) characteristic load combination (𝐷𝑅 =1.1).
5
Contribution of membrane action
5.1
Compressive membrane action
It is well known that compressive membrane action (CMA) can result in a significant increase in strength and stiffness of concrete flat slabs, although its inclusion into theoretical models is not straightforward (Qian and Li, 2016; Sagaseta et al., 2011). Einpaul et al. (2015) showed that CMA can develop around the supports in continuous flat slabs with low amounts of hogging (negative) flexural reinforcement due to cracking and subsequent restraint dilatancy. Einpaul (2016) also showed that this effect, which leads to a stiffer loadrotation response, can be taken into account by introducing a factor in the parabolic loadrotation equation in Model Code 2010 that depends on the ratio between the cracking moment and the flexural strength in the support strip. For column removal situations, the compressive membrane action in the slab around the removed column can be significant (Keyvani et al., 2014) due to restraint dilatancy after substantial cracking which is expected as the slab is not designed to resist sagging (positive) moments in this region. This effect can cause a relative horizontal movement between the adjacent supports pushing them away from each other. Keyvani et al. (2014) obtained compressive membrane forces at the adjacent column of the order of 700 kN/m (48 kip/ft) for span lengths after column removal of 𝐿F =12 m (39 ft); the flat plate studied by Olmati et al. (2017) shown in Fig. 5 predicted compressive membrane forces along line (2) at column B2 of around 550 kN/m (37 kip/ft) for the frequent load combination and around 700 kN/m (48 kip/ft) near the removed column. The increase in flexural capacity in such cases can be relevant and it can contribute significantly to arresting progressive collapse. An example of the arrest of progressive collapse due to CMA of a real building is shown by Keyvani et al. (2014), in which the column removal in a posttensioned concrete flat slab structure resisted progressive collapse with a relatively small permanent maximum vertical deflection 𝑢"0Z ≈ 𝐿F /300. Further research is needed on CMA and its modelling for general slab configurations.
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5.2
Tensile membrane action and activation of tying forces
The development of tensile membrane action due to geometry nonlinearity after large deformations has often been claimed as an effective defense mechanism to arrest progressive collapse (e.g. Hawkins and Mitchell, 1979). In fact, in many cases the tying force approach adopted in codes of practice is intended to provide resistance to gravity loads relying solely on tensile membrane action. As discussed in (Izzudin and Nethercot, 2009) and (Arup, 2011), an ideal efficient design for normal load cases in which the suddenly applied gravity loads exceeds the plastic resistance of the system would require a significant development of membrane action. This is shown schematically in Fig. 9(a) from (Izzudin and Nethercot, 2009) and (Arup, 2011) by the shaded areas representing the work done by the load and the internal energy required for stability of the system. In flat slabs, lateral constraint can be provided by the continuity of the slab, however the development of tensile membrane action might be preceded by punching around the supports.
(a)
(b) Figure 9:
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Tensile membrane action: (a) balance between work done and internal energy for gravity static load above the plastic resistance (Izzudin and Nethercot, 2009; Arup, 2011); and (b) calculation of membrane forces to resist gravity loads.
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The limitation of tensile membrane action due to punching has not been well defined in the past, as the development of tensile membrane forces and punching has not been investigated in combination. Qian and Li (2013) showed experimentally that in order to develop significant tensile membrane action, mitigating actions need to be put in place to avoid punching (e.g. column drops). In this section, the development of tensile membrane forces is investigated using two widely accepted analytical membrane models developed by Park (1964) and Hawkins and Mitchell (1979), both providing relatively similar results of membrane forces. These models are used in combination with the CSCT, since membrane action and punching capacity can therefore be written in terms of the slab rotation near the column using the compatibility relationship 𝑢 ≈ 1.5𝐿( 𝜓. Park’s model is given by Eq. (6) for a square plate with equal yield membrane forces of the reinforcement in each direction, which is placed over the whole area of the slab. 𝑞𝐿O = 𝑇𝑢
𝜋N 4
1 o jpF,N,a… 𝑛N
jkF (−1) O
𝑛𝜋 1 − 𝑐𝑜𝑠ℎ 2
≈ 13.6
(6)
where 𝑞 is the uniformly distributed load in the slab and 𝑇 is the tensile membrane force. Similarly, Hawkins and Mitchell’s model is given by Equations (6) and (7) as a function of the strain along the membrane 𝜀: 𝑇= 𝑢=
q𝐿 4𝑠𝑖𝑛 6𝜀
(7)
3𝐿𝜀 2𝑠𝑖𝑛 6𝜀
(8)
Both membrane models predict (from basic equilibrium considerations) that for larger slab rotations at the boundaries, lower tensile membrane forces are required to balance the gravity loads (Fig. 9(b)). Conversely, if the slab rotation at punching is small, the corresponding tensile membrane forces required will be significant and potentially larger than the prescriptive tying forces used in design. A systematic analysis was carried out to assess the minimum tensile forces that would need to be activated in the slab to resist gravity loads by means of membrane action alone prior to punching. Different case studies were investigated for a regular column layout changing the span length 𝐿( (from 4 m [13 ft] to 8 m [26 ft]), slab thickness (from 200 mm [8 in.] to 350 mm [14 in.]), column size (from 200 mm [8 in.] to 400 mm [16 in.]), concrete strength (from 20 MPa [2900 psi] to 50 MPa [7250 psi]) and live loads (from 1 kN/m2 [21 psf] to 5 kN/m2 [105 psf]). The connections were designed systematically for each case according to Eurocode 2 to obtain the required area of flexural and shear reinforcement for the factored live load considered (1.35D+1.5L). For the amount of shear reinforcement obtained in each case, the corresponding punching strengthrotation curve was obtained according to the failure criterion of the CSCT, as shown in Fig. 9(b). This curve was then used to obtain the maximum slab rotation that a slab could ever achieve near the connection 𝜓^\2 for an accidental gravity load combination (frequent combination D+0.5L). The rotation 𝜓^\2 was then used to assess the corresponding minimum tensile membrane force required according to each membrane model as shown in Fig. 9(b).
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Figure 9(b) shows the results from this analysis for a flat slab of span lengths equal to 6 m (20 ft), 250 mm (10 in.) thick, imposed loading of 4 kN/m2 (84 psf) and 400 mm (16 in.) square columns in which the minimum tensile force obtained to resist the accidental load combination prior to punching was 680 kN/m (47 kip/ft) according to Park’s model and 578 kN/m (40 kip/ft) according to Hawkins and Mitchell’s model. In this case the corresponding maximum rotation 𝜓^\2 was 36 mRad. The tensile force obtained is significantly larger than the tying forces required in UK provisions in the National Annex to Eurocode 2 (tying force varies in this case from 43 kN/m (3 kip/ft) to 108 kN/m (7 kip/ft) depending on the number of stories); tying force requirements in Eurocode 2 give even lower values than those in the UK National Annex. For this case DoD provisions for tying forces of flat slabs give a value near 200 kN/m (14 kip/ft), which is again significantly lower than the membrane force needed to balance the gravity loads. The parametric analysis showed that in the large majority of the cases considered, punching would occur prior to the required level of rotation to form a pure tensile membrane with the required specified tying forces. Although under a true column loss event other mechanisms may work to prevent progressive failure, the use of prescriptive tie requirements may provide a false sense of security to a designer if they neglect to check ductility capacity. It is apparent that further work is needed to investigate the tying force provisions in codes and its applicability to flat slab systems. In particular, the introduction of ductility conditions would need to be considered for different types of construction. With this respect, guidelines by DoD in its latest version (2009) seem to go in this direction by giving rotation limits at the connections for frame structures; however, for flat slabs further work is needed. As discussed in next section, the assessment of ductility of the connections in flat slabs is related to the postpunching response, which is an underresearched field.
6
Modelling postpunching response
The postpunching response of the connection can contribute significantly towards arresting progressive collapse due to the residual capacity and ductility, which could potentially allow the development of alternative load paths such as tensile membrane action (Fig. 9(a)). Work on postpunching is scarce and has traditionally been focused on the residual capacity (Hawkins and Mitchell, 1979; Regan, 1981; Fernández Ruiz et al., 2013; Melo and Regan, 1998). Hawkins and Mitchell (1979) or Melo and Regan (1998) demonstrated that the tensile reinforcement at the top of the slab in the connection has a negligible contribution to the postpunching residual strength. Tests by Fernández Ruiz et al. (2013) showed that slabs with tensile reinforcement alone could develop postpunching strengths of around 30% of the punching strength, whereas slabs with tensile reinforcement and integrity reinforcement at the compression zone could have a residual strength of 70%; these results are consistent to those given in the CIRIA report (Regan, 1981). The early work by Regan led to the recommendation that is also adopted in ACI 31814 detailing rules for structural integrity of using at least two reinforcement bars in each direction passing through the column at the bottom of the slab. The development of analytical models for postpunching is problematic, as discussed by (Fernández Ruiz et al., 2013) and (Habibi, 2012), due to the large number of parameters affecting the localized damage of the concrete around the reinforcement bars, which often requires several assumptions based on experimental evidence. The use of numerical models can provide additional information regarding the deformation capacity of the connection after 292
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failure, which can be critical in the assessment of structural robustness. The development of numerical tools to model postpunching behavior requires significant validation and verification against experimental data. Some postpunching modelling techniques were presented by Keyvani et al. (2014) in which the flexural and integrity reinforcement was modelled explicitly using connectors (Cartesian–Cardan elements) between the slab and the assumed punching cone. Liu (2014) also used beam connectors to model postpunching in the system model. A difficulty of such techniques is the definition of the constitutive relationships used in the connectors based on prepunching and postpunching shear transfer mechanisms, which are uncertain and difficult to assess individually. Alternative techniques include the use of solid elements with discrete modelling of the reinforcement, as presented in this section.
(a)
(b)
(c)
Figure 10: Numerical modelling of postpunching using solid FE models: (a) mesh of one quarter of specimen and deformations after punching, (b) and (c) comparison between experimental and numerical predictions of residual strength for tests PM4 and PM12 respectively (tests by Fernández Ruiz et al. [2013]).
Tests PM4 and PM12 by Fernández Ruiz et al. (2013) without and with integrity reinforcement, respectively, were modelled by the authors using LSDYNA with solid elements, as shown in Fig. 10. The FE models included only one quarter of the test specimens due to symmetry. An explicit solver was adopted in the analysis with a displacement control approach (gradual increase of vertical displacement 𝑤 applied). The solid element consisted of eightnoded hexahedral elements with a constant stress solid element formulation. The constitutive model adopted for the concrete was the Winfrith plasticity concrete model with linear strain softening in tension based on the fracture energy. The reinforcement was
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modelled explicitly as one dimensional beam elements embedded in the concrete elements with a Lagrange solid constraint (perfect bond), which provided reasonable predictions as the reinforcement of the specimens considered was wellanchored and the localized damaged of the concrete near the reinforcement was taken into account directly in the model. Figures 10(b) and (c) show that FE models captured the punching and postpunching strength of the tests considered and were also able to capture the increase in postpunching capacity and stiffness due to the provision of integrity reinforcement. Whilst complex, the modelling of postpunching using solid elements is a promising tool and could potentially be used in the future to assess different detailing configurations proposed in design against progressive collapse.
7
Conclusions
Structural robustness of reinforced concrete flat slabs and their susceptibility to progressive collapse is highly influenced by the structural performance of the columnslab connections. This paper presents a simplified approach to assess punching of adjacent columns in a sudden column removal scenario, which is commonly used in the assessment of structural robustness. The main conclusions from this work are: 1. Using loaddeformationbased formulae for punching (e.g. CSCT) is particularly useful for accidental loading and dynamic situations, as shown in this work. Such approaches allow consideration of inertial effects and apply energybased principles that can be used to assess dynamic punching cases governed either by the local or global response. These situations are commonly found in progressive collapse analyses of flat slab buildings for different triggering events (e.g. column damage, punching around the connection, slab impacting from above and blast). 2. The sudden column removal scenario, used in practice as a simple dynamic test of the structural robustness of the system, introduces a significant increase in the demandcapacity ratio of the adjacent columns due to dynamic amplification of the reactions (demand) and reduction of the punching capacity due to the dynamic amplification of the deformations (global response). It is shown that adopting a constant dynamic amplification factor for the reactions equal to 2, as recommended in practice for concrete structures, can lead to overly conservative results, which justifies the need to carry out a more rigorous dynamic assessment. 3. The proposed formulae for punching assessment under sudden column removal situations based on the DRA and CSCT give a simple but theoretically sound solution that takes into account the main parameters involved (i.e. dynamic effects on demand and capacity, increase in both moment transfer and slenderness after column removal). The proposed pseudostatic relationship enables the introduction of analogous concepts for quasistatic punching, such as the definition of an increased pseudostatic slenderness in the shear demand to rotation formula in Model Code 2010. The proposed method can take into account strainrate effects (in demand and capacity) and can also be formulated in a LoA format so that it could easily be implemented in codes of practice. 4. The punching predictions for the case study presented using the proposed approach are consistent with those obtained using a more refined dynamic NLFEA. The case study investigated confirms that column removal is not always critical. The load dynamic amplification factor formula obtained from energy balance considerations provides
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results similar to those observed experimentally and numerically for flat slabs with similar levels of gravity loads. The proposed formula for λ" is only valid for punching assessment of an individual connection and does not consider the development of tensile membrane action (highductility). 5. The analysis carried out in this paper using different widely accepted tensile membrane models in combination with the CSCT confirms that punching occurs prior to the required level of slab rotation to form a pure tensile membrane with the tying forces specified in codes of practice. Further work is needed to confirm the potential development of tensile membrane action after punching in flat slabs, which is challenging as it is highly influenced by detailing of the reinforcement in the slab crossing the column. The numerical models presented in this work using solid FE models provide a promising tool to address this problem, although they are complex as the flexural and integrity reinforcement needs to be modelled explicitly. 6. The analytical and numerical tools presented in this work can be used in different alternative load path analyses, which are commonly used in the assessment of structural robustness. The use of the DRA and the CSCT is promising in terms of assessing the arrest of progressive collapse of concrete flat slab structures, as both approaches are highly compatible. These approaches can potentially be used for more realistic scenariodependent analyses.
8
Acknowledgments
The work is a continuation of a research project financially supported by the Engineering and Physical Sciences Research Council EPSRC, Impact Acceleration Account (IAA) held by the University of Surrey (grant ref: EP/K503939) linked with a previous project funded by EPSRC of the UK (grant ref: EP/K008153/1). The authors would also like to acknowledge project collaborators at Arup in the UK (Tony Jones and David Cormie) and EPFL in Switzerland (Aurelio Muttoni and Miguel Fernández Ruiz) for their feedback and technical discussions on the topic. The valuable feedback from Bassam Izzudin from Imperial College London is also acknowledged.
9
References
ACI Committee 318 (2014) Building Code Requirements for Structural Concrete (ACI 31814) and Commentary (ACI 318R14), American Concrete Institute, Farmington Hills, MI, 519 pp. Arup
(2011) Review of International Research on Structural Robustness and Disproportionate Collapse, Department for Communities and Local Government, 198 pp.
BSI (British Standards Institution) (1997) BS 8110 Part 1: Structural use of concrete: Code of Practice for Design and Construction, BSI. CEN (European Committee for Standardization) (2002) Eurocode – Basis of Structural Design, EN 1990:2002, Brussels, Belgium, 115 pp.
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CEN (European Committee for Standardization) (2004) Design of Concrete Structures— Part 1: General Rules and Rules for Buildings, EN 199211:2004, Brussels, Belgium, 225 pp. Collings, D.; and Sagaseta, J. (2015) ”A Review of Arching and Compressive Membrane Action in Concrete Bridges”, Proceedings of the Institution of Civil Engineers – Bridge Engineering. 10.1680/bren.14.00039 Concrete Society (2007) Guide to the Design and Construction of Reinforced Concrete Flat Slabs, Technical Report, No. 64, 101 pp. COST (European Cooperation in the field of Scientific and Technical Research) (2008) Robustness of Structures – Proceedings of the 1st Workshop, ETH Zurich, Switzerland. Delhomme, F.; Mommessin, M.; Mougin, J. P.; and Perrotin, P. (2007) “Simulation of a Block Impacting a Reinforced Concrete Slab with a Finite Element Model and a Massspring System”, Engineering Structures, V.29, No.11, pp. 2844–2852 Department of Communities and Local Government (2010) The Building Regulations 2010 Structure: Approved Document A, HM Government, UK. DoD (Department of Defense) (2009) Design of Buildings to Resist Progressive Collapse (UFC 402303), Unified Facilities Criteria, Washington DC. Einpaul, J. (2016) “Punching Strength of Continuous Slabs”, PhD Thesis, EPFL, No. 6928, Lausanne, Switzerland, 178 pp. Einpaul, J.; Bujnak, J.; Fernández Ruiz, M.; and Muttoni, A. (2016) “Study of Influence of Column Size and Slab Slenderness on Punching Strength”, ACI Structural Journal, V. 113, No. 1, pp. 135–145. doi: 10.14359/51687945 Einpaul, J.; Fernández Ruiz, M.; and Muttoni, A. (2015) “Influence of Moment Redistribution and Compressive Membrane Action on Punching Strength of Flat Slabs”, Engineering Structures, V. 86, pp. 43–57. doi: 10.1016/j.engstruct.2014.12.032 fib (Fédération internationale du béton) (2013) fib Model Code for Concrete Structures 2010, Ernst & Sohn, Berlin, Germany, 434 pp. Fernández Ruiz, M.; Mirzaei, Y.; and Muttoni, A. (2013) “Postpunching Behavior of Flat Slabs”, ACI Structural Journal, V. 110, No. 5, pp. 812–801. General Services Administration (GSA) (2000) Progressive Collapse Analysis and Design Guidelines for New Federal Office Buildings and Major Modernization Projects, Office of Chief Architects, Washington DC. Habibi, F. (2012) “PostPunching Shear Response of TwoWay Slabs”, PhD Thesis, McGill University, Montreal, Canada. Hawkins, N. M., and Mitchell, D. (1979) “Progressive Collapse of Flat Plate Structures”, ACI J. Proceedings, V. 76, No. 7, pp. 775–808.
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Izzudin, B. A.; Vlassis, A. G.; Elghazouli, A. Y.; and Nethercot, D. (2008) “Progressive Collapse of Multistorey Buildings due to Sudden Column Loss  Part I: Simplified Assessment Framework”, Engineering Structures, V. 30, pp. 1308–1318. doi:10.1016/j.engstruct.2007.07.011 Izzudin, B. A; and Nethercot, D. (2009) “DesignOriented Approaches for Progressive Collapse Assessment: LoadFactor vs DuctilityCentred Methods”, ASCE Structures Congress, Austin, pp. 800–1791. doi: 10.1061/41031(341)198 Keyvani, L.; and Sasani, M. (2014) ”Analytical and Experimental Evaluation of Progressive Collapse Resistance of a FlatSlab Posttensioned Parking Garage”, ASCE Journal of Structural Engineering, V. 141, No. 11. Keyvani, L.; Sasani, M.; and Mirzaei, Y. (2014) “Compressive Membrane Action in Progressive Collapse Resistance of RC Flat Plates”, Engineering Structures, V. 59, pp. 554–564. Liu, J. (2014) “Progressive Collapse Analysis of Older Reinforced Concrete Flat Plate Buildings Using Macro Model”, PhD Thesis, University of Nevada, Las Vegas. Melo, G. S.; and Regan, P. E. (1998) “PostPunching Resistance of Connections between Flat Slabs and Interior Columns,” Magazine of Concrete Research, V. 50, No. 4, pp. 319– 327. Merola, R. (2009) “Ductility and Robustness of Concrete Structures Under Accidental and Malicious Load Cases”, PhD Thesis, University of Birmingham, UK. Micallef, K.; Sagaseta, J.; Fernández Ruiz, M.; and Muttoni, A. (2014) “Assessing Punching Shear Failure in Reinforced Concrete Flat Slabs Subjected to Localized Impact Loading”, International Journal of Impact Engineering, V. 71, pp. 17–33. Minister of Housing and Local Government (1970) The Building (Fifth Amendment) Regulations 1970, (S.I. 1970, No. 109), Statutory Instruments of the UK. Mitchell, D.; and Cook, W. D. (1984) “Preventing Progressive Collapse of Slab Structures”, Journal of Structural Engineering ASCE, V.110, No.7, pp. 1513–1532. Muttoni, A. (2008) “Punching Shear Strength of Reinforced Concrete Slabs without Transverse Reinforcement”, ACI Structural Journal, V. 105, No. 4, pp. 440–450. Olmati, P.; Sagaseta, J.; Cormie D.; and Jones, A. E. K. (2017) “Simplified Reliability Analysis of Punching in Reinforced Concrete Flat Slab Buildings Under Accidental Actions”, Engineering Structures, V. 130, No. 1, pp. 83–98. dx.doi.org/10.1016/j.engstruct.2016.09.061 Park, R. (1964) “Tensile Membrane Behaviour of Uniformly Loaded Rectangular Reinforced Concrete Slabs with Fully Restraint Edges”, Magazine of Concrete Research, V. 16, No. 46, pp. 39–44. Qian, K.; and Li, B. (2013) “Experimental Study of DropPanel Effects on Response of Reinforced Concrete Flat Slabs after Loss of Corner Column”, ACI Structural Journal, V. 110, No. 2, pp. 319–330.
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Qian, K.; and Li, B. (2016) “Resilience of Flat Slab Structures in Different Phases of Progressive Collapse”, ACI Structural Journal, V. 113, No. 3, pp. 537–548. Regan, P. E. (1981) Behaviour of Reinforced Concrete Flat Slabs, Construction Industry Research and Information Association (CIRIA), Report No. 89, 88 pp. Russell, J. M.; Owen, J. S.; Hajirasouliha, I. (2015) “Experimental Investigation on the Dynamic Response of RC Flat Slabs After a Sudden Column Loss”, Engineering Structures, V. 99, pp. 28–41. Sagaseta, J.; Muttoni, A.; Fernández Ruiz, M.; and Tassinari, L. (2011) “Nonaxissymmetrical Punching Shear Around Internal Columns of RC Slabs Without Transverse Reinforcement”, Magazine of Concrete Research, V.63, No.6, pp. 441457. doi: 10.1680/macr.10.00098 Sagaseta, J.; Olmati, P.; Micallef, K.; and Cormie, D. (under review) “Punching Shear Failure in Blastloaded RC Slabs and Panels”, Engineering Structures. Szyniszewski, S. T.; Krauthammer, T.; and Yimk, H. C. (2009) Energy Flow Based Progressive Collapse Studies of Moment Resisting Steel Framed Buildings, Final report to US army ERDC, CIPPSTR0032008. Center for Infrastructure Protection and Physical Security, University of Florida. Vlassis, A. G.; Izzudin, B. A.; Elghazouli, A. Y.; and Nethercot, D. (2008) “Progressive Collapse of Multistorey Buildings due to Sudden Column Loss  Part II: Application”, Engineering Structures, V. 30, pp. 1424–1438. doi:10.1016/j.engstruct.2007.08.011
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Influence of flexural continuity on punching resistance at edge columns Luis F. S. Soares, Robert L. Vollum Imperial College London, UK
Abstract This paper examines the influence of flexural continuity on punching resistance at edge columns of braced flat slabs under gravity loading, making use of experimental data, nonlinear finite element analysis (NLFEA) and the Critical Shear Crack Theory (CSCT) as presented in the fib Model Code 2010 (MC2010). According to the CSCT, punching resistance reduces with increasing rotation y of the slab relative to its support area due to loss of aggregate interlock in the critical shear crack. NLFEA shows that as loads are increased to failure, moment redistribution from edge column supports to the span causes the loading eccentricity at edge columns to reduce below its initial elastic value. The resulting rotation y and peak shear stress are less than they are in comparable isolated test specimens with fixed loading eccentricity. Consequently, the CSCT predicts punching resistance at edge columns of flat slabs to be significantly influenced by flexural continuity, which is unaccounted for in the design methods of ACI 318 and EC2. Both NLFEA and the CSCT suggest that providing surplus flexural reinforcement in the span can be more effective at increasing punching resistance at edge columns than the common UK practice of providing surplus hogging flexural reinforcement.
Keywords Continuity, edge columns, flat slabs, punching shear
1
Introduction
This paper is concerned with punching resistance at edge columns of flat slabs in braced structures subject to gravity loading. Punching resistance is much less researched at edge columns than internal slabcolumn connections, even though flat slabs typically have more edge columns than internal columns. Significantly, in typical flat slab buildings, moment transfer from the slab to column has a greater influence on the design punching resistance of edge and corner columns than internal slabcolumn connections. Flat slabs are normally designed using either elastic finite element analysis (FEA) or equivalent frame analysis. In UK practice, slabcolumn connections of equivalent frames are modelled as rigid (The Concrete Society, 2007). Conversely, ACI 31814 (ACI, 2014) introduces torsional members between the slab and column to account for the flexibility of the slabcolumn connection. Equivalent frame analysis according to UK practice tends to overestimate column moments, since the flexibility of the slabcolumn connection is not modelled. Consequently, UK practice allows moments about an axis parallel to the slab edge 299
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(i.e. normal moments) to be redistributed by up to 50% at edge columns of flat slabs designed using the equivalent frame method. Only 30% moment redistribution is allowed if the design moments are calculated with elastic finite element or grillage analysis. Punching failure of slabedge column connections has been studied by testing both isolated and continuous subassemblies of the types shown in Figures 1a and b. In each case, the loading eccentricity with respect to the column centre equals M/V, where M is the slab moment resisted by the column (i.e. unbalanced moment about the column centreline) and V is the punching shear force. A fundamental difference between isolated and continuous tests is that the eccentricity M/V is typically fixed in tests of isolated but not continuous specimens. In tests with constant M/V, the ultimate column load is limited by the least of the punching and flexural resistances. Conversely, in continuous specimens, like flat slabs, the punching shear force can increase after the moment at the column face reaches its ultimate resistance due to support moments being redistributed back into the span (Soares and Vollum, 2016). The critical design case for punching shear in flat slabs dimensioned in accordance with UK practice is commonly the maximum possible design punching resistance VRmax at edge columns. Both EC2 (BSI, 2004) and the fib Model Code 2010 (MC2010) (fib, 2013), which is based on the CSCT of Muttoni (2008), limit the maximum possible punching resistance VRmax to a multiple of the shear resistance provided by the concrete alone, Vc. According to EC2 and MC2010, Vc can be increased by increasing the provided area of flexural reinforcement over that required for strength. MC2010 attributes the increase in Vc to a reduction in the rotation of the critical shear crack, which increases the shear resistance provided by aggregate interlock. As well as enhancing Vc, providing surplus flexural reinforcement normal to the slab edge potentially increases the moment transferred to the column at the design ultimate load, which is detrimental due to the consequent increase in maximum shear stress. This paper uses NLFEA in conjunction with the CSCT, as implemented in MC2010 Level IV, to assess the effect on punching resistance of providing surplus flexural reinforcement in continuous flat slabedge column subassemblies with the geometry tested by Regan (1993) shown in Figure 1b.
(a) Figure 1:
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(b) Slab edge column punching specimens a) isolated slabs and b) continuous slabs of Regan (1993). (Note: dimensions in mm; 1 mm = 0.0394 in.)
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2
Review of ACI 318, EC2 and MC2010 punching design methods at edge columns
2.1
ACI 31814
ACI 31814 (ACI, 2014) adopts a rectangular control perimeter of length b0 located at a distance of 0.5d from the perimeter of the concentrated load. In most cases, ACI 31814 takes the shear resistance provided by the concrete in slabs without shear reinforcement as 𝑣" = 1/3 𝑓" MPa (4 𝑓" psi). This practice is questionable because it neglects the influence on shear resistance of both the flexural reinforcement ratio and the socalled size effect where the shear stress at failure reduces with increasing slab depth. For edge columns with normal moments (i.e. about an axis parallel to the slab edge), which are the subject of this paper, ACI 318 calculates the maximum applied shear stress as the greatest of: 𝛾; 𝑀=" 𝑐56 𝐽" 𝛾; 𝑀=" 𝑐@A 𝑣(@A) = 𝑣)* − 𝐽" 𝑣(56) = 𝑣)* +
where 𝑣)* =
+ , .
(1)
is the design ultimate shear stress due to gravity load without moment
transfer, Jc is a property of the critical section analogous to polar moment of inertia. Figure 2 shows the dimensions cAB and cCD as well as the stress distribution on the critical section. Shear reinforcement is required if the maximum shear stress is greater than 𝑣" . The coefficient γv defines the proportion of the moment Msc transferred from the slab to column about the centroid of the critical perimeter that is resisted by eccentric shear with the remainder resisted by flexure. 𝛾; = 1 −
1 1+
2 3
𝑏E /𝑏F
(2)
where b1 and b2 are the dimensions of the critical section b0 measured parallel and perpendicular to the slab edge, respectively. Moehle (1988) showed that for the span direction perpendicular to the slab edge the interaction between Msc and punching resistance at edge columns can be neglected if 𝑣)* ≤ 0.75𝑣" . In this case, ACI 31814 allows γv to be taken as 0. Additionally, sufficient flexural reinforcement needs to be provided within a width of c2+3h, centred on the column to resist Msc. Ghali et al. (2015) argue that this practice is unsafe and should be discontinued because nonlinear finite element analysis shows that γv is not zero in reality. In this context, it is pertinent to note that Alexander and Simmonds (2003) developed a simple limit analysis for moment transfer at internal slabcolumn connections, in which the greater proportion of moment transfer was attributed to horizontal rather than vertical shear. Alexander and Simmonds concluded that Equation (2) overestimates gv. The present paper reviews the recommendations of ACI 318 and Ghali et al. (2015) for unbalanced moment transfer at edge columns.
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Figure 2:
2.2
Design shear stress distribution on ACI 318 critical perimeter.
EC2
EC2 locates the basic control perimeter u1 at a constant distance of 2d from the column face and calculates the design punching shear stress as: 𝑣 = 𝛽
𝑉 𝑢E 𝑑
(3)
where the multiple β accounts for the effects of uneven shear, and d is the average effective depth of the tension reinforcement. The concrete contribution to punching shear resistance is calculated as follows: E
𝑣" = 0.18 100𝜌𝑓" I (1 + 200 𝑑 where 𝜌 = 𝜌MN 𝜌ON 5PQ
K.L
K.L
)/𝛾"
(4)
≤ 0.02 in which 𝜌MN and 𝜌ON are the flexural tension reinforcement
ratios within a slab width equal to the column width plus 3d on each side. 𝑓" is the ,. characteristic concrete cylinder strength in MPa (1 MPa = 145 psi) and 𝛾" is the partial factor for concrete, which equals 1.5 for design. The EC2 design procedure for punching shear at edge columns is based on the work of Regan (1999) who proposed the interaction diagram in Figure 3 for the interaction between punching resistance and unbalanced normal moments about the column centreline. The unbalanced moment at point C where V = 0 is Mcf, which is the maximum moment that can be transferred from the slab into the column through flexure. The moment is enhanced along the line BC by eccentric shear. Regan (1981) showed that for practical purposes Mcf can be assumed to be resisted by reinforcement anchored within a width of be = c2 + 2y (where c2 is the column width parallel to the slab edge and y is the perpendicular distance from the inner column face to the slab edge) centred on the column. Regan suggested that the design punching resistance should be taken as that at point B in Figure 3, which is considered to be the minimum punching resistance because points along the line BC below B correspond to 302
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flexural failure. Regan justified his assumption by noting that punching resistance at edge columns is typically preceded by yielding of reinforcement at the column face. Points along the line BC are critical for isolated specimens with fixed loading eccentricity M/V (e.g. Figure 1a) but not for indeterminate slabs (e.g. Figure 1b), where equilibrium can be maintained by moment being redistributed back into the span. Point A in Figure 3 corresponds to the maximum possible shear resistance, which according to Regan (1999) occurs when the shear stress is uniformly distributed along the three column sides in contact with the slab. Conversely, MC2010 and ACI 31814 predict the maximum punching resistance to occur when the shear stress is uniform along the critical shear perimeter at 0.5d from the column face.
Figure 3:
Interaction between punching resistance and normal unbalanced moment (adapted from Regan, 1999).
For normal moments, EC2 assumes the punching resistance to be independent of eccentricity and considers shear stress to be uniformly distributed along the reduced control perimeter u1* depicted in Figure 4. This is equivalent to taking β as u1/u1* in Equation (3) and has a similar effect to taking γv = 0 for 𝑣)* ≤ 0.75𝑣" in ACI 31814. EC2 requires the bending moment at the column face Mcf to be resisted by reinforcement centred on the column within a width c2+y. However, it is common UK practice (The Concrete Society, 2007) to provide flexural reinforcement within an effective width be = c2 + 2y. Additionally, Annex I of EC2 limits the unbalanced moment transferred to the column to Mtmax = 0.255(c2+y)fc𝑑UF /γc (where γc = 1.5), ostensibly to prevent over reinforcement.
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Figure 4:
2.3
EC2 reduced control perimeter u1*.
MC2010
MC2010 (fib, 2013) locates the basic control perimeter u at a constant distance of 0.5d from the column face. It has four levels of approximation, of which I to III are intended for design and IV for assessment. Level II is intended for standard design unless the geometry is irregular in which case Level III is required. The shear resistance is calculated in terms of the slab rotation ψ relative to the support area, which is calculated in Levels II and III as follows: 𝑟= 𝑓O. 𝑚= 𝜓=𝛼 𝑑 𝐸= 𝑚[
E.L
(5)
where rs denotes the position where the radial bending moment is zero with respect to the column axis, ms is the average design moment for the reinforcement per unit width in the support strip and mR is the design average flexural strength per unit width of the support strip. The coefficient α in Equation (5) is 1.5 for Level II and 1.2 for Level III. The width of the support strip is taken as bsr = c2+2y and 0.5c1+0.5bs respectively for reinforcement perpendicular and parallel to the slab where 𝑏= = 1.5 𝑟=M 𝑟=O . In Level I and II, rs is estimated as 0.22L where L is the slab span in the direction being considered. In Level II, moments resisted by reinforcement normal to the slab edge ms are estimated as: 1 𝑒] 𝑚= = 𝑉 + 8 𝑏=^ where 𝑒 ] is the eccentricity of V with respect to the centroid of the control perimeter.
(6)
Moments resisted by reinforcement parallel to the slab edge are estimated as: 1 𝑒] 𝑉 𝑚= = 𝑉 + ≥ 8 2𝑏= 4
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(7)
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In Level III, rs is calculated with linear elastic FEA (LFEA) but should not be taken as less than 0.67bsr at edge columns for reinforcement perpendicular to the slab edge. The design moments ms are determined with elastic FEA. In Level IV the rotations are determined with nonlinear analysis. MC2010 bases the punching resistance on the greater of the rotations relative to the support area about axes normal and parallel to the slab edge. The punching resistance provided by concrete is calculated as: √𝑓" 𝑉" = 𝑘a 𝑘b 𝑢𝑑 (8) 𝛾" in which fc is in MPa (1 MPa = 145 psi), 𝑑 is the average effective depth of the tension reinforcement, which in this paper is assumed to equal the shear resisting effective depth. The parameter kψ depends on the slab rotation and is calculated as: 1 𝑘a = ≤ 0.6 (9) 1.5 + 0.9𝑘.* 𝜓𝑑 32 𝑘.* = ≥ 0.75 (10) 16 + 𝑑* where dg is the maximum aggregate size. The coefficient 𝑘b is a reduction factor for eccentric shear that can be approximated as 0.7 for braced frames where the adjacent spans do not differ in length by more than 25%. Alternatively, ke can be determined from linear finite element analysis as the ratio of the average to peak shear stress or calculated as: 𝑒′ 𝑘b = 1/(1 + ) (11) 𝑏) where e' is the eccentricity of V with respect to the centroid of the basic control perimeter u, and bu is the diameter of a circle with the same surface area as enclosed by u.
3
Assessment of continuous specimens of Regan (1993)
Consideration of the design methods of ACI 318, EC2 and MC2010 shows that ACI 318 is unique in not relating shear resistance to the area of flexural reinforcement provided at the edge column. This assumption is well known to be overly simplistic for isolated slabcolumn connections, but recent research (Einpaul et al., 2014; Soares and Vollum, 2015, 2016) into the influence of flexural continuity on punching resistance shows it to be more reasonable for continuous slabs. According to MC2010 Level IV, this is because punching resistance depends on the rotation of the critical shear crack, which is influenced by the span as well as support flexural reinforcement. Hence, the punching resistance at internal columns is less affected by redistribution of design moments between the span and support than implied by EC2 or MC2010 Levels II and III. This paper investigates the influence of flexural continuity and flexural reinforcement ratio on punching resistance at edge columns making use of a series of five fullscale continuous edge column punching shear specimens tested by Regan (1993). The slabs were chosen because they are almost unique in having a relatively large thickness of 200 mm (7.87 in.). The slabs measured 3000 mm (118.1 in.) wide and 5784 mm (227.7 in.) long, and had 300 305
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mm (11.8 in.) square columns at the centres of their short edges as shown in Figure 1b. The bottom longitudinal flexural reinforcement consisted of 23 number 12 mm (0.47 in.) diameter bars with an effective depth of 174 mm (6.85 in.), which equates to a reinforcement ratio of 0.5%. The cover was 20 mm (0.79 in.) to the top and bottom longitudinal reinforcement that was in the outer layer. The flexural tension reinforcement ratio normal to the short slab edge, within a width be = 2c1 + c2 centred on the column, rsup was 0.8% in slabs S1 to S3, 1.0% in S4 end 1 and 0.5% in S4 end 2. The reinforcement rsup was provided in the form of U bars and distributed over a width of 500 mm (19.7 in.) in slabs S1 to S3 and S4 end 2. In S4 end 1, rsup was distributed over a width of 600 mm (23.6 in.). The U bar diameter was 12 mm (0.47 in.) with the exception of S4 end 1, where 16 mm (0.63 in.) diameter bars were used. No other longitudinal hogging reinforcement was provided within 500 mm (19.7 in.) of the column centreline. The top transverse reinforcement at the ends of the slabs consisted of 10 number 12 mm (0.47 in.) diameter bars positioned over a width of 1350 mm (53.1 in.), with three bars passing through the column. There were five different arrangements of bottom transverse reinforcement within the support strip as shown in Figure 5, which also shows the top transverse reinforcement. Top transverse reinforcement consisting of 14 number 8 mm (0.31 in.) diameter bars were positioned over the central region of the slab. Further details of the slabs are summarised in Table 1, including the measured failure loads and the ratio of the moment of resistance at the column face McfR to the ultimate elastic moment Mcfel across the slab width at the column face. The ultimate elastic moment Mcfel was derived from reactions calculated with FEA at the measured failure load. The moment of resistance at the column face McfR was calculated in terms of the flexural reinforcement provided within a width be = c2 + 2y centred on the column. The elastic moment is typically greater than the moment of resistance, indicating that moment redistribution is implicit in the calculation of reinforcement design moments. Table 1 also gives concrete cylinder strengths, calculated as 0.8 times the cube strength of specimens cured with the slabs, and reinforcement yield strengths. The maximum aggregate size was 20 mm (0.787 in.). Table 1:
Details of Regan slabs Experimental
Author
Slabs
Calculated
c1 c2 h d fc fy Vtest Mtest Mcf R(2) McfR r mm mm mm mm MPa MPa % kN kNm kNm /Mcfel(3) 1(1) 300 300 200 168 35.4 507 0.56 282 118 100.4 0.63 1(2) 300 300 200 168 35.4 507 0.56 264 138 100.4 0.67 2(1) 300 300 200 168 35.4 507 0.56 256 124 100.4 0.69 2(2) 300 300 200 168 35.4 507 0.56 285 129 100.4 0.62 Regan 3(1) 300 300 200 168 41.0 507 0.56 416 73 101.7 3(2) 300 300 200 168 41.0 507 0.56 233 148 101.7 0.77 (1993) 4(1) 300 300 200 165 42.7 507 0.66 289 149 127.3 0.78 4(2) 300 300 200 168 42.7 507 0.45 281 111 66.7 0.42 5(1) 300 300 200 168 38.4 507 0.61 327 84 98.2 5(2) 300 300 200 168 38.4 507 0.60 234 86 46.3 (1) Calculated in accordance with EC2 punching shear requirements; (2) McfR is moment of resistance at column face; (3) Mcfel is elastic moment across slab width at column face. 1 mm = 0.0394 in; 1 MPa = 145 psi; 1 kN = 0.2248 kip; 1 kNm = 737.6 lbf.ft
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(1)
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Slab 1 Ends 1 & 2
Slabs 2 & 3 Ends 1& 2
Slabs 1, 3 & 4 Ends 1& 2
Slab 2 End 1
Slab 2 End 2
Slab 4 End 1 Figure 5:
Slab 4 End 2
Reinforcement details for Regan slabs. (Note: dimensions in mm; 1 mm = 0.0394 in.) (Figure continues on next page.)
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Slab 5 End 1
Slab 5 End 2
Figure 5 (cont.): Reinforcement details for Regan slabs. (Note: dimensions in mm; 1 mm = 0.0394 in.)
The two ends of each slab were tested separately, with the column providing the support at one end and the other end simply supported across the slab width just inside from the column face. With the exception of slab S3, at each loading stage equal and opposite horizontal forces were applied to the column at the end under test “so as to keep the column free of rotation” (Regan, 1993). In slabs S1 and S2 with rsup = 0.8%, the column transfer moments had become constant before the bottom steel yielded in the span and punching occurred. In slab S4 end 1, with rsup = 1.0%, the span steel had not visibly yielded when punching occurred. Midspan yield was well developed in slab S4 end 2, with rsup = 0.5%, at punching failure. In test S3 end 1, the slab was loaded to failure with four equal loads positioned at the loading points closest to the supported column (points a to d in Figure 1b). The effect of this was to reduce the eccentricity M/V as well as the maximum span moment for a given total applied load relative to that for the standard loading arrangement. In the case of slab S3 end 2, a relatively small vertical load was applied, distributed as for slabs 1 and 2, with the column prevented from rotating. The vertical load was then held constant and the column moment increased until a plastic hinge formed at the slabcolumn connection. The vertical load was then increased to failure whilst maintaining the column moment. Specimen S5 investigated the influence of the column being outboard of the slab. At end 1, the centreline of the column was flush with the slab edge, whereas at edge 2 the inner face of the column was flush with the slab edge. End 1 of slab S5 was loaded following the general procedure until hinges formed at the slabcolumn connection and in the span. Subsequently, the specimen was unloaded and reloaded with all the actuators applying a constant load of 25 kN (5.62 kip) apart from those at the slab edge (points a and b in Figure 1b), where the loads were increased until failure. In 308
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all specimens, strains were measured in the top bars perpendicular to the slab edge close to the inner column face. The strain measurements indicate that the bars passing into the column generally yielded. The strains in the outer bars to either side of the column were typically around 2/3 of those in the bars going into the column. These bars did not generally yield, although yielding occurred in ends 2 of slabs S3 and S4. All the slabs are reported to have failed in punching. However, flexural failure appears to have been imminent in both tests of slabs S1 and S2 as well as end 2 of S4.
4
Strength assessment of Regan slabs
Slabs S1 end 1 and S4 ends 1 and 2 were modelled using the commercially available finite element program Atena version 5.1.2 (Cervenka et al., 2016). Concrete was modelled using the fractureplastic model CC3DNonLinCementitious2, which combines constitutive models for tensile (fracturing) and compressive (plastic) behaviour. The total strain fixed crack model was used in which strains and stresses are converted into the material directions given by the principal directions at the onset of cracking. A Rankine tensile failure criterion was adopted in conjunction with an exponential softening curve. The MenétreyWillam failure surface was used for concrete in compression with a hardening/softening law based on the uniaxial compressive test. The ascending branch of the hardening curve is elliptical and strain based but the softening branch is linear and displacement based to introduce mesh objectivity into the finite element solution. The shear stiffness after cracking was assumed to be 20 times the normal crack stiffness. The shear strength of cracked concrete was determined in accordance with the ‘Modified Compression Field Theory’ of Vechio and Collins (1986). An arc length solution procedure was adopted in which the stiffness was updated at each iteration. Error tolerances were adopted of energy (104) and displacement (102). In order to reduce computational time, which varied between 1 and 2 days depending on the desktop computer used, only half of the slab was modelled with eightnode brick elements using the mesh shown in Figure 6. The ends of the columns were laterally restrained to keep the column vertical. Figure 7 compares the measured and predicted slab rotations relative to the column for each modelled slab. Rotations are shown about an axis parallel to the 3000 mm (118.1 in.) slab edge as these were greatest and hence govern punching resistance according to MC2010. The NLFEA is seen to give good predictions of the measured rotations and failure loads of the slabs. Figure 7 also shows rotations calculated for each test with MC2010 Levels II and III. The Level II rotations were calculated with ms from Equation (6) using the measured ultimate eccentricity M/V, which is significantly less than the elastic eccentricity that would be used in design. Figure 7 also shows resistances calculated with MC2010 for ke = 0.7 and ke derived from elastic finite element analysis with shell elements as the ratio of the average to peak shear stress. Levels II and III are seen to significantly overestimate the rotation particularly for S4 end 2 with rsup = 0.5%. The MC2010 failure loads are given by the intersection of the rotation and resistance curves in Figure 7. MC2010 Level IV, with rotations from the NLFEA, is seen to give reasonable estimates of the measured punching resistances if ke = 0.7 but resistances are significantly underestimated if ke is derived from elastic FEA or calculated with Equation (11) using the elastic eccentricity as would be done in design. Figure 7 also shows that MC2010 Levels II and III significantly underestimate punching resistance particularly if ke is derived from elastic FEA. EC2 is shown to give good estimates of punching resistance.
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Figure 6:
Finite element mesh of Regan slabs used in Atena analysis.
Resistance (ke=0.7)
Slab 1 End 1
Resistance (ke = 1/(1+eu/bu)) Resistance (ke = Shell LFEA)
350
Experimental
Load (kN)
300
Vtest EC2 Vshear
250
NLFEA Rotation  Level II (MC2010)
200
Rotation  Level III (MC2010)
150 100 50 0 0
0.01
0.02
0.03
Rotation (radians)
(a) Figure 7:
310
Load rotation relationships for Regan slabs: a) S1 end 1. (Figure continues on next page.)
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Slab 4 End 1
Resistance (ke=0.7) Resistance (ke = 1/(1+eu/bu)) Resistance (ke = Shell LFEA) Experimental
400 350
EC2 Vshear Vtest
Load (kN)
300 250
NLFEA Rotation  Level II (MC2010) Rotation  Level III (MC2010)
200 150 100 50 0 0
0.01
0.02
0.03
Rotation (radians)
(b)
Slab 4 End 2
Resistance (ke=0.7)
350 300
Vtest
EC2 Vshear NLFEA
250
Load (kN)
Resistance (ke = 1/(1+eu/bu)) Resistance (ke = Shell LFEA) Experimental
Rotation  Level II (MC2010) Rotation  Level III (MC2010)
200 150 100 50 0 0
0.01
0.02
0.03
Rotation (radians)
(c) Figure 7 (cont.): Load rotation relationships for Regan slabs: b) S4 end 1, and c) S4 end 2. (Note: 1 kN = 0.2248 kip)
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Table 2 shows the ratio of the measured Vtest to calculated flexural Vflex and punching resistances Vshear according to MC2010 Level II, EC2 and ACI 318, with gv = 0, as permitted by the code, and gv from Equation (2), as suggested by Ghali et al. (2015). The flexural resistance Vflex is the column load at flexural failure resulting from the development of plastic hinges in the span and at the column face. It was calculated neglecting strain hardening and the widths of the loading plates. The moment of resistance at the column face McfR was calculated assuming that reinforcement was only effective if positioned within a width of be = c2+2y centred on the column, which can be conservative as shown by the third column of Table 2. This assumption follows UK practice (The Concrete Society, 2007), which is based on research by Regan (1981, 1993) and is also consistent with the recommendations of ACIASCE Committee 352 (2011). The ratio Mcftest/McfR is more critical for ACI 318 than the ratio of the moment about the centroid of the critical section to the moment of resistance provided by reinforcement within an effective width of c2+3h centred on the column. All the slabs are reported as having failed in punching, though flexural failure appears to have been imminent in slabs where the ratio Vtest/Vflex is greater than 1.0. The measured ultimate eccentricity was used in the assessments with MC2010 Level II and ACI 318 with gv from Equation (2). Table 2 shows that EC2 gives the best predictions of the failure load of the Regan slabs S1 to S5. ACI 318 with gv = 0 tends to give higher shear resistances than EC2 and can significantly overestimate resistance if not limited by McfR. However, it should be noted that flexural failure (Vtest/Vflex) is calculated to be critical for slabs S1, S2 and S4 than Vtest/Vcalc. Consequently, it is possible that the measured shear resistances were reduced by flexural failure. ACI 318 gives reasonable but conservative predictions of shear resistance when gv is calculated with Equation (2), but the predictions are overly conservative if the initial elastic eccentricities are used to calculate the peak shear stress. The MC2010 Level II resistances are overly conservative due to the rotation being overestimated. Better estimates of shear resistance are obtained with ke = 0.7 than for ke calculated with Equation (11). Table 2: Slabs 1(1) 1(2) 2(1) 2(2) 3(1) 3(2) 4(1) 4(2) 5(1) 5(2)
Comparison of measured and predicted failure loads of Regan slabs
Flexure EC2 ACI 318 MC2010 LII Vtest/Vshear Vtest /Vshear Vtest/Vshear Mcftest/McfR Vtest/Vflex Vtest/Vshear c2+2y gv = 0 gv Eq (2) ke = 0.7 ke Eq. (11) 1.12 0.81 1.04 0.91 1.21 1.38 1.49 1.05 1.03 0.97 0.85 1.30 1.44 1.65 1.02 0.91 0.94 0.83 1.20 1.34 1.51 1.13 0.92 1.05 0.92 1.28 1.45 1.60 0.50 0.40 1.46 1.25 1.08 1.33 1.17 0.92 1.16 0.82 0.70 1.22 1.34 1.63 1.07 0.88 0.98 0.87 1.32 1.32 1.52 1.21 1.12 1.05 0.83 1.06 1.57 1.67 0.76 0.36 1.25 1.60 1.62 1.24 1.12 1.05 1.10 1.00 1.41 1.39 1.64 1.65 Average 1.06 1.02 1.28 1.41 1.50 Standard deviation 0.18 0.30 0.15 0.12 0.20 5%(1) 0.76 0.53 1.03 1.20 1.18 (2) COV 0.17 0.29 0.12 0.09 0.13
(1) 5% lower characteristic value, (2) COV = coefficient of variation
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5
Influence of reinforcement arrangement
A series of parametric studies were carried out to investigate the influence on punching resistance of varying the flexural reinforcement ratio in slabs with the same geometry and loading arrangement as those tested by Regan (1993). The reinforcement arrangement was similar to slab S1. The concrete strength was taken as 35.4 MPa (5133 psi) as in specimens S1 and S2. The motivation is that EC2 and MC2010 Levels II and III relate the punching resistance provided by concrete to the area of flexural tension reinforcement at the edge column. In the case of EC2, increasing the reinforcement ratio increases the shear resistance vc according to Equation (4). In the case of MC2010 Levels II and III, increasing the area of flexural tension reinforcement over the column increases mR and hence reduces rotation for given ms, which in turn increases Vc. Analyses were carried out with longitudinal hogging reinforcement ratios rsup within an effective width be = 2c1 + c2 at the column support equal to 0.43%, 1.0% and 1.6%. The reinforcement ratio rsup = 1.6% gives a moment of resistance of approximately 0.17𝑏b 𝑑 F 𝑓" , which is the maximum design unbalanced moment permitted by Annex I of EC2. For each of these ratios, the longitudinal reinforcement ratio in the span rspan was taken as 0.25%, 0.5%, which corresponds to Regan’s slabs, and 1.0%. The analyses were initially carried out with the area of transverse reinforcement provided in the Regan slabs, which corresponds to a reinforcement ratio of 0.5% within a width of 1350 mm (53.1 in.) from the slab edge. Subsequently, the analyses with rspan equal to 0.5% and 1.0% were repeated with the hogging transverse reinforcement ratio doubled to 1.0%. In all cases, the NLFEA rotations relative to the support area about an axis parallel to the short slab edge were greatest and hence critical. The resulting load rotation diagrams are presented in Figure 8a for 0.5% transverse reinforcement as in the Regan slabs. Figure 8a also shows the punching resistance calculated with MC2010 for ke = 0.7. Figure 8a shows that the slab rotation relative to the column is essentially governed by the span reinforcement and is almost independent of rsupport. The MC2010 Level IV resistances are given by the intersection of the load rotation curves with the resistance curve. Figure 8a shows that the Level IV punching resistances are slightly lower but broadly consistent with the NLFEA predictions. In the absence of test data, it is unclear which of the two are most accurate. The slabs with rspan = 0.25% failed in flexure. Figure 8a also shows rotations calculated with MC2010 Level III for rsup = 0.43%, 1.0% and 1.6%, which do not correlate with the NLFEA rotations owing to the dependence of the Level III rotations on elastic moments.
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Resistance (ke = 0.7) ρsup = 1.6% ρsup = 1.0% ρsup = 0.43% Test MC2010 LIII ρsup = 1.6% MC2010 LIII ρsup = 1.0% MC2010 LIII ρsup = 0.43%
500 450 400
ρspan = 1.0%
Load (kN)
350 300
ρspan = 0.50%
250 200
ρspan = 0.25%
150 100 50 0 0
0.005
0.01
0.015
0.02
0.025
0.03
Rotation (radians)
(a) ρsup = 1.6% ρspan = 0.5% 0.7
ρsup = 1.0% ρspan = 0.5% 0.6
ρsup = 0.43% ρspan = 0.5%
M/V (m)
0.5
0.4
0.3
0.2
0.1
0 0
50
100
150
200
250
300
V (kN)
(b) Figure 8:
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Influence of varying reinforcement arrangement on: a) load rotation relationships for Regan slabs, and b) eccentricity M/V rspan = 0.5%. (Figure continues on next page.)
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ρsup = 1.6% ρspan = 1.0% 0.7
ρsup = 1.0% ρspan = 1.0% 0.6
ρsup = 0.43% ρspan = 1.0%
M/V (m)
0.5
0.4
0.3
0.2
0.1
0 0
100
200
300
400
V (kN)
(c) Figure 8 (cont.): Influence of varying reinforcement arrangement on: c) eccentricity M/V rspan = 1.0%. (Note: dimensions in meters; 1 m = 39.37 in.; 1 kN = 0.2248 kip)
The relative independence of the calculated punching resistance on rsupport in Figure 8a is significant because it is contrary to the predictions of EC2 and MC2010 Levels II and III. As previously discussed, these methods relate punching resistance to the area of tension reinforcement provided within the support strips adjacent to the column with punching resistance being independent of span reinforcement. The Atena analysis suggests that increasing the span reinforcement ratio is more effective at reducing the maximum crack strain adjacent to the column than increasing rsupport above that required for flexure. This is consistent with the prediction of the CSCT that the width of the critical shear crack is related to the rotation y, which Figure 8a shows to reduce with increasing rspan. Table 3 shows the punching resistances of the slabs considered in the parametric study, with rspan equal to 0.5% and 1.0%, according to EC2, ACI 318 and MC2010 Level II, with ke = 0.7 and from Equation (11), and Level III with ke = 0.7. Additionally, Table 3 shows the ratio of the moment of resistance at the column face McfR to the elastic moment at the NLFEA failure load across the slab width at the column face Mcfel. The ratios of McfR/Mcfel corresponding to rsup = 0.43% correspond to a larger degree of moment redistribution than the maximum of 70% allowed by EC2 and UK practice for edge column moments calculated with elastic FEA (The Concrete Society, 2007). All the punching resistances in Table 3 were calculated using the elastic eccentricity M/V as would be done in design. Consequently, the underestimation of resistance for MC2010 Level II and ACI 318 with gv from Equation (2) is greater in Table 3 than in Table 2, where the measured ultimate eccentricity M/V was used in calculations of resistance. Table 3 shows that increasing the transverse reinforcement ratio to 1.0% at the edge column marginally increased the punching resistances given by NLFEA with 315
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the increase greatest for rspan = 1.0% and rsup = 1.6%. Table 3 also shows that EC2 gives the most accurate predictions of VNLFEA. MC2010 Level II and ACI 318 with gv from Equation (2) are overly conservative partly because the eccentricity M/V reduces with loading, as shown in Figures 8b and 8c for selected analyses from Figure 8a. The reduction in M/V occurs as a result of moment being redistributed into the span following torsional cracking and yielding of flexural reinforcement at the support. The practical consequence is that calculating the punching resistance at edge columns on the basis of the elastic eccentricity M/V is overly conservative because it overestimates the design shear stress. It is suggested that when using MC2010, the effectiveness coefficient ke is taken as 0.7 at the edge columns of braced frames rather than being calculated with Equation (11). ACI 318 with gv = 0 tends to overestimate the shear resistance of the slabs considered in this paper partly due to its neglect of the size effect and flexural reinforcement ratio in the calculation of vc. Neither ACI 318 nor EC2 capture the influence of the span reinforcement on punching resistance. It is however notable in Table 3 that the EC2 predictions of the NLFEA column failure loads VNLFEA become less conservative as rsup increases, whereas the accuracy of the ACI 318 predictions with gv = 0 are almost independent of rsup for any given span reinforcement ratio. The MC2010 Level II predictions are particularly poor for these slabs because the rotation y is significantly overestimated. Table 3: Comparison of calculated punching resistance for slabs in parametric study.
rtran
% 0.5 0.5 0.5 0.5 0.5 0.5 1 1 1 1 1 1
rsupport rspan VNLFEA
%
kN
ACI 318 VNLFEA/Vshear gv = 0
0.5 273 0.49 1.12 0.88 0.5 275 1.06 0.98 0.89 0.5 269.8 1.61 0.89 0.87 1.0 352.4 0.38 1.28 1.14 1.0 379.2 0.77 1.20 1.23 1.0 364.6 1.19 1.07 1.18 0.5 273.2 0.49 1.12 0.88 0.5 277.6 1.05 0.99 0.90 0.5 273.6 1.59 0.90 0.89 1.0 379.8 0.35 1.38 1.23 1.0 391.4 0.75 1.24 1.27 1.0 419.6 1.03 1.23 1.36 Average 1.11 1.06 Standard deviation 0.16 0.19 5% 0.86 0.75 COV 0.14 0.18 1 kN = 0.2248 kip; COV = coefficient of variation
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0.43 1 1.6 0.43 1 1.6 0.43 1 1.6 0.43 1 1.6
%
EC2 Mcf R/ Mcf el V NLFEA /Vshear
gv Eq (2)
1.39 1.40 1.38 1.80 1.94 1.86 1.39 1.42 1.40 1.94 2.00 2.14 1.67 0.30 1.18 0.18
MC2010 VNLFEA/Vshear Level II
Level III
ke= 0.7 ke Eq (11) 2.07 2.39 1.38 1.62 1.29 1.51 2.67 3.08 1.91 2.24 1.74 2.05 2.07 2.39 1.40 1.64 1.13 1.34 2.88 3.32 1.97 2.31 1.73 2.05 1.85 2.16 0.53 0.60 0.98 1.17 0.29 0.28
ke= 0.7 1.76 1.17 0.95 2.27 1.61 1.28 1.76 1.18 0.96 2.45 1.67 1.47 1.54 0.48 0.76 0.33
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6
Conclusions
This paper reviews design for punching shear at edge columns of continuous flat slabs with normal moments. The research supports the rotational failure criterion of MC2010 and justifies the practice of redistributing normal moments at edge columns within normal code limits. It is shown that according to finite element analysis with Atena 5.1.2 and MC2010 Level IV the punching resistance of continuous slabcolumn subassemblages is significantly affected by the flexural reinforcement ratio in the span due to support moments being redistributed from edge columns into the adjacent span. As shown in Figure 8b, moment redistribution can cause the loading eccentricity M/V to reduce significantly below its elastic value, which is used to define the punching resistance in MC2010 Levels II and III. Consequently, MC2010 Levels II and III can be overly conservative if slab rotations and ke are calculated using elastic eccentricities, particularly if support moments are redistributed in the reinforcement design. The good predictions of EC2 and MC2010 Level IV with ke = 0.7 provide support for the practice of considering the strength reduction due to eccentricity normal to the slab edge to be independent of the unbalanced moment, as suggested by Regan (1981) and Moehle (1988). ACI 318 gives reasonable predictions of the shear resistance of Regan’s specimens if gv is calculated with Equation (2), as suggested by Ghali et al. (2015), using the measured eccentricity, but can significantly underestimate resistance if the elastic eccentricity is used as shown in Table 3. However, ACI 318 tends to overestimate capacity with gv = 0 as shown in Tables 2 and 3 in part due to its neglect of the size effect and reinforcement ratio in the calculation of vc. Both NLFEA and MC2010 Level IV suggest that providing surplus flexural reinforcement in the span normal to the slab edge can be more effective at increasing vc at edge columns than providing surplus flexural reinforcement normal to the slab edge. The is contrary to the predictions of ACI 318, EC2 and MC2010 Levels II and III, which assume punching resistance to be independent of the span reinforcement ratio. In UK practice, surplus reinforcement is often provided in the span for control of deflections at the serviceability limit state, but its potential benefit on punching resistance is neglected and merits further study. The findings of this research are not considered to be a substitute for the provision of shear reinforcement which is commonly required in economically proportioned flat slabs. Increasing Vc is beneficial because it reduces the required area of shear reinforcement as well as increasing VRmax if considered to be a multiple of Vc as in EC2, MC2010 and ACI. Of the design methods considered in this paper, EC2 and MC2010 Level IV give the best predictions of punching resistance with EC2 being preferred by the authors for design due to its simplicity. MC2010 Level IV is however a powerful tool for assessment of existing structures and gives valuable insights into structural behaviour.
7
Acknowledgements
This research was supported by the Science Without Borders Program, through the National Council for Scientific and Technological Development (CNPq), funded by the Brazilian Ministry of Science and Technology.
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8
References
ACI Committee 318 (2014) Building Code Requirements for Structural Concrete (ACI 31814) and Commentary, American Concrete Institute; Farmington Hills, MI, 503 pp. Alexander, S. D. B. and Simmonds, S. H. (2003) “Moment transfer at interior slabcolumn connections,” ACI Structural Journal, V. 100, No. 2, MarApr, pp. 197202. BSI (2004) EN 199211:2004, Eurocode 2, Design of Concrete Structures – Part 11: General Rules and Rules for Buildings, BSI, London. Červenka, V., Jendele, L, and Červenka, J. (2016) ATENA Program Documentation Part 1 Theory, Prague. Einpaul, J., Fernández Ruiz, M. and Muttoni, A. (2015) “Influence of moment redistribution and compressive membrane action on punching strength of flat slabs,” Engineering structures, V. 86, pp. 4357. fib (Fédération international du béton) (2013) fib Model Code for Concrete Structures 2010, Ernst & Sohn, Berlin. Ghali, A., Gayed, R. B. and Dilger, W. (2015) “Design of Concrete Slabs for Punching Shear: Controversial Concepts,” ACI Structural Journal, V 112, No. 4, pp. 505514. Joint ACIASCE Committee 352 (2011) Guide for Design of Slab Column Connections in Monolithic Concrete Structures (ACI 352.1R11), American Concrete Institute, Farmington Hills, MI, 28 pp. Moehle, J. (1988) “Strength of SlabColumn Edge Connections,” ACI Structural Journal V. 85, No. 1, pp. 8998. Muttoni, A. (2008) “Punching shear strength of reinforced concrete slabs without transverse reinforcement,” ACI Structural Journal, V. 105, No. 4, pp. 440–450. Regan, P. E. (1981) Behaviour of reinforced concrete flat slabs, Ciria Report 89, London. Regan, P. E. (1993) Tests of Connections between flat slabs and edge columns, School of Architecture and Engineering, University of Westminster. London. Regan, P. E. (1999) “Ultimate limit state principles,” in fib Bulletin 2, Structural Concrete: Textbook on Behaviour, Design and Performance; Volume 2: Basis of design, Fédération international du béton, Lausanne, Switzerland. Soares, L. F. S. and Vollum R. L. (2015) “Comparison of punching shear requirements in BS8110, EC2 and MC2010,” Magazine of Concrete Research, V. 67, No. 24, pp. 13151328. Soares, L. F. S. and Vollum, R. L. (2016) “Influence of continuity on punching resistance at edge columns”, Magazine of Concrete Research, V. 68, No. 23, pp. 12251239. The Concrete Society (2007) Guide to the Design and Construction of Reinforced Concrete Flat Slabs, Technical Report 64, Camberley, UK, ISBN 1904482333. Vecchio, F. J. and Collins, M. P. (1986) “The modified compressionfield theory for reinforced concrete elements subjected to shear,” ACI Structural Journal, V. 83, No. 2, pp. 219231.
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Seismic retrofit of concrete slabs against punching shear: Testing and modelling Dritan Topuzi1, Maria Anna Polak2, Sriram Narasimhan2 1
: Delta Innoveering Inc., Canada
2
: University of Waterloo, Canada
Abstract The focus of this research is on developing new punching shear retrofit techniques for slabcolumn connections to improve the seismic response of flatplate systems. Previous tests have shown the effectiveness of using shear reinforcement to enhance the shear strength and ductility of individual slabcolumn connections. However, the advantage of ductility in reducing the earthquake impact on structures is accompanied by an increase in the base shear, due to increased stiffness. Herein, a new type of punching shear retrofit element, shear bolts with flexible washers, is introduced. The flexible washers allow for shear crack opening during the lateral displacements, while at the same time providing control of the crack width by controlling the washer thickness and/or stiffness. The results show that this technique increases the ductility of the connections, without a commensurate increase in stiffness. The effect of this type of shear reinforcement on the response of an assembled structure is investigated through dynamic analysis, to check how energy dissipation within individual connections affects the overall energy dissipation of a flatplate system. The presented system was designed for slab retrofit. However, it can be anticipated that similar concepts could be used in the construction of new slabs in seismic zones.
Keywords Flat slab, punching reinforcement, reinforced concrete flatplate system
1
Introduction
Reinforced concrete flatplate structures are widely used as structural systems. They consist of a flat slab and columns with no beams to support the slab. The flatplate system is popular in construction because it offers low storey height, ease of construction, better architectural appearance, and low construction costs. However, when exposed to large vertical forces or horizontal deformations, especially during an earthquake, this type of structure is vulnerable to punching shear failure at the slabcolumn connections. Cracks can occur inside the slab, in the vicinity of the column perimeter, and may propagate at an angle of 20 to 45 degrees, leading to punching failure of the joint This type of failure is usually brittle, particularly if no shear reinforcement is used in the slab around the column. Brittle failures are sudden and catastrophic for the safety of building occupants. In some cases, the failure of a joint may cause the failure of the adjacent joints, triggering a progressive collapse of part or even the entire building (Feld and Carper, 1997) (Fig. 1). 319
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Figure 1:
Skyline Plaza collapse, Fairfax County, VA, USA, 1973 (Courtesy of NIST).
The most typical methods of mitigating punching shear failure at a slabcolumn connection are the following: (i) increasing the area of concrete resisting shear stresses, which can be achieved by increasing the thickness of the slab, providing a drop panel or capital, or increasing the dimensions of the column; (ii) providing concrete of higher strength; and (iii) providing additional shear strength through shear reinforcement within the slab around the column perimeter. The first two methods are effective in increasing punching strength, but not ductility (Megally and Ghali, 2000). Adding shear reinforcement increases both strength and ductility, and it is also the most practical way of retrofitting existing slabcolumn connections, which is the primary motivation of this research. Research on including shear reinforcement in new slabs was initiated by Graf in 1933 and Wheeler in 1936, while research on retrofitting techniques was initiated by Ghali et al. in 1974. In these studies, strengthening was provided by prestressing the slab around the column through tensile bolts placed in holes near the column. Later research was done by Ramos et al. (2000), who introduced shear heads consisting of steel Ibeams around the column, and Ebead and Marzouk (2002), who introduced a similar approach using external steel plates and steel bolts bonded to the slab surface. Both these techniques improve the behaviour of the connection, changing the failure type from brittle to flexural, but they are elaborate and change the aesthetics of the slab significantly.
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Several test programs related to punching shear of reinforced concrete slabcolumn connections have been carried out at the University of Waterloo, where an effective retrofit through steel bolts, called shear bolts, was developed (ElSalakawy et al., 2003). They consist of a headed rod, threaded at the other end for anchoring using a washer and a nut. The bolts (Fig. 2) are installed in holes drilled in the slab around the column perimeter, which makes them suitable for retrofits.
Figure 2:
Shear bolt.
In 2003, ElSalakawy et al. published the results of tests on four edge slabcolumn specimens strengthened using the aforementioned shear bolts, concluding that shear bolts can increase the capacity and ductility, and change the failure from brittle punching shear mode to a more favourable flexural mode. In 2005, Adetifa and Polak tested six simply supported interior slab column connections reinforced with shear bolts, subjected to vertical loading only. They concluded that retrofitting through shear bolts increased the ultimate punching shear load by over 42% and displacement ductility by 229%, compared to the specimens without shear bolts. As a continuation of this work, in 2008, Bu tested nine slabcolumn connections reinforced with shear bolts, subjected to lateral cyclic loading. Steel shear bolts were shown to be an effective method for retrofitting slabs in seismic zones, changing the failure mode from brittle punching to ductile flexural. In 2008, Lawler tested slabcolumn connections reinforced with shear bolts made of glassfibrereinforced polymer (GFRP). In comparison to slabcolumn connections reinforced with steel bolts tested by Bu, the FRP connections showed lower strength, but also less pinching resulting in higher energy dissipation. This happened because the GFRP bolts were not fully secured against the slab surface, allowing for opening of cracks within the slab and the friction between the crack faces under cyclic loading, resulting in significant energy dissipation. The opening of cracks was also the reason for the lower strength observed compared to the connections reinforced with steel bolts. The installation of shear reinforcement introduces a high level of pinching in the lateral loaddisplacement response, as a result of increased strength and stiffness at the slabcolumn connection (Bu and Polak, 2009). Such stiff steel reinforcements keep the connection within a relatively elastic range, limiting the amount of plastic deformations. As a result, such connections would attract higher seismic forces due to their high stiffness and strength, transferring a higher base shear force at the respective column. As such, reinforcing and strengthening of slabs against punching shear from seismic forces could result in a system that would require strengthening of the columns as well. A dual structural system, with special lateral force resisting elements to withstand seismic forces and slender columns to withstand gravitational forces, would be architecturally more appealing for flatplate systems. This would require slabcolumn connections with a low stiffness and strength and high deformability, to allow for transfer of seismic forces onto laterally stiffer and stronger 321
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elements.
The desirable characteristics of the flatplate structure in an earthquake is to have sufficient strength and stiffness for it to withstand moderateintensity shaking, and sufficient ductility to act in parallel with a more rigid structural system for strong base motions. Such design of slabcolumn connections would be in agreement with the philosophy of “capacity design,” where the designer “tells the structure what it should do in the event of a major earthquake” (Paulay, 1999; Paulay and Priestley, 1992). Following on previous research (Bu and Polak, 2009), herein, a new type of punching shear retrofit elements, shear bolts with flexible washers, is introduced. The flexible washers allow for shear crack opening during the lateral displacements, while at the same time provide control of the crack width by using the appropriate washer thickness and/or stiffness. This study focuses on how anchoragecontrolled shear reinforcement can improve punching shear capacity of flat slabs, while introducing lower pinching to the lateral loaddeformation response of slabcolumn connection. Such connection response would allow for more efficient capacity design of flatplate systems by distributing strength and stiffness more appropriately (Paulay and Priestley, 1992). Previous work (Bu and Polak, 2009) has demonstrated the ability of shear reinforcement to retrofit the shear capacity and ductility of individual slabcolumn connections. However, quantifying the effect of these types of reinforcement on the behaviour of a complete structure is important in order to understand and advance this technology. As it is not practical to test a fullscale structure in the lab, a key objective of this study is the investigation of system behaviour using a computerbased model, which incorporates the joint behaviour (determined experimentally and/or analytically) into the evaluation of the seismic response of flatplate frames. The effect of the proposed shear reinforcement on the crack interface behaviour is investigated first. This response is then incorporated into the slabcolumn joint model, to investigate the effect on the lateral response of the joint. Modelling of frames, as assemblages of these joints, enables the investigation of the performance of flatplate systems, under seismic forces. The presented system was designed for slab retrofit; however, it can be anticipated that similar concepts could be used in construction of new slabs in seismic zones as well. Further research is necessary prior to being considered for practical applications.
2
Research significance
Reinforced concrete flatplate systems are a popular structural system because of their many advantages discussed in the introduction. Improving punching shear strength and ductility of slabcolumn connections remains an important issue, especially in seismic areas. Use of traditional shear bolts has been successful in increasing punching strength and ductility, but makes the connection very strong and stiff, which results in higher pinching of the loaddeformation response and most importantly, higher seismic forces attracted by these connections; these forces are then transferred onto the columns. In this research, slabcolumn connections retrofitted with a combination of stiff bolts and flexible washers were tested. The results show that the retrofitted slabs can undergo large deformations with a limited pinching of the response and without experiencing considerable increase in strength and stiffness. Such joint response would introduce higher energy dissipation and better distribution of seismic forces from the weaker elements to the special moment resisting frames or shear walls of a flatplate system. Models were developed to prove the effectiveness of such shear reinforcement at a microlevel, focused at the crack interface response, and at a macrolevel,
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focused on the performance of frames, as assemblage of these joints. These models involved the integration of other original or modified models, adjusted for the slabcolumn joints and flatplate systems.
3
Test program
The test program focused on how anchoragecontrolled shear reinforcement can improve the punching shear capacity and ductility of slabcolumn connections, without a commensurate increase in strength and stiffness. Six fullscale specimens were tested: SD01 being the control specimen, with no shear reinforcement, and SD03 and SD06 shear retrofitted with the configurations shown in Table 1. The details of the test program are described in Topuzi et al. (forthcoming). Herein, only brief description of the tests is provided. Table 1: Test specimens Spec. SD01 SD03 SD06
Dimensions, m (in) (between supports) 1.5 x 1.5 x 0.12 (59.1 x 59.1 x 4.72) 1.5 x 1.5 x 0.12 (59.1 x 59.1 x 4.72) 1.5 x 1.5 x 0.12 (59.1 x 59.1 x 4.72)
Vert. load, kN (kips)
𝑓"# , MPa (psi)
Concrete ft, MPa (psi) (split)
Gravity shear ratio
110 (24.7)
50 (7250)
3.2 (464)
0.45
110 (24.7)
41 (5950)
3.0 (435)
0.50
110 (24.7)
41 (5950)
3.0 (435)
0.50
Type of shear reinforcement No shear reinforcement SB + 1 × 3mm (1/8in.) nylon washer Steel bolts only
Since this research builds upon previous experimental work done at the University of Waterloo by Bu and Polak (2009), the same specimen configuration was considered for comparison purposes. All specimens have slab dimensions of 1,800 x 1,800 x 120mm (70.9 x 70.9 x 4.72in.) with top and bottom column stubs with a section of 200 x 200mm (8 x 8in.) and a length of 700mm (27.6in.) going through the centre of the slab (Fig. 3). They were simply supported at a 1,500 x 1,500mm (59.1 x 59.1in.) perimeter on the bottom face of the slab. Thick neoprene pads were provided on top and bottom of the slab to allow for rotations. The neoprene pads were 25mm (1in.) thick and 50mm (2in.) wide, and installed along the lines of support, as shown in Fig. 3. All specimens were first subjected to a vertical load of V=110kN (24.7kips), which was kept constant while the connection was subsequently subjected to lateral cyclic displacements, until failure, defined by a significant drop in the lateral strength. On the bottom of the slab, the tension flexural reinforcement ratio was 1.05% for the outer bars (10M at 100 mm (3.94in.)) and 1.3% for the inner bars (10M at 90 mm (3.54in.)), to ensure the same flexural capacities in both orthogonal directions. On the top face of the slab the reinforcement ratio was 0.58% (10M at 200 mm (7.87in.)) in both directions. The reinforcement of the columns consisted of 825M bars (three bars on each face) with 10M at 100 mm (3.94in.) closed ties. The columns were designed to transfer shear force and cyclic moment to the slab, while experiencing negligible deformations. The configuration of the reinforcement is shown in Fig. 4.
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(24.7 kips)
Figure 3:
Specimen dimensions, loading and support conditions. (Note: units in mm; 1mm = 0.0394in.)
Figure 4:
Slab and column reinforcement details. (Note: 1mm=0.0394in.)
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Shear bolts were installed into drilled holes on the existing slab structure. Each shear bolt assembly was made of a threaded steel rod with thick hardened steel washers and nuts on each end. Additional flexible washers were added between the steel washers, as shown in Fig. 5(a) and 5(b). The following shear bolts setup was used for the tested specimens: • • •
Specimen SD01 had no shear reinforcement. Specimen SD03 had a single nylon washer of 3mm (1/8 in.) thickness, on one side of the slabs. Specimen SD06 had no flexible washers; steel bolts only.
(a) Schematic of steel bolt and washers assembly
(b) Installed bolt assembly Figure 5:
(c) Bolts plan configuration (4x2x6)
Shear retrofit details. (Note: 1mm=0.0394in.)
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Washers properties (material and thickness) were determined based on the expected size of punching shear cracks in the slab, considering previously tested similar slabs (Bu and Polak, 2009). Materials considerably more flexible than the steel bolts were used, for the washers to attract practically all the deformation occurring within the bolt assembly. Their thickness was determined such that the opening of cracks would be limited by the amount of their squeezing. A peak compressive force, equal to the yielding of the steel bolts, was considered for these calculations, based on previous experiments where the bolts close to the column yielded. The threaded bolts were of a nominal diameter of 9.5 mm (3/8 in.) and the holes in the concrete were drilled using a 12.7 mm (1/2 in.) drill bit. Six rows of bolts were installed on each of the five retrofitted specimens and no bolts on the first specimen. The plan configuration of shear bolts, applicable to specimens SD03 and SD06, is shown in Fig. 5(c). The specimens were cast using readymixed concrete with a 35 MPa (5075psi) specified strength and max. aggregate size of 9.5mm (3/8in.), supplied in two batches. Tested material properties are shown on Tables 1 and 2. Table 2: Flexural and shear reinforcement material properties Flexural reinforcement Modulus of Yield stress, elasticity, MPa (ksi) GPa (ksi) 460 (66.7) 200 (29000)
Shear reinforcement Yield stress, MPa (ksi)
Modulus of elasticity, GPa (ksi)
Modulus of elasticity, kN/mm (kips/in)
400 (58.0)
180 (26100)
50 (285)
(a) Test frame Figure 6:
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Nylon washer
Test frame and Specimen SD03.
(b) Crack pattern for SD03
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The test frame and specimen SD03 are shown in Fig. 6. All specimens failed in punching (observed from the respective hysteresis curves), as expected, considering they were overreinforced in flexure. Moment vs. horizontal drift ratios are shown in Fig. 7. As expected, the strength and the deformability of the connections were increased by the use of shear reinforcement of any type. The flexible washers, however, improved the ductility of the connection without considerably increasing its strength. Fig 7(a) shows the specimen without shear reinforcement that failed in punching shear, in a brittle manner. Specimen SD03 had nylon washers, which improved the strength of the specimen at low momentdrift. The specimen experienced a better plateau and lower pinching, in its momentdrift response, compared to Specimen SD06, which was reinforced with steel bolts and washers. Specimen SD06, having the stiffest reinforcement, had the highest strength and also highest pinching. The steel bolts restrained the opening of cracks up to a drift of 4%, introducing a relatively linear behaviour of the connection. Since this is a large drift for practical structural applications, the joint would practically remain within the elastic range in a seismic event, without taking advantage of plastic deformations, introduced in the case of flexible washers. Washers that allow large opening of cracks would not be very efficient because they would allow for higher strength degradation instead of introducing a flat plateau in the joint response curve. The experimental work showed that the use of anchoragecontrolled shear reinforcement increases the ductility of slabcolumn joints without a commensurate increase in strength. Such connections improve the seismic performance of flatplate systems.
Figure 7:
Momentdrift response of the three tested specimens. (Note: 1kNm=8.86 kipin)
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4
Punching crack interface model
In this section, the Walraven model (Walraven, 1981, 1994), which captures the shear stressslip behaviour along a crack interface using an aggregate interlock model, is used to investigate the effect of the shear reinforcement on the response along the punching crack interface in a slab. Fig. 8 shows the case of a particle (e.g. aggregate) in the cement matrix. The springs represent the reinforcement crossing the crack. During a shear displacement, the particle penetrates into the cement matrix, causing crushing of the matrix and friction between the matrix and the particles. An additional opening of the crack size generates tensile forces on the springs that are equivalent to the generated compressive stress of the crack plane. For the current crack opening, the compressive stress is calculated from the restraining stiffness of the extended springs, which is given as an input parameter depending on the reinforcement properties crossing the crack. For linear springs, the compressive stress will be proportional to the stiffness; for nonlinear springs, the stress can be calculated incrementally.
Figure 8:
Walraven model.
The original Walraven model considers the reinforcement crossing the crack as being perpendicular to the shear crack interfaces and of linear elastic behaviour, while in this analysis the following modifications are considered for the punching shear crack case: • •
The shear reinforcement crosses the crack at different angles (depending on the assumed punching crack angle) which may be considered accordingly in determining the stiffness of the representing spring. Different reinforcement behaviours have been considered for the springs to consider the effect of different shear reinforcement types.
Following the first cycle, the opening of cracks does not considerably increase, based on the test by Laible et al. (1977). For this reason, and considering that the calculation of cyclic opening is computationally expensive, in this research only the first cycle was considered to determine the properties of the Interface Shear Spring.
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4.1
Steel bolts as shear reinforcement
Steel reinforcement only was considered first as the reinforcement crossing the crack. The effect of the reinforcement stiffness and yielding was considered for the parameter analyses. The change in reinforcement stiffness would be achieved in practice by changing the bolt stem diameter and/or bolt spacing. Fig. 9 shows the effect of the reinforcement stiffness on the shear stressstrain behaviour of the crack interface with regards to shear slip and crack opening. The values of k = 2, 3, 4 and 5MPa/mm (7370, 11100, 14700 and 18400 psi/in.) represent the stiffness of the reinforcement crossing the crack. These values are calculated as the linear stiffness of the steel bolts (kN/mm (kips/in.)) divided by the crack interface area, for an assumed crack angle of 45°. This parameter analysis shows that flexible shear reinforcement would introduce higher ductility through larger opening of cracks, as well as larger shear slip resulting in higher interface friction. Since excessive opening of cracks would also result in failure, the following section introduces the flexibility through flexible washers, which allow larger, but limited, opening of cracks. It should be noted that the proposed crack interface model may also be used to consider the effect of other parameters, such as: aggregate size, concrete strength, etc.
Figure 9:
The effect of the reinforcement stiffness on the crack interface response (crack interface model). (Note: 1MPa = 145psi, 1mm=0.0394in.)
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4.2
Steel bolts and flexible washers as shear reinforcement
A combination of steel bolts and flexible washers results in springs connected in series crossing the crack, as shown in Fig. 10. Consequently, the stiffness of the equivalent spring is: 𝑘% =
𝑘' ∙ 𝑘* 𝑘' + 𝑘*
where kb is the bolt stiffness and kw is the washer stiffness, determined from testing the washer in compression.
Figure 10: Springs configuration representing the shear reinforcement.
Figure 11 shows the effect of washers on the crack interface response. In this case, thin washers of thickness t = 0.2mm (0.008in.) and t = 0.3mm (0.012in.), with negligible stiffness (assumed to be zero in the model), were considered. The equivalent stiffness of the shear reinforcement is equal to zero until a deformation equal to the washer thickness is reached. After this point, the equivalent stiffness is equal to the stiffness of the steel bolts. The stiffness of the bolts, per unit of area, is 3MPa/mm (11100psi/in.). It may be observed that there is an increase in crack opening and shear slip, which would also introduce a higher ductility of the crack interface response. The main advantage provided by the washers is in allowing for an additional crack opening, which is then “locked” by the stiff bolts, blocking any further considerable opening of cracks.
5
Slabcolumn joint modelling
This section proceeds with the investigation of the effect on the slabcolumn joint response through a largescale model of the joint. The joint model is calibrated based on the tested slabcolumn connections and is then used for the analytical investigation of frames, which cannot be practically tested in the laboratory. Shear transfer from the slab to the connection is modelled by the Walraven model (Walraven, 1981, 1994), while shear within the joint itself is modelled by the Modified Compression Field Theory (Vecchio and Collins, 1986). A computer model of the joint enables the study of the effect of various connection hysteretic responses on the energy dissipation of the whole structural system and therefore the effect on the structural response, in terms of distribution of forces and total base shear. 330
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Figure 11: The effect of the washers on the crack interface response (crack interface model). (Note: 1MPa = 145psi, 1mm=0.0394in.)
In order to simulate the behaviour of the tested slabcolumn connection, a nonlinear simulation was carried out in OpenSees. The connection model is composed of the following elements: (i) two nonlinear beams (representing the slabbeams); (ii) two nonlinear columns, and (iii) one joint superelement, as shown in Fig. 12, which is composed of: (a) one shear panel spring, intended to simulate strength and stiffness loss associated with shear failure of the joint core (Modified Compression Field Theory); (b) four interface shear springs, which are intended to simulate loss of sheartransfer capacity at the jointslab and jointcolumn interface (Walraven model), and (c) eight barslip springs, which are intended to simulate stiffness and strength loss associated with bondstrength deterioration for slab and column longitudinal reinforcement embedded in the joint core.
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Figure 12: Joint element (with 13 DOFs).
5.1
Modelling of slabbeams and columns
Columns and slabbeams were modelled using displacementbased nonlinear beamcolumn elements, with corotational geometric transformation and five GaussLobatto integration points along the element length. Concrete was modelled by a uniaxial constitutive model with tension softening (Concrete02 material, available in OpenSees). For the columns, concrete properties for the fibres confined by stirrups were computed using the modified KentPark procedure (Kent and Park, 1971). For the slabbeams, the confinement was neglected considering the configuration of reinforcement within the slab. Reinforcement steel was modelled by the GiuffreMenegottoPinto model (Filippou et al., 1983) (Steel02 material, available in OpenSees). A fibre model was used for the cross section (Spacone et al., 1996). 5.2
Constitutive models of joint springs
The response of the springs of the superelement was determined based on the guidelines given by Lowes et al. (2004), as described in the following. The barslip spring is intended to simulate the bond response of the slab and column longitudinal reinforcement embedded in the joint core. The BarSlip material available in OpenSees, calibrated on test results from Eligehausen et al. (1983) and Hawkins et al. (1982), was used. The shear panel is modelled using a symmetric multilinear momentrotation relationship, using the Pinching4 material available in OpenSees. The envelope properties (four typical (M−θ)) points are determined based on the Modified Compression Field Theory (MCFT), proposed by Vecchio and Collins (1986). MCFT provides shear stress vs. shear strain (τγ) response for a reinforced concrete panel, subjected to pure shear only. The panel, in this case, is defined by the vertical plane of the slabcolumn joint. The (τγ) relationship can be converted into a momentrotation (M−θ) relationship according to M = τ * Voljoint and tan(θ) = γ, where Voljoint = colheight × colwidth × slabheight is the volume of the intersection of the slab and column. For the Interface Shear Spring, the Walraven model was considered as described above. A shear forceslip response is calculated with this model considering the shear reinforcement technique used. Figure 13 shows the tested and analytical responses for two joints: one reinforced with steel bolts only and one reinforced with anchoragecontrolled shear reinforcement. Being a macrolevel model, the investigation is mostly qualitative, to determine the effect of the stiffness and flexible washers on the joint response.
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(a) Joint with steel bolts only
(b) Joint with anchoragecontrolled reinforcement
Figure 13: Momentdrift response of joints (model vs. test). (Note: 1kNm=8.86 kipin)
6
Modelling and analysis of flatplate systems
This part is focused on the analytical investigation of the effect of the slabcolumn joint response on the global response of a frame. The information obtained in the tests program is now used in a largescale frame analysis with a simplified joint depicted in Fig. 14. Two typical joint responses, shown in Fig. 15, were considered in this model, following the experimental part of this research. The grey lines in Fig. 15 show momentdrift ratio envelopes for the tested specimens. The responses captured from the tested specimens have been simplified into one envelope elasticperfectly plastic response for the case with flexible washers – Joint I, and a quadrilinear curve for the stiffer response, with steel bolts only – Joint II. A failure criterion was used such that the material representing the joint behaviour fails if a rotation of 0.06 is achieved (based on the failure drift observed from the tested specimens). From that point on, values of 0.0 are returned for the tangent stiffness and stress. This also enables the simulation of the progressive collapse of the frames. The frame analyses presented here are based on joint responses captured from the tested slabcolumn specimens. Otherwise, when such test results are not available, the necessary joint response can be analytically determined, as described above. The internal flexible columns were modelled by displacementbased nonlinear beamcolumn elements. Slabbeams were divided into three segments. The two end segments were modelled as very stiff, with an artificially high modulus of elasticity, and the central segment was modelled as a linear elastic beam element. The dimensions of the slabbeam were determined based on the effective slab width theory. The rigid segments were used because that portion of the slab had already been included in the response of the slabcolumn joints, represented by the rotational spring in Fig. 14. The column did not need such adjustments, since the column of the tested specimen was overdimensioned and overreinforced, resulting in negligible deformations. Two rotational springs were used on each beamcolumn node, one on each side of the column (Fig. 14). The tested momentrotation response was used for the springs behaviour for the threestorey frame split into the two springs in series.
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Figure 14: Joint spring in the frame model.
A threestorey structure was subjected to ground acceleration using the ElCentro 1968 record. Rayleigh damping proportional to the initial stiffness matrix was considered, except for the springs for which damping was set to zero due to their negative stiffness at large deformations. The recorded base shear force is shown in Fig. 16 for the two types of spring responses. From this graph it may be observed that the peak base shear for the frame with stiff joints is 17% higher than in the case with flexible joints. For the initial cycles, no difference is observed, since the behaviour of the springs is the same up to 2% lateral drift. It is beyond the drift of 2% that the structure behaves better in the case of flexible joints due to the ‘yielding’ of the slabcolumn connections. For the same reason, the difference in base shear becomes higher as the peak load becomes higher, making this a more efficient seismically responsive structure when subjected to an earthquake.
Figure 15: Envelope joint responses considered in the frame model.
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Figure 16: Time history: base shear force. (Note: 1kN=0.225kips)
7
Conclusions
The experimental program showed that shear retrofitting existing reinforced concrete flat slabs with anchoragecontrolled reinforcement increases the ductility of slabcolumn connections without a commensurate increase in lateral strength. This results in a better response of flatplate systems in seismic areas, as concluded by the analytical investigations. The anchoragecontrolled shear reinforcement allowed for larger, but limited, opening of punching shear cracks without hindering the deformability of the connection. The achieved lower stiffness results in lower seismic forces attracted by the connection. Compared to traditionally retrofitted slabcolumn joints, larger opening of cracks also resulted in higher plastic deformations, which may be observed as reduced pinching in the joint loaddeformation response. The macroscale nature of these models enables the analysis of slabcolumn joints and flatplate systems, which would otherwise be computationally expensive if 3D finite elements were to be used. However, this comes at a cost to accuracy, due to the various assumptions accepted in the modelling process. The models are not intended to accurately predict the behaviour of slabcolumn joints, but may be used successfully for parametric analyses, especially if test results of a similar slabcolumn specimen are available.
8
References
Adetifa B. and Polak M. A. (2005) “Retrofit of slab column interior connections using shear bolts,” ACI Structural Journal, V. 102, No. 2, pp. 268–274. Bu W. and Polak M. A. (2009) “Seismic retrofit of reinforced concrete slabcolumn connections using shear bolts,” ACI Structural Journal, V. 106, No. 4, pp. 514–522. Bu W. (2008) Punching Shear Retrofit Method Using Shear Bolts for Reinforced Concrete Slabs under Seismic Loading, PhD thesis, University of Waterloo, Waterloo, ON, Canada. Ebead U. and Marzouk H (2002) “Strengthening of twoway slabs subjected to moment and cyclic loading,” ACI Structural Journal, V. 99, No. 4, pp. 435–444.
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ElSalakawy E. F., Polak M. A. and Soudki K. (2003) “New shear strengthening technique for concrete slabcolumn connections,” ACI Structural Journal, V. 100, No. 3, pp. 297–304. Eligehausen R., Popov E. P., and Bertero V. V. (1983) EERC report 83/23: Local bond stressslip relationship of deformed bars under generalized excitations, Technical report, University of California, Berkeley, 169pp. Feld J. and Carper K. L. (1997) Construction Failure, John Wiley and Sons, 2nd ed., New York, 528pp. Filippou, F. C., Popov, E. P., Bertero, V. V. (1983) Effects of Bond Deterioration on Hysteretic Behavior of Reinforced Concrete Joints, Technical report 8319, Earthquake Engineering Research Center, pp. 184. Ghali A., Sargious M. and Huizer A. (2011) “Vertical prestressing of flat plates around columns,” Shear in Reinforced Concrete, ACISP
4238, American Concrete Institute, Farmington Hills, MI, 1974, pp. 905–920. Graf O. (1933) “Versuche über die widerstandsfaehigkeit von eisenbettonplatten unter konzentrierten last nahe einem auflager,” Deutscher Ausschuss fuer Eisenbeton, V. 73, pp. 1–16. Hawkins N. M., Lin I. J., and Jeang F L. (1982) “Local bond strength of concrete for cyclic reversed loadings,” Bond in Concrete, P. Bartos (ed.), Applied Science Publishers Ltd., London, pp 151161. Kent D. C. and Park R. (1971) “Flexural members with confined concrete” Journal of the Structural Division, V. 97, No. 7, pp. 1969–1990. Laible J. P., White R. N., and Gergely P. (1977) Experimental investigation of seismic shear transfer across cracks in concrete nuclear containment vessels, ACI Special Publication, V. 53, pp. 203–226. Lawler N. (2008) Punching shear retrofit of concrete slabcolumn connections with GFRP shear bolts, Master’s thesis, University of Waterloo, Waterloo, ON, Canada. Lowes L. N., Mitra N., and Altoontash A. (2004) A beamcolumn joint model for simulating the earthquake response of reinforced concrete frames, Technical report, Pacific Earthquake Engineering Research Center, pp. 59. Megally S. and Ghali A. (2000) “Seismic behavior of slabcolumn connections,” Canadian Journal of Civil Engineering, V. 27, No. 1, pp. 84–100. Paulay T. and Priestley M. J. N. (1992) Seismic Design of Reinforced Concrete and Masonry Buildings,
WileyInterscience, New York, 768pp. Paulay T. (1999) “A simple seismic design strategy based on displacement and ductility compatibility,” Earthquake Engineering and Engineering Seismology, V. 1, No. 1, pp. 51–67. Ramos A. M., Lucio V. J. and Regan P. E. (2000) “Repair and strengthening methods of flat slabs for punching,” Proceedings, International Workshop on Punching Shear Capacity on RC Slabs, Stockholm, pp. 125–130. Spacone E., Filippou F. C., and Taucer F. F. (1996) “Fibre beamcolumn model for nonlinear analysis of r/c frames: Part I. formulation,” Earthquake Engineering and Structural Dynamics, V. 25, No. 7, pp. 711–725. Topuzi D., Polak M., and Narasimhan S. (forthcoming) “A new technique for the seismic retrofit of slabcolumn connections,” ACI Structural Journal. Vecchio F. J. and Collins M. P. (1986) “The modifiedcompression field theory for reinforced concrete elements subjected to shear,” Journal of the American Concrete Institute, V. 83, No. 2, pp. 219–231. Walraven J. (1981) “Fundamental analysis of aggregate interlock,” Journal of the Structural Division of the American Society of Civil Engineers, V. 107, No. 11, pp. 2245–2270. Walraven J. (1994) “Rough cracks subjected to earthquake loading,” Structural Journal of the American Society of Civil Engineers, V. 120, No. 5, pp. 1510–1524. Wheeler W. H. (1936) “Thin flatslab floors prove rigid under test,” Engineering News Record, V. 116, No. 2, pp. 49–50.
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A new method for postinstalled punching shear reinforcement Rupert Walkner, Mathias Spiegl, Jürgen Feix University of Innsbruck, Austria
Abstract The assessment of existing flat slabs and bridges often shows insufficient punching shear resistance in the area of the support regions. The reasons for this are modified requirements of use or more restrictive design rules. There are many different methods to increase the punching shear resistance, but most of them are expensive and require access to the upper surface of the structure. Thus the construction work is only possible under restricted operation. In addition, difficult detail issues arise with regard to the rearrangement of the structure sealing. This paper deals with a new strengthening system using concrete screws, which are installed vertically into predrilled holes from the soffit of the slab. The results of two test series with a total of nine specimens are presented in this paper. It turns out that this strengthening method leads to a significant increase in the shear punching capacity and to a less brittle failure mode.
Keywords Punching shear reinforcement, strengthening, concrete screws, postinstalled shear reinforcement, punching tests
1
Introduction
The verification of sufficient punching shear strength is often decisive for the design of flat slabs. This applies to flat slabs in buildings as well as to bridges. Many concrete bridges in Europe were built in the 1960s, 70s and 80s. Recalculations of existing slab bridges frequently indicate insufficient punching shear resistance in the area of the support regions. The reasons for this are modified terms of use, caused by a greater volume of traffic in combination with increasing axle loads of vehicles, as well as more restrictive design rules with the introduction of the European standardization. In addition, many engineering structures are weakened as a result of their advanced age and partly due to insufficient maintenance. Therefore, the upgrading and strengthening of existing structures is of great importance not only in present but also in the future.
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Figure 1:
Methods to increase the punching shear capacity of existing slabs (adapted from Fernández Ruiz et al. [2010]).
Punching shear strength is substantially influenced by the size of the supporting area, the effective depth of the slab, the concrete strength, the flexural reinforcement ratio and the presence of punching shear reinforcement. Methods to increase punching shear capacity aim to change these parameters in a positive way. A conventional solution is to enlarge the supported area by concrete or steel reinforced column caps or by widening the cross section of the whole column (see Fig. 1(a)). But ensuring sufficient rigidity of the supplemented support area is rather complex. Other solutions are to reinforce the slab tension zone by additional reinforced concrete layers (refer to Fig. 1(b) and Amsler et al. [2014]) or glued carbonfiberreinforced polymer strips (see Fig. 1 (c), Faria et al. [2014]). CFRPstrips can also be installed in predrilled holes through the punching shear zone, prestressed and anchored from the lower side of the slab and act as a punching shear reinforcement (refer to Fig. 1(d), Keller et al. [2013]). Other methods to reinforce the shear zone are the installation of prestressed vertical bolts (refer to Fig. 1(e), Inácio et al. [2012]) or postinstalled inclined bonded anchors (refer to Fig. 1(f), Fernández Ruiz et al. [2010]). To assess the strengthening systems, the strengthening efficiency should not only be considered. The economic efficiency, limitation of usage of the structure during and after the construction activities or the practicability of the strengthening method also have to be included in the evaluation. For example, widening of the column head or the whole column might be undesirable due to space requirements or for architectural reasons. In the cases of solutions (a), (b) and (c), the behavior of the punching failure becomes much more brittle. Especially in combination with missing shear reinforcement, the limited deformation capacity, reduced warning signs and decreased redistribution of internal forces can lead to a very dangerous failure mode determined by a progressive collapse of the entire structure. Therefore, solutions to reinforce the punching shear zone, like in (d), (e) or (f), should be preferred. They improve not only the punching shear strength but also the deformation capacity of the slab (Fernández Ruiz et al., 2010). Regarding the usability of the structure during the construction activities, solutions (b) to (e) have the great disadvantage of needing an accessible upper face of the slab. Method (f) combines the advantages of improved ductility behavior and an untouched top face of the slab. The bars are installed into 45° inclined hammerdrilled holes and bonded by a highperformance epoxy adhesive. The inclined anchors lead on one
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hand to large anchor lengths, and on the other hand to a favorable acting direction. Thereby vertically holes would be much easier to drill. Furthermore it should also be considered that epoxy adhesive, and consequently the whole strengthening system, loses strength in case of fire. Therefore an anchorage system would be beneficial, which depends not only on the adhesive bond between steel anchors and the surrounding concrete but also guarantees a mechanical interlock.
2
Strengthening method with postinstalled concrete screws
Concrete screws are screws made of steel that are installed into hammer or diamond core drilled holes in the concrete. The diameter of the predrilled holes is specified by the screw manufacturer. After cleaning the boreholes from drill dust, a composite mortar is inserted. Finally, the concrete screws are installed into the boreholes. The screws have a sharp thread at their front part and a diameter slightly larger than the diameter of the predrilled hole. Thereby the screws cut themselves into the sidewalls during insertion and ensure a mechanical interlock with the surrounding concrete. A standardized ISO thread at the other end of the screws allows mounting of heavy loads.
Figure 2:
Punching shear strengthening with postinstalled concrete screws.
The idea of the new strengthening method is to use these screws as a postinstalled shear or punching shear reinforcement (refer to Fig. 2 and Feix et al. [2012], Wörle [2014], Feix and Lechner [2014]). The system is able to transfer forces between the tensile and the compressive zone of beams or slabs and thus significantly increase the shear or punching shear strength and the deformation capacity of the member. To facilitate the preparation of the holes, a vertical installation is provided.
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3
Experimental program
Two series of punching shear tests were performed at the University of Innsbruck to investigate the efficiency and behavior of the strengthening method with postinstalled concrete screws. The first series includes four slabs and was conducted in the year 2011 (refer to Feix et al. [2012] and Wörle [2014]), whereas the second series includes five slabs and was carried out in 2016. In order to provide clarity and better comparability, all nine tests are presented in this paper.
3.1
Test setup, test program and specimens
The punching shear tests were performed on circular slabs with a diameter of 2700 mm (106.30 in) and a slab thickness h of 200 mm (7.87 in). All specimens had a circular column stub. The diameter bc of the column stub was 300 mm (11.81 in) in the series of 2011 (four slabs) and 250 mm (9.84 in) in the series of 2016 (five slabs). The column stubs had a height of 100 mm (3.94 in) and were poured simultaneously with the slabs. The punching shear load was applied by a centrically arranged hydraulic jack that was supported between the specimens and a massive reinforced concrete plate. The compressive force of the hydraulic jack was compensated by 12 circular arranged tensile steel bars with a distance of 1200 mm (47.24 in) from the center of the test setup (see Fig. 3).
Figure 3:
Test setup. (Dimensions in mm (in).)
Endhooked deformed steel bars were used for the flexural tension reinforcement. In series 2011 the nominal value of the concrete cover cnom was 24 mm (0.94 in). The diameter of the tensile reinforcement in this series was 20 mm (0.79 in) in the punching shear zone and 16 mm (0.63 in) at the periphery of the slab. The bar distance was equal to 100 mm (3.94 in) in both directions. Two additional bars with a diameter of 16 mm (0.63 in) were placed at the center of the slab in each direction (ref. Fig. 4). The nominal value of the effective depth dnom
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was 156 mm (6.14 in) and of the flexural reinforcement ratio rm,nom=2,23%. This uncommonly high flexural reinforcement ratio was chosen to avoid any flexural failure. The reinforcement in the compression zone consisted of straight deformed bars with a diameter of 10 mm (0.39 in) and a spacing of 100 mm (3.94 in).
Figure 4:
Layout of flexural reinforcement for specimens. (Dimensions in mm (in).)
In series 2016, the bending reinforcement was significantly reduced. The reinforcement in the tension zone of the slab consisted of deformed steel bars with 16 mm (0.63 in) in diameter and 90 mm (3.54 in) in space. The nominal value of the concrete cover was 20 mm (0.79 in), of the effective depth 164 mm (6.46 in), and of the flexural reinforcement ratio 1.37%. Straight bars with a diameter of 8 mm (0.31 in) were arranged at a distance of 90 mm (3.54 in) at the lower side of the slab. The actual location of the flexural reinforcement was determined before pouring the concrete, as well as after testing, on the basis of saw cuts through the slabs. The main geometrical parameters are summarized in Table 1. The first specimen in series 2011 and the first and last specimens in series 2016 served as reference tests to determine the punching strength of similar slabs without shear reinforcement. Fig. 5 shows the arrangement and Fig. 6 the geometrical dimensions of the concrete screws used. The number following the letter B in the denotation of the screws describes the borehole diameter in mm. The number following the letter M stands for the diameter of the ISO thread. The last number describes the length of the concrete screws in mm. With the exception of specimen P03, a composite mortar was injected before tightening the screws into the drill holes. Specimen P02 and P03 were strengthened with TSM 341
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B16 screws and specimen P04, S01P01, S01P02 and S01P03 with TSM B22 screws. In all slabs the anchors were installed up to the upper layer of the flexural tensile reinforcement. In specimen S01P02 only, the screws were screwed up to the lower side of the flexural reinforcement. Slab S01P03 was reinforced with 48 screws, and the others with 32 screws. The diameters of the washers were 39 mm (1.54 in) and 44 mm (1.73) in series 2011 and 60 mm (2.36 in) in series of 2016. Two different screw types were used in series 2016, depending on the installation depth, so that the ISO thread for both installation types reached nearly equally in to the slab. After drilling the screws into the holes, the supernatant anchors were cropped. The nuts were tightened with about 60 Nm (5.31 kip in) before the adhesive hardened.
Figure 5:
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Arrangement of the postinstalled concrete screws. (Dimensions in mm (in).)
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Table 1: Year
Geometric details of specimens
Test ID
bs mm (in)
bq mm (in)
h mm (in)
bc mm (in)
dnom mm (in)
d mm (in)
dx mm (in)
dy mm (in)
155 (6.10)
165 (6.50)
145 (5.71)
154 (6.06) 161 (6.34) 160 (6.30) 164 (6.46)
164 (6.46) 169 (6.65) 168 (6.61) 172 (6.77)
144 (5.67) 153 (6.02) 152 (5.98) 156 (6.14)
161 (6.34)
169 (6.65)
153 (6.02)
P01 P02 2011
300 (11.81)
P03
156 (6.14)
P04 S01P00
2700 (106.30)
2400 (94.49)
200 (7.87)
S01P01 2016
250 (9.84)
S01P02
164 (6.46)
S01P03 S01P04 Year
Test ID
f x, f y mm (in)
s x, s y mm (in)
rx
ry
rm
P01 P02 2011
2016
P03
Shear reinforcement no
20 (0.79)
91 (3.58)
0.0210
0.0239
0.0225
32 TSMB16M18220 mm 32 TSMB16M18220 mm without adhesive
P04
0.0212
0.0241
0.0226
32 TSMB22M24315 mm
S01P00
0.0132
0.0146
0.0139
no
S01P01
0.0133
0.0147
0.0140
32 TSMB22M20635 mm
0.0130
0.0143
0.0137
32 TSMB22M20335 mm
0.0132
0.0146
0.0139
S01P02 S01P03 S01P04
16 (0.63)
90 (3.54)
48 TSMB22M20635 mm no
bs = diameter of the slab; bq = diameter of the support circle; h = slab thickness; bc = diameter of the column stub; dnom = nominal value of the effective depth in both directions of the flexural reinforcement; d = measured mean value of the effective depth in both directions of the flexural reinforcement; dx = measured effective depth of the flexural reinforcement in the xdirection (first layer); dy = measured effective depth of the flexural reinforcement in the ydirection (second layer); fx = diameter of the flexural tensile reinforcement bars in the xdirection; fy = diameter of the flexural tensile reinforcement bars in the ydirection; sx = average spacing of the bars in the xdirection over bc + 6d; sy = average spacing of the bars in ydirection over bc + 6d; rx = flexural reinforcement ratio of tensile bars in the xdirection; ry = flexural reinforcement ratio of tensile bars in the ydirection; rm = mean value of the flexural reinforcement ratio in both directions.
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Figure 6a: Geometric dimensions of the used concrete screws in series 2011. (Dimensions in mm (in).)
Figure 6b: Geometric dimensions of the used concrete screws in series 2016. (Dimensions in mm (in).)
3.2
Material properties
The mechanical concrete and steel properties are listed in Table 2. All specimens of both series were poured simultaneously with concrete from the same batch and a maximum aggregate size of 16 mm (0.63 in). The mechanical properties of the flexural reinforcement were not tested in series 2011. The given values are the characteristic values guaranteed by the steel factory.
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Table 2: Year
Material properties
Test ID
Age on testing day
P01
94
P02
95
P03
96
P04
102
S01P00
70
S01P01
72
S01P02
77
S01P03
79
S01P04
84
2011
2016
Year
Test ID
fy MPa (ksi)
fc,cube,28 MPa (psi)
42.6 (6177)
40.6 (5889)
fc,cube MPa (psi) 47.2 (6839) 47.3 (6857) 49.0 (7104) 45.4 (6583) 40.7 (5899) 40.7 (5899) 40.8 (5912) 40.9 (5931) 41.8 (6055)
ft MPa (ksi)
fc MPa (psi)
fct,sp MPa (psi)
32.2 (4666)
3.7 (529)
32.9 (4772)
3.0 (439)
34.6 (5024)
3.0 (432)
Es GPa (ksi)
fyw MPa (ksi)
fct,f MPa (psi)
Ec GPa (ksi)
4.7 (682)
27.7 (4017)
28.1 (4076)
ftw MPa (ksi)
Esw GPa (ksi)
P01 P02 ~620 ~550 ~200 558 657 210 2011 (~79.8) (~29008) (80.93) (95.29) (30458) (~89.9) P03 P04 S01P00 S01P01 513 650 199 522 729 200 2016 S01P02 (28862) (75.71) (105.73) (29008) (74.40) (94.27) S01P03 S01P04 fc.cube.28 = cube (150 mm (5.91 in)) compressive strength 28 days after concreting (standard storage); fc.cube = cube (150 mm (5.91 in)) compressive strength at time of slab testing; fc = cylinder (150 mm (5.91 in) / 300 mm (11.81 in)) compressive strength at time of slab testing; fct.sp = splitting tensile strength at time of slab testing; fct.f = tensile flexural strength (modulus of rupture) at time of slab testing; Ec = modulus of elasticity of concrete at time of slab testing; fy = yield strength of flexural reinforcement; ft = tensile strength of flexural reinforcement; Es = modulus of elasticity of flexural reinforcement at time of slab testing; fyw = yield strength of concrete screws; ftw = tensile strength of concrete screws; Esw = modulus of elasticity of concrete screws at time of slab testing.
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3.3
Procedure of the shear punching tests
In series 2011, the manually controlled load was applied at a rate of approximately 1.5 kN/s (0.34 kip/s). The 2016 tests were automatically displacement controlled with 0.6 mm/min (0.02 in/min). In two defined load stages, the displacements were kept constant while the crack patterns were recorded at the top surface. After a decrease to a load level of about 100150 kN (2233 kip), the tests were loaded up to failure. The measurements include the tension forces of the 12 threadbars with integrated load cells, the load and displacement of the hydraulic cylinder, vertical deformation measurements of the top surface of the slab, strain measurements of the flexural reinforcement, as well as strain measurements of eight concrete screws for each specimen.
3.4
Test results and interpretation
Table 3 summarizes the main results of the punching shear tests. The displacement wm describes the vertical displacement between the center of the slab’s top surface and the 12 surrounding support points at the highest load. The values yx,out and yy,out describe the slabs rotation outside the punching shear zone in the x and ydirections. The rotation y0 is the rotation between the center of the slab and the support points and is defined by wm and bq in the form wm/(0,5 bq). Vu is the highest force reached at the column stub. In all tests, the crack pattern and the saw cuts indicated the development of clear punching shear cones. The recorded strain values of the flexural reinforcement of series 2011 nearly reached the yield limit, while in series of 2016 the reinforcement started to yield close to the column face at a load level between 500 kN (112 kip) and 600 kN (135 kip). Nevertheless, comparison of the failure loads Vu with the flexural capacities of the slabs Vflex according to the yieldline theory (refer to Equation (1) and Muttoni [2008]) confirms that all specimens failed in punching. V flex = 2p ×
bs × mR bq  bc
(1)
In Equation (1), mR describes the nominal moment capacity of the slab per unit width. Table 3:
Test results
Year
Test ID
wm mm (in)
yx,out mrad
yy,out mrad
y0 mrad
Vu kN (kip)
Vflex/Vu
2011
P01
8.33 (0.33)
7.78
9.37
6.94
612 (138)
2,59
P02
12.83 (0.51)
12.99
14.19
10.69
906 (204)
1,75
P03
12.15 (0.48)
12.17
12.33
10.13
793 (178)
2,01
P04
15.93 (0.63)
16.42
17.41
13.28
937 (211)
1,67
S01P00
12.65 (0.50)
11.36
11.62
10.55
720 (162)
1,79
S01P01
17.29 (0.68)
14.78
15.50
14.41
858 (193)
1,50
S01P02
14.82 (0.58)
13.02
14.21
12.35
843 (190)
1,57
S01P03
23.75 (0.94)
20.06
21.44
19.80
986 (222)
1,32
S01P04
10.73 (0.42)
9.92
9.67
8.94
677 (152)
1,92
2016
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Figure 7 shows the relationship between the shear force V and the displacement wc of the hydraulic cylinder. The specimens without any shear reinforcement (P01 in series 2011 and S01P00 as well S01P04 in series 2016) show a very brittle behavior. After reaching the punching resistance, the load decreased abruptly with a loud noise. With exception of test S01P02, the use of postinstalled concrete screws led to a less brittle failure mode. In test S01P02, the anchors were screwed only up to the lower face of the bending reinforcement. In this experiment, the shear crack went steeply up and continued along the bending reinforcement without crossing the screws (refer to Figure 8). The resistance reached was only slightly lower than that of the corresponding test with screws up to the top layer of the flexural reinforcement (S01P01). However, the saw cuts of the unstrengthened tests show typical punching shear cracks with an average angle of approximately 30°. The punching shear cone of specimen S01P00 was clearly asymmetric. Nevertheless, the resistance achieved was higher than expected. For this reason, we decided to also keep specimen 0104 unstrengthened. The mean value of both failure loads was 699 kN (157 kip). This load level is used to describe the reinforcement effect in series 2016. Despite the higher reinforcement ratio and perhaps due to the smaller effective depth, the failure load of test P01 in series 2011 was lower.
Series 2011
Figure 7:
Series 2016
Loaddeflection curves of the punching shear tests.
In series 2011, the highest strengthening rate was reached with test P04. With 32 screws of type TSM B22, the punching strength increased up to 53%. Using the thinner screws (test P02 with 32 TSM B16), the strengthening effect was only slightly lower. Even without adhesive material (test P03), the punching strength was at least 30% higher than for the unstrengthened test P01. In series 2016, the postinstalled concrete screws also led to a significant increase in the failure load, although the strengthening effect was lower. The reason for this could be the lower flexural reinforcement ratio, which leads to a larger slab rotation and thereby to a re
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duced anchorage strength of the screws at their upper end. Nevertheless, with the use of more screws a strengthening effect of 41% was achieved (S01P03 with 48 TSM B22). In this experiment, the load almost reached the nominal load of the hydraulic jack (1000 kN (225 kip)). Due to the preset tolerance range, the control system automatically reduced the load. After resetting the system and reloading the specimen, punching failure occurred close to the prior maximum load stage. Therefore, it is assumed that the specimen was very close to punching failure when the load automatically took off. The saw cuts of specimens P02, P03 and P04 indicate a crushing of the compressive zone of the slab close to the column. The anchor heads of single concrete screws of the first strengthening ring were simultaneously pulled into the slab (refer to Fig. 8, Fig. 9 and Wörle [2014]). For this reason, significantly larger washers were used in series 2016. Those prevented the anchor heads successfully from being pulled into the slab. The shear cracks in series 2016 were steeper and less branched than in series 2011. The punching shear cone therefore crossed the screws only in the first row and in the upper section and anchor failure of the screws occurred above the shear crack in case of specimen S01P01 and S01P03.
Figure 8:
Sawcuts of the specimens in series 2011 (left) and in series 2016 (right).
Figure 9:
Detailed pictures from the soffit of the specimens in series 2011 after failure (refer to Wörle [2014]).
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Series 2011
Series 2016
Figure 10: Measured strains of the concrete screws.
Figure 10 shows the measured strains of the concrete screws. For each specimen, eight screws were applied with two opposed strain gauges at the smooth part of the screws, close to the end of the ISO thread. The measured strains were significantly below the yield strain (fyw≈2.6‰) and decreased strongly with increasing distance between the anchors and the column stub. Therefore, the screws in the first row (close to the column) received much more 349
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strain than the anchors of the second row. The lowest strains were measured in row three and four. The strains are also influenced by the screw diameter. According to the tests, thinner screws TSMB16 (P02 and P03 in series 2011) are more utilized than the thicker screws TSMB22. However, the activated forces were similar, about 3040 kN (79 kip). Furthermore, the reduced installation depth of the anchors in test S01P02 and the larger number of screws in test S01P03 led to lower strains. The highest strains of the recorded concrete screws were measured after reaching the maximum punching shear force. This was particularly apparent in series 2011. This leads to the conclusion that after the formation of the critical shear crack, stresses are rearranged by the concrete on the anchors until they fail in bond.
4
Conclusion and outlook
The effectiveness of the strengthening method with postinstalled concrete screws was investigated with two test series. The first series was executed in the year 2011 and was characterized by a very high flexural reinforcement ratio with more than 2%. The second series, with similar material properties, was performed in 2016. Compared to series 2011, the reinforcement ratio was much lower and the diameter of the loaded area was smaller, but the effective depth was slightly higher. The main conclusions that can be drawn from the experimental program are: 1. 2. 3. 4. 5. 6. 7. 8.
The postinstalled concrete screws lead to a significant higher punching shear resistance. If the anchors are screwed up to the upper layer of the bending reinforcement, the failure mode is less brittle. Even without the additional use of a composite mortar, the system ensures a substantial increase in the punching shear capacity. Reducing the screw diameter from TSM B22 to TSM B16 causes only a slight reduction of the punching shear resistance. Reducing the installation depth of the screws, to only the bottom face of the flexural reinforcement, leads to a small decrease in the strengthening effect. However, in that case the failure mode remains quite brittle. Comparison of the two series shows that a high flexural reinforcement ratio has a favorable impact on the strengthening effect. The reason for this is assumed to be the smaller slab rotation and thus smaller transverse strains at the screw ends. The installation of the strengthening system is very easy, not least because of the vertical orientation of the screws. Furthermore, the system does not need an accessible upper face of the slab. In contrast to gluedonly solutions, anchorage by mechanical interlock is very robust.
In addition to variations regarding the number and arrangement of the anchor elements, further tests under cyclic load are intended to clarify possible usage in bridge construction. Further tests with lower flexural reinforcement ratios are intended to clarify the influence of the flexural reinforcement ratio on the strengthening efficiency. For the development of a design approach, numerical analyses and statistical evaluations will follow.
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5
Acknowledgement
The 2016 test series was supported by grants from the Austrian Ministry for Transport, Innovation and Technology (bmvit), the Austrian Railway Company (ÖBB) and the Austrian Reliable Motorway Operator (ASFINAG) to the Unit of Concrete Structures and Bridge Design, University of Innsbruck.
6
References
Amsler M.; Thoma K.; Heinzmann D. (2014) “Mit Aufbeton verstärkte Durchstanzplatte – Versuch und Nachrechnungen.” Beton und Stahlbetonbau. Vol. 109, No. 6, pp. 394402 (in German). Faria D. M. V.; Einpaul J.; P. Ramos A. M. et al. (2014) “On the efficiency of flat slabs strengthening against punching using externally bonded fibre reinforced polymers.” Construction and Building Materials, Vol. 73, pp. 366377. Feix J.; Wörle P.; Gerhard A. (2012) “Ein neuer Ansatz zur Steigerung der Durchstanztragfähigkeit bestehender Stahlbetonbauteile.” Bauingenieur, Vol. 87, No. 4, pp. 149155 (in German). Feix J.; Lechner J. (2014) “Development of a new shear strengthening method for existing concrete bridges,” 10th Japanese German Bridge Symposium, Munich. Fernández Ruiz M.; Muttoni A.; Kunz J. (2010) “Strengthening of flat slabs against punching shear using postinstalled shear reinforcement.” ACI Structural Journal, Vol. 107, No. 4, pp. 434442. Inácio M. M. G.; Ramos A. P.; Faria D. M. V. (2012) “Strengthening of flat slabs with transverse reinforcement by introduction of steel bolts using different anchorage approaches.” Engineering Structures, Vol. 44, pp. 6377. Keller T.; Kenel A.; Koppitz R. (2013) “Carbon FiberReinforced polymer punching reinforcement and strengthening for concrete flat slabs.” ACI Structural Journal, Vol. 110, No. 6, pp. 919927. Muttoni A. (2008) “Punching shear strength of reinforced concrete slabs without transverse reinforcement.” ACI Structural Journal, Vol. 105, No. 4, pp. 440450. Wörle P. (2014) “Enhanced shear punching capacity by the use of post installed concrete screws.” Engineering Structures, Vol. 60, pp. 4151.
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Punching failure of slabcolumn connections reinforced with headed shear studs Thai X. Dam1, James K. Wight2, Gustavo J. ParraMontesinos3, Alex DaCosta4 1
: Wiss, Janney, Elstner Associates, Inc., USA
2
: University of Michigan, USA
3
: University of WisconsinMadison, USA
4
: RS&H, Inc., USA
Abstract Seventeen largescale interior reinforced concrete slabcolumn connections were tested to study the effect of different shear stud layouts and the percentage of slab flexural reinforcement. They were divided into two series M (twelve specimens) and S (five specimens) based on their dimensions. Each specimen in Series M had a 6 ft by 6 ft (1830 mm by 1830 mm) and 8 in. (200 mm) thick slab and a 6 in. by 6 in. (150 mm by 150 mm) column crosssection, while each specimen in Series S had a 10 ft by 10 ft (3050 mm by 3050 mm) and 10 in. (250 mm) thick slab and a 12 in. by 12 in. (300 mm by 300 mm) column crosssection. The percentage of slab flexural tension reinforcement was approximately either 0.8% or 1.2%, and shear studs were arranged in either an orthogonal or radial layout. Test results showed that shear strength equations in the ACI Building Code (ACI 318, 2014) overestimated the strength of some test specimens. Also, specimens with a radial layout of shear studs typically had higher strength and more ductile behavior than specimens with an orthogonal stud layout. Recommendations to improve the design of flat plate systems are presented.
Keywords Punching shear, slab reinforcement ratio, twoway slab, shear studs, shear reinforcement layout, integrity reinforcement
1
Introduction
Punching shear strength of slabcolumn connections without shear reinforcement has been shown to be affected by slab flexural reinforcement ratio (𝜌) (Hognestad, 1953; Moe, 1961; Ospina et al., 2012; ACI 421.3R, 2015). It has been observed that significant yielding of slab flexural tension reinforcement near a column allows a wider opening of shear cracks close to the column, which reduces aggregate interlock along these cracks (Hatcher et al., 1961; Guralnick and La Fraugh, 1963; Kinnunen and Nylander, 1960; Hawkins et al., 1974), and thus, shear strength of slabcolumn connections may decrease substantially below the nominal shear strength given by the ACI Code (ACI 318, 2014). The percentage of slab flexural reinforcement ( 𝜌 ) may have the similar effects on slabcolumn connections reinforced with shear reinforcement, e.g. headed shear studs, which are often designed to transfer high shear forces. Prior research investigations (ACI 421.1R, 1999) on shear strength of slabcolumn connections 353
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reinforced with shear studs, however, often used a relatively high value for 𝜌 . Thus, the empirical equations for punching shear strength in the ACI Code, which have been developed based on the results from those prior investigations, may overestimate shear strength of slabcolumn connections that are reinforced with shear studs and have a relatively low 𝜌. Shear studs, or stud rails, are a popular form of shear reinforcement used at slabcolumn connections (Langohr et al., 1976; Mokhtar, 1985; ACIASCE Committee 426, 1974; fib, 2001; ACI SP232, 2005), and they can be assembled from singleheaded or doubleheaded studs. For a singleheaded stud rail, the steel rail serves as the other head for the studs (ASTM A1040, 2010). Stud rails are often placed in either an orthogonal layout (in North America) or a radial layout (in Europe), as shown in Figs. 1(a) and 1(b), respectively. For an orthogonal (or cruciform) layout, stud rails are placed perpendicular to column faces (parallel to slab reinforcing bars) to reduce interferences with slab flexural reinforcement, but the absence of diagonal stud rails in this layout may cause large regions of the slab extending out from the corners of the columns to be essentially unreinforced in shear (Figs. 1a). Some research investigations (Seible et al., 1980; Birkle and Dilger, 2009; Dilger, 2000; Ferreira et al., 2014) have indicated that stud layout (radial versus orthogonal) has no effect on shear strength of slabcolumn connections. However, other research investigations (Gomes and Regan, 1999; Broms, 2007) have indicated that there may be a significant difference in behavior and shear strength of slabcolumn connections with a radial versus an orthogonal layout of shear studs. A reason for these apparently conflicting research results may be related to the percentage of flexural reinforcement in the slab near the slabcolumn connection.
a) S08O and S12O Figure 1:
b) S08R and S12R
Shear stud layouts: orthogonal (a) and radial (b).
The primary objectives of the research investigation presented herein were to experimentally study the effect of the layout of stud rails (radial vs. orthogonal) and the percentage of slab flexural reinforcement (𝜌) on the behavior and shear strength of slabcolumn connections reinforced with headed shear stud reinforcement. Notation—For the sake of convenience, the term slab shear stress in this paper, shown as 𝑣, refers to the average shear stress calculated at a critical section located a distance 𝑑/2 (𝑑 is the average effective flexural depth of a slab) away from the column faces, as defined in the ACI Building Code (ACI 318, 2014). If 𝑉 is the shear force transferred between the slab and column, and 𝑏' is the perimeter of the critical section, slab shear stress 𝑣 = 𝑉/𝑏' 𝑑.
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2
Experimental program
2.1
Test specimen descriptions
Seventeen largescale interior slabcolumn specimens in this program were tested in two series (M and S) consisting of twelve and five specimens, respectively. The test specimens in each series were identical in geometry. Each specimen in Series M has a 6 ft by 6 ft (1830 mm by 1830 mm) and 8 in. (200 mm) thick slab and a 6 in. by 6 in. (150 mm by 150 mm) column section extending 6 in. (150 mm) above the center of the slab, as shown in Fig. 2 (a). Each specimen in Series S has a 10 ft by 10 ft (3050 mm by 3050 mm) and 10 in. (250 mm) thick slab and a 12 in. by 12 in. (300 mm by 300 mm) column section extending 15 in. below and 11 in. above (380 mm and 280 mm) the center of the slab, as shown in Fig. 2 (b). The primary parameters for the tests were the average slab flexural reinforcement ratio (𝜌), the layout of shear studs (orthogonal vs. radial), the type of shear studs (singleheaded vs. doubleheaded), the location of stud rails (compression vs. tension regions), and the shear stud spacing (𝑠).
a) Specimens in Series M Figure 2:
b) Specimens in Series S
Specimen dimensions and reinforcement details.
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Twelve specimens in Series M were labeled as M1 to M12, corresponding to the order in which they were tested. For convenience, suffixes (in parenthesis) may be added to these specimen labels in later discussions to indicate information about the primary specimen parameters, as given in Table 1. The first two numbers in those suffixes are either “08” or “12” representing 𝜌 ≅ 0.8% or 1.2%, respectively. The following letter indicates shear design and stud layout information: “C” for Control specimens built without shear reinforcement, including M3 (08C) and M7 (12C), and “O” and “R” for specimens that were reinforced with shear studs in Orthogonal and Radial layouts, respectively. The type of shear studs and rail locations are indicated using a forth letter, which is “D” for Doubleheaded studs, “C” for rails placed in Compression regions (top of slabs), or “T” for rails placed in Tension regions (bottom of test slabs). The last numbers in those suffixes are either “40” or “75” indicating the approximate stud spacing of 0.40𝑑 or 0.75𝑑, respectively. Five test specimens in Series S were labeled as S08C, S08O, S08R, S12O, and S12R. For these specimen labels, the first letter “S” stands for Specimen, the numbers 08 and 12 represent the approximate slab flexural reinforcement ratio (0.87% and 1.25%), and the last letter indicates shear design and stud layouts: “C” for Control specimens without shear reinforcement (S08C) and “O” and “R” represent specimens with an Orthogonal and a Radial layout of studs, respectively. Table 1: Specimen label
Test specimen description and measured material properties 𝜌
𝑑
% (2)
in. (3)
(1) Series M M1 (08OC40) (08OT40) 0.77 M2 0.77 M3 (08C) 0.77 M4 (08OC75) 0.77 M5 (08OD75) 0.80 M6 (12OD75) 1.27 M7 (12C) 1.22 M8 (12OD40) 1.27 M9 (08RD40) 0.80 M10 (08RD75) 0.80 M11 (12RD40) 1.27 M12 (08RD40) 0.80 Series S S08C 0.87 S08O 0.87 S08R 0.87 S12O 1.25 S12R 1.25
𝑛,
(4) (5)
61/2 61/2 8 8 61/2 61/2 8 61/4 8 61/8 8 63/8 61/8 8 61/4 8 61/4 8 61/8 8 61/4 16 85/8 85/8 85/8 81/2 81/2
𝑛
12 12 12 12
Shear reinforcement Material properties 𝑓1 𝑓12 𝑠' 𝑠 Layout Heads Rail loc. 𝑓/0 in. in. (psi) (ksi) (ksi) (6) (7) (8) (9) (10) (11) (12) (13)
12 12 7 7 7 12 12 7 12 912
22 2 2 2 2 11/4 23/8 11/4 11/4
21/2 21/2 43/4 43/4 43/4 21/2 21/2 43/4 21/2 21/2
Ortho. Ortho. Ortho. Ortho. Ortho. Ortho. Radial Radial Radial Radial
Single Single Tension Comp. 7813 6934 6520 Single Comp. 6180 Double 5550 Double 6200 4890 Double 6120 Double 5440 Double 5930 Double 5790 Double 5590
60.8 60.8 60.8 62.2 62.2 59.9 59.9 59.9 62.5 62.5 64.4 64.4
77.9 77.9 77.9 77.9 77.9 77.9 77.9 63.5 63.5 62.5
8 8 8 8
33/4 33/4 33/4 33/4
41/4 41/4 41/4 41/4
Ortho. Radial Ortho. Radial
Single Single Single Single
66.5 66.5 66.5 65.0 65.0
71.1 71.1 71.1 71.1
Comp. Comp. Comp. Comp.
6100 5050 5360 4510 4790
(Unit conversions: 1 in. = 25.4 mm, 1 ksi = 6.9 MPa, 1 k = 4.45 kN, 1 𝒇0𝐜 psi = 0.083 𝒇0𝐜 MPa)
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2.2
Specimen design
It was assumed that no unbalanced moments were transferred from the slabs to the columns. Flexural reinforcement was Grade 60, and the specified concrete compressive strengths (𝑓/0 ) were 5000 psi (34.5 MPa) and 4000 psi (27.6 MPa) for the Series M and S specimens, respectively. Slab flexural reinforcement—the shear required to develop a flexural mechanism in the slabs (𝑉5678 ) for the Series M and S specimens (using different support systems) was calculated using yield line analysis (Wight, 2015; Dam and Wight, 2015; Johansen, 1962) and results are given in Eqs. (1) and (2), respectively. In these equations, ℎ: is the column side dimension, 𝑙 is the specimen span length, and 𝑚 is the slab moment strength per unit width given in Eq. (3). The slab flexural tension reinforcement for each test specimen was designed so that the slab shear stresses (𝑣) corresponding to 𝑉5678 were approximately either 6 𝑓:0 or 8 𝑓:0 psi (0.5 𝑓:0 or 0.66 𝑓:0 MPa). Thus, for the Series M specimens, the two corresponding slab flexural tension reinforcement solutions were #4 and #5 (𝜙13 and 𝜙16 mm) bars spaced at 4 in. (100 mm). The average effective depth (for two directions), 𝑑, varied from 61/4 in. to 61/2 in. (155 to 165 mm), resulting in an average percentage of slab flexural reinforcement of 0.77% to 1.27%. For the Series S specimens, two slab flexural tension reinforcement solutions corresponding to the two design values of 𝑣5678 were #5 and #6 bars (𝜙16 and 𝜙19 mm) spaced at 41/8 in. (105 mm), and the average effective depths (𝑑) were 8.63 and 8.5 in. (220 and 215 mm), respectively. The average percentages of slab flexural reinforcement were 0.87% (S08 specimens) and 1.25% (S12 specimens). Because the specimens were tested upsidedown (for testing convenience), the tension (primary) flexural reinforcement was placed at the bottom of the slabs. 1 − 3 + 2 2 𝑚 1 − ℎ: 𝑙 4 2 𝑉5678 = 𝑚 ℎ: 2 𝜋 cos ( ) − 8 𝑙 𝜌𝑓M 𝑚 ≅ 1− 𝜌𝑓M 𝑑 P 1.7𝑓:0 𝑉5678 = 8
1
(1)
2
(2)
3
(3)
The compression (top) reinforcement in the slabs for Series S specimens consisted of #4 bars (𝜙13 mm) at a spacing of 6.5 in. (165 mm), with two bars passing through the column core to satisfy the structural integrity requirement in the ACI Code (ACI, 2014) and reflect general construction practice. For the Series M specimens, on the other hand, slab compression reinforcement was not used to eliminate possible dowel effects of this reinforcement on behavior and shear strength of these test specimens. The bars for the top and bottom layers were placed symmetrically about the center of the slabs (Fig. 2). The column longitudinal reinforcement consisted of 4 #4 bars and 8 #8 bars for the Series M and S specimens, respectively, and they were equally distributed around the column core. Ties were #2 and #3 bars (𝜙6 and 𝜙10 mm) at a spacing of 3 in. (75 mm) along the entire column length for the Series M and S specimens, respectively (Fig. 2). A description of the flexural design for all test specimens is given in Table 1. Slab shear reinforcement—Specimens M3 (08C), M7 (12C), and S08C were built without shear reinforcement in the slabs, and thus their nominal shear strength, as given by the ACI 357
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Code (ACI 318, 2014), was 𝑉QR7ST = 4 𝑓:0 𝑏' 𝑑 lb (0.33 𝑓/0 𝑏' 𝑑 kN). The other specimens were reinforced with headed shear studs and their nominal shear strength, as given by the ACI Code, was computed using Eq. (4), 𝑉QR7ST = (𝑣: + 𝑣Q )𝑏' 𝑑
4
(4)
where 𝑣: and 𝑣Q are shear strength, expressed as a stress, provided by the concrete and shear stud reinforcement, respectively. Headed shear studs (ASTM A1044, 2010) with a shaft diameter of 3/8 in. (10 mm) were used in this investigation. The studs were designed so that 𝑣: + 𝑣Q ≅ 6 𝑓:0 psi (0.5 𝑓:0 MPa) for Specimens M4, M5, M6, M10, and the Series S specimens (except S08C) and 8 𝑓:0 psi (0.66 𝑓:0 MPa) for Specimens M1, M2, M8, M9, M11, and M12. To obtain the lower design shear strength, eight stud rails (nT = 8) with a stud spacing (𝑠) of approximately 0.75𝑑 were used for Specimens M4, M5, M6, and M10, and twelve stud rails with a stud spacing of approximately 0.5𝑑 were used for the Series S shearreinforced specimens. To reach the higher design shear value of 8 𝑓:0 psi (0.66 𝑓:0 MPa), eight stud rails (𝑛T = 12) with a stud spacing (𝑠) of approximately 0.4𝑑 were used for in Specimens M1, M2, M8, M9, M11, and M12. Eight additional stud rails with the same stud spacing of 0.4𝑑 were used in Specimen M12. The number of studs per rail (n ) and the distance from the first studs to the column faces (s' ) for all test specimens are given in Table 1. Configurations of the stud rails is shown in Fig. 1 and Fig. 3 for specimens with shear studs in Series S and M, respectively. The total heights of the stud rails for the Series M and S specimens were 6 in. and 8.5 in. (150 mm and 215 mm), respectively. The rails of shear studs for Specimens M1 and M2 and the Series S specimens (except S08C) were 1 in. wide and 3/16 in. thick (25.4 and 4.8 mm), and the rails of shear studs for other shearreinforced specimens were 11/4 in. wide and 1/4 in. thick (32 and 6.4 mm). For Specimens M5, M6, and M8 to M12, the rails of shear studs extending approximately 1.5ℎ (ℎ is slab thickness) from the column faces were removed, except for short portions that extended approximately 0.5 in. (13 mm) from both side of the studs (Fig. 3c to Fig. 3g). The discontinuation of the rails in these regions was to eliminate possible dowel effects caused by the rails on shear strength and behavior of these test specimens. The studs with the short portions (approximately 13/8 in. or 35 mm long) of the rails were equivalent to doubleheaded shear studs, as shown in Fig. 2 (a). Details for the shear design of all test specimens are given in Table 1.
2.3
Material properties
Concrete—Specimens M1 and M2 were cast using concrete mixed in the laboratory and the other specimens were cast using readymix concrete. The concrete was normal weight and was mixed using the same concrete mixture design for all specimens in each series to ensure consistency. The concrete was specified to have a 28day compressive strength of 5000 psi (34.5 MPa) for the Series M specimens and 4000 psi (27.6 MPa) for the Series S specimens, a slump of 6 in. (150 mm), and a maximum aggregate size of ½ in. (13 mm). The compressive strengths of the slab concrete (𝑓:0 ) for all specimens, measured with 4 in. x 8 in. (100 mm x 200 mm) cylinders on the day of testing, are given in Table 1. Reinforcement—Yield strengths of the slab flexural tension reinforcing bars and shear studs, measured using uniaxial tensile tests, are given in Table 1. The results in this table show that measured yield strength of shear studs were higher than the upper strength limit of 60,000 psi (415 MPa) specified by the ACI Code (ACI 318, 2014), and thus the upper strength limit was 358
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used to calculate 𝑉QR7ST for the test specimens. The calculated values for 𝑉QR7ST (Eq. 4) and 𝑉5678 (Eqs. 1 and 2) using the measured material properties for all specimens are given in Table 2.
Figure 3:
2.4
Stud configurations and strain gauge locations for Series M specimens.
Test setup
Support system—The slabs in the Series M specimens were supported along their perimeter (slab corners were free to lift) by a continuous steel frame with a bearing width of 4 in. (100 mm), as shown in Fig. 4 (a), while the slabs in the Series S specimens were supported by eight 6 in. (150 mm) square bearing pads that were placed on eight reinforced concrete blocks and arranged in a symmetrical pattern around the center of the slab, as shown in Fig. 4 (b). Photographs of the test setups for the Series M and S specimens are shown in Fig. 5 (a) and (b), respectively. Between the slabs and the support systems, neoprene strips (for Series M) and pads (for Series S) were used to distribute loads equally from the slab to the supports and to provide free rotation and negligible inplane restraint at the edges of the slabs.
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Loading method—A vertical downward force was applied at the top of the column by a 500 kip (2.2 MN) hydraulic jack (Fig. 5). The applied force was measured with a load cell placed between the hydraulic jack and the reaction steel frame. Initial loading increments of 20 k (90 kN) were used until the load approached the predicted strength of the specimen. After each loading step, the applied load was held constant so the development of cracks in the slab could be recorded. Smaller load increments were then used to capture the peak load resisted by the specimen. When the specimen started to fail, the column was continuously pushed downward until the load decreased below 60% of the peak specimen strength. The total testing time for each specimen was approximately 45 minutes.
a) Series M specimens Figure 4:
b) Series S specimens
Support systems (top view).
A (see Figure 16)
a) Series M specimens Figure 5:
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Test setup.
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Measurement apparatus—Strains in slab reinforcement and shear studs were measured through 0.2 in. (5 mm) long foil strain gauges at the locations shown in Fig. 6, and Figs. 1 and 3, respectively. Displacement of the slabs and columns was monitored using an Optotrak Certus System (NDI Measurement Sciences). This system, which uses highresolution infrared cameras, detects signals emitting from markers glued on the specimens (Fig. 5). Local x, y, and z coordinates of each marker were recorded at 10 Hz with an accuracy of 0.004 in. (0.1 mm).
a) Series M specimens Figure 6:
b) Series S specimens
Strain gauge locations () in slab flexural tension reinforcing bars.
3
Experimental results
3.1
Overall test specimen behavior
All seventeen test specimens in this study failed in punching shear, in which the columns punched through the slabs. Punching failure surfaces were observed in the slabs after the tests. The maximum measured applied load (𝑉VS8 ) and a ratio of 𝑉VS8 to the calculated shear strength (𝑉QR7ST ), calculated using the ACI Code, are given in Table 2 for all test specimens (these applied load values did not include the specimens’ selfweight, which was below 5.5 percent of 𝑉VS8 ). It shows that the measured shear strengths (𝑉VS8 ) for Specimens M3 (08C), M7 (08C), and S08C (control specimens) were approximately 30, 65, and 5 percent higher than the calculated 𝑉QR7ST values, respectively. These results indicate the ACI Code was conservative for the test specimens without shear reinforcement. Table 2 also shows that the measured shear strengths for Specimens M6, M10, S12O, and S12R were close to the corresponding 𝑉QR7ST values. The measured shear strengths for the other ten specimens (M1, M2, M4, M5, M8, M9, M11, M12, S08O, and S08R), which were reinforced with shear studs, were significantly smaller (up to 40%) than their corresponding 𝑉QR7ST values. Thus, the equation for punching shear strength in the ACI Code (Eq. 4) overestimated the shear strength of these ten slabcolumn connections, which had slab flexural reinforcement ratios of approximately 0.8% and 1.2%.
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Table 2:
Calculated and measured strengths
Specimen label
(1) Series M M1 (08OT40) M2 (08OC40) M3 (08C) M4 (08OC75) M5 (08OD75) M6 (12OD75) M7 (12C) M8 (12OD40) M9 (08RD40) M10 (08RD75) M11 (12RD40) M12 (08RD40) Series S S08C S08O S08R S12O S12R
𝑉WXYZ 𝑉[Y\, (k) (2) 141 140 140 143 136 195 200 195 137 137 208 141 285 285 285 390 390
(k) (3)
𝑉WXYZ 𝑉]\Z 𝑣]\Z 𝑉]\Z 𝑉]\Z 𝑓/0 𝑉[Y\, 𝑉WXYZ 𝑉[Y\,
𝜇
Failure mode
(4)
(k) (5)
(psi) (6)
(7)
(8)
(9)
(10)
223 * 217 105 149 138 138 88 ( ) * 186 ( ) * 181 140 ( ) * 181 ( ) * 183
0.63 0.65 1.33 0.96 0.99 1.41 2.26 1.05 0.76 0.98 1.15 0.77
135 131 135 133 127 151 147 166 140 149 173 151
4.69 4.86 5.16 5.21 5.55 6.45 6.65 7.16 6.19 6.31 7.66 6.60
0.60 0.61 1.29 0.89 0.92 1.09 1.66 0.90 0.77 1.06 0.96 0.83
0.95 0.94 0.97 0.93 0.93 0.77 0.73 0.85 1.02 1.08 0.83 1.07
1.7 2.5 1.5 3.0 2.2 1.8 1.2 1.7 4.1 2.8 2.8 4.0
Flexurallytriggered P.S. Flexurallytriggered P.S. Punching shear (P.S) Flexurallytriggered P.S. Flexurallytriggered P.S. Punching shear (P.S) Punching shear (P.S) Flexurallytriggered P.S. Flexurallytriggered P.S. Flexurallytriggered P.S. Flexurallytriggered P.S. Flexurallytriggered P.S.
222 317 322 304 308
1.28 0.90 0.89 1.28 1.27
233 287 293 301 314
4.20 5.67 5.62 6.44 6.51
1.05 0.91 0.91 0.99 1.02
0.82 1.01 1.03 0.77 0.81
1.3 Punching shear (P.S) 2.8 Flexurallytriggered P.S. 3.6 Flexurallytriggered P.S. 2.2 Punching shear (P.S) 2.2 Punching shear (P.S)
( )
* The limit of 8 𝑓:0 𝑏' 𝑑 lb (0.66 𝑓/0 𝑏' 𝑑 kN) governs. (Unit conversions: 1 in. = 25.4 mm, 1 ksi = 6.9 MPa, 1 k = 4.45 kN, 1 𝑓:0 psi = 0.083 𝑓:0 MPa)
( )
Measured applied load versus column displacement relationships for all test specimens are shown in Fig. 7 and Fig. 8 for the Series M and S specimens, respectively. These figures show that: 1) the three control specimens without shear reinforcement (M3, M7, and S08C) exhibited brittle behavior, 2) the specimens reinforced with shear studs had more ductile behavior than the corresponding control specimen, 3) shear studs did not significantly improve shear strength for the Series M specimens that had 𝜌 ≅ 0.8% (Fig. 7a), and 4) the test specimens with shear studs arranged in a radial layout (presented as solid lines) had measured strengths and displacement capacities equal to or higher than the test specimens with shear studs arranged in an orthogonal layout (shown as thin dashed lines).
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a) For specimens with 𝜌 ≅ 0.8%
b) For specimens with 𝜌 ≅ 1.2%
Figure 7:
Load versus displacement for Series M specimens.
Figure 8:
Load versus displacement for Series S specimens.
To study the effects of shear stud layout, slab flexural reinforcement ratio, shear stud type, and rail location, the measured load vs. displacement relationships for all test specimens are presented in six separate plots, shown in Figs. 9 and Fig. 10. Within each plot the specimens had a similar flexural and/or shear design. Fig. 9 presents nine test specimens that had 𝑉5678 𝑉QR7ST ≤ 1, and thus, the applied loads (ordinate) for these specimens were normalized by 𝑉5678 . For the other eight test specimens presented in Fig. 10, 𝑉5678 𝑉QR7ST > 1, the applied loads (ordinate) for those specimens were normalized by 𝑉QR7ST . In all of these plots, except Fig. 10 (c), the dashed and solid lines represent specimens with either an orthogonal or radial layout of shear studs, respectively.
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Figure 9:
3.2
Normalized load versus displacement for test specimens with 𝑉WXYZ 𝑉[Y\, ≤ 1.
Behavior of test specimens with 𝑽𝐟𝐥𝐞𝐱 𝑽𝐬𝐡𝐞𝐚𝐫 ≤ 𝟏
The effect of two different stud layouts (radial vs. orthogonal) on shear strength and behavior of test specimens that had 𝑉5678 𝑉QR7ST ≤ 1 can be studied through three pairs of specimens, including M1 (08OT40) or M2 (08OC40) vs. M9 (08RD40) shown in Fig. 9 (a), S08O vs. S08R shown in Fig. 9 (b), and M5 (08OD75) vs. M10 (08RD75) shown in Fig. 9 (c). The results from these pairs of specimens show that: 1) measured shear strengths for test specimens that had shear studs arranged in an orthogonal layout (except S08O) were approximately 10 percent lower than the corresponding shear required to develop a flexural mechanism in the slabs (𝑉5678 ), 2) measured load carrying capacities for all test specimens with shear studs arranged in a radial layout reached or exceeded the corresponding 𝑉5678 , 3) test specimens with radial stud layout had more ductile behavior than test specimens with an orthogonal stud layout, and 4) the behavior of test specimens with a radial stud layout became less ductile as 𝑉5678 𝑉QR7ST increased. The effect of single vs. doubleheaded studs can be studied in Fig. 9 (c) for Specimens M4 (08OC75) vs. M5 (08OD75), respectively. The results from this pair of specimens show that although measured shear strengths provided from the two different types of shear studs were similar, Specimen M4 with singleheaded shear studs (continuous rails) had more ductile behavior than Specimens M5 with doubleheaded shear studs. This difference in the behavior of the two specimens was likely related to the dowel effects from the steel rails used in Specimen M4. The effect of rail locations can be studied in Fig. 9 (a) for Specimens M1 (08OT40) vs. M2 (08OC40). The rails for Specimen M1 were placed in the flexural tension region (bottom of the slab), while the rails for Specimen M2 were placed in the compression region (top of the slab). Fig. 9 (a) shows that the two specimens had similar measured shear strengths, but the specimen with the rails placed in the compression region (M2) had slightly more ductile behavior than the specimen with rails placed in the tension region (M1).
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3.3
Behavior of test specimens with 𝑽𝐟𝐥𝐞𝐱 𝑽𝐬𝐡𝐞𝐚𝐫 > 𝟏
Figure 10: Normalized load versus displacement for test specimens with 𝑉WXYZ 𝑉[Y\, > 1.
Figure 10 (a) presents results from Specimens M8 (12OD40) and M11 (12RD40). Similar to other pairs of specimens with different stud layouts in Fig. 9, the specimen with shear studs arranged in a radial layout (M11) had better results compared to the specimen with shear studs arranged in an orthogonal layout (M8) in terms of shear strength and ductile behavior. The results in Fig. 10 (a) also show that the measured shear strength of the two specimens, with 𝑉5678 𝑉QR7ST ≅ 1.1, were lower (4% for M11 and 10% for M8) than the corresponding nominal shear strength given by the ACI Code (𝑉QR7ST ). Figure 10 (b) presents results for Specimen M6 (12OD75) and a pair of Specimens S12O and S12R, which had 𝑉5678 𝑉QR7ST ≅ 1.2. The results in Fig. 10 (b) show that the measured shear strengths for all three specimens were equal to or higher than the nominal shear strength given by the ACI Code, and the orthogonal and radial layouts of shear studs provided similar shear strength and load vs. displacement behavior for Specimens S12O and S12R. Figure 10 (c) shows test results for the three control specimens (M3, M7, and S08C). It can be seen that the measured shear strengths of these specimens were higher than the calculated shear strength given by the ACI Code, and the ratio 𝑉 𝑉QR7ST was largest for Specimen M7, which had the largest slab flexural reinforcement ratio of these three slabs.
3.4
Measured strains in slab flexural reinforcement
Measured strains in slab flexural reinforcement at the maximum load for all specimens are shown in Fig. 11 and Fig. 12. These figures show the spread of the flexural yielding away from the north and south faces of the column for one northsouth bar near the center of the slab for the Series M (Fig. 11) and S (Fig. 12), respectively. The measured strains indicate that slab flexural tension reinforcement near the columns yielded and plastic hinging regions in the test specimens extended approximately to 0.15𝑙 to 0.3𝑙 (𝑙 is the span length of specimens), or 2𝑑 to 3.5𝑑, from the center of columns. The strains in the slab flexural reinforcement in Specimens M6, S12O, S12R, and three control specimens (M3, M7 and S08C), which have 𝑉5678 𝑉QR7ST ≥ 1.3, were significantly smaller than for the other specimens, which have 𝑉5678 𝑉QR7ST < 1.2. 365
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The measured strains in slab reinforcement were highest in Specimens M9 and M12 for Series M and in Specimen S08R for Series S, which corresponds to the more ductile behavior for those specimens (all had a radial layout of studs), as shown in Fig. 9. Significant yielding of slab flexural reinforcement near the columns in test specimens with 𝑉5678 𝑉QR7ST < 1.2 is believed to have been the primary cause of their lower measured shear strengths (𝑉VS8 𝑉QR7ST ≤ 1), and their failure mode is thus labelled “flexurallytriggered punching shear” in Table 2.
a) Specimens with 𝑉WXYZ 𝑉[Y\, > 1
b) Specimens with 𝑉WXYZ 𝑉[Y\, ≤ 1
Figure 11: Measured strains in slab reinforcement for Series M specimens.
Figure 12: Measured strains in slab reinforcement for Series S specimens.
3.5
Measured strains in shear studs
Strains measured in the instrumented shear studs at six load stages, S1 to S6, during the tests of specimens that had shear studs arranged in orthogonal and radial layouts are shown in Fig. 13 and Fig. 14, respectively. The dashedlines in these figures represent averages of the measured strains for each specimen. For each load stage in these figures, measured strains are plotted as dots with vertical bars. The plot furthest to the left for each load stage represents the first row of studs from the columns, and the following plotted strains represent the studs in rows further from the column faces (Fig. 1 and Fig. 3). The results in Fig. 13 and Fig. 14 show some significant differences in the measured stud strains for different shear stud layouts, slab flexural reinforcement ratios, and stud spacing. For test specimens reinforced with orthogonal layouts (Fig. 13), the measured average strains in shear studs increased as the applied load reached stage S3, when all of the specimens 366
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experienced a slight drop or leveling off in load capacity. The measured strains then remained nearly constant and below the yield strain for the studs. Almost all of the shear studs in these specimens did not yield, except for a few studs in the second and third rows from the column faces in Specimens M4, M5, and M6, as shown in Fig. 13 (b). The yielding of those shear studs was likely related to the stud spacing of 0.75𝑑 for these specimens, which was larger than the stud spacing for the other test specimens (0.4𝑑 and 0.5𝑑).
a) For S08O (similar to S12O, M2, M8) b) For M5 (similar to M4 and M6) (Unit conversion:1.0 𝑓/0 𝑝𝑠𝑖 = 0.083 𝑓/0 𝑀𝑃𝑎) Figure 13: Measured strains in shear studs for test specimens with orthogonal stud layouts.
For test specimens reinforced with a radial layout (Fig. 14), except for Specimens M11 and M12, measured strains in most of the shear studs reached or exceeded the yield strain of the studs, as the load reached stage S3. Averages of the measured strains in shear studs for these specimens increased rapidly beyond stage S3, especially for the shear studs in the first two peripheral lines of studs from the column faces in Specimens S12R and S08R (Fig. 14a) and for the third and fourth peripheral lines of studs from the column in Specimens M9 and M10 (Fig. 14b). Some shear studs close to the columns faces in Specimens S08R and S12R fractured at load stages S5 and S6. For Specimens M11 and M12, development of measured strains in shear studs, as shown in Fig. 14 (c), were similar to that for Specimens M9 and M10 (Fig. 14b) up to the load stage S3. After that, the measured strains in these two specimens remained constant and below the yield strain of the studs. The differences in measured strains in the shear studs for Specimens M11 (12RD40) and M12 (08RD40) vs. M9 (08RD40) were probably related to a higher slab reinforcement ratio (1.27%) used in Specimen M11 and to the additional stud rails added to Specimen M12. Development of strains in shear studs is often related to cracks in the slabs. Thus, investigations on cracks and failure surfaces in the slabs were conducted after the completion of the tests and the results from these investigations are presented in the following sections.
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a) For S12R (similar to S08R)
b) For M10 (similar to M9)
c) For M11 (similar to M12) (Unit conversion:1.0 𝑓/0 𝑝𝑠𝑖 = 0.083 𝑓/0 𝑀𝑃𝑎) Figure 14: Measured strains in shear studs for test specimens with radial stud layouts.
3.6
Inclined and splitting cracks
After the completion of the tests, cracks inside the slabs were observed by cutting the slabs in the Series S specimens along a line close to the north face of the column and removing any loose concrete at the top of the slabs in the Series M specimens. The typical crack patterns for the slabs are presented in Fig. 15. Two main cracks, including inclined cracks (labeled as 1 and 2) and horizontal cracks located above the tops of the shear studs (referred to as spitting cracks, labeled as 3), were observed.
Figure 15: Observed cracks in the slabs.
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Inclined cracks—For Specimens M3, M7, and S08C, which were built without shear reinforcement, a single wideopening inclined (shear) crack was observed near the column (crack 1 in Fig. 15a). This inclined crack developed around the column to form a truncated pyramid failure surface. For the other specimens with shear studs, several inclined cracks can be observed within the regions reinforced with shear studs (cracks 1 in Fig. 15b) and beyond the outermost periphery of shear studs (cracks 2 in Fig. 15c). The first inclined cracks were found to initiate near the column when slab shear stresses reached from 1.5 𝑓/0 to 2.0 𝑓/0 psi (0.13 𝑓/0 to 0.17 𝑓/0 MPa). Other inclined cracks further away from the column formed later as the applied load increased. Details about the development of the inclined cracks have been reported elsewhere by the authors (Dam and Wight, 2015; Dam et al., 2017; Dam 2016). Splitting cracks—Horizontal cracks located above the shear studs, labeled (3) in Figs. 15(b) to 15(d), were observed in all test specimens with shear studs. These cracks split the concrete cover off the top of the slabs (compression surface), and thus, they are referred to as splitting cracks. The splitting cracks were found to be significant parts of the failure surfaces in the test specimens with shear studs arranged in an orthogonal layout (Fig. 15c) and some of test specimens with a radial layout of studs. Thus, studying the development of these splitting cracks is important for understanding the failure mechanisms for the test specimens. Region A in Figure 5 (b)
Figure 16: Measurement of slab throughthickness expansions and stud elongations.
The splitting cracks in the shearreinforced test specimens were not observed during the tests because the top of the slabs remained intact during testing. The development of splitting cracks, however, can be studied through the measurements of shear stud elongations and slab throughthickness expansion, as illustrated in Fig. 16. An elongation of a shear stud was computed from the measured strain in the stud, assuming that the measured strain was constant along the stud’s smooth shaft length. The slab throughthickness expansion adjacent to a shear stud was measured as the relative displacement between two positiontracking markers that were attached to the top and the bottom of the slabs, using the assembly shown in Fig. 16. Because the stud elongation, labeled (1) in Fig. 16, indicated the growth of inclined cracks, and the slab expansion, labeled (2) in Fig. 16, indicated the growth of all cracks (inclined and splitting cracks), the difference between these two measurements indicated the growth of splitting cracks.
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Fig. 17 shows the calculated stud elongations (labeled (1)) and slab expansions (labeled (2)) at locations approximately 𝑑/2 from the column faces for the Series S specimens, which were reinforced with shear studs, as the applied loads increased to the maximum load. The results in Fig. 17 show that these displacement measurements for the test specimens started to vary significantly when the measured slab shear stresses reached approximately 3 𝑓/0 to 4 𝑓/0 psi (0.25 𝑓/0 to 0.33 𝑓/0 MPa). At the maximum slab shear stress, Fig. 17 indicates that the calculated stud elongations were smaller than 20 percent of the corresponding measured slab expansions, and thus approximately 80% of the slab expansion of the test specimens could be attributed to the horizontal splitting cracks. This phenomenon indicated the initiation and growth of the splitting cracks above the shear studs before the punching shear failures developed. Results from similar calculations at other locations further from the column faces indicate the splitting cracks started to form above the first peripheral lines of studs and then propagated horizontally away from the columns (Dam, 2016). The splitting cracks were found to be a significant part of the failure surface in some of the test specimens.
Figure 17: Measured slab shear stress versus slab expansions and stud elongations.
3.7
Failure surfaces
Observations of the slabs of the test specimens after the completion of the tests showed that failure surfaces in the specimens reinforced with shear studs arranged in orthogonal and radial layouts were significantly different. Stud spacings and the amount of studs were also found to have some influence on the observed failure surfaces. Failure surfaces in specimens with an orthogonal layout of studs—Observed failure surfaces for these specimens were nearly cruciformshaped surfaces, as shown in Fig. 18 (a). These failure surfaces consisted of horizontal splitting cracks (labeled (3) in Fig. 15 and Fig. 18) and inclined cracks outside the edges of shear stud regions. These inclined surfaces were shown to develop in the slab’s diagonal regions adjacent to the corners of the columns and then extend away from the column faces, remaining parallel to the stud rails (labeled (4) in Fig. 18) (Dam et al., 2017). It can be seen from Fig. 18 (a) to Fig. 18 (c) that the propagation of the inclined surfaces (4) was not restrained because of the absence of shear reinforcement in the diagonal regions. The size of the observed cruciformshaped punching surfaces was found to depend on the spacing between peripheral lines of shear studs (stud spacing). 370
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For the test specimens with the stud spacing of 0.5𝑑 or smaller (Specimens S08O, S12O, M1, M2, and M8), the observed failure surfaces extended away from the column to the outermost line of shear studs (Fig. 18b and Fig. 18c). During this process, none of the shear studs were engaged by the observed failure surfaces, and thus, the measured strains in the studs remained relatively low and constant up to failure, as shown in Fig. 13 (a). Extending the stud rails further away from the columns may not have improved the behavior and strength of these specimens.
Figure 18: Observed failure surfaces for test specimens with an orthogonal stud layout.
For the test specimens with the larger stud spacing of 0.75𝑑 (M4, M5, and M6), some sides of the observed cruciformshaped cones did not extend to the outermost stud peripheral lines. The observed splitting surfaces (3) for those specimens only extended over three peripheral stud lines away from the column, and then the failure surfaces extended downward to the bottom of the slab in regions between the peripheral lines of studs, as shown as surfaces (5) in Fig. 18 (d). The inclined surfaces (5) had an angle of approximately 45 degrees with the horizontal plane and crossed some of shear studs near their heads. These failure surfaces would have caused increases in the measured strains for some studs in the third peripheral line of studs from the columns in those specimens, as shown in Fig. 13 (b). Failure surfaces in specimens with a radial layout of studs—Observed failure surfaces for these specimens were nearly the typical truncated pyramid cones extending from the columns for Specimens S08R and S12R (Fig. 19a) or enlarged punching cones with different base dimensions for Specimens M9, M10, M11, and M12 (Fig. 19b). For Specimens S08R and S12R, 371
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the observed failure surfaces crossed the three peripheral lines of shear studs closest the columns (Fig. 19a), causing significant yielding for those studs, as shown in Fig. 14 (a). Splitting cracks were found to develop in regions extending to approximately 2.5𝑑 from the column faces (Dam and Wight, 2015), but they were not a significant part of the observed failure surfaces. For Specimens M9 and M10, the observed splitting cracks extended above the stud regions to approximately 2.5 𝑑 from the column faces (Fig. 14b); then these cracks extended downward to the bottom of the slabs, crossing the 3rd and 4th peripheral lines of studs from the column in Specimen M10 (or the 5th and 6th peripheral lines of studs for Specimen M9). This failure surface would have caused the increases in the measured strains for those shear studs, as shown in Fig. 14 (b). For Specimens M11 and M12, the observed failure surface had a similar shape to that for Specimens M9 and M10, but the top base of the enlarged punching cones extended to the approximately 4𝑑 (crossed the 11th peripheral line of studs), as shown in Fig. 19 (c). Because the observed failure surfaces for M11 and M12 did not engage the instrumented shear studs, which were located in the first six peripheral line of studs from the columns (Fig. 3), the measured strains in those shear studs remained constant during the failure progress of these specimens, as shown in Fig. 14 (c). Observations of failure surfaces for Specimens M9, M10, M11, and M12 indicated that the increase in slab flexural reinforcement ratio (M11) or the number of shear studs (M12) helped to extend the failure surfaces away from the columns.
Figure 19: Observed failure surfaces for test specimens with a radial stud layout.
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4
Key findings
4.1
Shear stud layouts
The results from the seventeen test specimens presented in this paper show that shear studs arranged in a radial layout performed better than shear studs arranged in an orthogonal layout in terms of shear strength and ductile behavior for slabcolumn connections. This was especially true for connections that had significant yielding of slab flexural reinforcement before punching failure. To make a quantifiable comparison of ductile behavior for slabcolumn connections with two different stud layouts, the displacement ductility, 𝜇 , taken as the ratio of the displacement when the applied load decreased to 90% of the maximum load to an equivalent displacement corresponding to initial yielding of slab flexural reinforcement (Dam et al., 2017; Dam, 2016), was computed for all of the test specimens in this study, as well as other punching tests of slabcolumn connections reinforced with shear studs, or studlike shear reinforcement, reported by other research investigations (Birkle and Dilger, 2009; Ferreira et al., 2014; Gomes and Regan, 1999; Broms, 2007; Einpaul et al., 2016; Lips et al., 2012; Beutel, 2002; Regan and Samadian, 2001; Birkle, 2004).The calculated ductility was plotted versus the corresponding ratio 𝑉5678 𝑉QR7ST in Fig. 20. It can be seen that for specimens with 𝑉5678 𝑉QR7ST < 1 (relatively low 𝜌), the calculated ductility for a radial layout of shear studs was higher than that for an orthogonal layout. Also, the calculated ductility provided by a radial layout increased when the relative slab flexural reinforcement ratio 𝑉5678 𝑉QR7ST decreased. The solid line in Fig. 20 shows a bilinear relationship (Eq. 5) between the calculated 𝑉5678 𝑉QR7ST ratio and the measured ductility of slabcolumn specimens reinforced shear studs in a radial layout. For specimens with an orthogonal layout, however, the scattered test results indicate that there was no clear improvement in the measured ductility as the calculated ratio 𝑉5678 𝑉QR7ST decreased.
Figure 20: Ductility of test specimens reinforced with shear stud reinforcement.
𝜇 =
7 − 4 𝑉5678 𝑉QR7ST if 1.5 > 𝑉5678 𝑉QR7ST > 0.7 1 if 𝑉5678 𝑉QR7ST ≥ 1.5
5
(5)
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Observed failure surfaces in the slabs from this study indicated that the differences in shear strength and behavior provided by a radial vs. an orthogonal layout of shear studs were likely caused by inclined cracks extending from the tops of the shear studs near the column to the bottom of the slabs in the diagonal regions. Because of the absence of shear studs in the diagonal regions, the inclined cracks were able to continuously extend away from the columns and remain parallel to the stud rails in an orthogonal layout. As these inclined cracks developed, shear forces at the connections were mainly transferred through the four beamtype strips of shear studs extending from the columns. Concrete in diagonal regions of the slabs was essentially not involved in resisting shear, and thus, those concrete regions were nearly intact after punching failures (Fig. 18). For specimens with shear studs arranged in a radial layout, the observed failure surfaces, which were enlarged punching cones with nearly circle bases, indicated that shear forces were transferred more uniformly in all directions (Fig. 19), resulting in higher shear strength and more ductile behavior.
4.2
Slab flexural reinforcement ratio, 𝝆
A ratio of the measured shear strength (𝑉VS8 ) to the calculated nominal shear strength given by the ACI Code (𝑉QR7ST ) for test specimens reinforced with shear studs in this study, as well as other punching tests of slabcolumn connections reinforced with shear studs, or studlike shear reinforcement, from other research investigations (Birkle and Dilger, 2009; Ferreira et al., 2014; Gomes and Regan, 1999; Broms, 2007; Einpaul et al., 2016; Lips et al., 2012; Beutel, 2002; Regan and Samadian, 2001; Birkle, 2004), was plotted versus the corresponding calculated ratio 𝑉5678 𝑉QR7ST in Fig. 21. It shows that the equation for punching shear strength in the ACI Code overestimates the shear strength of slabcolumn connections reinforced with shear studs and with 𝑉5678 𝑉QR7ST