Progressive Collapse Resilience of Concrete Structures: Mechanisms, Simulations and Experiments 9819907713, 9789819907717

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Kai Qian Qin Fang

Progressive Collapse Resilience of Concrete Structures: Mechanisms, Simulations and Experiments

Progressive Collapse Resilience of Concrete Structures: Mechanisms, Simulations and Experiments

Kai Qian · Qin Fang

Progressive Collapse Resilience of Concrete Structures: Mechanisms, Simulations and Experiments

Kai Qian College of Civil Engineering and Architecture Guilin University of Technology Guilin, Guangxi, China

Qin Fang Army Engineering University Nanjing, Jiangsu, China

ISBN 978-981-99-0771-7 ISBN 978-981-99-0772-4 (eBook) https://doi.org/10.1007/978-981-99-0772-4 Jointly published with China Architecture & Building Press The print edition is not for sale in China (Mainland). Customers from China (Mainland) please order the print book from: China Architecture & Building Press. © China Architecture & Building Press 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publishers, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publishers nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

Progressive collapse is defined as the spread of local damage, from an initiating event, from element to element resulting, eventually, in the collapse of an entire structure or a disproportionately large part of it; also known as disproportionate collapse. Historically, progressive collapse is a low probability event, but the consequence of which is disastrous. For example, the collapse of A. P. Murrah Federal Building in Oklahoma City in 1995 kills 168 people. The collapse of the Twin Towers in the World Trade Center in 2001 claims 2996 deaths and more than one trillion losses to global economy. The disastrous consequences of such events attract widespread attention and great interest from researchers and engineers. To date, research on progressive collapse has lingered into its sixth decade, and extensive meaningful efforts have been performed to promote the development of theory and design method to prevent progressive collapse. This work is done in response to the need to comprehensively introduce the main load resisting mechanisms of reinforced concrete (RC) frames to resist progressive collapse and the critical parameters of them. In Chap. 2, the main load resisting mechanisms of RC frames are introduced. In Chap. 3, the dynamic increase factor of RC frames under a sudden column removal scenario is presented. In Chap. 4, spatial and slab effects on RC frames to resist progressive collapse are quantified. In Chap. 5, the load resisting mechanisms of flat slab structures to mitigate progressive collapse are revealed. In Chap. 6, the effects of infill walls with or without openings on progressive collapse resistance of RC frames are investigated. In Chap. 7, the vulnerability and robustness of RC frames to resist progressive collapse are investigated. This book is built on the experimental studies conducted by the authors and their research group. In this book, the new compressive arch action model and an equivalent single degree of freedom model are proposed. Moreover, the experimental results are compared with the updated provisions of main guidelines and standards for progressive collapse.

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Main contributing authors to this book include me and Prof. Qin Fang. I would like to sincerely thank my friend Xiao-Hui Yu and my students Zhi Li, Yun-Hao Weng, Hai-Ning Hu, Dong-Qiu Lan, Zhi-Qiang Huang, Yang Yu, and Zong-Ze Li for their contributions to this work. Finally, we would be grateful for any useful comments or criticisms that readers may have and for notification of any errors that they will inevitably detect (E-mail: [email protected]). Guilin, China

Kai Qian

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Research Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Definitions of Progressive Collapse and Well-Known Progressive Collapse Events . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Design Methods to Resist Progressive Collapse . . . . . . . . . 1.2 Standards and Code Provisions for Progressive Collapse . . . . . . . . . 1.2.1 GSA (2003) or (2016) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 DoD (2003) or (2009) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 CECS392 (2014) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Advance in Progressive Collapse Study . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Advances in Progressive Collapse Study on RC Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Advances in Progressive Collapse Study on PC Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Load Resisting Mechanisms of Concrete Structures to Resist Progressive Collapse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Flexural Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Compressive Arch Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Existing Models for CAA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 A New CAA Model Based on Moment–Curvature Relationship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Catenary Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Existing Catenary Action Models . . . . . . . . . . . . . . . . . . . . . 2.3.2 Reliability of Existing Catenary Action Models . . . . . . . . . 2.4 Compressive Membrane Action in Two-Way Slab with All Edges Restrained . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Tensile Membrane Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 5 10 10 11 12 12 13 19 21 27 27 29 30 35 52 52 54 56 58 58 59

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3 Dynamic Increase Factor of Concrete Structures . . . . . . . . . . . . . . . . . . 61 3.1 Studies on DIF of Beam-Column Sub-structures . . . . . . . . . . . . . . . . 61 3.1.1 Specimen Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.1.2 Test Setup and Instrumentation . . . . . . . . . . . . . . . . . . . . . . . 65 3.1.3 Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.1.4 Quasi-static Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.1.5 Discussion of the Test Results . . . . . . . . . . . . . . . . . . . . . . . . 76 3.1.6 Dynamic Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.1.7 Discussion of Dynamic Test Results . . . . . . . . . . . . . . . . . . . 87 3.1.8 Dynamic Load Increase Factor . . . . . . . . . . . . . . . . . . . . . . . . 89 3.2 Study on DIF of Beam-Column-Slab Sub-structures . . . . . . . . . . . . . 90 3.2.1 Specimen Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.2.2 Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.2.3 Test Setup and Instrumentation . . . . . . . . . . . . . . . . . . . . . . . 93 3.2.4 Quasi-static Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.2.5 Dynamic Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.2.6 Single Degree of Freedom Model . . . . . . . . . . . . . . . . . . . . . 101 3.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4 Spatial and Slab Effects on Concrete Structures . . . . . . . . . . . . . . . . . . . 4.1 Spatial Effects on Concrete Structures . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Specimen Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Test Setup and Instrumentation . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.5 Load Redistribution Mechanisms . . . . . . . . . . . . . . . . . . . . . . 4.1.6 Spatial Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.7 Dynamic Load Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.8 Analytical Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Slab Effects on Concrete Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Specimen Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Test Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Slab Effects Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5 Theoretical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Load Resisting Mechanisms of Flat Slab Structures to Resist Progressive Collapse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Progressive Collapse Resistance of RC Flat Slabs After the Loss of a Corner Column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Specimen Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Test Setup and Instrumentation . . . . . . . . . . . . . . . . . . . . . . .

111 111 111 112 113 115 120 121 121 121 125 125 130 131 142 149 154 155 157 158 158 160

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5.1.3 Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4 Discussion of the Test Results . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Progressive Collapse Resistance of RC Flat Slabs After the Loss of a Middle Column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Design of Test Specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Design Variables in Test Specimens . . . . . . . . . . . . . . . . . . . 5.2.3 Test Setup and Instrumentation . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.5 Analysis and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Progressive Collapse Performance of Infilled Frames . . . . . . . . . . . . . . 6.1 Performance of Frames with Full-Height Infill Walls . . . . . . . . . . . . 6.1.1 Specimen Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3 Test Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.4 Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.5 Discussion of MI Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Analytical Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 YL and CAA of Bare Frames . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 FPL of Infilled Specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Dynamic Capacity Curves of Tested Specimens . . . . . . . . . 6.3 Performance of Frames with Punctured Infill Walls . . . . . . . . . . . . . . 6.3.1 Design of Test Specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Test Setup and Instrumentation . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.5 Results Analysis and Discussion . . . . . . . . . . . . . . . . . . . . . . 6.3.6 De-composition of the Load Resistance . . . . . . . . . . . . . . . . 6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

193 193 193 196 197 200 213 214 214 215 217 217 218 220 222 222 234 236 238 239

7 Vulnerability and Robustness of RC Frames to Resist Progressive Collapse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Numerical Study on Progressive Collapse Behavior of RC Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Characteristics of the Case Study Buildings . . . . . . . . . . . . . 7.1.2 Validation of Numerical Model . . . . . . . . . . . . . . . . . . . . . . . 7.2 Study on Vulnerability of RC Frames to Resist Progressive Collapse Considering Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Pushdown Analysis and Damage Criteria . . . . . . . . . . . . . . . 7.2.2 Determination of Uncertainty Parameters . . . . . . . . . . . . . . . 7.2.3 Correlation-Controlled Latin Hypercube Sampling Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7.2.4 7.2.5 7.2.6 7.2.7 7.2.8 7.2.9 7.2.10 7.2.11 References

Methodology and Procedure of Random Pushdown Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Probabilistic Assessment Using Random Pushdown Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Confidence Intervals of Progressive Collapse Fragility of RC Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effects of Location of the Removed Column . . . . . . . . . . . . Effect of Story Number on Vulnerability of Progressive Collapse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Empirical Fragility Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . Sensitivity Analysis on Uncertainty Parameters . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .....................................................

254 256 260 260 264 267 270 276 277

About the Authors

Dr. Kai Qian is a Professor, Doctoral Supervisor at Guilin University of Technology (GLUT). He is the Chair of the College of Civil Engineering and Architecture at GLUT. He is also the recipient of National Natural Science Funds for Excellent Youth Scholars, a backbone member of Innovative Research Groups of the National Natural Science Foundation of China, a Committee Board Member of ASCE, Singapore Section, Vice-Chairman of the Singapore Institute of Engineering Technologists, Vice-Chairman of the Youth Committee of Earthquake Prevention and Disaster Reduction in Infrastructure of Seismological Society of China, member of ASCE Multihazard Mitigation Committee, and member of ASCE Disproportionate Collapse Committee. His research interests include basic theory of structures to resist progressive collapse, innovative precast concrete structures and their application in practical projects, new materials, and solid waste re-utilization. He has presided several National or provincial research projects including three projects from National Natural Science Foundation of China. Professor Qian has published 2 academic books as well as 100 SCI refereed articles (h-index = 34) in a wide variety of high-level journals in the field of structural engineering. Two papers were included in the top 1% of cited papers in ESI and 16 papers published in ASCE, Journal of Structural Engineering. The total citation of his publication has exceeded 3000 times (Google Scholar). He has 10 authorized invention patents and software copyrights and published 2 books either Chinese or English. He has participated in the compilation of one national standard. Additionally, he is a reviewer for 20 high-level leading journals, such as ASCE Journal of Structural Engineering, Engineering Structures, International Journal of Impact Engineering, etc. Dr. Qin Fang is a Professor of Civil Engineering at the Army Engineering University of PLA, China. His research interests include dynamic responses and reinforcement measures of engineering structures under severe loads, such as blast, impact, fire, and earthquake. The academic achievements have been widely applied in Chinese codes for protective structures. He has published three academic books entitled xi

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“Concrete Structures under Projectile Impact”, “Underground protective structure” (in Chinese), and “Analysis and design of protective structure” (in Chinese), as well as over 400 journal and conference papers. He has been funded by many research projects, such as the Science Fund for Creative Research Groups of the National Natural Science Foundation of China, the National Science Foundation for Distinguished Youth Scholar, the National Basic Research Program of China, and so on. Currently, he is a member of the executive board of International Association of Protective Structures, the vice chairman of the Protective Engineering Division of China Civil Engineering Society, as well as a member of the editorial board of the International Journal of Protective Structures.

Chapter 1

Introduction

1.1 Research Background 1.1.1 Definitions of Progressive Collapse and Well-Known Progressive Collapse Events The definitions of progressive collapse or disproportionate collapse in existing design guidelines and literatures are similar. As tabulated in Table 1.1, Allen and Schriever (1972) defines progressive collapse as “A situation where local failure is followed by collapse of adjoining members, which in turn causes additional collapse.” Ellingwood (2006) defines progressive collapse as “a result of local structural damage and developed in a chain reaction mechanism, which resulted in a failure disproportionate to the initial local damage.” ASCE 7-05 (2005) defines progressive collapse as “the spread of local damage, from an initiating event, from element to element, eventually, in the collapse of an entire structure or a disproportionately large part of it.” In CECS392 (2014), the term “progressive collapse” is used to describe a situation where the final damage of a building is disproportionate to the local damage that initiates it. In summary, progressive collapse refers to a situation in which the initiated local damage caused by abnormal loads (e.g., gas explosions, vehicular collisions, and sabotage) results in the collapse of an entire building or a disproportionately large part of it. Typical unexpected events that may cause initiate local damage include blast, impact, fire, huge earthquake, construction error, foundation settlement, and abnormal snow, etc. Although progressive collapse is a relatively low probability event, it can lead to catastrophic consequences including severe life and property loss. Due to the low probability but high consequences nature, progressive collapse obtained considerable attentions from researchers and practical engineers. Historically, there are three research upsurges on progressive collapse since 1960s. On May 16, 1968, a 22-story precast concrete building, called Ronan Point Apartment, was © China Architecture & Building Press 2023 K. Qian and Q. Fang, Progressive Collapse Resilience of Concrete Structures: Mechanisms, Simulations and Experiments, https://doi.org/10.1007/978-981-99-0772-4_1

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1 Introduction

Table 1.1 Progressive collapse or disproportionate collapse in definition Source

Definition

Allen and Schriever (1972)

Progressive collapse is a situation where local failure is followed by collapse of adjoining members, which in turn causes additional collapse

Gross and McGuire (1983)

Progressive collapse is a situation in which a localized failure in a structure, caused by an abnormal load, triggers a cascade of failure affecting a major portion of the structure

GSA (2003)

An extent of damage or collapse that is disproportionate to the magnitude of the initiating event

ASCE 7-05 (2005)

The spread of local damage, from an initiating event, from element to element, eventually, in the collapse of an entire structure or a disproportionately large part of it

Ellingwood (2006)

A progressive collapse of a building is initiated by an event that causes local damage that the structural system cannot absorb or contain, and that subsequently propagates throughout the structural system, or a major portion of it, leading to a final damage state that is disproportionate to the local damage that initiated it

Canisius et al. (2007)

Disproportionate failure of a structural system can be described as the situation where the total damage (or risks) resulting from an action is much greater than the initial damage caused by the action which acted upon only a local region or a component of the structure system. Progressive collapse, where the initial failure of one or more components results in a series of subsequent failures of components not directly affected by the original action is a mode of failure that can give rise to disproportionate failure

NISTIR 7396*(2007)

The term “progressive collapse” has been used to describe the spread of an initial local failure in a manner analogous to a chain reaction that leads to partial or total collapse of a building

Agarwal and England (2008)

Disproportionate collapse results from small damage or a minor action leading to the collapse of a relatively large part of the structure. Progressive collapse is the spread of damage through a chain reaction, for example through neighboring members or storey by storey. Often progressive collapse is disproportionate but the converse may not be true

Krauthammer (2008)

Progressive collapse is a failure sequence that relates local damage to large scale collapse in a structure

Starossek and Haberland (2010) If there is a pronounced disproportion between a relatively minor event and the ensuing collapse of a major part or even the whole of a structure, then this is a disproportionate collapse. When the collapse commences with the failure, induced by a triggering initial event, of one or a few structural components which then in turn triggers a successive failure of other components not directly affected by the initial event, then this is a progressive collapse (continued)

1.1 Research Background

3

Table 1.1 (continued) Source

Definition

Kokot and Solomos (2012)

Progressive collapse of a building can be regarded as the situation where local failure of a primary structural component leads to the collapse of adjoining members and to an overall damage which is disproportionate to the initial cause

collapsed due to a gas explosion in 18th story, the explosion wave blew out one of exterior walls in this floor. The upper stories failed to achieve new balance and began to fall. The falling stories impacted the lower stories and, eventually, resulted in the collapse of lower stories as domino effects. The final collapse was disproportionate to the initial damage and thus the term “progressive collapse” or “disproportionate collapse” was proposed. The research interests regarding progressive collapse were first triggered after this event. Academic research communities and design engineers began to look for methods to enhance the robustness of the building and reduce the likelihood of progressive collapse of a building in the event of abnormal loads. After the collapse of Ronan Point Apartment, the tie-force approach was first proposed by the design standards in British. In 1975, the special provisions on progressive collapse were also included in Canada design codes. Then, several workshops were held in the United States regarding progressive collapse design Dusenberry and Juneja (2002). Breen (1975) reviews a three-day workshop in Nov. 1975 to discuss the regulatory approaches, priorities, and research needs in the field of progressive collapse in the USA. For the collapse of Ronan Point Apartment, it was pointed out that the collapse was mainly due to the insufficient integrity of the connections and the lax construction. The critical issues on precast concrete buildings against progressive collapse were discussed in detail. Similarly, Popoff (1975) also pointed out that the highvulnerability of precast concrete buildings against progressive collapse is mainly due to relatively weak integrity of the connections. In addition, the strengthening methods to enhance the resilience of precast concrete buildings to resist progressive collapse were proposed. Taylor (1975) addressed the necessity to enhance the integrity of buildings and to reduce the collapse risk of the buildings during construction. Moreover, Taylor (1975) emphasized the importance to increase the ductility, alternate load path, and local resistance for progressive collapse resistance. As it is very difficult to predict the abnormal loads and their intensity in practical design, the load combination factor and robustness of the building should be determined by probabilistic reliability analysis. Ellingwood and Leyendecker (1978) and Ellingwood et al. (1983) suggested a reliability method to determine the probability of abnormal loads in which dead load, live load, and snow load were considered. As the majority of the building collapse occurred during construction, Breen (1975), Baldridge and Humay (2003), Monsted (1979), and Webster (1980) investigated the importance of the alternative load path and catenary action when one of the load bearing components was lost. Webster (1980) investigated the overload issue of flat slab structures during construction. It was found that the construction load of the floor during construction often exceeded the design value, which is prone to result

4

1 Introduction

in punching shear failure at the slab-column connections. The local punching shear failure may propagate to adjacent connections horizontally, resulting in the falling of entire floor. Then, the falling of upper floor may impact on the lower one and result in progressive collapse. Bennett (1988) investigated the robustness of precast slab-wall structures, which are similar to that of Ronan Point Apartment, to resist progressive collapse. It was found that enhance the continuity of the critical components, such as connections, is an effective way to enhance the robustness of buildings to resist progressive collapse. Actually, in 1970s, the necessity of including progressive collapse evaluation in conventional structural design is controversial in academic community. Breen and Siess (1979) discussed the typical progressive collapse events and tried to mitigate the progressive collapse risks of buildings by generating a series of safety criteria for design. The second research upsurge on progressive collapse after the collapse of Alfred P. Murrah Federal Building. In 1995, a truck containing more than two tons of TNT was detonated in front of the A. P. Murrah Federal Building, resulting in the collapse of one third of the building. This event claimed 168 deaths and near one thousand injuries, which was the most well-known terrorist attack in America before “9.11” event. The design of transfer beam was realized as the reason of such catastrophic collapse. The explosive wave caused substantial damage of the two columns supporting the transfer beam. The loss of supporting columns resulted in the span of the transfer beams suddenly amplified into three times, which resulted in the collapse of transfer beam and the columns above the transfer beam lost their support and lead to collapse one-third of the building, which is disproportionate to the initial damage. It was a consensus that such catastrophic disaster may be avoided if the transfer beams were not designed. Moreover, the collapse of the Alfred P. Murrah Federal Building indicated that the likelihood of abnormal load should not be ignored in structural design. It was necessary to include the design provisions for progressive collapse in building design codes. Afterward, studies on progressive collapse were surge and the results help code-writers to refine design guidelines. Prendergast (1995) pointed out that setting necessary barriers, such as bollards, to ensure enough standoff distance could reduce the peak pressure of the explosives and reduce the collapse risks. Moreover, the development of cantilever beam action or catenary action could reduce the loss of life and property effectively. Erling (1995) investigated the progressive collapse behavior of flat plate structures subjected to snow loads and a design equation for alternative load path method was proposed. Similarly, Duthinh (2004) indicated that the load combination for progressive collapse design should include dead load, live load, and wind load. The third research upsurge on progressive collapse after the “911” event. On Sep. 11, 2001, the twin towers in World Trade Center were subjected to impact of airplane and followed by fire due to fuel leakage after explosive. The South tower began to collapse after fire of 56 min while the North tower was collapsed after fire of 102 min. Following studies indicated that the collapse of twin towers mainly due to the impact loading with following high temperature. This event result in the loss of 2996 lives and over 200 billion US dollars. Actually, over 1000 billion US dollars were lost in the World. Moreover, the research intensity reached highest level

1.1 Research Background

5

after the “911” event. The numerical analysis from National Institute of Standards and Technology explored the collapse process of twin towers. Firstly, the aircraft impact on several perimeter columns and couple interior columns. The loss of these columns resulted in load redistribution. Secondly, the loads initially suffered by the lost columns were redistributed into remaining intact columns via the truss and floor; Thirdly, the fuel leakage resulted in fire and high temperature, which softening the remaining floor and columns and lead to the falling of the upper floors. Finally, the falling down of the debris from upper floors impact on the lower floors and the floor collapse one by one until final entire collapse of the whole building. Progressive collapse design methods consist of Linear Elastic Static Analysis, Linear Elastic Dynamic Analysis, Nonlinear Static Analysis, and Nonlinear Dynamic Analysis. Marjanishvili (2004) compared these analysis procedures and discussed their advantages and disadvantages, limitations, and main steps.

1.1.2 Design Methods to Resist Progressive Collapse Figure 1.1 illustrates strategies to mitigate progressive collapse, including: (1) event control; (2) specific design methods. The event control strategy is normally a costeffective method for risk reduction. It puts efforts to minimize the possibility of the abnormal events. This strategy requires to predict the magnitude of the abnormal loads accurately. The commonly means including controlling the access or keeping the stand-off distance by using perimeter barriers for example. The specific design method includes indirect method and direct method. The former design buildings to mitigate progressive collapse by requirements of the minimum levels of ductility, strength, and continuity. The latter can be further categorized into: (a) key element method and (b) alternative load path method. The key element method predicts the vulnerability of specific column or partial walls by assuming an abnormal load. This is a threat-dependent method and is often the most rational method for strengthening existing buildings. The alternative load path method designing the buildings to resist progressive collapse mainly focused on the “bridge capacity” of the residual building when serious local damage occurred. This method has advantage of threat-independent, and it is also effective for a specific threat.

1.1.2.1

Indirect Method

1. Concept Design Indirect design methods include concept design and tie-force approach. Concept design enhances the structural robustness through provision of lowest requirements on strength, ductility, continuity, and integrity, such as (a) regular plan layout, (b) integrity, (c) special detailing. The concept design had been incorporated in most of

6

1 Introduction Event control key element method Strategies to mitigate progressive collapse

Direct approach

Indirect approach

Alternative load path method

Fig. 1.1 Progressive collapse design methods

the building codes. However, it relies on the experience of the designers and difficult to operate quantitatively, it is unable to adopted in special design accurately. 2. Tie-Force Method For tie-force approach, the building is tied together mechanically to improve its strength, ductility, continuity, integrity, and development of alternative load paths. The tie includes vertical and horizontal ties. The horizontal ties include internal longitudinal tie, internal transverse tie, and peripheral tie. Vertical ties should be checked for columns and load-bearing walls. Table 1.2 tabulates the requirement for tie forces of RC frame structures in BS 8110-1 (1997), EN 1991-1-7 (2006), and DoD (2003). In comparison, the requirement in BS 8110-1 (1997) and EN 1991-1-7 (2006) is more arbitrary. However, that in DoD (2003) is enhanced by analytical results. Moreover, tie-forces are allowed by beams in both BS 8110-1 (1997) and EN 1991-1-7 (2006), but not in DoD (2003) unless they are proved could provide tensile force when the beam end rotation exceeded 0.20 rad. This is mainly due to the beams may failed before beam end rotation reached 0.20 rad (Stevens et al. 2011). However, recent tests had indicated that non-seismically designed RC frames or precast concrete frames with emulative connections could provide tensile force even the rotation reached 0.20 rad (Qian et al. 2021). For steel frames with pin connection or semi-rigid connections could provide tensile force even the rotation had reached 0.20 rad (Qian et al. 2021).

1.1.2.2

Direct Method

Direct method includes alternative load path method and specific local resistance method 1. Alternative Load Path Method Alternate load path (ALP) method is also named as load path method in Eurocode. The ALP method assumes a hypothetical local damage (the notional removal of one or several critical vertical load bearing elements) but ignores the potential damage of other surrounding structural elements. Then, the beam or column spacing increased double and shear force and bending moment increased significantly, which may

1.1 Research Background

7

Table 1.2 Required tie forces in different codes Types of ties

BS 8110-1 (1997)

BS EN 1991-1-7 (2006)

DoD (2003)

Highest designed ultimate dead load and live load received by the column from each story

Column and wall should be able to resist tensile force due to accidental action

Vertical ties should have design tensile strength larger than or equal to the largest vertical load received by the column from each story

The larger of 75 kN and Ft , in and Ti = 0.8(gk + ψqk )s L; which Ft represents in which s is the the lower of spacing of columns or (20 + 40n 0 ) and 60 walls and L is the same kN/m; lr is the larger as lr in BS 8110-1: of column spacing, or 1997; ψ is a factor in wall spacing in the the expression for direction of the tie combination of the under consideration loads

Fi = 3w F L 1 , in which w F =1.2gk + 0.5qk , and L 1 is the same as lr in BS 8110-1: 1997

Vertical ties The larger of 1.0 Ft gk +qk lr 7.5 5

Horizontal interior ties

1.0 Ft

The larger of 75 kN F p = 6w F L 1 L p , in and which L p =0.91 m; Ti = 0.4(gk + ψqk )s L w and L are the F 1 same as those for interior ties

Horizontal peripheral ties Note gk is the norminal dead load; qk is the imposed live load; n 0 is the story number including ground floor

result in damage of the slab or beams for load transfer. In sequence, the damage is propagated and the final collapse region is disproportionate to the initial damage. As the initial damage is commonly caused by explosive or vehicular impact, the transition from the original structural configuration to the damaged one is assumed to be instantaneous and involved dynamic effects. Thus, the ALP method is focused on evaluating the load redistribution ability of the remaining (or residual) building to bridge over the local damage. This method is independent of the initial damage and thus, it is easily used in practical design. There are four computational procedures were used for the ALP analysis: Linear Static procedure (LSP); Linear Dynamic procedure (LDP); Nonlinear Static procedure (NSP); and Nonlinear Dynamic procedure (NDP). All these computational procedures were conducted by finite element software, such as SAP2000, Etabs, OpenSees, among which the NDP has greatest accuracy. However, the accuracy of NDP not only requires considerable computational resources, but also requires the high-level experience and skill of design engineers. Relied on the merit of ALP

8

1 Introduction

method, nonlinear static loading regime and nonlinear dynamic loading regime are utilized in laboratory studies. Marjanishvili and Agnew (2006) compared all these computational procedures by case study based on commercial software SAP2000. They concluded that the LSP was the simplest. However, it could not simulate the nonlinearity and dynamic effects directly. NSP was much better as the nonlinearity of materials could be considered explicitly in the model. Due to the nonlinear dynamic nature of progressive collapse event, a load increase factor should be incorporated in the LSP to account for inertial effects and nonlinear behavior, while a dynamic increase factor should be incorporate in the NSP to account for inertial effects only. Exiting studies shows that the dynamic increase factor is related to rotational capacity of the structure. The authors proposed a general ductility-based function (displacement-based ductility or rotation-based ductility) to predict the dynamic increase factor, details of which can be found in Chap. 3. In BS 8110-1 (1997), BS EN 1991-1-7 (2006), DoD (2003), and GSA (2003), the initial damage in ALP method commonly expressed by the notional removal of a single column or partial of wall. As illustrated in Fig. 1.2, the removal scenarios include the corner column, penultimate column, internal column, near penultimate column, edge column, and near edge column. Potential load resisting mechanisms of RC frames comprise flexural action, compressive arch action, catenary action, and compressive/tensile membrane action (Qian and Li 2015a, b). However, majority of existing design guidelines never discussed the development and critical characteristics of these load resisting mechanisms. The few codes include these load resisting mechanisms did not provide specific design formula for those mechanisms. To date, the majority of codes or guidelines still take flexural action as the main load resisting mechanism, ignored the enhancement of the compressive arch action. Existing tests had indicated that RC frames subjected to the loss of an interior or a penultimate column, considerable compressive arch action could increase the flexural strength significantly. In summary, existing codes or guidelines for progressive collapse design are prone to rely on flexural action or yield load capacity of the frame, which is over-conservative. To deeply understand the load resisting mechanisms of concrete structures, this work presents a comprehensive discussion in Chaps. 3 and 5. Fig. 1.2 Column removal scenarios

Corner Penultimate

Edge Near penultimate

Near edge Interior

1.1 Research Background

9

Qian and Li (2013a, b) proposed an innovative test method, which could investigate the behavior of multi-story prototype frame by a series of single-story substructures. Then, it was proved that the beam-column-slab sub-assemblages can be used to reflect the behavior of the single-story structure. Based on quasi-static responses, Izzuddin and Nethercot (2009) proposed an energy-based method to predict the dynamic response of the fame based on quasi-static responses. The quasi-static load function (load–displacement curve) of the substructure can be obtained through simplified macro numerical model and high-fidelity numerical model. Moreover, Izzuddin and Nethercot (2009) noted that catenary action should be considered in progressive collapse design. 2. Specific Local Resistance Method Specific local resistance method is also called enhanced local resistance method or key element method. The key element method could be used when the remaining structure could not find the new balance when ALP method was adopted, or strengthening the remaining structure is extremely costly based on ALP method. Moreover, the key element method might allow the owner to increase the resistance of critical structural elements, such as peripheral perimeter columns of a building, to resist any reasonable threat. In BS 8110-1 (1997), for the key element method, the key components (such as columns) should be proved to be able to sustain the pressure of 34 kPa. Table 1.3 compares the required load combinations from various standards. In Yu et al. (2017), based on vulnerability analysis, the importance of each component in a building was determined. The greater vulnerability coefficient, the more importance of the structural component. Moreover, the influences of each parameter, such as beam span, depth, reinforcement ratio, etc. were quantified by uncertainty analysis. Table 1.3 Load combinations for progressive collapse analysis Codes

Load combinations after notional member removal

BS 8110-1 (1997)

1.05(D + L/3 + Wn /3)

ASCE 7-98 (1998); ASCE 7-02 (2002)

(0.9 or 1.2)D + (0.5L or 0.2S) + 0.2Wn (ALP method) 1.2D + Ak + (0.5L or 0.2S) (Key element method) (0.9 or 1.2)D + Ak + 0.2Wn (Key element method)

GSA (2013)

Ω L [1.2D + (0.5L + 0.2S)] (LSP or NSP) 1.2D + (0.5L + 0.2S) (LDP or NDP)

DoD (2003)

Ω L [(0.9 or 1.2)D + (0.5L or 0.2S)] + 0.2Wn (LSP) Ω N [(0.9 or 1.2)D + (0.5L or 0.2S)] + 0.2Wn (NSP) (0.9 or 1.2)D + (0.5L or 0.2S) + 0.2Wn (NDP)

D is dead load, L is live load, Wn is wind load, S is snow load; Ω L is load increase factor (see Tables 3–4 of DoD 2003), Ω N is dynamic increase factor (see Table 3–5 of DoD 2003)

10

1 Introduction

1.2 Standards and Code Provisions for Progressive Collapse In order to upgrade the robustness of new and existing building, design requirements for progressive collapse have been incorporated in majority of existing design codes. For example: ASCE 07-10 (2010), Eurocode EN 1991-1-7 (2006), BS 8110 (2002), GSA (2013, 2016), DoD (2003, 2009), CECS 392 (2014), etc. Among which, GSA (2003, 2016), DoD (2009), and CECS 392 (2014) provide step-by-step procedures for analysis. Thus, the analysis procedures in GSA (2003), DoD (2009), and CECS 392 (2014) are summarized and presented below.

1.2.1 GSA (2003) or (2016) The purpose of GSA guidelines is to diminish the risk of progressive collapse of the new and renovated Federal buildings. The requirements for progressive collapse have been updated in GSA (2003) to be consistent with that in Interagency Security Committee Physical Security Criteria. In GSA (2003), an exemption process is evaluated first for a building under progressive collapse evaluation. The exemption process is primarily based on whether the building is 4-story or higher, with identical buildings exempt based on their function. Alternatively, the applicability of GSA (2016) is categorized by the protection level in accordance with the facility security level. The threshold of progressive collapse triggered by the number of floors is consistent with that of GSA (2003). It should be noted that DoD (2003) differs in that this threshold requires consideration of progressive collapse of all buildings of 3 stories or more. LSP, NSP, LDP, and NDP are also adopted by GSA (2003). LSP can be used only for structures with 10 or less stories that meet the requirements of non-standard and demand to capacity ratio (DCRs). If there is no structural irregularity, the LSP can be used without calculating DCR. If the structure is irregular, linear static treatment should be performed when all DCRs are less than or equal to 2.0. LSP cannot be used if the structure is irregular and one or more DCRs exceed 2.0. In GSA (2003), the recommended design strategy is to theoretically remove a support column or partial of wall in order to estimate the load redistribution capacity of the remaining building. Facilities with unregulated parking or public areas: at least one vertical load bearing component was removed from the perimeter (near the edge and near the second-to-last floor) from the ground floor to the top roof. The choice of elements to remove depends on the putative threat, building type, and layout. The duration of column removal has marginal effect on static analysis, but significant effect on dynamic analysis. GSA (2003) requires that the duration of column removal must be less than 1/10 of the free vibration period of the remaining building. Furthermore, it is assumed that beam-beam continuity is maintained on the removed column.

1.2 Standards and Code Provisions for Progressive Collapse

11

As summarized in Table 1.2, for a static analysis, the combination of gravity loads for the bay above removed column or wall is Ω L [1.2D + (0.5L + 0.2S)], in which the term Ω L indicates load increase factor to account for inertial effects and material and geometric nonlinearities. For floor areas away from removed column or wall, the combination of gravity load is 1.2D + (0.5L + 0.2S). For dynamic analysis, combination of gravity load of 1.2D + (0.5L + 0.2S) is imposed to entire structure since the inertial effects are included intrinsically. Only 25% of live load is considered in the load combination to reflect the small probability of a joint occurrence of the abnormal load and live load. The GSA (2003) accepted allowable extents of collapse owing to removal of a vertical structural member for both new and existing buildings. The extent of collapse was defined as the bays associated with the removed member at the floor level above the member, not to exceed 1,800 ft2 or 3600 ft2 for exterior and interior removal scenarios, respectively. Previous versions of the UFC adopted a similar approach which stipulated the allowable extent of collapse to be limited to 15 and 30% of the floor area above the removed member for exterior and interior removal scenarios, respectively. In the last version of the UFC, removed any allowance of collapsed area, requiring that all members, including those directly above the removed element, be designed to meet the defined acceptance criteria. For LSP, Demand-Capacity Ratio (DCR) is used to assess progressive collapse risk. The DCR shall not exceed the acceptance criteria. DC R =

Q U DLim Ω L · Q US = Q CE Q CE

(1.1)

Demand of the structural members or joints considering load increase factor of ΩL . Capacity of the structural members or joints Q CE . Demand of the structural members or joints excluded the load increase factor Q CE .

1.2.2 DoD (2003) or (2009) DoD (2009) is the refined version of DoD (2003). Different from DoD (2003), DoD (2009) focused on the facilities of three stories or more. The difference in threshold of story number leads to a different study objective. The difference between the DoD (2009) and DoD (2003) was emphasized here. (a) Replacement of protection level with risk categories, to determine the proper design method; (b) Added an Appendix B. Definitions, including descriptions of key terms and structural analysis concepts; (c) Revision of the level of protection for progressive collapse designs, including the option of using ALP methods to replace tension forces for type II risks; (d) Elimination of floor upward loading and the requirement of double column height; (e) Revised tie-force method, including the size and location of tie-forces;

12

1 Introduction

(f) Modeling parameters and acceptance criteria from ASCE 41-06 (2006) seismic repair of existing buildings; (g) Implementation of the “M-factor” method for LSP; (h) Including load increase factors for LSP model and DIFs for NSP model; (I) Removing the requirement for peer review of ALP designs; (j) Clarification of the dimensions and location of the dismantled load-bearing walls; (k) Replace additional ductility requirements with key element approach; (l) Three example problems (RC, steel, and wood) have been revised to reflect updated DoD (2003). Similar to other major guidelines for progressive collapse, both indirect and direct design methods are incorporated into DoD (2009). The analysis and design procedures are consistent with other major guidelines.

1.2.3 CECS392 (2014) CECS392 (2014) is the first standard for progressive collapse design of buildings in China, before which only general requirements for progressive collapse are addressed in other building codes. In CECS392 (2014), for the LSP, the bearing capacity of the remaining structural members shall meet Sd < Rd . While for the NSP or NDP, the plastic rotation angle [ ] of the horizontal members of the residual structure should be satisfied θp,e < θp,e where Sd is the demands based on load combination of, Rd is the load resisting capacity of the residual structure, [ ] θp,e is the plastic rotation of the remaining structure horizontal member, and θp,e is the allowable plastic rotation of the horizontal members.

1.3 Advance in Progressive Collapse Study Since the collapse of Ronan point in 1968, there are growing interests in progressive collapse behavior of building subjected to extreme events. A number of researchers devote themselves to this area and improves the understanding on progressive collapse design. As introduced above, many design guidelines have been proposed and upgraded due to the increasing of the risk of extreme event. To date, study on progressive collapse is still very hot, a lot of useful findings had been reported. Advances in progressive collapse investigations are briefly reviewed in this chapter, the advance in progressive collapse of RC and precast concrete structures are reviewed individually.

1.3 Advance in Progressive Collapse Study

13

1.3.1 Advances in Progressive Collapse Study on RC Structures To date, studies on progressive collapse has lingered into its fifth decade, most of which focused on RC structures (Adam et al. 2018; Kiakojouri et al. 2020). Thus, the numerical and experimental studies on the RC structures to resist progressive collapse will be briefly summarized in below.

1.3.1.1

Numerical Study

In general, existing numerical studies could be categorized into two folders: finite element method (FEM) and applied element method (AEM). Moreover, discrete element method (DEM) is used in a few studies. Most of the studies adopt general commercial finite element analysis software, such as ABAQUS (Fu 2009; Sun et al. 2012), LS-DYNA (Kwasniewski 2010; Pham et al. 2017) and DIANA (Bao et al. 2008, Yu et al. 2017). In addition, fiber-based macro models are built by OpenSees (Kim et al. 2011; Jiang et al. 2015) and SAP2000 (Marjanishvili 2004; Marjanishvili and Agnew 2006). Regarding AEM, researchers often used extreme loading for structures software (Grunwald et al. 2018; Dinu et al. 2016; Elshear et al. 2017). In the majority of the FEM studies, extensive assumptions and simplifications are usually adopted (Alashker et al. 2011). In the past, most researchers preferred two-dimension (2D) beam fiber elements or shell elements. However, with the development of computers and computer-aided design software, high-fidelity 3-dimension (3D) finite element models have become more popular now. In general, 2D finite element model is able to reasonably assess the structural behavior, but the 2D finite element models may misestimate the load resisting mechanisms, such as flexural action, compressive arch action, tensile catenary action, and membrane action due to ignored spatial and slab effects (Qian and Li 2012). As RC slabs can significantly increase the flexural and compressive arch action capacity, the spatial and slab effects should be considered in progressive collapse assessment and design. At the element level, the global response of models made of beam elements can be similar to that made of shell or solid elements. However, models with beam elements are unable to simulate some structural responses, such as buckling of web and flange, these responses can be important in the event of blast and thermal loads. Based on different research purposes of the numerical studies, entire structure, sub-structure, single element and connection should be modeled. For an entire structure, beam elements or fiber-element based FE techniques are recommended for the beams and columns while shell elements are suitable for slabs. For micro-model, solid elements are preferred for modeling the members or connections. Some researchers adopted multi-scale models in which the parts near initial local damage are modeled by solid or shell elements while other parts are modeled by beam elements (Li and Hao 2013; Wang et al. 2017). The multi-scale model compromised the accuracy and efficiency. This method can also be used at the material level, such that the

14

1 Introduction

nonlinear and damage properties are only considered for the postulated damaged zones, while only linear behavior is considered in the other zones to achieve time saving. Moreover, this method can be applied to solving schemes, (Brun et al. 2012) used explicit/implicit multi-time step co-computations to study blast resistance of RC structures. Multi-scale modeling is more useful in steel structures. The majority of the available numerical studies focused on steel and RC momentresisting frames with or without bracings. However, a few studies focused on structures with friction and viscous dampers (Kim et al. 2011), base isolation systems (Tavakoli et al. 2015) and infilled walls (Helmy et al. 2015). The collapse type and extreme event should be considered in choosing the numerical analysis methods and proper software. When the progressive collapse analyses follow the steps recommended by design guidelines, most of the software could be used, including nonlinear dynamic analysis. However, for a specific threat, such as blast or fire, general commercial finite element software may be more effective as the effects of high strain rate and high temperature on material properties can be simulated directly. When the collapse mode was required, AEM and DEM were suggested as FEM considers the structure as an integrity system that must satisfy the requirement of force balance. Deleting of partial of building or free fall of the components (such as element removal) may break the mechanical equilibrium. Thus, FEM may be not suitable to study the free fall and impact of the frame components. Differently, AEM and DEM solvers are based on displacement compatibility of each discrete structural part, which is connected together by links. Once the links fail, the displacement solution is still solved and the procedure of progressive collapse could be tracked. As reported in Grunwald et al. (2018), FEM is unable to predict the collapse mode, especially for the sequence of collapse. Finite element method Both explicit and implicit finite element methods are used for the analysis of progressive collapse of structures because they enable the assessment of the likelihood of progressive collapse, especially in code-based approaches (Fu and Parke 2018; Kiakojouri and Sheidaii 2018). The results of the two methods are comparable. In contrast, the explicit approach is more advantageous in threat-related progressive collapse studies that consider specific triggering events. In addition, for collapse sequence problems, the explicit method has better performance under extreme loads, high nonlinearity and complex contact conditions, so it is recommended to adopt the explicit method. Implicit legalization, on the other hand, is straightforward and therefore recommended for practical use by design engineers. In evaluating the calculation time, consideration should be given to the different techniques available to simulate initial local failures in different finite element programs, and the technology chosen will affect the calculation time. Regarding the element type and mesh size of the FE models for progressive collapse analysis, Kiakojouri and Sheidaii (2018) gives their suggestions. However, these recommendations are not always effective, as the accuracy of the results is highly dependent on triggering events in threat-related events and on modeling variables such as column removal duration in threat-independent progressive collapse

1.3 Advance in Progressive Collapse Study

15

analysis. Therefore, grid dependency analysis is necessary for any collapse scenario. Analyzing a detailed three-dimensional model with a large number of elements is very time-consuming. Kwasniewski (2010) reported that the parallel computation of 60 processors took 19 days. In these cases, techniques such as multi-scale modeling or large-scale scaling are unavoidable. Simplified truss-filled walls have been used in many studies to reduce computational time. Numerical models should be validated by comparing with test outcomes. However, due to lack of true dynamic testing, Verifying the accuracy of finite element analysis is a challenge. Most researchers use the results of quasi-static push-down test to verify (Fu 2009; Szyniszewski and Krauthammer 2012; Kiakojouri and Sheidaii 2018; Tavakoli and Hasani 2017). Most of the FE studies focused on assessing the ALP of the remaining structure after key element removal. The progressive collapse analysis based on the ALP method can be used for both static analysis and dynamic analysis, and nonlinear analysis can be directly considered. Nonlinear static analysis has been widely accepted in progressive collapse analysis, while nonlinear dynamic analysis has received more attention. In recent years, incremental dynamic analysis (IDA) (Vamvatsikos and Cornell 2002) has been used to study progressive collapse (Tsai and Lin 2008; Brunesi et al. 2015). A push-down analysis program was also developed for the progressive collapse assessment of steel frame buildings under column removal scenarios (Ferraioli 2019). An innovative non-iterative progressive collapse design method based on virtual thermal push-down analysis, inspired by progressive collapse induced by fire, is proposed to evaluate the progressive collapse behavior of RC structures (He et al. 2019). In addition, some researchers have applied probabilistic methods for progressive collapse assessment (Javidan et al. 2018). According to the recommendations of GSA (2013), the initial damage was caused by sudden column removal without causing any damage or overload to the adjacent members. It should be noted that the above hypothesis only applies to the damage caused by near-field small explosions. In other words, triggering events in design guidelines and specifications are implicitly limited to near-field explosions. For other cases, such as far-field explosions, earthquakes, and fires, such simulations are unrealistic and should be studied in greater depth separately. Many methods have been proposed to simulate dynamic column removal. Kim and Kim (2009) proposed a force-based method that was widely adopted by other researchers. This method can be easily implemented in any finite element software. In their method, a column is removed from the position being considered and a LSP is performed by applying a gravitational load to the bay from which the column is removed. Then, check the demand to capacity ratio in each structural member. If the demand capacity ratio of a member exceeded the acceptance criteria in shear, the member is considered as failed. If the demand to capacity ratio of a member end exceeded the acceptance criteria in bending, a hinge is placed at the member end. If hinge formation causes a member to fail, it is removed from the model and its load is redistributed to adjacent members. At each inserted hinge, an equal but opposite bending moment is applied corresponding to the expected bending strength of the member. The above steps are repeated until the required capacity ratio of any member did not exceed the limit state. If the bending moment is redistributed

16

1 Introduction

Table 1.4 Existing equation for DIF Source

Equation

Note

Qian and Li (2015a, b)

1+

It depends only on the empirical formula of ductility

Stevens et al. (2008)

1.44 m−0.12

Empirical formulas for steel structures that rely only on ductility

McKay et al. (2012)

1.08 +

Empirical formulas for steel structures that rely only on ductility

Tsai and Lin (2009)

2μ[1+α(μ−1)] 1+α(μ−1)2 +2(μ−1)

Analytical formulas for ductility and post-elastic stiffness ratios

Liu (2013)

0.84 +

The model based on the empirical formula of residual bearing capacity considers the gravity load and ductility requirements

0.5 μ−0.5

0.76 +0.83

θall θ yield

1.23 M 2.95 max( M dp )−0.28

Mashhadi and Saffari (2016) 2 − 2.54ζ −

Mashhadi and Saffari (2017) (1.1 + 2η) +

θp ) yield θp (0.84−2.15ζ )+( θ yield

(0.9−1.81ζ )( θ

0.56−η θ 0.65+( θ p ) yield

)

Empirical formulas for ductility and damping ratio of steel structures The empirical formula depends on the post-elastic stiffness ratio

μ = ductility demand;m = the ratio of plastic rotation to yield rotation; θall = the lowest nonlinear acceptance criteria; θ yiel = yield rotation; α = post-elastic stiffness ratio; Md = factored moment demand under original un-amplified gravity load; M p = plastic bending moment; θ p = plastic rotation; ζ = damping ratio; η = post-elastic stiffness ratio.

throughout the building and the area outside the allowable collapse area defined in the guidelines still exceeded the demand-capacity ratio, the structure will be considered to have a high potential for progressive collapse. Fu (2009) proposed direct element removal to simulate sudden column removal while Tavakoli and Kiakojouri (2013) proposed a material degradation method. The results of these methods are similar and can simulate the dynamic effects reasonably. Investigations on column removal duration and strain rate can be found in Arshian and Morgenthal (2017). Although NDP is the best method to study progressive collapse, it requires a lot of calculation time. To overcome this problem, an appropriate dynamic increasing factor (DIF) should be used for nonlinear static analysis to equivalent consider the dynamic response indirectly. To this end, different DIF determination methods are proposed, some of which are related to ductility, others related to the post-elastic stiffness ratio, duration, and damping ratio. Table 1.4 summarizes some important formulas for calculating DIF.

1.3 Advance in Progressive Collapse Study

17

Applied element method Applied element method was first developed by Meguro and Tagel-Din (2000) to avoid the limitation of FEM. The most important distinguishing feature of AEM is the elements connected by links (1D nonlinear springs), which only considering the comparable of displacements (Kim and Wee 2016). For this reason, AEM is a perfect approach to simulate the collapse procedure and collapse mode of the building under extreme loading events such as blast or impact. Although AEM is more efficient than FEM, high-fidelity finite element models are still used in most of the published research work. The AEM method is very useful in the study of progressive collapse with specific threats such as explosion-induced or earthquake-induced progressive collapse. No reliable results have been reported for progressive collapse caused by fire, other than the study by Elkholy et al. (2003). AEM is able to analyze a detailed 3D multi-story model in a few hours, whereas FEM may take days. The characteristics of AEM make it suitable for modeling infilled masonry walls (Helmy et al. 2015; Zerin et al. 2017). Therefore, many researchers began to pay attention to the development of AEM. For example, El-Kholy et al. (2012) proposed multilayer elements to simulate non-uniform cross sections. Multilayer elements can effectively simulate composite cross sections. Further improvements are needed to make the AEM more effective in threat-based collapse studies, especially under fire conditions. Discrete element method DEM was first used to simulate the movement of rock slopes and has been widely used in soil mechanics and successfully applied to the study of granular and discontinuous materials (Lu et al. 2018). DEM is more effective than FEM in analyzing large displacement discontinuities because it only needs equilibrium equations and boundary conditions. Hakuno and Meguro (1993) first apply DEM, which is commonly used in seismic analysis, in progressive collapse analysis. However, DEM needs considerable computational time, especially for multi-story prototype frames. In recent years, with the increase of computational efficiency, using DEM for progressive collapse become possible. Munjiza et al. (2004) decoupled FEMDEM to increase the efficiency and decrease the computational time. DEM was also successfully applied in blast-induced and earthquake-induced progressive collapse studies.

1.3.1.2

Experimental Study

Despite costly, recently, a number of experimental studies on progressive collapse were conducted, especially for frame structures. These tests can be categorized into connection tests (Yang and Tan 2012; Culache et al. 2017; Dinu et al. 2017), beamcolumn sub-structure and beam-slab tests (Pham et al. 2017; Choi and Kim 2011; Yu et al. 2018), and in-situ tests (Sasani 2008; Sasani and Sagiroglu 2010).

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To date, there are many attempts to simulate column removal in laboratory, but their effectiveness to capture the real dynamic response is still an issue. Sasani (2008) used explosive to remove the column, the explosive was installed into pre-drilled hole in the target column. Then the column was well wrapped with a layer of protective materials to prevent air blast and flying fragments to affect the surrounding structural components and instrumentation devices. However, as part of service load was removed before test, only elastic behavior was captured. Thus, advanced dynamic tests should be invented to capture the dynamic response of the structure with considerable plastic deformation. Moreover, the effects of column removal duration and critical parameters, which could not be systematically investigated by on-site tests and other methods should be adopted. For steel frames, Song et al. (2014) suddenly pulled out the targeted removed column by a bulldozer to simulate sudden column removal. The middle segment between the removed sections is suddenly pulled out by a bulldozer. However, column removal duration is unclear. Test results demonstrated that the most of structural members of the tested building exceeded the demand to capacity ratio limit when the second column was removed. However, the building did not experience a collapse even after the four columns were removed. Chen et al. (2011) proposed a method to suddenly remove a column by instantaneous pulling down of a column by a chain block. To ensure the column removal fast enough, one end of the removed column was pinned to the ground while the other end contacted with the upper building via a roller. The column removal was implemented by suddenly pulling down the column. Li et al. (2018) used a three-hinged steel assembly with a glass locking rod through an additional hole at the middle hinge to sustain its stability before tests. During the tests, a pendulum hammer was released to strike the three-hinged assembly. As a result, the glass locking rod was broken and the assembly could rotate freely. Xiao et al. (2013) carried out dynamic column movement by impinging a concrete block inserted in the middle of the column with pellets fired by hydrogen guns. Russell et al. (2015) reported a dynamic column removal in flat slab structures with a innovative temporary support. In another study, Pham and Tan (2017) used an innovative quick-release device whose ropes were suddenly yanked after temporary supports were removed. Under the gravity of the applied load, the specimen immediately moves downward. In Bermejo et al. (2017)’s work, dynamic column removal was carried out during the construction phase with explosive charges located within the column, similar to the method used by Sasani (2008) for the existing building. Qian (2012) designed an instantaneous column moving device to simulate the sudden column moving out. In the test process, the column is not actually moved out, but rotates to release the axial force. Other experimental studies in this area include testing the effects of infilled walls (Yu et al. 2019; Shan et al. 2016), using video measurement and photogrammetry (Liu et al. 2015), and using extreme point symmetric modal decomposition (Liu et al. 2018). In addition, in recent years, researchers have focused on strengthening existing structures with FRP to resist progressive collapse (Liu et al. 2017; Qian and Li 2019).

1.3 Advance in Progressive Collapse Study

19

1.3.2 Advances in Progressive Collapse Study on PC Structures With the development of building industrialization, more and more buildings were built by PC forms. PC structure is a form of structure, of which partial or all structural elements are precast in factory and then transported to site for assembling. Compared with conventional in-situ RC structures, PC structures have advantages of effective construction, good quality control, environment friendly, and sustainable development. However, PC structures may be deficient in structural integrity, construction control, and design difficulty. PC structures categorized into two folders: fullassembled (dry connection) and monolithic assembled (wet connection) depending on whether cast-in-situ casting is required. For the monolithic assembled PC structures, structural elements are partially constructed in the factory and assembled in the site before cast-in-situ casting the topping layers. Differently, for full-assembled PC structures, the entire structural components were constructed in the factory. After transported into the site, the components were connected by bolts, welds, or prestressing tendons, etc. No cast-in-situ casting is required. Post-earthquake investigation indicated that the collapse of PC structures is mainly attributed to insufficient strength and ductility of beam-column or beam-slab connections. To date, most of the studies on behavior of PC structures focused on their seismic performance. However, the seismic behavior of PC structures subject to cyclical load, rather than gravity, which is the load for progressive collapse study. Compared with RC structures, PC structures are deficient in continuity and integrity. Thus, the effectiveness of alternate load transfer paths of PC structures under the loss of structural members are unclear. In past decade, studies on progressive collapse behavior of PC structures are slowly increased. Cleland (2008) evaluated the structural integrity and performance of large panel structures to resist progressive collapse and proposed that abnormal load should be considered in building design. Tohidi et al. (2014) simulated the bond–slip behaviour at the steel–concrete interface by using the “translator” element embedded in ABAQUS. The validated numerical models were used to simulate the ductility behaviour of PC floor joints in the absence of underlying wall supports and to study the tie force developed during progressive collapse. The discrepancies in the tie force between the numerical and the codified specifications suggest that an underestimate based on the tie force method may lead to an unsafe design. Nimse et al. (2014, 2015) carried out an experimental study on five 1/3 scale specimens including one RC beam-column assembly and four PC beam-column assemblies, among which three adopted wet connections while the other one adopted dry connection. It was found that the PC connection showed good performance at the elastic stage. However, the strength of PC assembly with dry connections declined more quickly than that of RC assembly. The PC assemblies with wet connections had commensurate performance with the RC assembly. Moreover, it was found that the performance of the PC connections with RC corbel was better than that with steel billet.

20

1 Introduction

Kang and Tan (2015a) and Kang et al. (2015) conducted pushdown tests on six half-scaled PC assemblies with cast-in-situ topping layers. The studied parameters included anchorage method for the beam bottom rebars in the beam-column joints, rebar ratio, preparation for interfaces between PC units and cast-in-situ topping layers, and material for the cast-in-situ topping layers (normal strength concrete and engineered cementitious composites). It was found that, under a middle column removal scenario, compressive arch action and catenary action in the beams were able to increase load resistance of the specimens. The development of the compressive arch action and catenary action is highly dependent on the detailing of the joints and the rebar ratio of the beams. In addition, it was concluded that PC specimen using engineered cementitious composites as cast-in-situ topping layers can achieve similar performance as RC specimen. Kang and Tan (2015b) proposed an analytical method for predicting bond-slip relations of steel bars. In this method, the bond stress in the post-yielding stage of reinforcement is calculated based on the existing test data of the deformed reinforcement embedded well in the constrained concrete under the pulled-up load. The method is calibrated by the test results of different lengths of embedded reinforcement. According to the distribution of bond stress along the embedding length and the average bond stress, a simplified method is proposed. Finally, the model of precast concrete beam-column joint based on component is established, and the force-slip relationship of reinforcement is deduced. The combined model can well predict the buckling capacity and catenary capacity of beam-column sub-assemblage under progressive collapse. Main et al. (2015) tested two PC moment-resisting frames subjected to a middle column removal scenario. The PC beams belong to deep beam, while the beam-column connections were welded connections. Test results demonstrated that the PC moment-resisting frames suffered a shear failure. The bottom welded plates at the connections fractured during the test. Kang and Tan (2016) conducted an experimental investigation on four PC beamcolumn assemblies. The research parameters include the details of reinforcement and size of side column. The test results showed that the PC beam-column assembly can produce compressive arch action in the initial loading stage. However, the beamcolumn assembly with smaller side columns appeared obvious shear failure at the side joints, which hinders the further development of catenary action. Increasing the size of the side column will increase the catenary action and eventually lead to the bending failure of the side column under horizontal tension in the beam. In the design of resisting progressive collapse, the horizontal forces caused by the compressive arch action and catenary should be considered. Elsanadedy et al. (2017) developed a nonlinear finite element model using LSDYNA to simulate the performance of PC substructures assembled by bolts, each consists of three columns and two beams, under a column removal scenario. The nonlinear behavior of concrete and steel, the effect of strain rate on material properties and the contact between surfaces at joints are considered in this model. Model calibration was carried out for three half-scale specimens in the middle column removal scenario. The test consists of two PC specimens and one RC specimen. The validated finite element model was further extended to the potential progressive collapse

References

21

of seven modified PC connections. Kang and Tan (2017) tested four PC frames to investigate their structural performance under the middle column removal scenario. The reinforcement is anchored at the middle beam-column joints by lapping with 90° bending hook. PC frames have similar properties in the compressive arch action, and the compressive arch action capacity of all four frames is roughly the same. However, different reinforcement arrangements and horizontal constraints cause different the catenary action capacities for frames. In contrast to the 90° bending of the rebar at the bottom of the beam, the lap-splice rebar at the joint allows the development of higher catenary action in PC frames. According to the test results, it is suggested to adopt the lap-splice connections to prevent progressive collapse. However, for developing greater catenary actions in external frames, it is necessary to prevent potential bending and shear failure of side columns at the large deformation stage. Qian and Li (2018) tested three 1/3 scaled beam-slab substructures, including one cast-in-situ RC specimen and two PC specimens. The results show that the welded PC specimen achieves brittle failure and has the lowest ultimate bearing capacity and deformation capacity. Although the lowest initial stiffness and the first peak load capacity in the PC specimen with a pin connection, its large rotating capacity ensures that the specimen fails in a ductile manner, which allows for the development of considerable tensile membrane action. Qian et al. (2019a, b) tested several groups of PC beam-column assemblies assembled by unbonded prestressed strands. It was found that these PC assemblies had desirable robustness to resist progressive collapse. Catenary action in the strands is able to develop once the beams are deformed. The load-resisting mechanisms of these PC assemblies were quite different from that of RC assemblies, compressive arch action can even make negative contribution. Feng et al. (2018) numerically investigated the progressive collapse resistance of PC components. Based on open-source finite element software, an efficient numerical model of PC component was established. Fiber beam element was used for beam and column, and Joint 2D element was used for beam-column connection. In order to consider the significant bond-slip effects in the core of PC joints, the bond-slip relationship of reinforcement with different embedded lengths was deduced and used to generate the force–deformation relationship of spring in Joint2D element. The effectiveness of the numerical model is verified by comparing with the experimental results of PC component in the scenario of column removal. Based on the validated model, the influence of typical parameters on the progressive collapse resistance of beam-column sub-assemblages is studied.

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Chapter 2

Load Resisting Mechanisms of Concrete Structures to Resist Progressive Collapse

When a column is removed, the beam span is double and the vertical load initially resisted by the removed column needs to be redistributed into adjacent columns. If a new balance could be achieved, progressive collapse can be prevented. Otherwise, progressive collapse may happen. The potential load resisting mechanisms for concrete structures to resist progressive collapse comprise flexural action or bending action, compressive arch action (CAA), catenary action (CA) of the beams as well as compressive/tensile membrane action (CMA/TMA) of the slab. To deep understand the nature of those mechanisms, this chapter presents a comprehensive introduction individually. In addition, the reliability of existing models to capture the load resisting capacity of each load resisting mechanism is evaluated.

2.1 Flexural Action As illustrated in Fig. 2.1, at the very beginning of the load redistribution process, the beams only respond in elastic range. At this stage, the most significant feature is that the beams are in pure flexure, i.e., axial force of the beams is close to zero. As the first defense line for the beams to resist progressive collapse, the flexural action capacity is predicated based on the yield bending moment capacity of the beams when the section reached yield status. The flexural action capacity is obtained when the yield strength of tensile reinforcements is reached, and therefore, it is also called yield load capacity. As illustrated in Fig. 2.1, the flexural capacity or yield load capacity can be expressed as follows ) ( 2 My1 + My2 Py = ln © China Architecture & Building Press 2023 K. Qian and Q. Fang, Progressive Collapse Resilience of Concrete Structures: Mechanisms, Simulations and Experiments, https://doi.org/10.1007/978-981-99-0772-4_2

(2.1)

27

28

2 Load Resisting Mechanisms of Concrete Structures to Resist …

Py 1

2

N=0 M y1

1

N=0

M y2 2

M y1

M y2

Fig. 2.1 Illustration of flexural action

where My1 and My2 are yield moment of the Sects. 1.1 and 2.2, respectively. In general, the two sections will not yield simultaneously and may have different yield bending moment capacities due to different reinforcement ratios installed. Thus, the yield load Êpacity will be dominated by the yield bending moment capacity of the sections. In the work of Park and Paulay (1975), a calculation procedure is derived for determination of elastic bending moment capacity of a beam section. As the yield bending moment is a special status of the elastic bending moment, the proposed calculation procedure could be adopted to predict the yield bending moment capacity of the critical sections. As illustrated in Fig. 2.2, within elastic stage, the axial force of a section is close to zero. The compressive stress block of concrete presents a triangle distribution, rather than a parabolic distribution that often presented at plastic stage. Based on the equilibrium condition and geometrical relationship between strain of rebar and concrete, the neutral axis depth k1 (k2 ) can be determined (refer to Eqs. 2.2a and ( , factor ) , are area of tension and compression rebar, respecA A b), where As1 ( As2 ) and ( ) s1 s2 tively; ρ1 (ρ2 ) and ρ1, ρ2, are ratio of tension and compression rebar, respectively; n is the ratio of elastic modulus of rebar to concrete. [ k1 = [ k2 = ρ1 =

(

(

ρ1,

) ] 21 , ) ( as1 n − ρ1 + ρ1, n h − as1

(2.2a)

( )2 ρ2 + ρ2, n 2 + 2 ρ2 + ρ2,

) ] 21 , ) ( as2 n − ρ2 + ρ2, n h − as2

(2.2b)

(

ρ1 +

)2 ρ1, n 2

+ 2 ρ1 +

As2 A,s1 A,s2 As1 Es ρ2 = ρ1, = ρ2, = n= b(h − as1 ) b(h − as2 ) b(h − as1 ) b(h − as2 ) Ec (2.2c)

Based on geometric relationship between stress of rebar ( , )and concrete, the , fibre of the concrete f compressive stress in the extreme c1 f c2 can be expressed ( ) by yield stress of the rebar f y1 f y2 as follows f c1, =

k1 · f y1 n(1 − k1 )

f c2, =

k2 · f y2 n(1 − k2 )

(2.2d)

2.2 Compressive Arch Action

29

f

T

S1

S1

S1

a'S1

d1

h

aS1

ε

b

N 1=0

φ ε' ε 1

c1=k1d1

S1

S1

C

M1

f'

C1

C

C1

Section 1

Strain

Stress

a'S2

ε c2=k 2d 2

S1

C1

Resultant force f'

C

C2

C1

ε'

f'

S2

φ

S2

S2

C

C2

2

N 2=0

aS2

d2

h

f

'

b

Section 2

ε

Strain

f

S1

S2

Stress

T

M2

S2

Resultant force

Fig. 2.2 Strain, stress, and resultant force diagram of critical beam sections at the elastic stage

f s1, =

, k1 (h − as1 ) − as1 · n f c1, k1 (h − as1 )

f s2, =

, k2 (h − as2 ) − as2 · n f c2, k2 (h − as2 )

(2.2f)

Then, by taking moment about the centroid of tension of the rebar, the yield bending moment is determined as following. My1

My2

] [ ( ) k1 (h − as1 ) , = 0.5 + f s1, A,s1 h − as1 − as1 − as1 ) (h − as1 ) − 3 (2.3a) ] [ ( ) k2 (h − as2 ) , = 0.5 f c2, bk2 (h − as2 ) (h − as2 ) − + f s2, A,s2 h − as2 − as2 3 (2.3b) f c1, bk1 (h

For convenience, the two sections can be considered to yield simultaneously. There, substituting Eqs. 2.3a, b into Eq. 2.1 could calculate the yield load capacity.

2.2 Compressive Arch Action After achieving the yield load capacity, as the second defense line against progressive collapse, CAA is often mobilized as a result of strong boundary condition and deformation compatibility. In general, there is no distinct demarcation between flexural action and CAA, the end of flexural action is commonly regarded as the commencement of CAA. Or CAA could be taken as the N-M interaction, which amplified yield bending moment of the section. If the boundary of a two-span beam is restrained, compressive axial force will be induced as the vertical movement and neutral axis

30

2 Load Resisting Mechanisms of Concrete Structures to Resist …

varying resulted in the outward movement of the beam section, which is restrained by the boundary. The compressive axial force could increase the yield bending moment capacity of the beam sections due to M–N interaction. Therefore, CAA is a load resisting mechanism associated with axial compression (diagonal thrust in the beams), while the nature of CAA capacity exceeded yield load capacity can be attributed to the additional plastic bending moment induce by the axial compression. The CAA is not used in conventional design. Quantitatively, CAA has been found to improve the structural resistance by 30–150% (Lu et al. 2018). Compared with CA, the mobilization of CAA requires much lower demand in continuity and rotation capacity of beam-column connection. Thus, it is preferred to resist progressive collapse relying on CAA.

2.2.1 Existing Models for CAA Park and Gamble’s model In the work of Park and Gamble (2000), a model for assessment of compressive membrane action (CMA) of one-way RC slab under a concentrated load is proposed. Since the merit of CAA is similar to that of CMA, the Park and Gamble (2000)’s model is often used to determine the CAA capacity of beams. Details of this model are presented below. As illustrated in Fig. 2.3, the projection of the deformed beam segment onto the original beam configuration to achieve the compatibility equation. [βl + 0.5ε(1 − 2β)l + t] sec φ = (h − c, ) tan φ + (1 − ε)βl − c tan φ h−c, −c =

2βl sin2 (φ/2) + εβl cos φ + 0.5ε(1 − 2β)l + t sin φ

(2.4a)

(2.4b)

where l is clear span of two-span beam; β is ratio of clear span of beam to two-span beam; ε is axial strain of beam; t is lateral drift of boundary; φ is inclination of beam; h is beam depth; c, and c are neutral-axis depth of critical section near the side column and near the middle column, respectively. Since φ and ε are so small that following simplification was done: sin φ = 2 sin

δ φ = , cos φ = 1 2 βl

where δ is middle column displacement. Therefore, Eq. 2.5 can be further expressed as: c, + c = h −

δ βl 2 2t − (ε+ ) 2 2δ l

(2.5)

2.2 Compressive Arch Action

31

Fig. 2.3 Compatibility condition of One-way slab at CMA stage in Park and Gamble (2000)

Based on force equilibrium, the arching forces acting on the critical sections are expressed as: Cc, + Cs, − T , = Cc + Cs − T

(2.6)

where Cc, and Cc are the concrete compressive forces; Cs, and Cs are steel compressive forces; T , and T are steel tensile forces, acting on the critical sections, respectively. Using the ACI concrete compressive stress block, the concrete compressive forces can be written as: Cc, = 0.85 f c, βl c,

(2.7)

Cc = 0.85 f c, βl c

(2.8)

where f c, is the concrete cylinder strength, and βl is the ratio of the depth of the equivalent rectangular stress block to the neutral-axis depth, as defined in ACI 31895 (βl = 0.85 for f c, ≤ 30 MPa, and for f c, ≥ 30 MPa, βl reduces linearly by 0.05 for each 7 MPa, but βl must not less than 0.65) c, − c =

T , − T − Cs, + Cs 0.85 f c, βl

(2.9)

solving Eqs. 2.5 and 2.9 gives c, =

h δ βl 2 2t T , − T − Cs, + Cs − − (ε+ ) + 2 4 4δ l 1.7 f c, βl

(2.10)

c=

h δ βl 2 2t T , − T − Cs, + Cs − − (ε+ )− 2 4 4δ l 1.7 f c, βl

(2.11)

32

2 Load Resisting Mechanisms of Concrete Structures to Resist … 0.85fc'

a

c

Cs Cc

nu

d

h

Mu

h/2

d'

a/2

a=?1c T

Fig. 2.4 Condition at positive moment yield section

Hence, based on the geometry compatibility and force equilibrium, the neutralaxis depth at the critical section could be determined. Note from Eqs. 2.10 and 2.11 that if the rebar in the top and the bottom layer is similar (T , = T = C , = C) and ε = 0 and t = 0, the neutral-axis depth at the critical sections varies between 0.5 h and 0.25 h when the middle column displacement δ varies from 0 to h. The greater ε and t, the lower the neutral-axis depth. If the δ is very limited, ε and t are greater than zero, based on Eqs. 2.10 and 2.11, the neutral-axis depth trends to infinite. These equations apply properly only when the deflection of the beam is significant. Actually, the initial deflections are governed by elastic behavior. The plastic theory discussed above is proper only at and after the ultimate load stage. Figure 2.4 shows condition at a positive-moment yield section. Based on force equilibrium, the axial force and bending moment acting on the middle depth of the critical sections is given as: n u = Cc + Cs − T = 0.85 f c, βl c + Cs − T m u = 0.85 f c, βl c(0.5h − 0.5βl c) + Cs (0.5h − d , ) + T (d − 0.5h)

(2.12) (2.13)

Take the beam as an isolation body, the sum of the bending moment at the section subjected to negative bending moment is m ,u + m u − n u δ. Shear forces were neglected since their net contribution to the analysis by virtual work will be zero. After substituting Eqs. 2.7 and 2.8 into the equation m ,u + m u − n u δ, it is found that: ⎤ ( ) ) ( βl 2 βl δ 2t h ⎥ ⎢ 2 1 − 2 + 4 (βl − 3) + 4δ (βl − 1) ε+ l ⎥ ⎢ m ,u + m u − n u δ = 0.85 f c, βl h ⎢ ⎥ ( ) ( )( ) ) ( ⎣ δ2 βl βl 2 βl 2t βl β2 l 4 2t 2 ⎦ + 2− + 1− ε+ − ε+ 8h 2 4h 2 l l 16hδ2 ( ( ) ) ) ) ( ) ( h δ h δ 1 ( , 2 T − T − Cs, + Cs + Cs, + Cs − d, − + T, + T d − + − 2 2 2 2 3.4 f c, ⎡

(2.14)

where

2.2 Compressive Arch Action

( 2t ε+ = l

1 h Ec

+

2 lS

)[

33

0.85 f c, βl

(

h 2

−

1 + 0.2125

δ 4

T , −T −Cs, +Cs 1.7 f c, βl

− ( 2

f c, βl βl δ

1 h Ec

+

2 lS

)

)

+ Cs − T

] (2.15)

Based on virtual work principle, the applied load can be obtained as: P=

2 (m , + m u − n u δ) βl u

(2.16)

Note that, the middle column displacement δ is unknown. In the work of Park and Gamble (2000), δ is assumed to be 0.5 h. In above analysis, unit width is used for slab. In this study, the beam width, rather than the unit width should be used in calculation. Yu and Tan (2014a, b)’ model Yu and Tan (2014a, b) pointed out some shortcomings of the Park and Gamble (2000)’s model. Park and Gamble (2000)’s model did not take into account the actual stress state of the reinforced bars along the critical section of the beam. This leads to inaccurate predictions of CAA capacity. In the CAA state, the compressive reinforcement in the intermediate joint area either yields or did not yield. Therefore, the compressive stress of reinforcement cannot simply be specified to achieve yield strength or zero stress state, as suggested by Park and Gamble (2000)’s model. In addition, Park and Gamble (2000)’s model only considers the effect of partial axial constraints at the beam end, and did not consider other imperfect boundary conditions. Therefore, Park and Gamble’s model cannot take into account the effect of rotational constraints on the beam end and axial joint clearance on CAA. To address the problems in Park and Gamble (2000)’s model, Yu and Tan (2014a, b) proposed a new model that takes into account the actual stress states of compressive reinforcement K a , rotational constraints K r , and axial joint gaps t0 . The model uses axial constraint stiffness, rotational constraint stiffness and initial joint clearance to consider the imperfect boundary conditions (Fig. 2.5).

c=

( ( ) ) )( 2 δ βl 2 1 h βlt0 Mu1 βl h + − − + − c1 1 − (2.17) N− 2 2 2δ bh E c l K a δ 2 Krδ

Unlike Park and Gamble (2000)’s model, the sum of the depth of the neutral axis c, and c in Yu and Tan (2014a, b)’s model cannot be determined by analytical methods. Therefore, Yu and Tan (2014a, b)’s model should rely on iterative analysis. Previous models focused only on the capacity and displacement of CAA. In contrast, Yu and Tan (2014a, b)’s model can predict the evolution of CAA. Although the model of Yu and Tan (2014a, b) considers the influence of imperfect boundary conditions, it still has limitations. As shown in Fig. 2.6a, it is assumed that the ultimate compressive strain of concrete is constant. However, it can be seen from Fig. 2.6b that the strain of tensile reinforcement εs increased continuously, and the

34

2 Load Resisting Mechanisms of Concrete Structures to Resist …

Fig. 2.5 Compatibility condition of beam-column sub-assemblage in Yu and Tan (2014a, b)

compressive strain of both concrete εcu and reinforcement εs, increased first and then decreases, reaching the peak value. Therefore, the neutral axis depth C1 increased first and then decreases. If it is assumed that the concrete strain εcu remains the same even when it starts to break, then the change of εs, and c1 is quite different from the actual CAA process, as shown in Fig. 2.6a. Above unreasonable assumption will induce unreliable analytical results comparing with that from tests. Lu et al. (2018)’s model To address the unreasonable assumption in Park and Gamble (2000)’s model, Lu et al. (2018) refined Park and Gamble (2000)’s model by proposing a formula to determine the middle column displacement at CAA stage. Recalling, in Park and Gamble (2000)’s model, it is assuming the middle column displacement equal to 0.5 h, which is constant whatever the reinforcement ratio, span, etc. In Lu et al. (2018), finite element models are built to investigate the influence of compressive strength of concrete, reinforcement ratio, and geometry of beam on the middle column displacement at the stage of CAA. Based on regression fitting, it is concluded that the middle column displacement at CAA capacity δ can be estimated by l 2 / h with a reasonable correlation coefficient as follows δ = 0.00050l 2 / h

(2.18)

2.2 Compressive Arch Action

35

(a) Strain variations based on assumed constant ultimate concrete strain

(b) Actual strain variations at CAA stage Fig. 2.6 Comparisons of strain variations in actual CAA and the Yu and Tan (2014a, b)’s model

where l is the total length of the two-span beam; h the beam height. Moreover, for beam-slab substructures, which includes the slab effects, Lu et al. (2018)’s proposed an equation for middle column displacement at CAA stage based on parametric studies and regression analysis. As shown in Eq. 2.20, it was found that the refined model could improve the accuracy of CAA model significantly. (

δbeam−slab

) ( ) bf l2 = 0.000276 × 0.0023 × + 0.9875 × bb h f + 0.5h b

(2.19)

2.2.2 A New CAA Model Based on Moment–Curvature Relationship A new model for CAA is proposed to address the limitations in Yu and Tan (2014a, b)’s model by consideration of the moment–curvature (M−ϕ) relationship of the critical sections.

36

2 Load Resisting Mechanisms of Concrete Structures to Resist …

V1 N1 M1

φ

P

q

1

ln

1

δ 2 M2 2

V 2L N2

N2L

φ V2

M2L

V 2R N2R M2R

Fig. 2.7 Force equilibrium of left side-bay beam and middle column at the CAA stage

2.2.2.1

Development of New CAA Model by Authors

CAA can be regarded as the bending action of beams with sufficient boundary constraints, and its nature is the bending action of post-elastic stage. As can be seen from Fig. 2.7, plastic hinge occurred at the beam ends of the two-span beams, and the load is concentrated at the middle joint. Due to symmetry, only half of the two-span beams are shown here. Since the axial constraints are at both ends of the beam, considerable axial compression forces are mobilized throughout the length of the beam. Assuming that the load resistance of the sub-assembly based on shear failure is greater than that based on bending failure, considering the influence, the vertical anti-load P is determined according to the vertical force balance at the middle joint. Because the deformation of the beam is relatively small, the rotation of the critical section can be determined by δ/ln , where ln is clear beam span. For a given rotation, the external work done by the concentrated load can be expressed as Eq. 2.21a while the increased internal energy due to CAA can be expresses as Eq. 2.21b. Considering virtual work principle, CAA capacity can be obtained by Eq. 2.21c. Wexternal = (V2L + V2R )δ = PCAA · δ ) qln2 ϕi 2 i ) ( 2 M1 + M2 − N δ − qln2 /2 = ln

Winternal =

PCAA

2.2.2.2

L ,R ( Σ

M1i + M2i − Ni δ −

(2.20a)

(2.20b)

(2.20c)

Assumptions of the New CAA Model

The analytical analysis and derivation of the model based on the assumption of rigidplastic body, in which the two-span beam is regarded as a system consisted of two rigid beams and four zero-length hinges. Moreover, assuming plastic hinges occur at the beam ends while the length of the plastic hinges is neglected. The stiffness of imperfect axial and rotational restraints are linear elastic and symmetrical at both

2.2 Compressive Arch Action

37

beam ends, and beams are fully restrained against vertical translation at the ends. To consider the imperfect boundary condition, t0 and ω0 are used to indicate connection gaps and free rotation at the boundary. Two assumptions are introduced for calculation of the internal forces. The first one is the plane-section assumption. The second one is the concrete compressive forces at plastic stage can be replaced by equivalent rectangular stress block as defined in ACI 318-14 (2014). Therefore the concrete compressive forces are f c , where 0.85 f c bβ1 c is compressive strength of concrete based on cylinder tests; β1 is the ratio of the depth of the equivalent rectangular stress block to the neutral-axis depth, (β1 = 0.85 for f c ≤ 30 MPa, and for f c ≥ 30 MPa, β1 reduces linearly by 0.05 for each 7 MPa, but β1 must not less than 0.65).

2.2.2.3

Compatibility Conditions

Figure 2.8 illustrates the compatibility condition of the beam. Due to mobilization of axial compressive forces in beams, the compressive force at middle column is expressed as 0.5bj εb = 0.5εb (1 − 2β)l, the outward movement of the boundary has an amount of t +t0 . t indicates horizontal movement of beam due to axial compressive forces. (1 − εb )βl indicates the beam length due to compressive force. εb indicates axial compression due to axial compressive forces. E c is elastic modulus of concrete.

t=

E Ts1 ω

Φ C c1 C s1 Rotated support (beam end)

c1

(2.21a)

b j = (1-2β)l

βl +0.5εb(1-2β)l+(t+t0)-0.5htanω

Δ1

Sect ion

N Ka

O ω +Φ

(1-ε b)

1

βl

δ h

Top Bott o

mb

Δ2

ba r

ar

M c2

C s2 C c2

S ect ion 2

beam net span length : ln=βl middle column width : b j= (1-2β)l

ω+Φ

Ts2

Middle Joint

Fig. 2.8 Compatibility condition schematic diagram of beam-column assemblage

38

2 Load Resisting Mechanisms of Concrete Structures to Resist …

εb =

N bh E c

(2.21b)

The cracking status and concrete crushing of the beam could be expressed by Eqs. 2.22(c) and (d). ω and Φ present rotation at the boundary and rotation of the critical section with respect to the boundary. Δ1 = (h − c1 ) tan Φ

(2.21c)

Δ2 = c2 tan(ω + Φ)

(2.21d)

ω = ω0 +

M1 Kr

(2.21e)

Since the rotations are extremely small, triangle transformation can be used: δ M1 , tan ω = ω = ω0 + βl Kr ( ) δ M1 δ Φ = tan Φ = −ω = − ω0 + (2.21f) βl βl Kr

sin(ω + Φ) = tan(ω + Φ) = ω + Φ =

As shown in Fig. 2.8, the projection of the deformed beam segment onto the original beam configuration yields Eq. 2.22g. Segment EM can be expressed as Eq. 2.22h. The geometric relationship in ΔO E M yields Eq. 2.22i. βl + 0.5εb (1 − 2β)l + (t + t0 ) − 0.5h tan ω

(2.21g)

(1 − εb )βl + Δ1 − Δ2 = (1 − εb )βl + (h − c1 ) tan Φ − c2 tan(ω + Φ) (2.21h) βl + 0.5εb (1 − 2β)l + (t + t0 ) − 0.5h tan ω = [(1 − εb )βl + (h − c1 ) tan Φ − c2 tan(ω + Φ)] · cos(ω + Φ) ⇒ 0.5εb l + (t + t0 ) − 0.5h tan ω = (1 − εb )βl[cos(ω + Φ) − 1] + (h − c1 ) tan Φ cos(ω + Φ) − c2 sin(ω + Φ) (2.21i) To eliminate trigonometric function in the compatibility function, they are replaced by equivalent infinitesimal mathematical terms. cos x = 1 −

( ) ( ) x3 x2 + o x 2 , sin x = x − + o x3 2 3!

2.2 Compressive Arch Action

39

h βl 1 c2 = − (0.5εb l + (t + t0 )) − (1 − εb )δ+ 2 δ 2 ( ] ( ( ) ) )) [ ( M1 βl h 1 δ 2 ω0 + · (h − c1 ) 1 − 1− − δ Kr 2 βl 2

(2.21j)

Substituting Eqs. 2.21a, b into Eq. 2.21j gives the following compatibility function: ( ( ) )[ ( ) ] h 2 M1 βl δ βl 2 1 h βlt0 + + − c1 1 − ω0 + N− c2 = − − 2 2 2δ bh E c l K a δ 2 Kr δ (2.22)

2.2.2.4

Introduction of Moment–Curvature Relationship

As illustrated in Fig. 2.2, the equations defining the bending moment and curvature at first yield are: ϕy1 =

εy1 f y1 /E s = d1 − k1 d1 d1 (1 − k1 )

(2.23a)

ϕy2 =

εy2 f y2 /E s = d2 − k2 d2 d2 (1 − k2 )

(2.23b)

where ϕyi , εyi , and f yi are the curvature, yield strain, and yield stress of the rebar at critical sections; d the neutral axis depth factor; E s the elastic modulus of the rebar. Curvature of the critical sections can be expressed as Eqs. 2.25a, b, where I0 indicates initiate inertia moment of the gross section, the equivalent elastic modulus of the section is set to the elastic modulus of the concrete E c . Thus, E c I0 indicates flexural rigidity of the gross section. Bsi = αi E c I0 (i = 1, 2) indicates effective flexural rigidity, αi is a reduction coefficient, the determination of which will be discussed hereafter.

2.2.2.5

ϕ1 =

M1 M1 = Bs1 α1 E c I0

(2.24a)

ϕ2 =

M2 M2 = Bs2 α2 E c I0

(2.24b)

Equilibrium Condition

For a beam undergoing CAA, rotations of beam sections are very small, so that there is no obvious difference between horizontal reaction forces (H ) and axial forces of the beam (Yu and Tan 2013). Therefore, the axial forces acting at the beam ends (N1 ) and the joint interfaces (N2 ) are equal.

40

2 Load Resisting Mechanisms of Concrete Structures to Resist …

f

T

S1

S1

S1

a'S1

d1

h

aS1

ε

b

N1

φ ε' ε 1

S1

0.85f

C1

C

M1

C

'

C

a'S2

ε c2=k 2d 2

0.85f '

S2

φ

C

C

C1

ε'

f

'

β1c2

C

C2

S2

aS2 b

ε

f

S1

S2

S1

C1

S2

2

d2

h

f'

c1=k1d1 β1c1

S1

T

N2 M2

S2

Fig. 2.9 Strain, stress, and resultant force diagram of critical beam sections at the plastic stage

N = N1 = N2 = H

(2.25)

At CAA stage, the distribution of compressive tress of concrete is non-linear. For easy calculation, equivalent rectangular stress block is used. Figure 2.9 shows strain and stress distribution at the critical sections. As shown in Fig. 2.10, where N1 and N2 are axial forces, Cc1 and Cc2 are concrete compressive forces. N1 = Cc1 + Cs1 − Ts1

(2.26a)

N2 = Cc2 + Cs2 − Ts2

(2.26b)

Cc1 = 0.85 f c, bβ1 c1

(2.27a)

Cc2 = 0.85 f c, bβ1 c2

(2.27b)

As shown in Eq. 2.29a, b, the strain of the extreme compressive fiber of concrete can be determined by using a similar triangle. Similarly, the compressive and tensile strain of rebar are given by Eqs. 2.30a, b and 2.31a, b, respectively. εc1 = c1 ϕ1

(2.28a)

εc2 = c2 ϕ2

(2.28b)

2.2 Compressive Arch Action

41

Fig. 2.10 Constitutive models of tension/compression steel bar at the CAA stage

fy

ε'y 0 εy

f 'y

( ) a, , εs1 = 1 − s1 εc1 c1 ( ) a, , = 1 − s2 εc2 εs2 c2 ( ) h − as1 − 1 εc1 εs1 = c1 ( ) h − as2 − 1 εc2 εs2 = c2

(2.29a) (2.29b) (2.30a) (2.30b)

As shown in Fig. 2.10, the material property of rebar is set to be idealized elastic– plastic, and the unloading slope after yielding is equal to the elastic modulus. Thus, the constitutive relationship of rebar is given by (

) εs1 < εy ) Ts1 = εs1 ≥ εy f y As1 ( ) εs2 E s As2 εs2 < εy ) Ts2 = εs2 ≥ εy f y As2 ( ) ⎧ , , , , , , ⎪ E A < ε & ε < ε ε i f ε s ⎪ s1 s1 s1 y max1 y ⎪ ⎨ ( ) , , , , , , i f εs1 ≥ ε y & εmax1 ≤ εs1 Cs1 = f y As1 ⎪ [ ( ) ⎪ )] , ⎪ , , , , ⎩ f , − E (ε, < εmax1 & εmax1 ≥ εy, i f εs1 s max1 − εs1 As1 y εs1 E s As1

( if ( if ( if ( if

(2.31a)

(2.31b)

(2.32a)

42

2 Load Resisting Mechanisms of Concrete Structures to Resist …

( ) ⎧ , , , , , , ⎪ E A < ε & ε < ε i f ε ε s ⎪ s2 s2 s2 y max2 y ⎪ ⎪ ⎨ ( ) , , , Cs2 = f y, A,s2 ≥ εy, & εmax2 ≤ εs2 i f εs2 ⎪ ⎪ ( ) [ ⎪ )] , ⎪ , , , , ⎩ f , − E s (ε, < εmax2 & εmax2 ≥ εy, i f εs2 y max2 − εs2 As2

(2.32b)

By taking moment about the centroid of critical section, the following equations can be obtained ) ( ( ) ( ) β1 c1 h h h , − + Cs1 − as1 + Ts1 − as1 (2.33a) M1 = Cc1 2 2 2 2 ) ( ( ) ( ) β1 c2 h h h , − + Cs2 − as2 − as2 + Ts2 (2.33b) M2 = Cc2 2 2 2 2

2.2.2.6

Flowchart for Application of the Model

Based on the compatibility condition, moment–curvature relationship, force equilibrium condition, a flowchart is illustrated, as shown in Fig. 2.11. For a given displacement array δ = {δi }T (δi > δi−1 ), the model will output corresponding load resistance array {Pi }T and axial force array {Ni }T . The maximum value of {Pi }T is the CAA capacity, while the maximum value of {Ni }T is the maximum axial force at CAA stage.

2.2.2.7

Reduction Coefficient for Effective Flexural Rigidity and Modified M-ϕ Relationship

In this Chapter, calculation based on Xtract is conducted to determine the effective flexural rigidity of a beam section. Then, the reduction coefficient α = Bs /E c I0 of the effective flexural rigidity is determined. After that, comprehensive parameter studies are performed to regress an empirical model for the reduction coefficient. Based on the results of parameter studies in Table 2.1, an empirical function for the reduction coefficient α in terms of f cu , ρ, ρ , is proposed. α=

( )− 3 0.032 ρ , − 1 + ρ 0.506 f cu0.1374 (ρ , − 0.3549)2

(2.34)

Noted that Eq. 2.35 applies properly only when ρ and ρ , change from 0.3 to 2.0%, while f cu varies from 30 to 80 MPa. As seen in M = α E c I0 · ϕ, the bending moment resistance increased with the increase of curvature. However, in reality, moment resistance increased first and then decreases after reaching the maximum value. To overcome the limitation of this equation, a modified M − ϕ relationship is proposed.

2.2 Compressive Arch Action

43

1. Input geometric and material properties, boundary condition as well as reduction coefficient for flexural rigidity.

i=1 2. Input

4. Solve Eqs. (2-25 b) to (2-33b) gives , ，substitute

3. Solve Eqs. Eqs. (2-25 a) to (2-33a) gives , ，substitute ，and

, into Eq. 18a gives

5. Substitute

，and

, .

,

into Eq. 18b gives

, and

6. Substitute

,

into (2-27 a) gives , then input

, then input

.

into

gives

7. Substitute

8. Solve for the given

, and

into (2-27 b) gives

into

gives

.

give

gives .

.

,

into Eq. 2-23

, then substitute gives

9. Substitute

into

，

and

and input the results into Eq. 2-23 gives Then substitute

and

Eq. 9 gives

and

.

into .

No Increase

10. Judge

into .

Yes i=i+1

Output load resistance

, internal forces (

concrete and the rebars (

,

,

, ,

, ,

), strains of the ,

) at

.

Fig. 2.11 Procedure for calculating the vertical resistance and internal forces

As shown in Fig. 2.12, a bilinear model is used to describe the development of M − ϕ relationship with introduction of linear degradation assumption for effective flexural rigidity. At ascending phase, M = Bs · ϕ = α E c I0 · ϕ is used for the M − ϕ relationship. After the axial force reached the peak value Nmax , it is assumed that the bending moment decreases linearly while the declining phase is symmetric to the ascending phase with respect to ϕ = ϕmax . Therefore, M = Mmax − α , E c I0 · (ϕ − ϕmax ) is used to describe the M − ϕ relationship at the declining phase.

44

2 Load Resisting Mechanisms of Concrete Structures to Resist …

Table 2.1 Short-term flexural rigidity reduction factor for selected cross-section based on Xtract b×h (mm × mm)

f cu (MPa)

E c I 0 (kN·m2 )

Tension rebar ratio ρ (%)

Compression rebar ratio ρ , (%)

Bs (kN·m2 )

α

150 × 300

30 (C30)

1.005E+04

0.593

0.395

0.210E+04

0.209

1.005E+04

0.854

0.569

0.277E+04

0.275

1.005E+04

1.162

0.775

0.354E+04

0.352

1.005E+04

1.517

1.012

0.437E+04

0.434

1.005E+04

1.517

1.517

0.455E+04

0.452

1.136E+04

0.593

0.395

0.215E+04

0.189

1.136E+04

0.854

0.569

0.282E+04

0.248

1.136E+04

1.162

0.775

0.370E+04

0.325

1.136E+04

1.517

1.012

0.466E+04

0.410

1.136E+04

1.517

1.517

0.473E+04

0.416

1.215E+04

0.593

0.395

0.226E+04

0.186

1.215E+04

0.854

0.569

0.301E+04

0.247

1.215E+04

1.162

0.775

0.391E+04

0.322

1.215E+04

1.517

1.012

0.473E+04

0.389

1.215E+04

1.517

1.517

0.483E+04

0.397

1.282E+04

0.593

0.395

0.238E+04

0.186

1.282E+04

0.854

0.569

0.306E+04

0.239

1.282E+04

1.162

0.775

0.405E+04

0.316

1.282E+04

1.517

1.012

0.491E+04

0.383

1.282E+04

1.517

1.517

0.499E+04

0.389

16.087E+04

0.611

0.362

3.315E+04

0.206

16.087E+04

0.913

0.611

4.961E+04

0.308

16.087E+04

1.132

0.755

6.043E+04

0.376

16.087E+04

1.370

0.913

7.064E+04

0.439

16.087E+04

1.598

1.370

7.692E+04

0.478

16.087E+04

1.598

1.598

8.025E+04

0.499

18.175E+04

0.611

0.362

3.825E+04

0.210

18.175E+04

0.913

0.611

5.338E+04

0.294

18.175E+04

1.132

0.755

6.480E+04

0.357

18.175E+04

1.370

0.913

7.626E+04

0.420

18.175E+04

1.598

1.370

8.295E+04

0.456

18.175E+04

1.598

1.598

8.565E+04

0.471

19.436E+04

0.611

0.362

3.822E+04

0.197

19.436E+04

0.913

0.611

5.506E+04

45 (C45)

60 (C60)

80 (C80)

300 × 600

30 (C30)

45 (C45)

60 (C60)

0.283 (continued)

2.2 Compressive Arch Action

45

Table 2.1 (continued) b×h (mm × mm)

f cu (MPa)

80 (C80)

300 × 800

30 (C30)

45 (C45)

60 (C60)

60 (C60)

80 (C80)

E c I 0 (kN·m2 )

Tension rebar ratio ρ (%)

Compression rebar ratio ρ , (%)

Bs (kN·m2 )

α

19.436E+04

1.132

0.755

6.878E+04

0.354

19.436E+04

1.370

0.913

8.053E+04

0.414

19.436E+04

1.598

1.370

8.437E+04

0.434

19.436E+04

1.598

1.598

8.718E+04

0.449

20.503E+04

0.611

0.362

3.985E+04

0.197

20.503E+04

0.913

0.611

5.610E+04

0.283

20.503E+04

1.132

0.755

6.784E+04

0.354

20.503E+04

1.370

0.913

8.052E+04

0.414

20.503E+04

1.598

1.370

8.760E+04

0.434

20.503E+04

1.598

1.598

9.009E+04

0.449

38.133E+04

0.570

0.346

8.898E+04

0.233

38.133E+04

0.862

0.570

12.55E+04

0.326

38.133E+04

1.207

0.712

15.38E+04

0.403

38.133E+04

1.436

1.026

17.86E+04

0.468

38.133E+04

1.926

1.641

21.50E+04

0.564

38.133E+04

1.926

1.926

22.27E+04

0.581

43.082E+04

0.570

0.346

8.886E+04

0.206

43.082E+04

0.862

0.570

12.89E+04

0.299

43.082E+04

1.207

0.712

16.28E+04

0.378

43.082E+04

1.436

1.026

18.91E+04

0.439

43.082E+04

1.926

1.641

22.77E+04

0.528

43.082E+04

1.926

1.926

23.36E+04

0.542

46.071E+04

0.570

0.346

9.086E+04

0.197

46.071E+04

0.862

0.570

13.23E+04

0.287

46.071E+04

1.207

0.712

16.82E+04

0.365

46.071E+04

1.436

1.026

18.84E+04

0.409

46.071E+04

1.926

1.641

23.08E+04

0.501

46.071E+04

1.926

1.926

23.64E+04

0.513

48.600E+04

0.570

0.346

9.408E+04

0.194

48.600E+04

0.862

0.570

13.61E+04

0.280

48.600E+04

1.207

0.712

16.69E+04

0.343

48.600E+04

1.436

1.026

19.64E+04

0.404

48.600E+04

1.926

1.641

24.03E+04

0.494

48.600E+04

1.926

1.926

24.52E+04

0.505

Note f cu is cubic concrete compressive strength

46

2 Load Resisting Mechanisms of Concrete Structures to Resist …

Fig. 2.12 Modified moment–curvature relationship

Therefore, Eqs. 2.25 and 2.36 are used to describe the M − ϕ relationship at ascending and declining phases, respectively. ϕ1 =

M1max − M1 M1max − M1 + ϕ1max = + ϕ1max , Bs1 α1, E c I0

(2.35a)

ϕ2 =

M2max − M2 M2max − M2 + ϕ2max = + ϕ2max , Bs2 α2, E c I0

(2.35b)

, , = α1, E c I0 and Bs2 = α2, E c I0 are the effective flexural rigidity of the where Bs1 critical sections at declining phase. Based on the linear degradation assumption for effective flexural rigidity, the reduction coefficients at declining phase are consistent with that at ascending phase (i.e., α1, = α1 and α2, = α2 ).

2.2.2.8

Validation of the Proposed Model

Table 2.2 lists the information of specimens that used to validate the proposed CAA model. The predicted results from the proposed model are compared with those from Lu et al. (2018)’s model. As shown in Table 2.2, the model proposed by the authors and Lu et al. (2018) is able to predict the CAA capacity and the maximum axial force well. Figures 2.13, 2.14, 2.15 compare the predicted results with the measured ones. It

2750

2750

2750

2750

1225

1225

1225

1225

1225

1225

1975

2725

150 × 250

150 × 250

150 × 250

150 × 250

150 × 300

150 × 300

150 × 300

150 × 300

150 × 300

150 × 300

150 × 300

150 × 300

S4

S5

Su A1 et al. A2 (2009) A3

A4

A5

A6

B1

B2

3T14

3T14

3T14

3T12

2T12

3T14

3T12

2T12

3T13

3T13

3T13

3T10

3T14

3T14

2T14

2T12

1T14

3T14

3T12

2T12

3T13

2T13

2T10

2T10

2T10

2750

150 × 250

Yu and S1 Tan S2 (2013) S3 2T10+1T13

Bottom

Longitudinal rebar

b × h(mm2 ) ln (mm) Top

Source

Table 2.2 Specimen information

19.3

18.6

28.6

26.5

23.1

31.2

28.2

25.8

38.15

38.15

38.15

31.24

31.24

f c, (MPa)

10.0 × 105

0.0

0.0

× 105 10.0

0.0

10.0 × 105

10.0 × 105

0.0

0.0

× 105

10.0

0.0

10.0 × 105

0.0

0.0

× 105

350 (T12) 10.0 340 (T14) 10.0 × 105

0.8

4.29 × 105

0.8

1.0

4.29 × 105 4.29 × 105

1.2

0.5

1.75 × 1010

1.75 × 1010

1.75 × 1010

1.75 × 1010

1.75 × 1010

1.75 × 1010

1.75 × 1010

1.75 × 1010

3.00 × 1010

3.00 × 1010

3.00

× 1010

1.00 × 1010

1.00 × 1010

(continued)

0

0

0

0

0

0

0

0

0

0

0

0

0

K a (N/mm) t0 (mm) K r (N · mm/rad) ω0 (rad)

Boundary conditions 511 (T10) 1.06 × 105 494 (T13) 1.06 × 105

f y (MPa)

2.2 Compressive Arch Action 47

2750

2750

2750

5385

5233

150 × 250

150 × 250

711 × 508

864 × 660

SS-4

Sadek IMF et al. SMF (2011)

2750

150 × 250

150 × 250

2T16+1T13

2750

Alogla SS-1 et al. SS-2 (2016) SS-3

3T10

2750

Kang MJ-B-0.52/0.35S 150 × 300 and CMJ-B-1.19/0.59 150 × 300 Tan (2015) 2T10

7 #8

4 #8

5T10

3T10

6 #8

2 #9

2T10

4T10

3T10(+2T10) 2T10

3T10

2T13

2T10

2T14

3T14

2725

150 × 300

B3

Bottom

Longitudinal rebar

b × h(mm2 ) ln (mm) Top

Source

Table 2.2 (continued)

35.9

32.5

27.5

27.5

27.5

26.8

36.1

35.8

21.1

f c, (MPa)

476 462

510

549

462

f y (MPa)

1.2

0.2

2.65 × 105 8.65 × 105

1.2

0.85 × 105

0.6

0.4

0.85 × 105

0.2

0.9

1.06 × 105

0.85 × 105

1.1

1.51 × 105

0.85 × 105

0.0

10.0 × 105

7.35 × 1010

3.0

× 1010

1.50 × 1010

1.50 × 1010

1.50

× 1010

1.50 × 1010

1.99 × 1010

2.14 × 1010

1.75 × 1010

0.015

0.010

0.012

0.005

0.003

0.004

0.010

0.005

0

K a (N/mm) t0 (mm) K r (N · mm/rad) ω0 (rad)

Boundary conditions

48 2 Load Resisting Mechanisms of Concrete Structures to Resist …

Alogla et al. (2016)

Kang and Tan (2015)

Su et al. (2009)

Yu and Tan (2013)

Source

125.0 82.90 74.70

B1

B2

B3

34.00 37.90

SS-1

SS-2

90.40

226.0

A6

CMJ-B-1.19/0.59

198.0

A5

50.50

147.0

A4

MJ-B-0.52/0.35S

246.0

A3

70.33

S5 221.0

63.22

S4

A2

54.47

S3

168.0

38.38

S2

A1

41.64

S1

38.8

37.6

92.71

53.33

70.6

81.63

116.5

203.9

183.9

143.8

232.4

212.6

162.6

73.01

64.04

53.23

36.12

43.33

Pa

1.024

1.106

1.026

1.056

0.945

0.985

0.932

0.902

0.929

0.978

0.944

0.962

0.968

1.038

1.013

0.977

0.941

1.041

Pa PCAA

39.50

39.50

N/A

N/A

78.69

86.88

125.4

224.5

189.3

149.0

256.9

214.0

171.4

81.89

71.86

62.42

33.65

42.67

Pa

1.042

1.162

N/A

N/A

1.053

1.048

1.003

0.993

0.956

1.014

1.044

0.968

1.021

1.164

1.137

1.144

0.877

1.025

Pa PCAA

64.30

63.80

281.1

231.3

210.0

210.0

225.0

191.0

393.0

344.0

305.0

324.0

388.0

238.4

212.7

221.0

155.9

177.9

Measured Nmax

77.5

73.8

198.0

226.5

232.7

215.6

226.9

369.4

359.7

310.7

394.2

346.7

328.9

195.1

199.7

210.7

129.3

148.7

Na

1.205

1.157

0.704

0.979

1.108

1.027

1.008

1.934

0.915

0.903

1.292

1.070

0.848

0.818

0.939

0.953

0.894

0.836

Na Nmax

128.3

128.3

N/A

N/A

294.4

270.1

267.6

384.8

360.7

320.8

408.9

380.4

353.5

275.9

275.9

259.8

141.2

189.8

Na

(continued)

1.995

2.011

N/A

N/A

1.402

1.286

1.189

2.015

0.918

0.933

1.341

1.174

0.911

1.157

1.297

1.176

0.906

1.067

Na Nmax

Proposed model Lu et al. (2018)’s model

Maximum axial force Nmax (kN)

Measured PCAA

Proposed model Lu et al. (2018)’s model

CAA capacity PCAA (kN)

Table 2.3 Comparison of predicted results from different CAA models

2.2 Compressive Arch Action 49

1.062

Note “—” indicates unreliable data

0.069

954.5

1.010

1.147

1.172

Pa PCAA

1.007

899.0

SMF

298.7

42.2

43.6

Pa

Variable coefficient

295.6

36.70

SS-4

IMF

37.20

SS-3

N/A

N/A

43.83

45.68

Pa

0.085

1.060

N/A

N/A

1.194

1.228

Pa PCAA

2448

1097

69.60

62.70

Measured Nmax

2783

1129

81.6

67.9

Na

0.146

1.001

1.137

1.029

1.172

1.083

Na Nmax

N/A

N/A

129.0

129.0

Na

0.148

1.135

N/A

N/A

1.853

2.057

Na Nmax

Proposed model Lu et al. (2018)’s model

Maximum axial force Nmax (kN)

Measured PCAA

Proposed model Lu et al. (2018)’s model

CAA capacity PCAA (kN)

Mean value

Sadek et al. (2011)

Source

Table 2.3 (continued)

50 2 Load Resisting Mechanisms of Concrete Structures to Resist …

2.2 Compressive Arch Action

51

60

0

50

-50

40 30 20 Test Result Lu et al. (2018) Proposed Model

10 0 0

100

200

Axial Force (kN)

CAA Capacity (kN)

is found that the proposed model predicts the varying of CAA and axial force with the vertical displacement reasonable. In summary, the proposed CAA model is able to predict the CAA capacity reasonably. Compared with Park and Gamble (2000)’s model and Lu et al. (2018)’s model, the proposed model is not only able to predict the CAA capacity, but also to predict the varying of CAA with the vertical displacement.

-100 -150 -200 -250

Test Result Proposed Model

-300 0

300

100

200

300

Vertical Displacement (mm)

Vertical Displacement (mm)

(a) CAA capacity

(b) beam axial force

150

0

120

-50

90 60 Test Result Lu et al. (2018) Proposed Model

30 0 0

100

200

300

Axial Force (kN)

CAA Capacity (kN)

Fig. 2.13 Comparison of predicted results of Specimen MJ-B-0.52_0.35S with the measured ones

-100 -150 -200 -250

Test Result Proposed Model

-300 0

100

200

Vertical Displacement (mm)

Vertical Displacement (mm)

(a) CAA capacity

(b) beam axial force

Fig. 2.14 Comparison of predicted results of Specimen B1 with the measured ones

300

52

2 Load Resisting Mechanisms of Concrete Structures to Resist … 350 300

800

CAA Capacity (kN)

CAA Capacity (kN)

1000

600 400 200

Test Result Proposed Model

0 0

250 200 150 100 50

Test Result Proposed Model

0

100 200 300 400 500 600 Vertical Displacement (mm)

(a) SMF

0

100 200 300 400 500 600 Vertical Displacement (mm)

(b) IMF

Fig. 2.15 Comparison of predicted results of Specimens SMF and IMF with the measured ones

2.3 Catenary Action As shown in Fig. 2.16, when the middle column displacement exceeded about one beam depth, the axial force began to change from compression to tension. At this stage, the load resistance is mainly ascribed to the tensile force developed in longitudinal reinforcements. Therefore, catenary action is a mechanism associated with tensile axial force developed in beams.

2.3.1 Existing Catenary Action Models As the last defense line against progressive collapse, the importance of catenary action is understood. The mobilization of catenary action highly relies on longitudinal reinforcement ratio, material properties of the reinforcement, and rotational capacity of beam-column connection. In this chapter, the reliability of existing catenary action models is evaluated.

P T1

T1

T2

T2

Fig. 2.16 Catenary action

2.3 Catenary Action

53

Yi et al. (2008)’s model Based on the tests on a 4-bays and 3-story planar RC frame, Yi et al. (2008) proposed a model for catenary action as Eq. 2.38. In their model, all continuous longitudinal reinforcements are considered to contribute catenary action capacity PTCA = 2ψ(Ast f u + Asb f u, ) sin α

(2.36)

where ψ is a strain adjustment coefficient, and ψ = 0.85 herein; f u and f u, are the ultimate strength of top and bottom rebars, respectively; Ast and Asb are the area of top and bottom rebars, respectively; α is chord rotation of the beam (Fig. 2.17). Su et al. (2009)’s model Su et al. (2009) tests twelve two-span beam specimens and propose a catenary action model. As illustrated in Fig. 2.18, different from Yi et al. (2008)’s model, only bottom rebars are considered to provide catenary action capacity. Moreover, the rotation of beam end in Su et al. (2009)’s model is also different from that in Yi et al. (2008)’s model. PTCA = 2 Asb f u, sin ϕ

(2.37)

P f u Ast

f u Ast

f u' Asb

f u' Asb

α

α

Fig. 2.17 Yi et al. (2008)’s model

P f u' Asb

f u' Asb φ

Fig. 2.18 Su et al. (2009)’s model

φ

54

2 Load Resisting Mechanisms of Concrete Structures to Resist …

P f y Ast

f y Ast α

α

Fig. 2.19 Yu and Tan (2013)’s model

where ϕ is the angle between the line connected the top rebar near to the side column and bottom rebar near to the middle column and the horizontal line. Yu and Tan ( 2013)’s model In contrary to Su et al. (2009)’s model, Yu and Tan (2013)’s model only considers the contribution of the top rebars in their model as shown in Fig. 2.19. Similar to Yi et al. (2008)’s model, chord rotation is adopted by Yu and Tan (2013)’s model. PTCA = 2 Ast f y sin α

(2.38)

2.3.2 Reliability of Existing Catenary Action Models Based on the database, which consists of test results from 30 specimens (from important experimental data), the reliability of existing catenary action models is conducted, as shown in Fig. 2.20. Noted that, the deformation capacity of each specimen is assumed to be 10% of the total span of the double-bay beam, in accordance to DoD (2009). As shown in the Figure, the mean ratio of the measured catenary action capacity to the calculated one based on the models of Yi et al. (2008), Su et al. (2009), and Yu and Tan (2013) were 1.06, 1.43 and 1.60, respectively. The standard deviation was 0.28, 0.42 and 0.53, respectively. Thus, among them, the model of Yi et al. (2008) gives the best prediction. The model of Su et al. (2009) neglected the contribution from top rebars resulted in conservative prediction. However, in the model of Yu and Tan (2013), the bottom rebar was assumed fractured completely, which is not true in reality, the model may result in conservative results.

2.3 Catenary Action

Analitycal TCA Capacity (kN)

200

Liu 2019 Su et al. 2009 Yu and Tan 2013b Yu and Tan 2014 Qian et al. 2015 Ren et al. 2016 Alogla et al. 2016

150

100

50

MN=1.06 SD=0.28 CV=0.27

0 0

50

100

150

200

Measured TCA Capacity (kN)

(a) Yi et al. (2008)’s model Analitycal TCA Capacity (kN)

200

Liu 2019 Su et al. 2009 Yu and Tan 2013b Yu and Tan 2014 Qian et al. 2015 Ren et al. 2016 Alogla et al. 2016

150

100

50

MN=1.43 SD=0.42 CV=0.29

0 0

50

100

150

200

Measured TCA Capacity (kN)

(b) Yi et al. (2008)’s model 200

Analitycal TCA Capacity (kN)

Fig. 2.20 Comparison of analytical catenary action capacity with the measured one

55

Liu 2019 Su et al. 2009 Yu and Tan 2013b Yu and Tan 2014 Qian et al. 2015 Ren et al. 2016 Alogla et al. 2016

150

100

50

MN=1.60 SD=0.53 CV=0.33

0 0

50

100

150

Measured TCA Capacity (kN)

(c) Yu and Tan (2013)’s model

200

56

2 Load Resisting Mechanisms of Concrete Structures to Resist …

2.4 Compressive Membrane Action in Two-Way Slab with All Edges Restrained The model for predicting compressive membrane action of two-way RC slab with all edges restrained, which is proposed by Park and Gamble (2000), is introduced in below. The following assumptions were made:

0.5l y

0.5l y

45°

45°

ly

y

A o

B

x

45°

Sagging moment Hogging moment

lx Fig. 2.21 Yield line pattern for slab

45°

0.5l y

1. The slab is composed of strips running in the x- and y-directions which have the same depth as the slab. Assuming the x-direction strips only contain the steel reinforcements in x-direction while the y-direction strips only contain the steel reinforcements in y-direction. 2. The yield-line pattern of the slab is shown in Fig. 2.21. The yield sections of the strips lie on the yield lines and have the same deflection as the slab. 3. The yield sections of the strip perpendicular to the slab with long side. At the yield sections the torsional moments are zero, the tensile strength of concrete is neglected herein. 4. The strip between the critical sections kept straight. 5. The steel is arranged symmetrically along the x- and y-directions and the area of steel per unit width in the same direction remains unchanged. 6. The total axial strain in the same direction kept constant, but it can be different in different directions. 7. The total horizontal drift in the same direction kept identical, but it can be different in different directions. 8. The ultimate load achieved when the central deflection reached half slab thickness.

2.4 Compressive Membrane Action in Two-Way Slab with All Edges …

57

When unit displacement occurred in the yield line A-B, the rotation of the critical section near the edge is 2/l y . Based on virtual work principle, following equation is given: ¨ wu d xd y = 4

( 0.5ly 0

(m ,ux + m ux − n ux δ)

+ 2(m ,uy + m uy − n uy δ)

( 0.5ly 2 2 dy + 4 (m ,uy + m uy − n uy δ) d x ly ly 0

2 (lx − ly ) ly

(2.39) where wu is ultimate uniform load per unit area. The term on the right-hand side can be determined by Eq. 2.14. The term on the left-hand side equal to: wu ly 1 1 wu ly2 + wu (lx − ly )ly = (3lx − ly ) 3 2 6

(2.40)

On performing integration on both sides of Eq. 2.40 gives: ( ( ) [ ) ( lx lx wu lu2 εx l y 2 l x 3 − 1 = 0.85 f c βl h 2s (0.188 − 0.141β1 ) + (0.479 − 0.245β1 ) + (3.5β1 − 3) 24 ly ly 4 hs ly ( ) [ ( ) [ ]) ] εy l y 2 lx lx l y 4 l x ( )2 εy 2 (1.5β1 − 1) + (0.5β1 − 1) − β1 + (εx )2 4 hs ly l y hs ly [ ] )2 l x ( )2 ( 1 − Ty − Ty − Csy + Csy Tx − Ty − Csx + Csx + 3.4 f c ly ( [ ( ) ] ) ) ( ( ) lx h 3h hs 3h s + Csy + Csy + (Csx + Csx ) − dx + (Tx + Tx ) dx − − dy + d 8 ly 4 8 ) ] [ ( hs lx hs − dy − + (Ty + Ty ) ly 4 8

(2.41) where h s is slab height; lx and ly are the clear span in x- and y-direction, respectively; Tx and Tx, are the tensile force of the reinforcement per unit length on the sections with positive and negative moments in x-direction, respectively; Ty and Ty, are the tensile force of reinforcement per unit length on yield sections with positive and negative moments in y-direction, respectively; Cx and Cx, are the compressive force of concrete per unit length on yield sections with positive and negative moments in x-direction, respectively; Cy and Cy, the compressive force of concrete per unit length on yield sections with positive and negative moments in y-direction, respectively; εx, and εy, the axial strain of slab in x- and y-direction, respectively.

58

2 Load Resisting Mechanisms of Concrete Structures to Resist …

Fig. 2.22 Tensile membrane action

Tensile zone

Compressive ring

2.5 Tensile Membrane Action When the slab undergoes significant deflection, tensile forces in rebar net began to contribute to progressive collapse resistance. Different from the compressive membrane action of RC slab, the rigid transverse constraint of slab edge is not the necessary condition for the tensile membrane action. Some previous studies, such as Bailey (2001), have found that significant tensile membrane effects can be developed in reinforced concrete slabs, even with simple supporting edges. As shown in Fig. 2.22, the formation of the outer compressive ring can provide considerable lateral constraints on the deflector plate, resulting in the development of tensile membrane force in the central region. Detailed discussion on tensile membrane action please refer to Chap. 4. To avoid duplication, it is not discussed in this chapter.

2.6 Summary In this chapter, potential load resisting mechanisms for concrete structures against progressive collapse, including flexural action, compressive arch action, catenary action of beam, and compressive/tensile membrane action of slab, are introduced. Moreover, existing models for those mechanism are introduced, while their reliability is validated by a database from available works.

References

59

References Alogla K, Weekes L, Augusthus NL (2016) A new mitigation scheme to resist progressive collapse of RC structures. Constr Build Mater 125:533–545 Bailey CG (2001) Membrane action of unrestrained lightly reinforced concrete slabs at large displacement. Eng Struct 23(5):470–483 DoD (2009) Design of building to resist progressive collapse. Unified Facility Criteria. UFC 4-02309. US Department of Defense, Washington (DC) Kang SB, Tan KH (2015) Behaviour of precast concrete beam–column sub-assemblages subject to column removal. Eng Struct 93:85–96 Lu Z, He X, Zhou Y (2018) Discrete element method-based collapse simulation, validation and application to frame structures. Struct Infrastruct Eng 14(5):538–549 Park R, Gamble WL (2000) Reinforced concrete slabs. Wiley, New York, p 716 Park R, Paulay T (1975) Reinforced concrete structures. John Wiley & Sons Inc., New York Sadek F, Main JA, Lew HS et al (2011) Testing and analysis of steel and concrete beam-column assemblies under a column removal scenario. J Struct Eng 137(9):881–892 Su YP, Tian Y, Xiao S (2009) Progressive collapse resistance of axially-restrained frame beams. ACI Struct J 106(5):600–607 Yi WJ, He QF, Xiao Y, Kunnath SK (2008) Experimental study on progressive collapse-resistant behavior of reinforced concrete frame structures. ACI Struct J 105(4):433–439 Yu J, Tan KH (2013) Structural behavior of RC beam-column subassemblages under a middle column removal scenario. J Struct Eng 139(2):233–250 Yu J, Tan KH (2014a) Analytical model for the capacity of compressive arch action of reinforced concrete sub-assemblages. Mag Concr Res 66(3):109–126 Yu J, Tan KH (2014b) Special detailing techniques to improve structural resistance against progressive collapse. J Struct Eng 140(3):4013077

Chapter 3

Dynamic Increase Factor of Concrete Structures

Existing analysis procedures for progressive collapse study include Linear Static procedure (LSP), Nonlinear Static procedure (NSP), Linear Dynamic procedure (LDP), and Nonlinear Dynamic procedure (NDP). LSP and NSP analyses are easy to perform as static analysis with much easier definition of material properties. However, they fall short in capturing dynamic nature of progressive collapse, which are normally caused by vehicular impact, blast or huge earthquake. The LDP analysis can take into account the inertia effect, but not the nonlinear characteristics of the building. Therefore, NDP analysis has the highest precision and is the best choice for progressive collapse analysis. However, NDP analysis requires a lot of computing resources, which brings inconvenience to engineering applications. For comprehensive advantages, load increasing factor (LIF) and dynamic increasing factor (DIF) are proposed to convert LSP results and NSP results into NDP behaviors, respectively.

3.1 Studies on DIF of Beam-Column Sub-structures In this chapter, quasi-static and dynamic studies on RC beam-column sub-structures subject to a corner column removal scenario are performed. Static ultimate strength of the sub-structures is measured through quasi-static tests, and then DIF of the sub-structures is obtained by comparing the static test results with dynamic test results.

3.1.1 Specimen Design In the present study, 9-story RC prototype buildings with non-seismic or seismic design were designed according to the Singapore Standard CP a 65(1999) and © China Architecture & Building Press 2023 K. Qian and Q. Fang, Progressive Collapse Resilience of Concrete Structures: Mechanisms, Simulations and Experiments, https://doi.org/10.1007/978-981-99-0772-4_3

61

62

3 Dynamic Increase Factor of Concrete Structures

the American Concrete Institute (ACI) 318-08(2008), respectively. For analysis purposes, the test target is assumed to be a regular frame. It is worth noting that the seismic design prototype is located in a class D site (rigid soil profile), and the designed spectral response acceleration parameters SD1 and SDs are 0.32 and 0.47, respectively. The dead load acting on the prototype structure is set to be 5.1 kPa including the gravity load of 210 mm thick plate. It is assumed that the additional dead load caused by the dead weight of the ceiling is 1.0 kPa. The equivalent additional dead loads of the filled wall and beam are 2.25 and 1.59 kPa, respectively. Therefore, the design dead load (DL) is 9.94 kPa. The live load (LL) was set at 2.0 kPa. Figure 3.1 shows the critical structural information of the prototype frame based on a typical non-seismic design sample F3. The dimensions and reinforcement details of other prototype frames are shown in Table 3.1. Taking into account laboratory space constraints and transportation difficulties, a one-third scaled model was conducted. Eleven specimens were designed and tested, among which F2, F3, F4, F5 and F6 were used for static test, and DF1, DF2, DF3, DF4, DF5 and DF6 were used for static test. The dynamic test sample is the same as the static test sample. It should be noted that specimen DF1 and specimen DF3 (F3) are the same except for loading conditions. The design axial force of each specimen Angle column is determined according to the full load combination or load reduction combination recommended in DoD (2009), which is defined as the DoD (2009) load combination amplified with a fraction factor. As tabulated in Table 3.2, Specimens DF3 and DF4 bear full load, ignoring wind and snow loads, with the load combination of 1.2DL + 0.5LL. Specimens DF1 and DF2 were subjected to a load reduction combination of 0.9(1.2DL + 0.5LL) (Table 3.2).

Fig. 3.1 Plane and top view of prototype frame of Specimen F3

540 × 300

540 × 300

720 × 300

720 × 300

540 × 300

540 × 300

720 × 300

540 × 300

F3

F4

F5

F6

2T20+T32

2T25+T32

2T20+T32

2T20+T32

2T25+T32

2T25+T32

2T20+T32

2T20+T32

2T25+T32

2T25+T32

2T20+T32

2T20+T32

3T32

Bottom

180 × 100

240 × 100

180 × 100

180 × 100

180 × 100

Beam-T

240 × 100

240 × 100

180 × 100

180 × 100

180 × 100

Beam-L

Dimensions of the model beams (mm)

4T10

4T10

4T10

4T10

4T13

Beam-T

4T10

4T10

4T10

4T10

4T13

Beam-L

Longitudinal rebar in the model beams

Note T32 = Deformed bar of 32 mm diameter, T25 = Deformed bar of 25 mm diameter, T20 = Deformed bar of 20 mm diameter, T13 = Deformed bar of 13 mm diameter, T10 = Deformed bar of 10 mm diameter, Beam-L = Longitudinal beam; Beam-T = Transverse beam

2T20+T32

2T25+T32

2T20+T32

2T20+T32

3T32

3T32

Top

Top

540 × 300

540 × 300

F2

Bottom

Beam-L

Beam-T

Beam-L

Beam-T

3T32

Longitudinal rebar in the prototype beams

Dimensions of prototype beams (mm)

Test

Table 3.1 Relationship between the prototype frames and the test specimens

3.1 Studies on DIF of Beam-Column Sub-structures 63

Type a*

Type a*

Type a*

DF3 (F3)

Type b*

Type a*

DF6 (F6)

4-T10

4-T10

4-T10

4-T10

4-T10

4-T10

4-T10

4-T10

4-T13

Stirrup

None

None

None

None

R6@55

None

Joint

R6@180

R6@160

R6@80

R6@180

R6@60

R6@180

Beam-T

R6@160

R6@160

R6@80

R6@180

R6@60

R6@180

Beam-L

−18.8 −23.2 −23.2

λ = 1.0 λ = 1.0

−18.7

λ = 1.0 λ = 0.8

−16.9 −16.9

λ = 0.9

Axial force (kN)

λ = 0.9

Load case λ(1.2DL + 0.5LL)

Note Type a*: Clear span = 2175 mm, cross-section = 180 × 100; Type b*: Clear span = 2775 mm, cross-section = 240 × 100

Type b*

Type a*

Type b*

DF4 (F4)

DF5 (F5)

4-T10

4-T13

4-T10

Type a*

Type a*

Type a*

Type a*

DF2 (F2)

Beam-L

Longitudinal rebar

Beam-T

Beam-L

Beams

Beam-T

DF1 (F3)

Specimen

Table 3.2 Specimen properties (unit: mm)

64 3 Dynamic Increase Factor of Concrete Structures

3.1 Studies on DIF of Beam-Column Sub-structures

65

Fig. 3.2 Details of DF2 and DF3

3.1.2 Test Setup and Instrumentation Mohamed (2009) observed that appreciable deformation of typical RC frames was concentrated in the corner panels, while the deformation of other panels could be ignored. Therefore, we extracted a typical key panel (the corner panel of the second story) from the prototype framework for our study. A schematic diagram of the quasi-static test device is shown in Fig. 3.3. Settings can be broken down into three components. In component one, vertical, axial and rotational constraints are

66

3 Dynamic Increase Factor of Concrete Structures 1 2

3

5 9

4 6 7

11 1

8 10

Fig. 3.3 Quasi-static test setup

provided at the magnification of adjacent columns to simulate the fixed boundary conditions provided by surrounding structural elements. In component two, downward displacement was applied to the short end of the corner column by a hydraulic jack with a 600 mm stroke to simulate axial loading before the corner column was damaged. Previous pertinent studies (Sasani and Sagiroglu 2008) have demonstrated the reversal of the direction of the bending moment at the beam end near the Angle joint (BENC) after the removal of the corner column, resulting in the formation of a considerable positive bending moment (soffit tension) in the BENC due to Vierendeel action after the removal of the corner column. However, Sasani and Sagiroglu (2008) observed in the deformation shape of corner joints that the corner joints moved slightly horizontally as well as vertically after the removal of ground corner columns, suggesting that BENC’s rotation constraint was not sufficient. The positive bending moment is applied to the BENC test substructure using Component three. A sturdy steel column is attached to the corner of the reinforced concrete specimen using anchor bolts. Four high strength and high stiffness steel pins are used to apply specified partial rotation and horizontal constraints in each direction. The positive bending moment is applied to the BENC test substructure using Component three. A strong steel column is attached to the corner of the reinforced concrete specimen using anchor bolts. Four high strength and high stiffness steel pins are used to apply specified partial rotation and horizontal constraints in each direction. That is, the steel column can move freely in the vertical direction, but its rotation and horizontal freedom are partially limited. A finite element model of the component was established using commercial finite element analysis software ABAQUS (2006) to help design the allowance between the steel pin and the hole. The validity of the model was verified by comparing the numerical results with those of Sasani et al. (2007). The relationship between horizontal displacement and vertical deflection of the center of the corner joint is predicted by using this model for nappe analysis. The finite element results show that the joint center is just above the loss column, and the maximum horizontal movement outward is about 7.2 mm, while the vertical displacement (D1) is about 180.0 mm. The allowance between the steel pin and hole is designed to

3.1 Studies on DIF of Beam-Column Sub-structures

67

be 3 mm. That is, the steel column can move freely in the vertical direction, but its rotation and horizontal freedom are partially limited. A finite element model of the component was established using commercial finite element analysis software ABAQUS (2006) to help design the allowance between the steel pin and the hole. The validity of the model was verified by comparing the numerical results with those of Sasani et al. (2007). The relationship between horizontal displacement and vertical deflection of the center of the corner joint is predicted by using this model for nappe analysis. The finite element results show that the joint center is just above the loss column, and the maximum horizontal movement outward is about 7.2 mm, while the vertical displacement (D1) is about 180.0 mm. The allowance between the steel pin and hole is designed to be 3 mm. To measure the structural response, extensive measuring devices were installed symmetrically. A total of 100 data channels were activated during the test. A load sensor was used to measure the force on the corner end, and a series of LVDTs were used to monitor the deflection shape of the beam. Three compression/tension sensors were mounted on each mounting bracket. Two of them are vertical and were used to determine the vertical reaction force and the moment at the fixed support. The remaining horizontal load sensors were used to measure the horizontal binding force at the fixed bracket. A series of LVDT and linear potentiometers are placed at different positions in the substructure to measure different types of internal deformations such as fixed support rotation, curvature, and diagonal deformations. It should be noted that two LVDTs with 25 mm stroke were placed on each bracket to monitor the rigid body rotation of the bracket (see items 8 and 9 in Fig. 3.3). The rotational response of each fixed support in each specimen was recorded during the test, and the additional vertical deflection of corner joints caused by rigid body rotation was determined by assuming that the beam was a cantilever beam. The error in the cantilever hypothesis can be ignored because the rigid-body rotation recorded is finite (for example, the maximum rigid-body rotation in the F3 transverse fixing bracket is 0.00183 rad). It should be noted that the displacement results of the following sections are net displacement, which is defined as the total deflection minus the additional deflection caused by the rotation of the rigid body at the fixed support. Approximately 60 resistance strain gauges were installed on the reinforcement bars at critical locations to monitor strain changes along beams, corner columns and joints during the test. The schematic diagram of dynamic test device is shown in Fig. 3.4. At the top of the test apparatus in Fig. 3.3, a column shifting apparatus consisting of a specially designed steel column, a pin support and a load cell (Item 6 in Fig. 3.4) is designed to simulate the sudden removal of the corner column under extreme loads. In addition, a series of concentrated weight assemblies were suspended from the beams to simulate service loads. Two link blocks (Item 2 in Fig. 3.4) are suspended from the steel column to prevent the corner pile from moving downward when simulated gravity is added. Six specific concentrated weight assemblies (Item 7 in Fig. 3.4) are slowly applied to the beam in a symmetrical order. It should be noted that the weight of the concentrated weight assembly is designed according to the service load of each specimen. Raise the special steel column (Item 2 in Fig. 3.5) and adjust the height of the pin holder (Item 3 in Fig. 3.5) until the tip of the hemispherical steel ball

68

3 Dynamic Increase Factor of Concrete Structures

1

2 3 4

11

5 9

10

12

Fig. 3.4 Dynamic test setup Fig. 3.5 Column removal apparatus

7 6

13

7

8

3.1 Studies on DIF of Beam-Column Sub-structures

69

touches the bottom of the corner pile. The chain block is then released to slightly adjust the height of the pin support until the reaction force on the corner column reached the design axial force as shown in Table 3.2. The static data recorder was used to record the reinforcement strain, axial load and vertical reaction in the process of static load. In this static process, the crack development of beam and corner joints is also monitored. Several selected cables originally connected to the static data recorder are switched to the dynamic strain gauge via a special bridge head after all the specific concentrated weight components are applied. Due to the limitation of the channel, the dynamic strain gauge can only connect 10 cables. In order to obtain more data, different channel distribution strategies were set up for different specimens. At last, the support in the corner was suddenly knocked down by a heavy hammer. In order to monitor the response of the sample, a large number of measuring devices are installed both internally and externally. A total of 25 data channels are active in the static data logger during static loading. However, only 10 data channels are active during dynamic loading. Several accelerometers are connected to a digital high-speed data recorder to monitor the acceleration distribution along the beam after initial damage. The sampling frequencies of dynamic strain gauge and high-speed data recorder are 2000 and 1000 Hz respectively. A force measuring element (Item 4 in Fig. 3.5) is placed below the pin support to monitor the increase of axial force of the corner column in the static process and record the change of axial force in the dynamic process. Four LVDTs were placed along the beam to monitor the dynamic displacement distribution of DF1 and DF2. For the remaining specimens, only an LVDT with a stroke of 300 mm was installed to measure the vertical displacement response of the corner column. Two compression/tension load sensors (Items 8 and 10 in Fig. 3.4) mounted vertically measure the vertical reaction force and determine the bending moment on the fixed support. A compression/tension load sensor (Item 9 in Fig. 3.4) was installed horizontally to measure the horizontal reaction at the fixed support. Prior to casting, a total of 12 strain gauges were installed at critical positions of the reinforcement.

3.1.3 Material Properties The target compressive strength of concrete is 30 MPa. The average compressive strength of F2, F3, F4, F5, F6 and F7 are 32.1, 31.9, 32.5, 33.1 and 32.8 MPa, respectively. The average compressive strength of DF1, DF2, DF3, DF4, DF5 and DF6 concrete cylinder specimens is 32.8, 31.4, 31.8, 33.1, 33.8 and 33.1 MPa, respectively. Table 3.3 shows the measured tensile properties of the rebar in the test.

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3 Dynamic Increase Factor of Concrete Structures

Table 3.3 Properties of rebars

Item

Yield strength (MPa)

Ultimate strength (MPa)

Elongation (%)

R6

530

613

20.3

T10

575

695

21.7

T13

520

637

22.6

T16

556

635

21.1

Note R6 = ordinary round rod with diameter of 6 mm; T10 = deformed bar with diameter of 10 mm; T13 = deformed bar with diameter of 13 mm; T16 = deformed bar with diameter of 16 mm

3.1.4 Quasi-static Test Results A total of five beam-column substructures with different design details and span lengths are built and tested to assess the behavior of reinforced concrete frames withstanding ground corner column losses. The test results of the five samples are summarized in Table 3.4 and discussed in the following chapters. The vertical/horizontal load–displacement curves are shown in Fig. 3.6.

3.1.4.1

The Relationship Between Vertical Load and Horizontal Reaction and Deflection

Influence of transverse reinforcement ratio in plastic hinge zone of beam In order to relate the test results with the performance of each specimen, the axial force division method of corner column design was used to normalize the test results. The failure mode of F3 is shown in Fig. 3.7. Under the load of 4.3 kN, the first crack appeared at the beam end near the fixed support (BENF). The value of 0.23 indicates that the crack of Specimen F3 began when the load reached 23% of the designed Table 3.4 Test results Test

Yield load kN

Ultimate load kN

MCHR Beam-T kN

MCHR Beam-L kN

MBM Beam-T kN·m

TMBM Beam-T kN·m

MBM Beam-L kN·m

MBM Beam-L kN·m

F2

29.1

36.5

27.3

27.9

24.8

25.6

25.6

25.6

F3

22.5

25.8

19.6

19.8

15.7

16.6

16.6

16.6

F4

23.2

27.5

20.2

20.7

16.5

16.6

16.6

16.6

F5

25.2

26.8

20.5

20.3

20.8

23.4

23.4

23.4

F6

21.5

26.0

19.3

20.9

16.4

16.6

16.6

23.4

Note MCHR is the maximum horizontal compression reaction; MBM and TMBM are the maximum bending moment and theoretical maximum bending moment respectively. The final failure stage of FF 5 is defined as the total loss of resistance capacity

Horizontal reaction (kN) Load on sub-frames (kN)

3.1 Studies on DIF of Beam-Column Sub-structures

71

40 F2 F3 F4 F5 F6 HT2 HT3 HT4 HT5 HT6 HL6

30 20 10 0 -10 -20 -30 0

100

200

300

400

500

Vertical displacement (mm)

Fig. 3.6 Vertical/horizontal load–displacement curves

Fig. 3.7 Failure mode of specimen F3

axial force. However, BENC formed its first bending crack at a load of 10.0 kN. This indicates that fasting action is the main load distribution mechanism within the elastic response range of specimens. Shear cracks appeared in the joints under 21.0 kN load, while plastic hinges appeared in the joints under 22.5 kN load. The deflection of the yield load is 28.9 mm. The ultimate bearing capacity of F3 is 25.8 kN and the displacement is 44.0 mm. With the further increase of the vertical displacement, the vertical load resistance begins to decrease. When the joint shear crack expands, the strain of the longitudinal reinforcement at BENC begins to decrease, while that at BENF increased rapidly. This indicates that the joint resistance mechanism is changing to the cantilever mechanism, indicating that the cantilever redistribution mechanism dominates the load redistribution after the severe shear failure of the joint.

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3 Dynamic Increase Factor of Concrete Structures

When the vertical deflection reached 275.9 mm, the vertical load resistance begins to rise again, which is caused by the catenary action of the internal development of the beam. At 456.2 mm deflection, the load resistance is 11.9 kN, while at 461.3 mm, the load resistance is suddenly reduced to 0.0 kN as the longitudinal reinforcement at the top of the beam near the end of the fixed support is fractured. The maximum horizontal compressions of the transverse and longitudinal beams of F3 were 19.6 and 19.8 kN, respectively. However, the horizontal tension is small, indicating that catenary action is developed marginally. Seismic design details F2 is designed with seismic resistance, and details of dimensions and reinforcement are shown in Table 3.2. The crack width at the bottom of BENC did not change after serious cracks occurred in the joints for F3. For F2, more cracks were formed at the bottom of BENC, and the more serious of the corner joint was, the wider the crack was. Another significant difference in failure mode between F2 and F3 is that when the displacement is 280 mm, the core concrete of the joints remains relatively intact due to the effective restriction of the transverse reinforcement of the joints in F2. The higher the ratio of longitudinal reinforcement in the beam, the higher the first yield load and ultimate bearing capacity of the specimen, and the higher the ratio of transverse reinforcement in the plastic hinge zone of the beam, the later the spalling of the concrete and buckling of the compressive longitudinal reinforcement at the bottom of the BENF. The ultimate bearing capacity of F2 is 36.5 kN, which is 141.5% of F3. The maximum horizontal compressions of the transverse and longitudinal beams of F2 are 27.3 and 27.9 kN, respectively. The failure mode of F3 is shown in Fig. 3.7. Design span length F5 has a net span of 2775 mm, while the F3 has a net span of 2175 mm. The span ratio of the two specimens is 1.0. Dimensions and reinforcement details are shown in Table 3.2. F5 has a higher initial stiffness than F3. The diagonal shear cracks of joints appeared when the load on F5 is 14.3 kN, which is much lower than that on F3. As shear cracks in F5 joints develop earlier and faster than those in F3 joints, limited bending cracks can be seen at the bottom of BENC. The ultimate bearing capacity of F5 is 26.8kN, 3.9% higher than that of F3. However, it should be emphasized that the design axial force of the corner column in F5 is 29.1 kN based on DoD (2009). Thus, even with a dynamic magnification factor of 1.0, F5 cannot survive without the corner column. The failure mode of F5 is shown in Fig. 3.8. Aspect ratio The beam span of F6 is different in different directions. Dimensions and reinforcement details are shown in Table 3.2. For F6, the crack development of longitudinal and transverse beams is significantly different and needs to be described separately. The first crack appeared in the transverse and longitudinal beam under the loads of 5.9 kN and 10.0 kN. Bending cracks first appeared in BENC under the loads of 10.0 kN and 20.0 kN. Asymmetric joint shear cracks were observed. Transverse shear

3.1 Studies on DIF of Beam-Column Sub-structures

73

Fig. 3.8 Failure mode of specimen F5

Fig. 3.9 Failure mode of specimen F6

cracks appeared at the joint under the loads of 17.8 kN, and longitudinal shear cracks appeared at the joint under the load of 19.6 kN. Although the occurrence time of longitudinal crack is later than that of transverse crack, the development speed of longitudinal crack is faster than that of transverse crack. The ultimate load bearing capacity of F6 is 26.0 kN, which is 100.8% of that of F3. The maximum transverse and longitudinal horizontal compressive reactions are 19.3 and 20.9kN, respectively. When the vertical displacement is further increased to 120 mm, concrete spalling is observed in the transverse BENF, while the bottom compressive zone of the longitudinal beam remains intact. Concrete spalling was first observed on the longer beam with a deflection of 200 mm. The failure mode of F6 is shown in Fig. 3.9.

3.1.4.2

Results of Strain Gauge

Figure 3.10 shows the strain distribution of longitudinal reinforcement of the beams of F3 at different performance levels. PL1, PL2, PL3, PL4 and PL5 in the figure

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3 Dynamic Increase Factor of Concrete Structures

6000

Top strain gauges 250

Strain gauge (με)

5000

250

250

250

250

250

4000 3000 PL1 PL2 PL3 PL4

2000 1000 0 -1000

F3-Top-Rebar

-2000

0

250

500

750

1000 1250

1500

1750 2000

Displacement from beam-corner column interface (mm)

(a) stain of the top rebars 4000 3000

Strain gauge (με)

2000

250

250

250

250

Bottom strain gauges

250

250

1000 0 PL1 PL2 PL3 PL4

-1000 -2000 -3000

F3-Bottom-Rebar

-4000 0

250

500

750

1000 1250 1500 1750 2000

Displacement from beam-corner column interface (mm)

(b) strain of the bottom rebars Fig. 3.10 Strain gauge results of specimen F3

respectively represent the first bending crack, the first yield of longitudinal reinforcement of the beam, the ultimate bearing capacity, the normal failure stage (the bearing capacity drops to 75.0% of the ultimate bearing capacity) and the vertical load resistance starts to rise again. For F3, at BENF, the strain of the top longitudinal reinforcement is tensile, while at BENC, the strain of the bottom longitudinal reinforcement is compressive. This is consistent with the results of crack pattern, which indicates that flexural action dominates the redistribution of load when the specimen is in elastic response. After PL3, the inflection point (zero strain point) of the upper and lower longitudinal bars moves to the corner joint. This indicates that the load resisting mechanism of the specimens has changed to the redistribution mechanism

3.1 Studies on DIF of Beam-Column Sub-structures

75

of the cantilever beam after the severe damage of the corner joints. During the test, the strain of the bottom longitudinal reinforcement of BENF yielded at PL4, while the strain of the bottom longitudinal reinforcement of BENC never yielded during the test. In general, the strain distribution of F2 is similar to that of F3. For F2, the bottom longitudinal reinforcement at BENC yielded at PL3. In F3, however, the strain never yielded at BENC. Figure 3.11 shows the strain measurement results of column longitudinal reinforcement and joint shear reinforcement of F2. It should be noted that rebar C2 is a compressive bar if a 2D longitudinal frame is considered, while it is a tensile bar if a 2D transverse frame is considered. Thus, the strain of C2 was limited. In contrast, C1 and C4 are tensile and compressive in two 2D frames, respectively. Therefore, when only3D frames are considered, the net strain is much larger than that when only 2D frames are considered. As shown in the figure, the strain of the transverse reinforcement of the joint is initially limited. The strain increased rapidly after the first diagonal shear crack appeared at the corner joints. One consequence of shear force is the expansion of the core concrete. The lateral reinforcement of the joint restrains the expansion of the joint to some extent and increased the strain. Finally, the strain of transverse bars of joints remains constant and the displacement increased further. The maximum strain of the transverse bars of the joints is 2380 μE. This indicates that the transverse reinforcement of the joint has not yielded. For other specimens, due to non-seismic design, there is no transverse reinforcement at the joints. 5000

C1 C2 C3 C4 J1

Strain gauge (με)

4000 3000

C3 J1

C4 C2 C1

2000 1000 0

-1000 -2000 0

100

200

300

400

Vertical displacement (mm) Fig. 3.11 Strain gauge results of specimen F2

500

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3 Dynamic Increase Factor of Concrete Structures

3.1.5 Discussion of the Test Results 3.1.5.1

Tie-Strength Approach

The Department of Defense (2009) design guidance is a revised version of the previous Department of Defense (2009) guidance, incorporating some improvements. A significant modification of DoD (2009) from DoD (2005) is that horizontal tie-forces (internal and external) are no longer allowed to be concentrated on beams, girders, and spandrels (unless the designer can demonstrate that these members can withstand tensile loads when subjected to large rotations (i.e. 0.2 radians). The final rotations of most beams in the tested specimens were close to 0.2 rad. Thus, the beams can be used to carry the required peripheral tie strength. The peripheral tie strength required is: Fp = 6wF L1 Lp

(3.1)

Allowable tie strength refers to the maximum horizontal tension that can be generated when only the reinforcement bar on the top of the beam is considered. According to DoD (2009), sufficient tie force can be provided to meet the requirements of the peripheral tie strength. However, due to the partial rotation constraints of the corner joints, the measured maximum tensile force is significantly less than the required peripheral tensile strength, and the corner joints can only provide limited horizontal constraints. Therefore, it is very unsafe to use the tensile strength method to resist the progressive collapse of reinforced concrete frames caused by the loss of corner columns. Enhancing the local resistance of the corner column may be an effective alternative (Table 3.5). Table 3.5 Tie-force from test and DoD (2009) Test

RTTB (kN)

RTLB (kN)

ATTB (kN)

ATLB (kN)

MTTB (kN)

MTLB (kN)

F2

55.7

55.7

122.1

122.1

11.1

11.3

F3

55.7

55.7

72.2

72.2

7.9

7.5

F4

55.7

55.7

72.2

72.2

7.5

7.5

F5

69.7

69.7

72.2

72.2

4.3

3.1

F6

55.7

69.7

72.2

72.2

6.9

1.1

Note RTTB and RTLB = required tie-force in the beams in transverse and longitudinal direction, respectively; ATTB and ATLB = allowable tie-force in beams in transverse and longitudinal direction, respectively; MTTB and MTLB = measured tie-force in beams in transverse and longitudinal direction, respectively

3.1 Studies on DIF of Beam-Column Sub-structures

Bending moment (kN.m)

25

77

Transverse fixed support Longitudinal fixed support

20

Theoretical Ultimate Moment =16.6 kN·m

15 a

10

b

5 0 0

100

200 300 400 Vertical displacement (mm)

500

Fig. 3.12 Bending moment at fixed support of F3 versus vertical displacement

3.1.5.2

Properties of Plastic Hinge

Figure 3.12 shows the relationship of the bending moment of the beam ends near the fixed support and the vertical displacement of F3. One can see from the figure, with the increase of vertical displacement, the bending moments of the two beam ends begin to increase. When the displacement reached 28.9 mm, the bending moment of the two beam ends is almost unchanged with the increase of the vertical displacement. The measured bending moments of the lateral and longitudinal beam ends near the fixed supports begin to decrease when the displacements up to 130.0 and 140.0 mm, respectively. This is ascribed to the concrete crushing occurred of the beam ends near the fixed supports. The measured and theoretical maximum bending moment of each beam end is shown in Fig. 3.12. The comparison between the measured maximum bending moment of each beam and the theoretical value obtained by ASCE 41-06 (2006) shows that the overstrength factor recommended by DoD (2009) in ASCE 41-06 (2006) is slightly overestimated. In addition, both nonlinear static processes (NSP) and nonlinear dynamic processes (NDP) require a correct definition of the performance of plastic hinges. The current version of DoD (2009) adopts the plastic hinge modeling parameters from ASCE 41-06 (2006) for beam elements. Modeling parameters of the test specimens were compared with DoD (2009)’s recommendations. As shown in Table 3.6, the measured values of parameter a are close to those recommended by DoD (2009). However, the suggested value of parameter B in DoD (2009) is very conservative. The definitions of parameters a and b are shown in Fig. 3.12.

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3 Dynamic Increase Factor of Concrete Structures

Table 3.6 Comparison of the parameter of plastic hinge Test

a (rad) Beam-T

b (rad) Beam-L

DoD

Beam-T

Beam-L

DoD

F2

0.058

0.063

0.063

0.198

0.198

0.10

F3

0.046

0.046

0.05

0.146

0.177

0.06

F4

0.052

0.052

0.05

0.143

0.161

0.06

F5

0.042

0.042

0.05

0.134

0.134

0.06

F6

0.040

0.038

0.05

0.137

0.140

0.06

3.1.6 Dynamic Test Results A total of six beam-column specimens with different design details and span lengths were tested to evaluate the dynamic redistribution performance of reinforced concrete frames at the background of a ground corner column is suddenly removed. The test results of the six specimens are as follows.

3.1.6.1

Reliability of the Column Removal Device

The column removal device designed in this study involves the simulation of the sudden removal of the column under extreme loads with a heavy hammer. The removal time of corner supports has a great influence on the accuracy of dynamic response of the substructure, so the effectiveness of the design device must be guaranteed. The axial force–time history of the corner column is monitored by a force transducer (Item 4 in Fig. 3.13). As shown in Fig. 3.13, the axial force on corner columns DF1 and DF2 was −16.9 kN initially (negative represents compression force). At 0.2 s, the axial force begins to release (the compression force begins to decrease). The axial forces of the corner columns of DF1 and DF2 are fully released at 0.2035 and 0.2030 s, respectively. Thus, the duration of the force release was 0.0035 and 0.0030 s for DF1 and DF2, respectively. This proves that the designed device meets the requirements of the DoD (2009) guideline, which requires that the duration of column removal is less than one-tenth of the natural period of vertical movement of the structure after column loss (the natural period of vibration of DF1 and DF2 is about 0.15 s).

3.1.6.2

Crack Pattern and Failure Mode

As shown in Fig. 3.14, asymmetrical failure modes were observed for DF1. More severe diagonal shear cracks and slight concrete spalling appeared at the beam end near the fixed support (BENF), while narrower diagonal shear cracks appeared at the longitudinal BENF. Hairline bending cracks were observed in BENC, while

3.1 Studies on DIF of Beam-Column Sub-structures

79

Axial force in the corner column (kN)

0.0

DF1 DF2

-3.0 -6.0 -9.0 -12.0 -15.0 -18.0 0.18

0.19

0.20

0.21

0.22

0.23

0.24

0.25

Time (s) Fig. 3.13 Recorded reaction force history in the corner column during test

Fig. 3.14 Crack patterns and failure mode of specimen DF1

symmetrical hairline shear cracks were observed in corner joints. This shows that the direction of the bending moment in BENC, which was initially negative under gravity, reversed after removing the corner support. For DF2, only some flexural cracks appeared in BENF (see Fig. 3.15). the reversal of direction of bending moment was also observed in BENC, similar to DF1. However, there is no crack at the corner joint of DF2. Note that, DF1 and DF2

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3 Dynamic Increase Factor of Concrete Structures

Fig. 3.15 Crack patterns and failure mode of specimen DF2

with the same load combination of 0.9(1.2DL + 0.5LL) prior to the sudden removal of corner supports. The dimensions and reinforcement details of DF3 were similar to that of DF1. For DF3, the load combination of 1.0(1.2DL + 0.5LL) was used. As shown in Fig. 3.16, not only severe bending but also shear cracks were observed in the BENF. However, only mild concrete cracking at the bottom of BENF. Compared with DF1, more flexural cracks in BENCs and more serious diagonal shear cracks developed in DF3. Both DF1 and DF3 gained new balance after the tests, although DF3 suffered more severe damage owing to the higher load combination. Compared with DF3, the transverse reinforcement ratio in the beam ends of the beam of DF4 was higher. As shown in Fig. 3.17, similar crack patterns were observed for DF4 and DF3. However, in the BENFs and corner joints, the crack width is slightly smaller than DF3. DF5 has a larger design span than DF3. Dimensions and reinforcement details are shown in Table 3.2. DF5 was subjected to a 0.8(1.2DL + 0.5LL) load combination before corner support was removed. As shown in Fig. 3.18, DF5 was severely damaged after the corner support was removed. Extremely wide shear cracks and bending cracks appeared in BENF, and corner joints showed obvious spalling. Compared with BENF, the damage of BENC was negligible. It should be noted that the collapse of DF5 is stopped by the pin support of the column removal device when the maximum displacement exceeded the allowable displacement of approximately 360 mm. The DF5 can be predicted to collapse without pin support. DF6 has unequal longitudinal and transverse spans. Dimensions and reinforcement details are shown in Table 3.2. The DF6 was subjected to a load combination of 1.0(1.2DL + 0.5LL) before the sudden corner support removal. As shown in

3.1 Studies on DIF of Beam-Column Sub-structures

81

Fig. 3.16 Crack patterns and failure mode of specimen DF3

Fig. 3.17 Crack patterns and failure mode of specimen DF4

Fig. 3.19, DF6 collapsed after removing the corner supports. Asymmetrical damage mode was observed for longitudinal and transverse beams. Extreme damage was concentrated in transverse BENF. The longitudinal BENF damage of DF6 was much milder than DF5. As shown in Fig. 3.19, serious cracks appeared at the corner joint. These cracks were asymmetrical due to the unequal spans of the beams.

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3 Dynamic Increase Factor of Concrete Structures

Fig. 3.18 Crack patterns and failure mode of specimen DF5

Fig. 3.19 Crack patterns and failure mode of specimen DF6

3.1.6.3

Displacement Response

Figure 3.20 presents the time histories of vertical displacement of corner joints. Negative values indicate downward displacement. All corner supports were removed at 0.2 s. At 0.38 s, the first peak vertical displacement of −83.9 mm was measured for DF1. The downward displacement was decreased to −79.3 mm, at 0.48 s, but

3.1 Studies on DIF of Beam-Column Sub-structures

83

Vertical Displacement (mm)

50

0 -50 -100 DF1 DF2 DF3 DF4 DF5 DF6

-150 -200

-250 0.00

0.25

0.50

0.75

1.00

1.25

1.50

Time (s) Fig. 3.20 Recorded history of the vertical displacement of the corner joint of each specimen

then rose to −82.6 mm at 0.55 s. The vibration is basically eliminated within 1.5 s, and the permanent downward displacement of the substructure is −81.3 mm, 96.9% of the first peak displacement. The first peak displacement measured for DF2, DF3 and DF4 is −29.0, −146.6 and −99.4 mm, respectively. It should be noted that the peak displacement of DF5 and DF6 exceeded −300 mm, which is the measurement capability of the LVDT placed at the corner joint. In addition, the vertical movement of the two specimens was stopped by the pin support, and the distance from the bottom of the corner column to the pin support was measured to be about 360 mm. The vertical displacement distribution of DF1 along the transverse beam is shown in Fig. 3.21. It was found that the maximum displacements of DF1 at D1, D2, D3 and D4 are −83.9, −73.4, −46.8 and −20.0 mm, respectively. This shows that the distribution of deformation is nonlinear, and the deformation mode of the beam is different from that of the cantilever beam because the corner joints exert partial rotational constraints.

3.1.6.4

Vertical Reaction Force Response

Figure 3.22 shows the recorded time-history of total vertical reactions for DF3 and DF4. For DF3, the initial total vertical reaction measured in the longitudinal and transverse fixed supports was −20.4 kN. The vertical reaction force suddenly increased to −44.5 kN at 0.29 s. After 0.70 s, the total reaction force vibrates along a straight line, and the value is −39.1 kN. Similar behavior was observed in DF4. It can be seen from Table 3.2 that the axial forces of DF3 and DF4 in the corner column are −18.7 and −18.8 kN respectively. The maximum reaction force increased in the fixed supports was −24.1 and −25.8 kN for DF3 and DF4, respectively. In other words, the increased maximum reaction force is significantly greater than the initial

84

3 Dynamic Increase Factor of Concrete Structures 20

DF1-D1 DF1-D2 DF1-D3 DF1-D4

Vertical displacement (mm)

0 -20 -40 -60 -80 -100 0.00

0.25

0.50

0.75

1.00

1.25

1.50

Time (s) Fig. 3.21 Recorded vertical displacement distribution of DF1

-10

Total reaction force (kN)

DF3

DF4

-20

-30

-40

-50 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Time (s) Fig. 3.22 Recoded history of the total vertical reaction force of DF3 and DF4

axial force in the corner support. The measured inertia forces of DF3 and DF4 are 5.4 and 7.0 kN, respectively. In addition, if the dynamic effect of total reaction force enhancement is concerned, the dynamic increase coefficients of DF3 and DF4 are 1.29 and 1.37, respectively. A further discussion of dynamic effects is shown below.

3.1 Studies on DIF of Beam-Column Sub-structures

85

10

Horizontal reaction force (kN)

DF3 DF4 0

-10

-20

-30 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Time (s) Fig. 3.23 The history of horizontal reaction of DF3 and DF4 beams is recorded

3.1.6.5

Horizontal Reaction Force

Figure 3.23 shows the history of measured horizontal reactions in the beam fixed supports of DF3 and DF4. For DF3, a tensile horizontal reaction of 4.1kN was measured before removing the corner column. After the sudden removal of the support, the horizontal reaction plummets to the peak of −20.4 kN at 0.37 s. The compressive reaction force was kept at −14.7kN after vibration. Similar behavior has been documented in DF4. Therefore, it can be concluded that no horizontal tension was measured during the dynamic process, and therefore DF3 and DF4 did not form catenary action against progressive collapse in the test. This may be because the deformation of both specimens was not sufficient to exert the catenary to resist progressive collapse. Alternatively, the corner column did not provide sufficient horizontal constraints to develop catenary action in the beam.

3.1.6.6

Bending Moment Response

The bending moment history of DF3 and DF4 at the longitudinal and transverse fixed supports are shown in Figs. 3.24 and 3.25 respectively. As stated in the instrument section, the bending moment is calculated according to the axial force history of the two vertical tension/compression load sensors in each fixed support, and its expression is: M = (R1 − R2 ) × 0.5D

(3.2)

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3 Dynamic Increase Factor of Concrete Structures

Bending moment (kN.m)

25

20

15

10 Longitudinal beam 5

Transverse beam

0 0

0.5

1

1.5

2

2.5

3

Time (s) Fig. 3.24 Bending moment at supports of specimen DF3

Bending moment (kN.m)

25

20

15

10

5

Longitudinal beam Transverse beam

0 0

0.5

1

1.5

2

2.5

3

Time (s) Fig. 3.25 Bending moment at supports of specimen DF4

For DF3, it can be seen from Fig. 3.25 that the initial bending moments at the longitudinal and transverse supports are 2.66 and 2.45 kN·m respectively. After the removal of the corner support, the bending moments of the longitudinal and transverse fixed supports suddenly increased to 20.8 and 20.1 kN·m, respectively. After free vibration, the permanent bending moments of the longitudinal and transverse fixed supports are 16.6 and 16.3 kN·m, respectively. It can be seen that after the removal of the corner support, the bending moment of the longitudinal and transverse

3.1 Studies on DIF of Beam-Column Sub-structures

87

fixed bracket increased sharply by 682.0% and 720.5%. For DF4, the initial bending moment of longitudinal and transverse supports is 2.79 and 2.51 kN·m respectively. After removing the corner support, the bending moment of the longitudinal and transverse fixed support increased sharply to 20.1 and 19.5 kN·m. After free vibration, the permanent bending moment recorded on the longitudinal and transverse fixed supports is 14.0 and 14.8 kN·m. It can be seen that the bending moments of the longitudinal and transverse fixed supports increase sharply by 620.0 and 676.9%, respectively.

3.1.6.7

Strain of Rebar

In general, the variation trend of strain gauge results of all specimens are similar. Therefore, only the results of DF1 are explained in detail. As shown in Fig. 3.26, the strain at the top of BENC (BSTL1) is 360 με. At 0.41 s, the tensile strain decreases gradually and finally reached the maximum compressive strain of −1001 με. The initial strain of BSBL1 is −163 με. After removing the corner support, the compressive strain of BSBL1 suddenly drops to zero, and finally reached the maximum tensile strain of 2488 με at 0.42 s. Both results verify that the direction of the bending moment in BENC reverses after removing the corner support. It can also be seen that the strain reading decreases significantly after 0.5 s. The permanent strains of BSTL1 and BSBL1 at the end of vibration are −723 and 1731 με, respectively. The initial strain readings of BSTL8 and BSBL8 are 578 and −205 με, respectively. After removing the corner support, the readings of BSTL8 and BSBL8 increased to 4487 and −1817 με at 0.41 and 0.42 s, respectively. This shows that the bending moment increased by a considerable amount for BENF (agree with the bending moment result). Similar behavior has been observed in other specimens. Figure 3.27 shows the strain gauge readings for DF1 and DF2 transverse reinforcements. As shown in the figure, the strain of BST4 is 430 με at the beginning and increased to 1807 με at 0.42 s. The permanent tensile strain of BST4 at the end of vibration is 1425 με. The yield strain of steel bar R6 is 2650 με. Therefore, the transverse reinforcement did not yield in the test. The peak strain of BST4 is only 1451 με, which is much smaller than that of BST4 under DF1, this can be attributed to to the higher transverse reinforcement ratio in the beam ends.

3.1.7 Discussion of Dynamic Test Results 3.1.7.1

Seismic Details

The non-seismically designed specimen DF1 had serious flexural and shear cracks in BENFs, while the seismically designed specimen DF2 only had limited flexural cracks in BENFs. In addition, no cracks appeared at the corner joints of DF2, while some small diagonal cracks appeared at the corner joints of DF1. By comparing the

88

3 Dynamic Increase Factor of Concrete Structures 6000 5000

BSTL8

BSTL1

BST4

BST1

Strain ( με )

4000

BSBL8

BSBL1

3000

BSTL8

2000 1000

BSBL1

0

BSTL1

-1000 -2000 -3000 0.0

BSBL8

0.5

1.0

1.5

2.0

2.5

3.0

2.5

3.0

Time (s)

Strain (με )

Fig. 3.26 Strain gauge recording of the flexural reinforcement in DF1

2000 1800 1600 1400 1200 1000 800 600 400 200 0

BST4 in DF1

BST4 in DF2

BSTL8

BSTL1

BST4

BST1

BSBL8

BSBL1

0.0

0.5

1.0

1.5

2.0

Time (s) Fig. 3.27 Strain gauge recording of the transverse reinforcement in DF1 and DF2

peak vertical displacement of the two specimens, it can be seen that the peak vertical displacement of DF2 is only 34.6% of that of DF1. A similar trend was observed for acceleration and velocity results.

3.1.7.2

Various Service Load Conditions

Before the corner supports were removed, DF3 was subjected to a full load of 1.0(1.2DL + 0.5LL) while DF1 was subjected to a reduced load combination of 0.9(1.2DL + 0.5LL). By comparing the crack pattern of DF1 and DF3, it is found that the damage of the corner joints and longitudinal BENF of DF3 was more severe.

3.1 Studies on DIF of Beam-Column Sub-structures

89

The shear cracks in transverse BENF of DF1 are much more than those in DF3. This is an unexpected phenomenon, which may be ascribed to the inherent defects of DF1, such as the formation of honeycomb structure in BENF due to insufficient vibration during casting. DF3 increased the peak vertical displacement by 74.7%.

3.1.7.3

Improved Details

Compared with DF3, the potential plastic hinge zone of the beam of DF4 has a higher transverse reinforcement ratio. Similar to DF3, DF4 carried a full-service load of 1.0(1.2DL + 0.5LL) before removing the corner support. By comparing the crack pattern of DF4 and DF3, it can be seen that the crasks in the BENF and corner joint of DF4 are mild, but the crack patterns of DF4 and DF3 are similar in general. In addition, the peak vertical displacement of DF4 decreased by 32.2%.

3.1.7.4

Design Span

DF5 has a net span of 2775 mm and carried a service load of 0.8(1.2DL + 0.5LL). As shown in Fig. 3.18, DF5 failed and extensive damage occurred in BENF and corner joints. The peak vertical displacement exceeded 360 mm. In fact, if there is no pin support for the column removal device, or if there is enough room for the specimen to fall free, the specimen will fail completely.

3.1.7.5

Span Aspect Ratio

The two beams of DF6 have different span lengths, and its service load was reduced by 0.9(1.2DL + 0.5LL). Figure 3.19 shows extensive damage to the transverse BENF. Although severe cracking occurred in the longitudinal BENF, it was much less severe than in the transverse beam. Compared with DF3, DF6 collapse due to the larger longitudinal beam span. The axial force, previously resisted by the corner column, attempted to be redistributed to the two fixed supports after the sudden removal of the corner support. However, due to the similar stiffness of the longitudinal and transverse beams of DF3 and DF6, almost half of the axial force will be redistributed to the transverse fixed support. Therefore, compared with DF3, more axial force was redistributed to the transverse fixed support. This caused the beams of DF6 to collapse completely, while DF3 suffered less damage.

3.1.8 Dynamic Load Increase Factor As can be seen from Table 3.4 and Fig. 3.28, the static load capacity of F2, F3, F4, F5 and F6 are 36.5, 25.8, 27.5, 26.8 and 26.0 kN, respectively. The axial forces of DF1,

90

3 Dynamic Increase Factor of Concrete Structures 40 F2(DF2) F3(DF1,DF3) F4(DF4) F5(DF5) F6(DF6)

Load on substructures (kN)

35 30 25 20 15 10 5 0 0

100

200

300

400

500

Vertical displacement (mm) Fig. 3.28 Load–displacement curves of the F-series specimens

DF2, DF3, DF4, DF5 and DF6 before removing the corner column are 16.9, 16.9, 18.7, 18.8, 23.3 and 23.2 kN, respectively. The dynamic test results showed that DF1, DF2, DF3 and DF4 survived the removal of corner support. It can be concluded that the dynamic load increase factor (DLIF) of F2, F3 and F4 is less than 2.16, 1.38 and 1.46, respectively. It should be noted that the DLIF is defined as the ratio of the static load capacity to the dynamic load capacity. Conversely, DF5 and DF6 completely collapsed after the removal of the support supports. Therefore, we found that the DLIF of F5 and F6 specimens was greater than 1.15 and 1.12, respectively.

3.2 Study on DIF of Beam-Column-Slab Sub-structures Only beam-column sub-structures are considered in previous chapter, note that the absence of RC slab may lead to over-conservative estimation. Therefore, it is inevitable to study the behavior of beam-column-slab sub-structures under progressive collapse.

3.2.1 Specimen Design As shown in Fig. 3.29, based on beam-column sub-structure specimens F3 (named as F1 hereafter for easy comparison) and F2, RC slabs are designed for specimens S1 and S2. Moreover, specimens DS1 and DS2, which are the same as specimens S1 and S2, are cast to conduct dynamic tests. In summary, a total of eight beam-column sub-structure or beam-column-slab specimens are introduced in this chapter, among

3.2 Study on DIF of Beam-Column-Slab Sub-structures

91

Fig. 3.29 Dimensions, cross-section details, and strain gauge locations of Specimens S1 and S2 (in mm)

which specimens F1, F2, S1, and S2 are cast to conduct static test, while their counterpart specimens DF1, DF2, DS1, and DS2 are cast to conduct dynamic test. Specimens F1 and S1 are designed non-seismically while Specimens F2 and S2 follow seismic design. All those specimens are one-third scaled. Detailed geometric properties and reinforcing layout of the beam-column-slab sub-structures are shown in Figs. 3.29 and 3.30. The sectional dimensions of the beam and column are the same as that in Sect. 3.1.1, whereas the thickness of the slab is 70 mm.

92

3 Dynamic Increase Factor of Concrete Structures

Fig. 3.30 Details of plate bars and strain gauge positions of specimens S1 and S2 (unit: mm)

(a) slab top reinforcement

(b) slab bottom reinforcement

3.2 Study on DIF of Beam-Column-Slab Sub-structures

93

3.2.2 Material Properties The average compressive strength of concrete obtained from the concrete cylinder samples was found to be 32.0, 32.0, 31.9, 32.1 MPa, 31.6, 31.5, 32.8, and 31.4 MPa for S1, S2, F1, F2, DS1, DS2, DF1, and DF2, respectively. Properties of Rears are listed in Table 3.4.

3.2.3 Test Setup and Instrumentation Test setup and instrumentation are the same as that introduced in Sect. 3.2.3 and as shown in Fig. 3.31. Different from beam-column specimens, the columns of beamcolumn-slab specimens are fixed to strong steel legs by bolts. In order to simulate a load case of 1.0(1.0DL + 0.25LL), a total of 7000 kg weights is placed on the slab surface of specimens DS1 and DS2.

3.2.4 Quasi-static Test Results 3.2.4.1

Global Behavior and Failure Mode

The global behavior and failure mode of specimens F1 and F2 are discussed in Sect. 3.1.2, and therefore, they are not presented in this chapter for the sake of Fig. 3.31 Test setup of DS-series of specimens

1

2 7 3 8

5 6

4

94

3 Dynamic Increase Factor of Concrete Structures

brief. Critical results of quasi-static test specimens F1, F2, S1, and S2 are listed in Table 3.4. S1—The measured vertical load and horizontal reaction forces vary with the vertical displacement of corner joint as shown in Fig. 3.32. Bending cracks occurred when the load was 18.0 kN. No cracks were observed in the beam and corner joint areas. Under the load of 22.0 kN, a small number of bending cracks appeared in the beam end near the adjacent column (BENAC), and the first diagonal crack in the RC slab formed and passed through the center of the slab. Under the load of 24.0 kN, the first shear crack appeared in the corner joints. When the applied load increased further, more diagonal cracks are formed in the plate. When the vertical displacement reached 43.0 mm, which is equivalent to 2.0% of the beam top displacement ratio (TDR), which is defined as the ratio of the vertical displacement of the center of the corner column to the net span of the beam. The specimen achieved the maximum horizontal compression load of 21.0 kN. When the vertical displacement reached 56.0 mm, equivalent to 2.6% of TDR, the ultimate bearing capacity of specimens reached 39.1 kN, equivalent to 116.0% of the load required to resist progressive collapse stipulated in GSA (2003). When the applied load increased further, more diagonal cracks are formed in the slab. When the vertical displacement reached up to 43.0 mm, which is equivalent to 2.0% of the TDR. When displacement reached 200 mm, equivalent to 9.2% of the TDR, diagonal cracks in the slab penetrated the depth of the slab and the load–displacement curve began to rise again owing to tensile membrane action. Figure 3.33 shows the development of crack pattern of S1 slab at different performance levels.

Vertical load(kN)

50 40 30 20

Horizontal reaction (kN)

10 0 -10 F1 -20

S1

-30 0

100

200

300

400

500

Vertical displacement (mm)

Fig. 3.32 Curve of vertical load and horizontal reaction force of F1 and S1 with displacement

3.2 Study on DIF of Beam-Column-Slab Sub-structures

Corner Joint Stub

Corner Joint Stub

Corner Joint Stub

95

Corner Joint Stub

Corner Joint Stub

Fig. 3.33 Cracking patterns at different performance levels of S1

S2—The relationship between measured vertical load and horizontal reaction forces and the vertical displacement of S2 corner joint is shown in Fig. 3.34. In general, the crack development of S2 is similar to that of S1, and the key points of the test results are shown in Table 3.4. Therefore, only the most important differences between these two specimens are highlighted. For S1, the slab forms the first diagonal crack under a load of 22.0 kN. However, for S2, under a load of 34.0 kN, diagonal cracks in the slab formed and passed through the center of the plate. For S2, the concrete of the joint began to crack at a displacement of 260.0 mm, much later than S1, because the shear bars in the joint maintained the strength of the diagonal compression strut by passively constrained the jointed concrete core. When the vertical displacement reached69.4 mm, equivalent to 3.2% of TDR, the ultimate bearing capacity of specimens reached 52.0 kN, equivalent to 154.0% of the load resistance specified in GSA (2003). When the vertical displacement reached 218.0 mm, the tensile membrane action is observed in the load–displacement curve. Figure 3.35 shows the crack pattern development of S2 at different performance levels.

3.2.4.2

Comparison of Test Results

The ultimate load bearing capacity of F1, F2, S1 and S2 were 25.8, 36.5, 39.1 and 52.0 kN, respectively. The ultimate load bearing capacity of S1 and S2 (with slab) was increased by 51.6 and 42.5%, respectively, compared with that of F1 and F2 (without slab), indicating that it is extremely conservative to ignore the contribution of RC slabs to progressive collapse resistance. In the S-series specimens, the flanges of the L beam significantly increased the flexural capacity of the beam. Figure 3.36 shows the contribution of slabs and beams of the S-series specimens. For S1, reinforced concrete slabs initially bear only 4.8% resistance. With the increase of displacement, the bearing capacity contribution of slab increased to 31.2%. After that, as cracks appeared in the slab for the first time, the percentage of slab contribution fell to about 26.2%. Secondly, due to plastic hinge forming, the slab contribution increased to 38.6%. After displacement reached 200 mm, the slab contribution increased significantly due to the development of tensile membrane action in the slab. In general, S2 had similar behavior to S1.

96

3 Dynamic Increase Factor of Concrete Structures

Horizontal reaction (kN)

Vertical load (kN)

60 50 40 30 20 10 0 -10

F2

-20

S2

-30 0

100

200

300

400

500

Vertical displacement (mm) Fig. 3.34 Curves of vertical load and horizontal reaction of F2 and S2 with displacement

Corner

Joint

Corner Joint Stub

Corner Joint Stub

Corner Joint Stub

Corner Joint Stub

Fig. 3.35 Cracking patterns at different performance levels of S2 100

90

Frame-contribution

80

Resistance decomposition (%)

Resistance decomposition (%)

90

100

Slab-contribution

70 60 50 40 30 20 10

S1

0

Slab-contribution Frame-contribution

80 70 60 50 40 30 20 10

S2

0 Displacement (mm)

(a) Specimens S1 Fig. 3.36 Decomposition of load resistance

Displacement (mm)

(b) Specimens S1

3.2 Study on DIF of Beam-Column-Slab Sub-structures

3.2.4.3

97

Rebar Strain

Figure 3.37a shows the relationship between the strain of the slab rebar and the vertical displacement of S1. The position of strain gauge is shown in Fig. 3.30. As shown in Fig. 3.37a, ST1 is initially compressed, but it began to decrease when the vertical displacement reached 22.4 mm. In this vertical displacement, diagonal shear cracks appeared in corner joints. When the vertical displacement up to 200.0 mm, ST1 converted to tensile strain. Tensile membrane action developed at this stage. When the displacement reached 90.0 mm, the strain of ST3 suddenly increased beyond the maximum limit. While the strain of ST6 was close to zero during the test, indicating that most of the force received by the corner column at the initial stage was transferred to the adjacent column, and the force transmitted to the inner column can be ignored. Figure 3.37b shows the relationship between the strain of plate bottom reinforcement and vertical displacement. During the test, all the bottom reinforcement strains except SB1 and SB2 were tensile. The strain at SB3, SB5 and SB8 was much larger than that at other strain points because the deformation was mainly concentrated at diagonal crack. The strain distributions of SB5, SB8, SB9 and SB1 indicated that the slab reinforcement strain decreased with the increase of the distance from the adjacent column when the slab developed tensile membrane force. This is one of the main differences between the membrane action formed in the current slab and that in the plate previously tested by the researchers. In the test, it was assumed that all the reinforcing bars at the bottom of the center slab yield after the formation of the tensile membrane. Further experimental and analytical studies are needed to analyze this behavior. The strain of SB6 and SB7 was very small, which is consistent with the observed failure mode, and the damage of the inner semi-triangular slab was limited during the test.

3.2.5 Dynamic Test Results 3.2.5.1

Duration of Corner Support Removal

The removal speed of corner support has a great influence on the accuracy of dynamic response of specimen, so the effectiveness of column moving device must be guaranteed. The axial force history of the corner column was monitored by a force transducer (Item 4 in Fig. 3.5). The change of axial force in the corner support was measured, as shown in Fig. 3.38. Taking specimen DS1 as an example, the axial force initially resisted by the corner support of DS1 was −16.9kN (negative value indicated compression force) The axial force started to release at 0.2 s and decreased to about 0.0 kN at 0.205 s. Therefore, the force release duration of specimen DS1 was 0.005 s. The duration of other specimens was also recorded: specimens DF1, DF2 and DS2 were 0.0035, 0.003 and 0.004 s, respectively. This proved that the designed column moving device meet the requirement of DoD (2009), which required that the duration of column removal is less than one-tenth of the natural period of vertical

98

3 Dynamic Increase Factor of Concrete Structures

4500

ST1 ST2 ST3 ST4 ST5 ST6

4000 3500

Strain (με)

3000 2500 2000 1500 1000 500 0 -500 0

100

200

300

400

500

Vertical displacement (mm)

(a) Top rebar 4000

3000

Strain (με)

SB2 SB5 SB8

SB1 SB4 SB7

3500

SB3 SB6 SB9

2500 2000

1500 1000 500 0 0

100

200

300

400

500

Vertical displacement (mm)

(b) Bottom rebar Fig. 3.37 Strain of slab rebar versus vertical displacement in S1

vibration of the structure (the measured natural period of vertical vibration is about 0.15 s).

3.2 Study on DIF of Beam-Column-Slab Sub-structures

99

Axial force in the corner column (kN)

0.0 DF1 DF2 DS1 DS2

-3.0 -6.0 -9.0 -12.0

-15.0 -18.0 0.18

0.2

0.22

0.24

Time (s) Fig. 3.38 History of axial force in corner support variation with time of test specimens

3.2.5.2

Displacement Behavior

Figure 3.39 shows the vertical displacement response of corner joints of each specimen. Negative values indicated downward displacement. The first peak displacement of DS1, DS2, DF1 and DF2 was −23.0, −17.6, −83.8 and −29.0 mm, respectively. The test results showed that the maximum displacement of DS1 and DS2 was increased by 264.4 and 64.8% respectively without the help of slab to redistribute the force.

3.2.5.3

Crack Pattern

The crack pattern and failure mode of specimen DS1 after dynamic test are shown in Fig. 3.40. Diagonal cracks were observed at the beam-adjacent column interface after the sudden removal of the corner supports. Several bending and diagonal shear cracks appeared at the beam ends near the enlarged adjacent columns. Hairline bending cracks were found at the bottom of the beam end near the corner column. Slight shear cracks appeared at the corner joints. In DF1, severe diagonal shear cracks and slight concrete spalling occurred at the beam end near the fixed support (BENF), while relatively light diagonal shear cracks occurred in the longitudinal BENF (See Fig. 3.14). Thinner bending cracks also appeared in BENC. This showed that the direction of the bending moment in BENC, which was initially negative under gravity (top pull), changed after removing the corner support. Slight shear cracks appeared at corner column joints.

100

3 Dynamic Increase Factor of Concrete Structures

Vertical displacement (mm)

20 0 -20

-40 DF1

-60

DF2 DF3

-80

DF4

-100 0

0.25

0.5

0.75

1

1.25

1.5

Time (s) Fig. 3.39 Displacement response of corner joint

Fig. 3.40 Failure mode of specimen DS1

For DS2, as shown in Fig. 3.41, no significant diagonal crack was developed on the slab after the sudden removal of corner supports. In addition, no crack was found in the BENC and corner joints of DS2. Overall, the damage observed in specimen DS2 was very mild. For DF2, slight bending cracks appeared in the BENF (see Fig. 3.15). Similar to DS2, there was no crack in the corner joint. By comparing the crack pattern of DS-series specimens with the corresponding DF-series specimens, it was further proved that the reinforced concrete slab can achieve a considerable amount of force redistribution and significantly reduced the damage of beams. In addition, the effect of reinforced concrete slabs on non-seismic design frames was more significant than that on seismic design frames.

3.2 Study on DIF of Beam-Column-Slab Sub-structures

101

Fig. 3.41 Failure mode of specimen DS2

3.2.5.4

Rebar Strain

Figure 3.42 shows the comparison of strain gauge responses of DS1 and DF1. The maximum strains of BSTL1, BSBL1, BSTL8 and BSBL8 were −420, 748, 1736 and −1111 με, respectively. This indicated that the longitudinal reinforcement of the beam of DS1 was always in the state of elastic response after the removal of corner support. The maximum strain of DF1 strain BSTL8 was 4487 με, which was significantly higher than the yield strain 2895 με. This is in good agreement with the failure mode and displacement response (significant plastic deformation was observed in DF1, while only limited elastic response was measured in DS1). In addition, the strain at the specific position of the bar was recorded in the dynamic test. The initial compression strain of SB1 was −56 με. After removing the corner support, tensile strain was generated in the beam bottom rebar near the corner joint. This further proved that the bending moment direction of the plate near the corner column changed. For ST3, an initial strain of 86με was measured, which was amplified to 545 με after the sudden removal of the corner support. Both strain gauges showed that the redistribution of axial force in the corner support by reinforced concrete slab cannot be ignored.

3.2.6 Single Degree of Freedom Model A dynamic increasing factor (DIF) is proposed in this chapter, which is defined as the ratio of static ultimate strength (SUS) to dynamic ultimate strength (DUS). SUS for test samples is captured in static tests, but DUS for each sample cannot be determined from a single dynamic test. Therefore, the single degree of freedom (SDOF) model was validated and incremental dynamic analysis was performed on each specimen.

3 Dynamic Increase Factor of Concrete Structures 600 400 200 0 -200 -400 -600 -800 -1000 -1200

6000

BSTL1 BSTL8

BSTL1 BST1

0.5

1

1.5

BSTL1

4000

BSBL1

2

2.5

2000 1000

DF1 DS1

0

3

0

0.5

1

(a) BSTL1

2

2.5

6000

BSBL1

BSBL8

5000

BSTL8

BSTL1 BSBL1

BST1

Strain (με)

BSBL8

1000 500

DF1 DS1

0

BSTL8

BSTL1

BST4

1500

4000

BST4

BSBL1

BSBL8

3000 2000 1000

DF1 DS1

0 0

-500 0

0.5

1

1.5

3

(c) BSTL8

BST1

Strain (με)

1.5

Time (s)

3000

2000

BST4 BSBL8

3000

Time (s)

2500

BSTL8

BST1

BSBL8

DF1 DS1

0

BSTL8

5000

BST4

BSBL1

Strain (με)

Strain (με)

102

2

2.5

3

0.5

1

1.5

2

2.5

3

Time (s)

Time (s) (b) BSBL1

(d) BSBL8

Fig. 3.42 Comparison of the beam reinforcement strain gauge recording of specimens DS1 and DF1

3.2.6.1

Definition of DIF

DIF comes in two types: displacement-based and force-based. Displacement-based DIF is usually defined as: DIF D =

/\d /\st

(3.3)

where DIF D is the displacement-based DIF; /\d and /\st are respectively the dynamic and static displacement under the same load. Force-based DIF is defined as: DIF F =

Fst Fd

(3.4)

where DIF F is the force-based DIF; Fd and Fst are respectively the required static and dynamic force under same displacement. Since progressive collapse is low probability events, preventing complete collapse is the primary consideration, and large deformation is acceptable. Therefore, forcebased DIF is proposed in this study. However, it is understandable that DIF is a function of target displacement (degree of nonlinear behavior). The closer the response is to “elastic” (or close to yield), the higher the force-based DIF value is (close to 2.0). To simplify this problem, the authors propose the use of dynamic load increasing

3.2 Study on DIF of Beam-Column-Slab Sub-structures

103

factor (DLIF) as an alternative measure. Considering the goal of preventing collapse, DLIF can be used as force-based DIF with infinite target displacement. DLIF =

3.2.6.2

SUS DUS

(3.5)

Description of the SDOF Model

As shown in Fig. 3.43, a simple mechanical model of a single-degree-of-freedom (SDOF) vibration system consists of a single mass element, which is connected to a rigid bracket by a linear elastic spring and a viscous damper. The equation of motion of the system can be expressed as me x¨ + ce x˙ + ke x = P(t) − R(t)

(3.6)

where me is the equivalent mass; ce the equivalent viscous damping coefficient; ke the effective stiffness; P(t) and R(t) the applied force and reaction force at corner joint, respectively; and x, x˙ , and x¨ the displacement, velocity, and acceleration of the mass, respectively.

3.2.6.3

Equivalent Mass

The mathematical expression of equivalent mass is: P(t)

Fig. 3.43 Schematic of the SDOF model

P . cex(t)

kex(t)

0

me x(t)

me P(t)

t(s)

R(t) P

R(t) 0

t

t0+Δt

t(s)

104

3 Dynamic Increase Factor of Concrete Structures

{ me =

m(z)[ψ(z)]2 dz +

E

mk [ψ(zk )]2

(3.7)

k

where m(z) is the distributed mass function; ψ(z) is the shape function; mk is the concentrated mass k at location of zk ; ψ(zk ) is the shape function value at point zk .

3.2.6.4

Effective Stiffness

This analytical study presents another challenge, which is to qualify the effective stiffness at different stages. Large deformation is acceptable for progressive collapseresistant design. Therefore, the stiffness of structures with large plastic deformation should be fully considered. There are two ways to consider the plastic stiffness of a structure: Sasani and Sagiroglu (2008) assumed the load–displacement bilinear relationship of a single degree of freedom system with yield force and corresponding yield displacement. The plastic stiffness of the structure is evaluated by strain hardening ratio. Under the bilinear hypothesis, the structure resistance increased gradually with the increase of displacement until it reached the maximum displacement. The maximum displacement is defined differently, including displacement at peak strength, peak degradation of 20 or 50% (nominal strength), and displacement at initial fracture of transverse reinforcement (Priestley and Grant 2005). Calvi et al. (2008) characterized the structure by secant stiffness at the maximum displacement to consider the performance of the structure after yield. Because the above two methods are simple and easy to interpret the results of single degree of freedom analysis, they are often used. However, in order to reproduce the better displacement–time response of each specimen, a more rigorous definition of effective stiffness is required. Figure 3.44 shows the simplified load–displacement curve of the specimen (load resistance function of single-degree-of-freedom analysis). Key points A, B, C, D and E correspond to the first yield, the static ultimate strength, the normal failure stage, the beginning of the development of tensile membrane action or catenary action, and the overall failure stage respectively. As shown in Fig. 3.44, the initial stiffness (segment 0-A) was set as the secant stiffness at the first yield of the structure (Paulay and Priestley 1992). The post-yield stiffness at point “B” is calculated from yield strength, displacement at yield, and ultimate strength. For the stiffness of the midpoint of segment A-B, interpolation method is adopted. The equivalent stiffness of the specimen beyond the ultimate strength (point B) is secant stiffness. The equivalent stiffness of each specimen at each key point is determined as shown in Table 3.7.

3.2.6.5

Equivalent Viscous Damping Coefficient

The damping coefficients and damping forces depend on the stiffness values used. In most inelastic analyses, the stiffness value is the initial stiffness value. However,

3.2 Study on DIF of Beam-Column-Slab Sub-structures

Load resistance (kN)

F Fy FNF

105

B A

C (drop by 20 %) KeC=FNF)/ΔNF

Ki

KeB=(F-F y)/(Δu-Δy) D

0

Δy

E

Δu ΔNF Vertical displacement (mm)

Fig. 3.44 A method to determine the post-yielding stiffness of substructures is proposed

when the response is inelastic, this results in large and spurious damping forces, which Priestley and Grant (2005) consider inappropriate. In fact, the calculation of elastic damping should be based on secant stiffness. Therefore, the equivalent damping coefficient adopted in this paper is based on the corresponding equivalent stiffness and changes with the corresponding equivalent stiffness value. / ce = ζ × 2 Ke me

(3.8)

It should be emphasized that the damping coefficient varies with the equivalent stiffness. Due to the inevitable gap between the bolt and the hole on the boundary, the rigid body of the specimen moves and absorbs a large amount of kinetic energy in the natural vibration stage. Therefore, a large damping ratio (ζ = 15%) is used in the current SDOF analysis because of the large damping ratio observed in the test results. The equivalent damping coefficients of each specimen corresponding to each key point are also listed in Table 3.7.

3.2.6.6

Load and Reaction Force Applied at the Corner Stub

As shown in Fig. 3.43, the weight of gravity transferred to the substructure can produce an equivalent external load P(t) at the corner stub. Since the corner support restrains the vertical movement of the corner support, the corner support generates an equal amount of reaction force R(t) before removing the corner support. At t0 ,

932

1690

DF1

344

293

375

387

keB (kN/m)

252

211

297

338

keC (kN/m)

74

48

158

73.7

keD (kN/m)

kN·s m

11.9

10.2

12.6

12.3

CeA

7.2

6.2

7.6

7.7

CeB

kN·s m

6.2

5.3

6.7

7.2

CeC kN·s m

3.2

2.5

4.9

3.2

CeD

kN·s m

−24.1 −17.9 −81.3 −29.4

−23.0 −17.6 −83.8 −29.0

FPD-SDOF (mm)

FPD-Test (mm)

1.01

1.01

0.97

1.02

1.05

FPD-SDOF FPD-Test

Note FPD = First peak displacement; keA is the equivalent stiffness at point A of the load–displacement curve. It should be noted that A, B, C, D are corresponding to the stages of first yield, ultimate strength, normal failure and began to develop tensile catenary or membrane action, respectively

Mean

DF2

796

1690

DS2

994.4

1058

1690

1690

DS1

keA (kN/m)

me (kg)

Item

Table 3.7 Parameters in the SDOF Models

106 3 Dynamic Increase Factor of Concrete Structures

3.3 Conclusions

107

the reaction force begins to decrease and is zero at t0 + /\t. The release time (/\t) is 0.005 s, which is the maximum value of the release duration of axial force in all specimens. As shown in the previous sections, the analytical equations for equivalent mass, equivalent stiffness, and equivalent damping ratio are independent of the case of the loss column. Therefore, the proposed single degree of freedom model is not only suitable for predicting the dynamic response of reinforced concrete frames without corner columns, but also for capturing the dynamic response of reinforced concrete frames without interior or exterior columns.

3.2.6.7

Model Validation

The predicted first peak displacement of each specimen is compared with the test results in Table 3.7. The predicted first peak displacement is very close to the measured results. The results show that the SDOF model can accurately and reliably capture the dynamic response of RC specimens during progressive collapse, and can be used for a series of incremental dynamic analysis to determine their DUS.

3.2.6.8

Dynamic Ultimate Capacity and Dynamic Increase Factors

As mentioned previously, the validated SDOF model was used to conduct increment dynamic analyses for predicting their DUS. As shown in Fig. 3.45, the dynamic responses of DS1 with different initial axial forces in the corner support were predicted through the Validated SDOF model is used to perform incremental dynamic analysis and predict its DUS as mentioned above. As shown in Fig. 3.45, the dynamic response of DS1 in the corner support under different initial axial forces is predicted through the verified SDOF. As can be seen from the figure, when the initial axial force of angular support is 29.7 kN, the first peak displacement of DS1 is about 140.0 mm, while when the initial axial force of corner support increased to 29.8 kN, the movement of corner stub cannot be stopped (the displacement is infinite). Therefore, The DUS of DS1 is 29.7 kN, while the SUS of this specimen is 39.1 kN. Therefore, the DLIF value of DS1 can be determined as 1.32 according to Eqs. 3.3–3.8. DUS and DLIF values of the remaining specimens, similar to DS1, were measured and listed in Table 3.8. The DLIF values tested ranged from 1.30 to 1.34.

3.3 Conclusions In this chapter, the dynamic response tests of reinforced concrete beam-column and beam-column slab substructure under the condition of corner column removal are carried out. The test results show that the arch and catenary are not fully used due to the lack of boundary stiffness. At the stage of large deformation, the resistance mechanism of the specimen changes to the cantilever beam. The test of beam-column

108

3 Dynamic Increase Factor of Concrete Structures 20

Vertical displacement (mm)

0 -20 -40

-60 -80 -100

DS1-16.6kN DS1-16.9kN DS1-23.0kN DS1-25.5kN DS1-27.5kN DS1-28.0kN DS1-29.0kN DS1-29.7kN DS1-29.8kN

-120

-140 -160 -180 -200 0

0.5

1

1.5

2

2.5

3

Time (s) Fig. 3.45 Effects of the axial force in the corner support on the dynamic displacement response of Specimen DS1

Table 3.8 Predicted dynamic ultimate strength and values of DLIF of the tested specimens Test

Static ultimate strength (kN)

Dynamic ultimate strength (kN)

DLIF

DIF suggested in DoD (2009)

DS1

39.1

29.7

1.32

1.25

DS2

52.0

39.4

1.32

1.22

DF1

25.8

19.8

1.30

1.21

DF2

36.5

27.3

1.34

1.21

substructure shows that the stiffened plate can significantly improve the structure resistance. In order to obtain the ultimate dynamic strength of substructure, a SDOF is proposed to predict the dynamic response of substructure. Based on the tests and analyses on beam-column substructures, the following main conclusions can be drawn: 1. In dynamic tests, the column removal device proved to be effective, with measured release time not exceeding 0.0035 s, less than 10% of the natural period of the substructure. 2. The peak value of vertical total reaction force measured by fixed support is greater than the axial force before the removal of corner column. This is due to the inertial forces resulting from the sudden removal of corner supports. 3. There is no tension in the horizontal reaction of DF3 and DF4 fixed supports. This indicates that catenary action is not developed to resist progressive collapse in DF3 and DF4 tests.

References

109

4. Experimental results show that the seismic design and design details have successfully improved the progressive collapse resistance of the structure. Based on the tests and analyses on beam-column-slab substructures, the following main conclusions can be drawn: 1. In the absence of 70 mm thick plate, the first peak displacement of specimens DS1 and DS2 was increased by 264.4 and 64.8%, respectively, to redistribute the axial force of the initial corner column. The yield strength and initial stiffness of DS1 are increased by 48.9 and 27.6% respectively, ensuring that DS1 responds only in the elastic region. The specimen DF1 showed a large plastic deformation after the sudden removal of corner support. In addition, including plates may significantly increase the ductility of the specimen (as in the case of reinforced concrete frames without inner columns), which may greatly reduce the dynamic effect. 2. The test results show that the seismic design can significantly reduce the first peak displacement of RC frame, df series and DS series by 65.4 and 23.5%, respectively. 3. The DLIF value ranges from 1.30 to 1.34 for the tested specimens.

References ACI 318-08 (2008) Building code requirements for structural concrete and commentary. American Concrete Institute, Farmington Hills American Society of Civil Engineers (ASCE) (2006) Seismic rehabilitation of existing buildings. ASCE 41-06, American Society of Civil Engineers, Reston VA CP 65 (1999) Structural use of concrete, part 1. Code of practice for design and construction, Singapore Standard Department of Defense (DoD) (2009) Design of building to resist progressive collapse. Unified facility criteria, UFC 4-023-03, Washington, DC General Services Administration (GSA) (2003) Progressive collapse analysis and design guidelines for new federal office buildings and major modernization projects. Office of Chief Architects, Washington, DC Sasani M, Sagiroglu S (2008) Progressive collapse resistance of Hotel San Diego. J Struct Eng 134(3):478–488 Sasani M, Bazan M, Sagiroglu S (2007) Experimental and analytical progressive collapse evaluation of actual reinforced concrete structure. ACI Struct J 104(6):731–739 Priestley MJN, Grant DN (2005) Viscous damping in seismic design and analysis. J Earthquake Eng 9(2): 229–255 Calvi GM, Priestley MJN, Kowalsky MJ (2008) Displacement-based seismic design of structures. IUSS Press, Pavia, Italy Paulay T, Priestley MJN (1992) Seismic design of reinforced concrete and masonry buildings. Wiley, New York, p 769 Mohamed OA (2009) Assessment of progressive collapse potential in corner panels of reinforced concrete buildings. Eng Struct 31(3):749–757

Chapter 4

Spatial and Slab Effects on Concrete Structures

Existing studies on concrete structures were mainly based on planar beam-column substructures due to laboratory capacity limitation. In typical cast-in-situ construction, transverse and longitudinal beams, columns, and slabs act as a single structural unit. Ignoring the contribution from the transverse beam and slab will result in a significant underestimation of the structural resistance. Thus, it is inevitable to study the spatial and slab effects on concrete structures to resist progressive collapse.

4.1 Spatial Effects on Concrete Structures To study spatial effects (transverse beams) on concrete structures, planar beamcolumn specimens (refer to Fig. 4.1a) and 3D beam-column specimens including transverse beams (refer to Fig. 4.1b) are extracted from the prototype building for test.

4.1.1 Specimen Design According to the seismic requirements of ACI 318-08 (2008), two prototype structures with different length-length ratios were designed. The distributed dead load on the prototype structure under the gravity load of 220 mm thick slab is 5.2 kPa. The additional dead load of ceiling, mechanical piping, electrical and piping systems is assumed to be 1.0 kPa. The live load is assumed to be 3.0 kPa. As Singapore is located in a low seismicity zone, the test specimens are assumed to be located on the hard soil profile of site class D. The acceleration parameters of the designed spectral response are 0.45 and 0.34 g for short period and 1 s period, respectively. In consideration of cost and laboratory space, a quarter proportion beam-column substructure (S1 and © China Architecture & Building Press 2023 K. Qian and Q. Fang, Progressive Collapse Resilience of Concrete Structures: Mechanisms, Simulations and Experiments, https://doi.org/10.1007/978-981-99-0772-4_4

111

112

4 Spatial and Slab Effects on Concrete Structures

Transverse direction Longitudinal direction

Longitudinal direction

(a) Planar beam-column specimen

(b) 3D beam-column specimen

Fig. 4.1 Plan view of prototype building

S2) was tested in this study. Four additional specimens (T1, T2, P1 and P2) were also tested in order to quantify the impact of slabs and beams on the load-bearing capacity of reinforced concrete buildings to resist progressive collapse. See Table 4.1 for details of geometric characteristics and reinforcement of specimens. Figures 4.2 and 4.3 show the drawing of the T-series and P-series, respectively.

4.1.2 Test Setup and Instrumentation In this study, a quasi-static push-down loading scheme was used. As shown in Fig. 4.4, a hydraulic jack with a stroke of 600 mm (Item 2 in Fig. 4.4) was installed above the removed column to achieve displacement-controlled loading. A steel assembly (Item 3 in Fig. 4.4) was installed below the hydraulic jack to ensure concentric loading in order for the specimen to form a symmetrical failure mode. It is understandable that the failure of the beam connected to the inner column usually did not occur simultaneously. If steel members are not installed, most of the rotation will be concentrated at the beam-column interface as the displacement increased further when the reinforcement fracture occurred. This is different from the actual situation of reinforced concrete frames without inner columns. In a multi-storey frame, the upper beams provide horizontal constraints on the lower columns, ensuring that the inner columns cannot move vertically until the inner columns have been removed. A large number of measuring devices are installed both internally and externally to monitor the structure behavior. The vertical load applied to the inner column was measured with a load cell (Item 1 in Fig. 4.4). Tension/compression load sensors

4.1 Spatial Effects on Concrete Structures

113

Table 4.1 Specimen details (Unit: mm) Test

Elements

Longitudinal rebar

Stirrup ratio

Slab rebar ratio

Beam-T Beam-L Beam-T Beam-L Joint Beam-T Beam-L Top (%)

Bottom

P1

Null

Beam b*

Null

4T10

0.8

Null

0.5%

Null

Null

P2

Beam a*

Null

4T10

Null

0.8

0.6%

Null

Null

Null

T1

Beam a*

Beam b*

4T10

4T10

0.8

0.6%

0.5%

Null

Null

T2

Beam a*

Beam a*

4T10

4T10

0.8

0.6%

0.6%

Null

Null

SM1 Beam a*

Beam b*

4T10

4T10

0.8

0.6%

0.5%

0.25%

0.25%

SM2 Beam a*

Beam a*

4T10

4T10

0.8

0.6%

0.6%

0.25%

0.25%

Note Beam a*: Clear span = 1300 mm, cross-section = 140 × 80 mm2 ; Beam b*: Clear span = 1900 mm, cross-section = 180 × 100 mm2 ; T10 = Deformed bar of 10 mm diameter; Beam-L = Longitudinal beam; Beam-T = Transverse beam; Column-I = Interior Column

(Item 5 in Fig. 4.4) are mounted on three steel legs to monitor the load redistribution behavior of the specimen. A series of linear variable deformation transformers (LVDTs) and line sensors are placed at various positions to measure the deformation along the beam. In addition, a series of LVDTs were used to monitor the horizontal movement of adjacent columns and the lifting movement of the plate edge to determine the constraint stiffness and quantify the degree of horizontal constraint. In order to obtain the variation of steel bar strain under different loading mechanisms, strain gauges were mounted on the steel bar surface before casting.

4.1.3 Materials The target compressive strength of concrete at 28 d is 25 MPa. The average compressive strength of concrete in P1, P2, T1, T2, S1 and S2 is 19.9, 20.8 MPa, 21.5, 22.7, 21.4 and 23.3 MPa, respectively. Table 4.2 shows the measured tensile properties of the rebar in the test.

4 Spatial and Slab Effects on Concrete Structures

R6@140

Weld

R6@140

114

R6@140

R6@140 Weld

(a) T1 Fig. 4.2 Dimensions and reinforcement details of T-series specimen

4.1 Spatial Effects on Concrete Structures

115

R6@ 60 R6T60 R6@140

4T13

8T16

R6@140

R6@140

R6@140

R6@ 140 4T10

(b) T2 Fig. 4.2 (continued)

4.1.4 Experimental Results 4.1.4.1

Specimen P1

The total span of the planar or 2D beam-column sub-assemblage is 4200 mm, assuming that the middle column has been lost. The load–displacement curve of P1 is shown in Fig. 4.5. Under loading of 8 and 11 kN respectively, cracks first appeared at the beam end near the adjacent column (BENA) and the beam end near the inner column (BENI). The measured yield load and first peak load were 24 and 32 kN, respectively. CAA improved load-carrying capacity by 33.3%. Assuming that the difference between the first peak load was entirely attributed to the CAA, the constraints of the stirrup during this displacement phase were limited. In addition,

116

4 Spatial and Slab Effects on Concrete Structures R6@140

R6@140

Steel Plate 2 with 10 mm thickness

22 mm diameter bolt with 170 mm length

R6@140

R6@140

Weld Steel Plate 1 with 10 mm thickness

(a) P1 R6@140

R6@140

R6@140

R6@140

Weld Steel Plate 1 with 10 mm thickness 25 mm diameter bolt with 200 mm length

(b) P2 Fig. 4.3 Dimensions and reinforcement details of P-series specimen

concrete crushing was observed for the first time in BENI with a displacement of 58 mm. However, limited concrete crushing was observed in BENA until displacement reached 182 mm. In addition, it should be noted that the first reinforcement fracture was observed at BENI with a displacement of 191 mm. Figure 4.6 shows the horizontal motion of the adjacent column versus the vertical displacement of the column. The outward motion was initially observed in the adjacent column. When the vertical displacement reached 152 mm, the inward movement was observed, indicating that CAA in the beam has been transformed into TCA. With the further increase of vertical displacement, the inward motion increased obviously. The test was stopped when the displacement reached 370 mm (19.5% of the net beam span), because the horizontal displacement of the adjacent columns increased significantly with the further increase of the vertical displacement (See Fig. 4.6). The failure mode of the specimen is shown in Fig. 4.7.

4.1 Spatial Effects on Concrete Structures

117

Load cell

1 Pin

2

Hydraulic jack

3 7

4 5

6

1. Load cell 2. Hydraulic jack 3. Steel assembly 4. Specimen 5. Tension/Comp. load cell 6. Steel plate 7. Displacement transducer

Fig. 4.4 Test setup and instrumentation

Table 4.2 Material properties of reinforcement Types

Yield strength (MPa)

Ultimate strength (MPa)

Yield strain (10–6 )

Elongation (%)

Elongation (%)

R6

355

465

1910

17.5

17.5

T10

437

568

2273

13.1

13.1

T13

535

611

2605

11.6

11.6

T16

529

608

2663

14.3

14.3

Notes R6 = Ordinary round rebar with diameter of 6 mm; T10 = Deformed bar with diameter of 10 mm; T13 = Deformed bar with diameter of 13 mm; T16 = Deformed bar with diameter of 16 mm

4.1.4.2

Specimen P2

The 2D beam-column sub-assemblage has a span of 3000 mm. Since P2 has similar performance to P1, this chapter only gives a brief description. The yield and first peak loads of P2 were 26 and 36 kN, respectively. Therefore, CAA increased the yield load by 38.5%. Similar to P1, the outward motion was initially observed in adjacent columns. The inward motion was observed when the vertical displacement reached 134 mm.

118

4 Spatial and Slab Effects on Concrete Structures

Fig. 4.5 Relationship between load and displacement

Fig. 4.6 Relationship between horizontal movement and displacement

Fig. 4.7 Failure mode of specimen P1

4.1 Spatial Effects on Concrete Structures

4.1.4.3

119

Specimen T1

As shown in Table 4.1, the transverse and longitudinal beams of this specimen are similar to those of P1 and P2 respectively. The measured load–displacement curve of T1 is shown in Fig. 4.5. Under the load of 15 kN, the first bending crack appeared in the transverse BENA, and under the load of 20 kN, the first bending crack appeared in the transverse BENI. In addition, bending cracks also formed in the longitudinal BENA during the loading stage. When the load reached 48 kN, the yield at the longitudinal reinforcement of the beam was observed according to the strain measurement results. The first peak load T1 (67 kN) was recorded at a displacement of 32 mm. As a result, CAA increased its load resistance by 39.6%. With the further increase of displacement, more cracks appeared on the side of the beam. However, the crack width of the longitudinal beam is much narrower than that of the transverse beam. When the displacement reached 80 mm, one of the transverse BENI formed a severe shear crack. With the further increase of vertical displacement, the damage was mainly concentrated at the shear crack. When the displacement reached 126 mm, the load resistance began to rise again. The horizontal movements of the adjacent column indicated that the initial outward motion changed to inward motion at this loading stage, as shown in Fig. 4.8. The test was stopped after the displacement reached 250 mm because of the concentration of severe shear cracks in a transverse BENI. The failure mode of T1 is shown in Fig. 4.9. Fig. 4.8 Horizontal movement of adjacent column versus vertical displacement

Fig. 4.9 Failure mode of specimen T1

120

4.1.4.4

4 Spatial and Slab Effects on Concrete Structures

Specimen T2

This specimen has transverse and longitudinal beams similar to P2. When the longitudinal aspect ratio (the ratio of the longitudinal beam to the span length of the beam) is 1.0, the damage of the longitudinal beam and the beam occurred almost simultaneously. Therefore, only transverse beam behavior is presented. The first crack appeared in BENI and BENA under the loads of 18 and 22 kN, respectively. The yield load and the first peak load were 48 and 64 kN, respectively. In addition, when the displacement reached 80 mm, serious concrete crushing occurred in BENI and BENA. When the displacement reached 120 mm, the reinforcement fracture was observed at one of the transverse BENIs. After this displacement, catenary action developed and concrete in BENI and BENA crushed more severely. Test was stopped when the displacement reached 289 mm, about 20% of the beam span.

4.1.5 Load Redistribution Mechanisms When the columns of a building are suddenly removed, the forces initially supported by the lost columns are redistributed to adjacent columns. As described earlier in Sect. 4.2.1 “Test Setup and Instrumentation”, three tension/compression load cells were installed on three steel legs to monitor changes in load redistribution during the test. Figure 4.10 shows the change of load redistribution as vertical displacement increased. For T1, at this time of loading, 27.6 and 22.0% of the load was distributed on the transverse and longitudinal adjacent columns. At T2, because similar beams are designed in the transverse and longitudinal directions, loads are distributed in similar proportions in the transverse and longitudinal adjacent columns. Fig. 4.10 Varying of the load redistribution in columns of specimen T1

4.1 Spatial Effects on Concrete Structures

121

4.1.6 Spatial Effect As shown in Fig. 4.5 and Table 4.3, T1 increased the initial stiffness, yield load, first peak load and ultimate load of P1 by 126.7, 100.0, 109.4 and 68.1%, respectively. For T2 and P2, the 3d effect increased the initial stiffness, yield load, first peak load and ultimate load of P2 by 116.7, 84.6, 77.7 and 52.5%, respectively. Therefore, it is too conservative to ignore the resistance of transverse beams to reinforced concrete frames, and a 3D model should be established in the future analysis and numerical analysis.

4.1.7 Dynamic Load Resistance Since progressive collapse is a dynamic event, it is necessary to use the capacity curve method to evaluate the impact of reinforced concrete slab on the dynamic resistance. Its mathematical expression is: PCC (u d ) =

1 ud



ud

PNS (u)du

(4.1)

0

where PCC (u d ) and PNS (u) are the bearing capacity function and the nonlinear static load estimated at displacement demand u, respectively. Capacity curve method was first proposed by Abruzzo et al. (2006) based on the principle of energy conservation. For details of the derivation of this method, see Abruzzo et al. (2006). It should be noted that strain rate and damping are not considered in this method. Tsai (2010) verified the accuracy of the capacity curve method. Dynamic load bearing capacity curve of specimens is shown in Fig. 4.11. According to the figure, the dynamic ultimate loads of P1, P2, T1 and T2 were 28.9, 34.0, 56.4 and 62.1 kN, respectively. Therefore, these two three-dimensional structures could greatly improve the dynamic bearing capacity of reinforced concrete frame.

4.1.8 Analytical Study 4.1.8.1

Yield Load

To quantify the role of each mechanism in reinforced concrete buildings to resist progressive collapse, a series of analytical methods are proposed, which are primarily based on previous work (Bailey 2001; Park and Gamble 2000).

48

Initial stiffness (kN/mm)

3.9

3.4

1.8

1.5

Fu * (kN)

64

67

36

32

VDLR* (mm)

120

131

134

182

Ft * (kN)

90

79

59

47

VDFT* (mm)

289

250

299

370

Fu /Fy

1.33

1.40

1.38

1.33

Ft /Fu

1.41

1.18

1.64

1.47

Note Fcr*, Fy*, Fu*, and Ft* = the first cracking load, yield load, the first peak load, and ultimate load due to tensile catenary or tensile membrane actions, respectively. VDLR* and VDFT* = the vertical displacement at load re-ascending and vertical displacement at final test, respectively

18

T2

48

26

11

15

P2

T1

8

Fy * (kN)

24

Fcr * (kN)

Test

P1

Table 4.3 Test results of P- and T-series specimens

122 4 Spatial and Slab Effects on Concrete Structures

4.1 Spatial Effects on Concrete Structures

123

Fig. 4.11 Dynamic performance of test specimens

Fypr edicted = 2 ×

MbP + MbN ln

(4.2)

pr edicted

where Fy is the predicted yield load, ln is the clear span of the beam, MbP is the positive bending moment of the beam at first yield stage, and MbN is the negative bending moment of the beam at first yield stage. The yield bending moment of the beam section could be calculated by Eq. 4.3 (Park and Paulay 1975).  Mb = Abt f y db − 0.59Abt

fy  fc × b

 (4.3)

where Abt is the area of tensile reinforcement in beam section, f y is yield strength of beam longitudinal reinforcement, f c is compressive strength of concrete, db is the beam effective depth, and b is the beam width. Based on Eq. 4.3, the yield bending moment of beam is 11.7 and 8.4 kN m for P1 and P2, respectively. Thus, in accordance with Eq. 4.2, the yield load of P1, P2, T1, and T2 are 24.6, 25.9, 49.2, and 51.8 kN, respectively. As the measured yield load of P1, P2, T1, and T2 respectively are 24, 26, 48, and 48 kN, the analytical model generally predicts the capacity of beam action to resist progressive collapse of RC frames well.

4.1.8.2

Compressive Arch Action

As shown in Fig. 4.12, if the edge of the slab or beam is constrained by rigid boundary element and there is no lateral movement, when the slab or beam deflection, the change of the geometry can make the edge of the slab or beam tend to move outward, while the rigid boundary element prevents this trend, thus developed compressive force or thrust in the slab and beam. Induced compression forces increase the yield loads of beams and slabs through moment-axial force interaction. It should be noted that the horizontal compression force is never large enough to prevent the tensile steel

124

4 Spatial and Slab Effects on Concrete Structures

Fig. 4.12 Schematic of CAA or CMA

from yielding, thus always resulting in an increase in the ultimate moment capacity of the beam or slab section. According to the Park and Gamble (2000)’s model introduced in Chap. 2, the first peak loads of P1, P2, T1 and T2 were 34.8, 32.2, 68.4 and 63.2 kN, respectively. The measured values of P1, P2, T1 and T2 were 32, 36, 67 and 64 kN, respectively. Thus, in general, the Park and Gamble (2000)’s model can predict CAA well for double-span beams under fixed boundary conditions.

4.1.8.3

Catenary Action

With the further increase of vertical displacement, concrete spalling becomes serious and flexural cracks may penetrate the whole beam thickness and slab depth. Then, the load is mainly resisted by the rebar acting as a tensioning net or chain, as shown in Fig. 4.13. The catenary action behavior of P series and T series specimens was predicted by using the equation proposed by Yi et al. (2008), as shown in Fig. 4.5, and compared with the measured performance. It can be seen that, in general, the predicted response of TCA matches well with the measured response of P1 and T1, but the predicted response is slightly conservative.

Fig. 4.13 Schematic of TCA or TMA

4.2 Slab Effects on Concrete Structures

125

4.2 Slab Effects on Concrete Structures In this paper, the performance of substructures with or without RC slabs is evaluated. The main purpose of this study is to study the effect of slab on vertical load– displacement relationship, crack pattern, and failure mechanism of substructure by comparing the results. Two column removal scenarios are considered, i.e., corner column removal and internal column removal. The specimens subjected to corner column removal include three beam-column specimens (FC1, FC2, FC3) and three beam-column-slab specimens (SC1, SC2, SC3). The specimens designed to simulate the internal column removal include two beam-column specimens (T1, T2) and three beam-column-slab specimens (SM1, SM2).

4.2.1 Specimen Design Non-aseismic detail RC bending moment frame is designed according to Singapore standard CP65, while aseismic detail frame is designed according to ACI 318-08 (2008). The dead load of 210.0 mm thick slab of prototype structure is 5.1 kPa. The additional dead load is assumed to be 1.0 kPa. The live load is assumed to be 2.0 kPa. A one-third scale substructure was cast and tested in this study. A uniform pressure of 6.6 kPa was applied to the the top surface of the slab with a load combination (DL + 0.25LL). A uniform load of 3.1 kN/m was applied to the edge beam to simulate the line load induced by the filled wall. The required load resistance recommended by the GSA (2013) guidelines for each specimen are listed in Table 4.4. Figure 4.14 shows the typical size and reinforcement diagram of beams and columns of SC1 and SC2 specimens, and Fig. 4.15 shows the reinforcement diagram of slabs of SC1 and SC2 specimens. Concrete cover thicknesses of beams, columns and slabs are 10, 20 and 7 mm respectively. For SC series specimens, there are 1 corner column, 3 adjacent enlarged columns and 4 RC beams. For all specimens, the dimension of corner column representing the removed column is 200.0 mm square and the adjacent magnifying column is 250.0 mm square to ensure that these adjacent magnifying columns will not be damaged. Four 25 mm diameter bolts were combined in the adjacent column and connected with steel legs. See Table 4.4 for details of the size and reinforcement of each specimen. SC1 and SC3 adopt non-seismic design. Therefore, the four T10 beams are reinforced in double longitudinal direction, and bending 90°stirrups are used for transverse reinforcement, and no transverse reinforcement is designed in the joint zone. However, SC2 is designed to be able to resist seismic action, so 4T13 was used for the longitudinal reinforcement in the beam and transverse reinforcement is bent to 135°, and transverse reinforcement was designed in the joint zone. It should be noted that SC1 and SC2 have equal longitudinal and transverse spans, while SC3 has longitudinal and transverse spans of 2775 and 2175 mm, respectively. FC1, FC2, and FC3 correspond to SC1, SC2, and SC3 respectively. As shown in Table 4.4, the beams and columns of the F-series specimens have similar details

Beam b*

Beam a*

Beam a*

Beam a*

Beam a*

Beam a*

FC3

SC1

SC2

SC3

0.87

1.47

0.87

0.87

1.47

0.87

0.75

1.47

0.87

0.75

1.47

0.87

Beam-L (%)

None

0.49%

None

None

0.49%

None

Joint

0.31

0.95

0.31

0.31

0.95

0.31

Beam-T (%)

Transverse rebar

Note Beam-T, Beam-L = Transverse beam and longitudinal beam respectively Type a*: Clear span = 2175 mm cross-section = 180 × 100 Type b*: Clear span = 2775 mm cross-section = 210 × 100

Beam b*

Beam a*

Beam a*

Beam a*

FC2

Beam a*

Beam-T (%)

Beam-L

Beam-T

Beam a*

Longitudinal rebar

Elements

FC1

Test

Table 4.4 Specimen details

0.36

0.95

0.31

0.36

0.95

0.31

Beam-L (%)

40.4

33.7

33.7

40.4

33.7

33.7

Required resistance (kN)

0.4%

0.4%

0.4%

Null

Null

Null

Slab rebar

126 4 Spatial and Slab Effects on Concrete Structures

127

Transverse Interior Beam 8T16 R6@ 125

R6@ 125 R6@ 180 Detail A-A

R6@ 60

4T10

Detail C-C

4T13

R6@ 125 R6@ 150 A

Detail B-B

4T10

Longitudinal Interior Beam

Longitudinal Edge Beam Adjacent Column

4.2 Slab Effects on Concrete Structures

4T16 Detail D-D

B

Transverse Edge Beam R6@180

R6@250

Corner Joint Stub

R6@180

Adjacent Column

R6@55

Plan View of Specimen SC1 R6@180

R6@55

D

C

R6@250

R6@180

R6@200

Strain gauge

Elevation View of the Specimen S1

R6@60

R6@125

R6@60

R6@200

Elevation View of the Specimen S2 Fig. 4.14 Dimensions, cross-section details, and strain gauge locations of specimens SC1 and SC2

as the corresponding SC series specimens, while no reinforced concrete slabs are incorporated. The beam cross-section in the longitudinal beam of SC3 is 210 × 100 mm2 . In addition, for FC-series specimens, no slabs are added and a fixed end is assumed in the adjacent expanded column. Therefore, longitudinal and transverse internal beams are excluded from the model. High yield strength steel (T16, T13, T10) is used for longitudinal reinforcement, and low carbon steel (R10, R6) is used for transverse reinforcement. The average compressive strength of concrete is about 32.0 MPa.

128

4 Spatial and Slab Effects on Concrete Structures

ST6

SB6 SB7

ln/4

SB8

SB9

ST5 ST1

ST2

18 R6@ 125

SB4 SB2

ST3

(a) Top rebar

SB5

SB1

ST4

4 R6@ 125

SB3

18 R6@ 125

4 R6@ 125

(b) Bottom rebar

Fig. 4.15 Slab reinforcement details and strain gauge locations of specimens SC1 and SC2

Typical size and reinforcement of SM1 are shown in Fig. 4.16. SM1 is a 2 × 2 bay single layer beam-slab substructure. Nine columns, twelve beams and a 55 mm slab are cast in one piece. The column size is 200 mm × 200 mm. To prevent column damage, the column size is increased. The central span of transverse and longitudinal beams is 1500 and 2100 mm, respectively. The transverse and longitudinal beam section sizes are 140 mm × 80 mm and 180 mm × 100 mm respectively. Due to the small size of the specimen, continuous 4T10 was was installed in the longitudinal and transverse beams. T10 represents a deformed steel bar with a diameter of 10 mm. According to the actual design, two layers of reinforcement are arranged on the floor. The bottom layer is placed with a continuous R6 with a spacing of 250 mm. R6 stands for round steel bar with a diameter of 6 mm. However, for the top layer, the rebar in the center of each panel is cut off in accordance with the design details of ACI 318-08 (2008), the steel bar spacing of the top plate is also 250 mm. The net cover plates of column, beam and plate are 10, 7 and 7 mm respectively. For SC2, the dimensions of columns, beams and slabs, and details of reinforcement are generally similar to those of SC1, as shown in Table 4.1. The only difference between SM2 and SM1 is that SM2 has unequal beam span in longitudinal and transverse directions.. For T1 and T2, their beams are the same as SM1 and SM2, respectively. The only difference between T series and SM-series specimens is that RC slabs are not added. The vertical displacement of the inner column is imposed by displacement control method to limit the action of the side beam not connected with the inner column. Therefore, only four inner beams connected to the inner column were cast for the T-series specimens. The target compressive strength of concrete at 28d is 25 MPa. The average compressive strength of T1, T2, SC1, and SC2 concrete is 21.5, 22.7, 21.4, and 23.3 MPa, respectively. A series of LVDTs and line sensors were placed at different positions to measure the deformation of internal joint and deformation shapes along the slab and beam, as shown in Fig. 4.17. In addition, a

Top rebar

4.2 Slab Effects on Concrete Structures

129

R6@250

Fig. 4.16 Dimensions and reinforcement details of SM1 (units: mm)

series of LVDTs were used to monitor the horizontal movement of adjacent columns and the lifting movement of the slab edge to determine the constraint stiffness and quantify the degree of horizontal constraint. In order to obtain the variation of steel bar strain under different loading mechanisms, strain gauges were pasted on the steel bar surface before casting. Similar instruments were installed for T-series specimens.

130

4 Spatial and Slab Effects on Concrete Structures

Fig. 4.17 Instrumentation layout in SM1

4.2.2 Test Setup The schematic diagram of test setup of SC-series specimens is shown in Fig. 4.18. The floor is supported by three rigid steel legs, each of which is connected to a 75.0 mm thick strong steel plate by four 27 mm diameter bolts. Steel plates are fixed to strong floors with pre-tensioned steel bars. The influence of the continuity of the surrounding slab on the response of SC-series specimens was simulated by applying seven steel self-weight components to the extension part of the slab. A hydraulic jack with 600 mm stroke exerted downward displacement on the top of the corner column to simulate the axial load before the corner column lost. The test setup of FC-series specimen is similar to Fig. 3.3. As shown in Fig. 4.19, the side columns of SM-series specimens were designed to be fixed on strong steel legs to simulate the fixed boundary conditions. The horizontal and rotational constraints on the edge of the fixed column support plate were mainly provided by the torsional stiffness of the edge beam. In addition, a specified weight (380 kg) is suspended over an extension of the slab to apply additional rotational constraints. A concentrated load is then applied to the top of the inner column by a hydraulic jack with a stroke of about 600 mm. It should be noted that the displacement control method was used in this push-down test. To enable the specimen to form a symmetrical failure mode, a steel assembly was installed below the hydraulic jack to ensure concentric loading. It is understandable that the failure of the beam connected to the inner column usually did not occur simultaneously. If steel members are not installed, most of the rotation will be concentrated at the beam-column interface as the displacement increased further when the reinforcement fracture. RC frames without inner columns are different

4.2 Slab Effects on Concrete Structures

131 1. Load cell measure applied load 2. Hydraulic Jack with 600 mm stroke 3. Steel column 4. Comp/tension load cell 5. Steel assembly 6. RC substructure 7-1, 7-1 and 7-3. Rigid steel leg 1, 2 and 3 8. Steel weight 9. Line LVDTs to measure deflection of the slab 10. LVDTs with 300 mm travel 11. LVDTs with 25 mm travel to monitor the deformation of steel legs

1

2

3 5

4 10

8 6

8 8

6

11 7-1

7-3

9

7-2

Fig. 4.18 Test setup of SC-series specimens

from the actual situation. In a multi-storey frame, the upper beam provides horizontal restraint on the lower column, ensuring that the inner column can not move vertically until the inner column has been removed. The test setup of FC-series specimens is shown in Fig. 4.4.

4.2.3 Test Results 4.2.3.1

Specimen FC1

The test results of FC-series and SC-series specimens are shown in Table 4.5. The vertical load and horizontal reaction forces of FC1 corner joints under different performance levels are shown in Fig. 4.20a. Five performance levels were identified, PL1, PL2, PL3, PL4 and PL5 respectively represent the first bending crack, first yield, ultimate bearing capacity and normal failure stage of the longitudinal reinforcement of the beam, which is defined as the bearing capacity decreases by more than 20% of the ultimate bearing capacity and begins to be affected by catenary action/membrane action. Under the load of 4.3 kN, the first crack appeared near the fixed end of the beam. After the first cracking, shear cracking appeared in the joints under the load of 22.1 kN, and shear cracking appeared in the BENF reinforcement under the load

132

4 Spatial and Slab Effects on Concrete Structures

Load cell 1

Pin

2

Jack 3

4

6 5

1. Load cell 2. Hydraulic jack 3. Steel assembly 4. Specimen 5.Tension/compression load cell 6. Steel plates (weights)

Fig. 4.19 Test setup of SM-series specimens

of 22.5 kN. After the plastic hinge is formed in the beam end, the tensile strain of the longitudinal reinforcement at the bottom of the beam near the corner joint reached the maximum. Shear cracks in corner joints widen and the ultimate bearing capacity Pcu of 25.8 kN was reached, which is equivalent to 77.6% of the load required by GSA (2013) to resist progressive collapse. The compressive reaction of the fixed end was measured by a horizontal tension and pressure sensor. Before the first crack appeared, the compressive reaction was little, but after the first crack loading, the compressive reaction increased significantly. The relationship between horizontal reaction force and deflection is similar to that between vertical load and deflection. When the deflection is 54.0 mm, the maximum horizontal compression load was 19.7 kN. This corresponded to 2.5% of the TDR. However, the ultimate load capacity achieved at deflection of 44.0 mm, equivalent to 2.0% TDR. The horizontal compression reaction began to decline due to severe shear cracks at the corner joints. Severe shear cracks reduced the BENC’s rotational constraints, as can be seen from the measured decrease in vertical load resistance and horizontal compressive reaction. When the joint shear crack expanded, the strain of longitudinal reinforcement in BENC beam de increased rapidly. This indicates that the resistance mechanism of the substructure has changed to the cantilever beam. With the further increase of deflection, concrete cracks appeared in BENF, and concrete cracks appeared in corner joints at the deflection of 160 and 240 mm, respectively. When the beam reinforcement near the fixed end completely fractured, the vertical load resistance decreased to zero. The FC1 failure mode is shown in Fig. 4.21.

29.1

5.0

3.9

22.0

18.0

17.0

FC3

SC1

SC2

SC3

32.8

45.8

33.5

21.0

22.5

4.3

First yield load kN

FC2

First beam crack load kN

FC1

Test

17.0

22.0

24.0

16.1

25.3

21.0

First joint shear crack kN

20.0

34.0

22.0

None

None

None

First diagonal slab crack kN

Table 4.5 Test results of FC- and SC-series specimens

37.5

52.0

39.1

23.0

36.5

25.8

Ultimate load Pcu, kN

22.0

28.7

22.3

19.6

27.9

19.8

MCHR in Beam-T kN

23.3

27.3

21.0

18.4

27.3

19.6

MCHR in Beam-L kN

0.054

0.057

0.047

0.058

0.061

0.051

Beam-T rotation at PL4 (rads)

0.049

0.050

0.044

0.047

0.056

0.049

Beam-L rotation at PL4 (rads)

260.0

218.0

200.0

299.2

275.9

332.8

Start to develop tensile membrane mm

4.2 Slab Effects on Concrete Structures 133

4 Spatial and Slab Effects on Concrete Structures Horizontal reaction (kN) Load on sub-frames (kN)

134 50 PL3

40

PL2 PL4

PL3·

30 PL1

PL5

PL2

20

PL4

10

PL5

PL1

0 -10 -20

FC1 SC1

-30 0

100

200 300 Vertical displacement (mm)

400

500

Horizontal reaction (kN) Load on sub-frames (kN)

(a) FC1 and SC1 60

PL3

50

PL4

PL2 PL3·

40

PL5 PL4

30

PL2

PL1

20

PL5

10

PL1

0 -10 FC2 SC2

-20 -30 0

100

200 300 Vertical displacement (mm)

400

500

Horizontal reaction (kN)

Load on sub-frames (kN)

(b) FC2 and SC2 50 PL3

40 PL2

PL4

30

PL3· PL1

20

PL5

PL2 PL4

10

PL5

PL1

0 -10 FC3 SC3

-20 -30 0

100

200 300 Vertical displacement (mm)

400

(c) FC3 and SC3

Fig. 4.20 Load–displacement curves of FC- and SC-series specimens

500

4.2 Slab Effects on Concrete Structures

135

Fig. 4.21 Failure mode of specimen FC1

4.2.3.2

Specimen SC1

The measured vertical load and horizontal reaction forces of SC1 are shown in Fig. 4.20a. Bending cracks occurred when the load was 18.0 kN. No cracks were observed in the beam and corner joint areas. Under the load of 22.0 kN, a small number of bending cracks appeared in the beam end near the adjacent column (BENAC), and the first diagonal crack in the plate formed and passed through the center of the slab. Under the load of 24.0 kN, the first diagonal shear crack appeared in the corner joints. When the applied load increased further, more diagonal cracks were formed in the slab. When the vertical displacement reached 43.0 mm, equivalent to 2.0% of TDR, the specimen achieved the maximum horizontal compression load of 21.0 kN. When the vertical displacement reached 56.0 mm, equivalent to 2.6% of TDR, the ultimate bearing capacity of specimens reached 39.1 kN, equivalent to 116.0% of the progressive collapse load stipulated by GSA (2013). With the increase of vertical displacement, the main diagonal crack and corner joint diagonal crack of the slab became wider, and appeared larger torsional deformation appeared in the side beam. When the vertical displacement reached 90 mm, concrete breakage occurred in BENAC and concrete splitting occurred in corner joints with a deflection of 120 mm. When the displacement reached 200 mm, equivalent to 9.2% TDR, the diagonal crack in the slab penetrated the depth of the slab, and the load–displacement curve started to rise again (mainly due to the tensile membrane action). Figure 4.22 shows the development of crack pattern of SC1 slab at different performance levels.

4.2.3.3

Specimen FC2

Figure 4.20b shows the relationship between the measured vertical and horizontal reaction forces and the vertical displacement. In general, the crack development of FC2 was similar to that of FC1. Therefore, only the most important differences

136

4 Spatial and Slab Effects on Concrete Structures

SC1-PL1

Corner Joint Stub

SC1-PL2

SC1-PL3

Corner Joint Stub

Corner Joint Stub

SC1-PL4

Corner Joint Stub

SC1-PL5

Corner Joint Stub

Fig. 4.22 Observed cracking patterns at different performance levels of the slab of SC1

between the two specimens are highlighted here. In FC1, the crack width of the beam bottom near the corner joint did not widen after serious cracks appeared in the joint. FC2 developed more cracks at the bottom of BENC, and these cracks became wider after serious joint shear cracks appeared. Another difference in failure mode between FC2 and FC1 was that the core joint concrete remained relatively intact because transverse reinforcement effectively confined the core concrete after the concrete cover is split at a deflection of 280.0 mm. When the vertical displacement reached 53.0 mm, corresponding to TDR of 2.4%, the ultimate bearing capacity of the specimen reached 36.5 kN, which was equivalent to 108.0% of the progressive collapse required load resistance stipulated by GSA (2013). When the beam top reinforcement near BENF finally fractured, the vertical load resistance droped to zero. FC2 has a similar failure mode to FC1, so it did was not presented.

4.2.3.4

Specimen SC2

The measured vertical and horizontal reaction forces were shown in Fig. 4.20b. The key points of the test results are shown in Table 4.5. In general, the crack development of SC2 was similar to that of SC1. Therefore, only the most important differences between these two specimens are highlighted. For SC1, the first diagonal crack formed in the slab under the load of 22.0 kN. However, under the load of 34.0 kN, inclined cracks were formed in SC2 and passed through the center of the slab. In SC2, the core concrete spalling occurred at the displacement of 260.0 mm, much later than SC1. When the vertical displacement reached 69.4 mm, equivalent to 3.2% of TDR, the specimen reached the ultimate bearing capacity of 52.0 kN, equivalent to 154.0% of the progressive collapserequired load resistance stipulated by GSA (2013). When the vertical displacement reached 218.0 mm, the load–displacement curve appeared tensile membrane action. Figure 4.23 shows the development of crack pattern of SC2 plate under different performance levels.

4.2 Slab Effects on Concrete Structures

SC2-PL1

SC2-PL2

Corner Joint Stub

Corner Joint Stub

137

SC2-PL3

Corner Joint Stub

SC2-PL4

Corner Joint Stub

SC2-PL5

Corner Joint Stub

Fig. 4.23 Observed cracking patterns at different performance levels of the slab of SC2

4.2.3.5

Specimen FC3

Figure 4.20c shows the relationship between the measured vertical and horizontal reaction forces and the vertical displacement of the corner joint of FC3. Because the design span of longitudinal and transverse beams was not equal, the longitudinal and transverse beams showed asymmetric crack patterns. The first bending crack appeared in the longitudinal beam under 3.9 kN load. Under 10.0 kN load, bending cracks appeared at the top of transverse BENF and at the bottom of longitudinal BENF. In this loading stage, the first shear crack appeared simultaneously on both sides of the corner joint. However, the first bending crack at the bottom of the transverse BENC was observed at a load of 16.0 kN. When the vertical displacement reached 38.9 mm, the ultimate bearing capacity of the specimen reached 23.0 kN, which was equivalent to 57.0% of the progressive collapse load resistance stipulated by GSA (2013).

4.2.3.6

Specimen SC3

Figure 4.20c shows the relationship between vertical and horizontal reaction forces and vertical displacement. Because the design span of longitudinal and transverse frames was not equal, the longitudinal and transverse beams appeared demonstrated asymmetric crack patterns. When the load was 20.0 kN, the slab has the first diagonal crack. However, the first diagonal crack did not connect to the two interfaces between the beam and the adjacent column. It’s like an arc, and the radius of the arc is the net span of the short beam. When the loads were 17.0 and 20.0 kN respectively, bending cracks appeared at the transverse and longitudinal ends of beams near adjacent columns first. However, the first bending crack in BENC occurred under a load of 24.0 kN. Under the loading of 17.0 and 20.0 kN, the first shear crack along the transverse and longitudinal joints appeared. Under a load of 24.0 kN, a new branch crack was formed, connecting the first diagonal crack to the interface of the adjacent longitudinal and transverse columns. When the vertical displacement reached 60.2 mm, the ultimate bearing capacity of the specimen reached 37.5 kN, which was equivalent to 92.8% of the progressive collapse load resistance stipulated by GSA (2013). When the vertical displacement reached 70.0 mm, the joint shear

138

4 Spatial and Slab Effects on Concrete Structures

SC3-PL1

Corner Joint Stub

SC3-PL2

SC3-PL3

Corner Joint Stub

Corner Joint Stub

SC3-PL4

Corner Joint Stub

SC3-PL5

Corner Joint Stub

Fig. 4.24 Observed cracking patterns at different performance levels of the slab of SC3

crack widened significantly, and concrete cracks appeared at the bottom of the transverse BENAC. When the vertical displacement reached 100.0 mm, concrete cracks appeared at the corner joints and concrete cracks appeared at the bottom of the longitudinal BENAC. The further increase of displacement will lead to the increase of the width of the main inclined crack. Figure 4.24 shows the development of crack pattern of SC3 plate under different performance levels.

4.2.3.7

Specimen SM1

The test results of T series and SM series specimens are listed in Table 4.6. The measured vertical load and displacement curves are shown in Fig. 4.25. The span ratio of specimen SM1 is 1.4. It has the same beam as T1, but a 55 mm thick RC slab is incorporated. Under the load of 44 kN, the first crack appeared on the top surface of slab. When the vertical load was increased to 53 kN, the cracks on the top surface of slab developed and joined into an ellipse. With the further increase of vertical load, oblique cracks formed on the bottom surface of the plate. Further increasing the load to 80 kN, the longitudinal reinforcement of the beam yield. After reaching the first peak load of 115 kN (corresponding to a displacement of 47 mm), the beams and slabs suffered severe concrete crushing. This suggested that CMA and CAA became exhausted. When the displacement reached 75 mm, the load resistance began to rise again. This indicated that CMA has been transferred to TMA on the slab. The test results of T-series and P-series specimens showed that the catenary action began to develop normally when the displacement reached about 10% of beam span. Therefore, the re-rise of load resistance in this loading stage was mainly attributed to the development of TMA in the slab. As the vertical displacement further increased, the deformation was mainly concentrated in the central region (elliptic region), while the deformation in the peripheral region (compression ring) was limited, as shown in Fig. 4.26. However, when the displacement exceeded 215 mm, the force cannot be redistributed to the side column through the slab. This is because the concentration of severe concrete crushing in the top plate close to the inner column, damaging the integrity of the slabs and beams. At the same time, with the further increase of displacement, punching failure accelerated the separation of the slab and beam. When

18

T2

48

48

90

80

Fy * (kN)

3.9

3.4

6.2

4.8

Initial stiffness (kN/mm)

64

67

123

115

Fu * (kN)

120

131

58

75

90

79

165

169

VDLR* Ft * (kN) (mm)

289

250

231

305

VDFT* (mm)

1.33

1.40

1.37

1.44

Fu /Fy

1.41

1.18

1.34

1.47

Ft /Fu

Note Fcr*, Fy*, Fu*, and Ft* = the first cracking load, yield load, the first peak load, and ultimate load due to tensile catenary or tensile membrane actions, respectively. VDLR* and VDFT* = the vertical displacement at load re-ascending and vertical displacement at final test, respectively

56

15

SM2

44

SM1

T1

Fcr * (kN)

Test

Table 4.6 Test results of T- and SM-series specimens

4.2 Slab Effects on Concrete Structures 139

140

4 Spatial and Slab Effects on Concrete Structures

Vertical Load (kN)

250

Note:TCA=Tensile catenary action TMA=Tensile membeane action

200 S1(PTCA+PTMA)

T1 T2 S1 S2

150 100 50 0

T1-PTCA

0

100 200 300 Vertical Displacement (mm)

400

Fig. 4.25 Load–displacement curves of T- and SM-series specimens

the displacement reached 210 mm, the reinforcing bars at the bottom of the longitudinal BENI both fractured. The test was stopped at the displacement of 305 mm, and all beams suffered fracture of reinforcing bars, and the internal connections of the slabs and columns were severely damaged by punching. The failure modes of SM1 are shown in Fig. 4.27. As shown in Fig. 4.27a (top view), a series of concentric elliptical cracks, known as tensioning nets, were observed at the center of the plate. The outer area immediately adjacent to the support had less deflection and was called the pressure ring. As shown in Fig. 4.27b, the concrete was crushed at the slab near the interior column and the reinforcement was fractured at BENI.

4.2.3.8

Specimen SM2

The specimen has a span ratio of 1.0. It has the same beam as T1, but the 55 mm thick RC slab is included. Overall, SM2’s performance is similar to SM1’s. In addition, as the key results of the test sample are listed in Table 4.6, only the major differences between SM2 and SM1 are highlighted here. Different from S1(elliptical crack), the top surface of slab formed a circular crack under 56 kN load. Under the load of 90 kN, the longitudinal stiffeners yield first, and the yield value is slightly higher than S1. When the displacement reached 42 mm, SM2 has the first peak load, which is 123 kN. Compared with yield load, it is increased by 36.7%. Similar to SM1, punching failure occurred at 165 mm displacement. When the displacement reached 170 mm, one of the beams suffered reinforcement fracture. The vertical displacement was further increased to 200 mm and reinforcement fracture was observed on another beam. The test was stopped at a displacement 231 mm. The failure modes of SM2 are shown in Fig. 4.28.

4.2 Slab Effects on Concrete Structures

141

Fig. 4.26 The contour of deformation shape of SM1 at a displacement of 180 mm

(a) Top

(b) Bottom

Fig. 4.27 Failure mode of SM1

142

4 Spatial and Slab Effects on Concrete Structures

(a) Top

(b) Bottom

Fig. 4.28 Failure mode of SM2

4.2.4 Slab Effects Discussion 4.2.4.1

Comparison of Performance of FC-Series Specimens to SC-Series Specimens

Load–displacement curves Figure 4.20 shows the comparison of load–displacement relation between SC-series specimens and FC-series specimens. The TDR of F1 at PL4 was 4.2%, while that of SC1 at a similar stage was 4.1%. By comparing the ultimate load bearing capacity of the two specimens, it can be seen that FC1 can reach 77.6% of the load resistance required to resist progressive collapse specified in GSA (2013), while SC1 can reach 116.0% of the load required. Therefore, it can be concluded from the experimental results that SC1 can survive if the corner columns are lost, but FC1 cannot. The initial stiffness of FC1 and SC1 is 0.82 and 1.08 kN/mm, respectively. The TDR of FC2 at PL4 was 5.1%, while that of SC2 at PL4 was 6.5%. By comparing the ultimate bearing capacity of FC2 and SC2, it can be seen that FC2 and SC2 can reach 108.0 and 154.0% of the load resistance required to resist progressive collapse as stipulated by GSA (2013), respectively. Therefore, based on the test results, we can conclude that if one ground corner column is lost under extreme loads, both specimens can survive.

4.2 Slab Effects on Concrete Structures

143

The initial stiffness of FC2 and SC2 are 0.95 and 1.13 kN/mm, respectively. For FC3 and SC3, according to the test data, FC3 and SC3 can reach 57.0 and 92.8% of the load resistance required to resist progressive collapse specified by GSA (2013), respectively. The final capacity of SC3 is about 63% higher than that of FC3. However, neither specimen survives the progressive collapse. The initial stiffness of FC3 and SC3 is 0.75 and 1.04 kN/mm, respectively. The results show that the performance of SC-series is better than FC-series. This is because the slabs work as flanges of L beams in SC-series specimens significantly improve the bending moment capacity of the beam ends near the adjacent amplifying columns. However, it should be noted that because the slab flange is in the compression zone, the influence of L-beam action on the bending capacity of the beam end near the corner column is limited. Resistance capacity decomposition Figure 4.29 shows the contribution of RC slab and beam to the load bearing capacity of SC-series specimens. For SC1, RC slabs initially bear only 4.8% resistance. With the increase of displacement, the bearing capacity contribution of slab increased to 31.2%. After that, as cracks appeared in the plate for the first time, the contribution percentage of slabs fell to about 26.2%. Secondly, due to plastic hinge forming, the slab members increased to 38.6%. When the displacement reached 200 mm, the contribution of slab was greatly increased due to the development of the tensile membrane action. In general, SC2 and SC3 behaved similarly to SC1. The initial contribution of slab to load carrying capacity of SC2 and SC3 was 5.0 and 5.4%, respectively. When the vertical displacement of SC2 and SC3 reached 218.0 and 260.0 mm respectively, the contribution of slab member increased significantly due to the tensile membrane action developed in the slab. Rebar strains Figures 4.30 and 4.31 show the measured strain distributions of FC1 and SC1 at different performance levels, respectively. For FC1, the bottom longitudinal reinforcement of BENC beam showed strain reversal after the removal of the corner column. After PL3, the tensile strain of the top longitudinal reinforcement in BENF increased significantly, while that of the bottom longitudinal reinforcement began to decrease. After PL3, the inflection point (zero strain point) of the upper and lower longitudinal bars moved to the corner joints obviously, indicating that the resistance mechanism of the specimen changed to the cantilever beam after the serious damage of the corner joints. The bottom longitudinal bars in the BENF yielded at PL4, while the bottom longitudinal bars near the corner joints remained elastic during the test. In general, the strain distribution of SC1 was similar to that of FC1. As shown in Fig. 4.31, the difference between the two specimens is that the maximum compressive strain of the bottom longitudinal bars in BENAC of SC1 was much larger than that of FC1. This may be due to the flange effect of the slab on the beam section in SC1. The strain distribution of other specimens was similar, so it was not given. Figure 4.32 shows the relationship between slab strain and vertical displacement of SC1. The position of strain gauge is shown in Fig. 4.15. As shown in Fig. 4.32a, ST1 was in compression at the beginning, but begins to decrease when the vertical

144 Fig. 4.29 Resistance capacity decomposition results of SC-series specimens

4 Spatial and Slab Effects on Concrete Structures

5500 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0 -500 -1000 -1500

250

250

Top strain gauges

250

250

250

145

250

Strain (με)

Strain (με)

4.2 Slab Effects on Concrete Structures

PL1 PL2 PL3 PL4

F1-Top-Rebar 0

250

500

750

1000

1250

1500

1750

2000

3500 3000 2500 2000 1500 1000 500 0 -500 -1000 -1500 -2000 -2500 -3000 -3500

250

250

250

Bottom strain gauges

250

250

PL1 PL2 PL3 PL4

F1-Bottom-Rebar 0

Displacement from beam-corner column interface (mm)

250

250 500 750 1000 1250 1500 1750 2000 Displacement from beam-corner column interface (mm)

(b) Bottom rebar

(a) Top rebar Fig. 4.30 Strain profile of beam rebar of FC1 6000 5000 250

250

Top strain gauges

250

250

250

250

Strain (με)

Strain (με)

4000 3000 PL1 PL2 PL3 PL4

2000 1000 0

S1-Top-Rebar

-1000 0

250

500

750

1000

1250

1500

1750

2000

Displacement from beam-corner column interface (mm)

(a) Top rebar

6000 5000 4000 3000 2000 1000 0 -1000 -2000 -3000 -4000 -5000

250

250

250

250

Bottom strain gauges

250

250

PL1 PL2 PL3 PL4

S1-Bottom-Rebar 0

250

500

750

1000

1250

1500

1750

2000

Displacement from beam-corner column interface (mm)

(b) Bottom rebar

Fig. 4.31 Strain profile of beam rebar of SC1

displacement reached 22.4 mm. In this vertical displacement, diagonal shear cracks appeared in the corner joints. When the vertical displacement exceeded 200.0 mm, ST1 was transformed into tension. Tensile membrane action developed at this stage of displacement. When the displacement reached 90.0 mm, the strain of ST3 increased suddenly and exceeded the maximum limit. While the strain of ST6 was close to zero during the test, indicating that most of the force received by the corner column at the initial stage of damage is transferred to the adjacent column, and the force transmitted to the inner column can be ignored. Figure 4.32b shows the relationship between the strain of slab bottom bar and vertical displacement. During the test, all the bottom reinforcement strains except SB1 and SB2 were tensile. The strain at SB3, SB5 and SB8 was much larger than that at other strain points because the deformation was mainly concentrated at the main diagonal crack. The strain distributions of SB5, SB8, SB9 and SB1 indicated that the slab reinforcement strain decreased with the increase of the distance from the adjacent column when the slab developed tensile membrane action. This is one of the main differences in membrane action between current slabs and those tested by previous researchers (Park 1964). In the test, it is assumed that all the reinforcing bars at the bottom of the center slab yield after the formation of the tensile membrane action. Further experimental and analytical studies are needed to analyze this behavior. The strain of SB6 and SB7 was very

146

4 Spatial and Slab Effects on Concrete Structures 4500

3000 2500

SB1 SB4 SB7

3500

SB2 SB5 SB8

SB3 SB6 SB9

3000

Strain (με)

3500

Strain (με)

4000

ST1 ST2 ST3 ST4 ST5

4000

2000 1500 1000

2500 2000 1500 1000

500

500

0

0

-500 0

100

200

300

400

500

0

100

Vertical displacement (mm)

(a) Top rebar

200

300

400

500

Vertical displacement (mm)

(b) Bottom rebar

Fig. 4.32 Steel strain of slab versus vertical displacement in SC1

small, which was consistent with the observed failure mode, and the damage of the inner semi-triangular plate is limited during the test.

4.2.4.2

Comparison of Performance of T-series Specimens to SM-Series Specimens

Load–displacement relationship The slab effect was quantified by comparing the performance of SM-series specimens with T-series specimens. As shown in Fig. 4.25, SM1 increased the initial stiffness, yield load, first peak load and ultimate load of T1 by 41.1, 66.7, 71.6 and 114.0%, respectively. For SM2 and T2, the initial stiffness, yield load, first peak load and ultimate load of RC plate are increased by 59.0, 87.5, 92.2 and 83.3%, respectively. The results showed that RC slabs can further improve the static properties of RC frames and significantly resist progressive collapse. In order to decompose the contribution of TMA and TCA to the large displacement stage of RC beam-slab structure, the load resistance decomposition of SM-series specimens was carried out. As shown in Fig. 4.33, when SM1 reached the first peak load, the beam provides 54.0% load resistance. When the displacement within 126–215 mm, beams accounted for 42.0% of the load resistance, and slab accounted for 58.0% of the load resistance. After 215 mm, the contribution of the beam increased significantly due to punching failure of the internal connection of the plate column. Similar behavior was observed in SM2. Resistance capacity decomposition When the columns of a building are suddenly removed, the forces initially supported by the lost columns are redistributed to adjacent columns. Figure 4.34 shows the change of load redistribution as vertical displacement increased. In the yield load stage of SM1, 23.5, 18.8 and 3.8% of the load were distributed in transverse adjacent columns, longitudinal adjacent columns and corner columns. For T1, at this loading stage, 27.6 and 22.0% of the load was distributed to the transverse and longitudinal

100

Slab-contribution

First Peak Strength

80

147

Beam-contribution Punching 54 % Failure

60

42 %

40 20 0

0

30 60 90 120 150 180 210 240 Vertical displacement (mm)

Resistance decomposition (%)

Resistance decomposition (%)

4.2 Slab Effects on Concrete Structures

100 First Peak Strength

80

54 %

60

Slab-contribution Beam-contribution Punching Failure

32%

40 20 0

0

30 60 90 120 150 180 210

Vertical displacement (mm)

(a) SM1

(b) SM2

Fig. 4.33 Resistance capacity decomposition results of SM-series specimens 40.0%

20.0% FIRST PEAK STRENGTH

15.0% 10.0%

FIRST YIELD

LONGITUDINAL COLUMN

CORNER COLUMN

5.0% 0.0%

TRANSVERSE COLUMN

TRANSVERSE COLUMN

25.0%

Percentage of load redistribution

Percentage of load redistribution

30.0%

30.0% 20.0%

50

100

150

200

250

Vertical displacement (mm)

(a) SM1

300

350

LONGITUDINAL COLUMN

FIRST YIELD

0.0%

0

FIRST PEAK STRENGTH

10.0%

0

50

100

150

200

250

Vertical displacement (mm)

(b) T1

Fig. 4.34 Varying of the load redistribution with increasing the vertical displacement

adjacent columns Therefore, RC slab not only improved the strength of specimens, but also provided more load paths for load redistribution. Although the load redistribution of SM1 varied with the increase of vertical displacement, the proportion of corner columns was between 3.5 and 5.8%. For SM2 and T2, since similar beams were designed in both transverse and longitudinal directions, loads were distributed in similar proportions on adjacent transverse and longitudinal columns. Rebar Strains The reinforcement strain of beam and plate is measured. Since some strain gauges did not work at T1, we compared the strain gauge results of T2 and SM2. Figure 4.35 shows the variation of steel strain on T2 beam with different displacement stages. It can be seen that the top reinforcement near the adjacent column and the bottom reinforcement near the inner column both record the tensile strain, which is in good agreement with the crack morphology. When the displacement reached 126 mm (TCA began to develop), the compressive strains of the top reinforcement near the inner column and the bottom reinforcement near the inner column began to decrease. Figure 4.36 shows the variation of steel strain on SM2 beam. For SM2, the strain distribution of the beam is generally similar to that of T2. The compression strain of

148

4 Spatial and Slab Effects on Concrete Structures

12000

8000

6000 4000

εy

2000

16 mm 32 mm 126 mm 180 mm 240 mm

10000

Strain (με)

8000

Strain (με)

12000

16 mm 32 mm 126 mm 180 mm 240 mm

10000

6000

εy

4000 2000

0

0

-2000 0

200

400

600

800

1000

1200

-2000

1400

0

Distance from the adjacent column interface (mm)

200 400 600 800 1000 1200 Distance from adjacent column interface (mm)

(a) Top rebar

1400

(b) Bottom rebar

Fig. 4.35 Strain profile of beam longitudinal reinforcement of T2 8000

4000

15 mm 40 mm 120 mm 150 mm

10000 8000

Strain (με)

6000

Strain (με)

12000

15 mm 40 mm 120 mm 150 mm

εy

2000

6000

4000

εy

2000

0

0

-2000 0

200

400

600

800

1000

1200

Distance from the adjacent column interface (mm)

1400

-2000 0

200

400

600

800

1000

1200

1400

Distance from adjacent column interface (mm)

(a) Top rebar

(b) Bottom rebar

Fig. 4.36 Strain profile of beam longitudinal reinforcement of SM1

the top beam near the inner column was much smaller than that of T2, which was mainly due to the introduction of flanges in the compression zone. When the displacement exceeded 120 mm, the overall tension strain of the top beam reinforcement occurred due to catenary action. Figure 4.37 shows the relationship between slab strain and vertical displacement of SM2. It should be noted that “ST” and “SB” refer to the slab strain gauge at the top and bottom rebars, respectively. As shown in the figure, ST2, SB1 and SB2 were initially compressed due to the inverted bending moment and the slab flanges being in the compression zone of the beam section near the inner column. However, the compressive strains of the three strain gauges began to decline after the displacement reached 58 mm, probably due to the development of TMA in the slab. In addition, significant tensile strains were recorded at SB3 and SB7 during this loading phase, even though they were installed at the bottom of the bars. The strain measured in SB8 and SB10 is limited. This indicated limited damage to the outer area of the slab, which acted as a compressive ring to balance the tension in the middle of the slab due to the TMA. It should be noted that the increase of steel strain in the slab became slow after the displacement reached 165 mm, which was mainly caused by punching failure at the internal connection of the slab and column. For the remaining specimens, similar results were recorded and therefore not presented.

Strain (με)

4.2 Slab Effects on Concrete Structures 6000 5500 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0

ST2 SB5

149 ST3 SB6

SB1 SB7

SB2 SB8

SB3 SB10

εy

0

50

100

150

200

250

Vertical displacement (mm)

Fig. 4.37 Steel strain of slab versus vertical displacement of SM2

4.2.5 Theoretical Analysis 4.2.5.1

Yield Load of SM-Series Specimens

The yield load of S-series specimens is determined via principle of virtual work based on the observed yield-line configuration, as shown in Fig. 4.38. The internal virtual work due to rotation of the yield line in slab and plastic hinge in beams is: 

W I = (2m sP l y θ2 + 2m sN l y θ2 + 2m sP l x θ1 + 2m sN l x θ1 ) + 2MblP θ1 N + 2MblN θ1 + 2MbsP θ2 + 2Mbs θ2

(4.4)

where l x and l y are the total span of the slab in x and y directions, respectively, θ1 and θ2 are the rotations of yield line or plastic hinges perpendicular with y and x direction, respectively, MblP and MblN are the positive and negative bending moment N are the positive and negative bending of the beam with longer span, MbsP and Mbs moment of the beam with shorter span, and MsP and MsN are the positive and negative bending moment of the slab, which could be determined by Eq. 4.5 (Park and Paulay 1975).   fy Ms = As f y ds − 0.59As  fc

(4.5)

where As is the tensile slab reinforcement per unit width and ds is the slab effective depth. The external virtue work is:  W E = FyPr edicted × δ (4.6) As shown in Fig. 4.38, θ1 =

2δ lx

and θ2 =

2δ . ly

150

4 Spatial and Slab Effects on Concrete Structures

δ θ1

θ2

δ

Fig. 4.38 Assumed yield-line pattern for SM-series specimens

The predicted yield load of S1 and S2 are 77.3 and 88.2 kN while their measured values are 80 and 90 kN. Thus, the analytical models predict the yield load of S-series specimens well.

4.2.5.2

Tensile Membrane Action

Different from the compressive membrane action of reinforced concrete slab, the rigid transverse constraint of slab edge is not the necessary condition for the tensile membrane action of RC slab. Some previous studies, such as Bailey (2001), have found that significant tensile membrane action can be developed in reinforced concrete slabs, even with simple supporting edges. As shown in Fig. 4.39, the formation of the outer compressive ring provides considerable lateral constraints to the central slab, resulting in a tensile force developed in the central region. In the past, most of the researches on membrane action simplified the beam-slab structure as completely fixed edge slab or simply supported edge slab. A more realistic experimental model (beam—slab specimen) was used in this study. The lateral constraints imposed on the edge of the test slab are actually local constraints. For simplicity, the analytical model proposed by Bailey (2001) was modified to predict TMA development in the slabs of the S-series specimens. Bailey (2001) pointed out that when bidirectional slabs with simple supports at the edges undergo large deformation, the load they bear can be greater than the yield load calculated by yield line theory. The increase of load can be attributed to two sources: the in-plane tensile stress developed in the center of the slab and the increase of yield bending moment in the outer area of the slab (compression ring). Based on the failure mode shown in Fig. 4.28, the

4.2 Slab Effects on Concrete Structures

151

Fig. 4.39 In-plane membrane force of a laterally unrestrained beam-slab structures at TMA stage

critical distribution of internal force can be determined by rigid plastic assumption. The distribution of these forces is shown in Fig. 4.40. bK T0Bot T2 = 2(1 + k)

√

L2 + l2 2

kbKT0

kLd 1 k where

In-plane force

Ld 1 k

bKT0

Reinforcement: In longer span

Reinforcement: In shorter span

Yield force =

Yield force =

Ultimate force =

Ultimate force =

Yield moment =

Yield moment =

Fig. 4.40 Assumed in-plane stress distribution for membrane action

(4.7)

152

4 Spatial and Slab Effects on Concrete Structures

kbK T0Bot C= 2 S=



k 1+k

√

L2 + l2 2

1 bK T0m (C − T2 ) = × (k − 1) tan φ tan φ 2 Sin φ = √

L L2 + l2

(4.8) (4.9) (4.10)

where K T0 is the force in steel per unit width in short direction; L and l are longer and shorter span of rectangular slab; k is the parameter defining magnitude of membrane force (k = 1 based on force equilibrium in element 1); φ is the angle defining yield line pattern; b is a parameter defining magnitude of membrane force. Sawczuk and Winnicki (1965), Bailey (2001) and Bailey et al. (2008) pointed out that there are three possible failure modes for RC bidirectional slabs under uniform pressure, as shown in Fig. 4.41. However, after the removal of the inner column, the most likely failure mode for the RC beam-slab structure (similar to the SM series specimens) was concrete breakage at the slab corner, as shown in Fig. 4.41a. This is mainly due to the inner beam passing through the center of the slab, preventing failure modes 2 and 3 from occurring. Thus, the parameter b could be calculated based on equilibrium of compressive membrane actions (kbK T0 ), concrete compressive  2 ) and the tension reinforcement at the edge of the stress block (0.67 f cu 0.45 d1 +d 2 slab ( K T02+T0 ), as shown in Eq. 4.17. The maximum depth of the compressive depth   2 of the compressive stress block is assumed to be 0.45 of average 0.45d d1 +d 2     K T0 + T0 d1 + d2 − kbK T0 = 0.67 f cu 0.45 2 2

(4.11)

where d1 and d2 are the effective depth in short span and long span, respectively. Rearranging Eq. 4.11 results in      K +1 1 d1 + d2 − T0 0.67 f cu 0.45 b= k K T0 2 2

(4.12)

For details of the analytical model, see Bailey (2001). As shown in Fig. 4.25, the proposed analytical model slightly overestimates the capabilities of TCA and TMA in beam-slab structures because strain coordination between internal beams and slabs is not well considered. In addition, the model did not simulate the punching failure between the inner column and the slab due to slab concrete breaking.

4.2 Slab Effects on Concrete Structures

153

Fig. 4.41 Three possible failure modes

(a) concrete compression failure at the corner of the slab

(b) fracture of reinforcement across the center of slab

(c) fracture of reinforcement across the intersection of yield lines

154

4 Spatial and Slab Effects on Concrete Structures

4.3 Conclusions For specimens under middle column removal scenario: 1. The test results show that the three-dimensional effect without reinforced concrete slab can improve the action of frame beam by 100%, while the 3D effect with reinforced concrete slab can improve the action of frame beam by 246.2%. 2. The test results of SM-series specimens show that the anti-load behavior is bending behavior first, followed by the action of compressive membrane, compressive arch, tensile membrane, and tensile catenary. Tension membrane action developed at displacement corresponding to 4.5–5.7% of the beam span, well before catenary action (10% of the beam span). 3. The response of the catenary and membrane of RC beams and slabs under tension will significantly reduce the possibility of structural collapse. However, the test results show that RC slab is the main source of structural bearing capacity and can bear up to 68% of the load in the stage of large displacement. 4. The comparison between the analytical results and the experimental results shows that the established analytical model can better predict the yield load, the first peak load and the ultimate load of bare frame. However, existing analysis models tend to overestimate the first peak load and ultimate load of beam-slab structures. This is mainly because the strain coordination between beam and slab is not considered in the model. For specimens under corner column removal scenario: 1. Detailed seismic design can significantly improve the ultimate bearing capacity of underground structures. Non-seismic details FC1 and FC3 only reach 77.6 and 57.0% of the progressive collapse load specified by GSA (2013), respectively. However, seismic detailed F2 can achieve the 108.0% load specified by GSA (2013). 2. The test results show that it is very conservative to ignore the role of reinforced concrete slab in resisting progressive collapse, especially for cast-in-situ structure. The ultimate bearing capacity of SC1, SC2 and SC3(with slab) is 51.6, 40.7 and 63% higher than that of FC1, FC2 and FC3(without slab). 3. The second ascending branch of the load–displacement curve of SC-series specimens shows that in THE FC-series specimens, due to the limited catenary action, tensile membrane action occurred in the slab.

References

155

References ACI 318-08 (2008) Building code requirements for structural concrete and commentary. American concrete institute, Farmington Hills Abruzzo J, Matta A, Panariello G (2006) Study of mitigation strategies for progressive collapse of a reinforced concrete commercial building. J Perform Constr Facil 20(4):384–390 Bailey CG (2001) Membrane action of unrestrained lightly reinforced concrete slabs at large displacement. Eng Struct 23(5):470–483 Bailey CG, Toh WS, Chan BM (2008) Simplified and advanced analysis of membrane action of concrete slab. ACI Struct J 105(4):30–40 General Services Administration (GSA) (2013) Alternate path analysis and design guidelines for progressive collapse resistance. Office of Chief Architects, Washington, DC Park R (1964) Tensile membrane behaviour of uniformly loaded rectangular reinforced concrete slabs with fully restrained edges. Mag Concr Res 16(46):39–44 Park R, Gamble WL (2000) Reinforced concrete slabs. Wiley, New York, p 716 Park R, Paulay T (1975) Reinforced concrete structures. Wiley, New York, p 769 Sawczuk A, Winnicki L (1965) Plastic behavior of simply supported reinforced concrete plates at moderately large deflections. Int J Solids Struct 1(1):97–111 Tsai MH (2010) An analytical methodology for the dynamic amplification factor in progressive collapse evaluation of building structures. Mech Res Commun 37(1):61–66 Yi WJ, He QF, Xiao Y, Kunnath SK (2008) Experimental study on progressive collapse-resistant behavior of reinforced concrete frame structures. ACI Struct J 105(4):433–439

Chapter 5

Load Resisting Mechanisms of Flat Slab Structures to Resist Progressive Collapse

RC flat slab structures are being widely used in residential and industrial buildings around the world due to construction and decoration ease, short load transfer path, flexible column layout, and short story height. However, RC flat slab structures are vulnerable to progressive collapse, in which the relatively brittle failure mechanism attributable to punching shear failure may lead to catastrophic consequences. For example, the collapse of a 16-story building during construction in Boston, Massachusetts, in 1971 and Sampoong Department Store in Seoul, South Korea, in 1995 that claimed over 500 deaths. More recently, the collapse of a 12story residential building in Miami, Florida, on June 24, 2021, which claims near 100 deaths, draws the attention of public to progressive collapse of RC flat slab structures again. These disastrous tragedies clearly demonstrated the vulnerability of flat slab buildings to progressive collapse. Most of existing progressive collapse studies on RC flat slab structures focused on behavior of slab-column connections. However, studies on slab-column connections are unable to account for lateral restraints from adjacent bays, which may significantly influence the post-punching performance of RC flat slab structures. In this case, the second load resisting mechanism, namely tensile membrane action, is neglected. Therefore, to reflect the actual progressive collapse performance of RC plat slab system, multibay RC flat slab specimen should be tested. In this chapter, multibay RC flat slab specimens are designed and tested to investigate the structural behavior of RC flat slab subjected to a corner or interior column removal scenario. Based on test results, the failure modes and load resisting mechanisms of flat slab structures under progressive collapse are discussed. Moreover, the effects of drop-panel and loading method are quantified.

© China Architecture & Building Press 2023 K. Qian and Q. Fang, Progressive Collapse Resilience of Concrete Structures: Mechanisms, Simulations and Experiments, https://doi.org/10.1007/978-981-99-0772-4_5

157

158

5 Load Resisting Mechanisms of Flat Slab Structures to Resist …

5.1 Progressive Collapse Resistance of RC Flat Slabs After the Loss of a Corner Column In this chapter, “ND” series and “WD” series specimens, which were designed to simulate flat slab without drop panel and flat slab with drop panels, respectively— were tested experimentally to investigate the effects of reinforcement ratio of slab. Moreover, the additional contribution to load resistance provided the drop panels were quantified by comparing the test results of these two series of specimens.

5.1.1 Specimen Design Dimensions and reinforcement details are shown in Table 5.1. Different reinforcement ratios were designed for the slabs of WD1, WD2 and WD3. Figure 5.1 shows the geometrical and reinforcing details of WD1. The thickness of column and slab concrete cover slabs are 20.0 and 7.0 mm respectively. For WD-series samples, there is one corner column, three expanded columns, and four drop-slabs cast in one piece. For all specimens, the dimension of the corner column representing the removed column is 200.0 mm square and the enlarged column is 250.0 mm square to ensure that these enlarged columns will not be damaged. In addition, reinforcement is arranged on the top and bottom of the column-slab connection to prevent the brittle failure of the specimen in the column slab connection. The thickness of the drop slab is 40.0 mm, and the reinforcement in the drop slab is a layer of reinforcement with a spacing of 70.0 mm. Each expansion column is prefabricated with 4 bolts and connected to the steel leg. Figure 5.2 shows details of reinforcement for ND2 and WD2 slabs. The reinforcement in middle strip consists of two layers of R6 reinforcement at the top and bottom with spacing of 125.0 mm, and the column strip consists of two layers of R6 reinforcement at the top and bottom with spacings of 60.0 and 125.0 mm, respectively. Specimens ND1, ND2 and ND3 correspond to WD1, WD2 and WD3, respectively. As shown in Table 5.1, the columns and slabs of the ND series tests have similar details to the corresponding WD-series tests, but without the drop panel. High yield strength steel is used for longitudinal reinforcement (T16), while mild steel is used for transverse and slab reinforcement (R6). T16 and R6 represent deformed steel bars with diameter of 16 mm and plain steel bars with diameter of 6 mm respectively. The average compressive strength of concrete of ND-series and WD-series is 19.5 and 26.0 MPa, respectively. WD2 was designed according to ACI 318-08 (2008). The dead load (DL) of 210.0 mm slab is 5.1 kPa. The extra DL is assumed to be 1.0 kPa. The equivalent additional DL due to the weight of the filled wall is 2.25 kPa. The live load (LL) was set at 2.0 kPa. A one-third scale substructure was cast and tested. According to the load combination (1.2DL + 0.5LL) proposed in DoD (2009), a uniform pressure of 11.0 kPa was applied to the surface of the top of slabs. In order to make sure the same demand/capacity ratio was applied to the scale slab and the prototype slab, 11.0 kPa of uniform load was applied to the scale

Reinforcement ratio = 2.0%

70

R6@35 R6@60 R6@35

70

R6@125

70

70

R6@60

R6@125

R6@70

R6@125

R6@250

R6@70

R6@125

R6@250

R6@250

R6@125 R6@70

R6@70

R6@250

R6@70

R6@125

R6@125

R6@250

R6@70

R6@125

R6@250

Column strip (mm)

Mid strip (mm)

Slab bottom rebar

Column strip (mm)

Mid strip (mm)

Slab top rebar

Notes 1 mm = 0.0393 in.; R6 is plain reinforcing bar with diameter of 6 mm

WD3

WD2

WD1

ND3

Cross section = 200 × 200 mm2

70

70

Height = 400 mm

ND1

ND2

Slab thickness

Column stub

Test

Table 5.1 Specimen properties

40 mm

40 mm

40 mm

N/A

N/A

N/A

Drop-panel thickness (mm)

R6@70

R6@70

R6@70

N/A

N/A

N/A

Drop-panel rebar (mm)

15.9

15.9

15.9

15.9

15.9

15.9

Design axial force (kN)

5.1 Progressive Collapse Resistance of RC Flat Slabs After the Loss … 159

160

5 Load Resisting Mechanisms of Flat Slab Structures to Resist …

Fig. 5.1 Test setup

1 2

3

5

4

10

8

11

6 8 8

7

7 7

9

slab. The designed axial forces of all specimens are listed in Table 5.1 according to DoD (2009)’s recommendations.

5.1.2 Test Setup and Instrumentation As shown in Fig. 5.1, the test setup of this study are similar to those shown in Fig. 4.17. See Sect. 4.2.2 for details. A large amount of measuring equipment is installed both internally and externally to monitor the response of test specimens. The force on the corner stub was measured with a load cell. Two tensile and compression load cells are connected horizontally to the steel assembly to measure the horizontal response of the box in each direction. A linear variable differential transformer (LVDT) with a stroke of 300 mm was installed vertically to measure the vertical movement of the corner column during the test. During the test, a displacement sensor with a stroke of 120 mm was installed horizontally to monitor the horizontal movement of the corner joint. The remaining six displacement sensors are placed vertically to monitor the deflection of the slab. For the ND-series specimens, a total of 23 strain gauges were installed on the rebar at key locations to monitor strain along the corner column and slab during the test. For WD series samples, strain gauges are installed not only in the column and plate, but also in the drop panel. The position of the strain gauge placed on the drop panel and corner column is shown in Fig. 5.2, and the position of the strain gauge placed on the slab reinforcement is shown in Fig. 5.3.

13 R6 @ 70

Interior Column

Transverse Adjacent Column

5.1 Progressive Collapse Resistance of RC Flat Slabs After the Loss …

161

SD7

13 R6@70 R6@55

R6@55

4T16 8T16

Detail of C-C

SD2 SD1

13 R6@70

Strain Gauge 7 R6 @ 70

SD3

7 R6 @ 70

7 R6@70

Detail of D-D

Corner Joint Stub

SD6 SD5 SD4

Longitudinal Adjacent Column

R6@55

Plan View Strain Gauge

Elevation View

Fig. 5.2 Dimensions, cross-section details, and strain gauge locations of WD-series specimens

ST7

SB6

SB7

SB5

SB8

ST3 ST2 ST1

ST6 ST5 ST4

SB1

SB2

SB3

SB4

Integrity Rebar Bottom Slab Reinforcement Top Slab Reinforcement

Fig. 5.3 Slab reinforcement details and strain gauge locations of specimens ND2 and WD2

162

5 Load Resisting Mechanisms of Flat Slab Structures to Resist …

5.1.3 Test Results Two groups of one-third scale flat slab specimens (ND and WD) were experimentally studied to assess the behavior of drop-panel on the performance of the flat slab subjected to a corner column removal scenario. Table 5.2 summarizes the key points of the test results of the 6 specimens, which are discussed as follows. Figure 5.4a shows the changes of vertical load and horizontal reaction forces with the increasing vertical displacement of corner joints. Four performance levels were identified. The performance levels of PL1, PL2, PL3, and PL4 represent the first flexural crack, the first yield load, the first peak capacity, and the beginning of the development of the tensile membrane action, respectively. When the load was 1.8 kN, the first crack appeared in the interface between slab and adjacent enlarged column. After the first crack, several bending cracks appeared in the floor near the corner column under the Vierendeel action. Under a load of 6.3 kN, the first diagonal crack of the slab was formed and passed through the center of the slab. However, the first yield of the top reinforcement occurred under a load of 7.3 kN, corresponding to a vertical displacement of 30.9 mm. When the vertical displacement reached 70.3 mm, the first peak capacity of 8.5 kN was obtained, equivalent to 53.4% of the recommended design axial load in DoD (2009). During this loading phase, more oblique cracks are formed parallel to the first diagonal crack, and these diagonal cracks move towards the corner column. With the further increase of the vertical displacement, the yield of the slab is serious and the resistance of the specimen begins to decline. When the displacement reached 120.3 mm, equivalent to 4.2% of TDR, the load–displacement curve began to rise again (due to the tensile membrane action). With the further increase of vertical displacement, diagonal cracks run through the floor depth. At the end of the test, a punching crack appeared at the corner columnslab connection. However, no significant punching failure was observed in the roof around the adjacent columns. Figure 5.5a shows the crack pattern development of ND1 at different performance levels. It should be noted that no cracks were observed in the corner column and joint zone during the test. Both transverse and longitudinal horizontal reaction forces are measured by tension/compression load sensors connected to the steel assembly. As shown in Fig. 5.4a, the recorded horizontal compression force reached its limit before the first crack appeared in the specimen. However, after the first crack appeared, its strength increased significantly. The lateral response of the measured horizontal reaction is basically the same as that of the measured longitudinal response. At the displacements of 70.3 and 80.1 mm, the maximum compressive forces measured in transverse and longitudinal directions were 4.1 and 4.5 kN, respectively. It was important to note that the measured resistance did not represent the horizontal axial force developed at the center of the corner joints, and most of the resistance is applied to balance the positive bending moment at the slab-column connection. When the displacement reached 202.4 mm, equivalent to 8.4% of TDR, tensile reaction forces were recorded in both horizontal load cells. At the end of the test, the maximum horizontal tensile reactions measured were 6.0 kN in transverse direction and 6.1 kN in longitudinal

20.0

15.4

22.0

32.2

ND2

ND3

WD1

WD2

WD3

36.2

26.8

19.1

22.4

14.3

8.5

FPL kN

10.9

9.5

8.5

6.6

4.2

4.5

MCHR-T kN

11.9

10.4

9.0

5.2

5.7

4.5

MCHR-L kN

13.9

9.5

9.0

11.0

7.3

6.0

MTHR-T kN

12.5

9.0

7.9

9.5

8.1

6.1

MTHR-L kN

40.3

32.5

24.6

24.8

18.5

17.3

SPL kN

32.1

23.1

16.0

18.6

11.8

6.9

DS kN

1.13

1.16

1.19

1.20

1.21

1.23

DLIF

Notes FPL is first peak load; -T and -L indicate transverse and longitudinal direction, respectively; MCHR and MTHR are maximum compressive horizontal reaction force and maximum tensile horizontal reaction force, respectively; SPL is second peak load; DS is dynamic strength; DLIF is dynamic load increase factor, as ratio of static ultimate capacity to peak value measured in capacity curve

7.3

11.6

ND1

Yield load kN

Test

Table 5.2 Test results

5.1 Progressive Collapse Resistance of RC Flat Slabs After the Loss … 163

Vertical load (kN)

40 30

Note: LH=Longitudinal horizontal reaction force TH=Transverse horizontal reaction force

20

PL2

Horizontal reaction (kN)

10

PL3 PL4

PL2

PL1

PL3

PL1 PL4

0

WD1 WD1-TH WD1-LH ND1 ND1-LH ND1-TH

-10 -20 0

100 200 300 400 Vertical displacement (mm)

500

Vertical load (kN)

(a) ND1 and WD1 50

Note: LH=Longitudinal horizontal reaction force TH=Transverse horizontal reaction force

40 30

PL3

PL2

PL4

20

Horizontal reaction (kN)

PL3 PL4

10 PL2 PL1 PL1 0

WD2 WD2-TH WD2-LH ND2 ND2-LH ND2-TH

-10 -20 0

100 200 300 400 Vertical displacement (mm)

500

(b) ND2 and WD2 Vertical load (kN)

Fig. 5.4 Comparison of vertical load and horizontal reaction force versus vertical deflection

5 Load Resisting Mechanisms of Flat Slab Structures to Resist …

50 40

Note: LH=Longitudinal horizontal reaction force TH=Transverse horizontal reaction force PL4 PL3 PL2 PL3 PL4

30 20

PL2

PL1

Horizontal reaction (kN)

164

10 PL1

0

WD3 WD3-TH WD3-LH ND3 ND3-LH ND3-TH

-10 -20 0

100 200 300 400 Vertical displacement (mm)

(c) ND3 and WD3

500

5.1 Progressive Collapse Resistance of RC Flat Slabs After the Loss … ND1-PL1

Corner

ND1-PL2

Corner

ND1-PL3

Corner

ND1-PL4

Corner

(a) ND1

WD1-PL1

Corner

WD1-PL2

Corner

WD1-PL3

Corner

WD1-PL4

ND2-PL1

Corner

ND2-PL2

Corner

ND2-PL3

Corner

ND2-PL4

WD2-PL1

Corner

WD2-PL2

Corner

WD2-PL3

Corner

WD2-PL4

ND3-PL1

165 WD3-PL1

Corner

ND3-PL2

WD3-PL2

Corner

ND3-PL3

WD3-PL3

Corner

ND3-PL4

Corner

Corner

Corner

Corner

(b) WD1

(c) ND2

(d) WD2

(e) ND3

WD3-PL4

(f) WD3

Fig. 5.5 Cracking patterns at different performance levels of test specimens

direction. The measured vertical and horizontal reaction forces of WD1 specimens are shown in Fig. 5.4a. Bending cracks occurred on the column-slab interface under the load of 3.2 kN. Under the Vierendeel action, a few bending cracks appeared at the bottom of the vertical slab around the corner column under the load of 7.6 kN. Under the load of 10.2 kN, the first diagonal crack formed in the slab. It should be noted that this diagonal crack is connected to the drop-panel edge around the adjacent longitudinal and transverse columns, which increased the bending capacity of the slab section near the adjacent column and moves the most critical section from the slab-adjacent column interface to the edge of drop-panel. Under the load of 15.4 kN, the top reinforcement yielded for the first time, and the corresponding vertical displacement was 30.0 mm. When vertical displacement reached 110.7 mm, the first peak capacity Pcu was achieved at a load of 19.1 kN, corresponding to 120.1% of the design axial load recommended by the DoD (2009). With the further increase of vertical displacement, the main diagonal crack in the slab gradually widened. When the vertical displacement reached 130.0 mm, concrete is broken at the top slab near the corner column. When the displacement reached 221.3 mm, equivalent to 9.2% of TDR, the load–displacement curve began to rise again (due to tensile membrane action). Compared with ND1 specimen, there was no obvious punching failure at

166

5 Load Resisting Mechanisms of Flat Slab Structures to Resist …

the corner column-slab connection during the test, because the drop-panel significantly increased the effective depth of the slab. Similar to ND1, when the vertical displacement reached 350.0 mm, serious bending cracks are formed on the bottom surface of the corner drop-panel. In addition, similar to ND1, no cracks appeared on the corner columns and joints during the test. Figure 5.5b shows the crack pattern development of WD1 at different performance levels. At displacements of 90.8 mm and 100.7 mm, the measured maximum lateral and longitudinal compression forces were 8.5 and 9.0 kN, respectively. In the final phase of the test, the maximum horizontal tensile reaction forces were measured to be 9.0 and 7.9 kN, respectively. The vertical and horizontal reaction forces of sample ND2 vary with the vertical displacement of corner joints as shown in Fig. 5.4b. In general, the crack development of ND2 is similar to that of ND1. The key points of the test results are shown in Table 5.2. Therefore, only the most important differences between the two specimens are highlighted here. For ND1, the first diagonal crack of the slab is formed under a load of 6.3 kN. For ND2, under the load of 9.1 kN, diagonal cracks were formed in the slab and passed through the center of the slab. Another difference in fracture mode between ND2 and ND1 is that a punching failure occurred at the corner column-slab connection of ND2 with a displacement of 380.9 mm, while the punching failure of ND1 occurred at the last stage of the test (410.9 mm). In general, the crack pattern of ND2 is much finer than that of ND1. In ND1, only a few discrete diagonal cracks are formed. In ND2, a large number of cracks were found between diagonal cracks. The first yield and the first peak bearing capacity of the specimen are significantly improved by the higher slab reinforcement ratio. The failure mode of ND2 is shown in Fig. 5.6, and the crack pattern development of ND2 is shown in Fig. 5.5c. The reaction force of WD2 specimen in vertical and horizontal directions varies with the vertical displacement of corner joints, as shown in Fig. 5.4b. The key points of the test results are shown in Table 5.2. In general, the crack development of WD2 is similar to that of WD1. Therefore, only the most important differences between the two specimens are emphasized. WD1 forms the first diagonal crack in the slab Fig. 5.6 Failure mode of specimen ND2

5.1 Progressive Collapse Resistance of RC Flat Slabs After the Loss …

167

Fig. 5.7 Failure mode of specimen WD2

under the load of 10.2 kN. WD2 first forms diagonal cracks under 14.3 kN load and connects with the edges of adjacent columns. Similar to ND2, the crack pattern of WD2 is much finer than that of WD1. Figure 5.7 shows the failure mode of WD2, and Fig. 5.5d shows the crack pattern development corresponding to different performance levels of WD2. The vertical load and horizontal reaction forces versus displacement histories specimen ND3 are shown in Fig. 5.4c. In general, the crack development of ND3 was similar to that of ND1. For ND1, the first diagonal crack of the slab was formed under a load of 6.3 kN. While diagonal cracks formed and passed through the center of the slab of ND3 under the load of 13.4 kN. Another difference between ND3 and ND1 in failure mode is that the punching failure occurred in the corner column-slab connection of ND3 at the displacement of 50.4 mm, which has not yet reached the first peak bearing capacity. However, although signs of shear failure were observed before the first peak bearing capacity was reached, this failure deteriorated slowly and did not prevent further redistribution of load. This is probably due to the special design-complete reinforcement installed on both the top and bottom of the slab. In general, the crack pattern of ND3 was much finer than that of ND1 and ND2. The failure pattern of ND3 was similar to that of ND2, so it was not present. The crack pattern development of ND3 is shown in Fig. 5.5e. The changes of vertical and horizontal reaction forces of WD3 specimens with vertical displacement of corner joints are shown in Fig. 5.4c. The first diagonal crack formed in the slabs of Specimens WD1 and WD2 under the load of 10.2 and 14.3 kN, respectively. Inclined cracks first formed in the slab of WD3 under 17.1 kN load. It should be noted that in the final stage of the test, slight cracks were also observed in the corner joints and corner columns. WD3 has a similar failure mode to WD2, so it didn’t appear. Figure 5.5f shows the development of crack pattern corresponding to different performance levels of WD3.

168

5 Load Resisting Mechanisms of Flat Slab Structures to Resist …

5.1.4 Discussion of the Test Results 5.1.4.1

Comparison of Performance of ND- & WD-Series Specimens

The comparison of load–displacement relationship between WD series specimens and ND series specimens is shown in Fig. 5.4. By comparing the first peak capacity of the two specimens, it can be seen that specimen ND1 can only reached 53.4% of the designed axial load recommended by the Ministry of National Defense, while specimen WD1 can reach 120.1% of the designed axial load. Compared with ND1, the first peak capacity of WD1 increased by 124.7%. According to the test results, ND2 and WD2 can achieve 89.9% and 168.6% of the axial load recommended by the DoD (2009), respectively. Compared with ND2, the first peak capacity of WD2 increased by 87.5%. ND2 and WD2 reached the second peak load capacity of 18.5 and 32.5 kN respectively in the final stage of the test. According to the test results, the ND3 and WD3 specimens can reach 140.9 and 227.7% of the design axial load recommended by DoD (2009). Compared with ND3, the first peak capacity of WD3 increased by about 61.6%. ND3 and WD3 reached the second peak load capacity of 24.8 and 40.3 kN, respectively, in the final stage of the test. In this study, the initial stiffness is defined as secant stiffness at the first yield strength. The initial stiffness of ND1, ND2 and ND3 were 0.24, 0.39 and 0.61 kN/mm, respectively. The initial stiffness of WD1, WD2 and WD3 were 0.51, 0.63 and 0.86 kN/mm, respectively. Therefore, the initial stiffness can be improved 112.5% by adding the drop-panel. In the case of column loss, the survival of structures is related to their ability to dissipate input energy. In this study, the energy consumption is defined as the area under the load–displacement curve. The dissipative energy of ND1 and WD1 in the final stage of the test is 4.1 and 7.6 kN, respectively. The dissipation energy of ND2 and WD2 was 6.3 and 11.3 kN, respectively. The dissipated energy of ND3 and WD3 was 8.4 and 14.7 kN, respectively. Therefore, the energy dissipation capacity of WD1, WD2 and WD3 increased by 85.4, 79.4 and 75.0%, respectively, after the drop-panel was added. Figure 5.8 shows the relationship between slab strain and vertical displacement in specimen ND1. As shown in Fig. 5.8a, strain gauges ST1 and ST2 are in compression state during the test. The maximum compressive strains of ST1 and ST2 are −561 and −298 με, respectively. This confirms that the direction of the bending moment of the beam end connected to the corner joint changes due to the equivalent Vierendeel action after the corner column is removed. The corner columns and joints were relatively intact during the experiment, indicating that the degree of fasting effect was not weakened during the experiment. For the beam-column-slab substructure, the Vierendeel effect is sluggish with the increase of corner joint damage. The strain gauge ST4 recorded the tensile strain of 2331 με and yielded at 30.9 mm vertical displacement. The maximum tensile strains of ST5 and ST6 are 2143 and 1799 με, respectively. They are close to yield strain, though they are not yielded. However, the maximum tensile strain of ST7 is 120 με, indicating that most of the initial force received by the corner column after damage is transmitted to the adjacent column, while the force transmitted to the inner column is negligible. Figure 5.8b shows

5.1 Progressive Collapse Resistance of RC Flat Slabs After the Loss … 12000

8000

ST2

ST4

ST5

ST6

ST7

Strain (με)

Strain (με)

10000

ST1

6000 4000 2000 0 -2000 0

3500 3000 2500 2000 1500 1000 500 0 -500

SB1 SB4 SB7

0

50 100 150 200 250 300 350 400 450 Vertical displacement (mm)

(a) Top layer

169 SB2 SB5 SB8

SB3 SB6

50 100 150 200 250 300 350 400 450 Vertical displacement (mm)

(b) Bottom layer

Fig. 5.8 Strain of slab reinforcement versus vertical displacement in specimen ND1

the relationship between the strain of slab bottom bars and vertical displacement. With the exception of SB5 and SB6 strain gauges, the strain gauges at the bottom of the reinforcement are initially in a state of compression, but with the increase of the vertical displacement, the strain gauges change to a state of tension. When the displacement reached 100.2 mm (3.94 in.), the tensile strains of SB1, SB2, SB3 and SB4 increase significantly. In this displacement stage, the load–displacement curve appeared to rise again. The maximum tensile strains of SB7 and SB8 are 2143 and 191 με, respectively. This is due to the fact that for ND series specimens, the main diagonal crack passes through the center of the slab. It should be emphasized that the strain of the ND1 column longitudinal reinforcement was also measured. The maximum tensile strain is 363 με and the maximum compressive strain is −209 με. This is consistent with the observed crack pattern; no cracks were observed in the column and joint region (elastic region) in the ND1 test. For WD1, in general, the variation trend of strain curve is similar to that of ND1. Figure 5.9 shows the variation of reinforcement strain in drop-panel of WD1. The position of these strain gauges is shown in Fig. 5.2. The maximum tensile strains of SD1, SD2 and SD3 under Vierendeel action are 2965 με (over yield strain), 1674 and 988 με, respectively. The maximum compressive strains of SD4, SD5 and SD6 are −295, −254 and −261 με, respectively.

Strain (με)

Fig. 5.9 Strain of reinforcing bar in drop panels of specimen WD1

4000 3500 3000 2500 2000 1500 1000 500 0 -500

SD1 SD4

0

SD2 SD5

SD3 SD6

50 100 150 200 250 300 350 400 450 Vertical displacement (mm)

170

5.1.4.2

5 Load Resisting Mechanisms of Flat Slab Structures to Resist …

Discussion of Punching Shear Strength of Corner Column-Slab Connection

It can be seen from the crack pattern that the unpredictable punching shear cracks are formed in ND series specimens. As mentioned above, in order to better study the mechanical properties of specimens at the large deformation, the test slab is designed to prevent brittle shear failure. Table 5.3 summarizes the comparison of the measured punching shear strength of test specimens with the predicted values of ACI 318-08 (2008), Eurocode 2 (2004a, b), CEB-FIP MC90 (1978) and DIN 1045-1 (2001). The design formula used to predict punching capacity in the above design specifications can be found in Gardner (2011). According to ACI 318-08 (2008) and Eurocode 2 (2004a, b), it is assumed that the critical section has both rectcorner and circular perimeters to calculate the punching shear capacity of corner slab connections. It can be seen from the table that in ACI 318-08 (2008), regardless of assuming rectcorner or circular perimeters, the punching shear capacity of corner slab-column connections is significantly overestimated. However, the predictions of the European specification, particularly those of CEB-FIP MC90 (1978) and DIN 1045-1 (2001), are closer to the measured values than those of ACI 318-08 (2008). This is mainly due to the fact that the ACI 318-08 (2008) punching and shearing formula neither considers the role of reinforcement ratio nor the size of components. Guandalini et al. (2009) concluded that the nominal punching shear strength decreased with the decrease of flexural reinforcement ratio. In addition, in general, the assumed circular perimeter is closer to the calculated result than the rectcorner perimeter.

5.1.4.3

Dynamic Effect

With the improvement of the design criterion, the dynamic ultimate strength of each specimen is predicted by using the simplified analysis model-bearing capacity curve method. Then, the corresponding dynamic load increasing factor (DLIF) of each specimen is determined. DLIF is defined as the ratio of the static ultimate bearing capacity to the dynamic ultimate bearing capacity of each specimen (i.e., the peak value of the bearing capacity curve). The capacity curve method was proposed by Abruzzo et al. (2006) based on the principle of energy conservation. Through nonlinear pushover analysis, the load–displacement curve of the structure can be obtained, and the area under the curve represents the strain energy of the structure. At the moment when the system reaches equilibrium, this internal energy will equal the external work, defined as the product of the constant applied load (the column axial force before failure) and the resulting displacement. If the system is not deformable enough to dissipate the required energy, the internal and external work will never balance each other, resulting in collapse. Thus, the capacity curve can be constructed by dividing the accumulated stored energy by the corresponding displacement. However, it should be noted that in this simplified model, the energy dissipated by damping is not taken into account. It’s mathematically expressed as

31.9

31.9

31.9

17.3

17.2

21.1

ND1

ND2

40.6

40.6

40.6

S kN VACI

25.6

21.1

16.7

R kN VEC2

32.6

26.9

21.3

S kN VEC2

21.7

17.9

14.2

S VCEB kN

23.0

18.9

15.0

S kN VDIN

0.58

0.66

0.54

0.54

VTest R VACI

kN

0.46

0.52

0.43

0.43

VTest S VACI

kN

0.89

0.82

0.82

1.04

VTest R VCE2

kN

0.70

0.65

0.64

0.81

VTest S VCE2

kN

1.05

0.97

0.96

1.22

VTest S VCEB

kN

0.99

0.92

0.91

1.15

VTest S VDIN

kN

S , V S are punching shear strength according to CEB-FIP MC90 and DIN 1045-1 by considering control perimeters with rounded corners and straight VCEB DIN corners, respectively

R , V S are punching shear strength according to Eurocode 2 by considering control perimeters with rounded corners and straight corners, respectively VCE2 CE2

R , V S are punching shear strength according to ACI 318-08 by considering control perimeters with rounded corners and straight corners, respectively Note VACI ACI

Mean

ND3

R kN VACI

VTest kN

Test

Table 5.3 Comparison of measured punching shear resistance with design codes

5.1 Progressive Collapse Resistance of RC Flat Slabs After the Loss … 171

172

5 Load Resisting Mechanisms of Flat Slab Structures to Resist …

Strain (με)

Fig. 5.10 Illustration of capacity curve and load curve of each specimen

45 40 35 30 25 20 15 10 5 0

Load curve Capacity curve

1 ud



ud

WD2 ND3

WD1 ND2 ND1

Pushover curve of ND1

0

PCC (u d ) =

WD3

50 100 150 200 250 300 350 400 450 Vertical displacement (mm)

PN S (u)du

(5.1)

0

where PCC (u) and PN S (u) are the capacity function and the nonlinear static loading estimated at the displacement demand u, respectively. The bearing capacity curve and load curve of each specimen are shown in Fig. 5.10. The load curves of WD1, ND3, WD2, and WD3 at 137.8, 59.2, 50.8 and 24.8 mm (5.43, 2.33, 2.00 and 0.98 in.) were the intersection points of the capacity curves. Therefore, the four specimens will not collapse because energy balance can be achieved. However, the load curves of ND1 and ND2 are larger than the dynamic ultimate bearing capacity of the corresponding specimens. Therefore, if the corner supports were suddenly removed, both ND1 and ND2 would collapse completely. Taking ND1 as an example, obvious tensile membrane action is observed on load–displacement curves. However, the increased dynamic limit capacity due to this tensile membrane action is very limited. Therefore, the contribution of tensile membrane action in resistance to real dynamic progressive collapse events may not be very reliable. The predicted value of DLIF of each specimen is between 1.13 and 1.23. The effect of dynamics on ultimate bearing capacity is much lower than assumed in GSA (2013), which may be explained by the relative ductility properties exhibited by the test specimens.

5.2 Progressive Collapse Resistance of RC Flat Slabs After the Loss of a Middle Column Two groups of RC slab substructures with multiple panels were tested with two different loading devices. The load redistribution behavior and residual load resistance capacity of the slab structures under two different collapse stages were studied. The main test results such as load–displacement response, crack pattern, failure mode and local strain gauge readings were introduced and discussed. Based on the experimental results, the influence of design parameters was further analyzed.

5.2 Progressive Collapse Resistance of RC Flat Slabs After the Loss …

173

5.2.1 Design of Test Specimens The prototype structure with thickness of 280 mm was non-seismically designed. The average live load was 3.0 kPa, and the additional constant load was 1.0 kPa. According to the direct design method (ACI 318-11 2011), the flexural reinforcement of floor slab was designed. The flexural reinforcement ratios of the slab in x and y directions were both 0.25%. The top tensile reinforcement is cut from the surface of the support at 0.22 ln according to ACI 318-11 (2011), where ln is the clear span length. Two integrity reinforcement with 25 mm diameter steel bars, determined according to Eq. 5.2, meet the requirements of ACI 352.1R-11 (2011). Asmi =

0.5wu l x l y φ fy

(5.2)

where Asmi is the minimum area of the integrity reinforcement in normal directions passing through the column; wu is the factored uniformly distributed load; f y is the yield strength of the integrity reinforcements; φ = 0.9; and l x and l y are the column spacing in the x- and y-directions.

5.2.2 Design Variables in Test Specimens The control prototype slab was scaled to the test specimen P1-70-1.0, which belonged to P1 series. It has a slab thickness of 70 mm and a span aspect ratio of 1.0. The specimen names of the remaining specimens are shown in Table 5.4. Due to the need for multi-panel panels, the prototype panels had to be scaled down due to the limitations of laboratory testing facilities. The dimensions of the components in the prototype and model structures are shown in Table 5.5, and the reinforcement ratio remains the same. In this study, six 1/4 scale specimens were divided into two series (P1 and P2) to investigate the performance of RC slab structures at different collapse stages. Therefore, one of the most important variables is the loading state. Four specimens of P1 series were loaded by centralized pushdown loading, and the other two specimens of P2 series were loaded by 12-point loading system to simulate uniform distributed pressure (UDP) loading. In P1 series, the variables of span aspect ratio and slab thickness are studied. The details of P1-100-1.4 are shown in Fig. 5.11. P1-100-1.4 is supported by 8 columns (C1 to C8) along the periphery of the affected area. The central column C9 is notionally lost before the load is applied. The extension of the slab in P1-100 1.4 is 375 mm, lx/4 and 525 mm, ly/4 in the x and y directions, respectively, to simulate the surrounding bay near the test structure. As shown in Table 5.4, the thickness and reinforcement ratio of P1-100 1.0 are similar to that of P1-100 1.4, but the span aspect ratio is 1.0 (lx/ly = 1.0). The only difference between P1-70 1.0, P1-100-1.0 and P1-55-1.0 is the slab thickness. In the P2-series, the influences of drop panel were investigated by comparing the

174

5 Load Resisting Mechanisms of Flat Slab Structures to Resist …

Table 5.4 Test specimen designation and properties Specimen

Panel dimension

Slab top reinforcement

Slab bottom reinforcement

Drop panel reinforcement

P1-100-1.0

1500 mm × 1500 mm

R6@130 (ρtx,y = 0.24%)

R6@130 (ρbx,y = R6@80 (ρdx,y = 0.24%) 1.4%)

P1-100-1.4

1500 mm × 2100 mm

R6@130 (ρtx,y = 0.24%)

R6@130 (ρbx,y = R6@80 (ρdx,y = 0.24%) 1.4%)

P1-70-1.0

1500 mm × 1500 mm

R6@190 (ρtx,y = 0.25%)

R6@190 (ρbx,y = R6@80 (ρdx,y = 0.25%) 1.4%)

P1-55-1.0

1500 mm × 1500 mm

R6@250 (ρtx,y = 0.25%)

R6@250 (ρbx,y = R6@80 (ρdx,y = 0.25%) 1.4%)

P2-55-1.0

1500 mm × 1500 mm

R6@250 (ρtx,y = 0.24%)

R6@250 (ρbx,y = R6@80 (ρdx,y = 0.25%) 1.4%)

P2-P-55-1.0

1500 mm × 1500 mm

R6@250 (ρtx,y = 0.24%)

R6@250 (ρbx,y = N/A 0.25%)

Table 5.5 Dimensions of structural components of P1-70-1.0 Dimension

Prototype building (mm)

Specimens (mm)

Slab thickness

400

100

Column section

600 × 600

200 × 200 (enlarged for fixed boundary condition)

Column spacing

6000/6000 (x-/y-direction) 1500/1500 (x-/y-direction)

Concrete cover thickness 25

7

behavior of P2-55-1.0 and P2-P-55-1.0. Specimens P1-100-1.4, P1-100-1.0, P1-701.0, P1-55-1.0 and P2-55-1.0 had a floor thickness of 35 mm and were mounted above the support column. Single layer rebar with reinforcement ratio of 1.4% was designed in drop-panels. However, no drop-panel designed in Specimen P2-P-55-1.0. The characteristics of the material are shown in Table 5.6.

5.2.3 Test Setup and Instrumentation Two different test setups were designed for the P1- and P2-series. The schematic diagram of P1-series test setup is shown in Fig. 5.12. Eight strong steel supports were used to support the peripheral edges of the test specimen. However, there is no support for installation below the center column as it is nominally removed. A downward displacement was applied at the center column through a 600 mm stroke hydraulic jack to simulate the existing axial load before the loss of the center column. To ensure a symmetric failure mode and to simulate the horizontal constraints from the upper floor of the building, a special steel assembly was designed (Item 4 in Fig. 5.12). The steel column (Item 3 in Fig. 5.12) can only move freely in the vertical

5.2 Progressive Collapse Resistance of RC Flat Slabs After the Loss …

175

3

C5C C5

C6 C6

C7 C7

R6@130

SLAB

C8 C8

C9 C9

R6@130

R6@130

SECTION 3-3

C4 C4

R6@130

R6@130

R6@130

100mm

C3 C3

0

C1 C1

C2 C2

1

1

3

X

R6@80

R6@130

R6@130 2

R6@80

Y

R6@130

R6@130

R6@130

2 SECTION 1-1

SECTION 2-2

Fig. 5.11 Details of typical specimen P1-100-1.4 (Unit in mm)

direction, but the horizontal and rotational freedoms are limited. The P2-series is designed to study the residual load resistance (RLRC) of the remaining structure after column removal, and to design a 12-point loading system (Item 6 in Fig. 5.13) to equivalently simulate the application of UDP on the slab. A large number of measuring devices are installed internally and externally to monitor the response of the specimen. A load cell (Item 1 in Fig. 5.12) was used to measure the applied force. A series of displacement sensors are placed at different locations below the floor slab

176

5 Load Resisting Mechanisms of Flat Slab Structures to Resist …

Table 5.6 Material properties

Property

Value

Concrete compressive strength P1-100-1.0

25.5 MPa

P1-100-1.4

26.0 MPa

P1-70-1.0

24.1 MPa

P1-55-1.0

25.6 MPa

P2-55-1.0

22.3 MPa

P2-P-55-1.0

25.2 MPa

R6 (round rebar with diameter of 6 mm) Yield strength

500 MPa

Ultimate strength

617 MPa

Yield strain

2650 με

Elongation

0.21

T13 (deformed rebar with diameter of 13 mm) Yield strength

529 MPa

Ultimate strength

608 MPa

Yield strain

2711 με

Elongation

0.14

to measure the vertical displacement distribution and deflection shape. Three tensile load sensors (Item 6 in Fig. 5.12 and Item 8 in Fig. 5.13) were used to monitor the load redistribution behavior of the specimen at different stages. For the P1-series, tension/compression load cells are installed in three steel supports (C1, C2, and C8 in Fig. 5.11), which are located on the same panel. However, for the P2-series, the load cells are mounted on three steel supports (C1, C3, and C7 in Fig. 5.11) located at the corner. A series of strain gauges were mounted on the slab reinforcement to track the development of different load resistance mechanisms during the test.

5.2.4 Experimental Results 5.2.4.1

Global Behavior and Failure Modes

The load–displacement curves of P1-series specimens are shown in Fig. 5.14a. As shown in the figure, when the vertical displacement of P1-100-1.0 is 10.2 mm, the maximum load is 93.1 kN. After reaching the maximum load, the load resistance capacity suddenly drops to 22.0 kN, which is about 21.5% of the first peak load (FPL). The load resistance remains almost constant until failure. The failure mode of P1-100-1.0 is shown in Fig. 5.15. Most of the deformation is concentrated in the inner part of the slab-column connection due to the failure of punching shear. There

5.2 Progressive Collapse Resistance of RC Flat Slabs After the Loss … Fig. 5.12 Test setup for specimen P1

177

1 2

3 4

5 8

6

6

7

6

Fig. 5.13 Test setup for specimen P2

1 2

3 4

6

7

9 8

178

5 Load Resisting Mechanisms of Flat Slab Structures to Resist … 30 P1-100-1.4 P1-100-1.0 P1-70-1.0 P1-55-1.0

80 60 40 20

Applied pressure (kPa)

Vertical load (kN)

100

25 20 15 10 P2-55-1.0

5

P2-P-55-1.0

0

0 0

0

30 60 90 120 150 180 210 Vertical displacement (mm)

(a) P1-series specimens

30 60 90 120 150 Vertical displacement (mm)

180

(b) P2-series specimens

Fig. 5.14 Load–displacement curve of test specimens

is no obvious bending deformation of the slab, but small bending cracks are formed in the slab. The strain gauge reading of the slab reinforcement showed that the slab reinforcement did not yield during the test. Since the failure is caused by the short section of the top column penetrating into the slab surface, the impact of drop-panel on the punching shear resistance of the connection is very limited, which is very different from the failure mode of plate slab under gravity or cyclic load. It should be noted that no damage was observed in the surrounding slab-column connections. Similar to P1-100-1.0, the load–displacement response of P1-100-1.4 is basically linear before the punching shear failure of the internal slab-column connection. Different from P1-100-1.0, many elliptical cracks are formed on the top surface of the slab at first. Above 88.9 kN, punching shear failure begins to appeared and the load resistance begins to decline abruptly. It should be noted that the displacement corresponding to the maximum load bearing capacity is 13.4 mm, which is slightly larger than P1-100-1.0 (10.2 mm) due to the large longitudinal column spacing between P1-100-1.4. The residual load resistance of the specimen is 29.4 kN, which is only 33.1% of its FPL. The failure mode of P1-100-1.4 is shown in Fig. 5.16, which is generally very similar to that of P1-100-1.0. According to the measurement, the perimeter of the critical section is rectangular, the distance is 150 mm, 1.74d, where d is the effective depth from slab to cylinder. Therefore, the perimeter of the punching area is uniform in the longitudinal direction. In the loading process of P1-70-1.0, under 17.9 kN, the first circular bending crack appeared on the top surface of the slab, accompanied by the reduction of load–displacement stiffness. Different from P1-100-1.0 and P1-100 1.4, P1-70-1.0 showed yielding of slab reinforcement at 43.2 kN, which led to further reduction of the stiffness of load–displacement curve. After the yield load was reached, the relative ductility behavior was accompanied. The displacement ductility ratio was 4.0. Due to strain hardening and compressive membrane action, the load resistance increased continuously until the vertical displacement reached 52.8 mm, and the internal connection of the slab and column was damaged by punching shear. It should be noted that the punching shear failure occurred after the slab yielded. Therefore, it is more accurate to call it “secondary punching shear failure” to distinguish it from

5.2 Progressive Collapse Resistance of RC Flat Slabs After the Loss …

Fig. 5.15 Failure mode of specimen P1-100-1.0

179

180

5 Load Resisting Mechanisms of Flat Slab Structures to Resist …

Fig. 5.16 Failure mode of specimen P1-100-1.4

general punching shear failure, such as P1-100-1.0 and P1-100-1.4. The failure mode of P1-70-1.0 is shown in Fig. 5.17. The bending crack of P1-70-1.0 is wider than that of P1-100-1.0 and P1-100-1.4. The bending deformation of P1-70-1.0 is much larger than that of the above two specimens, although it is still not obvious compared with column spacing. The perimeter of critical section of P1-70-1.0 is 102 mm, 1.78d away from the column-slab interface. Similar to P1-70-1.0, it’s the failure mode of P1-55-1.0 is initially dominated by bending. The measured yield load was 28.1 kN, and the displacement was 15.6 mm. The failure mode of P1-55-1.0 is shown in Fig. 5.18. It can be seen that a perceptible bending deformation occurred in the specimen. A series of circular bending cracks are formed on the top surface of the slab, and serious diagonal bending cracks appeared on the bottom surface. The maximum crack width measured in top and bottom exceeded 1.5 and 1.1 mm, respectively. Although serious secondary punching shear failure also occurred at the distance of 89.0 mm away from the cylinder surface,

Fig. 5.17 Failure mode of specimen P1-70-1.0

5.2 Progressive Collapse Resistance of RC Flat Slabs After the Loss …

181

Fig. 5.18 Failure mode of specimen P1-55-1.0

the perimeter of the punching shear zone was much smaller than that of the above specimens. For P2 series specimens, UDP was applied to the top surface of the slab. Therefore, in this series of tests, behavior is described in terms of pressure rather than load. Figure 5.14b shows the pressure–displacement response of P2 series specimens. It should be noted that the dimensions and reinforcement details of P2-55-1.0 are identical to those of P1-55-1.0. Under the load of P2-55-1.0 and the pressure of 2.0 kPa, the bending cracks of the inner column and the corner column are the first to appeared on the bottom surface of the slab. And at this time a circular crack formed in the top surface of the slab. Note that the outermost circular crack with the widest crack width did not connect to the interface of the surrounding column (the predicted maximum bending moment), but to the vertical slab edge of the surrounding column. The yield of the bottom steel bar near the inner column was measured at a pressure of 18.7 kPa. Further increase in the applied pressure caused the top reinforcement near the surrounding column to yield. When the displacement was 80.2 mm, the first peak pressure (FPP) was 24.9 kPa, accompanied by severe concrete spalling. Considerable load resistance was observed in the large-displacement stage, accompanied by wider bending cracks. The measured ultimate pressure capacity of P2-55-1.0 at large displacement stage was 26.0 kPa, which was about 104.4% of its FPL. Figure 5.19 shows the failure modes of P2-55-1.0. In general, the crack pattern of P2-55-1.0 was similar to that of P1-55-1.0. However, only serious flexural cracks appeared on the bottom surface of the slab near the inner column, and there was no punching shear failure in the inner connection of the slab and column. The dimensions and reinforcement details of the P2-P-55-1.0 are similar to those of the P2-55-1.0, but no drop-panels are constructed above the columns. The stiffness and yield pressure of P2-P-55-1.0 are lower than that of P2-P-55, which is 13.0 kPa. When the pressure exceeded 18.2 kPa, the shear failure of the surrounding slab-column connections began to appear, and the load resisting ability decreased obviously. When the vertical displacement was more than 80.4 mm (1.9d), the tensile load of the specimen was also detected to rise again. Figure 5.20 shows the failure mode of P2-P-55-1.0 at the end of the test. The failure mode of the top surface is

182

5 Load Resisting Mechanisms of Flat Slab Structures to Resist …

Fig. 5.19 Failure mode of specimen P2-55-1.0

Fig. 5.20 Failure mode of specimen P2-P-55-1.0

very similar to that of P2-55-1.0, except that the widest crack is formed at the interface of the surrounding column rather than at the edge of the drop-panels. All the surrounding slab-column joints appeared serious secondary punching shear failure.

5.2.4.2

Load Redistribution Behavior

Figure 5.21 shows the load redistribution response of the specimen at the critical stage. P1-100-1.0 at the FPL, approximately 19% of the axial load (initially borne by the lost center column C9) was transferred to the adjacent columns C2 and C8. At this stage, only 6.4% of the load was transferred to corner column C1. After punching failure, more force was transferred to the adjacent column while less load was transferred to the corner column, which may be because the load resistance after punching shear failure is mainly provided by the local catenary action of the whole bar passing through the cage of the adjacent column, as shown in Fig. 5.22a. In general, similar behavior was observed for P1-100-1.4. However, different from

5.2 Progressive Collapse Resistance of RC Flat Slabs After the Loss … 30.0%

Percentages of Load Transfer (%)

Percentages of Load Transfer (%)

25.0% 20.0% 15.0% 10.0% 5.0% 0.0%

First Peak Load After Punching Failure C2

25.0% 20.0% 15.0% 10.0% 5.0% 0.0%

C8 Column Supports

First Peak Load After Punching Failure C2

C1

(a) P1-100-1.0 Percentages of Load Transfer (%)

Percentages of Load Transfer (%)

15.0% 10.0%

Yield Load First Peak Load After Punching Failure C2

20.0% 15.0% 10.0% 5.0% 0.0%

C8 Column Supports

C1

C1

12.0% Percentages of Load Transfer (%)

Percentages of Load Transfer (%)

C8 Column Supports

(d) P1-55-1.0

10.0% 8.0% 6.0% 4.0%

0.0%

Yield Load First Peak Load After Punching Failure C2

(c) P1-70-1.0

2.0%

C1

25.0%

20.0%

0.0%

C8 Column Supports

(b) P1-100-1.4

25.0%

5.0%

183

Yield Load First Peak Load Load Re-ascending

10.0% 8.0% 6.0% 4.0% 2.0% 0.0%

C7

C5 Column Supports

(e) P2-55-1.0

Yield Load First Peak Load Load Re-ascending C7

C5 Column Supports

(f) P2-P-55-1.0

Fig. 5.21 Load re-distribution response of P1-series of specimens in critical stages

P1-100-1.0, the load transmitted to adjacent columns C2 and C8 were unequal due to their different longitudinal and transverse spacing and stiffness. Slab reinforcement of P1-70-1.0 and P1-55-1.0 yielded during the tests, the figure also contains the force transfer behavior under yield load. As shown in Fig. 5.21c, d, when the load is increased from yield load to FPL, more force was transferred to the adjacent column, while less force is transferred to the corner column. This is because the compressive membrane action of P1-70-1.0 and P1-55-1.0 is similar to the compressive arch action in both longitudinal and transverse aspects, as shown in Fig. 5.22b. Due to the installation of tension/pressure load cells at corner columns C1, C5, and C7 of the P2 series specimens, only the load redistribution response of the corner columns was monitored. Different from P1-70-1.0 and P1-55-1.0, the force on the corner column is greater with the increase from yield load to FPP load. This may

184

5 Load Resisting Mechanisms of Flat Slab Structures to Resist …

(a) dowel action

(b) compressive arch/membrane action Fig. 5.22 Load resisting mechanism

be because a larger area was involved in the development of compressive membrane action in the P2 series specimens with UDP application. When the load reached the re-ascending stage, the force transferred to the corner columns increased. This is because P2-P-55-1.0 suffered more serious damage at the slab proximity of adjacent columns at this stage, that is, severe punching shear failure occurred at the connection between adjacent columns and slab.

5.2.4.3

Strain Gauge Results

Figure 5.23 shows the strain distribution of slab reinforcement in FPL stage. It can be seen that the slab radar ofP1-100-1.0 did not yield at FPL. Similar results were observed for P1-100-1.4. The failure modes of the two specimens are basically the same (the main failure mode is punching shear). P1-55-1.0’s top and bottom reinforcements appeared serious yield. Compared with P1-55-1.0, P2-55-1.0 obtained more steel yield strain, finer crack and more uniform deformation due to the simulated multi-point loading mode. Despite the severe punching shear failure observed in the adjacent slab-column connections, the yield was achieved in the P2-55-1.0.

5.2 Progressive Collapse Resistance of RC Flat Slabs After the Loss … Fig. 5.23 Strain distribution in slab reinforcements of typical specimens at first peak load stage

1551

-97

185

825

534

509

710

-23

60

N/A

Top layer

Bottom layer

N/A 85

-63 -21 66

-67 -55

(a) P1-100-1.0

3356

-210

775

1810 3197

809

-49

1113

1656

Top layer

1445

Bottom layer

687 164 56 262 362

868

(b) P1-55-1.0

>660

-122

3554

938

2687

N/A

-59

2433

N/A

Top layer

1882

Bottom layer

2358 N/A N/A 1435 3775

3325

(c) P2-55-1.0

3490

-136

2648

89

2040

N/A

-122

665

1985

Top layer

N/A

Bottom layer

2015 604 N/A 312 1202

(d) P2-P-55-1.0

N/A

186

5 Load Resisting Mechanisms of Flat Slab Structures to Resist …

5.2.5 Analysis and Discussion 5.2.5.1

Yield Line Prediction

The yield-line prediction method is used to predict the flexural strength of specimens. Figure 5.24a shows the yield line pattern of the specimen with drop-panel based on the observed crack pattern. For a square specimen with an aspect ratio of 1.0, the internal virtual work caused by yield line rotation is: 

     γ1 h1 β γ2 α + 2m sP + sin + sin W1 = 4δ m sN h1 h2 h2 2 2

(5.3)

where W1 is the internal work due to strain energy; δ is virtual displacement at the center column; h 1 and h 2 are the length of the centerline of Segments A and B, respectively; γ1 = 2h 1 tg(α/2) and γ2 = 2h 2 tg(β/2) are the length of hogging yield line in Segments A and B, respectively; the measured angle of α and β range from 65 to 73 degrees and 17 to 25 degrees, respectively. For simplicity, the values of α and β are assumed to be 69 and 21 degrees, respectively; and m sP and m sN are, respectively, the yield moment of the sagging and hogging yield line, which could be determined by Eq. 5.4

γ1 Se gm

Segment A

h1

γ2 en tB

h1

Segment C

γ1 γ2

θ

Seg. A h2

h1

α β

Seg. B

β

Lost

γ3

α

Hogging yield line

Sagging

(a) P1-100-1.0

yield

Hogging yield line

Sagging yield line

(b) P1-100-1.4

Fig. 5.24 Yield line patterns considered to determine flexural resistance of specimens with drop panels

5.2 Progressive Collapse Resistance of RC Flat Slabs After the Loss …

  fy m s = As f y ds − 0.59As  fc

187

(5.4)

where As is the tensile slab reinforcement per unit width; and ds is the slab effective depth. As shown in Fig. 5.24b, for P1-100-1.4 which has an aspect ratio of 1.0, the internal virtual work due to rotations of the yield lines is   γ1 γ2 γ3 + W1 = δm sN 2 +4 +2 h1 h2 h3     h1 h3 α θ β β P 4m s δ sin + sin sin + sin + h2 2 2 h2 2 2



(5.5)

where h 3 is the length of the centerline of Segment C; γ3 = 2h 3 tg(θ/2) is the length of hogging yield line of Segment C; and the measured angles of α, β and θ are 46, 16, and 102 degrees, respectively, in P1-100-1.4. For P1-series specimens with concentrated loads, the external virtual work is 

WE = Pδ

(5.6)

where WE is the external work due to applied external load; and P is the external concentrate load. For P2-series specimens, the external virtual work is 

WE = q

2δ (γ1 h 1 + γ2 h 2 ) 3

(5.7)

where q is the external pressure. The analytical yield loads of specimens P1-55-1.0, P1-70-1.0, P1-100-1.4, and P1-100-1.0, are 28.1, 50.1, 114.3, and 112.5, kN, respectively. The FPLs of P1-1001.4 and P1-100-1.0 were 88.9 kN and 93.1 kN, respectively. Therefore, the yield line method further demonstrated that the structural behavior of P1-100-1.4 and P1-1001.0 was controlled by punching shear failure rather than bending failure. According to EN1992-1-1:2005, the punching shear strengths of P1-100-1.4 and P1-100-1.0 inner slab-column connections were 85.7 kN. By comparing the FPLs of these two specimens, it was concluded that the design equation proposed by EN1992-1-1:2005 is reliable.  1/3 VEC2 = 0.18b0,EC2 dξ 100ρ f c

(5.8)

where d is the effective depth; b0,EC2 is the control perimeter of the critical section (2d distance from the cylinder); f c is the specified compressive strength of concrete; ρ is flexural reinforcement ratio; ξ is a factor considering size effect (nominal shear strength decreases with the increase of component size), and its value is:

188

5 Load Resisting Mechanisms of Flat Slab Structures to Resist …

ξ =1+

200mm  2.0 d

(5.9)

In addition, the analytical yield pressure of specimens P2-P-55-1.0 and P2-551.0 are 14.1 and 20.9 kPa, respectively, which agreed well with the test results (the measured yield pressure of P2-55-1.0 and P2-P-55-1.0 are 18.7 and 13.0 kPa, respectively).

5.2.5.2

P2 Residual Load Resistance of P2-Series

P2 series specimens were subjected to gravity pressure after removing the central column. The measured behavior was defined as residual behavior, and the recorded resistance was called residual resistance (RLRC). Figure 5.25 shows the yield line configuration of P2-55-1.0 in the service state with no columns missing. For simplicity, the values of α and β are set to be 69 and 21 degrees, respectively. The yield pressures of P2-55-1.0 and P2-P-55-1.0 are 80.4 and 52.3 kPa, respectively. Thus, the RLRC of P2-55-1.0 and P2-55-1.0 in normal service phase is only 25.9 and 27.0% of their flexural capacity, respectively. However, the failure of P2-55-1.0 and P2-P-55-1.0 under normal service conditions may be controlled by punching shear failure rather than bending failure. According to Eq. 5.10, the punching shear strength of the internal slab-column connections of P2-55 1.0 and P2-P-55-1.0 are 59.5 and 31.6 kN, respectively. Therefore, the maximum pressure of P2-55-1.0 and P2-P-55-1.0 under normal operation is 26.4 and 14.0 kPa respectively. Therefore, the measured RLRC of P2-55-1.0 is 79.2% of its load resistance capacity under normal working conditions. The RLRC of the P2-P-55-1.0 specimen was even greater than the load resistance of the specimen under normal service conditions. The reason is that the failure mechanism of the specimen will change from the brittle punching shear failure at the slab-column connection to the bending failure of the slab due to the double-column spacing of the specimen after column removal.

5.2.5.3

Effects of Slab Thickness and Span Aspect Ratio

Due to different slab thicknesses, the load resistance of P1-100-1.0 was 85.8 and 150.9% higher than that of P1-70-1.0 and P1-55-1.0. However, the dynamic load resistance of P1-100-1.0 was only 45.3 and 99.7%than that of P1-70-1.0 and P155-1.0, respectively. This is because, with the increase of thickness, the failure mode changes from ductile bending failure to brittle punching shear failure, and the deformation capacity was significantly reduced. The dynamic load resistance of the specimen can be calculated by Eq. 5.10 1 PCC (u d ) = ud

 0

ud

PNS (u)du

(5.10)

5.2 Progressive Collapse Resistance of RC Flat Slabs After the Loss …

189

Fig. 5.25 Typical yield-line patterns of specimens with drop panels under normal service

Hogging yield line

Sagging yield line

where PCC (u) and PNS (u) are the capacity function and the nonlinear static  u loading estimated at the displacement demand u, respectively. In Eq. 5.10, 0 d PNS (u)du represents the accumulated area under the nonlinear static load–displacement curve at displacement u d . Thus, the capacity curve method could be understood as the dividing the accumulated area under the nonlinear static load–displacement curve by its corresponding displacement ud . P1-100-1.4 has identical thickness to P1-100-1.0 but but different span aspect ratio of 1.4. No significant difference was observed between these two specimens, as both specimens were controlled by the punching shear failure of the interior slab-column connection.

5.2.5.4

Effects of Loading Method

P1-55-1.0 and P2-55-1.0 have the same dimensions and reinforcement details, but have different failure stages (using two different loading regimes). Both specimens failed after steel bar yielded, but the crack of P2-55 1.0 specimen was finer and the deformation capacity was greater. P1-55-1.0 had secondary punching failure capacity, but P2-55-1.0 has no secondary punching failure. This is because for P155-1.0, when the penetration range is from the top surface of the slab to the bottom of the slab, drop-panel cannot improve the punching shear resistance of the slabcolumn connection. However, the performance comparison between P2-P-55-1.0 and P2-55-1.0 shows that the drop-panel did significantly improved the punching

190

5 Load Resisting Mechanisms of Flat Slab Structures to Resist …

shear resistance of adjacent slab-column connections and effectively prevented the occurrence of punching shear failure. The experimental results further confirm that different from the moment-resistant frame, different test devices and loading methods should be designed when studying the different collapse stages of the slab structure.

5.3 Conclusions In this chapter, the progressive collapse behavior of plate slab is studied when the corner and inner column are removed. The influence of rebar ratio, slab thickness, span length ratio, loading mode and drop-panel were quantitatively analyzed. According to the test results and analysis, the following conclusions are drawn: For corner column removal scenario 1. The experimental observation shows that one of the possible failure modes of ND series specimens in the process of resisting the progressive collapse caused by the loss of the ground corner column is the punching shear failure, and the punching shear failure occurred at the corner slab-column connections. The deterioration of the punching shear failure was mild and the test could continue because of the integral reinforcement installed at both the top and bottom of the slab. In addition, vertical slabs significantly reduce the likelihood of this brittle failure mode. No punching shear cracks were observed in WD series specimens. 2. The experimental results show that adding drop-slab can significantly improve the overall performance of concrete against progressive collapse. The first peak bearing capacity of WD1, WD2 and WD3(with drop-slab) is increased by 124.7, 87.5 and 61.6% compared with ND1, ND2 and ND3(without drop-slab), respectively. 3. The experimental results show that the initial stiffness and energy dissipation capacity of the flat slab structure can be increased by 117.4 and 85.4%, respectively. 4. The reinforcement quantity of floor slab has significant influence on the progressive collapse resistance of floor slab structure. Compared with ND1, the first load peak of ND2 and ND3 increased by 68.2 and 163.5%, respectively. Compared with WD1, the first load peak of WD2 and WD3 increased by 40.3 and 89.5%, respectively. 5. The load–displacement curve of the specimen showed a re-ascending branch, indicating that the tensile membrane action occurred in the slab. The second peak bearing capacity of each specimen exceeded the first peak bearing capacity. However, punching failure may prevent the development of tensile membrane action 6. The prediction range of dynamic effects of the specimens is 1.13–1.23. Due to the relative ductility of the specimens, the prediction results are obviously less than “2.0” assumed in the design criteria.

References

191

For interior column removal scenario 1. Thicker slab can significantly improve the initial stiffness and load resistance of specimens. However, the failure mechanism of thick slabs (P1-100-1.0 and P1-100-1.4) may be dominated by brittle punching shear failure, which reduces their deformation ability. 2. For relatively thinner slabs (P1-70-1.0 and P1-55 1.0), secondary punching shear failure occurred after the slab reached the yield strain. Due to strain hardening and compressive membrane action, the load–displacement response of the slab is significantly enhanced after yielding. All specimens of P1 series obtained similar residual bearing capacity after punching shear failure, which was due to the similar degree of local catenary action in the integrity reinforcement. 3. By comparing the behaviors of P1-55-1.0, P2-55-1.0 and P2-P-55 1.0, the influence of different loading protocols on different collapse stages was evaluated. The internal slab-column connections suffer from secondary punching shear failure under the condition of concentrated pushdown load. The results show that the impact resistance of the drop-panel to the internal connection of the slab-column is limited. However, no significant punching shear failure was observed at any of the connections of P2-55-1.0, which were under multi-point push-down loading. The failure mode of P2-P-55-1.0 under multi-point push-down loading, which indicates that punching failure may occur at the adjacent slab-column connection of the specimen in the second stage. The reason why there is no punching shear failure in P2-55-1.0 is that drop-panels significantly improve the punching shear resistance of adjacent slab-column connections. 4. The results show that the yield-line method can predict the yield load of the specimen well, and it is controlled by bending failure well. The results show that P2-55-1.0 and P2-P-55-1.0 can reach 79.2 and 100.7% of the yield capacity of the corresponding specimens under normal service condition (without column removal), respectively. This important finding is that the failure mode of the specimen changes from brittle punching shear failure to ductile bending failure due to the increase of column spacing.

References Abruzzo J, Matta A, Panariello G (2006) Study of mitigation strategies for progressive collapse of a reinforced concrete commercial building. J Perform Constr Facil 20(4):384–390 ACI 318-08 (2008) Building code requirements for structural concrete and commentary. American Concrete Institute, Farmington Hills ACI 318-11 (2011) Building code requirements for structural concrete and commentary. American Concrete Institute, Farmington Hills Comité Européen de Normalisation (CEN) (2004a) EN 1992-1-1. Eurocode 2: design of concrete structures. Part 1: general rules and rules for buildings. European Committee for Standardization, Brussels

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5 Load Resisting Mechanisms of Flat Slab Structures to Resist …

Comité Européen de Normalisation (CEN) (2004b) EN 1992-1-2. Eurocode 2: design of concrete structures. Part 1-2: general rules - structural fire design. European Committee for Standardization, Brussels Comité Eurointernational du Béton (CEB/FIP) (1978) Code Modèle pour les structures en béton. CEB Bulletin 124/125F, Paris (in French) DIN 1045-1 (2001) Ausgabe: 2001-07. Tragwerke aus Beton, Stahlbeton und Spannbeton. Teil 1: Bemessung und Konstruktion DoD (2009) Design of building to resist progressive collapse. Unified Facility Criteria. UFC 4-02309. US Department of Defense, Washington (DC) General Services Administration (GSA) (2013) Alternate path analysis and design guidelines for progressive collapse resistance. Office of Chief Architects, Washington, DC Guandalini S, Olivier B, Aurelio M (2009) Punching tests of slabs with low reinforcement ratios. ACI Struct J 106(1):87–95

Chapter 6

Progressive Collapse Performance of Infilled Frames

Masonry infilled (MI) panels are often considered non-structural elements, however, the weight of which is often considered in the progressive collapse analysis. Consequently the resistance of the MI panels are usually neglected in the collapse-resistant design. Based on simple structural analysis, it is found that the behavior MI panels are equivalent to compressive struts under a column removal scenario, which not only enhance the global stiffness, but also improve the vertical load resistance of the frame. Since progressive collapse is a low-probability event, it is inevitable to quantify the merits of MI walls to the robustness of RC frames to mitigate progressive collapse. Unfortunately, there are few related studies. To fill this gap, experimental and numerical studies have been conducted to quantify the effect of MI in the event of progressive collapse.

6.1 Performance of Frames with Full-Height Infill Walls 6.1.1 Specimen Design In this experiment, six one-quarter scale multi-storey RC frames were designed and fabricated. As listed in Table 6.1, the six specimens are divided into two series: the first series includes three bare frames without MI panels (BNS, BSS, BNL); the second series consists of three infilled frames with MI panels (WNS, WSS, WNL). Design variables include with/without MI wall, non-seismic/seismic design details, and span to depth ratio of the beam. Take BNS as an example, it represents a bare frame has non-seismic design and a short beam span (low span to depth ratio). Similarly, WNL stands for an infilled frame with non-seismic design details and long beam span. As shown in Fig. 6.1, the planar internal frame is extracted from the prototype RC building and experimentally studied after removing the penultimate column. © China Architecture & Building Press 2023 K. Qian and Q. Fang, Progressive Collapse Resilience of Concrete Structures: Mechanisms, Simulations and Experiments, https://doi.org/10.1007/978-981-99-0772-4_6

193

194

6 Progressive Collapse Performance of Infilled Frames

Table 6.1 Specimen properties for investigation of solid wall effects Element size

Reinforcements

Span (mm)

Infill wall

Beam

BNS

1800

BSS

Test ID

Specimen description

Column

Longi positive (%)

Longi negative (%)

Trans. in ends (%)

Trans. in ends (%)

Joint

None

0.73

0.97

0.32

0.32

N/A

Nonseismic designed bare frame with short span

1800

None

1.40

1.40

1.20

1.00

1.00%

Seismic designed bare frame with short span

BNL

1800

None

0.73

0.97

0.32

0.32

N/A

Nonseismic designed bare frame with long span

WNS

1800

Infilled

0.73

0.97

0.32

0.32

N/A

Nonseismic designed infilled frame with short span

WSS

2400

Infilled

1.40

1.40

1.20

1.00

1.00%

Seismic designed infilled frame with short span

WSL

2400

Infilled

0.73

0.97

0.32

0.32

N/A

Nonseismic designed infilled frame with long span

As shown in the picture, the penultimate column is assumed to be removed due to explosion or vehicle impact. Therefore, the impact of MI panel on the performance of external and internal joints can be evaluated simultaneously. The prototype building has eight stories, the story heights of the ground floor and the other floors are 3,600 and 3,300 mm, respectively. The spans in both longitudinal and transverse directions are 7,200 mm. The design live load and Dead load including dead weight are 2.0 and 6.4 kN/m2 , respectively. For the seismic design specimens BSS and WSS, it is assumed that the prototype building is located in the class D site, and the short-period and one-second periodic spectral accelerations SDS and SD1 are 0.48 and 0.35 g, respectively. Figure 6.2 shows the typical reinforcement layout of WSS. One can see,

6.1 Performance of Frames with Full-Height Infill Walls

195

Column Loss

Region Studied

(a) Plan view

Region Studied

(b) Elevation view

Fig. 6.1 Location of the extracted frame in the prototype building

the transverse reinforcement is stirrup with a 135-degree bend. Two transverse links are placed in the joint area. In addition, the longitudinal reinforcement of the beam is double reinforced and continuously without considering the cut of reinforcement. Considering the capacity of the laboratory facilities, the allowable scale is one-quarter because the construction is a three-story, two-span specimen. As shown in Fig. 6.2, the span of the specimen is 1800 mm. The height of the first story of test specimen was 900 mm, and the upper story was 825 mm. To fixed the specimens to the ground, a 400 × 300 mm2 foundation is prefabricated at the column foundation. The beam and column sections are 90 × 140 mm2 and 150 × 150 mm2 , respectively. The concrete clear cover for beam and column sections is 7 and 10 mm, respectively. All three floors have side columns, while the penultimate column is built for only upper two floors to simulate the removal of the ground column due to explosion or vehicle impact. A plain steel bar R3 with a diameter of 3 mm is used as a reinforcement to strengthen the MI wall. Holes are punched on the inner surface of the column and the tie rod is inserted into the hole with an epoxy resin fixation. In addition, before casting MI wall, roughness treatment is carried out at the interface between frame members and MI wall. For non-seismic design specimens WNS, the same span, storey height, beam and column dimensions were selected to facilitate for comparison. However, the reinforcement details are quite different from the seismic specimens. As shown in Fig. 6.3, a 90-degree bending hoop is used for transverse reinforcement. There is no transverse reinforcement in the joint area. In addition, rebar reduction is designed according to ACI 318-08 (2008). The non-seismic design specimen WNL has the same floor height, beam and column dimensions, and reinforcement details as the specimen WNS. The only difference between WNL and WNS is the ratio of beam span to depth. As shown in Table 6.1, the span/depth ratio is 17 for WNL and 13 for WNS. For bare specimen BNS, BNL and BSS, the design and construction of the RC frames are identical to the corresponding infilled specimen WNS, WNL and WSS, except that there is no MI wall construction.

196

6 Progressive Collapse Performance of Infilled Frames

Exterior

Interior

(a) Elevation view

(b) Cross section of RC frame Fig. 6.2 Reinforcement layout of the Specimen WSS (Note Unit in mm, T = Deform reinforcing bar; R = Plain reinforcing bar)

6.1.2 Material Properties The target compressive strength of concrete for 28 days is 30 MPa. The average cylinder compressive strengths of BNS, WNS, BSS, WSS, BNL and WNL on the test day were 32.1, 33.8, 33.9, 34.3, 32.1 and 33.3 MPa, respectively. The masonry size used in this study is 195 × 90 × 65 mm. The compressive strength of masonry and mortar is 17.3 and 16.5 MPa, respectively. The measured compressive strength and shear strength of masonry prism (brick and mortar combination) are 10.5 and 1.1 MPa, respectively. It can be seen that the compressive strength of masonry prisms is much lower than that of individual masonry and mortar. The yield strengths of R3, R6, T10 and T13 are 417, 449, 515 and 534 MPa, and the tensile strengths are 479,

6.1 Performance of Frames with Full-Height Infill Walls

197

Exterior column

Interior column

Third story

Second story

First story

(a) Elevation view,

(b) Cross section of RC frame Fig. 6.3 Reinforcement layout of the Specimen WNS (Note Unit in mm, T = Deform reinforcing bar; R = Plain reinforcing bar)

537, 594 and 618 MPa, respectively. The stress–strain curve of reinforcement is shown in Fig. 6.4.

6.1.3 Test Setup Figures 6.5 and 6.6 illustrate the location of a typical experimental setup and instrument, respectively. As shown in Fig. 6.5, the specimen is fixed on the lab. floor via strong foundation, which is integrated with the side column. Vertical loads are applied at the top of the penultimate column using a 600 mm stroke hydraulic jack. Load cells mounted above hydraulic jacks are used to measure the applied vertical load. Steel

198

6 Progressive Collapse Performance of Infilled Frames 700

Tensile Stress(MPa)

600 500 400 300

R3 R6 T10 T13

200 100 0 0

5

10

15

20

Strain (%) Fig. 6.4 Stress–strain curves of the reinforcements

1. Load cell 2. Hydraulic Jack 3. Steel column 4. Steel assembly 5. LVDT (25 mm) 6. LVDT (100 mm) 7. LVDT (300 mm)

1 2

3 5

4 6

7

Fig. 6.5 Test setup

6.1 Performance of Frames with Full-Height Infill Walls

199

Fig. 6.6 Layout of instrumentation

columns and specially designed steel assemblies are installed to prevent unwanted out-of-plane which can be prevent by slab in a real building. As mentioned above, the conceptually removed column is a penultimate column. In a realistic frame, there are no perimeter bay outside the external joints while the internal joints are constrained by the significant level of the surrounding bay. In addition, due to space limitations, the surrounding bays were not completely constructed in this study. Therefore, in order to simulate the horizontal constraints of the beams in surrounding bays, three rollers are installed horizontally in the frame extension, as suggested by Yu and Tan (2014a, b). Each roller was fitted with a tension/compression load sensor to measure the horizontal reaction during the test. As shown in Fig. 6.6, a series of linear variable deformation sensors (LVDTs) are installed on the joints, beams as well as columns to measure the shear deformation of the joints, deformation shape and horizontal movement of the side columns, respectively. The strain gauge is mounted on the steel bar in the important position.

200

6 Progressive Collapse Performance of Infilled Frames

6.1.4 Test Results 6.1.4.1

Global Behavior and Failure Modes

In order to study the progressive collapse characteristics of the filled frame, the RC frame with or without MI wall was experimentally studied. The key results are listed in Table 6.2 and discussed below. Specimen BNS-Typical crack patterns of Specimen BNS at a vertical displacement of 50 mm are shown in Fig. 6.7a. The first crack is formed at the end of the beam near the middle column (BENC). Then, cracks are formed at the curtialment of the top longitudinal reinforcement of the beam. In general, the first-floor cracks earlier than the third floor. When the vertical displacement reached 300 mm, shear cracks appeared in the external joints of the first story, but no cracks appeared in the internal joints. The load–displacement curve of the BNS specimen is shown in Fig. 6.8. As shown in the figure, when the displacement was 21.3 mm, the yield load (YL) of the specimen was 28.4 kN, and when the displacement was 38.1 mm, the first peak load (FPL) of the specimen was 31.9 kN. As a result, FPL increased by only 12.3%, indicating that the contribution of compressive arch action (CAA) was not important in this test. When the displacement reached 153.0 mm (0.1ln , l n was the clear span of the beam), the load resistance of the specimen began to improve again. However, the re-rising loads is quite modest. When the displacement reached 285.6 mm (0.19 ln ), the ultimate bearing capacity (UL) of the specimen reached the large deformation stage. However, UL measures only 28.9 kN, about 90.6% of its FPL. The failure mode of the BNS specimen is shown in Fig. 6.9. The fracture of reinforcement was observed at BENI and curtailment of the top longitudinal reinforcement at each story. Severe shear cracks were formed at the exterior joints in the first story. No shear cracks were found in the internal joints. Specimen BSS—The typical crack pattern of BSS specimen at a vertical displacement of 50 mm is shown in Fig. 6.7c. Compared with BNS, the cracks of BSS beam with the same vertical displacement are more uniform and the crack width is much thinner. Similar to the BNS specimen, the crack pattern of the beam in each story is basically the same. Figure 6.10 shows the load–displacement curve of specimen BSS. As shown in the figure, when the displacement was 24.8 mm, the YL of Specimen BNS was 68.6 kN, which was 241.5% of BNS. When the displacement reached 41.6 mm, the measured FPL was 82.7 kN, which was 259.2% of the BNS specimen. When the displacement reached 151.8 mm or 0.1ln , the load resistance tended to be stable with the further increase of the displacement. At a vertical displacement of 316.7 mm (0.21ln ), UL was 51.0 kN. The failure mode of the specimen is shown in Fig. 6.11. As shown in the figure, the reinforcement fracture occurred in the beam, and shear cracks also appeared in the external joints of the first and second floors. Different from the Specimen BNS, with the continuity of the longitudinal reinforcement, the fracture of the longitudinal reinforcement at the top of the beam occurred at the interface between the side column and the beam. In addition, the width of the shear crack of the external joint is much smaller than that of the Specimen BNS

13.6

122.5

5.6

62.3

11.1

8.5

2.0

1.8

1.8

1.8

2.4

1.1

1.0

1.3

WNS

BSS

WSS

BNL

WNL

WNS/BNS

WSS/BSS

WNL/BNL 8.4

9.0

10.0

26.0

3.1

68.1

7.6

47.8

4.8

E c (×103 kN/m)

N/A

N/A

N/A

N/A

27.0

N/A

24.8

N/A

21.3

uYL (mm)

N/A

N/A

N/A

N/A

19.6

N/A

68.6

N/A

28.4

F YL (kN)

0.12

0.07

0.08

5.7

47.9

2.9

41.6

2.9

38.1

uFPL (mm)

3.3

1.5

3.4

79.9

24.4

123.7

82.7

108.9

31.9

F FPL (kN)

0.90

0.93

0.99

400.1

445.0

294.5

316.7

292.1

297.8

uUL (mm)

1.10

1.32

1.82

29.3

26.6

78.4

59.3

52.5

28.9

F UL (kN)

Note F c , F FY, F FPL , and F UL , = crack load resisting capacity, first yield load, first peak load capacity, and ultimate load capacity, respectively. uc , uFY uFPL , and uUL = displacements corresponding the first crack, first yield load, first peak load capacity, and ultimate load capacity. E c = stiffness corresponding the first crack

11.1

95.6

8.6

1.8

BNS

F c (kN)

uc (mm)

Specimen

Table 6.2 Test results for investigation of solid wall effects

6.1 Performance of Frames with Full-Height Infill Walls 201

202

6 Progressive Collapse Performance of Infilled Frames

#2 #1

#2

#2

#2

#1

#3 #2

#2

(a) Specimen BNS

#3

#2

#2

(b) Specimen WNS #2

#2

#2

#2

#1

#1 #4 #3

(c) Specimen BSS

#2

#3

#4

(d) Specimen WSS #2

#3 #3

#2

(e) Specimen BNL

#3

#3

#2 #3

#3

#1

#1

#2

(f) Specimen WNL

Fig. 6.7 Crack pattern of the specimens at the vertical displacement of 50 mm

because the transverse reinforcement of the joint helps to resist the unbalance shear force and prevent the crack expansion. Specimen BNL—The typical crack pattern of BNL at vertical displacement of 50 mm is shown in Fig. 6.7e. Similar to the BNS specimen, the cracks of the beam are concentrated at the curtailment of the top longitudinal reinforcement of the BENC. The rest of the beam, especially the beam end near the side columns, was intact with little deformation. The load–displacement curve of the specimen is shown in Fig. 6.12. As can be seen from the figure, the YL of the specimen was 19.6 kN, about 69% of the Specimen BNS. When the displacement reached 47.9 mm, the FPL of 24.3 kN can be obtained, which was about 76.2% of that of the BNS specimen. The displacement was further increased to 205.4 mm (0.1 ln ) and the loading capacity rose again until the displacement reached 445.0 mm (0.21ln ). The UL of the specimen was 26.6 kN, which was 97.8% of the Specimen BNS. The failure mode of Specimen

6.1 Performance of Frames with Full-Height Infill Walls 120

203

FPL

BNS WNS

Applied load (kN)

100 80 CA

60 FPL

40

UL UL

CA

YL

20 0 0

50 100 150 200 250 300 350 400 450 Vertical displacement (mm)

Fig. 6.8 Comparison of the load–displacement curves of the bare frame BNS to infilled frame WNS

Fig. 6.9 Failure mode of bare frame BNS 140

FPL

BSS WSS

120

Applied load (kN)

Fig. 6.10 Comparison of the load–displacement curves of the bare frame BSS to infilled frame WSS

100

FPL

80

CA

60

UL

YL CA

40

UL

20 0 0

50 100 150 200 250 300 350 400 450 Vertical displacement (mm)

204

6 Progressive Collapse Performance of Infilled Frames

Fig. 6.11 Failure mode of bare frame BSS

100

Applied load (kN)

Fig. 6.12 Comparison of the load–displacement curves of the bare frame BNL to infilled frame WNL

FPL

BNL WNL

80 60 40

UL UL

CA YL

20

FPL

CA

0 0

50 100 150 200 250 300 350 400 450 Vertical displacement (mm)

BNL is shown in Fig. 6.13. In general, the failure mode of the specimen is very similar to that of the BNS specimen. The fracture of the longitudinal bars of the top beam is concentrated at the cut-off point, and the fracture of the longitudinal bars of the bottom beam occurred in the BENC. The external joints also show shear failure, but the internal joints do not show shear cracks, because the horizontal roller can resist the axial force generated by the beam reinforcement during the TCA stage. Specimen WNS—The typical crack pattern of the WNS specimen at a vertical displacement of 50 mm is shown in Fig. 6.7b, and the load–displacement curve of the specimen is shown in Fig. 6.8. When the vertical displacement of the central column is applied up to 1.1 mm, cracks appeared first in the MI wall and first in the RC frame. The measured bearing capacity of the specimen at this displacement stage was 93.0 kN. When the displacement reached 2.0 mm, cracks also appeared at the cut-off point of longitudinal reinforcement of the beams in the three stories. To easy comparison with bare specimen BNS, the initial stiffness was 47.8 × 103 kN/m in this state

6.1 Performance of Frames with Full-Height Infill Walls

205

Fig. 6.13 Failure mode of bare frame BNL

(where cracks first appeared in the RC frame). At this displacement stage, the MI wall crack is connected with the three-story inclined step crack (#1 in Fig. 6.7b). The vertical displacement further increased to 2.9 mm, and more step cracks appeared in the second and third stories (#2 in Fig. 6.7b). During this displacement phase, the BENC reached its first peak load (FPL) of 113.9 kN and cracks appeared at the bottom of the BENC. The crack is located at the interface of the beams and columns in the three stories, and the crack is formed at the cut-off point of the bottom reinforcement of the beam in the second story. When the displacement reached 4.8 mm, No. 3 step crack appeared on the second floor. Due to the slow development of cracks in the RC frame and the small development of beam and reinforcement strain, it can be concluded that the load resistance of the specimen is mainly attributed to the MI wall rather than the RC frame at the initial stage of the test. When the displacement is further increased to 9.8 mm, the bed-joint slippage occurred and the load resisting ability decreases significantly. It should be noted that the slippage formation at the second-story is earlier than that of the three-story. When the vertical displacement is larger than 50.7 mm, with the further increase of the vertical displacement, the decrease slope of the load resisting ability becomes smaller and basically remains unchanged. This may be due to the tensile failure of the main diagonal strut and the formation of several new compression struts outside the central zone. It can be seen that the new compressive strut has little effect on the further increase of displacement. With the further increase of displacement, the slippage of bed joint is aggravated and some bricks fall off completely. When the displacement reached 298.9 mm (0.2ln ), the steel bar fractured successively from the first story to the third story. The failure mode of specimen WNS is shown in Fig. 6.14. There are serious steps crack and block fall off in the diagonal area of MI wall. In addition, the steel bars in the beams were fractured and severe shear cracks appeared at the exterior joints located on the first and second floors. The failure mode is discussed further in the next section to highlight the influence of the MI wall.

206

6 Progressive Collapse Performance of Infilled Frames

Fig. 6.14 Failure mode of bare frame WNS

Specimen WSS–The typical crack pattern of the WSS specimen at a vertical displacement of 50 mm is shown in Fig. 6.7d. Similar to the Specimen WNS, the first crack was formed in the MI wall rather than in the surrounding frame. When the displacement only reached 1.1 mm, #1 step crack appeared on the third story of the specimen. Further increasing the displacement to 1.8 mm, cracks #2 and #3 form in the walls of the third and second stories. At this stage, cracks are also formed in the beam in the first story. The measured bearing capacity was 122.5 kN, and the initial stiffness was 68.1 × 103 kN/m. When the vertical displacement reached 2.9 mm, the FPL of 127.7 kN was obtained, and cracks appeared in the beam at the second and third stories. After that, the carrying capacity starts to decline sharply until the displacement reached 7.9 mm. At this displacement stage, with the further increase of the displacement, the bearing capacity of the specimen basically remains unchanged to 156.0 mm, but the bed joint slippage and detachment are more developed. However, compared with WNS specimens, the bed joint slippage of WSS specimens is only more serious in the second story. When the displacement reached 156 mm, the bed joint detaches, resulting in a sudden decline in the load bearing capacity. A UL of 72.0 kN was obtained at a displacement of 299.9 mm (02ln ). The failure mode of WSS specimen is shown in Fig. 6.15. In general, the integrity of the masonry wall is better than that of the test Specimen WNS, but some beam sections also have steel fracture. One of the interesting findings is that the shear crack width in the external joints is much smaller than that of the WNS and BSS specimens. Compared with WNS, the shear bars installed in the joint zone of WSS have less shear damage. The shear failure of WSS is milder than that of BSS, because the compressive struts of

6.1 Performance of Frames with Full-Height Infill Walls

207

Fig. 6.15 Failure mode of bare frame WSS

MI wall is relatively effective, which can resist part of the tensile force developed by the longitudinal reinforcement during the TCA stage. Specimen WNL–The typical crack pattern of specimen WNL at a vertical displacement of 50 mm is shown in Fig. 6.7f. Different from the Specimens WNS and WSS, the #1 crack appeared in the two-story MI wall. When the displacement reached 2.4 mm, # 2 step crack appeared on the wall, and cracks form on the beam in the first floor. The measured load capacity at this stage was 62.3 kN, which was only 65.2% of the WNS of the specimen. It can be seen that the initial stiffness of WNL is 26.0 × 103 kN/m, which is about 54.4% of that of WNS. When the displacement was further increased to 5.7 mm, the FPL reached 80.0 kN, which was 70.2% of the WNS of the specimen. Further increasing the displacement, more step cracks appear, and the load bearing capacity begins to decline sharply. The decrease of bearing capacity was accompanied by the slippage of the bed joints. When the displacement reached 45.8 mm, the downward trend of the bearing capacity tends to be gentle. The sudden decrease of bearing capacity at the displacement stage of 281.3, 361.2, and 409.8 mm is caused by the different fracture of reinforcement. Figure 6.16 shows the failure mode of WNL. In general, the wall integrity of WNL is the worst among the infilled

208

6 Progressive Collapse Performance of Infilled Frames

Fig. 6.16 Failure mode of bare frame WNL

specimens. Severe brick shedding was observed not only on the second floor but also on the third floor.

6.1.4.2

Deformation Shape of the Beams

Figure 6.17 shows the deformation shapes of the ground beam under different key states. As shown in Fig. 6.17a, in the FPL stage, the slope of the beam near the side column is smaller than that of the rest of the beam. The deformation of the center part of the beam is hyperbolic. With the increase of displacement, the slope of the beam near the side column increased less than that of the central part, so the main deformation is concentrated in the central part of the beam. Chord rotation, proposed in DoD (2009) and defined as the ratio of the mid-joint deflection (MJD) to the net span of the beam. As shown in the figure, the chord rotation would overestimate the slope of the beam near the side column, while the slope of the beam near the center column was better estimated. As shown in Fig. 6.17b, it is not clear in the figure because FPL is realized in the Specimen WNS under relatively small deformation. The deflection results measured in D3, D4 and D5 were −2.3, −2.9 and −2.1 mm, respectively. It can be seen from the figure that the deformation of the beam near the central column was almost the same after the CA state. Comparing the chord rotation curve proposed by DoD (2009), it can be seen that the chord rotation will underestimate the slope of the beam near the side column, but significantly overestimate the slope of the central part. Figure 6.17c shows the deformation results of specimen BSS. Different from the specimen BNS, the deformation of the specimen BSS was relatively smooth, initially hyperbolic, and then transited to a straight line after the UL state. Thus, in the final test, the measured slope was very close to the chord rotation and only slightly overestimated the slope of the beam near the side column. For specimen WSS, it was very similar to specimen WNS. The center part

0

0

-50

-50

-100 -150 -200

FPL

-250

CA UL

-300

Final Chord rotation

-350

D2

D1

D3

D4

D5

-400 0

450

D6

D7

Vertical dipalcement (mm)

Vertical dipalcement (mm)

6.1 Performance of Frames with Full-Height Infill Walls

209

FPL CA UL Final Chord rotation

-100 -150

-200 -250 -300

-350 -400

D1

0

900 1350 1800 2250 2700 3150 3600 LVDT Positions (mm)

0

0

-50

-50

-100 -150 FPL

-250

CA UL

-300

Final Chord rotation

-350

D1

-400 0

450

D2

D3

D4

D5

D6

D3

D4

D5

D6

D7

450

900 1350 1800 2250 2700 3150 3600 LVDT Positions (mm)

(b) Specimen WNS

D7

900 1350 1800 2250 2700 3150 3600 LVDT Positions (mm)

Vertical dipalcement (mm)

Vertical dipalcement (mm)

(a) Specimen BNS

-200

D2

-450

FPL CA UL Final Chord rotation

-100 -150 -200 -250

-300 -350 -400

D1

D2

D3

D4

D5

D6

D7

-450

0

(c) Specimen BSS

450

900 1350 1800 2250 2700 3150 3600 LVDT Positions (mm)

(d) Specimen WSS

Fig. 6.17 Overall deflection curve of beams in the first story

of the beam has a similar displacement and is deformed into a horizontal line. Chord rotation may underestimate the slope of the beam near the side column.

6.1.4.3

Horizontal Movement of the Exterior Joints

As shown in Fig. 6.18a, the horizontal motion of each story of the Specimen BNS is first inward and then outward. The maximum outward movement and inward movement were measured on the third floor, which were 10 and 54 mm, respectively. The specimen WNS was similar to the Specimen BNS. When the vertical displacement reached 230 mm, the outward motion changed to the inward motion, as shown in Fig. 6.18b. However, the maximum outward movement of the WNS speicmen was 16 mm, which was greater than that of the BNS specimen. This is because the MI wall acted as a compression strut, increasing the outward horizontal motion during the CAA phase. As shown in the figure, the increase in horizontal movement was mainly concentrated in the first and second stories. However, the maximum inward motion of BNS was 47 mm, which was smaller than that of BNS. This is mainly due to the presence of the secondary compressive struts, which resisted the inward movement of joints during the TCA stage, although the main diagonal compressive

210

6 Progressive Collapse Performance of Infilled Frames

60 Horizontal movement (mm)

LD2

40

LD3

30 20

10 0

LD1 LD2 LD3

-10 -20

0

50

Horizontal movement (mm)

60 LD1

50

LD1

50

LD2

40

LD3

30 20

10 0

LD1 LD2 LD3

-10 -20

100 150 200 250 300 350 400 450 Vertical displacement (mm)

0

50

(a) Specimen BNS

(b) Specimen WNS

70

60

LD1

LD1

60

LD2

50

LD3

40 30 20 10 0

LD1 LD2 LD3

-10 -20 0

50

100 150 200 250 300 350 400 450 Vertical displacement (mm)

(c) Specimen BSS

Horizontal movement (mm)

Horizontal movement (mm)

100 150 200 250 300 350 400 450 Vertical displacement (mm)

50

LD2

40

LD3

30 20 10 0

LD1 LD2 LD3

-10 -20

0

50

100 150 200 250 300 350 400 450 Vertical displacement (mm)

(d) Specimen WSS

Fig. 6.18 Horizontal movement of the exterior joints at different stories

struts have failed. For the seismic design bare specimen BSS, the maximum outward movement of 7 mm and inward movement of 63 mm were measured at three stories. The limited outward movement is due to the higher longitudinal reinforcement ratio of the column, which slightly increased the stiffness of the column. The larger inward motion is due to the larger longitudinal reinforcement ratio of the beam, resulting in a larger tensile force. However, the increase of inward motion is not proportional to the increase of longitudinal reinforcement ratio of the beam. This can be explained by the mild shear failure of the external joints of the BSS specimen, which can significantly reduce the horizontal movement in the TCA stage. As shown in Fig. 6.18d, similar to the Specimen WNS, the MI wall increased the outward motion of the external joint, but reduced the inward motion of the external joint. The trends of BNL and WNL were similar.

6.1.4.4

Strain Gauge Results

Figure 6.19 shows the variation of strain results along the longitudinal reinforcement of the beam. As shown in Fig. 6.19a, the maximum tensile strain of Specimen BNS

6.1 Performance of Frames with Full-Height Infill Walls

YL-3rd FPL-3rd UL-3rd TCA-3rd YL-2nd FPL-2nd TCA-2nd UL-2nd Center column εy interface

Strain (μɛ)

6000

4000

6000

2000 0 Side of exterior εy column

-2000

Side of interior column

4000

Side of exterior column εy

0

0

500

1000

1500

2000

2500

3000

0

4000

εy

2000

2500

Side of interior column

FC-3rd TCA-3rd FC-2nd TCA-2nd

Side of exterior column

6000

0

εy

1500

8000

FPL-3rd UL-3rd FPL-2nd UL-2nd Center column interface

2000

-2000

1000

3000

3500

(b) Bottom rebar in BNS

Strain (μɛ)

Strain (μɛ)

FC-3rd TCA-3rd FC-2nd TCA-2nd

Side of exterior column

6000

500

Distance from the adjacent column interface (mm)

(a) Top rebar in BNS 8000

Center column interface

εy

-4000

3500

Distance from the adjacent column interface (mm)

4000

FPL-3rd UL-3rd FPL-2nd UL-2nd

εy

Side of interior column

2000 0 Center column interface

εy

-2000

-4000

-4000

0

500

1000

1500

2000

2500

3000

3500

0

Distance from the adjacent column interface (mm)

500

YL-3rd TCA-3rd YL-2nd TCA-2nd

6000 4000

6000

Center column interface

0 Side of exterior column

-2000

Side of interior column

εy

-4000

0

500

1000

1500

2000

2500

3000

3500

Distance from the adjacent column interface (mm)

(e) Top rebar in BSS

YL-3rd TCA-3rd YL-2nd UL-2nd

8000

FPL-3rd UL-3rd FPL-2nd UL-2nd

εy

2000

1500

2000

2500

3000

3500

(d) Bottom rebar in WNS

Strain (μɛ)

8000

1000

Distance from the adjacent column interface (mm)

(c) Top rebar in WNS

Strain (μɛ)

Side of interior column

2000

-2000

-4000

FPL-3rd UL-3rd FPL-2nd UL-2nd

YL-3rd TCA-3rd YL-2nd TCA-2nd

8000

Strain (μɛ)

8000

211

4000

FPL-3rd UL-3rd TCA-2nd FPL-2nd Side of interior column

Side of exterior column

εy

2000 0 Center column interface

εy

-2000 -4000

0

500

1000

1500

2000

2500

3000

3500

Distance from the adjacent column interface (mm)

(f) Bottom rebar in BSS

Fig. 6.19 Variation of strain results along beam longitudinal reinforcements

was not at the beam end near the side column, but at the cut-off point of the top reinforcement, which was about 435 mm away from the side column. After the TCA stage, the compressive strain at the interface of the central column began to decrease, while the compressive strain at the UL stage remains. In addition, in general, the strain change of the second story was very close to that of the third story. For the bottom reinforcement, as shown in Fig. 6.19b, the maximum tensile strain and the maximum compressive strain were measured at the interface of the central column and the cut-off point of the top reinforcement, respectively. As shown in Fig. 6.19c, the maximum tensile strain of the reinforcement at the top of the WNS appeared at the strain gauge, which was 695 mm away from the side column, but did not appeared at the cut-off point of the reinforcement. However, the maximum compressive strain can be obtained at different positions of the beam reinforcements at the second

212

6 Progressive Collapse Performance of Infilled Frames

and third stories. For three story, the maximum compressive strain was measured at the central column interface. However, for the second story, it was measured at a strain gauge 395 mm from the interface due to the MI panel. The strain results of steel reinforcement at the bottom of the WNS specimen also showed a similar trend, as shown in Fig. 6.19d. For the seismically designed Specimen BSS, shown in Fig. 6.19d, e, the maximum compressive and tensile strains were measured at the beam ends, which were very close to the observations of existing studies (Sasani and Kropelnicki 2008; Yi et al. 2008; Qian et al. 2015). At the same time, the strain (BSS and WSS) of the longitudinal reinforcement of the column and the transverse reinforcement of the joint area were monitored. Figures 6.20 and 6.21 show the strain results of columns and joints of BSS and WSS specimens, respectively. As shown in Fig. 6.20, the longitudinal reinforcement of the column is not yielding. Larger tensile strain was measured for lateral bars of external joints, while limited strain was measured for internal bars, which was in good agreement with the failure mode of joints. When the TCA begins to develop, the strain in the steel bars of the columns and joints begins to rise again due to the large axial beam force applied to the columns and shear force applied to the joints. Comparing the strains on the second floor and the third floor, the strain readings of 2C1 and 2C2 are smaller than those of 3C1 and 3C2, but the strain readings of lateral reinforcement of nodes are similar. The strain measurement results of columns and joints of WSS specimens are shown in Fig. 6.21. The strain readings of column and joint bars in the specimen WSS were much smaller than in the specimen BSS, although a re-rise in readings was also observed. This can be explained from two aspects: (1) the compressive pressure applied on the joints by the compressive strut of the MI panel, which can balance partial of the axial force from the beam; (2) As shown in Fig. 6.15, the relative integrity of the masonry wall near the side column acts together with the side column to resist bending moment, shear force, and beam axial force. 1500

1500

3C1 3J1 3J3

3C2 3J2 3J4

Strain (με)

Strain (με)

2C1 2C3 2J1 2J3

1000

1000

500

Catenary action

2C2 2C4 2J2 2J4

Catenary action

500

0

0

-500

-500

0

50

100

150

200

250

300

350

0

50

Vertical displacement (mm)

(a) The third floor Fig. 6.20 Strain gauge results in the column and joints of BSS

100

150

200

250

Vertical displacement (mm)

(b) The second floor

300

350

6.1 Performance of Frames with Full-Height Infill Walls 600

400 3C1 3J1 3J3

3C2 3J2 3J4

200

Strain (με)

Strain (με)

213

Catenary action 0

-200

2C1

2C2

2C3

2J1

2J2

2J4

300 Catenary action

0

-300 0

50

100

150

200

250

300

350

0

50

Vertical displacement (mm)

(a) The third floor

100

150

200

250

300

350

Vertical displacement (mm)

(b) The second floor

Fig. 6.21 Strain gauge results in the column and joints of WSS

6.1.5 Discussion of MI Effects 6.1.5.1

Initial Stiffness, Load Resisting Capacity, and Deformation Capacity

As shown in Figs. 6.8, 6.10, 6.12 and Table 6.2, the initial stiffness of the infilled specimen was much greater than that of the bare specimen. The initial stiffness was defined as the ratio of cracking load to corresponding displacement, rather than the ratio of yield load to yield displacement. This is because when the surrounding steel bars begin to yield, the infilled specimen will be severely damaged. The results showed that the initial stiffness of WNS, WSS and WNL were 1000, 900 and 840% of that of BNS, BSS and BNL, respectively. The FPL capacities of BNS, WNS, BSS, WSS, BNL and WNL specimens were 31.9, 113.9, 82.7, 123.7, 24.4 and 88.5 kN, respectively. Thus, the MI panel improved the FPL of the bare frame by 260, 50 and 260%, respectively. Compared with aseismic design frame, MI panel had better enhancement effect on non-aseismic design frame. After MI panel failure, the ultimate strength of BNS, WNS, BSS, WSS, BNL and WNL were 28.9, 57.5, 59.3, 78.4, 26.6 and 29.3 kN, respectively. Thus, the MI panel can improve the UL of frames by 100, 30, and 10%, respectively. The increment of UL of the infilled specimen was not consistent, but at least larger than that of the bare frame. However, it should be noted that transverse beams and RC slabs were not included in the specimen construction. Therefore, it can be foreseen that if the transverse beam and the RC slab are merged, the load resistance of the specimen will be significantly improved (Qian et al. 2015). Therefore, considering 3D and slab effects in the model may reduce the efficiency of infilled walls. Assuming that the failure occurred when the load resistance decreases by 20%, the ultimate displacements of BNS, WNS, BSS, WSS, BNL and WNL were 132, 11, 89, 19, 82 and 18 mm, respectively. Therefore, the deformation capacities of WNS, WSS and WNL specimens were only 8, 21 and 22% of the corresponding bare specimens, respectively. The test results showed that the infilled specimens can still bear large loads in the large deformation stage, which was attributed to the secondary compression struts of the MI walls and the action of TCA. The measured deformations of BNS, WNS, BSS, WSS, BNL and WNL

214

6 Progressive Collapse Performance of Infilled Frames

corresponding to the UL stage were 298, 292, 317, 300, 445 and 400 mm, respectively. Therefore, the deformation capacities of the infilled specimens WNS, WSS and WNL were about 98, 95 and 90% of those of the bare specimens, respectively. In general, the deformation capacity of the infilled specimen was similar to that of the bare frame.

6.1.5.2

Failure Modes

The effect of MI panel on the failure mode of RC frame is discussed. As shown in Figs. 6.9, 6.11 and 6.13, the damage of the bare specimens (specimens BNS and BNL) was concentrated at the middle column interfaces or cut-off positions. For the continuous longitudinal reinforcement beam (specimen BSS), the damage is mainly concentrated at the interface between the side column and the middle column. In addition, the damage and deformation of beams on different floors are generally similar. Therefore, for the bare frame, the effect of MI wall is ignored, and the simplified single-story substructures (Choi and Kim 2011; Qian et al. 2015) can simulate multi-story frames well as long as the dimensions and reinforcement details of the beams and columns are similar. However, for the infilled specimens (WNS, WSS, and WNL), as shown in Figs. 6.14, 6.15 and 6.16, the deformation shape of the beams on different floors are quite different. As shown in the figure, compared with the first story, the beam suffered negative bending moment in thirdstory failed further away from the side column. On the contrary, when the beam is located in the first story to the third one, the damage of the beam under the action of positive bending moment is closer to the middle column. Therefore, using simplified one-or two-story test models to predict multi-story frames with MI walls may lead to inaccurate results. Shear failure occurred at the external joints of both bare and infilled specimens. However, the crack width of the external joints of the infilled specimens is much smaller than that of the bare specimens, because the local compression struts that damage the MI panel can provide additional lateral constraints for the joints to resist the tensile force of the beam at the TCA stage.

6.2 Analytical Study 6.2.1 YL and CAA of Bare Frames According to Eq. 6.1 (Yi et al. 2008), the YL values of Specimens BNS, BSS and BNL are respectively 33.6, 68.7 and 24.0 kN: Fypredicted = 3 × 2 ×

MbP + MbN Ln

(6.1)

6.2 Analytical Study

215

Fig. 6.22 Possible locations of plastic hinges in the beams of bare frame with rebar cutting predicted

where Fy = predicted yield strength; L n = clear span of the beam; MbP = positive yield moment of the beam end; and MbN = negative yield moment of the beam end. YL values of BNS, BSS and BNL specimens were 28.4, 68.6 and 19.6 kN, respectively. The overestimation of YL values of Specimens BNS and BNL is because the beam ends do not always form plastic hinge during design, as shown in Fig. 6.22. Therefore, for non-seismic design specimens with longitudinal reinforcement shear, the YL value of the specimen should also be estimated by assuming that plastic hinge is formed at the cut-off point. According to Eq. 6.2, the YL values of BNS and BNL are 28.0 kN and 19.8 kN, respectively. Fypredicted = 3 × 2 ×

N MbP + Mbc Ln − LX

(6.2)

N where L X = distance of the rebar cut-off points to the side column; and Mbc = negative yield moment of the beam section at the cutoff point. For bare frames, the FPL of BNS, BSS and BNL are about 112, 120 and 125% of the corresponding YL. The improvement of load resistance of beams can be attributed to strain hardening and CAA. To quantify the contribution of CAA in this upgrade, the load resistance of the specimens including the strain hardening effect was determined based on the Eqs. 6.1 and 6.2. According to the ultimate strength of reinforcement, the load resisting capacities of BNS, BSS and BNL were determined to be 32.0, 75.5 and 24.0 kN, respectively. Therefore, CAA is very limited for the tested specimens. This coincides with the greater horizontal movement of the external joint Wells.

6.2.2 FPL of Infilled Specimens An MI panel is equivalent to a single or multiple compressive struts in predicting its contribution to load resistance. Since the FPL capacity of infilled specimens is mainly considered in this analysis, the single compressive strut model is adopted, as shown in Fig. 6.23. Assuming that the failure of the MI panel is due to the breakage of the struts, the contribution of the compressive struts to the FPL can be determined according to the model modification proposed by Mainstone and Weeks (1970) and Mainstone (1974). The width of compressive strut w is recommended to be: w = 0.175(λh L  )−0.4 d

(6.3)

216

6 Progressive Collapse Performance of Infilled Frames

where L  = length of the beam; d = diagonal length of the strut, the parameter λh is determined based on Eq. (6.4)  λh =

E m t sin 2θ 4E f Ib L

0.25 (6.4)

where E m = elastic modulus of the MI panel; E f = elastic modulus of the surrounding frame; t = thickness of the MI panel; Ib = moment of inertia of beam; L = length of the infill wall; and θ = angle of the strut, as shown in Fig. 6.23. The E m is determined based on the suggestion of FEMA 356 (FEMA 2000) as E m = 550 f m , where f m is the compressive strength of the MI panel (10.5 MPa here),  and E f is determined to 5,000 f c (Park and Paulay 1975). The diagonal strut resistance based on crushing failure mode Rc in each MI panel can be determined: Rc = wt f m

(6.5)

V c The vertical component of the resistance Vc is determined based on Eq. (6.6) Vc = Rc cos θ

(6.6)

Due to the four MI panels in the infilled specimens, the predicted FPL of the WNS, WSS and WNL specimens without considering the contribution of the RC frame were 201.2, 201.2 and 136.2 kN, respectively. Therefore, assuming that the failure mode of the compressive strut crushing control MI panel will greatly overestimate the FPL. As mentioned above, as shown in Figs. 6.14, 6.15 and 6.16, due to the low high-span ratio of the MI panel, the diagonal compressive struts in the specimen were split and destroyed before crushing. Therefore, the modified analysis model proposed by

Fig. 6.23 Fundamental geometric parameters for the evaluation of the width of the equivalent strut

6.3 Performance of Frames with Punctured Infill Walls

217

Saneinejad and Hobbs (1995) should be used to predict the contribution of FPL in MI walls. The diagonal strut resistance based on tensile failure mode Rt in each MI panel can be determined: √ Rt = 2 2t L f t cos θ

(6.7)

The vertical component of the resistance Vt is determined based on Eq. (6.6) Vt = Rt cos θ

(6.8)

 where f t = 0.25φ f m as the crack occurred in the bed joints rather than in the brick, φ = 0.65. The load resistance of MI panel in WNS, WSS and WNL were 123.7, 123.7 and 95.5 kN, respectively. Considering the load resistance of RC frame, the estimated FPL of WNS, WSS and WNL are 139.8, 134.7 and 102.9 kN, respectively. As a result, the analytical predictions are likely to overestimate FPL by 13, 18, and 16%, respectively. Due to the complexity and uncertainty of the material properties of MI wall, the accuracy of the analytical model is still acceptable. In addition, the prediction accuracy may be further improved if more sophisticated compressive strut models are used.

6.2.3 Dynamic Capacity Curves of Tested Specimens Based on the dynamic capacity curve method introduced in previous chapters, dynamic resistance of the specimens is calculated. Figure 6.24 shows the comparison of dynamic bearing capacity curves of specimens with and without MI panel. The dynamic ultimate bearing capacities of BNS, WNS, BSS, WSS, BNL and WNL specimens were 27.0, 97.6, 66.0, 109.0, 20.0 and 25.9 kN, respectively. Thus, the MI panel improved the DUC of BNS, BSS and BNL by 260, 65 and 230%, respectively. Similar to the conclusion of the pushdown curve, MI panels can significantly reduce the progressive collapse risk of RC frames.

6.3 Performance of Frames with Punctured Infill Walls Among the existing progressive collapse tests on infilled frame, little attention was drawn to the behavior of infilled frames with openings. The load distribution characteristics of punctured infilled wall are different from that of solid infilled wall. In order to understand the effect of opening, a series of experiments are carried out on five kinds of infilled frames with different opening rates. The results of this study can lay a foundation for the performance of infilled frame with openings to alleviate

218

6 Progressive Collapse Performance of Infilled Frames 140 Dynamic Load Resistance(kN)

Fig. 6.24 Comparison of the dynamic capacity curves of the specimens

120

BNS-D

BSS-D

BNL-D

WNS-D

WSS-D

WNL-D

100 80 60 40 20 0 0

100

200

300

400

Vertical displacement (mm)

progressive collapse and provide a necessary basis for improving the existing design criteria.

6.3.1 Design of Test Specimens In this experiment, five 1/4 scale frames were designed and cast to study the effect of opening ratio on the performance of RC frames, so as to reduce the progressive collapse of frames. As shown in Table 6.3, a bare frame and a frame with solid filled walls were named BF and WF, respectively. The other three infilled frames with opening rates of 31, 16 and 11% were named WF-L, WF-M and WF-S, respectively. The symbols “L”, “M” and “S” stand for wide open, medium open and small open respectively. The prototype building is an eight-storey office building designed according to ACI 318-14 (2014) with a span of 7,200 mm and a height of 3,600 mm for the first floor and 3,300 mm for the upper floors. The design live load is 2.0 kN/m2 , and the constant load including the dead weight is 6.4KN/m2 . Figure 6.25 Table 6.3 Specimen properties for investigation of opening effects Specimen

Beam longitudinal reinforcement Middle span

Beam ends

Top

Top

Bottom

Characteristic and opening size

Opening rate

Bottom

BF

2R6

3R6

4R6

2R6

Bare frame

100%

WF-L

2R6

3R6

4R6

2R6

400 mm × 440 mm

31%

WF-M

2R6

3R6

4R6

2R6

300 mm × 300 mm

16%

WF-S

2R6

3R6

4R6

2R6

250 mm × 250 mm

11%

WF

2R6

3R6

4R6

2R6

Solid wall

0

6.3 Performance of Frames with Punctured Infill Walls

219 Strain gauge

Interior column

Exterior column

(a) Elevation view

(b) Cross sections Fig. 6.25 Dimension and reinforcement details of Specimen WF-L (Note unit in mm)

shows the geometric structure and reinforcement details of WF-L. It should be noted that the dimensions and reinforcement details of the RC frame are the same. The difference is the opening rate of the infilled wall. The beam span is 1800 mm, the height is 900 mm, and the second and third floors are 825 mm. The column section is 150 mm × 150 mm, and the beam section is 140 mm × 90 mm. The relationship between the prototype frame and the corresponding test model is shown in Table 6.4. To simulate the initial damage caused by an explosion or car crash, the central column of the first floor was removed. The column foundation adopts two 400 mm × 300 mm foundation cast as a whole and anchored by columns. As shown in Fig. 6.25, the infilled wall is reinforced with R3 as a tie rod. Insert the tie rod into the column and secure it with epoxy. It should be noted that R6 and R3 represent ordinary steel bars with diameters of 6 and 3 mm, respectively. The T10 steel bar is a deformed steel bar with a diameter of 10 mm. Figure 6.26 shows the typical size and opening details of the rest specimen.

220

6 Progressive Collapse Performance of Infilled Frames

Table 6.4 Relationship between prototype frames and corresponding test models

Prototype frame (mm)

Items Column size

Depth Width

600

150

Beam size

Depth

560

140

Width

360

90

Story height Concrete cover

600

Test model (mm) 150

First story

3600

900

Upper story

3300

825

Beam

25

7

Column

30

10

Note T = deformed reinforcing bar; R = plain reinforcing bar

(a) BF

(c) WF-M

(b) WF-S

(d) WF

Fig. 6.26 Design details of opening specimens (Note unit in mm)

6.3.2 Test Setup and Instrumentation Figures 6.27 and 6.28 show the test setup and instrument layout, respectively. As shown in Fig. 6.27, the specimen column foundation is fixed on the bottom plate by expanding the foundation. The ground center column was removed, and a hydraulic jack with a stroke of 600 mm was used to impose a concentrated load on the top of the removed column. The displacement control method is used for load application. An increment of 1 mm was selected before each specimen reached the first peak load. After this stage, the displacement increment is 5 mm. In order to prevent

6.3 Performance of Frames with Punctured Infill Walls

221

1 2 3

6

7

4

1. Hydraulic Jack 2. Steel column 3. Steel assembly 4. Tens. /Comp. load cell 5. LVDT (300 mm) 6. LVDT (100 mm) 7. LVDT (25 mm)

5

Fig. 6.27 Overview of WF-M in position ready for testing

H-Steel Load Cell (F)

Hydraulic Jack Steel Column Steel Assembly LVDT (100 mm capacity)

Tension/Compression load cell

LD1 LF1

I2

I1

E2

E1

LF2

I4

I3

E4

E3

LF3

I6

I5

E6

E5

LD2

LD3 D7

D6

Foundation

D5

D4

D3

D2

LVDT (300 mm capacity)

D1

Floor

Fig. 6.28 Schematic view of test setup and instrumentation layout

the out-of-plane damage of the specimen, steel columns and steel assemblies were specially designed. To simulate the horizontal constraints of the surrounding beams, the overhanging beams were connected to the A-frame via three rollers. In addition, to measure axial forces, tension/compression load sensors are installed in each roller. As shown in Fig. 6.28, a series of LVDTs was installed under the beam on the first

222

6 Progressive Collapse Performance of Infilled Frames

floor to measure the deformation shape of the beam. A series of LVDTs was set along the column height and joint to measure the lateral displacement of the column and shear deformation of the joint.

6.3.3 Material Properties On the test day, the average compressive strength BF, WF, WF-S, WF-M and WF-L of concrete cylinders were 32, 34, 31, 31 and 32 MPa, respectively. The compressive strength of masonry was 17.3 MPa. The measured compressive strength of mortar was 16.5 MPa. The measured compressive strength and shear strength of masonry unit (brick and mortar combination) were 10.5 and 1.1 MPa, respectively. The properties of the rebar are shown in Table 6.5.

6.3.4 Experimental Results 6.3.4.1

Global Behavior

Specimen BF: The crack pattern of bare frame BF at the center column displacement (CCD) 50 mm is shown in Fig. 6.29a. In fact, the first crack was observed at the beam end near the central column on a 10 mm CCD. When the CCD increased to 23 mm, cracks are also formed at the cut-off point of the beam top longitudinal reinforcement. Cracks appeared on the first and second floors much earlier than on the third floor. When the CCD reached 280 mm, diagonal shear cracks appeared at the exterior joint of the first floor. However, there was no shear crack at the internal joint before failure. When the CCD reached 358 mm, the maximum width of the shear crack in the external joint reached 8 mm. The relationship between load resistance and BF vertical CCD is shown in Fig. 6.30. The YL of 28 kN was measured with a 21 mm CCD. However, a first peak load (FPL) of 32 kN is obtained at a CCD of 38 mm, which is about 114% of YL. Unlike traditional RC frames with simplified fixed boundary conditions at the beam ends, CAA benefits are not obvious, mainly due to fewer horizontal constraints at external joints. When the CCD reached 153 mm, a re-ascent was observed due to TCA mobilization. The limit load (UL) of 29 kN was measured on a 298 mm of CCD. In this displacement stage, the reinforcement is fractured Table 6.5 Reinforcements properties Items

Nominal diameter (mm)

Yield strength (MPa)

Ultimate strength (MPa)

Elongation (%)

T10

10

515

594

16.9

R6

6

449

537

13.3

R3

3

417

479

9.7

6.3 Performance of Frames with Punctured Infill Walls

(a) BF

(c) WF-M

223

(b) WF-L

(d) WF-S

(e) WF Fig. 6.29 Crack pattern of the specimens at the CCD of 50 mm

at cut-off points. With the further increase of displacement, bottom reinforcement at BENC and top rebar at curtailment of rebar successively. Figure 6.31 shows the failure mode of BF. No shear cracks were observed in the internal joints, but severe shear cracks were observed in the external joints of the first story, similar to previous tests of full-scale joint resistance to progressive collapse. Specimen WF-L: Fig. 6.29b shows the crack mode of WF-L with 50 mm CCD. Unlike BF, the first crack was found at a CCD of 6 mm. When the CCD is further increased to 8 mm, the bending crack is also formed at the cut-off point of the top longitudinal reinforcement of the beam. When the CCD was increased to 21 mm, a

224

6 Progressive Collapse Performance of Infilled Frames 120

BF WF-L WF-M WF-S WF

100 Applied load (kN)

120

Applied load (kN)

100

80

BF WF-L WF-M WF-S WF

60 40 20

80

0 0

20 40 Vertical displacement (mm)

60

60

40 20 0 0

100 200 300 Vertical displacement (mm)

400

500

Fig. 6.30 Load resistance versus vertical displacement at center column

Fig. 6.31 Failure mode of BF

separation was observed in the middle of the wall between the two openings. With the further increase of CCD, more and more cracks are produced in the beam. Horizontal cracks appeared in the lower part of the infill wall (below the opening). Because the opening was so large, no direct diagonal cracks formed in the wall. In other words, each panel of the wall did not form the main diagonal support. As shown in Fig. 6.30,

6.3 Performance of Frames with Punctured Infill Walls

225

Fig. 6.32 Failure mode of WF-L

FPL of 60 kN was measured with a 9 mm CCD, which was 188% of BF. When the CCD is 14 mm, the load resistance begins to decline, while when the CCD was 203 mm, the load resistance began to rise again, which is related to the mobilization of TCA in the longitudinal reinforcement of the beam. Therefore, infill walls with large openings do not hinder the development of TCA. The UL was 40 kN measured at 380 mm CCD. The failure modes of WF-L are shown in Fig. 6.32. As you can see, the wall panels on the second floor are almost completely separated from the beams. Three-storey wall piers appeared obvious slippage. In addition, the lower part of the wall (below the opening) deformed with the beam, acting as a deep beam. The steel bars in the beams from the second to the third floors were fractured successively. However, it should be noted that when the beam is located in the first to third floors, the fracture of the steel bar at the top of the beam is far away from the side column. On the contrary, when the beam is located in the first story to the third one, the fracture position of the beam bottom reinforcement is closer to the middle column. Shear cracks with a maximum width of 3.8 mm were detected at the external joints of the first story. Specimen WF-M: Similar to WF-L, the infill wall of WF-M shows the first crack at 5 mm CCD. Unlike WF-L, the first diagonal step crack is observed in WF-M. When the CCD reached 15 mm, cracks were also observed in the beam. In this

226

6 Progressive Collapse Performance of Infilled Frames

displacement stage, the wall connected with the second story beam appeared bed joint slippage. With the increase of CCD, the diagonal step cracks of the middle wall pier become wider and wider, and horizontal cracks appeared in the continuous lower part of the wall. Figure 6.29c shows the crack pattern of WF-M under 50 mm CCD. As shown in Fig. 6.30, when the CCD reached 7 mm, the measured FPL was 85 kN, which was 266% of BF. When the CCD increased to 122 mm, the decrease of load resistance becomes smaller. However, when the CCD was larger than 182 mm, the load resistance rose again due to the mobilization of the TCA. When the CCD was further increased to 399 mm, a UL of 48 kN can be obtained, which was about 56% of its FPL. The failure modes of WF-M are shown in Fig. 6.33. In contrast to WF-L, the middle wall pier mainly formed diagonal step cracks rather than horizontal cracks and disengagement. In addition, the wall integrity of WF-M is greater than that of WF-L, although the infilled wall is severely damaged. Although the top longitudinal reinforcement at the three stories was broken, the damage of the beam was much lighter than that of the WF-L beam. Shear cracks with a maximum width of 6.2 mm were measured at the external joints of the first story. Specimen WF-S: The first crack was found in the infilled wall at 3 mm CCD, much earlier than WF-M and WF-L. Similar to WF-M, the first crack appeared in the middle wall panel, and the first crack is diagonal rather than horizontal. At 6 mm CCD, FPL of 99 kN was obtained, which was 309% of BF. Further increasing the CCD to 21 mm, bed joint slippage was observed in the infilled wall. With the increase

Fig. 6.33 Failure mode of WF-M

6.3 Performance of Frames with Punctured Infill Walls

227

of CCD, the inclined step crack became wider. Figure 6.29d shows the crack drawing of WF-S at 50 mm CCD.As shown in Fig. 6.30, when the CCD reached 21 mm, the load resistance of the specimen began to decrease. When the CCD was 198 mm, it rose again. The ultimate load was 54 kN, about 186% of BF. The failure modes of WF-S are shown in Fig. 6.34. As can be seen from the figure, the infill wall integrity of WF-S is worse than that of WF-M, which was unexpected. Unlike the WF-M, the middle wall panels were heavily detached. All beams suffered steel fracture. Shear cracks with a maximum width of 2.1 mm were detected at the external joints of the first story. Specimen WF: Fig. 6.29e shows the crack pattern of WF at 50 mm CCD. As shown in Fig. 6.30, the first crack appeared on the infilled wall at 1 mm CCD, much earlier than on the wall with the opening. Firstly, the cracks in the RC beam were observed at a CCD of 2 mm. When the CCD was further increased to 3 mm, the infilled wall produces more cracks. An FPL of 114 kN was achieved during this loading phase. In addition, the beam forms bending cracks. Unlike BF, cracks do not always form at BNEC and cut-off points. In fact, the location of flexural cracks varies. According to the observation of crack width, the main load resistance mechanism in the FPL stage is the compressive struts of the infilled wall (this conclusion is confirmed by strain measurements). When the CCD reached 10 mm, slippage occurred in the bed joint of the infilled wall, and the load resistance decreases obviously. With the further increase of CCD, the slippage of joint is aggravated. However, it is worth noting that

Fig. 6.34 Failure mode of WF-S

228

6 Progressive Collapse Performance of Infilled Frames

Fig. 6.35 Failure mode of WF

the slippage of bed joint on the second floor is much earlier and more serious than that on the third floor. When the CCD reached 51 mm, the load resistance of the specimen basically remains unchanged. After this, the main diagonal strut loses its resistance to loading and several secondary compression struts outside the central region come into play. With the increase of CCD, the slippage of bed joint becomes more serious. Some of the bricks have even been removed entirely. When the CCD reached 292 mm, the reinforcement continuously fractured from one story to the third story. The failure mode of WF is shown in Fig. 6.35. The main diagonal step cracks become wider, and some bed joint separated completely. Different from BF, the external joints of the second floor also have diagonal shear cracks. In addition, although the steel bar fracture occurred in the beam of each story, the fracture location of the steel bar is quite different from that of BF. In general, infilled walls in WF have the best integrity. Shear cracks with a maximum width of 7.6 mm were measured at the external joints of the first story.

6.3.4.2

Deflection Shape of the Beams in the First Floor

Figure 6.36 shows the deflection shapes of the beam in first floor of BF and WF-L at different stages. For BF, as shown in Fig. 6.36a, from the beginning of the test, the

6.3 Performance of Frames with Punctured Infill Walls

-100 -150 -200 25mm 100mm 200mm 300mm Final

-250

-300 Chord Rotation D1 D2

-350

D3

D4

D5

D6

D7

-400 0

450

900 1350 1800 2250 2700 3150 3600 LVDT Positions (mm)

Vertical Dipalcement (mm)

Vertical Dipalcement (mm)

0 -50

229

0 -50 -100 -150 -200 -250 -300 -350 -400 -450 -500

25mm 100mm 200mm 300mm Final

Chord Rotation

D1 0

450

(a) BF

D2

D3

D4

D5

D6

D7

900 1350 1800 2250 2700 3150 3600 LVDT Positions (mm)

(b) WF-L

Fig. 6.36 Comparison of the beam deflection from bare and infilled frames

slope of the beam near the side column was significantly smaller than that near the center column. In the final phase of the test, the chord rotation, defined as the ratio of the CCD to the clear span of the beam, coincided with the slope of the beam near the center column, but overestimated the slope of the beam near the side column. As shown in Fig. 6.36b, the beam deformation of WF-L has double curvature. The relatively small slope of the beam near the central column can be attributed to the combined action of the infilled wall and the beam. In the final phase of the test, in contrast to BF, the chord rotation will overestimate the rotation of the beam section near the central column and underestimate the rotation of the beam section near the side column. It should be noted that the deformation shapes of the beam in the first story of WF-M, WF-S and WF were very similar to those of WF-L.

6.3.4.3

Horizontal Movement of Exterior Joints

As described earlier, we mounted a series of LVDTs horizontally to measure the lateral motion of the external joint during the test. The relationship between horizontal motion of external joint and CCD is shown in Fig. 6.37. For BF, the maximum outward motion of −10 mm was measured at the third-story at CCD of 110 mm. When the CCD reached 233 mm, the external joint returns to its original position. The maximum inward motion of 54 mm was measured at 370 mm CCD on the third floor. Similar to BF, WF-L measured a maximum outward movement of −12 mm and an inward movement of 53 mm at the external joints in the third floor. The greater outward movement is mainly due to the infilled wall acting as a compression strut, which brings additional outward movement. The maximum outward movement of WFM, WF-S and WF was −13, −14 and −16 mm, respectively. The failure mode and load–displacement curve verify that the larger the opening is, the lower the equivalent efficiency of the compression rod is. The maximum inward motion of WF-M, WF-S and WF was 50, 48 and 47 mm, respectively. The main reason for the small inward movement of WF in the large deformation stage is that the equivalent compression

230

6 Progressive Collapse Performance of Infilled Frames

60 Horizontal movement (mm)

LD2

40

LD3

30 20

10 0

LD1 LD2 LD3

-10 -20

0

50

Horizontal movement (mm)

60 LD1

50

LD1

50

LD2

40

LD3

30 20 10 0

LD1 LD2 LD3

-10 -20

100 150 200 250 300 350 400 450 Vertical displacement (mm)

0

50

100 150 200 250 300 350 400 450 Vertical displacement (mm)

(a) BF 60

LD1

50

LD2

40

LD3

30 20 10 0

LD1 LD2 LD3

-10 -20 0

50

Horizontal movement (mm)

Horizontal movement (mm)

60

(b) WF-L LD1

50

LD2

40

LD3

30 20 10 0

LD1 LD2 LD3

-10

-20 0

100 150 200 250 300 350 400 450 Vertical displacement (mm)

50

100 150 200 250 300 350 400 450 Vertical displacement (mm)

(c) WF-M

(d) WF-S

Horizontal movement (mm)

60

LD1

50

LD2

40

LD3

30 20 10 0

LD1 LD2 LD3

-10 -20

0

50

100 150 200 250 300 350 400 450 Vertical displacement (mm)

(e) WF Fig. 6.37 Horizontal movement of the exterior joints at different story

6.3 Performance of Frames with Punctured Infill Walls

231

rod can provide additional horizontal constraints when the TCA is mobilized in the beam.

6.3.4.4

Strain Gauge Readings

In order to study the resistance mechanism of the frame during the test, a series of strain gauges are symmetrically pasted at the longitudinal reinforcement of the beams on the second and third floors, as shown in Fig. 6.25. Figure 6.38 shows the variation of strain gauge results. As shown in Fig. 6.38a, the maximum tensile strain of the top bar of the BF was measured at 435 mm away from the interface of the side columns of the second and third story. When TCA was mobilized, the compressive strain at the beam end near the middle column begins to decrease. However, even at the last stage of the test, compressive strain was observed at the beam end near the central column. As shown in Fig. 6.38b, the maximum tensile strain of the reinforcement at the bottom of BF is measured at the interface of the two central columns. In addition, when TCA begins to develop, the tensile strain of the bottom reinforcement continues to increase. Therefore, the study shows that both top and bottom reinforcement contribute to TCA. Finally, according to the BF strain measurement results, it was concluded that the single-story substructure can better characterize the performance of the bare two-dimensional multi-story frame. The maximum tensile strains of the top reinforcement of WF-L at different positions of the second and third stories were measured, which were in good agreement with the crack pattern of the specimen. In addition, different from BF, the maximum tensile strain of the bottom steel bar was also different from that of the second and third story. When the CCD was increased to 184 mm, the tensile strain further increased significantly with the further increase of the CCD. Thus, an infilled frame with openings can form a TCA that coincides with the horizontal motion of the joints. For WF-M, WF-S, and WF, similar trends were observed, although the location of the maximum tensile strain may be different.

6.3.4.5

Shear Rotation in Exterior Joints

To monitor the behavior of the joint zone, two LVDTS are mounted orthogonal at the side joints (Item 7 in Fig. 6.27). Figure 6.39 provides a method for determining the shear deformation of the joint. As shown in the figure, the two LVDT readings can convert the joint shear distortion γ . It is worth pointing out that 1 and 2 are tensile deformation and shortening deformation of two LVDTs respectively. a is the length of square joint panel. Figures 6.40, 6.41 and 6.42 show the changes of shear panel at different stories of different specimens. As shown in Fig. 6.40a, before the CCD reached 183 mm, the maximum distortion −14 × 10–4 rad of the external joint of first story was measured. Negative values indicate shear deformation opposite to Fig. 6.39. When the CCD exceeded 183 mm, the negative distortion starts to decrease due to the TCA generated in the beam, which tends to pull the joint inward.

232

6 Progressive Collapse Performance of Infilled Frames

6000

Strain (με)

4000

8000

FPL-3rd UL-3rd FPL-2nd UL-2nd

FY-3rd TCA-3rd FY-2nd TCA-2nd Center Column Interface

FPL-3rd UL-3rd FPL-2nd UL-2nd Side of Exterior Column

4000

εy

2000 0 Side of Exterior Column

εy

-2000

FY-3rd TCA-3rd FY-2nd TCA-2nd

6000

εy

Strain (με)

8000

2000 0 Center Column Interface

-2000

εy -4000

-4000 0 300 600 900 1200 1500 1800 Distance from the adjacent column interface (mm)

0 300 600 900 1200 1500 1800 Distance from the adjacent column interface (mm)

(a) Top rebar in BF 8000

FPL-3rd UL-3rd FPL-2nd UL-2nd

4000

εy

2000

0 Side of Exterior Column

Center Column Interface

-2000

FC-3rd TCA-3rd FC-2nd TCA-2nd

6000 Strain (με)

Strain (με)

4000

8000

FPL-3rd UL-3rd FPL-2nd UL-2nd

FC-3rd TCA-3rd FC-2nd TCA-2nd

6000

(b) Bottom rebar in BF

εy

2000 0 Center Column Interface

-2000

εy

εy -4000

-4000 0

0 300 600 900 1200 1500 1800 Distance from the adjacent column interface (mm)

300 600 900 1200 1500 1800 Distance from the adjacent column interface (mm)

(c) Top rebar in WF-L 8000

FC-3rd TCA-3rd FC-2nd TCA-2nd

6000

(d) Bottom rebar in WF-L 8000

FPL-3rd UL-3rd FPL-2nd UL-2nd

Center Column Interface

εy 2000 0 Side of Exterior Column

-2000

4000

0 Center Column Interface

Side of Exterior Column

εy

-4000

-4000 0 300 600 900 1200 1500 1800 Distance from the adjacent column interface (mm)

0

(e) Top rebar in WF-S

6000 Center Column Interface

FC-3rd TCA-3rd FC-2nd TCA-2nd

300 600 900 1200 1500 1800 Distance from the adjacent column interface (mm)

(f) Bottom rebar in WF-S 8000

FPL-3rd UL-3rd FPL-2nd UL-2nd

FC-3rd TCA-3rd FC-2nd TCA-2nd

6000

2000 0 Side of Exterior Column

-2000

εy -4000

0 300 600 900 1200 1500 1800 Distance from the adjacent column interface (mm)

(g) Top rebar in WF

FPL-3rd UL-3rd FPL-2nd UL-2nd Side of Exterior Column

4000

εy

Strain (με)

8000

4000

εy

2000

-2000

εy

FPL-3rd UL-3rd FPL-2nd UL-2nd

FC-3rd TCA-3rd FC-2nd TCA-2nd

6000 Strain (με)

Strain (με)

4000

Strain (με)

Side of Exterior Column

εy

2000 0 -2000

Center Column Interface

εy

-4000 0 300 600 900 1200 1500 1800 Distance from the adjacent column interface (mm)

(h) Bottom rebar in WF

Fig. 6.38 Variation of strain results along the third floor beam longitudinal reinforcements

6.3 Performance of Frames with Punctured Infill Walls

233

a=100

Original panel

1 /2

/2 Δ2 Δ

γ2

j

l

a=100

Deformed panel

/2 Δ2

θ Δ

1 /2

γ1

γ=γ1+γ2=

Δ1+Δ2 √2a

Fig. 6.39 Method to determine the distortion of shear panel in joints 6

1st Floor 2nd Floor 3rd Floor

70 60

Shear Distortion(10-4rad)

Shear Distortion(10-4rad)

80

50 40

30 20 10 0

1st Floor 2nd Floor 3rd Floor

4 2 0 -2 -4

-10

-6

-20 0

100

200

300

0

400

100

200

300

400

Vertical displacement (mm)

Vertical displacement (mm)

(a)

(b)

Fig. 6.40 Variation of shear panel distortions of the joints in different stories of BF: a exterior joints, b interior joint 6 1st Floor 2nd Floor 3rd Floor

30 25

Shear Distortion(10-4rad)

Shear Distortion(10-4rad)

35

20 15 10

5 0

1st Floor 2nd Floor 3rd Floor

4 2 0 -2 -4

-5

-10

-6 0

100

200

300

Vertical displacement (mm)

(a)

400

0

100

200

300

400

Vertical displacement (mm)

(b)

Fig. 6.41 Variation of shear panel distortions of the joints in different stories of WF-L: a exterior joints, b interior joint

234

6 Progressive Collapse Performance of Infilled Frames 6 1st Floor 2nd Floor 3rd Floor

70 60

Shear Distortion(10-4rad)

Shear Distortion(10-4rad)

80

50 40

30 20 10 0

1st Floor 2nd Floor 3rd Floor

4 2 0 -2 -4

-10

-20

-6 0

100

200

300

400

0

Vertical displacement (mm)

(a)

100

200

300

400

Vertical displacement (mm)

(b)

Fig. 6.42 Variation of shear panel distortions of the joints in different stories of WF: a exterior joints, b interior joint

At the end of the test, the maximum deformation of the joint outside the first story was 225 × 10–4 rad. The maximum deformation of the external joints in the second and third stories was 13 × 10–4 rad and 10 × 10–4 rad, respectively. For internal joints, as shown in Fig. 6.40b, the maximum negative and positive shear deformation range was −2 × 10–4 rad–1 × 10–4 rad. The small deformation of internal joints is mainly due to the horizontal reaction force provided by the cantilever beam, which significantly reduces the shear force of the internal joints. As shown in Figs. 6.41a and 6.42a, the maximum negative shear deformation of external joints in WF-L and WF was −7 × 10–4 rad and −9 × 10–4 rad, respectively. However, the maximum positive shear deformation of WF-L and WF internal joints was 30 × 10–4 rad and 57 × 10–4 rad, respectively. Therefore, even the infilled wall with a large opening ratio can significantly reduce the shear deformation of the external joint. However, it should be noted that the maximum shear deformation of WF-L was measured in the second story, while the maximum shear deformation of WF was measured in the first story. This is mainly due to the interaction of the infilled wall with the surrounding frame, which was very complex.

6.3.5 Results Analysis and Discussion 6.3.5.1

Discussion of the Effects of Infill Walls and Openings

According to Table 6.6, compared with FPL of WF and BF, FPL and UL can be increased by 256% and 100% respectively by solid-filled wall. The increase in FPL was mainly due to the formation of diagonal compressive struts in the wall (see Fig. 6.43a). During the large deformation phase, the larger UL values measured by the WF can be attributed to secondary compressive struts formed in the relatively intact wall (see Fig. 6.43b). Comparing Figs. 6.31 and 6.35, the interaction between the solid wall and the surrounding frame can change the position of the fracture of

6.3 Performance of Frames with Punctured Infill Walls

235

Table 6.6 Test results for investigation of opening effects Specimen ID

Critical load (kN) Critical displacement (mm) FPL

UL

FPL

UL

Deformation capacity (mm)

BF

32

29

38

298

370

WF-L

60

40

14

380

380

WF-M

85

48

7

399

399

WF-S

99

54

6

420

431

114

58

3

299

393

WF

Note FPL means first peak load capacity; UL represents ultimate load capacity

the reinforcement. The opening rates of WF-L, WF-M and WF-S were 31, 16 and 11%, respectively. The FPL of WF-L, WF-M, WF-S and WF was 60, 85, 99 and 114 kN, respectively. Therefore, the opening rates of 31, 16 and 11% reduced the FPL of WF by 47, 25 and 13%, respectively. The disadvantageous effect of opening on FPL was mainly because the opening weakens the load redistribution ability of the compressive strut of infilled wall. As shown in Fig. 6.29, for WF-L with an opening ratio of 31%, no diagonal compressive strut is formed in the infilled wall. WF-M with an opening ratio of 16% only forms a diagonal strut between openings. For WF-S with an opening ratio of 11%, diagonal struts similar to WF can be formed. The UL of WF-L, WF-M, WF-S and WF are 40, 48, 54 and 58 kN, respectively. Therefore, the UL of WF was reduced by 31, 17 and 7% at open rates of 31, 16 and 11%, respectively. Since the TCA contribution to UL of RC beams was similar, the adverse effect of opening on UL is mainly due to the weakening of secondary compressive struts by opening. Considering the complexity of stress distribution in the infilled walls with openings, the compressive strut can be accurately determined by numerical analysis. In addition, as mentioned before, the first crack was formed in the infill walls of WF, WF-S, WF-M, and WF-L at the 1, 3, 5, and 8 mm CCD positions. Therefore, the opening rate can reduce the stiffness of the wall and delay the formation of wall cracks. In addition, as shown in Figs. 6.32, 6.33 and 6.34, in the large deformation stage, especially when the opening rate reached 31%, the opening rate will reduce the integrity of the wall. The maximum width of shear cracks in the exterior wall joints of BF, WF-L, WF-M, WF-S and WF are 8.0, 3.8, 6.2, 2.1 and 7.6 mm, respectively. By comparing the failure modes of the specimens, it was found that the crack width of the external joints is controlled by the secondary compressive struts efficiency. For WF-M, the width of the complete wall panel near the outer column was smaller than that of WF-L and WF-S, which reduced the efficiency of the secondary compressive struts. For WF, there was an unexpected detachment between the wall and the lateral column during the large deformation stage, which may hinder the development of secondary struts. The deformation of BF, WF-L, WFM, WF-S and WF were 370, 380, 399, 431 and 393 mm, respectively. Fortunately, infilled walls and openings may not reduce the deformability of the frame. Therefore, we believe that the contribution of infilled walls should be taken into account when

236

6 Progressive Collapse Performance of Infilled Frames

Fig. 6.43 Schematic view of equivalent compressive strut of infill walls: a at small deformation stage, b at large deformation stage

Main strut at small deformation stage

(a)

Secondary strut at large deformation stage

(b)

assessing the load redistribution capacity of RC frames, since progressive collapse events are in fact low-probability events.

6.3.6 De-composition of the Load Resistance In order to quantify the contribution of the infilled frame to the gradual collapse resistance, the resistance of the infilled frame is decomposed by subtracting the resistance of the bare frame under the same displacement. As shown in Fig. 6.44, initially, the load contribution of the filled wall on WF-L was 82%. At the CCD 14 mm or FPL stage, 61% of the load resistance was attributed to the infilled wall with an open rate of 31%. With further increased in CCD, the infilled wall contribution dropped to 40%. After that, the ratio decreased continuously until the CCD reached 204 mm. At this stage, the infilled wall contributed only 7% of the resistance. This can be explained by the mobilization of TCA in RC beams. Subsequently, due to the

6.3 Performance of Frames with Punctured Infill Walls

237

100

100

RC Frames

RC Frames 90

Infill Walls

80

Resistance Decomposition (%)

80

Resistance Decomposition (%)

90

Infill Walls

70 60 50 40 30 20

70 60 50 40 30 20 10

10

WF-M

WF-L 0

0

Displacement (mm)

Displacement (mm)

(a) WF-L

(b) WF-M 100

100

RC Frames

RC Frames 90

90

Infill Walls

Resistance Decomposition (%)

Resistance Decomposition (%)

Infill Walls

80

80 70 60 50 40 30 20 10

WF-S

70 60 50 40 30 20 10

WF

0

0 Displacement (mm)

(c) WF-S

Displacement (mm)

(d) WF

Fig. 6.44 De-composition of the load resistance from RC frames and infill walls

fracture of the rebar, the infilled wall contribution kept rising again, which reduced the load resistance of the TCA. At the end of the test, although the opening rate of the infilled wall was 31%, its contribution was about 60%. Therefore, even in the stage of large deformation, the load resistance of the infilled wall can not be ignored. For WF-M, 87% of the initial load resistance came from the infilled wall. In the FPL stage, the infill wall accounted for 81% of the load resistance. Then, with the increase of CCD, the contribution of the infilled wall decreased continuously until the CCD reached 264 mm. The minimum proportion of infilled wall was 22%. Similar to WF-L, with the further increase of CCD, the contribution of the infilled wall also increased. At the end of the test, the infilling wall contributed about 70%. For WF-S and WF, more than 90% of the load resistance was initially caused by the infilled wall. The minimum contribution of infilled wall to WF-S and WF was 35 and 43%, respectively. It can be seen that compared with the wall with 31% opening rate, the

Fig. 6.45 Dynamic load resistance-displacement curves

6 Progressive Collapse Performance of Infilled Frames 100 Dynamic Load Resistance (kN)

238

BF WF WF-S WF-M WF-L

80 60 40 20 0

0

50

100 150 200 250 300 350 400 450 Vertical Displacement (mm)

wall with 11% opening rate and the solid infilled wall can provide greater additional load resistance during the large deformation stage. Based on the energy-based method proposed by Izzuddin et al. (2008), the influence of the opening on the dynamic load resistance of the frame was quantified. As shown in Fig. 6.45, the dynamic peak loads of BF, WF-L, WF-M, WF-S and WF were 27, 55, 80, 86 and 97 kN, respectively. As a result, the solid-infilled wall increased the dynamic peak load by 259%. When the opening rate was 31, 16 and 11%, the dynamic peak load of WF was reduced by 43, 18 and 11%, respectively. Compared with the negative effect on static peak load, the negative effect of opening on dynamic peak load was smaller.

6.4 Conclusions Through the experimental and analytical observation of this study, the following conclusions are drawn: The reduction of longitudinal reinforcement in the beam will change the failure mode of the bare frame, but will not prevent the development of TCA. The lack of horizontal constraints and the possible shear failure of external joints seriously affect the efficiency of CAA and TCA in multi-story bare frames. The initial stiffness, yield bearing capacity, first peak bearing capacity and ultimate bearing capacity of BSS are 158, 242, 259, and 205% of that of BNS, respectively. Therefore, seismic design can significantly improve the progressive collapse resistance of RC frames. The initial stiffness, yield bearing capacity, first peak bearing capacity and ultimate bearing capacity of BNL (span to depth ratio 17) were 35, 31, 24 and 8% lower than that of BNS (span to depth ratio 13), respectively. The results show that the first peak load and initial stiffness of the structure can be increased by 260 and 900%, respectively. In addition, compared with the bare frame, the infilled specimen has greater load resistance at the large deformation stage, and its ultimate deformation capacity is almost the same. In addition, the analysis results show that

References

239

the MI panel can increase the dynamic ultimate capacity of BNS, BSS and BNL by 260, 65 and 230%, respectively. For non-seismic design frames, plastic hinges may form at the cut-off point rather than at the beam ends when rebar cutting is considered in the design. Therefore, attention should be paid to the prediction of flexural strength in the design. The analysis and prediction of FPL of the infilled frame further confirmed that the tensile failure of the compressive strut may occur before the crushing of the compressive struts when the infilled frame suffers the loss of the middle column. Therefore, it is assumed that the crushing of the compressive struts in the infilled wall may lead to the overestimation of the strengthening efficiency of the infilled wall. When the opening ratio is 31, 16 and 11%, the static peak load of the solid wall infilled frame is reduced by 47, 25 and 13%, respectively. However, 31, 16 and 11% opening rates will reduce the dynamic ultimate load by 43, 18 and 11%, respectively. Thus, the adverse effects of openings on static and dynamic ultimate load resistance are similar.

References ACI 318-08 (2008) Building code requirements for structural concrete and commentary. American concrete institute, Farmington Hills Choi H, Kim J (2011) Progressive collapse-resisting capacity of RC beam–column sub-assemblage. Mag Concr Res 63(4):297–310 DoD (2009) Design of building to resist progressive collapse. Unified Facility Criteria. UFC 4-02309. US Department of Defense, Washington (DC) FEMA 356 (2000) Pre-standard and commentary for the seismic rehabilitation of buildings. American society of civil engineers, Washington, DC, Reston Izzuddin BA, Vlassis AG, Elghazouli AY et al (2008) Progressive collapse of multi-storey buildings due to sudden column loss — part I: simplified assessment framework. Eng Struct 30(5):1308– 1318 Mainstone RJ, Weeks GA (1970) The influence of bonding frame on the racking stiffness and strength of brick walls. In: Proceedings of 2nd international brick masonry conference. Building Research Establishment, Watford, pp 165–171 Mainstone RJ (1974) Supplementary note on the stiffness and strengths of infilled frames. Building Research Station, CP 13/74, Garston Park R, Paulay T (1975) Reinforced concrete structures. Wiley, New York, p 769 Qian K, Li B, Ma JX (2015) Load-carrying mechanism to resist progressive collapse of RC buildings. J Struct Eng 141(2):4014107 Saneinejad A, Hobbs B (1995) Inelastic design of infilled frames. J Struct Eng 121(4):634–650 Sasani M, Kropelnicki J (2008) Progressive collapse analysis of an RC structure. Struct Des Tall Spec Build 17(4):757–771 Yi WJ, He QF, Xiao Y, Kunnath SK (2008) Experimental study on progressive collapse-resistant behavior of reinforced concrete frame structures. ACI Struct J 105(4):433–439 Yu J, Tan KH (2014a) Analytical model for the capacity of compressive arch action of reinforced concrete sub-assemblages. Mag Concr Res 66(3):109–126 Yu J, Tan KH (2014b) Special detailing techniques to improve structural resistance against progressive collapse. J Struct Eng 140(3):4013077

Chapter 7

Vulnerability and Robustness of RC Frames to Resist Progressive Collapse

Up to now, the robustness of RC frames against progressive collapse has been studied extensively. Several definitions of technology application robustness can be found in the literature related to various research and technology areas, such as structural and software engineering, product development and quality control, ecosystems, design optimization, Bayesian decision making, and linguistics. For example, control theory defines robustness as the degree to which a system is insensitive to influences not considered in the design. Similarly, statistical techniques with robust statistical assumptions are not sensitive to small deviations in the assumptions. Although a general consensus is still lacking in the structural engineering community, robustness is often defined as the ability of a structure to avoid consequences disproportionate to the events that lead to failure. Thus, robustness refers to the minimum residual ability to maintain function after changes in the structure (such as initial failure) or its environment (such as slope modification, urban site configuration due to underground excavation). Table 7.1 provides some of the definitions of robustness available in the literature. The robustness of a structure depends heavily on redundancy, which refers to the ability of a structure to redistribute load after a single or few members are damaged. In this regard, a robust structure is usually able to develop alternative load paths (ALPs) through structural tie, strength, and ductility. Alternatively (if there is no ALP), the extent of damage and collapse spread can be controlled by introducing discontinuities into the structure (segments) or by designing some key elements to resist extreme events (key element design). In this case, robustness can be achieved through continuity or segmentation of the structural system, depending on the case and design approach adopted. In any case, it should be kept in mind that a robust structure is not overdesigned, but is able to activate potential resistance mechanisms that are not normally utilized to withstand normal loads. In the existing research, the deterministic parameters are usually selected, while the uncertainty of parameters is ignored. Therefore, the uncertainties of loading, material properties and geometric properties are investigated in this study. The random pushdown analysis method is © China Architecture & Building Press 2023 K. Qian and Q. Fang, Progressive Collapse Resilience of Concrete Structures: Mechanisms, Simulations and Experiments, https://doi.org/10.1007/978-981-99-0772-4_7

241

242

7 Vulnerability and Robustness of RC Frames to Resist Progressive Collapse

Table 7.1 Definitions of structural robustness Source

Definition

GSA (2003)

Robustness—Ability of a structure or structural components to resist damage without premature and/or brittle failure due to events like explosions, impacts, fire or consequences of human error, due to its vigorous strength and toughness

EC1-Part1-7 (2006)

Robustness: The ability of a structure to withstand events like fire, explosions, impact or the consequences of human error, without being damaged to an extent disproportionate to the original cause

Bontempi (2007)

The robustness of a structure, intended as its ability not to suffer disproportionate damages as a result of limited initial failure, is an intrinsic requirement, inherent to the structural system organization

Agarwal and England (2008)

Robustness is the ability of a structure to avoid disproportionate consequences in relation to the initial damage

Biondini et al. (2008)

Structural robustness can be viewed as the ability of the system to suffer an amount of damage not disproportionate with respect to the causes of the damage itself

Vrouwenvelder (2008)

The notion of robustness is that a structure should not be too sensitive to local damage, whatever the source of damage

Starossek and Haberland (2010)

Robustness. Insensitivity of a structure to initial damage. A structure is robust if an initial damage does not lead to disproportionate collapse

Fib model code (2010)

Robustness is a specific aspect of structural safety that refers to the ability of a system subject to accidental or exceptional loadings (such as fire, explosions, impact or consequences of human errors) to sustain local damage to some structural components without experiencing a disproportionate degree of overall distress or collapse

Brett and Lu (2013)

Ability of a structure in withstanding an abnormal event involving a localized failure with limited levels of consequences, or simply structural damages

used to quantify the vulnerability and robustness of RC frame structures against progressive collapse. The main contents are as follows: Based on the combination of the traditional pushdown method and the related simplified Latin hypercube sampling method, a random pushdown method considering the influence of uncertainty is proposed and applied to the evaluation of the vulnerability of RC frames to progressive collapse. The results show that the normal cumulative distribution can well describe the probability characteristics of gradual collapse resistance of RC frames. The loss risk of the exterior column is higher than that of the inner column. Buildings with larger floors are at higher risk of losing ground columns. Through regression analysis, the empirical prediction formula between the number of floors

7.1 Numerical Study on Progressive Collapse Behavior of RC Frames

243

and the maximum load coefficient is obtained. Based on “tornado diagram” method, the sensitivity of each uncertainty of RC frame structures for progressive collapse prevention was investigated. It was found that the uncertainties for structures to resist progressive collapse could not be ignored. Among them, the dead load, live load, yield strength and ultimate strength of the reinforcements, the compressive strength of concrete and reinforcement ratio have significant effects on RC frame structures to mitigate progressive collapse. The quantitative assessment of robustness of RC frames to resist progressive collapse was carried out. It was demonstrated that the robustness was increasing with the increase of the floor number. For a building, the loss of a ground column has lowest robustness while the loss of a top column achieved the greatest robustness. Reducing the load combination and increase the reinforcement strength could enhance the robustness significantly.

7.1 Numerical Study on Progressive Collapse Behavior of RC Frames As a preliminary of study on vulnerability and robustness of RC frames to resist progressive collapse, a numerical study using OpenSees is performed to reduplicate the test results from existing works. After verification, the numerical model is employed to conduct case study.

7.1.1 Characteristics of the Case Study Buildings According to the requirements of ACI 318-08 (2008), two RC moment resisting frames with different span aspect ratios were selected as the case study buildings. The design dead load and live load are set to be 5.0 and 3.0 kN/m2 , respectively. Seismic forces were also considered in the design, assuming that the low seismicity zone has a class D site and a rigid soil profile. The designed 1 s spectral response acceleration parameter is 0.34 g. TF and LF represent frames with spans of 6000 and 8400 mm, respectively. TF and LF use the same elevation. The height of the first floor is 3600 mm, and the height of the upper floor is 3300 mm. The plans, elevations and typical reinforcement details of the case study buildings are shown in Figs. 7.1, 7.2 and 7.3. The frame has the same column section (see section C–C), while the beam section of LF (see section A–A) is larger than that of TF (see section B–B) due to the larger beam span. Qian et al. (2015) tested two substructures of TF and LF at the ratio of one quarter to examine the load carrying mechanism of RC buildings against progressive collapse. Test data provides an opportunity to validate structural modeling. In accordance with Qian et al. (2015), the substructures of LF and TF are labeled as P1 and P2 respectively (also shown in Fig. 7.2). The nominal performance

244

7 Vulnerability and Robustness of RC Frames to Resist Progressive Collapse

Fig. 7.1 Frame plan view

of the material used in the design is the compressive strength of concrete (f c = 27.6 MPa); Young’s modulus of concrete (E c = 24.3 GPa); Reinforcement yield strength (f y = 413.8 MPa); Young’s modulus of reinforcement (Es = 200 GPa). The critical responses of the beam-column structures (P1, P2, T1 and T2) measured in Sect. 4.1 are used for verification. In the test, the middle column under the beam was removed in advance, and the increasing displacement was applied to the top of the middle column for the pushdown test. The edge column of the specimen is designed to be fixed on a strong leg to effectively simulate the fixed boundary conditions. To create a symmetrical failure mode for the specimen, a steel assembly was installed below the hydraulic jack to ensure concentric loading. The span of P1 and P2 was 2,100 mm and 1,500 mm, respectively. The cross section sizes of P1 and P2 were 180 mm × 90 mm and 140 mm × 90 mm, respectively. Figures 7.4 and 7.5 show the size of the specimen and the details of reinforcement. R6 is a plain steel bar with a diameter of 6 mm, and T10 is a deformed steel bar with a diameter of 10 mm.

7.1 Numerical Study on Progressive Collapse Behavior of RC Frames

245

Fig. 7.2 Elevation of frames for case study

Fig. 7.3 Reinforcement arrangement of beam and column

7.1.2 Validation of Numerical Model In this study, a numerical model is established using the finite element software OpenSees. The beams and columns are modeled using displacements based nonlinear beam-column elements, namely dispbeamcolumns, with five Gaus-Legendre integration points along the element length. The hysteretic material is used to simulate the steel bar. The material MinMax was used to define the failure strain of reinforcement. Concrete material model concrete02 can consider the linear tensile properties of concrete. Constraints are specified by the stress–strain relationship of the KentPark model modified by Scott et al. (1982) The simulated force and displacement responses were compared with the experimental results, as shown in Fig. 7.6. It can be seen that the simulation results are in good agreement with the experimental results, which proves the reliability of the macro modeling technique adopted. Therefore, the numerical model can be used for case study.

246

7 Vulnerability and Robustness of RC Frames to Resist Progressive Collapse

(a) P1 R6@140

R6@140

R6@140

R6@140

Weld 25 mm diameter bolt with 200 mm length

Steel Plate 1 with 10 mm thickness

(b) P2 Fig. 7.4 Dimensions and reinforcement details of P-series specimens

In this part, a numerical model is established using OpenSees. The modeling details such as element type and material constitutive relation are introduced. The experimental results verify the reliability and accuracy of the numerical model, and the numerical model can reproduce the experimental results well, which can be used for case study.

7.2 Study on Vulnerability of RC Frames to Resist Progressive Collapse Considering Uncertainties Study on vulnerability of RC frames to resist progressive collapse considering uncertainties is conducted in this Chapter. A total of 16 uncertain parameters involves design loads, material properties, and geometric properties are selected. And then, structural models with random parameters are built for pushdown analyses to obtain probability distribution functions. Moreover, based on regression analysis, empiric prediction formula is obtained to correlate the floor number and the maximum load

7.2 Study on Vulnerability of RC Frames to Resist Progressive Collapse …

Fig. 7.5 Dimensions and reinforcement details of T-series specimens

247

248

7 Vulnerability and Robustness of RC Frames to Resist Progressive Collapse

Fig. 7.5 (continued)

coefficient. Finally, pushdown analyses are performed by testing the uncertainty parameters to study their sensitivity to progressive collapse.

7.2.1 Pushdown Analysis and Damage Criteria Pushdown analysis begins with the removal of critical members and then studies the complete behavior of the damaged structure by increasing gravity loads proportionally until the system fails. Only the gravity load on the damaged compartment is raised, while the rest of the damaged structure is subjected to a nominal gravity load. The gravity load is the combination of dead load (DL) and live load (LL), and

7.2 Study on Vulnerability of RC Frames to Resist Progressive Collapse … 60

Test FEM

Test FEM

50

40

Vertical load (kN)

Vertical load (kN)

50

30

20 10

40 30

20 10

0

0 0

100 200 300 Vertical displacement (mm)

400

0

50

100 150 200 250 300 Vertical displacement (mm)

(a) Specimen P1 90

350

(b) Specimen P2 100

Test FEM Vertical load (kN)

Vertical load (kN)

249

60

30

Test FEM

80 60

40 20 0

0 0

50 100 150 200 Vertical displacement (mm)

(c) Specimen T1

250

0

100 200 Vertical displacement (mm)

300

(d) Specimen T2

Fig. 7.6 Comparison of the load–displacement relationship of test specimens

the nominal combination is determined as (GSA 2013; DoD 2013). The displacement control method is adopted to apply vertical displacement increment  at the joint and record the gravity load applied in the damaged bay, as shown in Fig. 7.7. The recorded gravity load is expressed as α(1.2 × DL+0.5L L), where α = the load factor calculated by the ratio of the recorded gravity load to the nominal gravity load. According to DoD (2013) recommendations, under normal use, the applied pressure shall be, with a value of 1.0. However, due to the quasi-static push-down loading mode (without considering the dynamic effect), the load resistance changes with the increase of the vertical displacement at the lost column. Therefore, α varies with change of  and the load–displacement relationship in and coordinates is used as a “pushdown curve”. Using it to indicate the load resistance of the building can help determine whether the building can cross the bridge when the loss of the column is easier. In particular, the load factor is equal to the dynamic increase factor (DIF), n which corresponds to the state where the applied vertical load reaches the load specified in the nonlinear static process (NSP) of the Alternative path Method (APM). n (1.2 × DL+0.5L L) (GSA 2013; Department of Defense 2013). Less than the maximum load factor n means that the structure may collapse due to column loss.

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7 Vulnerability and Robustness of RC Frames to Resist Progressive Collapse

α(1.2DL+0.5LL

(1.2DL+0.5L

(a) removal of an exterior column

(1.2DL+0.5LL)

α(1.2DL+0.5LL)

(1.2DL+0.5LL)

(b) removal of an interior column

Fig. 7.7 Displacement-controlled pushdown procedure

According to the pushdown curve, two damage criteria (DC) are defined to describe the failure state of the frame. DC-I is defined as the first yield of a reinforced concrete beam, while DC-II is defined as the ultimate load resistance. Figure 7.8 shows a typical pushdown curve with the two damage criteria described above. DC-I marks the first plastic hinge formulation for beams. Beyond the DC-II, further increase in displacement will reduce its load resistance. Fig. 7.8 Typical pushdown curve and the considered damage criteria identified on it

7.2 Study on Vulnerability of RC Frames to Resist Progressive Collapse …

251

7.2.2 Determination of Uncertainty Parameters Progressive collapse is a process involves high geometric and material nonlinear, the presence of uncertainty will lead to randomness of structural performance. Therefore, it is inevitable to consider the effects of uncertainty on progressive collapse performance in vulnerability analysis. Sixteen uncertain parameters are selected as random variables, which can be divided into four categories: (1) gravity load, including dead load DL f and DL r of floor slab and roof and live load L L of floor slab; (2) properties of concrete materials, including compressive strength f c , tensile strength f t and elastic modulus E c ; (3) properties of reinforcement materials, including the yield strength of longitudinal reinforcement f y , the yield strength of transverse reinforcement f yt , the ultimate strength f u and elastic modulus of longitudinal reinforcement E s ; and (4) building geometry, including span length L, beam width Bb , beam height Hb , concrete cover t, longitudinal reinforcement area As,B and As,C in beam, column, respectively. The size of the column section is not included because the column contributes less than the beam performance to resist progressive collapse. Table 7.2 summarizes the statistical properties of the variables considered and the references used to quantify each variable. The total live load of the floor can be divided into two parts: continuous live load and abnormal live load (Ellingwood et al. 1980). Sustained live loads, also known as “live loads at any point in time,” remain relatively constant within a specific occupancy containing furniture and heavy equipment, while abnormal live loads, also known as “intermittent live loads” (JCSS 2001), usually last for a short time as people gather together, for example, in a crowded room during special events. In this study, only continuous live-load variability is considered, as the loss of a column can occur at any time during the lifetime of the structure.

7.2.3 Correlation-Controlled Latin Hypercube Sampling Technique Monte Carlo simulation technique is widespread used in engineering practice, in which probabilistic design parameters are sampled and a number of deterministic computations are performed to provide information about the distribution, or some statistics of response parameters. This is an accurate, simple, and indeed general approach, hence its popularity. However, when the probabilities of failure are small, as they usually are in reliability analysis, such an analysis is extremely time-consuming and expensive in terms of computer resources. Alternative methods should therefore be considered. Latin hypercube sampling (LHS) method for computational planning is first proposed by McKay et al. (2000), the sampling strategy of LHS method is improved so as reliable sampling precision can be obtained based on relatively small sampling scale. The important steps in LHS method include stratified sampling and disrupted sort. In the work of Olsson et al. (2003), it is shown that more than 50% of

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7 Vulnerability and Robustness of RC Frames to Resist Progressive Collapse

Table 7.2 Summary of the considered uncertainty parameters Uncertainty parameters

Mean

Coefficient of variation

Standard deviation

Distributions

Sources

DL f (kN/m2 )

5.25

0.100

0.525

Normal

Ellingwood et al. (1980)

DL r (kN/m2 )

1.05

0.100

0.105

Normal

Ellingwood et al. (1980)

LL (kN/m2 )

3.0

0.470

1.41

Lognormal

Park and Kim (2010)

L 1 /(mm)

6000

0.003

18

Normal

Mirza and Macgregor (1979a, b)

L 2 /(mm)

8000

0.00225

18

Normal

Mirza and Macgregor (1979a, b)

Bb1 /(mm)

323

0.012

3.876

Normal

Mirza and Macgregor (1979a, b)

H b1 /(mm)

557

0.013

7.241

Normal

Mirza and Macgregor (1979a, b)

Bb2 /(mm)

403

0.0099

3.990

Normal

Mirza and Macgregor (1979a, b)

H b2 /(mm)

717

0.0098

7.027

Normal

Mirza and Macgregor (1979a, b)

f y /(MPa)

490

0.093

45.57

Beta

Mirza and Macgregor (1979a, b)

f u /(MPa)

764.1

0.080

61.128

Beta

Mirza and Macgregor (1979a, b)

f yv /(MPa)

337

0.107

36.059

Beta

Mirza and Macgregor (1979a, b)

E s /(GPa)

200

0.033

6.6

Normal

Mirza and Macgregor (1979a, bb)

As, B /(mm2 )

647.7

0.04

25.908

Lognormal

Mirza and Macgregor (1979a, b)

As, c /(mm2 )

799.6

0.04

31.984

Lognormal

Mirza and Macgregor (1979a, b) (continued)

7.2 Study on Vulnerability of RC Frames to Resist Progressive Collapse …

253

Table 7.2 (continued) Uncertainty parameters

Mean

Coefficient of variation

Standard deviation

Distributions

Sources

f c /(MPa)

23.4

0.180

4.212

Normal

Ellingwood et al. (1980)

f t /(MPa)

2.34

0.180

0.412

Normal

Ellingwood et al. (1980)

E c /(GPa)

24.264

0.077

1.868

Normal

Mirza and Macgregor (1979a, b)

t/(mm)

21

0.145

3.045

Normal

Lee and Mosalam (2004)

the computer effort can be saved by using Latin hypercubes instead of simple Monte Carlo in importance sampling. Correlation-controlled Latin hypercube sampling (CLHS) technique is adopted in this study to conduct random pushdown analysis. The CLHS technique consists of sampling and correlation control. Note that the sample is spread over the entire sampling space as the generation of the Latin hypercube sampling plan requires one image from each row and each column. For standard Monte Carlo methods, the expected accuracy of the estimated distribution function determines the number of realizations required. Let N denote the number of realizations required and K the number of random variables. The sampling space is k-dimensional. An N × K matrix P, where each K column is 1, 2, … A random permutation of., N, and establish N × K matrix R of independent random numbers from uniform (0,1) distribution. These matrices constitute the basic sampling scheme, denoted by the matrix S as S=

1 ( P − R) N

(7.1)

Each element of S, sij , is then mapped according to its target marginal distribution as   x ij = Fxj−1 sij



(7.2)

where x ij denotes obtained sample; Fxj−1 represents the inverse of the target cumulative distribution function for variable j. x ij indicates jth variable of ith set of samples. Even if the marginal distribution of each variable is efficiently represented, it is possible that some spurious correlations may emerge. To solve this problem. The element of P, pi j , divided by the number of realizations plus one, is mapped to a Gaussian with mean zero and standard deviation one: 



254

7 Vulnerability and Robustness of RC Frames to Resist Progressive Collapse −1 yij = φ(0,1)



pij N +1

 (7.3)

Then the covariance matrix of Y is estimated and Cholesky decomposed as T

L L = C O V (Y )

(7.4)

where L is lower triangular. A new matrix Y* with a sample covariance equal to the identity is computed as  −1 T Y∗ = Y L

(7.5)

The rank of the column entries of Y* becomes the rank of the column entries of P*. If the elements of P in Eq. 7.1 are replaced by the elements of this matrix, the sampling matrix S will contain considerably less undesired correlation. However, it should be noted that the Cholesky decomposition of Eq. 7.4 requires Cov (Y) to be positive definite, which in turn requires the number of realizations to be higher than the number of random variables, that is, N > K.  −1 T L Y∗ = Y L

(7.6)

Considering progressive collapse is a low probability event, CLHS technique is adopted in this study to conduct random pushdown analysis. Olsson and Sandberg (2002) demonstrated that the CLHS with 100 samples could provide better estimation than the standard LHS with 1,000 samples. Therefore, the CLHS with Ntotal =100 samples is used in this study. By varying the structural parameters with the random samples, a population of Ntotal possible instances of the structure is created.

7.2.4 Methodology and Procedure of Random Pushdown Analysis Set X as random vector of the uncertainty parameters. While the capacity function is set to be C = g(X) = g(X 1 , X 2 , . . . , X n ). C is a random variable and C = g(X) is a function of the variable. The probability distribution of C = g(X) is the fragility function of progressive collapse. The determination of details of the capacity function requires extensive case studies for obtaining characteristic value of the probability distribution. For fragility analysis on earthquake, the ratio of collapse buildings Ncollapse and total studied buildings Ntotal is defined as collapse probability. This conception is introduced in this chapter through random pushdown analysis. Therefore, the empirical progressive collapse probability can be expressed as

7.2 Study on Vulnerability of RC Frames to Resist Progressive Collapse …

P[C|α = x ] =

Ni,collapse Ntotal

255

(7.7)

where Ni,collapse is item associated with αi ; Ntotal is number of total random samples. To simplify the analysis procedure, idealized normal distribution is employed to fit the empirical probability. Indicates the progressive collapse probability at α = x. Thus, the progressive collapse fragility function can be given by 

x − μc F(x)=P[C|α = x ] =  σc

 (7.8)

where and are mean and standard deviation (SD or σ) of the function, respectively. Random push-down curves were obtained by push-down analysis for each structural sample. The corresponding load factors of DC-I and DC-II are determined respectively on these curves. Random pushdown curves and determined load factors eventually form data pools for statistics. By summarizing the random push-down curve, the variation law of α-capacity with the increase of  is calculated, which provides a basis for further understanding the variation law of the whole push-down process. The α-capacitance samples corresponding to different DC are fitted appropriately to describe their probabilistic characteristics. Assuming a sufficient number of structural samples, the probabilistic performance of structures with columns removed can be reliably estimated. The purpose of sensitivity analysis is to study the influence of uncertain parameters on the performance of damaged structures. In order to achieve this goal, a simple method based on probability theory is adopted. In this method, random variables are represented in three levels: upper and lower bounds corresponding to the base value and the mean, which correspond to 1 standard deviation above and below the mean, respectively. Pushdown analysis is performed on structural samples, changing only the lower and upper bounds of each parameter, while the other variables remain unchanged at their base values. The sensitivity of this parameter is measured by the absolute difference between the corresponding responses at its upper and lower bounds. A special bar chart (Kim et al. 2011a, b) widely known as the “tornado chart” is used to illustrate the importance of each parameter. The importance of this parameter is determined by the width of the rod. For each bar, the end points of the bar are calculated based on the upper and lower bounds of the parameters. Thus, a total of structural samples for sensitivity analysis were generated 2 × N var + 1, where 2 × N var are the lower and upper bounds of the parameters, and an additional sample 1 was performed at their average. Figure 7.9 illustrates the methodology and main steps of this study.

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7 Vulnerability and Robustness of RC Frames to Resist Progressive Collapse

Fig. 7.9 Analysis steps of uncertainty and sensitivity evaluation under pushdown analysis

7.2.5 Probabilistic Assessment Using Random Pushdown Analysis According to the pushdown curve, two damage criteria (DC) are defined to describe the failure state of the frame. DC-I is defined as the first yield of a RC beam, while DCII is defined as the ultimate load resistance. Figure 7.10 shows a typical pushdown curve with the two damage criteria described above. DC-I marks the first plastic hinge formulation for beams. Beyond the DC-II, further increase in displacement will reduce its load resistance. For pushdown analysis, a uniform target displacement,  = 1,800 mm, is applied to joint other than columns for the stochastic structural model. As shown in Fig. 7.10, due to the variability of the structural models, some of the structural models lose their load resistance before reaching the target displacement, while others are retained. In addition, a significant change in α-capacity under  is evident. These observations reveal a significant effect of structural uncertainty on

7.2 Study on Vulnerability of RC Frames to Resist Progressive Collapse …

257

Fig. 7.10 Random pushdown curves

load resistance. Therefore, in the study of progressive collapse, it is very important to test the randomness of the pushdown curves and the idea of introducing probability. To obtain distribution characteristic of failure probabilities. The α-capacities for DC-I and DC-II are determined from each of the random pushdown curves, as shown in Fig. 7.11. The corresponding empirical failure probabilities are then calculated and idealized by a continuous normal distribution, as shown in Fig. 7.12. It is apparent that the idealized normal distributions fit the empirical probabilities well. Thus, it is viable to using a normal distribution to depict the probabilistic characteristics of the progressive collapse load-resisting capacity of the structure subjected to the loss of a column. Figure 7.13 compares fragility analysis results at criterion DC-II of frames subjected to interior and exterior column removal. It can be found that, for a given α, the progressive collapse probability of frame subjected to removal of exterior column is greater than that subjected to removal of interior column. This is because only Vierendeel action can be developed to resist progressive collapse when exterior column removal is considered. In comparison, both compressive arch action and catenary action can be developed to resist progressive collapse.

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7 Vulnerability and Robustness of RC Frames to Resist Progressive Collapse

Fig. 7.11 Random pushdown curves conditioned on different scenarios of column removal together with the identified DC

As indicated in Table 7.3, the mean and median are quite close to each other. The maximum SD of 0.35 shows some extent of discreteness. The mean of DC-II is 1.5– 1.6 times greater than that of DC-I, showing greater security reserve for frames to resist progressive collapse relying on DC-II. The maximum SD of DC-I and DC-II is 0.24 and 0.35 respectively, indicating structural resistance is significantly affected by the uncertainty parameters. Thus, the effects of uncertainty on progressive collapse resistance of RC frame cannot be neglected. Variable coefficient (VC), which is the ratio of SD to mean, can be used for estimating discreteness of the load factor α. As listed in Table 7.4, the maximum VC reached 15%.

7.2 Study on Vulnerability of RC Frames to Resist Progressive Collapse …

(a) TF-I

(b) TF-E

(c) LF-I

(d) LF-E

259

Fig. 7.12 Empirical probability function of DS and the fitted normal distributions

Fig. 7.13 Comparison of the fragility curves of DC-II under interior and external columns removal scenario

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7 Vulnerability and Robustness of RC Frames to Resist Progressive Collapse

Table 7.3 Characteristic value of DC-I and DC-II under different scenarios Scenarios

DC-I

DC-II

Median

Mean

SD

Median

Mean

SD

TF-I

1.34

1.30

0.19

2.19

2.12

0.28

TF-E

1.40

1.36

0.20

2.12

2.06

0.29

LF-I

1.71

1.66

0.24

2.44

2.50

0.34

LF-E

1.55

1.58

0.24

2.40

2.42

0.35

Table 7.4 Variation coefficient Scenarios

DC-I

DC-II

Mean

SD

VC (%)

Mean

SD

VC (%)

TF-I

1.30

0.16

12

2.12

0.28

13

TF-E

1.36

0.20

15

2.06

0.29

14

LF-I

1.66

0.24

14

2.50

0.34

14

LF-E

1.58

0.24

15

2.42

0.35

14

7.2.6 Confidence Intervals of Progressive Collapse Fragility of RC Frame Figures 7.14 and 7.15 show three different fragility curves constructed for yielding (DC-I) and collapse (DC-II), respectively. As shown in Fig. 7.15, for a given confidence level of 95%, the load factor α with respect to yielding probability of 5% for random examples of TF-I, TF-E, LF-I, and LF-E are 0.936, 0.99, 1.282, and 1.146, respectively. Similarly, as shown in Fig. 7.15, for a given confidence level of 95%, the load factor α with respect to collapse probability of 5% for random examples of TF-I, TF-E, LF-I, and LF-E are 0.936, 0.99, 1.282, and 1.146, respectively.

7.2.7 Effects of Location of the Removed Column TF-series RC frames are selected to study the effect of position of the removed column on fragility of progressive collapse. As shown in Figs. 7.16 and 7.17, the first story, fourth store, and seventh story are selected as a representative of story in lower part, middle part, and upper part of the frame, respectively. The material, geometric, and load properties of other stories are the same as the first story to eliminate the effect of other uncertainty parameters except position of the removed column. A total of 100 pushdown analyses are conducted, and then the obtained empirical failure probabilities are calculated and idealized by a continuous normal

7.2 Study on Vulnerability of RC Frames to Resist Progressive Collapse …

Fig. 7.14 α limit for DC-I level

Fig. 7.15 α limit for DC-II level

261

262

7 Vulnerability and Robustness of RC Frames to Resist Progressive Collapse

distribution. In this chapter, the numeral before the name of the frame indicates position of the floor. As shown in Fig. 7.18, it is further demonstrated that progressive collapse probability can be described by a normal distribution. It is apparent that the greater the load factor α, the greater the progressive collapse risk. Based on 100 pushdown analyses, the DC-I and DC-II at each α capacity are obtained. Table 7.5 lists the median, mean, and SD of α. It is found that the security reserve is higher when removal of a column in seventh story is considered, but SD is also greater as the beams that contribute to load redistribution are fewer.

Fig. 7.16 Schematic view of removal of an internal column at floors 4 and 7

Fig. 7.17 Schematic view of removal of an exterior column at floors 4 and 7

7.2 Study on Vulnerability of RC Frames to Resist Progressive Collapse …

263

Fig. 7.18 Empirical probability function of DS and the fitted normal distributions of the frame under column missing in different stories

Table 7.5 Moments from fitting the empirical probabilistic distributions of DC-I and DC-II Scenarios

DC-I

DC-II

Median

Mean

SD

Median

Mean

SD

TF-I

1.34

1.30

0.19

2.19

2.12

0.28

4-TF-I

1.35

1.34

0.19

2.36

2.47

0.37

7-TF-I

1.50

1.48

0.19

2.70

2.81

0.42

TF-E

1.40

1.36

0.20

2.12

2.06

0.29

4-TF-E

1.38

1.40

0.20

2.27

2.36

0.39

7-TF-E

1.55

1.60

0.27

2.40

2.42

0.35

As shown in Figs. 7.19 and 7.20, position of the removed column has great impact on progressive collapse resistance of the frame. Compared with removal of a column in fourth and seventh story, the progressive collapse probability due to removal of a column in the first story is much higher at the same load factor α. As shown in Figs. 7.19a and 7.20a, for the first story and fourth story, the probabilities of DC-I are similar regardless of location of the removed column. As shown in Figs. 7.19b

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7 Vulnerability and Robustness of RC Frames to Resist Progressive Collapse

I

II

Fig. 7.19 Fragility curves for models with different stories under interior column removal

I

II

Fig. 7.20 Fragility curves for models with different stories under exterior column removal

and 7.20b, the lower the story of the removed column, the higher the progressive collapse probability.

7.2.8 Effect of Story Number on Vulnerability of Progressive Collapse To investigate the effect of number of story on progressive collapse resistance of RC frame, three models are built based on TF-series frames, the material, geometric, and load properties of which are the same, while their story number ranges from 4 to 12. For the designation in this chapter, the numerals of 4, 8, and 12 before the hyphen indicate story number while the letter behind the hyphen indicates position of the removed column. The studied scenarios are shown in Figs. 7.21 and 7.22.

7.2 Study on Vulnerability of RC Frames to Resist Progressive Collapse …

(a) 4-I

(b) 8-I

265

(c) 12-I

Fig. 7.21 Schematic view of removal of an interior column of model with different stories

(a) 4-E

(b) 8-E

(c) 12-E

Fig. 7.22 Schematic view of removal of an exterior column of model with different stories

As shown in Figs. 7.23 and 7.24, 100 pushdown analyses are conducted for each model, and then the obtained empirical failure probabilities are fitted by a normal distribution. The analysis results are shown in Fig. 7.25 and Table 7.6. As shown in Fig. 7.25, for the same load factor α, the failure probability of frame under a ground column removal scenario increases with the increase of story number. Theoretically, increasing story number can increase load transfer paths of the frame, however, the results shown in Fig. 7.25 indicate that the failure probability of frame increases with the increase of story number. It is worthwhile noting that,

266

7 Vulnerability and Robustness of RC Frames to Resist Progressive Collapse

(a) interior column removal

(b) exterior column removal

Fig. 7.23 Empirical probability function of DS-II and the fitted normal distributions for 4-story model

(a) interior column removal

(b) exterior column removal

Fig. 7.24 Empirical probability function of DS-II and the fitted normal distributions for 8-story model

(a) interior column removal

(b) exterior column removal

Fig. 7.25 Comparison of fragility curves at DC-II level of models with different stories

7.2 Study on Vulnerability of RC Frames to Resist Progressive Collapse … Table 7.6 Characteristic value of DC-II of models with different stories

267

Scenarios

DC-II Median

Mean

SD

4-I

2.43

2.49

0.34

8-I

2.19

2.12

0.28

12-I

1.92

1.98

0.24

4-E

2.36

2.43

0.37

8-E

2.12

2.06

0.29

12-E

1.89

1.85

0.24

the properties of the three studied models are the same except story number, indicating that the cross section and reinforcement ratio of column remain unchanged. Thus, compared with the original model (8-story), the 3-story model may result in a conservative result, while the result of the 12-story model is the opposite.

7.2.9 Empirical Fragility Curves It is found that the fragility curve parameters (i.e., mean and SD) change with respect to the story number of the models. Thus, to extend the fragility curves of 4-, 8- and 12-story buildings to the fragility curves for 5-, 6-, 7-, 10-, and 11-story buildings, regression analyses are performed. Based on previous works (Shinozuka et al. 2000; Kirçil and Polat 2006), the empirical models that are used to obtain the relationship between the fragility curve parameters and the story number are given below: μ = an + b

(7.9)

σ = cn 2 + dn + e

(7.10)

where n is the story number of the model, and a, b, c, and d are coefficients obtained from regression analyses. Figures 7.26 and 7.27 shows the regression results of mean and SD for DC-II with respect to an interior and exterior column removal scenario, respectively. Table 7.7 shows the regression coefficients obtained for DC-II capacities in terms of story number. It is worth noting that R2 values are very high because the number of data is the minimum number of data required for a regression analysis. Table 7.8 lists the mean and SD of α for the models with story range from 4 to 12. Using the regression results, empirical fragility curves are shown in Figs. 7.28 and 7.29. As shown in Figs. 7.28 and 7.29, the progressive collapse fragility of frame is significantly influenced by the story number. It is found that the probability of collapse

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7 Vulnerability and Robustness of RC Frames to Resist Progressive Collapse

Fig. 7.26 Regression analysis results for interior column removal scenario

Fig. 7.27 Regression analysis results for exterior column removal scenario Table 7.7 Regression coefficients Scenarios

μ = an + b

σ = cn 2 + dn + e

a

b

R2

c

d

e

R2

Interior column removal

−0.638

2.7067

0.9365

0.0017

−0.0406

0.489

1

Exterior column removal

−0.074

2.7000

0.9789

0.0009

−0.0305

0.478

1

Table 7.8 Characteristic value of different stories Scenarios

Index

Story number 4

5

6

7

8

9

10

11

12

Interior column removal

μ

2.49

2.39

2.32

2.26

2.12

2.13

2.07

2.00

1.98

σ

0.35

0.33

0.31

0.29

0.28

0.26

0.25

0.24

0.22

Exterior column removal

μ

2.43

2.33

2.26

2.19

2.06

2.04

1.96

1.89

1.85

σ

0.37

0.35

0.33

0.31

0.29

0.28

0.26

0.25

0.24

7.2 Study on Vulnerability of RC Frames to Resist Progressive Collapse …

269

Fig. 7.28 Fragility curves for DC-II with respect to load factor α under interior column removal

Fig. 7.29 Fragility curves for DC-II with respect to load factor α under exterior column removal

for the same load factor α increase with the increase of story member. Moreover, this phenomenon becomes obvious when the story number exceeds 7. For the regression analysis, linear equations are assumed to be sufficient for representing the relationship between the story number and the maximum load factor α. Figure 7.30 shows the regression analysis results and confidence intervals for a confidence level of 95%. The high value of R2 shows a well agreement.

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7 Vulnerability and Robustness of RC Frames to Resist Progressive Collapse

Fig. 7.30 Relationship between load factor α and story number of building

The following equations were obtained based on above regression analyses, which produce the load factor α for collapse prevention performance levels with respect to interior and exterior column removal scenario, respectively: α = −0.0637n + 2.7085

(7.11)

α = −0.0729n + 2.6959

(7.12)

The obtained equations can be used to estimate the maximum allowable load factor α for collapse prevention performance level, which can be calculated using the curve of a confidence level of 95% as a lower bound.

7.2.10 Sensitivity Analysis on Uncertainty Parameters Sensitivity analysis is often used to study the effect of uncertainty parameter with significant influence on structural response, with purpose to reduce calculation resources and economy cost. Sensitivity analysis is widely used in seismic study to estimate the influence of important uncertainty parameters on structures to resist earthquake. In the current study, Sensitivity analysis is used to study the influence of important uncertainty parameters on progressive collapse behavior of structures.

7.2.10.1

Sensitivity Analysis on Uncertainty Parameters Using Random Pushdown Method

Pushdown sensitivity analysis is performed by testing each uncertain parameter independently and at the mean or next standard deviation above it. The sensitivity was then

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assessed against the pushdown curve shown in Fig. 7.31. Curves with average parameters (base case curves) always appear as solid lines, while curves corresponding to parameter upgrades and downgrades are distributed on either side. In short, only the results of case TF-I are presented. Similar observations were made for the remaining cases. The figure shows that DL f , L L, f y , f u , Hb , and As,B have a relatively large influence, while other parameters have a very limited influence because the three curves are almost identical. As expected, the gravitational load has a significant

Fig. 7.31 Sensitivity of the pushdown curves to the considered uncertain parameters

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Fig. 7.31 (continued)

7.2 Study on Vulnerability of RC Frames to Resist Progressive Collapse …

273

Fig. 7.31 (continued)

effect on the pushdown results. The impact of roof dead load DL r is negligible as it is only a very limited percentage of the total dead load. By contrast, only considering the change of the continuous load component has little influence on the floor live load L L. What is significant at first is predictable. However, at the peak stage of α- capacity, its effect decreases with the increase of displacement, and the catenary becomes the main load resisting mechanism, which is controlled by the ultimate strength of reinforcement f u . The results further confirm the correctness of the prediction. In addition, the beam height Hb and reinforcement area As,B also have a great impact. In addition to the above parameters, other parameters have obvious influence on the ultimate deformation capacity of the frame. For example, increasing the stirrup-related parameters f yt to one standard deviation above the mean can significantly increase the final deformation of the structure, which is consistent with the results of Bao et al. (2008) and Qian and Li (2013). In their study, the increase of the transverse reinforcement ratio of the beam will greatly improve the ultimate deformation capacity of the RC frame to resist the progressive collapse.

7.2.10.2

DC-Based Sensitivity Analysis on Uncertainty Parameters

Tornado maps are used for sensitivity analysis in this chapter. In a tornado diagram, the importance of this parameter is determined by the width of the bar (swing). For each bar, the end points of the bar are calculated based on the upper and lower bounds of the parameters. The sensitivity of this parameter is determined by the absolute difference of the corresponding response at the upper and lower bounds. Porter et al. (2002) examines the question of which sources of uncertainty most strongly affect the repair cost of a building in a future earthquake. Sensitivity of the progressive collapse mechanism to uncertain design parameters of steel buildings was investigated using (1) Monte Carlo simulation, (2) Tornado Diagram Analysis, and (3) the First-Order Second Moment method by Kim et al. (2011a, b), it is found that the tornado diagram analysis has reasonable precision. In recent years, the tornado diagram analysis is widely used in sensitivity study.

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Fig. 7.32 Tornado diagram analysis procedure (Kim et al. 2011a, b)

The schematic view of process to perform tornado diagram analysis is depicted in Fig. 7.32. In TDA, the upper and lower bounds as well as reference value of a random variable are selected and the corresponding structural responses are obtained. The difference between such structural responses, referred to as swing, is considered as a measure of sensitivity. In this study, using the 16th and 84th percentiles of the Gaussian distribution as the lower and upper bounds. In one analysis, only one random parameter is changed while mean is input for the other random parameters. In the current study, a total of 2n+1 analyses (n is the number of random parameters) were performed, of which 2n were performed at the lower and upper bounds of the parameters, while an additional 1 was performed at their average. As a deterministic function of one or more uncertain inputs, the output variables are studied by a series of deterministic tests. First, set each input variable to its best estimate, and then measure the output. This establishes a baseline output. One input is then set to an extreme value (low or high), and the output is measured again. The input is then set to another extreme value and the output is measured. The absolute value of the output difference between the two cases is a measure of the sensitivity of the output to the input variable. This difference is called the swing. The first input is then set to the best estimate and the process is repeated for the next input to determine the swing associated with the variability of that input. You can then sort the input variables according to swing. Greater volatility reflects more important input uncertainty. The sensitivity of each uncertain parameter on specific DC is investigated by the tornado diagrams, as shown in Figs. 7.33 and 7.34. Due to inherent variability, the effects of each random parameter on progressive collapse behavior of RC frames are different. It can be seen from the figure that the parameters of DL f , L L, and f y are of great importance to DC-I, while DL f and f u have the greatest influence on DC-II.

7.2 Study on Vulnerability of RC Frames to Resist Progressive Collapse …

(a) TF-I

(b) TF-E

(c) LF-I

(d) LF-E

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Fig. 7.33 Tornado diagrams for DC-I

In addition, the impact of As,B on both DC is also considerable. It is expected that DL f is the most influential parameter because it is closely related to α-capacity. Because DL r and L L account for only a small fraction of the total gravity load, they show less significance than DL f . Since DC-I is defined by the first yield of the steel bar in the beam, it has a significant effect. The influence of f u on DC-II is high because it contributes greatly to the peak load resistance of DC-II. Since f c is the key factor of the compressive arch action, the main load resisting mechanism corresponding to DC-I, f c has a considerable influence on the DC-I. However, when it comes to DC-II, the swing is small because the catenary plays a major role as the load resisting mechanism and the concrete plays a negligible role. Increasing Hb can significantly increase DC-I. The influence of elasticity modulus of the materials is marginal. Taking mean of each parameter as baseline, it is seen that the more obvious the asymmetry, the greater the variability of the parameter.

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(a) TF-I

(b) TF-E

(c) LF-I

( d ) L F -E

Fig. 7.34 Tornado diagrams for DC-II

7.2.11 Conclusions In this chapter, study on vulnerability and robustness of RC frames to resist progressive collapse considering uncertainty is conducted. Sixteen uncertain parameters are selected as random variables. Based on Correlation-controlled Latin hypercube sampling technique and validated finite element model, a series of case studies are conducted to obtain fragility curves of each uncertainty parameter. Based on the works in this chapter, the following general conclusions can be drawn: 1. A normal distribution was proven suitable to describe the probabilistic characteristics of progressive behavior of structure subjected to a column removal scenario. For the same load factor α, the progressive collapse probability of a building subjected to exterior column removal is higher than the one subjected to interior column removal. 2. For the same load factor α and column removal scenario, the lower the story of the removed column, the higher the progressive collapse probability. 3. The progressive collapse probability increases with the increase of story number. The empirical functions of load factor α and story number for collapse prevention

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performance levels with respect to interior and exterior column removal scenario are α = −0.0637n + 2.7085 and α = −0.0729n + 2.6959, respectively. 4. The dispersion of random pushdown curves could go beyond 0.15. The considerable variability reveals the significant effects of uncertainty on the ability of RC frame structures against progressive collapse. The parameters of dead load in floor (DL f ) and live load (LL), yield and ultimate strengths of rebar (f y and f u ), compressive strength of concrete (f c ), and reinforcement area of beams (As, B ) show apparent effects on the residual capacity of structures in resisting progressive collapse, while the influence of elasticity modulus of the materials is marginal.

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