Steel Plate Shear Walls with Gravity Load: Theory and Design 9811686939, 9789811686931

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Yang Lv

Steel Plate Shear Walls with Gravity Load: Theory and Design

Steel Plate Shear Walls with Gravity Load: Theory and Design

Yang Lv

Steel Plate Shear Walls with Gravity Load: Theory and Design

Yang Lv School of Civil Engineering Tianjin Chengjian University Xiqing, Tianjin, China

ISBN 978-981-16-8693-1 ISBN 978-981-16-8694-8 (eBook) https://doi.org/10.1007/978-981-16-8694-8 Jointly published with Science Press The print edition is not for sale in China (Mainland). Customers from China (Mainland) please order the print book from: Science Press. © Science Press 2022 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, express 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

Because of its excellent shear strength and stiffness, a properly designed Steel Plate Shear Wall (SPSW) will have considerable energy dissipation capacity, ductility, initial stiffness and ultimate strength. Furthermore, the steel walls are efficient in terms of cost and space due to their light weight, ease of construction and small footprint, and it has been extensively studied and used in a significant number of buildings in the past several decades. Previous SPSWs used stiffened or thick walls to prevent local buckling. In 1983, Thorburn et al. found out that unstiffened steel wall has a high ductility and strength even after buckling, and they proposed a strip model for estimating the shear strength of the walls. Their model has a high impact on the study of the SPSWs. In the strip model, the infill plate is divided into several strips to represent the tension field that develops as a result of applied horizontal load. The strips are cumulatively equal in area to the infill plate, inclined at an angle α and pin connected to the boundary members. The angle of inclination is determined based on the theory of least energy. The shear capacity of the infill plate is determined by the horizontal component of the yield capacity of the tension strips. Thereafter, the unstiffened SPSWs were examined by many researchers through large-scale tests, tests on construction details and analysis of design procedures. However, a thin rectangular steel wall in a SPSW structure always simultaneously sustains the lateral load and the gravity load. The gravity load can affect the shear strength of a SPSW. This effect is not considered in most of the research and standards, which will overestimate the shear capacity and may lead to potential danger in practice. This book gives an extensive study of the seismic design method of SPSWs by considering the effects of the gravity load. The structure of the book is as follows. Chapter 1 introduces the background of SPSWs and gives a brief literature review to the subject. Chapter 2 is devoted to studying the shear strength of SPSWs by considering the gravity load by the effective-width model. The vertical stress distribution of the wall under the gravity load was assumed to have a form of the effective-width model. The stress of the wall under gravity, gravity and shear were presented. A new inclination angle of the tension strip model by considering

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the effect of the gravity load was proposed. Finally, the shear strength and the sheardeformation relationship of SPSWs by considering the effect of the gravity load were proposed. In Chap. 3, a new vertical stress distribution (three-segment distribution) of the wall under uniform compression and in-plane bending was proposed. Then the influence of the gravity load on the shear strength of the SPSWs is considered by the three-segment distribution. The stress throughout the inclined tensile strip, considering the effect of different vertical stress distributions, is determined using the von Mises yield criterion. The shear strength is calculated by integrating the shear stress along the width. Chapter 4 gives the shear strength calculation method of the cross stiffened SPSWs by considering the effects of the gravity load. The stiffened SPSW is divided into several sub-plates by vertical stiffeners. For each sub-plate, a three-segment vertical stress distribution under gravity load is proposed. Then the shear strength of the wall is calculated as the sum of the shear strength of each subwall. Chapter 5 gives the shear strength of the corrugated SPSWs by considering the effects of the gravity load. Four scaled one-storey single-bay steel plate shear wall specimens with 2 flat panels and 2 corrugated panels were tested to examine their behavior under cyclic loadings. A finite element model adopting layered shell element model was built and verified by the test results. Then a parametric study was conducted to examine the influences of the gravity load on the shear capacity of the corrugated steel shear walls. Chapter 6 describes the main conclusions obtained in this book and corresponding limitations of the employed approaches, upon which future work in the field of SPSWs is summarized. The author gratefully acknowledge the partial support of this research by the National Natural Science Foundation of China (52178295 and 51508373), National Key Research and Development Program of China (2016YFC0701100), the Special Plan Young Top-notch Talent of Tianjin and Natural Science Foundation of Tianjin (16JCZDJC38900 and 17JCZDJC10010). In particular, special thanks go to my supervisor, Dr. Zhongxian Li, Chair Professor at Tianjin University. My sincere thanks go to him for the continuous support and instructions all the way and all the time. Tianjin, China

Yang Lv

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Recent Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Analytical Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Experimental Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Standards and Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Objectives and Organization of This Book . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Steel Plate Shear Walls with Considering the Gravity Load by the Effective-Width Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Gravity Load Effect on the Shear Strength of the Steel Plate Shear Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Numerical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Verifications Under Monotonic Tests . . . . . . . . . . . . . . . . . . . . 2.2.3 Verifications Under Cyclic Tests . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Pushover Analysis Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5 Cyclic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.6 Shear Strength Obtained from Existing Model . . . . . . . . . . . . 2.3 Vertical Stress Distribution Under Compression . . . . . . . . . . . . . . . . . 2.3.1 The Effective-Width Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Evaluation of the Effective-Width Model . . . . . . . . . . . . . . . . 2.3.3 Discusses on the Effective-Width Model . . . . . . . . . . . . . . . . . 2.4 Stresses Under Compression and Shear . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Stresses Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Discusses on Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Shear Capacity of the Infill Steel Plate . . . . . . . . . . . . . . . . . . 2.4.4 Load-Carrying Capacity of the Steel Plate Shear Wall . . . . . 2.5 Inclination Angle of the Steel Wall with Considering the Gravity Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 2 5 7 9 10 15 15 17 17 19 20 24 31 38 40 40 43 46 47 47 50 51 53 53

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2.5.1 2.5.2 2.5.3 2.5.4

Assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inclination Angles of the Infill Steel Plate . . . . . . . . . . . . . . . Discuss of the Inclination Angle . . . . . . . . . . . . . . . . . . . . . . . . Load-Carrying Capacity Considering the Inclination Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Shear-Displacement Relationship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Shear-Displacement Relationship Under Compression and Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Shear-Displacement Relationship Under Compression, Shear and Bending . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Experimental Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Steel Plate Shear Walls with Considering the Gravity Load by a Three-Segment Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Three-Segment Distribution Under Uniform Compression . . . . 3.2.1 The Cosine Distribution Under Uniform Compression . . . . . 3.2.2 The Proposed Stress Distribution Under Uniform Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Three-Segment Distribution Under Compression and In-plane Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 The Cosine Stress Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 The Proposed Three Segment Stress Distribution . . . . . . . . . 3.3.3 Finite Element (FE) Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Shear Strength Considering the Gravity Load . . . . . . . . . . . . . . . . . . . 3.4.1 Stresses Under Compression and Shear . . . . . . . . . . . . . . . . . . 3.4.2 Shear Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Discussion of the Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Parametric Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.5 Shear Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.6 Aspect Ratio and Slenderness Ratio . . . . . . . . . . . . . . . . . . . . . 3.5 Shear Strength Considering the Gravity Load and In-plane Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Parametric Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Shear Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Stress Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53 55 57 58 59 59 63 67 82 83 85 85 87 87 89 92 106 106 109 113 120 120 122 124 125 126 129 130 131 138 138 141 142

Contents

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4 Cross Stiffened Steel Plate Shear Walls Considering the Gravity Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Shear Strength Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Stress Under Gravity Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Stress Under Compression and Shear . . . . . . . . . . . . . . . . . . . 4.2.3 Shear Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Discussions of the Proposed Model . . . . . . . . . . . . . . . . . . . . . 4.3 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Finite Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Parametric Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Evaluation of the Vertical Stress Distribution . . . . . . . . . . . . . 4.3.4 Evaluation of the Shear Strength . . . . . . . . . . . . . . . . . . . . . . . 4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

145 145 147 147 149 151 152 155 155 157 157 158 161 162

5 Corrugated Steel Plate Shear Wall Considering the Gravity Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Experiment Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Hysteretic Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Envelope Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Specimens’ Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Finite Element Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Finite Element Model Verification . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Parametric Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Infill Steel Plate Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.4 Vertical Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

165 165 168 170 170 174 175 178 178 181 181 184 187 188

6 Summary and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 6.1 Main Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

Chapter 1

Introduction

1.1 Background and Motivation Steel plate shear wall, known as SPSW or SSW, has been extensively studied and used in a significant number of buildings in the past several decades. Applications of SPSWs in different countries like Japan, North America, and China have shown excellent lateral load-bearing capacity and high stiffness. The main components of a steel plate shear wall include the infill steel panel, the boundary columns and the boundary beams, sometimes, also horizontal or vertical or diagonal stiffeners for the stiffened steel shear walls. The first developed steel shear wall was prevented to buckle until the yielding stress achieved. Heavily stiffened or relatively thick infill steel panel were used. Later, after the post-buckling capacity from the diagonal tension field of the infill steel panel has been recognized, thin unstiffened steel shear wall became popular. Steel shear wall has high shear resistance and stiffness, stable hysteresis behaviour, high-energy dissipation capability, high ductility and redundancy, that have made the steel shear wall system an alternative to conventional lateral load-resisting systems in high-risk seismic and wind regions. Generally, steel shear walls can be divided into stiffened steel shear wall and unstiffened steel shear wall, in which stiffened steel shear walls were popular in China and Japan. It has high ductility and high-energy dissipation capacity without a need for thick infill steel panels. The disadvantages are using of stiffeners have been criticized due to the labor-intensive installation process and increased construction costs. Unstiffened steel shear walls are popular in North America. Some researchers treated the steel shear wall as a plate girder, i.e. the infill steel panel similar to the web, boundary columns as the flanges and boundary beams as the transverse stiffeners. The main advantages of steel shear wall include, as presented before, high initial stiffness and strength, and small thickness and reduced structural weight compared with reinforced concrete shear walls.

© Science Press 2022 Y. Lv, Steel Plate Shear Walls with Gravity Load: Theory and Design, https://doi.org/10.1007/978-981-16-8694-8_1

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1 Introduction

However, a thin rectangular steel wall in a steel shear wall structure always simultaneously sustains the lateral load and the gravity load. The gravity load can affect the shear strength of a steel shear wall. This effect is not considered in most of the research and standards, which will overestimate the shear capacity and may lead to potential danger in practice. To narrow the knowledge gap, this book gives an extensively study of the seismic design method of SPSWs with considering the effects of the gravity load. A vertical stress distribution of the infill steel plate under the gravity load is proposed. The stress and inclination angle of the inclined tensile strips along the width of the wall, considering the effect of different vertical stress distributions, are determined. The shear strength is calculated by integrating the shear stress along the width. The deformations of the wall at the elastic buckling of the infill steel plate, the yield of high vertical stress portion, the yield of low vertical stress portion, and the yield of the boundary frame were presented.

1.2 Recent Research 1.2.1 Analytical Studies A typical steel plate shear wall consists of horizontal and vertical boundary elements that may or may not carry gravity loads. The thin infill plates buckle in shear and form a diagonal tension filed to resist the later loads. Timler and Kulak [1] proposed an equation to determine the inclination angle of the tension field in 1983 based on an elastic strain energy formulation. This equation can reasonably consider the effects of the thickness of the infill plate, the story height, the bay width, the moment of inertia of the vertical boundary element, the cross-sectional area of the horizontal boundary element. Based on the inclination angle, the most famous analytical model, strip model, was developed by Thorburn et al. in 1983. The strip model assumes that the infill plates can be represented by a series of pin-ended, tension only strips. The strip model was later refined by Timler and Kulak in 1983. The form of the equation has been adopted in the Canadian and U.S. seismic design codes [2–4]. Many studies have reported reasonable match of the test results and the strip model results [5–9]. Elgaaly and Liu [10] used finite-element models to simulate the experimental results of Caccese et al. [6]. The finite element model used nonlinear material models and geometry. Each frame member was represented by 6 beam elements. The infill steel plate of each story was simulated by 6 × 6 shell elements. Moment-resisting beam-to-column connections were assumed. Lateral load was monotonically applied until failure of the wall. It was found that the finite element model using shell elements significantly overestimated the shear capacity and stiffness in comparison with the experimental results. The reason was due to difficulty in modeling initial imperfections in the plates and the inability to model out-of-plane deformations of the frame members. The tests were also simulated using the multi-strip method. Twelve stirps were used to represent the plate of each story. The inclination angle of the strips

1.2 Recent Research

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was calculated to be 42.8°. An elastic perfectly plastic material model was used to describe the behaviour of the strips, the model was found to be available to replicate the tests with respect to initial stiffness, ultimate strength, and displacement at the ultimate strength. Bruneau and Bhagwagar conducted nonlinear inelastic dynamic analyses to investigate how structural behaviour is affected when thin infills of steel, low-yield steel, and other nonmetallic materials are used to seismically retrofit steel frames located in regions of low and high seismicity. Fully rigid and perfectly flexible frame connection were considered to capture the extremes of frame behaviour. Thin steel infill panels were found to reduce story drifts without significant increases in floor accelerations, and low-yield steel was found to lead to slightly better seismic behaviour than the normal constructional steel under extreme seismic conditions. This study also indicated that the infill plate yields progressively across its width, as a function of the stiffness of its surrounding beams and columns. Only when the columns and beams are rigid and beam-to-column connections were pins, the entire infill plate across the width yield simultaneously. Kharrazi et al. [11] proposed a modified plate-frame interaction model for the analysis of shear and bending deformations and resulting forces in SPSW. This model can describe the interaction between frame and infill plate and characterize the respective contributions to the deformations and strength. The shear-displacement relationship is considered by a function with a trilinear form by considering the behaviour of the infill plate up to the points of shear buckling and perfectly plastic yielding of the tension field. The corresponding strength and displacements for both shear buckling and tension field yielding. Obviously, if the shear buckling is not considered, the derived equations for panel strength, stiffness and yield displacement reduce to the model proposed by Thorburn et al. The above-mentioned early strip model involved equally spaced tension-only truss elements, normally with perfectly plastic or bilinear material properties. Later, several modified strip models were proposed, including the use of supplemental degradation elements [12], multi-directional strip model [13], compression strut and deterioration strip [14] or semi-empirical material and hysteretic models to better match with the available test data. Other modification of strip models focused on the improvement of material properties of the strip elements [8, 10, 12, 15–18]. Most of the hysteretic models mentioned herein were developed based on the force–deformation behavior of the entire web panel, except that Webster’s and Choi and Park’s models [15, 18], which were developed based on the stress–strain behavior of a single finite-element of the panel. Zhang and Guo [19] established a reduction coefficient for the shear capacity of the system considering the pre-compression effect from boundary columns. Habashi and Alinia [20] examined the wall frame interaction for the system. They concluded that with practical steel plate shear wall dimensions, if the system is designed as per the AISC Design Guide 20 rules [21], the frame behavior is independent of the infill wall; therefore the shear capacity of the system can be calculated by simply adding frame and the infill wall capacities [20]. Hosseinzadeh and Tehranizadeh [22] reviewed the code-designed SPSW system and found that the boundary frames are effective in

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1 Introduction

resisting story shear only at a few of the lower stories, while a substantial portion of the story shear is taken by the infill plates at the upper stories. They also found that about 70–80% of the compressive axial force in the boundary columns results from plate tension field action [22]. Shi and Astaneh-Asl [23] investigated the design of steel plate shear wall system using different design philosophies. They found that the plate girder design procedures applied to steel plate shear walls can predict the shear strength of the walls reasonably well and can lead to more economical designs. Bruneau and his research associates studied the plastic design method of SPSWs and many other code-based design aspects of the steel plate shear walls, such as the capacity design of the boundary columns and boundary beams [24– 28]. Kharmale and Ghosh [29] proposed a performance-based plastic design method for steel plate shear walls with rigid beam-to-column connections. The strain rate and P-delta effects were considered by Bhowmick et al. [30]. They found that the loading rate increases flexural demand mostly at the base of the steel plate shear wall but imposes limited influences on the inelastic seismic demands for a suite of spectrum compatible earthquake records for Vancouver. They also pointed out the conservative nature of the current National Building Code of Canada stability factor approach to include the P-delta effects for steel plate shear walls, and that P-delta has small effects on seismic demand estimations. Research efforts have considered other design methods, such as the capacity spectrum method [31] and the design of the system based on inelastic drift demand [32]. Baldvins et al. [33] recently proposed a set fragility functions for steel plate shear walls for use in the performance-based design applications. Deng et al. [34] proposed a High Efficiency Analysis Model (HEAM) of the modular steel structure made of the corrugated steel shear wall. The skeleton curve and hysteretic regulation of walls are proposed. The general formula for determining the hysteresis model of corrugated SPSW is deduced. An effective cross bracing model to simulate the seismic performance of corrugated SPSW is proposed. Through the cyclic experimental data of corrugated SPSWs, the accuracy and rationality of the developed HEAM is verified that it can conservatively predict the dynamic response of corrugated SPSW under earthquake action, which helps to improve the analysis efficiency of corrugated SPSW and can be used for composite steel non-linear seismic analysis of the structure. Feng et al. [35] established a theoretical model composed of the orthotropic plates with elastic torsion constraint edges to predict the overall elastic buckling performance of trapezoidal corrugated steel shear walls reinforced by horizontal steel strips. In addition, a finite element model was established to verify the theoretical model, and on this basis, parameter studies were carried out to discuss the influence of design parameters such as rigidity constants, width-to-height ratio, and thickness of the trapezoidal corrugated steel plate on elastic buckling. Dou et al. [36] analyzed the elastic shear buckling performance of the infill plate in the sinusoidal corrugated steel plate shear wall. A correction formula for the bending stiffness of sinusoidal corrugated plates is proposed, and a fitting equation for predicting global and local shear buckling loads is established based on finite element analysis. It is found that for the sinusoidal corrugated-filled plate, only global buckling and local

1.2 Recent Research

5

buckling can be observed in the lowest buckling mode of the characteristic buckling analysis, and the interaction buckling behavior is not significant. Zhao et al. [37] proposed an improved empirical formula for the basic period of corrugated steel plate shear walls. Using the shear-flexure cantilever formulation, a simplified method for calculating the basic period of corrugated SPSW is proposed and verified by finite element analysis. The influence of major geometric properties including building height, wall plate aspect ratio, wall thickness, and corrugation depth on the fundamental period were also investigated. The results show that building height has the greatest impact on the basic cycle of corrugated SPSWs. The aspect ratio of the wall plate and the wall thickness also have a significant impact on the basic cycle, especially for high-rise buildings, and the wall corrugation depth has the least influence. Farzampour et al. [38] introduced a corrugated steel shear wall with reduced boundary beam section (RBS-CSSW), and proposed a step-by-step design procedure based on corrugated plate-frame interaction (CPFI). A new lateral-load-resisting system incorporating the corrugated plates with an RBS connection is proposed and computationally analyzed. Strength equations are developed to evaluate the new Lateral Resistance Systems’ (LRS) strength. Finally, two RBS-CSSWs were designed and analyzed to verify its accuracy through finite element models.

1.2.2 Experimental Studies In early applications of the steel plate shear wall system, relatively thick plates or closely spaced stiffeners were often used to prevent buckling and to utilize the shear yielding capacity of the steel plate. Takahashi et al. [39] published one of the earliest test results on steel plate shear walls, where the steel plate shear wall system was investigated as an alternative to concrete shear walls. Although incorporating stiffeners resulted in higher shear strength, stiffness, ductility, and energydissipation capacity of the system, the addition of the stiffeners has been criticized for being a relatively labor-intensive, time-consuming, and costly fabrication process that resulted in a wider footprint for the wall. In recent years, and particularly in North America, much of the research and development activities have focused on the post-buckling behavior of the unstiffened thin steel infill walls. Many benchmark research projects summarized below have shown that unstiffened steel plate shear walls can provide relatively large displacement ductility, stable hysteresis behavior, and desirable energy dissipation capability. Starting in the early 1980s, the post-buckling strength of steel plate shear walls was investigated at the University of Alberta, where Thorburn et al. [40], Timler and Kulak [1] and Tromposch and Kulak [41] tested several single and multi-story specimens under quasi-static cyclic load, and proposed the use of a strip model to compute the post-buckling shear strength of the steel plate shear walls. The initial strip model proposed by Thorburn et al. [40] was found to be capable of predicting the overall force–displacement response well but tended to overestimate the elastic stiffness. Based on test results, Timler and Kulak [1] modified the tension field

6

1 Introduction

action equation proposed by Lubell et al. [42] for multi-story systems and included the effect of flexural stiffness of the columns. Tromposch and Kulak [41] tested a similar specimen as the one tested by Timler and Kulak but with some modifications based on findings of the previous study. Lubell and his colleague [9, 42] tested two single panel specimens and one fourstory specimen under fully reversed cyclic quasi-static loading. The importance of sufficient boundary elements was clearly demonstrated by comparing the deformation of columns and the roof beam of the two single panel specimens. They found that the infill plates significantly reduced the rotational demand on the beam-to-column connections by providing a redundant lateral force resisting mechanism. The simplified strip model was again reported to be adequate in predicting post-yield strength, but not the elastic stiffness. Driver [7] tested a large-scale four-story specimen including gravity effects under cyclic loading. It was reported that the structure was still able to hold 85% of the ultimate strength even when local buckling of column flange below the first story and several tears in the first-story plate and fracture of the complete penetration welds at the base plate had occurred [7, 43]. This observation is a good example that shows the redundancy of the steel plate shear wall system. They also found that most of the energy was dissipated by the plates with limited yielding in the moment beam-to-column connections, and that the first-story plate absorbed most of the damage. The strip model was found to be adequate for predicting ultimate strength but underestimated the initial stiffness. Schumacher et al. [44] studied four infill plate-to-boundary element connection details. The load–displacement responses of all four specimens had similar force– displacement and energy-dissipation behavior regardless of the plate-to-boundary detailing. Tears were concentrated in the corner region of specimens B, C, and Modified B, but the deterioration of connection capacity was gradual and stable under increased load. Significant tearing was delayed in specimen Modified B compared to specimen B. Specimen A was found to be the least susceptible to tearing; however, it was regarded as impractical for actual construction considering the level of fabrication precision required. Zhao and Astaneh-Asl studied and cyclically tested two half-scale specimens of the system and established that the system has relatively high strength, stiffness, ductility, and energy dissipation-capacities, reaching larger than 0.03 rad inter-story drift and up to 15 inelastic cycles. Both specimens with different splice locations were shown to perform satisfactorily [45, 46]. Astaneh-Asl also developed seismic design procedures for this and traditional steel plate shear walls [47, 48]. Through the test of five single-bay three-story specimens, the system behavior and ductility behavior were classified to be shear-dominate for steel shear wall system with thin infill plate and flexure-dominate for thick infill plate by Choi and Park [49– 51]. By testing another set of SPSW specimens with ductile details, they reported higher displacement ductility and energy-dissipation capability of the shear dominated steel shear wall specimens compared to concentrically braced frames (CBF) and moment resisting frames (MRF). It was found that the shear strength and energy dissipation capacity of the steel plate walls increased in proportion to the width of

1.2 Recent Research

7

the infill steel plate, the overall displacement ductility increased with the ratio of the flexural capacity to the shear capacity, and the maximum inter-story drift ratio increased with the compactness of the column section. Different infill-to-boundary connections details were also studied and advantages and disadvantages of each alternative were discussed. The behavior of steel plate shear wall with reduced beam section connections and composite floors was studied by Qu et al. [52] through a two-phase experimental program on a full-scale two-story specimen. The buckled panels due to progressively increasing ground motions in Phase I were replaced by new panels before applying additional shakings in Phase II. The study verified the reparability and redundancy of the SPSW system such that the repaired specimen can survive a subsequent earthquake without severe boundary frame damages or overall strength degradation, and could achieve story drifts up to 5.2%. Steel shear walls with corrugated infill plate have also been extensively studied. Qiu et al. [53] used a quasi-static loading to conduct a cyclic quasi-static test on three CSPSW specimens of 1/3-scaled, two-story, single-compartment, different corrugation orientations and structures. A traditional flat steel plate was tested as a reference. The test shows that, compared with the SPSW specimen, the corrugated SPSW specimen has higher lateral stiffness and elastic buckling ability, and the pinch on the hysteresis curve is smaller. Different from the mechanism of the tension field of SPSW, the lateral-load-resisting mechanism of corrugated SPSWs is shear yield or inelastic shear buckling, which depends on the corrugated structure. Wang et al. [54] tested six shear wall specimens (flat, vertical and horizontal corrugated steel plate shear walls and corresponding steel plate reinforced concrete composite shear walls) with a ratio of 1:2. The deformation capacity and failure mode of the specimen under cyclic loading are studied. The failure modes, deformation and energy dissipation capacity, stiffness and bearing capacity degradation characteristics of the corrugated steel plate shear wall are analyzed, and the design formulas for the shear capacity of the corrugated steel plate shear wall are proposed. The test results found that the lateral stiffness, ductility, and energy dissipation capacity of the composite shear wall are better than those of the steel plate shear wall. Due to the mechanical constraints of concrete on steel, the initial stiffness of composite shear walls is much higher than that of prestressed steel concrete shear walls. Emami et al. [55] conducted hysteresis experiments on three 1/2-scale single-layer single-span non-stiffened steel plate shear walls and corrugated transversely arranged, and corrugated longitudinally arranged trapezoidal corrugated steel plate shear walls, and found a reasonable design The trapezoidal corrugated steel plate shear wall can consume energy after a certain level of lateral displacement, and the hysteresis curve is full.

1.2.3 Standards and Codes The strip model has been adopted in most of the current standards or codes. To do so, a large number of cyclic tests was carried out to verify the strip model [5, 6, 9, 12,

8

1 Introduction

24, 56–60], it formed the basis of the current design method in the North American design codes such as CAN/CSA S16-01 [2], FEMA 450 [3], AISC-341-10 [61] and the recently released AISC-341-16 [62]. Caccese et al. [6] conducted tests for a series of three-story quarter-scale specimens to study the effects of different parameters on the behavior of SPSWs. They recognized the difference in the governing limit state for thin and thick infill plates—the former is governed by yielding of infill plates and the latter is governed by the instability of columns. The basic design philosophy provided in the Canadian and U.S. seismic design codes are primarily strength-based: the infill walls are initially sized to carry the entire code-specified lateral forces, a capacity design concept is then applied for selecting the boundary beams and columns, and the system is checked in later stages against the drift limitations before moving on to connection designs. Canadian standard CAN/CSA-S16 [2], Limit States Design of Steel Structures, has included specifications for eh design of SPSW since 1994. In this standard, the equivalent truss model proposed by Thorburn et al. [40] is recommended for preliminary design purposes. The approach consists of first designing a tension-only braced frame, by using diagonal steel truss members where steel panels would be otherwise be sued in the SPSWs. The areas of the truss members are designed to resist the specified lateral loads and to meet the drift requirements. The thickness, t wi , of the steel panel at the ith story is given by: twi =

2 Ai sin θi sin 2θi L sin2 2αi

(1.1)

where Ai and θ i are respectively the area and the angle of the inclination of the equivalent truss member at the ith story. Certain phenomena cannot be captured by the this method such as the tension field effects on the columns and the compression resistance of the panel near the corners. The NEHRP Recommended Provisions for Seismic Regulations for New Buildings and Other Structures [3] and the 2005 Seismic Provisions for Structural Steel Buildings [4] include minimum design requirements for SPSW. In these documents, columns are designed as Vertical Boundary Elements (VBE), beams are referred to as Horizontal Boundary Elements (HBE), steel panels are denoted simply as webs. The nominal strength of a web is set equal to Vn = 0.42Fy tw L c f sin(2α)

(1.2)

where L cf is the clear distance between VBE flanges. α is the inclination of the tension field. Here, factor 0.42 is the theoretical factor 0.50 divided by an overstrength factor equal to 1.2. In this document, HBE and VBE are to remain essentially elastic under forces generated by fully yielded webs, but flexural hinges are allowed at the ends of HBE. Design Guide 20 [19] contains detailed examples that implements the code-design procedure. Bruneau and his research associates proposed a plastic design method for the SPSW system, and studied many other code-based design aspects of the steel plate

1.2 Recent Research

9

shear walls, such as the capacity design method for the boundary column, how to avoid in-span hinges in the boundary beams and the method to reduce the system overstrength by using the balanced design concept [24, 26–28, 63]. By using the proposed plastic and capacity design methods, it was shown that the behavior of the SPSW system could be improved. Currently, no explicit requirement for the service-level performance is included in the AISC Seismic Provisions [62]. Engineers are allowed to select and verify the service-level performance goals under the performance-based design guidelines, such as the guideline by the PEER Tall Building Initiative [64] and Los Angles Tall Building Structural Design Council [65].

1.3 Objectives and Organization of This Book This book is focused on the effects of the gravity load on the behaviour and analysis of the steel plate shear walls. The influence of the gravity load on the shear capacity was currently not much considered in the available strategies. Because researchers and engineers assumed that all of the gravity load is sustained by the boundary columns. In reality, the walls always take a large amount of the gravity load. The highest building using steel shear wall is the Tianjin World Financial Center (also called Jinta tower) in Tianjin, China. The 75-storey building has a height of 336.9 m. The wall thickness is 25 mm from the bottom to 22, 20 and 18 mm at the top. The slenderness ratio is between 170 and 230. In the construction process, the steel shear walls were installed 15 storeys slower than the outer frames. Consequently, it will take a lot of the gravity load. However, this effect is not considered in practice and may lead to potential danger. This book aims to narrow the gap between the practical design and theoretical calculation method. The book can be divided into six chapters. This chapter introduces the background of SPSWs and gives a brief literature review of the corresponding research on this field. Chapter 2 is devoted to study the shear strength of SPSWs with considering the gravity load by the effective-width model. A finite element model with high precise was developed and verified by both pushover tests and cyclic tests available in the literatures. Then the vertical stress distribution of the wall under the gravity load was assumed to have a form of the effective-width model. The stresses of the wall under gravity, gravity and shear were presented. A new inclination angle of the tension strip model with considering the effect of the gravity load was proposed. Finally, the shear strength and the shear-deformation relationship of the SPSWs with considering the effect of the gravity load was proposed. Chapter 3, in that a new vertical stress distribution, i.e. three-segment distribution, of the wall under uniform compression and compression and in-plane bending was proposed. The proposed distribution divided the wall into three segments, in both edge-segments, a combination of linear and cosine functions from the edge stresses to the minimum stress, while in the middle segment, the stress distribution is constant and equal to the minimum stress. Numerical simulation results show that the proposed three-segment stress distribution can well describe the behavior of thin walls of

10

1 Introduction

different slendernesses and stress gradients. Then the influence of the gravity load on the shear strength of the SPSWs is considered by the three-segment distribution. The stress throughout the inclined tensile strip, considering the effect of different vertical stress distributions, is determined using the von Mises yield criterion. The shear strength is calculated by integrating the shear stress along the width. Numerical simulation results show that the three-segment distribution is able to describe the effect of gravity load of the SSWs of different aspect ratios and slenderness ratios under different gravity load and in-plane bending combination loadings (considered as stress gradient). Chapter 4 gives the shear strength calculation method of the cross stiffened shear walls with considering the effects of the gravity load. The stiffened SPSW is divided into several sub plates by vertical stiffeners. For each sub plate a three-segment vertical stress distribution under gravity load is proposed. The tension filed stress of the inclination tension strips, considering the effect of the gravity load through the proposed three-segment distribution of the vertical stress, is determined through the von Mises yield criterion. Shear strength of the infill steel plate is calculated by integrating the shear stress along the width of each sub plate. Numerical simulation results show that the proposed approach is able to predict the shear strength of the walls of different stiffener configurations and the gravity loads. Chapter 5 gives the shear strength of the corrugated SPSWs with considering the effects of the gravity load. Four scaled one-storey single-bay steel plate shear wall (SPSW) specimens with 2 flat panel and 2 corrugated panel were tested to examine their behavior under cyclic loadings. A finite element model adopting layered shell element model was built and verified by the test results. Then parametric study was conducted to examine the influences of the gravity load on the shear capacity of the corrugated steel shear walls. Chapter 6 describes the main conclusions obtained in this book and corresponding limitations of the employed approaches, upon which future work in the field of SPSWs is summarized.

References 1. Timler PA, Kulak GL (1983) Experimental study of steel plate shear walls. Minn Med 69(5):268–270 2. CSA (2011) Limit state design of steel structures. CAN/CSA-S16-11, Canadian Standard Association, Toronto, Ontario 3. FEMA-450 (2003) NEHRP recommended provisions for seismic regulations for new buildings and other structures. FEMA report no. 450. Building Seismic Safety Council for Federal Emergency Management Agency, Washington, DC 4. AISC (2005) Seismic provisions for structural steel buildings. ANSI/AISC 341-05, American Institute of Steel Construction, Chicago (IL) 5. Behbahanifard MR, Grondin GY, Elwi AE (2003) Experimental and numerical investigation of steel plate shear wall. Struct Eng Rep 254 6. Caccese V, Elgaaly M, Chen R (1993) Experimental study of thin steel-plate shear walls under cyclic load. J Struct Eng-ASCE 119(2):573–587

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7. Driver RG (1997) Seismic behaviour of steel plate shear walls. Department of Civil and Environmental Engineering, University of Alberta 8. Elgaaly M, Caccese V, Du C (1993) Postbuckling behavior of steel-plate shear walls under cyclic loads. J Struct Eng-ASCE 119(2):588–605 9. Lubell AS (1997) Performance of unstiffened steel plate shear walls under cyclic quasi-static loading. University of British Columbia 10. Elgaaly M, Liu Y (1997) Analysis of thin-steel-plate shear walls. J Struct Eng-ASCE 123(11):1487–1496 11. Kharrazi M, Prion H, Ventura CE (2008) Implementation of M-PFI method in design of steel plate walls. J Constr Steel Res 64(4):465–479 12. Driver RG, Kulak GL, Elwi AE, Kennedy DL (1998) FE and simplified models of steel plate shear wall. J Struct Eng-ASCE 124(2):121–130 13. Rezai M, Ventura CE, Prion HG (2000) Numerical investigation of thin unstiffened steel plate shear walls. In: 12th World conference on earthquake engineering 14. Shishkin JJ, Driver RG, Grondin GY (2009) Analysis of steel plate shear walls using the modified strip model. J Struct Eng-ASCE 135(11):1357–1366 15. Choi IR, Park HG (2010) Hysteresis model of thin infill plate for cyclic nonlinear analysis of steel plate shear walls. J Struct Eng-ASCE 136(11):1423–1434 16. Purba R, Bruneau M (2015) Seismic performance of steel plate shear walls considering two different design philosophies of infill plates. I: Deterioration model development. J Struct Eng-ASCE 141(6):04014160 17. Sabouri-Ghomi S, Roberts MT (1992) Nonlinear dynamic analysis of steel plate shear walls including shear and bending deformations. Eng Struct 14(5):309–317 18. Webster DJ (2013) The inelastic seismic response of steel plate shear wall web plates and their interaction with the vertical boundary members. Doctor of Philosophy Doctor of Philosophy Dissertation, University of Washington 19. Zhang X, Guo Y (2014) Behavior of steel plate shear walls with pre-compression from adjacent frame columns. Thin-Walled Struct 77(4):17–25 20. Habashi HR, Alinia MM (2010) Characteristics of the wall–frame interaction in steel plate shear walls. J Constr Steel Res 66(2):150–158 21. AISC (2007) Steel design guide 20: steel plate shear walls. American Institute of Steel Construction, Chicago (IL) 22. Hosseinzadeh SAA, Tehranizadeh M (2014) Behavioral characteristics of code designed steel plate shear wall systems. J Constr Steel Res 99(8):72–84 23. Shi Y, Astaneh-Asl A (2008) Lateral stiffness of steel shear wall systems. Structures Congress, ASCE, Vancouver, Canada 24. Berman J, Bruneau M (2003) Plastic analysis and design of steel plate shear walls. J Struct Eng-ASCE 129(11):1448–1456 25. Bruneau M, Uang C, Sabelli R (2011) Material models. In: Ductile design of steel structures, 2nd edn. McGraw-Hill Professional, New York 26. Berman JW, Bruneau M (2008) Capacity design of vertical boundary elements in steel plate shear walls. Eng J Am Inst Steel Constr 45(1):57–71 27. Purba R, Bruneau M (2015) Seismic performance of steel plate shear walls considering two different design philosophies of infill plates. II: Assessment of collapse potential. J Struct Eng-ASCE 141(6):04014161 28. Qu B, Bruneau M (2009) Capacity design of intermediate horizontal boundary elements of steel plate shear walls. J Struct Eng-ASCE 136(6):665–675 29. Kharmale SB, Ghosh S (2013) Performance-based plastic design of steel plate shear walls. J Constr Steel Res 90(9):85–97 30. Bhowmick AK, Driver RG, Grondin GY (2009) Seismic analysis of steel plate shear walls considering strain rate and -delta effects. J Constr Steel Res 65(5):1149–1159 31. Shao JH, Gu Q, Shen YK (2008) Seismic performance evaluation of steel frame-steel plate shear walls system based on the capacity spectrum method. J Zhejiang Univ-Sci A 9(3):322–329

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32. Ghosh S, Adam F, Das A (2009) Design of steel plate shear walls considering inelastic drift demand. J Constr Steel Res 65(7):1431–1437 33. Baldvins NM, Berman JW, Lowes LN, Janes TM, Low NA (2012) Fragility functions for steel plate shear walls. Earthq Spectra 28(2):405–426 34. Deng EF, Zong L, Wang HP, Shi FW, Ding Y (2020) High efficiency analysis model for corrugated steel plate shear walls in modular steel construction. Thin-Walled Struct 156:106963 35. Feng L, Sun T, Ou J (2021) Elastic buckling analysis of steel-strip-stiffened trapezoidal corrugated steel plate shear walls. J Constr Steel Res 184:106833 36. Dou C, Jiang ZQ, Pi YL, Guo YL (2016) Elastic shear buckling of sinusoidally corrugated steel plate shear wall. Eng Struct 121(8):136–146 37. Zhao Q, Qiu J, Zhao Y, Yu C (2020) Estimating fundamental period of corrugated steel plate shear walls. KSCE J Civ Eng 24(2):3023–3033 38. Farzampour A, Mansouri I, Lee CH et al (2018) Analysis and design recommendations for corrugated steel plate shear walls with a reduced beam section. Thin-Walled Struct 132:658–666 39. Takahashi Y, Takemoto Y, Takeda T, Takagi M (1973) Experimental study on thin steel shear walls and particular bracings under alternative horizontal load (preliminary report). In: IABSE, Symposium on resistance and ultimate deformability of structures acted on by well-defined repeated loads, Lisbon, Portugal 40. Thorburn LJ, Kulak GL, Montgomery CJ (1983) Analysis of steel plate shear walls. Report No. 107. Department of Civil Engineering, University of Alberta, Edmonton 41. Tromposch E, Kulak GL (1987) Cyclic and static behaviour of thin panel steel plate shear walls. Department of Civil Engineering, University of Alberta, Edmonton, Canada 42. Lubell AS, Prion HG, Ventura CE, Rezai M (2000) Unstiffened steel plate shear wall performance under cyclic loading. J Struct Eng-ASCE 126(4):453–460 43. Driver RG, Kulak GL, Kennedy DL, Elwi AE (1998) Cyclic test of four-story steel plate shear wall. J Struct Eng-ASCE 124(2):112–120 44. Schumacher A, Grondin GY, Kulak GL (1999) Connection of infill panels in steel plate shear walls. Can J Civ Eng 26(5):549–563 45. Zhao QH, Astaneh-Asl A (2004) Cyclic behavior of an innovative steel shear wall system. In: 13th World conference on earthquake engineering, Vancouver, Canada 46. Zhao QH, Astaneh-Asl A (2008) Experimental and analytical studies of a steel plate shear wall system. Structures Congress, ASCE, Vancouver, Canada 47. Astaneh-Asl A (2001) Seismic behavior and design of steel shear walls—SEAONC seminar. SEAONC Seminar, Structural Engineers Association of Northern California, San Francisco 48. Astaneh-Asl A (2002) Seismic behavior and design of steel shear walls. Steel technical information and product services (Steel TIPS) report. Structural Steel Educational Council, CA 49. Choi IR, Park HG (2008) Ductility and energy dissipation capacity of shear-dominated steel plate walls. J Struct Eng 134(9):1495–1507 50. Choi IR, Park HG (2009) Steel plate shear walls with various infill plate designs. J Struct Eng 135(7):785–796 51. Park HG, Kwack JH, Jeon SW, Kim WK, Choi IR (2007) Framed steel plate wall behavior under cyclic lateral loading. J Struct Eng-ASCE 133(3):378–388 52. Qu B, Bruneau M, Lin CH, Tsai KC (2008) Testing of full-scale two-story steel plate shear wall with reduced beam section connections and composite floors. J Struct Eng-ASCE 134(3):364– 373 53. Qiu J, Zhao Q, Cheng Y, Li ZX (2018) Experimental studies on cyclic behavior of corrugated steel plate shear walls. J Struct Eng-ASCE 144(11):04018200 54. Wang W, Ren Y, Lu Z, Song J, Han B, Zhou Y (2019) Experimental study of the hysteretic behaviour of corrugated steel plate shear walls and steel plate reinforced concrete composite shear walls. J Constr Steel Res 160(9):136–152 55. Emami F, Mofid M, Vafai A (2013) Experimental study on cyclic behavior of trapezoidally corrugated steel shear walls. Eng Struct 48(3):750–762 56. Chen R (1991) Behavior of unstiffened thin steel plate shear walls. University of Maine

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57. Elgaaly M (1998) Thin steel plate shear walls behavior and analysis. Thin-Walled Struct 32(1– 3):151–180 58. Kharrazi MH, Ventura CE, Prion HG (2010) Analysis and design of steel plate walls: analytical model. Can J Civ Eng 38(1):49–59 59. Kharrazi MH, Ventura CE, Prion HG (2010) Analysis and design of steel plate walls: experimental evaluation. Can J Civ Eng 38(1):60–70 60. Sabouri-Ghomi S, Ventura CE, Kharrazi MH (2005) Shear analysis and design of ductile steel plate walls. J Struct Eng-ASCE 131(6):878–889 61. AISC (2010) Seismic provisions for structural steel buildings. ANSI/AISC 341-10, American Institute of Steel Construction, Chicago (IL) 62. AISC (2016) Seismic provisions for structural steel buildings. ANSI/AISC 341-16, American Institute of Steel Construction, Chicago (IL) 63. Berman JW (2011) Seismic behavior of code designed steel plate shear walls. Eng Struct 33(1):230–244 64. PEER (2010) Guidelines for performance-based seismic design of tall buildings (as part of the Tall Buildings Initiative). Report PEER-2010/05, Pacific Earthquake Engineering Research Center, University of California, Berkeley, CA 65. LATBSDC (2015) An alternative procedure for seismic analysis and design of tall buildings located in the Los Angeles Region (2014 edition with 2015 supplements). Los Angeles Tall Buildings Structural Design Council (LATBSDC), Los Angeles, CA

Chapter 2

Steel Plate Shear Walls with Considering the Gravity Load by the Effective-Width Model

Since a steel wall always simultaneously carries the gravity load and shear. The gravity load can affect the shear strength of a steel shear wall. However, this effect is not considered in most of the research and standards, which may lead to potential danger in practice. To investigate this effect, a nonlinear finite element model for SPSWs was developed and verified using monotonic tests and cyclic loading tests. To consider the effect of the gravity load on the shear capacity of SPSWs, the compression stress of the infill steel plate under the gravity load is divided into three zones through the effective-width model, and the wall under this compression is assumed to consider the negative effect of the gravity. The shear capacity of 21 SPSWs with different axial stresses of the boundary columns and different steel infill plate thicknesses is predicted and compared with the numerical simulation results. The shear-displacement diagram of the SPSW under compression, shear and in-plane bending was then obtained. The load-carrying capacities and deformations at elastic buckling of the infill steel plate, the yield of Zone I and Zone III, the yield of Zone II, and the yield of the boundary frame were presented. At the end of this chapter, cyclic loading test on four scaled one-story single bay unstiffened SPSWs under different axial forces at the top of the columns was carried out to verify the proposed analytical model.

2.1 Introduction SPSWs have been widely known and accepted as an excellent lateral load resisting system. It consists of vertical steel infill plates connected to surrounding beams and columns. Previous SPSWs in the United States, Canada, and Japan used stiffened thick plates to prevent local buckling of steel plates. However, the construction of stiffeners and thick plates was challenging and required significant labor cost. Therefore, this system has not been widely used in practice. In 1983, Thorburn et al. found [1] that steel plate walls with unstiffened thin plates presented high ductility © Science Press 2022 Y. Lv, Steel Plate Shear Walls with Gravity Load: Theory and Design, https://doi.org/10.1007/978-981-16-8694-8_2

15

16

2 Steel Plate Shear Walls with Considering the Gravity …

as well as high strength even after the thin infill plates experienced local buckling due to their tension-field action. After that, many researchers began to study the thin SPSWs, such as Driver et al. [2], Behbahanifard [3], Moghimi and Driver [4], and Qu et al. [5]. They conducted experimental studies on the performance of SPSWs subjected to cyclic loadings. Some other researchers studied the unstiffened SPSWs, which included the effects of simple versus rigid beam-to-column connections [6], the dynamic response of SPSWs [7], the effects of holes in the infill plates [8], the use of Light-Gauge SPSWs [9], the effects of bolted versus welded infill connections [10], the corrugated SPSWs [11, 12], two-side connected buckling restrained SPSWs [13], SPSWs with outriggers [14], and the self-centering SPSWs [15, 16]. Kang et al. [17] presented a comprehensive state-of-the-art review of researches on the unstiffened thin SPSWs. However, there has been little research considering the influence of compression stress on the performance of infill steel plates due to the gravity loads acting on the adjacent boundary columns. Some researchers [2–4] experimentally studied the performance of SPSWs when they were subjected to a combination of constant gravity loads acting on the top of the boundary columns and cyclic lateral reversed loading. However, they did not compare the difference of the responses with and without gravity loads, and the applied gravity loads on the boundary columns were small. Elgaaly and Liu [18] compared the shear capacity of a SPSW with and without gravity load using the finite element method. They concluded that the gravity load had little effect on the shear capacity, which may be due to that the magnitude of compression loads in the analysis was small and the infill plate was quite thin. Zhang and Guo [19] investigated the behavior of SPSWs with pre-compression from the adjacent columns using finite element method. Their research showed that the shear capacity of SPSWs significantly deteriorated by the pre-compression from the boundary columns. In this chapter, based on the theory of post buckling strength of four edges clamped supported plate, the infill steel plate was divided into three zones and the compression stress of each zone was proposed. By analyzing the stress state of each zone for the infill steel plate under compression-shear combination load, the shear capacity of the infill steel plate was proposed. A modified plate-frame interaction model for the shear capacity of the infill steel plate is proposed and verified. The shear-displacement diagram under shear-compression interaction, and the shear-displacement diagram involving the global bending effect were proposed successively. The load-carrying capacities and deformations at the state of elastic buckling of the infill steel plate, the yield of Zone I and Zone III, the yield of Zone II, and the yield of the boundary frame were presented. Four scaled SPSWs were designed and tested under compressionshear interaction to investigate the feasibility of the proposed analytical model.

2.2 The Gravity Load Effect on the Shear Strength of the Steel Plate Shear Walls

17

2.2 The Gravity Load Effect on the Shear Strength of the Steel Plate Shear Walls 2.2.1 Numerical Model A conventional SPSW comprises thin unstiffened steel plates and bounded components of steel columns and beams as shown in Fig. 2.1a. There are three strategies to simulate the SPSW, i.e. the shell element method which both the boundary components and the steel plates are simulated by the shell elements, the mixture method which the boundary steel members are simulated by fiber beam elements while the steel plates are simulated by shell elements, and the widely used strip method. The shell element model is difficult to build the finite element model and the computation cost is too high to simulate the high-rise building structures. The strip method has high accurate in simulating the hysteresis of SPSWs in plane force, but it cannot simulate the out-of-plane deformation, the combination of shear and axial force, and the forces interaction at the intersection of the shear walls. As shown in Fig. 2.1, the mixture method is used to simulate the SPSWs, i.e., the infill steel plates are simulated by layered shell elements, the boundary members are simulated by fiber beam elements. As shown in Fig. 2.1c, the steel plate is divided into several layers, and the strain and curvature of the neutral layer are firstly calculated, based on the plane-section assumption of strain and the curvature of the other layers are decided, and the element internal force is integrated through the element thickness corresponding to the material models of each layer. For the layered shell element in the LS-DYNA program, the shear strain can also be simulated, and based on the central difference method the numerical converge problem is avoided. The boundary columns and beams are simulated by the fiber beam element model employing a Hughes-Liu formulation, σ 1

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

Boundary column

Boundary beam

steel ε

(b) Fiber beam element model

Damage 1

z(w) y(v) η x(u)

(d) Steel material model

ξ ζ

Layer n Layer i Layer 1

(a) Unstiffened steel plate shear wall

(c) Layered shell element model

0 εth

εp

(e) Bonora damage model of the steel

Fig. 2.1 Schematic diagram of analytical finite element model of SPSW

18

2 Steel Plate Shear Walls with Considering the Gravity …

the neutral axis of the top beams and the boundary columns are translated until the flanges are coincident with the outer edges of the infill plates. Each boundary member is discretized into several sections, and each section is further divided into several fibers, as shown in Fig. 2.1b. The sections are located either at the center of the element or at its Caussian integration points or any artificially defined positions, that the behavior of each fiber can be tracked using a simple uniaxial material model allowing an easy and efficient implementation of the inelastic behavior. Finite transverse shear strains of the Hughes-Liu element formulation is also retained comparing to the general fiber beam elements adopting plane section assumption. To correctly model the connection between the shear wall and the boundary column, the original beam node (at the centroid of the beam cross-section) was offset using a beam offset option (see LS-DYNA manual regarding offset of ELEMENT_BEAM). The nodes of the beams and columns were offset that their flanges were coincide with the outer edges of the shear walls. It was assumed that the displacement of the wall and beam at the common nodes was the same. Figure 2.2 shows the node offset. Alinia and Shirazi [20] considered the initial imperfection from 1/100,000 to 1/500 of the plate width and observed only a little difference of the load-bearing capacity. The reason was that the plate was very slender. It buckled very early and therefore initial imperfection did not have a considerable influence on its nonlinear postbuckling behaviour. Consequently, the initial imperfection was also not considered in this work.

Fig. 2.2 Consideration of the shear wall-beam connection according to a common software and b LS-DYNA

2.2 The Gravity Load Effect on the Shear Strength of the Steel Plate Shear Walls

19

The mixed hardening material model of steel (Fig. 2.1d) combined with the Bonora [21] damage evaluation method (Fig. 2.1e) are used to track the mechanical properties of the steel plate and boundary frames. The plastic potential f p of the material model is defined as, f p = σeq − k(κ) −

3   α α − σy 4α∞ i j i j

(2.1)

where σ y is the initial uniaxial yield stress, α ∞ is kinematic hardening saturation value, σ eq is the equivalent stress, αi j is the kinematic hardening tensor, k is the isotropic hardening stress, which is defined through Osgood equation, k(κ) =

Eh [1 − exp(−βκ)] β

(2.2)

where E h is the isotropic hardening modulus, β is isotropic hardening parameter, which set β = 0 for linear isotropic hardening, κ is the isotropic hardening coefficient and is defined as the equivalent accumulated plastic strain,  κ = εp =

dε p

(2.3)

The damage dissipation potential [21] is expressed as,     1 Y 2 S0 (dcr − d)1−1/υ fd = − 2 S0 1 − d κ (2+n)/n

(2.4)

The kinetic law of damage evolution is given by, (dcr − d0 )1/υ ∂ fd = f d˙ = −dλ ∂Y ln(εu − εth )



 σm dκ (dcr − d)1−1/υ σeq κ

(2.5)

where Y is the variable associated to damage, υ and S 0 are material parameters, εu and εth are the critical and threshold equivalent accumulated plastic strain, d cr and d 0 are critical and initial damage of the material corresponding to εu and εth , respectively.

2.2.2 Verifications Under Monotonic Tests The test on a large-scale four-story steel plate shear wall conducted by Driver et al. [2] provided complete information of the structure undergoing very large cyclic deformations. The overall height and width of the specimen were 7.4 m and 3.4 m, respectively. The storey height was 1.83 m except the first storey 1.93 m. The columns

20

2 Steel Plate Shear Walls with Considering the Gravity …

Table 2.1 Material parameters Damage parameters εth

ε cr

dcr

Plastic parameters d0

0.202 1.00 0.10 0

α

A (MPa) B (MPa) n

0.198 313.0

280.0

c

D1

D2

D3

D4

0.228 0.0171 0.0705 1.732 −0.54 −0.015

were 3.05 m apart measured from centerline to centerline. The W310 × 118 columns ran the full height of the shear wall without splices. Beam sections at the first three floors were W310 × 60 and at the top floor was W530 × 82. Full moment connections were used at all beam-to-column joints. The thickness of the shear walls, from the first to the fourth storey, was 4.8 mm, 4.8 mm, 3.4 mm and 3.4 mm, respectively. The elastic–plastic material model with kinematic hardening was used to describe the steel behaviour. According to [3], the Young’s modulus, yield stress and tangent modulus are respectively 2.088 GPa, 355 MPa, 2.06 GPa for steel walls of the bottom two storeys, 210.8 GPa, 271 MPa and 2.06 GPa for the walls of the top two storeys, 208.8 GPa, 345 MPa and 2.06 GPa for the horizontal beams, and 203 GPa, 329 MPa and 2.06 GPa for the vertical columns. The gravity load of 720 kN was first applied at the top of each column and kept unchanged. The horizontal loads were evenly applied at the left beam-column joints. The specimen investigated by Behbahanifard [3] was the upper three storeys of the specimen of the test by Driver et al. [2]. Consequently, the material properties were the same. The infill plates are subdivided to 8 layers through the thickness, the boundary columns and beams are subdivided to 27 fibers, the fish plates were not considered in this simulation. The material parameters of the column are shown in Table 2.1, where, α is exponential parameter of the material, d 0 is the initial damage, d cr and εcr are the critical damage and equivalent plastic strain at failure, εth is the equivalent plasticity strain when damage starts to occur. D1 , D2 , D3 , D4 , D5 are material fracture parameters, A is the uniaxial yield strength, B and n are the plastic strain hardening coefficient and exponential. Three equal forces are applied at each level of the beam. The pushover curves obtained from this study and Driver et al. [2] are shown in Fig. 2.3a. Figure 2.3a indicates that the simulation method can well track the base shear of the test structure. Figure 2.3b shows the simulation results and the test results of Behbahanifard [3], the detail information of this test can be found by Behbahanifard [3]. Similarly, good agreement can be got by using the simulation method.

2.2.3 Verifications Under Cyclic Tests The tests carried out by Park et al. [22] are chosen to further verify the feasibility of the numerical method under cyclic loading. The test specimens were one-third scaled down models of a three-story prototype walls. The size of the infill plate was 1.5 m × 1.0 m, the boundary members were built-up sections made of SM490 steel.

2.2 The Gravity Load Effect on the Shear Strength of the Steel Plate Shear Walls

21

4000 3500

Base shear (kN)

3000 2500 2000

FEM(Driver, et al.,1997) Test(Driver, et al.,1997) This Study

1500 1000 500 0

0

30

60

90

120

150

Top displacement (mm)

4000 3500

Base shear (kN)

3000 2500

FEM(Behbahanifard,2003) Test(Behbahanifard,2003) This Study

2000 1500 1000 500 0

0

30

60

90

120

150

Top displacement (mm) Fig. 2.3 Base shear-top displacement pushover FEA compare to envelope of test

The beams at first and second stories were H-200 × 200 × 16 × 16 mm. The top beam connected to the actuator was H-400 × 200 × 16 × 16 mm. Only the strong column series of H-250 × 250 × 20 × 20 mm are simulated to verify the numerical precise. The infill plate thickness of the specimens SC2T, SC4T and SC6T were 2 mm, 4 mm and 6 mm. More detailed information and configurations of the test specimens can be found in the study by Park et al. [22]. The target displacements for the cyclic loading applied at the top of the specimens were ±3, ±6, ±10, ±20, ±3, ±45, ±60, ±90, and ±120 mm, each target displacement repeats three cycles.

22

2 Steel Plate Shear Walls with Considering the Gravity …

Figure 2.4 shows the hysteresis curves derived from the numerical model and the envelop curves of the tests results, good agreement is got. The maximum loading force of the tests for SC2T, SC4T, and SC6T are 1721, 2480, and 3020, and the simulation results for the same case are 1729, 2606, and 3041 kN. Figure 2.4a shows Fig. 2.4 Load-top displacement relationships

Load (kN)

3200 2400 1600 800

-120

-80

-40

0

0

40

80

120

Top Displacement (mm) -800

Max. displ.

-1600

Test by Park et al.

-2400 -3200

(a) SC2T Load (kN)

3200 2400

Test by Park et al.

1600 800

-120

-80

-40

0

0

40

80

120

Top Displacement (mm) -800

Max. displ.

-1600 -2400 -3200

(b) SC4T Load (kN)

3200 2400 1600

Test by Park et al.

800

-120

-80

-40

0

0

40

80

120

Top Displacement (mm) -800 -1600 -2400 -3200

(c) SC6T

Max. displ.

2.2 The Gravity Load Effect on the Shear Strength of the Steel Plate Shear Walls

23

Fig. 2.5 Deformation shape of specimens SC2T, SC4T, and SC6T

that the unloading strength of the numerical model is a little stronger than the test results from −80 to 80 mm, but in general, the numerical simulation results have sufficient precision. The deformation shapes of the SC specimens at the largest displacement are shown in Fig. 2.5. Figure 2.5 indicates that the tension field forms in all stories, and the deformation was distributed along the height of the specimens. While for specimen SC6T, because of the thick infill plate, the plastic deformation was concentrated at the first story with bending deformation. The simulated deformation shapes generally agree with the test results. Figure 2.6 shows the plastic hinges formed at end of the boundary members. The failure mode of SC2T in Fig. 2.6 shows shear-dominated behavior, the plastic hinges first formed at the base of the boundary columns, with the increasing of the top displacement, the ends of the boundary beams of the first and the second stories gradually yield and form plastic hinges, when approaching to the end of the test, the top ends of the columns yield. Therefore, the shear capacity can be defined as the sum of the shear capacity of the moment frame and the shear capacity of the infill steel plate. For SC4T and SC6T, the failure modes show flexure-dominated behavior. The plastic hinges mainly form at the base of the columns, and light plastic hinges form at the end of the beams of the first story, the shear capacity is controlled by the cantilever action. Meanwhile, the length of the plastic hinges at the base of the columns for SC6T are much larger than those of SC2T and SC4T, it is identical with the tests results. Generally, no matter the pushover analysis under monotonic loading or the reversal analysis under cyclic loading, the numerical simulation method can commendably track the load-deformation relationship and the failure modes, and can be used as a tool to analyze the properties of the strong column series specimens.

24

2 Steel Plate Shear Walls with Considering the Gravity …

Fig. 2.6 Plastic hinges of the boundary frames for SC2T, SC4T, and SC6T

2.2.4 Pushover Analysis Results 2.2.4.1

Pushover Curves

Adopting the numerical simulation method verified above, the pushover analysis for the specimens was performed. Except the tested three specimens, 9 more specimens with axial loads applied at the top of the boundary columns are analyzed. The finite element models are the same with the tested ones. Figure 2.7 shows the pushover curves of the specimens SC2T, SC4T, and SC6T with axial forces of each column equal to 0, 1420, 2840, and 4260 kN at the top of the column, the axial forces induce a stress of each column to be 0, 100, 200, and 300 MPa. The strengths of the SPSWs with thick infill plates are much larger than the thinner ones for the same axial force level. The axial force will decrease the shear strength significantly. For SC2T, the shear strength decreases 10.15, 15.17, and 35.26% with the axial stress of the columns increase from 0 to 100 MPa, 100 to 200 MPa, and 200 to 300 MPa. For SC4T and SC6T, these are 8.42%, 15.59%, and 28.82% and 11.14%, 20.33%, and 28.45%, respectively. The story drifts at the maximum strength of the SPSWs with zero stress are all greater than 4%, and the shear strength stably increase with the top displacement. The axial force will reduce the deformation capacity, for SC2T, the story drifts at the maximum shear strength are 3.4%, 2.1%, and 1.9% with the axial stress equal to 100 MPa, 200 MPa, and 300 MPa respectively.

2.2 The Gravity Load Effect on the Shear Strength of the Steel Plate Shear Walls Fig. 2.7 Pushover curves

25

1600

SC2T

Base shear (kN)

1400 1200 1000 800 600

000MPa 100MPa 200MPa 300MPa

400 200 0 0

30

60

90

120

150

120

150

120

150

Top displacement (mm)

(a) SC2T 2700 2400

SC4T

Base shear (kN)

2100 1800 1500 1200

000MPa 100MPa 200MPa 300MPa

900 600 300 0 0

30

60

90

Top displacement (mm)

(b) SC4T 3200

SC6T

Base shear (kN)

2800 2400 2000 1600 1200

000MPa 100MPa 200MPa 300MPa

800 400 0 0

30

60

90

Top displacement (mm)

26

2 Steel Plate Shear Walls with Considering the Gravity …

Pushover Curve Defined Yield Point

Pdefined

Idealized Elastic-plastic Curve

Initial Yield Point

Pinitial

K initial=K defined

0

dinitial ddefined

d max

Fig. 2.8 Definitions of yield point and initial yield point

Table 2.2 Definition of yield point and yield displacement Axial stress (MPa)

SC2T

SC4T

SC6T

Py (kN)

δ y (mm)

Py (kN)

δ y (mm)

Py (kN)

δ y (mm)

0

1474

25.5

2377

26.9

2844

25.9

100

1352

22.4

2196

23.8

2545

22.7

200

1160

22.7

1894

24.0

2080

20.9

300

727

16.8

1337

19.4

1454

16.4

The yield point was defined base on the concept of equal plastic energy, the pushover curve is idealized to an elastoplastic curve with the same enclosed area, as shown in Fig. 2.8. The defined yield point and yield displacement are summarized in Table 2.2. Figure 2.7 and Table 2.2 indicate that the axial forces of the boundary columns will reduce the yield shear strength and the stiffness of the SPSWs. For the analyzed specimens, the yield shear strength decreases about 8.3, 21.3, and 50.7% for SC2T, 7.6, 20.3, and 43.8% for SC4T, and 10.5, 26.9, and 48.9% for SC6T with the axial stress of the boundary columns increasing from 0 to 300 MPa.

2.2.4.2

Tension Strip Angle

Four more specimens with the same boundary members but the infill plates thicknesses equal to 1, 3, 5, and 7 mm are analyzed. The stresses of the shell element at the center of the first story are calculated, and the principal stresses and principal stress direction can be got by Eqs. (2.6) and (2.7). Considering the forming mechanism of

2.2 The Gravity Load Effect on the Shear Strength of the Steel Plate Shear Walls

27

δmax

Pg Gravity Load

Time

Top Displacement O

ķ

ĸ

Ĺ

ĺ

Fig. 2.9 Loading process

the tension strip of the infill plate, the principal stress angles are approximated to be the tension strip angles. σx + σ y σ1 ± = σ2 2



tan 2α = −

σx − σ y 2

τx y σx − σ y

2 + τx2y

(2.6) (2.7)

The loading process of the specimens are shown in Fig. 2.9. The principal stress directions of the elements versus the top displacement are shown in Figs. 2.10a, 2.11a, 2.12a and 2.13a, the corresponding tension strip shapes of SC6T at point ➀➁➂➃ are shown in Figs. 2.10b, 2.11b, 2.12b, and 2.13b. The loading points ➀➁➂➃ shown in Fig. 2.9 represent the time at beginning of the maximum gravity loads, the top displacement = 10 mm, the top displacement = 60 mm, and the top displacement = the maximum top displacement, respectively. Figures 2.10, 2.11, 2.12 and 2.13 indicate that the principal stress angles are significantly influenced by the axial forces acting at the boundary columns. Once the tension strips have been formed at about point ➁, the principal stress angles in Figs. 2.10b, 2.11b, and 2.12b tend to decrease with the increase of the top displacement, the reason is that the infill plates sustain part of the gravity load when the columns undergo large deformation concurrent with gravity loads. Figures 2.10, 2.11, 2.12 and 2.13 also show that the principal stress angles calculated from the shell element at the center of the first story seems underestimate the tension strip shapes at ➀➁➂➃ points. The tension strip shapes also show that the angles of the tension strips are not constant. To further examine the mechanism of the change of the tension strip angle with the change of gravity loads, Fig. 2.14 shows the principal stress angles at the center of the SC2T, SC4T and SC6T with different axial stress acted on the top of the

28

2 Steel Plate Shear Walls with Considering the Gravity …

Principal stress angle

50

Ĺ

ĸ

40

ĺ

30

SC1T SC2T SC3T SC4T SC5T SC6T SC7T

20

10

ķ

0 0

30

60

90

120

150

Top displacement (mm)

α

Fig. 2.10 Tension strips angle of SPSWs with the axial stress of the columns equal 0 MPa

Principal stress angle

50

40

30

Ĺ SC1T SC2T SC3T SC4T SC5T SC6T SC7T

ĸ

20

10

ķ 0 0

30

60

90

120

ĺ

150

Top displacement (mm)

α

Fig. 2.11 Tension strips angle of SPSWs with the axial stress of the columns equal 100 MPa

2.2 The Gravity Load Effect on the Shear Strength of the Steel Plate Shear Walls

29

Principal stress angle

50

SC1T SC5T

40

SC2T SC6T

SC3T SC7T

SC4T

30

ĸ

20

Ĺ ĺ

10 0 0

ķ 30

60

90

120

150

Displacement (mm)

α

Fig. 2.12 Tension strips angle of SPSWs with the axial stress of the columns equal 200 MPa

Principal stress angle

50

40

30

20

Ĺ

ĸ 10

ķ 0 0

SC1T SC5T 30

ĺ

SC2T SC6T 60

SC3T SC7T 90

SC4T 120

150

Top displacement (mm)

α

Fig. 2.13 Tension strips angle of SPSWs with the axial stress of the columns equal 300 MPa

50

Principal stress angle

40

30 000MPa 50MPa 100MPa 150MPa 200MPa 250MPa 300MPa

20

10

0 0

SC2T 30

60

90

120

150

120

150

Displacement (mm)

(a) Specimen SC2T 50

40

Principal stress angle

Fig. 2.14 Influences of axial stress of the columns on the tension strip angles for SC2T, SC4T and SC6T

2 Steel Plate Shear Walls with Considering the Gravity …

30 000MPa 50MPa 100MPa 150MPa 200MPa 250MPa 300MPa

20

10

0 0

SC4T 30

60

90

Displacement (mm)

(b) Specimen SC4T 50

40

Principal stress angle

30

SC6T

30 000MPa 50MPa 100MPa 150MPa 200MPa 250MPa 300MPa

20

10

0 0

30

60

90

Displacement (mm)

(c) Specimen SC6T

120

150

2.2 The Gravity Load Effect on the Shear Strength of the Steel Plate Shear Walls

31

Table 2.3 Plane shared axial force Specimens

50 MPa

100 MPa

150 MPa

200 MPa

250 MPa

300 MPa

SC2T

Plane (kN)

53.4

88

116

138

154

170

Ratio (%)

3.76

3.10

2.72

2.43

2.17

2.00

SC4T

Plane (kN)

228

344

480

506

600

662.2

Ratio (%)

16.06

12.11

11.27

8.91

7.89

7.77

Plane (kN)

373.8

542

806

1004

1102

1350

Ratio (%)

26.32

19.08

18.92

17.68

15.72

15.85

SC6T

boundary columns. Figure 2.14 indicate that the tension strip angles decrease with the increase of the axial stress, for the small axial stress cases, the tension strip angle of the SPSWs are between 40° and 45°, which is coincident with the predicted value by Timler and Kulak [23]. However, it is difficult to gain the quantitative relation between the tension strip angle and the gravity loads. The biggest factor of the structure influencing the tension strip angle is the strength ratio of the boundary columns and the infill steel plate, that is tL/2Ac in Timler and Kulak predict equation of the tension strip angle, which t = thickness of the infill plate, Ac = cross-sectional area of the vertical boundary column, L = bay width. Table 2.3 summarized the axial force that the infill steel plate undertook, it indicates that the smaller the gravity loads are, the higher ratio the infill steel plates undertook, because the greater gravity loads trigger out of plane buckling of the infill steel plate. The stronger infill steel plates shared much larger gravity loads, such as SC6T. Figure 2.15 shows the principal stress angles of the shell elements at different positions, the axial stress acted at the top of the column is 150 MPa, each element represents a tension strip of the infill steel plate. For SC2T and SC4T, the tension strip angles of the different positions are nearly the same, which means the tension strips are parallel to each other. While for SC6T, the principal stress angles at different positions are variable, in other words, the tension strip angles of SC6T change with the loading process and tension strip positions, and the strip model may not suitable to simulate the relative thick infill steel plate shear wall with large gravity loads.

2.2.5 Cyclic Results 2.2.5.1

Cyclic Curves

The cyclic performance of SC2T, SC4T, and SC6T are examined. Before the lateral loading process, the gravity loads are applied at the top of the columns and kept constant, then the target displacements for the cyclic loading are organized as ±3, 6, 10, 20, 30, 45, 60, 90, and 120 mm, each target displacement repeats three cycles. Figures 2.16, 2.17 and 2.18 show the base shear versus the top displacement relationships of SC2T, SC4T, and SC6T under different gravity loads.

50

Principal stress angle

40

M Md Rd Rm Ru Mu Lu Lm Ld

30

20

10

SC2T 0 0

30

60

90

120

150

Displacement (mm)

(a) Specimen SC2T 50

40

Principal stress angle

Fig. 2.15 Tension strip angles for SC2T, SC4T and SC6T at different positions with the axial stress equal to 150 MPa

2 Steel Plate Shear Walls with Considering the Gravity …

M Md Rd Rm Ru Mu Lu Lm Ld

30

20

10

SC4T 0 0

30

60

90

120

150

Displacement (mm)

(b) Specimen SC4T 50

40

Principal stress angle

32

M Md Rd Rm Ru Mu Lu Lm Ld

30

20

10

SC6T 0 0

30

60

90

Displacement (mm)

(c) Specimen SC6T

120

150

2.2 The Gravity Load Effect on the Shear Strength of the Steel Plate Shear Walls 2000 1500

Base shear (kN)

Fig. 2.16 Load versus top displacement relationships of SC2T under different gravity loads

33

Cyclic Pushover

1000 500 0 -500 -1000 -1500 -2000 -150

-100

SC2T Applied axial stress=000MPa 0 50 100 150

-50

Displaceme (mm)

(a) 0 MPa 2000

Base shear (kN)

1500 1000

Cyclic Pushover

500 0 -500 -1000 -1500 -2000 -150

-100

-50

SC2T Applied axial stress=100MPa 0 50 100 150

Top displacement (mm)

(b) 100 MPa

In the early loading cycles, the infill plates behave in an elastic manner, the gravity load acted at the top of the columns will decrease the initial stiffness of the SPSWs. With the increase of the loading displacement at the top of the specimen, the diagonal portion of the SPSWs yields firstly. As the load reversed, the stiffness at the first several cycles keep nearly consistent. After significant yielding of the infill panels occurred at the 30 mm cycles, unloading and reloading in the opposite direction redevelop tension field and form new tension strips, the load carrying capacity improve with the deformation of the SPSWs. All the following inelastic cycles have similar hysteresis characters except a little decrease of the shear stiffness. The repeat cycles with the same amplitude of the top deformation have little influences on the hysteresis characters for the early stages of the cycles, however, the strength decrease obviously when the top displacement repeated with large amplitude of top displacements for the last several cycles. The maximum loads achieved in each cycle increase slightly with each excursion to a new deflection level for specimens SC2T and SC4T with the applied axial stress equal to 0 and 100 MPa, and specimen SC6T without axial force, i.e. there

34

2 Steel Plate Shear Walls with Considering the Gravity …

Fig. 2.16 (continued)

2000

Base shear (kN)

1500

Cyclic Pushover

1000 500 0 -500 -1000 -1500 -2000 -150

SC2T Applied axial stress=200MPa -100

-50

0

50

100

150

Top displacement (mm)

(c) 200 MPa 2000

Base shear (kN)

1500

Cyclic Pushover

1000 500 0 -500 -1000 SC2T Applied axial stress=300MPa

-1500 -2000 -150

-100

-50

0

50

100

150

Top displacement (mm)

(d) 300 MPa

are no decreases of the strength until the end of the loading. While the load carrying capacities of the SPSWs of the other cases declined very gradually from cycle to cycle. Such as SC2T, SC4T and SC6T with 300 MPa axial stress reached the maximum shear carrying capacity at the 90 mm cycles, 60 mm cycles and 30 mm cycles, respectively. From Figs. 2.16, 2.17 and 2.18, for all cases, the specimens show excellent deformation capacity and high strength. The hysteresis loops are plumper and no severe pinching occur. All the specimens show stable ductile behavior. However, the gravity loads acted at the top of the columns decrease the load-carrying capacity for SC2T of 300 MPa, SC4T of 200 MPa and 300 MPa, and SC6T of 100–300 MPa. The pushover curves are redrawing in the figures for convenience. For all the cases, the pushover curves accurately predicted the maximum load-carrying capacities, except the SC2T series. The differences may come from the mixed hardening material model of the infill plates. Because under the repeat cyclic lateral displacements, equivalent accumulated plastic strain of the steel increase rapidly, and the yield strength of the steel will increase accordingly, as the isotropic hardening stress k defined in Eq. (2.2).

2.2 The Gravity Load Effect on the Shear Strength of the Steel Plate Shear Walls 2700 1800

Base shear (kN)

Fig. 2.17 Load versus top displacement relationships of SC4T under different gravity loads

35

Cyclic Pushover

900 0 -900 -1800 -2700 -150

SC4T Applied axial stress=000MPa -100

-50

0

50

100

150

Top displacement (mm)

(a) 0 MPa 2700

Base shear (kN)

1800

Cyclic Pushover

900 0 -900 -1800 -2700 -150

SC4T Applied axial stress=100MPa -100

-50

0

50

100

150

Top displacement (mm)

(b) 100 MPa

All the specimens survived from the combination actions of the gravity loads and lateral displacements except the SC6T of 300 MPa failed at the lateral displacement about 60 mm. The load-carrying capacity of SC2T under 100 MPa axial force even increases 5.56% comparing to the 0 MPa case. The load-carrying capacities of SC2T under 100, 200, and 300 MPa gravity loads decrease about −5.56, 10.35, and 23.07%. For SC4T and SC6T, these are 11.21%, 26.79%, and 42.28%, and 14.28%, 28.42%, and 46.28%, respectively. It means that the thicker the infill plates are, the greater influence on the lateral load-carrying capacity of the SPSWs have.

2.2.5.2

Energy Dissipation

The area enclosed by the hysteresis curves is a measurement of the energy dissipated by the system in resisting the load or displacement history imparted. Figures 2.16, 2.17 and 2.18 show that the hysteresis curves are wide, indicating the significant energy dissipation capacity during each cycle. The gravity loads acted at the top of

36

2 Steel Plate Shear Walls with Considering the Gravity …

Fig. 2.17 (continued)

2700

Base shear (kN)

1800

Cyclic Pushover

900 0 -900 -1800 -2700 -150

SC4T Applied axial stress=200MPa -100

-50

0

50

100

150

Top displacement (mm)

(c) 200 MPa 2700

Base shear (kN)

1800

Cyclic Pushover

900 0 -900 SC4T Applied axial stress=300MPa

-1800 -2700 -150

-100

-50

0

50

100

150

Top displacement (mm)

(d) 300 MPa

the boundary columns will decrease the deformation capacity and shear carrying capacity, and in turn decrease the energy dissipation capacity. Figure 2.19 shows the energy dissipation capacity per cycle of the specimens under different gravity load levels according to the plastic deformation. A steady increase in dissipated energy occurred for all the cases except the SC6T of 300 MPa, the dissipated energy increases linearly with the top displacement, which indicates well earthquake resisting capacity. The energy dissipation capacities with thicker infill plates are better than the thinner ones, the gravity loads will decrease the energy dissipation capacity for all specimens.

2.2 The Gravity Load Effect on the Shear Strength of the Steel Plate Shear Walls 3200

Base shear (kN)

2400

Cyclic Pushover

1600 800 0 -800 -1600 SC6T Applied axial stress=000MPa

-2400 -3200 -150

-100

-50

0

50

100

150

Top displacement (mm)

(a) 0 MPa 3200

Base shear (kN)

2400

Cyclic Pushover

1600 800 0 -800 -1600 SC6T Applied axial stress=100MPa

-2400 -3200 -150

-100

-50

0

50

100

150

Top displacement (mm)

(b) 100 MPa 3200

Base shear (kN)

2400

Cyclic Pushover

1600 800 0 -800 -1600 SC6T Applied axial stress=200MPa

-2400 -3200 -150

-100

-50

0

50

100

150

Top displacement (mm)

(c) 200 MPa

Fig. 2.18 Load versus top displacement relationships of SC6T under different gravity loads

37

38

2 Steel Plate Shear Walls with Considering the Gravity … 3200

Base shear (kN)

2400

Cyclic Pushover

1600 800 0 -800 -1600 SC6T Applied axial stress=300MPa

-2400 -3200 -150

-100

-50

0

50

100

150

Top displacement (mm)

(d) 300 MPa

Energy dissipationed per load cycle(kN.m)

Fig. 2.18 (continued)

700 600 500 400 300 200

SC2T of 000MPa SC2T of 100MPa SC2T of 200MPa SC2T of 300MPa SC4T of 000MPa SC4T of 100MPa SC4T of 200MPa SC4T of 300MPa SC6T of 000MPa SC6T of 100MPa SC6T of 200MPa SC6T of 300MPa

100 0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Story drift (%) Fig. 2.19 Energy dissipated per load cycle

2.2.6 Shear Strength Obtained from Existing Model The failure modes of the SPSWs can be classified into three types, the sheardominated mode like SC2T, the flexure-dominated mode like SC6T, and the shear and

2.2 The Gravity Load Effect on the Shear Strength of the Steel Plate Shear Walls

39

Pg

Pg

δ V sf M pc2

V ff

δ

M pb2 h f py M pb1

θ=δ/h

α Ft

M pc1 Shear-dominated action (V sf )

Tension field action (V sp)

yield zone

Fc

Flexue-dominated action(V ff )

Fig. 2.20 Analytical mechanism model of the SPSWs

flexure combined mode like SC4T. Park et al. [22] proposed an evaluation method for the deformation mode of the SPSWs, they calculate the lateral load-carrying capacities by shear-dominated deformation mode and flexure-dominated mode respectively, and the minor value is defined as the lateral load-carrying capacity and the corresponding deformation mode is judged as the failure mode. Figure 2.20 show the two analytical mechanism models of the SPSWs, for the shear-dominated deformation mode, the overall shear capacity V s is defined as the sum of the shear capacity of the moment-frame V sf and the shear capacity of the infill shear plate V sp , it can be calculated by the principal of virtual work as δ h (2.8)

Vs f δ = 2(M pc1 + M pc2 + M pb1 + M pb1 )θ = 2(M pc1 + M pc2 + M pb1 + M pb1 )

The shear capacity of the moment-frame can be calculated as Vs f = 2(M pc1 + M pc2 + M pb1 + M pb1 )θ = 2(M pc1 + M pc2 + M pb1 + M pb1 )/ h (2.9) The shear capacity of the infill steel plate can be calculated as Vsp = τlt p = f py lt p sin α cos α

(2.10)

The shear capacity of the shear-dominated deformation is Vs = Vs f + Vsp

(2.11)

40

2 Steel Plate Shear Walls with Considering the Gravity …

The flexural capacity developed by cantilever action can be calculated by V f f = Ac ( f cy − f cg )l/ h − Pg δ/ h

(2.12)

where h is the shear span, Ac is the area of the cross section of the boundary column, l is the effective depth of the infill steel plate, f py is the yield strength of the steel plate, f cy is the yield strength of the column, and f cg is the axial stress developed by gravity load. Pg is the gravity load applied at the top of the specimen, and δ is the lateral displacement at the top. α is the tension strip angle, the tension strip angle defined by the principal stress angle seems largely underestimate the tension strip angle, and the definition of Timler and Kulak [23] do not consider the influence of the gravity loads and will overestimate the value. The tension strip angles in this paper are defined by measuring the figures from 2.10b, 2.11b, 2.12b and 2.13b. From the deformation mechanism presented in Fig. 2.6 and tension strip shapes in Figs. 2.10, 2.11, 2.12 and 2.13, it is clearly that the infill plates of all specimens formed tension field strips. Therefore, in this paper, the overall shear capacity of the flexure-dominated deformation is defined as the sum of the flexural capacity and the shear capacity of the infill steel plate, but the tension strip angle is very small. The shear capacity of the flexure-dominated deformation is V f = V f f + Vsp

(2.13)

The predict shear capacity of the specimens are defined as the lesser of V s and V f. The strengths of the specimens are predicted by the proposed method. Table 2.4 shows the simulation results and the predicted results calculated in this paper. Table 2.4 indicate that the predict shear capacities of the flexure-dominate deformation considering the contribution of the infill plate are much closer to the simulation and tests results.

2.3 Vertical Stress Distribution Under Compression 2.3.1 The Effective-Width Model The specimen is fixed at the base of the boundary columns, with only axial force acted at the top of the boundary columns represent the gravity loads from the upper floors. The axial load sustained by the infill plate can be calculated as  Pg,p =

tσy,p (x)d x

(2.14)

43.2

37.7

28.0

44.4

40.3

32.3

25.0

43.0

33.9

21.0

19.0

42.0

42.0

SC2T100

SC2T200

SC2T300

SC4T000

SC4T100

SC4T200

SC4T300

SC6T000

SC6T100

SC6T200

SC6T300

Behbahanifard

Driver et al

78

60

68

101

150

150

85

130

150

150

69

77

123

150

d Vmax a (mm)

1645

2307

154

864

1595

2364

134

842

1595

2364

153

884

1606

2364

V ff (kN)

–

–

783

783

783

783

783

783

783

783

783

783

783

783

V sf (kN)

The top displacement at the maximum shear strength

43.5

SC2T000

a

α (°)

Specimen

1420

1420

1320

1435

1985

2139

1180

1392

1519

1540

611

713

736

736

V sp (kN)

Table 2.4 The simulation and predict shear capacity of the specimens

4735

4871

2103

2218

2769

2922

1963

2176

2303

2323

1394

1496

1519

1519

V s (kN)

3007

3669

1474

2299

3580

4503

1313

2234

3115

3904

764

1597

2341

3100

V f (kN)

3007

3669

1474

2218

2769

2922

1313

2176

2303

2323

764

1496

1519

1519

V predict (kN)

3183

3387

1528

2135

2680

3016

1399

1965

2328

2542

772

1193

1406

1565

Pushover (kN)

–

–

1634

2177

2607

3041

1504

1908

2314

2606

1330

1550

1825

1729

Cyclic (kN)

3080

3500

–

–

–

3020

–

–

–

2480

–

–

–

1721

Test (kN)

2.3 Vertical Stress Distribution Under Compression 41

42

2 Steel Plate Shear Walls with Considering the Gravity …

where t is the plate thickness, σy,p (x) is the stress of the infill plate in y direction, i.e. the compression stress. Because the compression stress of the infill plate is caused by the axial compression of the adjacent boundary columns which is not bigger than the axial stress of the columns, therefore, it is assumed that the stress distribution of the infill plate is symmetric and the maximum stress of the infill plate in y direction equal to the axial stress of the boundary columns, the stresses of the infill plate in y direction adjacent to the boundary columns at x = 0 and x = L are σy,p (0) = σy,p (L) = σg,c , where L is the infill plate width from centerline to centerline, σg,c is the axial compression stress of the columns under gravity loads. The infill plate can be assumed to be a rectangular plate clamped supported along all edges by the boundary columns and beams and subjected to a uniform longitudinal compressive stress, the elastic buckling stress σcry,p can be calculated by σcry,p = k

π2E 12(1 − μ2 )(L/t)2

(2.15)

where k [15] is the buckling coefficient of uniform compressed four edges clamped supported plate calculated by  2  2 L H k = 3.6 + 4.3 +2.5 H L

(2.16)

where H, t, μ, E are the infill steel plate height, infill steel plate thickness, Poisson’s ratio and modulus of elasticity. From Eq. (2.15), it is indicated that the elastic buckling stress is usually very small because of the large width to thickness ratio of the infill steel plate, therefore, the infill steel plate always undergoes post buckling loading stage. The true compression stress in y direction along x can be expressed as [24] average average + (σy,p − σcry,p ) cos 2π σy,p (x) = σy,p average

x L

(2.17)

σ max +σcry,p

max = y,p 2 , σy,p = σy,p (0) = σy,p (L) = σg,c ≤ f y,p , f y,p is the yield where σy,p stress of the infill steel plate. For the infill steel plate constrained by the boundary columns, the width to thickness ratio is far larger than the web plate in the shape steel member, and the constrained capacity is slight, so instead of the actual width of the infill steel plate, the effectiveness web plate width L e with uniform distribution max is used, as shown in Fig. 2.21, where [24] compression stress σy,p

 Le =

σcry,p ×L max σy,p

(2.18)

Then the axial load sustained by the infill steel plate can be calculated by max + t (L − L e )σcry,p Pg,p = t L e σy,p

(2.19)

2.3 Vertical Stress Distribution Under Compression

43

Le/2 Be equivalent to

σcr Le/2

σcr σave σc

σc

Fig. 2.21 Schematic diagram of the equivalent process of the compression stress

2.3.2 Evaluation of the Effective-Width Model To simplify the analysis process, choosing a simple one story SPSW structure as an example. The size of the infill plate is 1.5 m × 1.0 m made of Q235 steel, and the boundary members were built-up sections made of Q345 steel. The beam connected to the actuator is H-400 × 200 × 16 × 16 mm, and the column is H-250 × 250 × 20 × 20 mm. The infill plate thickness of the specimen is 2, 4 and 6 mm. Adopting the elastic-perfectly plastic material model, the yield strength of the infill plate and the boundary members are 235 MPa and 345 MPa, respectively. Using the numerical simulation method proposed in the previous section, Fig. 2.22 shows the simulation results of the axial stress distribution of the infill steel plate subjected to pure compression of adjacent columns about 100 and 200 MPa (the gravity loads applied at each of the boundary column are l420 and 2840 kN). From Fig. 2.22, it is indicated that the stress of most part of the infill plate is very small, the maximum stress appears near to the boundary columns, and the maximum compression stress is nearly equal to the maximum axial stress of the boundary columns produced by the gravity load. The theoretical distribution expressed in cosine form and the simplified distribution using effective length of the compression stress along x are shown in Fig. 2.22 as well. Figure 2.22 also indicate that the distribution of the compression stress of the 100 MPa case is wider than that of the 200 MPa case, which is consistent with Eq. (2.18). Adopting the previous numerical method, more simulation cases of the specimens with the axial stress applied at the top of the boundary columns equal to 50, 100 and 200 MPa, and the thicknesses of the infill steel plate equal to 2, 4 and 6 mm are analyzed. The compression force sustained by the infill steel plate calculated by Eqs. (2.15), (2.16), (2.18) and (2.19) are summarized in Table 2.5, in Table 2.5, Ps is the numerical simulation results of compression force sustained by the infill steel P −P plate, error is defined as g,pPg,p s,p .

44

2 Steel Plate Shear Walls with Considering the Gravity …

Pg=1420 kN

40 30

Le/2 y,p (L)

y,p (0)

x Le/2

400

Pg=1420 kN

y,p(x)

20 10

80

1000

50 60

90 100

4.76

250

Boundary column

70

Boundary beam 250

250

1500 Pg=2840 kN

Le/2

40

y,p(x)

20

y,p (L)

y,p(0)

x Le/2

400

Pg=2840 kN

1000

180 160

4.76

Boundary beam 250

1500

Fig. 2.22 Simulation results of compression stress distribution of the infill steel plate

250

Boundary column

80 200

250

2.3 Vertical Stress Distribution Under Compression

45

Table 2.5 Calculation of the compression force of the infill plate max (MPa) k σy,p

50

14.39

σcrx,p (MPa) H (m) L (m) T (mm) L e (m) Pg,p (kN) Ps,p (kN) Error (%) 4.76

1.0

1.50

2

0.463

60.6

60

0.99

100

14.39

4.76

1.0

1.50

2

0.327

79.7

86

−7.90

200

14.39

4.76

1.0

1.50

2

0.231

106.7

128

−19.96

50

14.39 19.04

1.0

1.50

4

0.925

185.1

196

−5.89

100

14.39 19.04

1.0

1.50

4

0.654

261.7

284

−8.52

200

14.39 19.04

1.0

1.50

4

0.463

370.2

448

−21.02

50

14.39 42.84

1.0

1.50

6

1.388

416.4

315

24.35

100

14.39 42.84

1.0

1.50

6

0.981

589.1

570

3.24

200

14.39 42.84

1.0

1.50

6

0.694

833.1

866

−3.95

Figure 2.23 shows the effective length and infill steel plate strength with the compression stress of the infill steel plate, the width to thickness ratio (L/t) vary from 300 to 1500, and it is indicated that the thin infill steel plate has little gravity resistance capacity, for t = 1 mm, the infill steel plate sustains about 0.69% (when the compression stress = 300 MPa) to 6.30% (when the compression stress = 5 MPa) of the total gravity load, while for the stocky infill steel plate such as t = 6 mm, the infill steel plate sustains about 28.68% (when the compression stress = 300 MPa) to 73.94% (when the compression stress = 5 MPa) of the total gravity load, the gravity 1250

1.6

Le of t=1mm

Pg,p of t=1mm

Le of t=2mm

Pg,p of t=2mm

1.4

Le of t=3mm

Pg,p of t=3mm

Le of t=4mm

Pg,p of t=4mm

Le of t=5mm

Pg,p of t=5mm

1.2

Pg,p of t=6mm

Le of t=6mm

1.0 0.8

1000

750

500

0.6 0.4

250

0.2 0.0

0

50

100

150

200

250

0 300

Gravity load sustained by infill plate (kN)

Effective length (m)

1.8

Compression stress (MPa) Fig. 2.23 Effective length and sustained gravity load for different thicknesses of infill steel plates

46

2 Steel Plate Shear Walls with Considering the Gravity …

load resistance capacity decrease with the increase of the gravity load because the effective length reduces with the increase of the compression stress.

2.3.3 Discusses on the Effective-Width Model To simplify the analysis process, a simple one-story SPSW structure was chosen as an example to illustrate the stress distribution of the infill steel plate due to the gravity load acting on boundary columns. The size of the infill plate is 1.5 m × 1.0 m made of Q235 steel, and the boundary members were built-up sections made of Q345 steel. The beam connected to the actuator is H-400 × 200 × 16 × 16 mm, and the column is H-250 × 250 × 20 × 20 mm. The infill plate thickness of the specimen is 2, 4 and 6 mm. Adopting the elastic-perfectly plastic material model, the yield strength of the infill plate and the boundary members are 235 MPa and 345 MPa, respectively. Using the numerical simulation method proposed and verified in reference [25], the effective length of the infill steel plate is shown in Fig. 2.24, and gravity load transferred between boundary columns and infill steel plate is shown in Fig. 2.25. The width to thickness ratio of infill steel plates (L/t) varies from 300 to 1500. These two figures show the relationships of the effective length and infill steel plate strength as a function of compression stress of the infill steel plate, it is found that the thin infill 1.8 Le for t=1mm Le for t=2mm Le for t=3mm Le for t=4mm Le for t=5mm Le for t=6mm

1.6

Effective length (m)

1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0

0

50

100

150

200

Compression stress (MPa) Fig. 2.24 Effective length for different thicknesses of infill steel plates

250

300

2.3 Vertical Stress Distribution Under Compression

47

1000

Gravity load (kN)

ratio for t=1mm ratio for t=2mm ratio for t=3mm ratio for t=4mm ratio for t=5mm ratio for t=6mm

Pg,p for t=1mm Pg,p for t=2mm Pg,p for t=3mm Pg,p for t=4mm Pg,p for t=5mm Pg,p for t=6mm

750

0.8

0.6

500

0.4

250

0.2

0

0

50

100

150

200

250

Gravity load transferring ratio

1.0

1250

0.0 300

Compression stress (MPa) Fig. 2.25 Gravity load transferring for different thicknesses of infill steel plates

steel plate has little gravity resistance capacity. For t = 1 mm, the infill steel plate sustains about 0.69% (when the compression stress = 300 MPa) to 6.30% (when the compression stress = 5 MPa) of the total gravity load. However, for the thick infill steel plate such as t = 6 mm, the infill steel plate sustains about 28.68% (when the compression stress = 300 MPa) to 73.94% (when the compression stress = 5 MPa) of the total gravity load. In addition, the gravity resistance capacity of infill steel plate decreases with the increase in the gravity load. This is because the effective length decreases faster than the increases of the compression stress of the infill steel plate due to the compression stress increases of the boundary columns.

2.4 Stresses Under Compression and Shear 2.4.1 Stresses Analysis A typical story of a SPSW building structure with ductile steel plate wall can be represented as an isolated panel. To account for the influence of compression on the load-carrying capacity of infill steel plates, the proposed analytical model, based on the tension strip model and previous analysis, the infill steel plate is partitioned into three zones, as shown in Fig. 2.26. The widths of zone I, II and III are L e /2, L − L e , and L e /2. The stresses state of zone I and III are assumed to be the same, and during buckling are

48

2 Steel Plate Shear Walls with Considering the Gravity …

u

G

G column

beam

V

zone II

zone I

zone III

H

y α1

x

α

Le/2

L-Le

Le/2

Fig. 2.26 Partition of the infill plate

σx x,I = 0 max σ yy,I = σv,p τx y,I = τcr

(2.20)

where τcr is the elastic shear buckling stress, which can be calculated by τcr = kshear

π2E 12(1 − μ2 )(L/t)2

(2.21)

where k shear is the elastic shear buckling coefficient under pure shear of the four edges clamped plate and is calculated by  kshear =

2 8.89 + 5.6 HL , for L ≥ H 2 5.6 + 8.89 HL , for L ≤ H

(2.22)

Stresses after buckling are σx x,I = σty,I sin2 αI max σ yy,I = σv,p + σty,I cos2 αI

τx y,I = τcr + 0.5σty,I sin 2αI

(2.23)

2.4 Stresses Under Compression and Shear

49

For the thin plates, the principal stress in z direction (the thickness direction) and the shear stresses in plane yz as well as xz are zero, so the Von Mises yield criterion is obtained as 2 2 + σx2x + 6τx2y − 2 f y,p =0 J (σ ) = (σx x − σ yy )2 + σ yy

(2.24)

Substituting equations of σ xx , σ yy and τ xy into Eq. (2.24), and assuming that for the thin plates τcr = 0, then the value of σty,I at which yielding of the tension strip occurs in zone I and III is defined as 2 max max 2 2 + σv,p (3 cos2 αI − 1)σty,I + (σv,p ) − f y,p =0 σty,I

(2.25)

then the yielding stress of the tension strip is expressed as

σty,I =

max (3 sin2 αI − 1) + σv,p



max (3 sin2 α − 1)]2 − 4[(σ max )2 − f 2 ] [σv,p I v,p y,p

2

(2.26)

The corresponding stresses of zone II during buckling are σx x,II = 0 σ yy,II = σcrv,p τx y,II = τcr

(2.27)

Stresses after buckling are σx x,II = σty,II sin2 αII σ yy,II = σcrv,p + σty,II cos2 αII τx y,II = τcr + 0.5σty,II sin 2αII

(2.28)

Substituting equations of σx x , σ yy and τx y into Eq. (2.24), and assuming τcr = 0, then the value of σty,II at which yielding of the tension strip occurs in Zone II is defined as  2 ] σcrv,p (3 sin2 αII − 1) + [σcrv,p (3 sin2 αII − 1)]2 − 4[(σcrv,p )2 − f y,p σty,II = 2 (2.29)

50

2 Steel Plate Shear Walls with Considering the Gravity …

2.4.2 Discusses on Stresses Figure 2.27 shows the relationship of the yielding stress as a function of the compression stress of the infill steel plate for different inclination angles in zone I and zone III. It is shown that the yielding stress decreases with the increase in the compression stress. Based on the analysis, it was found that the yield stress of the infill steel plate did not change significantly as the inclination angels changed from 39° to 42°. Therefore, the inclination angle within this range seems to have little effect on the yield stress of the infill steel plate. Also, the yielding stress decreases in square root relationship with the increase in the compression stress of the infill steel plate as expressed in Eqs. (2.26) and (2.29). Figure 2.28 shows the relationship of stress in x direction, stress in y direction and shear stress with the increasing compression stress of the infill steel plate in zone I and zone III when the inclination angle is equal to 42°. With the increasing of the compression stress from 0 to 300 MPa, the shear stress gradually decreases to zero, which means that when the compression stress reaches the yield stress of the infill steel plate, the shear capacity of the infill steel plate becomes zero. This observation implies that the shear capacity of infill steel plates reduces if the gravity loads on the infill steel plate cannot be avoided during the construction of the SPSW structures. Similarly, the stress in x direction gradually decreases to zero with the increase of the compression stress. Therefore,

Yield stress of the steel plate (MPa)

300 250 200

30 degree 37 degree 39 degree 42 degree 45 degree 50 degree

150 100 50 0

0

50

100 150 200 Pre-compression stress (MPa)

250

300

Fig. 2.27 The relationship of the yielding stress with the pre-compression of the infill steel plate for different inclination angles in zone I and zone III

2.4 Stresses Under Compression and Shear

51

Stresses of the steel plate (MPa)

300 200 100 0

50

100

150

200

250

300

-100 -200 -300

yield stress of the steel plate Stress in Y driection Stress in X direction Shear stress Pre-compression stress (MPa)

Fig. 2.28 The relationship of the stress state with the pre-compression of the infill steel plate inclination angle = 42° in zone I and zone III

the strength demand of the boundary beams and columns is decreased. The stress in y direction changes from tension stress to compression stress, which will enlarge the axial force of the compression column and decrease the axial force of the tension column, however, under the concurrent gravity load and lateral reversal load like earthquake, the strength demand will be enlarged if the gravity load on the infill steel plate is involved.

2.4.3 Shear Capacity of the Infill Steel Plate According to Thorburn et al. [1], infill steel plates that experience early buckling under cyclic shear loading can be modeled by a series of pin-ended tension strips with an inclination angle of α. For rigid connected walls, the inclination angle of the tension field to the vertical axis based on the researches of Timler and Kulak [23] is      −1 tL 1 H3 4 1+ 1 + tH + α = tan−1 2 Ac Ab 360Ic L 2

(2.30)

where Ac and Ab are the cross section of the boundary column and beam, I c is the moment of inertia of the boundary column. Although the inclination angles of zone

52

2 Steel Plate Shear Walls with Considering the Gravity …

I and zone III are different from that of zone II, previous researches show that this difference is small. Therefore, the inclination angle of the infill steel plate is assumed to be the same for zone I, zone II and zone III, which is equal to α. Three unstiffened steel plate shear walls simulated in the previous section are further studied here to discuss the shear capacity of the infill steel plate. The thicknesses of the infill steel plates are 2, 4, and 6 mm, the shear capacity of each zone for the infill steel plate is shown in Fig. 2.29. From Fig. 2.29, it is shown that the shear capacity of zone I decreases rapidly with the increase in the compression stress of the boundary columns. This is because the effective length of the infill steel plate decreases with the increase in the compression stress of the boundary columns (as shown in Fig. 2.21), also the yield stress decreases with the increase in the compression stress of the infill steel plate induced by the compression force of the boundary columns (as shown in Fig. 2.24). On the contrary, the shear capacity of zone II of the infill steel plate increases with the increase in the compression stress of the boundary columns, the main reason is that the width of zone II of the infill steel plate increases with the compression stress of the boundary columns. However, the overall shear capacity of the infill steel plate decreases with the increase in the compression stress of the boundary columns. For different thicknesses of the infill steel plates, the loss 2000

Vsp,I-2mm Vsp,I-4mm Vsp,I-6mm

1800 1600

Vsp,II-2mm Vsp,II-4mm Vsp,II-6mm

Vsp-2mm Vsp-4mm Vsp-6mm

Shear strenth (kN)

1400 1200 1000 800 600 400 200 0

0

50

100

150

200

250

300

350

Compression stress (MPa) Fig. 2.29 Shear capacity each zone of the infill steel plate with different compression stress acting at the top of the boundary columns

2.4 Stresses Under Compression and Shear

53

of the shear capacity of the thinner one is smaller than the thicker one’s, these are about 9.79% for the 2 mm infill steel plate, 19.6% for the 4 mm infill steel plate, and 29.4% for the 6 mm infill steel plate.

2.4.4 Load-Carrying Capacity of the Steel Plate Shear Wall The shear strength of the SPSWs can be obtained adopting the same method that proposed by Park et al. [22]. The lateral load-carrying capacities for shear-dominated deformation mode and flexure-dominated mode were calculated, respectively. The minor value was defined as the lateral load-carrying capacity and the corresponding deformation mode was determined. See Sect. 2.2.6. However, to consider the effect of the gravity load, the shear capacity of the infill steel plate is determined separately by the three zones as Vsp = τx y Lt = σty,I L e t sin α cos α + σty,II (L − L e )t sin α cos α

(2.31)

which σty,I and σty,II are the yield strength of the tension filed strip for Zone I and III and Zone II respectively. L e is the effective width. The strengths of the specimens are predicted by the proposed method. Table 2.6 shows the simulation results “Push over”, predicted by Park’s method and the predicted results calculated in this paper, and it is shown that the shear capacity predicted in this paper are much closer to the simulation results.

2.5 Inclination Angle of the Steel Wall with Considering the Gravity Load 2.5.1 Assumption A typical story of a SPSWs building structure with ductile steel plates wall can be represented as an isolated panel, to calculate the angle of the inclination of the SPSW, the following assumptions can be made: a.

b. c. d.

the influences of the pre-compression from the boundary columns on performance of the infill steel plate are only in a limit part, out of the part, the compression stress on the infill steel plate is neglected. the unbalance of the stress of the infill steel plate along the boundaries of the pre-compression influenced part and uninfluenced part is neglected. columns are rigid enough to neglect their deformation when calculation the shear deflection of the steel plate. the difference in tension-field intensity in adjacent stories is small and therefore bending of the floor beams due to the action of the tension field is neglected.

f py

351

351

351

351

351

351

351

392

392

392

392

392

392

392

377

377

377

377

377

377

377

Specimen

SC2T000

SC2T050

SC2T100

SC2T150

SC2T200

SC2T250

SC2T300

SC4T000

SC4T050

SC4T100

SC4T150

SC4T200

SC4T250

SC4T300

SC6T000

SC6T050

SC6T100

SC6T150

SC6T200

SC6T250

SC6T300

56

61

68

79

97

137

0

37

41

46

53

64

91

0

19

20

23

26

32

46

0

d max (cm)

42

42

42

42

42

42

42

43

43

43

43

43

43

43

44

44

44

44

44

44

44

α (°)

300

250

200

150

100

50

0

300

250

200

150

100

50

0

300

250

200

150

100

50

0

max σv, p

178

238

283

318

345

364

377

198

255

298

333

359

379

392

127

197

247

285

315

336

351

σty,I

56

61

68

79

97

137

0

37

41

46

53

64

91

0

19

20

23

26

32

46

0

L e (cm)

1222

1125

1015

886

723

480

320

521

480

434

382

317

228

95

122

112

101

89

74

54

12

Pg,p (kN)

Table 2.6 The predict shear capacity of the specimens (Unit of stresses: MPa)

1641

1720

1782

1835

1880

1921

1973

1226

1258

1285

1307

1327

1345

1370

572

582

590

597

602

607

614

V sp (kN)

67

406

758

1114

1490

1894

2299

26

361

713

1098

1490

1894

2299

65

429

797

1145

1511

1900

2299

V ff (kN)

783

783

783

783

783

783

783

783

783

783

783

783

783

783

783

783

783

783

783

783

783

V sf (kN)

2424

2503

2566

2618

2664

2704

2757

2009

2041

2068

2091

2111

2129

2153

1356

1366

1374

1380

1386

1391

1397

V s (kN)

1708

2126

2540

2949

3370

3815

4272

1252

1619

1998

2405

2817

3240

3668

638

1012

1387

1742

2113

2507

2913

V f (kN)

1708

2126

2540

2618

2664

2704

2757

1252

1619

1998

2091

2111

2129

2153

638

1012

1374

1380

1386

1391

1397

V (kN)

1528

1838

2135

2424

2680

2869

3016

1399

1705

1965

2164

2328

2451

2542

772

1022

1193

1313

1406

1493

1565

Pushover

54 2 Steel Plate Shear Walls with Considering the Gravity …

2.5 Inclination Angle of the Steel Wall with Considering the Gravity Load

55

u

u beam

column

zone I αI Le/2

column

2V sp,I+V sf

V

zone II

zone III

H

=

2×zone I

αII L-Le

beam u V sp,II

H

αI Le/2

Le

+

beam

zone II

H

αII L-Le

Fig. 2.30 Partition and recombination of the infill plate

e. f. g. h.

the effect of stress due to flexural behavior on shear buckling stress of the steel plate is neglected. it is assumed that a uniform tension field develops across each zone of the steel plate. the behavior of steel plate and frame are elastic and perfectly plastic. the principle of superposition applies.

To account for the influences of the pre-compression on the angle of inclination, the infill steel plate is partitioned into three zones, zone I and zone III are the precompression influenced area with the width equal to L e /2, and zone II is the precompression uninfluenced area with width L − L e /2, as shown in Fig. 2.30.

2.5.2 Inclination Angles of the Infill Steel Plate According to Thorburn et al. [1], infill plates that buckle early under cyclic shear can be modeled by a series of pin-ended tension strips inclined at angle α. For the walls with rigidly connected the inclination angle of the tension field to the vertical axis can be determined by Timler and Kulak [23]. However, the expression doesn’t consider the compression of the boundary columns under gravity loads. In this paper, the SPSW is divided into three parts bounded on the half of the effective length of the infill steel plate, because of the same mechanical property and symmetrical layout of zone I and zone III, these two parts are reconfigured to form a new SPSW with the width equal to the effective length with four edges clamped supported by the columns and beams. Zone II is assumed to be four edges simply supported. For zone I and zone III, assuming the internal work performed by the bending deformation is assigned to zone I and zone III, and the energy within the frame consists of contributions from the web, one beam, and two columns. The total internal work performed by the panel when subjected to a tension field and gravity load on the boundary columns is expressed as, Wtotal = 2 × Wweb,I + Wbeam + Wcolumns

(2.32)

56

2 Steel Plate Shear Walls with Considering the Gravity …

The expressions of W web,I and W beam are the same as those proposed by Timler and Kulak [23] except the infill steel plate width is replaced by the effective length L e . Assuming that the gravity load on the boundary column is γ times the axial force caused by the tension field action of the whole infill steel plate, and is approximately expressed as, 2Pg tan α0 V

γ =

(2.33)

Then the internal work performed by the columns contributing to the axial stresses including the gravity load, W c,axial is expressed as Wc,axial =

(γ + 1)2 V 2 H 4 Ac E tan2 αI

(2.34)

The total internal work is Wtotal =

(γ + 1)2 V 2 H V2H V 2 H 2 tan2 αI V 2 H 5 tan2 αI + + + 2 Ab EL 2EAc tan2 αI 720EIc L 2 2Et L e cos2 αI sin2 αI (2.35)

Minimizing this relationship by taking the first derivative with respect to α and setting the resulting equation equal to zero gives,  αI = tan

−1

4



  −1 1 t Le H3 2 1 + tH + (γ + 1) + 2 Ac Ab 360Ic L 2

(2.36)

To predict the shear capacity of the SPSW, the parameter γ in this paper is approximately further defined as the ratio of the gravity load and the force produced by the tension field action, γ =

Pg f y,p t L cos2 α0

(2.37)

For a typical SPSW structure, the width to thickness ratio is very large, and the constrained of the infill steel plate by the boundary columns is very slight. Therefore, the influence of the gravity load from the adjacent boundary columns on the angle of inclination in zone II is neglected, the definition originally postulated by Wagner and Thorburn is used as, αII = tan

−1

 4

1+

t (L − L e ) 2 Ac

 1+

tH Ab

−1 (2.38)

2.5 Inclination Angle of the Steel Wall with Considering the Gravity Load

57

2.5.3 Discuss of the Inclination Angle

Angle of inclination from vertical (degree)

The inclination angle of the infill steel plate is discussed in this section. Choosing the SPSWs tested by Park et al. [22] as examples, the size of the infill plate was 1.5 m × 1.0 m, the boundary members were built-up sections made of SM490 steel. The beams at first and second stories were H-200 × 200 × 16 × 16 mm. The top beam connected to the actuator was H-400 × 200 × 16 × 16 mm. In this paper, only the strong column series of H-250 × 250 × 20 × 20 mm are simulated to verify the numerical precise. The infill plate thicknesses of the specimens were 2, 4 and 6 mm, and denoted as SC2T, SC4T and SC6T. Three more specimens with the thicknesses of the infill steel plate 1, 3 and 5 mm are analyzed. The detailed information and configurations of the test specimens can be found in the study by Park et al. [22]. The inclination angle of the specimens from the vertical direction under increasing pre-compression stress from the adjacent columns from 0 to 300 MPa are shown in Fig. 2.31. From Fig. 2.31, it is indicated that the pre-compression stress of the boundary columns has significant influences on the inclination angles, generally, the inclination angle increase with the increase of the pre-compression stress. For different thicknesses of the infill steel plate, the thinner one increases faster than that of the thicker one, because the parameter γ of the thinner infill steel plate increase faster than that of the thicker one.

75 70 65 60 55 50

1mm 3mm 5mm

45 40

0

50

100

150

200

2mm 4mm 6mm 250

Pre-compression stress (MPa) Fig. 2.31 The variation of the angle of inclination with the pre-compression stress

300

58

2 Steel Plate Shear Walls with Considering the Gravity …

2000

2*Vsp,I of 2mm 2*Vsp,I of 4mm 2*Vsp,I of 6mm

Shear strenth (kN)

1600

Vsp,II of 2mm Vsp,II of 4mm Vsp,II of 6mm

Vsp of 2mm Vsp of 4mm Vsp of 6mm

1200

800

400

0

0

50

100

150

200

250

300

Pre-compression stress (MPa) Fig. 2.32 The shear strength of zone I, zone II and infill steel plate with the pre-compression stress

Adopting the shear capacity prediction method, the shear capacity of each zone of the infill steel plate under increasing of pre-compression stress form the adjacent columns is shown in Fig. 2.32. It is indicated that the shear strength of zone I and zone III decrease fast with the increase of the pre-compression, meanwhile, the shear strength of zone II increase with increase of the pre-compression, and the summation of zone I, zone II and zone III, i.e., the total shear strength of the infill steel plate decrease with the increase of the pre-compression.

2.5.4 Load-Carrying Capacity Considering the Inclination Angle As proposed in the previous section, the inclination angles of zone I and zone III are defined by Eq. (2.36), and that in zone II is defined by Eq. (2.38). The shear strength of the SPSWs can be obtained adopting the same method that proposed in Sects. 2.2.6 and 2.4.4. However, the inclination angle of the tension strips is considered due to the effect of the gravity load, the shear capacity of the infill steel plate is determined separately by the three zones as

2.5 Inclination Angle of the Steel Wall with Considering the Gravity Load

59

Vsp = Vsp,I + Vsp,III + Vsp,II = f y,p L e t sin αI cos αI + f y,p (L − L e )t sin αII cos αII (2.39) The notations are the same as those in Sect. 2.2.6. The predicted shear strength are listed in Table 2.7.

2.6 Shear-Displacement Relationship 2.6.1 Shear-Displacement Relationship Under Compression and Shear 2.6.1.1

Pre-buckling

The shear-displacement relationship of a typical unstiffened SPSW is shown in Fig. 2.33. The load–displacement relationship of the SPSW is superimposed by those of the plate and boundary frame. The points A0 , B0 , C0 and D0 are corresponding to the buckling limit of the infill steel plate, the yield point of Zone I and Zone III of the infill steel plate, the yield point of Zone II of the infill steel plate, and the yield point of the boundary steel frame. The stress states of Zone I, Zone II and Zone III are the same as those presented in Sect. 2.4.1. The critical shear force of the web plate F wcr is Fwcr = τcr Ltw

(2.40)

Then the shear displacement Dwcr is Dwcr = H τcr /G

(2.41)

From Eqs. (2.40) and (2.41), the stiffness of the infill steel plate in the pre-buckling state is K wcr = Ltw G/H

2.6.1.2

(2.42)

After Buckling

In the post-buckling state, as proposed in the previous sections, the infill steel plate is divided into three parts. Tension field inclined at an angle α I to the vertical is assumed to gradually develops throughout Zone I and Zone III of the web plates, and angle α II develops throughout Zone II of the wed plate [26]. The shear force of

f py (MPa)

351

351

351

351

351

351

351

392

392

392

392

392

392

392

377

377

377

377

377

377

377

Specimen

SC2T000

SC2T050

SC2T100

SC2T150

SC2T200

SC2T250

SC2T300

SC4T000

SC4T050

SC4T100

SC4T150

SC4T200

SC4T250

SC4T300

SC6T000

SC6T050

SC6T100

SC6T150

SC6T200

SC6T250

SC6T300

56

61

68

79

97

137

0

37

41

46

53

64

91

0

19

20

23

26

32

46

0

d max (cm)

56

55

53

51

48

46

43

61

59

57

55

52

48

43

69

67

65

63

59

54

44

α (°)

56

61

68

79

97

137

0

37

41

46

53

64

91

0

19

20

23

26

32

46

0

L e (cm)

584

653

744

874

1085

1545

1973

250

282

326

389

491

710

1370

44

51

60

75

99

152

614

2V sp,I (kN)

Table 2.7 The predicted shear capacity of the specimens

1345

1285

1204

1085

885

433

0

1079

1051

1014

959

866

657

0

549

543

534

522

501

454

0

V sp,II (kN)

1719

1775

1818

1854

1886

1919

1973

1268

1287

1303

1317

1331

1346

1370

585

589

592

595

599

604

614

V sp (kN)

67

406

758

1114

1490

1894

2299

26

361

713

1098

1490

1894

2299

65

429

797

1145

1511

1900

2299

V ff (kN)

783

783

783

783

783

783

783

783

783

783

783

783

783

783

783

783

783

783

783

783

783

V sf (kN)

2715

2724

2734

2744

2755

2762

2757

2113

2118

2124

2132

2141

2151

2153

1376

1376

1378

1380

1383

1390

1397

V s (kN)

1999

2346

2708

3075

3461

3873

4272

1356

1695

2054

2446

2848

3262

3668

658

1022

1391

1741

2111

2506

2912

V f (kN)

1999

2346

2708

2744

2755

2762

2757

1356

1695

2054

2132

2141

2151

2153

658

1022

1378

1380

1383

1390

1397

V (kN)

67

406

758

1114

1490

1894

2299

26

361

713

1098

1490

1894

2153

65

429

797

1145

1397

1397

1397

Park et al.

1528

1838

2135

2424

2680

2869

3016

1399

1705

1965

2164

2328

2451

2542

772

1022

1193

1313

1406

1493

1565

Push (kN)

60 2 Steel Plate Shear Walls with Considering the Gravity …

2.6 Shear-Displacement Relationship

F

Fcr-buckling of infill plate Fy,II-yielding of Zone II

F y,f

C

F y,II

B

F y,I

F wy,I

A

F cr

F wcr

Fy,I-yielding of Zone I Fy,f -yielding of frame

D

Panel

Post-yield stiffness Post-buckling stiffness

C0

F wy,II

F fy

61

B0

Yield point of Zone II Zone II Yield point of Zone I and Zone III Zone I&III

D0

A0

Frame

Yield point of frame Plate critical buckling

O Dwcr

Dwy,I Dwy,II Dfy

D

Fig. 2.33 Shear load–displacement of frame, infill steel plate and panel

the web plate F wy is the summarization of F wy,I , F wy,II and F wy,III , in which F wy,I and F wy,III are

Fwy,I = Fwy,III = 0.5σxy,I L e tw = 0.5 τcr + 0.5σty,I sin2αI L e tw

(2.43)

F wy,II is

Fwy,II = σxy,II (L − L e )tw = τcr + 0.5σty,II sin2αII (L − L e )tw

(2.44)

The shear force of the web plate is Fwy = 2Fwy,I + Fwy,III

(2.45)

The limiting elastic shear displacement at which Zone I and Zone III of the infill steel plate yield is [27]

62

2 Steel Plate Shear Walls with Considering the Gravity …

 Dwy,I =

 2σty,I τcr H + E sin 2αI G

(2.46)

where the shear displacement from the post-buckling component of the shear forces is determined by equating the work done by the post-buckling component of the shear forces to the strain energy produced by the tension field. Similarly, the limiting elastic shear displacement at which Zone II of the infill steel plate yields is  Dwy,II =

 2σty,II τcr H + E sin 2αII G

(2.47)

The failure modes of the SPSWs can be classified into three types, i.e., the sheardominated mode, the flexure-dominated mode and the shear and flexure combined mode. Park et al. proposed [22] an evaluation method for the deformation mode of the SPSWs. They calculated the lateral load-carrying capacities by shear-dominated deformation mode and flexure-dominated mode separately, and the minor value is defined as the lateral load-carrying capacity. For the shear-dominated deformation mode, assuming that the beam-to-column connections are infinitely rigid and the plastic hinges are developed at the beams (it should be noted that the practical plastic hinges distribution should be used to calculate the shear strength of the boundary frame), the ultimate shear strength [22] of the boundary frame is calculated by the principal of virtual work by the following equation: Fsy = 4Mpb /H

(2.48)

where M pb is the plastic moment for the boundary beam. For one story steel frame, the plastic moments may occur at the end of the columns because of the strengthen of the beams. Considering the compression stress of the columns, the plastic moment of the boundary columns are M pc = σ m S pc , where S pc is the plastic section modulus of the column. The limiting elastic shear displacement [27] of the boundary frame at which the frame yields is calculated by the following equation: Dsy =

Mpb H 2 12ρ + 4 6E Ic 12ρ+1

(2.49)

where I c is the moment of inertia of the boundary column, ρ is stiffness ratio [22] of the boundary beams and columns, it is defined by the following equation:  E Ib /L ρ= E Ic /H

(2.50)

where I b is the moment of inertia of the boundary beam. Then the stiffness of the boundary frame is

2.6 Shear-Displacement Relationship

63

K sy =

24E Ic 12ρ+1 H 3 12ρ + 4

(2.51)

The shear capacity of SPSW with the shear-dominated deformation is Fs = Fsf + Fsp

(2.52)

For the flexure-dominated deformation mode, the shear capacity of the boundary frame is calculated by cantilever action as [22] Fff = Ac ( f y,c − σc )L/H − Pg δ/H

(2.53)

where Pg is the gravity load applied at the top of the specimen, δ is the lateral top displacement of the SPSW at the maximum shear strength, f y,c is the yield strength of the column, and σc is the axial stress of the column produced by the gravity load. The limiting elastic shear displacement of the boundary frame when the frame yields is Dfy =

Mfy 2 H E Ify

(2.54)

where M fy is the plastic moment of the boundary frame, it is defined as M fy = Ac (f y,c − σ c )L − Pg δ, and I fy = Ac L 2 . The overall shear capacity of the flexure-dominated deformation is defined as the sum of the flexural capacity and the shear capacity of the infill steel plate as Ff = Fff + Fsp

(2.55)

The predict shear capacity of the specimens is defined as the minimum of F s and Ff. Generally, the points of A0 , B0 , C0 , and D0 in Fig. 2.33 are defined, the shear force–displacement diagram of the SPSW under shear-compression interaction can be obtained.

2.6.2 Shear-Displacement Relationship Under Compression, Shear and Bending 2.6.2.1

Pre-buckling

Considering the global bending of the SPSW, the stiffness and shear capacity will shift from points A, B, C and D to A’, B’, C’ and D’, as shown in Fig. 2.34. Assuming the effect of the compression on the infill steel plate is negligible on the shear buckling of the infill steel plate. The shear buckling capacity of SPSW under

64

2 Steel Plate Shear Walls with Considering the Gravity …

F

D Shear-compression C Shear-compression-bending D'(D'fy,F'fy) C'(D' wy,II,F' y,II)

F y,f F y,II

B

F y,I

Considering bending effect˖ F'cr buckling of infill plate F' y,I yielding of Zone I F' y,II yielding of Zone II F' y,f yielding of frame

B'(D' wy,I,F' y,I)

F cr

No bending effect˖ F cr buckling of infill plate F y,I yielding of Zone I F y,II yielding of Zone II F y,f yielding of frame

A A'(D' wcr,F'cr)

O Dwcr

Dwy,I

Dwy,II

Dfy

D

Fig. 2.34 Load–displacement diagram of the SPSW under compression-shear-bending interaction

shear-bending interaction is defined by [24] 

τcr τcr

2

 +

σcr σcr

2 =1

(2.56)

where the critical bending stress of the infill steel plate under pure bending is [24] σcr = kb

π2E 12(1 − μ2 )(L/tw )2

(2.57)

where [24]  kb =

23.9 , for 1 ≤ HL ≤ 1.5 2 2 15.87 + 1.87 HL + 8.6 HL , for

H L

≥ 1.5

(2.58)

2.6 Shear-Displacement Relationship

65

The critical moment of the infill steel plate is [27] Mcr = σcr Se =

2σcr Ie L

Ie = Ic + Ac (L/2)2 + tw L 30 /12 

(2.59) (2.60)



where τcr and σcr are the stresses calculated by the applied shear and bending moment,  the applied moment Mcr is calculated by the shear force as  = Fcr H Mcr

(2.61)

The buckling shear load Fcr is calculated by Fcr = τcr Ltw

(2.62)

Taking Eqs. (2.61) and (2.62) into Eq. (2.56) gives  

Fcr =

1 + Fcr2



H Mcr

2 −0.5 (2.63)

Then the displacement of the SPSW during the buckling is  = Dcr

2.6.2.2

σ L H2 τcr H + cr G 2E

(2.64)

After Buckling

Neglecting the decrease of the compression stress in the tension zone of the infill steel plate under global bending for safety, the stresses of Zone I and Zone III of the steel plate under compression-shear-bending interaction are  sin2 αI σx x,I = σty,I   σ yy,I = σc + σb + σty,I cos2 αI  σx y,I = τcr + 0.5σty,I sin 2αI

(2.65)

 and σx y,I into Eq. (2.24), and assuming τcr Substituting equations of σx x,I , σ yy,I  = 0, then the value of σty,I at which yielding of the steel plate occurs in Zone I and Zone III is defined as the following equation:

66

2 Steel Plate Shear Walls with Considering the Gravity …  = σty,I

(σc + σb )(3 sin2 αI − 1) +

 2 ] [(σc + σb )(3 sin2 αI − 1)]2 − 4[(σc + σb )2 − f y,p 2

(2.66)

where σ b is the compression stress produced by global bending moment. The ultimate yielding state of the infill steel plate during pure bending [27] were studied by Kharrazi et al. Considering the compression acted at the top of the boundary columns, the moment that the plate begins to yield is My = σm Seff =

σm Ieff (1 − η)L

(2.67)

where σ m is the allowable maximum stress, which is defined as σm = f y,c − σc

(2.68)

where f y,c is the yield strength of the column, and σ c is the axial stress developed by gravity load. η is the ratio of the distance between the neutral axis and the tension column to the width of the infill steel plate L, when the axial stress is 0, η = 0.5, i.e. the neutral axis is at the middle of the plate, when the axial stress is f y,c , the neutral axis is at the edge of the plate, it is defined as η=

f y,c − σc 2 f y,c

(2.69)

S eff and I eff are the effective section modulus and effective moment of inertia of the SPSW [27] Seff =

Ieff (1 − η)L 0 + 0.5h c

1 Ieff = Ac (ηL)2 + tw (ηL 0 )3 + Ac (1 − η)2 L 2 3

(2.70) (2.71)

The drift of Zone I and Zone III under compression-shear-bending interaction can be obtained from the following equation   Dwy,I

=

 τcr M 2 + H H+ E sin 2αI G E Ieff  2σty,I

(2.72)

The influences of bending on the stresses state of Zone II are not considered, because the moment of inertia of Zone II is insignificant and little bending moment can be resisted. Therefore, the drift of Zone II under compression-shear-bending interaction can be obtained from the following equation:

2.6 Shear-Displacement Relationship



 Dwy,II =

67

 2σty,II M 2 τcr H+ + H E sin 2αII G E Ieff 

(2.73) 

The corresponding shear force of the web plate Fwy is the summarization of Fwy,I ,    Fwy,II and Fwy,III , in which Fwy,I and Fwy,III are 





 sin2αI )L e tw Fwy,I = Fwy,III = 0.5(τcr + 0.5σty,I

(2.74)



Fwy,II is the same to the shear capacity without considering the moment effect. That is

 Fwy,II = σxy,II (L − L e )tw = τcr + 0.5σty,II sin2αII (L − L e )tw

(2.75)

The shear force of the web plate is 





Fwy = 2Fwy,I + Fwy,II

(2.76)

The shear capacity of the boundary frame is also calculated based on Eqs. (2.48) and (2.53). Generally, the load–displacement diagram of the SPSW under compression-shearbending interaction is shown in Fig. 2.34.

2.6.3 Experimental Verification 2.6.3.1

Specimens and Test Setup

The primary test parameter of this paper was the axial load from the boundary columns. The specimens were one story walls, the overall height of the specimen was 0.75 m, and the overall width was 1.1 m, the columns were 1.0 m apart from center to center. The sizes and configuration of the test specimens are shown in Fig. 2.35. The plate thickness was 2.1 mm. It is made of Q235 steel with the yield strength of 255 MPa. The size of the infill plate was 600 × 900 mm. The frame members were built-up sections made of Q345 steel with the yield strength of 460 MPa. The boundary columns were H-100 mm × 100 mm × 6 mm × 8 mm, i.e. H-overall depth (d) × flange width (bf ) × web thickness (t w ) × flange thickness (t f ). The top beam connected to the actuator was H-150 mm × 100 mm × 6 mm × 9 mm, which was a relative stiff and deep beam to anchor the tension field occurred below the beam. Moment connections were used at all beam-to-column joints. Connection of the beam flanges to the columns was made using complete penetration groove welds. The beam webs were connected to the column flange by two-sided fillet welds. The infill steel plate was connected to the boundary members using the fish plate connection as shown in Fig. 2.36. The fish plates were 50 mm wide and 3 mm thick and welded to

68

2 Steel Plate Shear Walls with Considering the Gravity …

Axial load 15

120

135

15

30

Load distribution beam

50 150

Cycling load

3mm Fish plate 600×900×2mm Steel plate

200

50

500

H100×6×100×8

HM 150×6×100×9

H100×6×100×8

20

Pin

265

400

265

HM200×6×150×9 20

140

20

870

40

100

40

0.6

Fig. 2.35 Dimensions of specimens (units: mm)

2.0 mm infill plate

3.0 mm fish plate

50 Fig. 2.36 Fish plate (units: mm)

2.6 Shear-Displacement Relationship

69

the beams and columns with fillet welds on both sides. The infill steel plate was fitted against one side of the fish plates, with a lap of approximately 20 mm all around, and the infill steel plate and the fish plates were welded with continuous fillets on both sides. To investigate the influences of gravity loads acting at the top of the boundary columns on the cycling behavior of the unstiffened SPSWs. As shown in Fig. 2.35, the axial loads applied at the load distribution beam were 300, 600, 900 and 1200 kN, and kept steady for the whole testing process. The load distribution beam was hinge connected to the top of the boundary columns, then the axial force for each column of the specimens S1, S2, S3 and S4 were 150 kN, 300 kN, 450 kN and 600 kN, respectively. Horizontal loads were applied at the centerline of the top beam. The reactions of the hydraulic jack used to apply the horizontal load were resisted by the laboratory reaction wall. The vertical loads at the top of the load distribution beam were applied using hydraulic jack resisted by the stiff steel frame. Rollers were located between the steel frame and the hydraulic jack to avoid the shear force. Two linear variable differential transformers (LVDT) were installed at the base of the boundary column and at the center line of the top beam. Keeping the vertical loads constant, prior to reaching the yield point, single loading cycle leading to shears in the specimen of ±50, ±100, ±150, and ±200 kN, and three cycles of deformation control stage with the shear deformation of ±4, ±8, ±12 mm, and et al. were applied to conduct the cyclic behavior of the SPSWs. The layout of the test set up is shown in Fig. 2.37. The results of the coupon tests for the materials used in this test are presented in Table 2.8. Three coupons were tested for each material, and the average value is used for the further analysis.

2.6.3.2

Specimens’ Behaviour

Under pure gravity loads, no buckling occurred in specimens A, B and C. However, in the case of specimen D, horizontal buckling of the infill steel plate took place. The following described the behaviour of the four specimens. In the case of specimen A, until the top displacement reached 4 mm there was no buckling in the infill steel plate. The tension strips, formed from the lower left corner to the upper right corner, has an inclination angle near to 45°. During the first displacement cycle of 8 mm was applied, the first loud bangs occurred. In the subsequent cycles, these noises continued to occur. With an increase of the top displacement, various parts of the infill steel plate progressed to yield. A large residual deformation formed at the end of each pull or push loading with a further increase of residual deformation in the subsequent cycles. The first tear was detected in the upper left corner between the fishplate and the infill steel plate during the first 12 mm displacement load. The tear gradually increased to nearly 30 mm at the end of the first 12 mm cycles, as shown in Fig. 2.38. At the end of the second 12 mm displacement cycle, new tears were detected at the two lower corners. All the tears extended with

70

2 Steel Plate Shear Walls with Considering the Gravity …

Fig. 2.37 Layout of the test

Table 2.8 Results of tensile coupon test Steel

Nominal thickness (mm)

Actual thickness (mm)

Yield stress (MPa)

Ultimate stress (MPa)

Gauge length

Elongation at Yield rupture (%) strength ratio

Q345

6/8

6/8

460

567

225

21.2

0.81

Q235

2

2.1

255

375

225

18.5

0.68

an increase of the top displacement, but no new tear was observed. During the first 20 mm displacement cycle, the shear resistance of the specimen did not decrease, but the tears grew faster and the specimen was pushed over. The ultimate deformation at the top of the specimen reached more than 50 mm. The force resistance only dropped by about 15%. At the end of the test, all the tears extended to be more than 60 mm. The tension boundary column failed due to the rupture of the weld at the bottom of the column, as shown in Fig. 2.39. Meanwhile, the compression column experienced local buckling in the flange, and only a slight out-of-plane deformation was observed. In the case of specimen B, i.e. under 600 kN vertical load, no buckling or yielding was detected. Prior to reaching the yield displacement, four load cycles

2.6 Shear-Displacement Relationship

71

Fig. 2.38 Tear at the interface between fish and infill steel plates at the end of 12 mm cyclic load

with increasing magnitude from ±50 to ±200 kN were necessary to cause the first yielding. At the cyclic load of 150 kN, the first loud bang occurred. These noises also occurred during the unloading phase in the following cycles. However, tension strips first only occurred during the second cycle of the 2 mm level, as indicated by the diagonal lines in Fig. 2.40. The first tear was detected at the upper left corner at the end of the third cycle of 4 mm. The length of the tear was about 10 mm, and this tear gradually grew in the following cycles. At the first 6 mm displacement cycle, new tears of about 15 mm length occurred at the upper right and lower left corners. The length of the tear at the upper left corner extended to about 20 mm. At the end of the second 10 mm displacement cycle, a slight buckling at the support of the boundary column under compression was observed. The length of tear at the upper right corner extended to about 50 mm. At the second 12 mm displacement

72

2 Steel Plate Shear Walls with Considering the Gravity …

Fig. 2.39 At the end of the test

(a) Tear at the upper left corner

(b) failure at the support of the column in tension

2.6 Shear-Displacement Relationship

73

Fig. 2.40 Formation of tension strips in infill steel plate

cycle, as shown in Fig. 2.41, the upper left tear extended to nearly 60 mm, and the shear force resistance did not decrease. While the first 16 mm displacement pull load, the specimen reached a maximum resistance at the top displacement of 13 mm. The shear resistance then began to decrease. At the second 16 mm displacement cycle, the shear resistance decreased rapidly. The failure started at the support of the column under compression where a significant buckling and yield occurred. An out-of-plane deformation increased very fast, resulting in a loss of the in-plane shear resistance. In the case of specimen C, soft noises occurred during the application of the vertical load of 900 kN, even prior to the horizontal load. However, no buckling and yielding were detected. Prior to reaching the yield displacement, three load cycles with increasing magnitude from ±50 to ±150 kN were necessary to cause the first yielding. The first loud bang occurred at the cycle of 150 kN. The tension strips occurred while pushing. After unloading, a large residual deformation was observed. At the first 6 mm displacement cycle, a first vertical tear occurred at the upper left corner. In the following load cycle a new tear occurred at the upper right corner at the compression direction. At the first 8 mm displacement cycle, local buckling of the compression column appeared. The shear resistance did not decrease. At the second 8 mm displacement cycle, the tear at the upper left corner extended to about 40 mm. The global out-of-plane buckling occurred, and the shear resistance of the specimen decreased rapidly to less than 100 kN. Specimen C failed because of the global out-of-plane buckling, as shown in Fig. 2.42.

74

2 Steel Plate Shear Walls with Considering the Gravity …

(a) Tear at the upper left during the second 12 mm load cycle

(b) local buckling of the column in compression Fig. 2.41 Failure mode of specimen B

2.6 Shear-Displacement Relationship

75

Fig. 2.42 Damage to the end of the test of specimen C

(a) Tear at the upper left corner

(b) the failure mode

76

2 Steel Plate Shear Walls with Considering the Gravity …

In the case of specimen D, a soft noise occurred when the vertical load reached 900 kN, and no buckling and yielding observed in the specimen. The vertical load gradually increased to 1200 kN. The noises raised but there was no loud bang. Slight horizontal waves due to buckling of the infill steel plate occurred. The first bang occurred at the 50 kN horizontal load cycle. These noises occurred several times during each cycle in the following load cycles. After the 150 kN load cycle, a displacement loading was applied. The shear resistance capacity was stable at 2 and 4 mm displacement cycles, and there were no tear. At the 6 mm displacement cycles, the shear resistance began to decrease during the pulling. The reason was the anchorage of the column under compression failed (Fig. 2.43a). The failure mode was shown in Fig. 2.43b.

2.6.3.3

Shear-Displacement Relationship

Figure 2.44 shows the shear-top displacement relationships of the test specimens. It should be noted that specimen S1 was pushed over after three times of 20 mm cycles. In the early loading cycles, the infill steel plate behaved in an elastic manner. As the deformation increased, portions of the infill steel plate yielded, resulting in a gradually reducing stiffness. After significant yielding of the infill steel plate, unloading and reloading in the opposite direction produced a consistent and characteristic hysteresis pattern. Generally, the specimens showed large initial stiffness and load-carrying capacity. During cycling loading, severe pinching did not occur, and the steel plate walls dissipated considerable energy. The deformation capacity was also very excellent, the maximum story drift δ max /H for S1 was 7%, the maximum displacement at the point of LVDT reached 51.4 mm. However, the deformation capacity was significantly decreased with the increase of the axial load applied at the top of the boundary columns. The maximum story drifts for S2, S3 and S4 were 2.42%, 1.39% and 1.22%, the maximum displacements at the point of LVDT were 18.14 mm, 9.07 mm and 6.41 mm, respectively. It should be noted that the failure modes of specimens S3 and S4 were due to the global out of plane buckling, if the lateral deformation were constrained, larger shear load-carrying capacity would be reached. The test results at the yield point, maximum load, and displacement according to the maximum load of the specimens are presented in Table 2.9. The yield point (δ y , F y ) was defined based on the concept of equal plastic energy so that the area enclosed by the idealized elastoplastic envelope curve was the same as that enclosed by the actual envelope curve. The initial yield point (δ yi , F yi ) was defined as the time when the permanent residual deformation was developed by cyclic loading for the first time. The displacement δ p was defined as the top displacement of the specimens at the maximum load. From Table 2.9, it is indicated that the maximum load-carrying capacity of the specimens decrease with the increase of the axial loads of the boundary columns. Specimen S1 experienced excellent deformation capacity, even the top displacement reached 51 mm, the load-carrying capacity only decreased about 15%. Specimen S2 behaved similar to S1 before the 16 mm cycles, during

2.6 Shear-Displacement Relationship

77

Fig. 2.43 Failure mode of specimen D

(a) Fracture at the support of the column under compression

(b) global out-of-plane buckling of the specimen

78

2 Steel Plate Shear Walls with Considering the Gravity …

Fig. 2.44 Shear-top displacement relationships of test specimens

350

S1

Load (kN)

300 250 200 150 100 50

-20

-10

0 -50

0

10

20

30

40

50

60

Top displacement (mm)

-100 -150 -200 -250 -300 -350

350 Load (kN) 300

S2

250 200 150 100 50 -20

-15

-10

-5

0 -50

0

5

10

15

20

Top displacement (mm)

-100 -150 -200 -250 -300 -350

the 16 mm cycle, the specimen experienced unstable and decreased load-carrying capacity. The failure of S2 was due to the buckling of the column base and large tears at the corners of the infill steel plate. For specimens S3 and S4, the initial stiffness were 88.1 and 57.8 kN/mm in the positive loading direction, and 79.5 and 62.1 kN/mm in the negative loading direction. The load-carrying capacity of S3 and S4 were much smaller than specimens S1 and S2. One reason was that the axial force acted at the top of the boundary columns decreased the yielding stress of the infill steel plate. The other reason was both specimens S3 and S4 were failed due to the out of plane buckling of the compressed column, and the large axial force acting at the top of the deflected boundary columns promoted the failure process.

2.6.3.4

Shear-Displacement Diagram for SPSWs

The envelope curves of the shear-top displacement relationships of the specimens are shown in Fig. 2.45. Figure 2.45 indicates that specimens S1 and S2 showed stable ductile behavior which the load-carrying capacity continuously increased with the increase of the lateral displacement, while specimens S3 and S4 showed gradually

2.6 Shear-Displacement Relationship

79

Fig. 2.44 (continued)

350 Load (kN) 300

S3

250 200 150 100 50 -20

-15

-10

-5

0 -50

0

5

10

15

20

Top displacement (mm)

-100 -150 -200 -250 -300 -350 350

S4

Load (kN)

300 250 200 150 100 50

-20

-15

-10

-5

0 -50

0

5

10

15

20

Top displacement (mm)

-100 -150 -200 -250 -300 -350

decreasing load-carrying capacity after the maximum load. These two specimens failed quickly due to the global out of plane buckling of the columns. Based on the formulations, the inclination angles of Zone I and Zone II, the compression stress distribution of the infill steel plate, the effective length of the infill steel plate, the yielding stress of the tension strips of Zone I and Zone II of the infill steel plate, and finally the shear force and displacement of the SPSW were determined. The calculation process is shown in Table 2.10. From Table 2.10, it is indicated that the predicted shear strength of specimen S1 is near to the test result, specimen S2 is lower than the testing result, while the predicted strengths of specimens S3 and S4 are larger than the testing results. The reasons are specimens S3 and S4 failed due to the global out of plane buckling, and this phenomenon cannot be considered in the proposed model. Comparing the predicted shear-displacement diagram and the envelope curves with the testing results in Fig. 2.39, it is shown that the stiffness of the predicted model is higher than that of the testing results, especially the stiffness prior to the yield of Zone II of the infill steel plate. The stiffness after the yielding of Zone II is similar to that of the testing results. Figure 2.46 shows the shear load–top displacement diagrams of specimens S1 and S2. It is indicated that considering the global bending of the SPSW will decrease the

334.0

320.5

230.9

146.9

S2

S3

S4

F max (kN)

3.49

8.10

16.65

18.32

δ max (mm)

Positive loading

S1

Specimen

Table 2.9 Test results

146.9

197.3

247.2

257.2

F y (kN)

3.49

4.7

6.53

5.91

δ y (mm)

51.4

55.5

102.2

80.0

F yi (kN)

0.89

0.63

0.95

0.93

δ yi (mm)

190.5

239.0

318.8

334.5

F max (kN)

5.73

6.94

18.03

20.15

δ max (mm)

Negative loading

151.3

190.6

265.3

246.0

F y (kN)

3.83

4.63

6.21

6.1

δ y (mm)

50.3

50.9

100.9

80.0

F yi (kN)

0.81

0.64

0.99

0.86

δ yi (mm)

80 2 Steel Plate Shear Walls with Considering the Gravity …

2.6 Shear-Displacement Relationship

Test-S1 Test-S2 Test-S3 Test-S4 Predict-S1 Predict-S2 Predict-S3 Predict-S4 -20

-15

-10

81

350 300 250 200 150 100 50 0

-5

Top displacement (mm) 0

5

10

15

20

-50 -100 -150 -200 -250 -300 -350 Fig. 2.45 Envelope curves of test specimens and predicted curves

Table 2.10 Comparison of the predicted shear capacity of the specimens with the test results Specimen

δ (mm)

α (°)

L e (mm)

2V sp,I (kN)

V sp,II (kN)

V sp (kN)

V ff (kN)

V sf (kN)

V s (kN)

V f (kN)

Test (kN)

S1

20.2

41.7

238

36

204

240

943

97.7

338

1180

334

S2

18.0

38.0

207

14

212

226

806

80.6

307

993

320

S3

8.1

35.0

192

0

217

217

635

63.5

280

849

249

S4

5.73

32.7

192

0

222

222

464

46.4

263

658

191

stiffness of SPSWs, and considering the effects of the bending stress on the tension yield stress of Zone I and Zone III of the infill steel plate will decrease the shear capacity of the SPSWs. Therefore, the shear-top displacement diagram considering both the compression-shear-bending interaction and decrease of the shear capacity of Zone I and Zone III of the infill steel plate is the most reasonable model, which can predict the shear capacity and yield deformation most precisely.

82

2 Steel Plate Shear Walls with Considering the Gravity … 360

Shear capacity (kN)

300

240

180

S1-compression-shear interaction S1-compression-shear-bending interaction S1-compression-shear interaction and strength decrease of panel S1-compression-shear-bending interaction and strength decrease of panel S2-compression-shear interaction S2-compression-shear-bending interaction S2-compression-shear interaction and strength decrease of panel S2-compression-shear-bending interaction and strength decrease of panel

120

60

0

0

2

4

6

8

10

12

14

16

Top displacement (mm) Fig. 2.46 Effects of global bending and bending compression stress on the shear capacity of specimens S1 and S2

2.7 Summary This paper investigates the shear capacity of unstiffened steel plate shear walls (SPSWs) with gravity loads from the adjacent boundary columns. The distribution of gravity loads between boundary columns and the infill steel plate are proposed, and a modified frame-plate interaction model based on the reduced yielding stress of the infill steel plate were proposed and verified. The shear-load displacement diagram and shear capacity of each zone are proposed and discussed. Four scaled SPSW specimens were tested to verify the feasibility of the proposed analytical model. The findings obtained in this study are summarized as follows. (1)

(2)

Based on the post plastic buckling strength of the infill steel plate and effective length concept, the infill steel plate is partitioned into three zones, for zone I and zone III, the compression stress of the infill steel plate is assumed to equal to the axial stress of the boundary column, and for zone II, the compression stress of the infill steel plate is elastic buckling stress. Then the compression stress distribution of the infill steel plate is proposed. Simulation results match well with the theoretical analysis results. The SPSWs with larger axial stress on the top of the boundary columns have narrower effective length. The compression force sustained by the infill steel plate increase with the increase of the axial stress of the boundary columns and the thickness of the infill steel plate.

2.7 Summary

(3)

(4)

(5)

(6)

(7)

(8)

83

Accounting for the tension–field action and reduction of the yielding stress of the infill steel plate, better shear strength prediction of the SPSWs with gravity loads acted at the boundary columns is obtained. The inclination angle of zone I and III measured from the vertical direction is larger than that of zone II, because the compression stress in zone I and III is greater than that of zone II. Shear-displacement diagram of SPSW under compression-shear interaction is obtained, and the load-carrying capacity and deformation according to the buckling of infill steel plate, yield of Zone I and Zone III, yield of Zone II and yield of the boundary frame are presented and verified. Comparing with the test results, the proposed analytical model will overestimate the stiffness of the SPSW, the shear load capacity of the SPSW will decrease with the increase of the axial load acted at the boundary columns. The failure mode of specimens S3 and S4 is due to global out of plane buckling, which is different from specimens S1 and S2, and can’t be predicted by the proposed model. The shear stress gradually decreases to zero with the compression stress increasing from 0 to yield stress, it is strongly recommended that the shear capacity and stiffness should be reduced by a factor if the gravity loads on the infill steel plate cannot be avoided during the construction of the SPSW structures. This factor mainly influenced by the depth-thickness ratio of the infill steel plate and the magnitude of gravity loads on the infill steel plate.

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Chapter 3

Steel Plate Shear Walls with Considering the Gravity Load by a Three-Segment Distribution

From the study in Chap. 2, the shear strength reduction not only influenced by the load magnitude but also the vertical stress distribution. Therefore, the vertical stress distribution is important in the determination of its shear capacity, e.g. under a wind load or an earthquake load. For a simply-supported thick square wall, i.e. width to thickness ratio smaller than 100, the stress distribution can be accurately described in a cosine form. However, for a thin wall under compression and in-plane bending, the cosine distribution will largely overestimate the vertical stress, especially when the walls enter the post-buckling condition. In this case, a cosine vertical stress distribution is no longer valid. In this chapter, firstly, a three-segment distribution under uniform compression is proposed. Secondly, the proposed three-segment distribution is improved to consider the gravity load and in-plane bending simultaneously. Then three vertical stress distributions, i.e. cosine distribution, effective width model and the three-segment distribution, of steel wall under this compression were assumed to consider the negative effect of the gravity and in-plane bending. To evaluate the proposed approach, an experimentally verified finite element model using the software LS-DYNA is developed. Steel walls with different gravity load and in-plane bending combination loadings (considered as stress gradient) and slenderness ratios are considered.

3.1 Introduction Steel shear walls have been proven to be an excellent lateral load resisting system. Previous pioneering research by Thorburn et al. [1] indicated that unstiffened steel wall has high ductility and strength even after buckling. Thereafter, many researchers began to study thin walls. Such as Driver et al. [2], Behbahanifard [3], Moghimi and Driver [4], Qu et al. [5] experimentally studied the wall performance under cyclic loadings. Other research focused on the effects of beam-to-column connections [6], the self-centering devices [7–9], the dynamic response [10], the behaviour of © Science Press 2022 Y. Lv, Steel Plate Shear Walls with Gravity Load: Theory and Design, https://doi.org/10.1007/978-981-16-8694-8_3

85

86

3 Steel Plate Shear Walls with Considering the Gravity Load …

perforated walls [11], the light-gauge walls [12], the unstiffened low yield point steel walls [13], the effects of bolted versus welded connections [14], the bucklingrestrained steel walls with inclined slots [15], and self-centering walls [16, 17]. Kang et al. [18] presented a comprehensive state-of-the-art review. However, the influence of gravity load on the shear capacity was not much considered. As shown in existing research [19–22] and practical projects [23–25], the walls always take a large amount of the gravity load. The highest steel wall building is the Tianjin World Financial Center (also called Jinta tower) [24] in Tianjin, China. The 75-storey building has a height of 336.9 m. In the construction process, the steel walls were installed 15 storeys slower than the outer frames. Consequently, it will take a lot of the gravity load. Another example was the Hyatt Regency Hotel [25] in Texas, USA. The steel walls were designed to take about 60% of the gravity load. Because of the load distribution by the boundary beams, the gravity load produces uniform compression on the wall. The research results by Lv et al. [19–21] and Zhang and Guo [22] showed that the shear capacity was significantly impaired by the gravity load. However, the shear wall used in the buildings [23–25] was designed without considering the gravity load influence. Consequently, the shear capacity is overestimated, and it can lead to potential danger. To find out a reduction effect of the gravity load on the shear capacity, the first step is to determine the stress distribution of a simple rectangular thin four-edgeclamped wall under uniform compression. If the axial forces of the boundary columns are different, the wall will concurrently experience uniform compression and inplane bending. Because the relationship between load and deformation is nonlinear, the wall experiences elastic buckling, post-buckling, and failure stage. An elastic buckling has been examined more than 100 years. The theoretical solution is accurate enough and has been applied frequently [26]. Previous post-buckling research mainly focused on the finding of theoretical, semi-theoretical and numerical solutions. A theoretical prediction of the load-bearing capacity requires a simultaneous solving of the compatibility and equilibrium equations. To avoid a difficulty in directly solving the two equations, researchers often assume an out-of-plane deflection. The two basic equations can then be expressed by the deflection and solved using Galerkin method, Ritz method [27], energy method [28], finite strip method [29] or superposition method [30]. The most widely postulated deflection is a cosine buckling shape. For loads smaller than twice of the buckling load, the buckling shape changes only a little with the load. However, analytical methods can only be used to deal with simplified problems like a simply-supported square wall. Bakker et al. [31] used a two-strip model to describe the post-buckling behaviour. Below twice of the buckling load, the authors obtained a lower bound of the load-bearing capacity. Later, more complicated boundary conditions were considered. For example, Scheperboer et al. [32] used a finite element method to examine a simply-supported wall subjected to a uniaxial compression. Komur and Sonmez [33] used a finite element method to study the buckling of perforated square and rectangular walls subjected to in-plane compressive load. Byklum et al. [34] proposed a computational model for a global post-buckling analysis of stiffened panels. Except the methods mentioned above, the most used formulation is the effective width model since it was firstly proposed by

3.1 Introduction

87

von Karman et al. [35] in 1932. After Karman’s model, Abdel-Sayed [36], Narayanan and Chow [37], Vilnay and Rockey [38], Bedair [39], and others proposed models for considering different walls like steel deck-plate in bridges, perforated walls and different steel channels. The effective width model is also adopted in the codes of CAS [40] and AISC [41]. A theoretical solution is suitable for simply-supported square walls with a slenderness ratio less than 100 or a post-buckling load less than three times of the elastic buckling stress. However, the most common slenderness [23–25, 42] used in steel walls ranges from 200 to 750 and the post-buckling load can be ten times larger than the buckling load. Also, if a lateral load is applied or the gravity loads of the boundary columns are different, the wall will experience compression and in-plane bending loads. Based on the buckling and post-buckling behaviour from previous research, a three-segment stress distribution is proposed, i.e., at two edge segments a cosine distribution from edge stress to buckling stress and in the middle segment a constant distribution of buckling stress. Furthermore, the proposed three-segment distribution is improved to consider the uniform compression and in-plane bending. Then the three vertical stress distributions, i.e. cosine distribution, effective width model and three-segment distribution, of steel wall under this compression were assumed to consider the negative effect of the gravity and in-plane bending.

3.2 The Three-Segment Distribution Under Uniform Compression 3.2.1 The Cosine Distribution Under Uniform Compression Figure 3.1 shows a typical steel wall under a gravity load G and the corresponding simplified mechanic model. In Fig. 3.1a, the dashed-dotted and solid lines respectively represent the deformation before and after the gravity load. Because of the load distribution by the boundary beams, part of the gravity load is sustained by the infill wall which conversely makes it experience a uniform compression p (see Fig. 3.1b). If the slenderness ratio is very large, the wall can enter the post-buckling stage. In fact, the most common slenderness [23–25] ranges between 200 and 500. FEAM 450 [42] suggested that it should not exceed 750. The commonly considered slenderness ratio will lead to a classic elastic post-buckling problem. Prior to buckling the compressive stresses are equally distributed. After buckling non-uniformly distributed stresses occurred because of the stiffness loss and stress redistribution. Due to the higher stiffness near the edges, the stresses at the edge portions are higher than that in the middle part. The theoretical stress distribution [26] can be obtained by solving the equilibrium and compatibility equations simultaneously. These two differential equations can be written in the following forms

88

3 Steel Plate Shear Walls with Considering the Gravity Load …

Fig. 3.1 A typical steel wall under gravity load

(a) deformation before and after the gravity load

(b) rectangular wall under uniaxial compression ∂4 F ∂4 F ∂4 F +2 2 2 + =E 4 ∂x ∂x ∂y ∂ y4



∂ 2w ∂ x∂ y

2

∂ 2w ∂ 2w − − ∂ x 2 ∂ y2



∂ 2 w0 ∂ x∂ y

2

∂ 2 w0 ∂ 2 w0 + ∂ x 2 ∂ y2



(3.1) ∂ 4w ∂ 4w ∂ 4w + 2 + ∂x4 ∂ x 2∂ y2 ∂ y4

3.2 The Three-Segment Distribution Under Uniform Compression

  t ∂ 2 F ∂ 2 (w + w0 ) ∂ 2 F ∂ 2 (w + w0 ) ∂ 2 F ∂ 2 (w + w0 ) + = −2 D ∂ y2 ∂x2 ∂ x∂ y ∂ x∂ y ∂x2 ∂ y2

89

(3.2)

where F is a function describing the membrane stress, w and w0 are the out-of-plane and initial deflections of the middle plane, D is the flexural rigidity factor, E is the Young’s modulus, x and y are the coordinates, and t is the thickness. One popularly used method to approximately solve these two equations is to assume a deflection w that satisfies the boundary conditions. Therefore, the accuracy of the solution depends significantly on the assumed deflection w. x* and y* are respectively the non-dimensional Cartesian coordinate along the length and width used in Sects. 3.4 and 3.5. The most widely used deflection function w and initial imperfection w0 are w = f sin

 mπ x

w0 = f 0 sin

a

sin

 mπ x a

πy

sin

b πy b

(3.3) (3.4)

where f and f 0 are the maximum deflections of the middle plane, a and b are respectively the height and width, m is the number of half-sines in the longitudinal direction. By solving the two basic equations simultaneously, the vertical stress along the width b changes following a cosine function and can be obtained as follows σcos (x) = p +

2m 4 ( p − σcr ) 2π x cos 4 4 β +m b

(3.5)

where p is the uniform compression at the loaded edge (see Fig. 3.1b), σ cr is the buckling stress under compression, and β = b/a is the aspect ratio. Equation (3.5) is called cosine distribution hereafter.

3.2.2 The Proposed Stress Distribution Under Uniform Compression As expressed in Eq. (3.5), after buckling the stress distribution along the width changes following a cosine function (see Fig. 3.2a). This analytical result is accurate enough for a square wall with the slenderness ratio below 100 or an edge stress lower than 3 times of the buckling stress. In the case of a heavily loaded thin wall, with an increase of the compression the vertical stress distribution along the width no longer follows a cosine function. Instead, a constant distribution close to buckling stress appeared in the middle portion. Equation (3.5) will also significantly overestimate the load-bearing capacity. The error increases with slenderness and compression load. In this work, a three-segment stress distribution is proposed (see Fig. 3.3). For

90

3 Steel Plate Shear Walls with Considering the Gravity Load …

Fig. 3.2 Distribution models

Fig. 3.3 Schematic of the proposed stress distribution

each considered portion the distribution is expressed by the following equations   ⎧ σe + σcr σe − σcr x ⎪ ⎪ , 0 ≤ x ≤ b1 + cos π ⎪ ⎪ 2 2 b1 ⎪ ⎨ σ (x) = σcr , b1 < x ≤ b − b1 ⎪   ⎪ ⎪ σe − σcr x −b σe + σcr ⎪ ⎪ ⎩ , b − b1 < x ≤ b + cos π 2 2 b1

(3.6)

where b1 is the edge portion width of the wall, σ e is the stress of the unloaded edge and should be smaller than the yield stress f y (see Figs. 3.2 and 3.3). It can be calculated through Eq. (3.5) by taking y = 0 or y = b. The two edge portions can be combined to form a new wall with a width equal to 2b1 and unchanged height a. The stress distribution in this part is then a full cosine function (see Fig. 3.3b). The stress of the middle portion is assumed to be constant and equals to the buckling stress of the original wall σ cr (see Fig. 3.3c). A cosine distribution (see Eq. (3.5)) and an effective width model [35] are also respectively shown in Fig. 3.2a, b. To determine b1 in Eq. (3.6), it is assumed that the load calculated by the proposed distribution is the same as that of the effective width model. The following equation then can be obtained

3.2 The Three-Segment Distribution Under Uniform Compression

91

b1 σe be = 2

σ (x)d x + (b − 2b1 )σcr

(3.7)

0

By substituting Eq. (3.6) into Eq. (3.7), b1 can be obtained as follows b1 =

σe be − bσcr σe − σcr

(3.8)

where be is the effective width. When b1 is 0.5b, the proposed distribution is then the same as the cosine distribution, i.e. the cosine distribution is the upper bound of the proposed distribution. For an extremely thin wall, the buckling stress σ cr is very small and can be assumed to be 0. In such a case, b1 equals to be . The effective width model assumes that the maximum edge stress acts uniformly over two strips with each the width of be /2 and the central portion is unstressed (see Fig. 3.2b). In this work, the von Karman’s equation [35] is used to determine be in Eq. (3.9)  be =

σcr ×b σe

(3.9)

where b is the width of the wall. σ cr is the buckling stress of a four-edge clamped wall under compression [26] σcr = kcr

π2E 12 (1 − μ2 )(b/t)2

(3.10)

where E and μ are respectively the modulus of elasticity and Poisson’s ratio, t is the thickness. k cr is the buckling coefficient [26] and has the value of  2  a 2 b kcr = 3.6 + 4.3 + 2.5 a b

(3.11)

Through Eqs. (3.8), (3.9) and (3.10), the proposed distribution is determined. Choosing a square wall of 1200 mm as an example, Fig. 3.4 shows the relative width by dashed-lines (b1 /b, right ordinate) and load by solid lines (left ordinate) under different edge stress σ e . The green dots represent the buckling points. For the thickness of 2 mm, 4 mm, 6 mm, and 8 mm, the slenderness ratio is 600, 300, 200 and 150, respectively. Under increasing edge stress σ e , it shows that prior to buckling, b1 = 0.5b. In this condition, the proposed distribution is the same as the cosine distribution. After buckling, the width decreases following a square root function of the edge stress (see Eqs. (3.8) and (3.9)). The width of the edge portion increases significantly with the thickness. When the edge stress is 200 MPa, the relative widths are 0.142, 0.249, 0.332 and 0.398 for the thickness of 2 mm, 4 mm, 6 mm, and 8 mm, respectively.

92

3 Steel Plate Shear Walls with Considering the Gravity Load …

Fig. 3.4 Relative width of the edge portion and load of the proposed distribution with edge stress

Keeping the width and height constant, the force increases linearly with the thickness. The load is very easy to obtain as  F = be σe = kcr

π 2 Eσe 12 (1 − μ2 )

0.5 t

(3.12)

Obviously, the load is proportional to the thickness t. Also, for a determined wall, the load is linear to the square root of the edge stress σ e . When the edge stress is 200 MPa, the load of the wall with the thickness of 2 mm, 4 mm, 6 mm, and 8 mm is 39.74 kN, 79.48 kN, 119.22 kN, and 158.95 kN, respectively. When the edge stress is 50 MPa, 100 MPa, 150 MPa, and 200 MPa, the load of the 6 mm wall is respectively 59.61 kN, 84.30 kN, 103.24 kN and 119.22 kN, which is linear to the square root of the edge stress.

3.2.3 Numerical Analysis 3.2.3.1

Finite Element Model

A finite element (FE) model was developed using the software ANSYS/LS-DYNA [43]. The results obtained from the verified numerical model are then used in the evaluation of the proposed distribution in a parametric study.

3.2 The Three-Segment Distribution Under Uniform Compression

93

In the FE model the wall is described by a layered shell element SHELL 163. Each element is divided into several layers along the thickness. The calculation starts with the determination of the strain and curvature of the neutral layer. Based on the planesection assumption, the strain and curvature of the other layers are determined. The internal force of the element is then integrated through the thickness. Alinia and Shirazi [44] considered the initial imperfection from 1/100,000 to 1/500 of the wall thickness and observed only a little difference of the results. The reason is that the wall is very slender. It buckled very early and therefore initial imperfection does not have a considerable influence on its nonlinear post-buckling behaviour. Consequently, the initial imperfection is also not considered in this work.

3.2.3.2

Verification of the FE Model

An experiment, carried out by the authors, on a steel wall with four-edge clamped by H-section beams under compression was used to verify the FE model. Figure 3.5 shows the dimensions of the test specimen. The wall thickness was 1.95 mm. The sections of the lateral boundary columns and top beam were respectively 100 mm × 100 mm × 6 mm × 10 mm (height × width × web × flange) and 150 mm × 100 mm × 6 mm × 10 mm. The base beam was fully constrained to the strong floor. A vertical load was applied at the top beam by a hydraulic actuator. The load was evenly distributed to the boundary columns. Two strain gauges were glued on the wall surface (see P1 and P2 in Fig. 3.5) to estimate the strains. The position of the strain gauges was 40 mm from the edge to avoid the location of a stress concentration. The yield stresses of steel plate and frame were respectively 305 MPa and 392 MPa and both have a Young’s modulus and Poisson’s ratio of 210 GPa and 0.3. The applied vertical load was respectively 100 kN, 200 kN, 300 kN, 400 kN and 500 kN, i.e. the axial stress of the boundary columns is respectively about 20 MPa, 40 MPa, 60 MPa, 80 MPa and 100 MPa. Figure 3.6 shows the strains obtained from the test (dots) and FE analysis (lines). Relatively good prediction was obtained. However, only two strain gauges can be used in the above experiment. It is hard to verify the strains of the relative width x* between 0.1 and 0.4. Another experiment of a wall by Zara´s et al. [45] was also adopted to verify the FE model. The wall has a width, height, and thickness of 250 mm, 350 mm and 1.06 mm, respectively. The Young’s modulus, Poisson’s ratio and yield stress were 208 GPa, 0.3 and 180 MPa, respectively. All the four edges were simple-supported. In Fig. 3.7, σ x * is the non-dimensional compressive stress. σ 1 * is the non-dimensional compression. x* and y* are the non-dimensional Cartesian coordinate along the length and width, respectively. A detailed definition of the notations can be found in [45]. Foil gauges were glued at the middle height on both surfaces to estimate the average stress under compressive load. The results obtained from the FE simulation (lines) were compared with those obtained from the experiment (dots). The comparison shows that the FE model can predict the stress reliably.

94 Fig. 3.5 Set-up of the experiment (unit: mm)

3 Steel Plate Shear Walls with Considering the Gravity Load …

3.2 The Three-Segment Distribution Under Uniform Compression

95

Fig. 3.6 Comparison between the results obtained from FE analysis and experiment

Fig. 3.7 Comparison with the test results by Zara´s et al. [45]

3.2.3.3

Parametric Analysis

Steel walls with different widths and thicknesses were chosen. To verify the applicability of the proposed distribution the results obtained from the FE model verified in Sect. 4.2 are used. The wall height of 1200 mm is constant. A varying widths of 1200, 1800 and 2400 mm are considered. The respective width to height ratio β is

96

3 Steel Plate Shear Walls with Considering the Gravity Load …

Fig. 3.8 FE model

Table 3.1 Configurations of the considered walls

Configuration Width b (m) Height a (m) Thickness t (mm) 1

1.2

1.2

2/4/6/8

2

1.8

1.2

2/4/6/8

3

2.4

1.2

2/4/6/8

then 1, 1.5 and 2, which are the most used ratio in practice. The considered thicknesses are 2, 4, 6 and 8 mm. Consequently, the height to thickness ratio is between 150 and 600. The walls were fully constrained at the bottom. For the vertical and top boundaries, all the degrees of freedom are constrained except the vertical one (see Fig. 3.8). A load is distributed uniformly along the top boundary. An elastic–plastic material model with kinematic hardening is used to describe the steel behaviour. The yield stress, Poisson’s ratio, elastic modulus, and tangent modulus are 235 MPa, 0.3, 210 GPa, and 2.1 GPa, respectively. The configuration of the analyzed cases is listed in Table 3.1.

3.2.3.4

Compressive Stress Distribution

The distribution of the vertical stress of each element along the middle height strip is considered (see Fig. 3.8). For the walls with a width of 1.2 m, 1.8 m, and 2.4 m, 40, 60 and 80 elements are used, respectively. The relative coordinates in the horizontal and vertical directions are defined respectively as x* = x/b and y* = y/a. The stress was normalized by the buckling stress σ cr σ∗ =

σ σcr

(3.13)

3.2 The Three-Segment Distribution Under Uniform Compression

97

For the thicknesses of 2 mm, 4 mm, 6 mm, and 8 mm, the slenderness ratios are respectively 600, 300, 200 and 150. The stress distributions along the width under an increasing edge compression are shown in Figs. 3.9, 3.10 and 3.11. No matter how large the compressive stress is, the proposed distribution can predict the vertical stress well. In the middle portion, the proposed distribution slightly underestimated the stress because the buckling stress obtained from FE analysis changes with the loading. In the edge portions, because the wall was fully clamped by the boundary columns, slight warping occurred under large compressive stress. The prediction is a little different from that obtained from FE analysis. The proposed approach can predict the stress distribution of 1.8 and 2.4 m cases better than that of 1.2 m. From Fig. 3.9, the width of the edge portion of the proposed distribution is underestimated. With an increase of the compression load σ e to about half of the maximum load, the proposed distribution can predict the vertical stress properly. Beyond it, the edge portion is severely buckled, and the maximum vertical stress transferred toward the nearby element. When the load approaches the ultimate stress about 200 MPa, the buckling mode will change from one-half sine to wrinkling. However, this is not considered in this work. In the case of different slenderness ratios, the accuracy of the predicted results changes little. The cosine distribution with the thickness equal to 8 mm is also shown in Figs. 3.9d, 3.10d and 3.11d. It can be seen the cosine distribution is similar to that of the proposed distribution when the slenderness, or the non-dimensional stress, i.e. the ratio of the edge stress to the buckling stress, is small. The width of the middle-flattened segment of the proposed distribution is then 0. With an increase of the width b, the flattened portion becomes larger. Therefore, for the rectangular wall, the cosine distribution cannot predict the stress distribution properly.

3.2.3.5

Effectiveness of the Proposed Approach

To examine the application scope of the proposed distribution, the cosine distribution and the effective width model the following standard deviation is used   N 1  [σ (i, j) − σFE (i, j)]2 d( j) =  N i=1

(3.14)

where σ FE (i, j) is the stress of the ith element at the jth time step obtained from the FE analysis, σ (i, j) is the corresponding stress predicted by the proposed distribution, cosine distribution or effective width model. For the width of 1.2 m, 1.8 m, and 2.4 m, N equals to 40, 60 and 80, respectively. Figures 3.12, 3.13 and 3.14 show the deviations of different approaches. For almost all the simulated cases, the proposed distribution has the smallest deviation, i.e. smaller than 10 MPa. Prior to buckling, the deviations of all the three approaches are the same and below 2 MPa.

98

3 Steel Plate Shear Walls with Considering the Gravity Load …

Fig. 3.9 Effect of wall thickness t on the stress distribution of the walls with height a = 1.2 m and width b = 1.2 m

(a) t = 2 mm

(b) t = 4 mm

(c) t = 6 mm

(d) t = 8 mm

3.2 The Three-Segment Distribution Under Uniform Compression Fig. 3.10 Effect of wall thickness t on stress distribution of the walls with height a = 1.2 m and width b = 1.8 m

(a) t = 2 mm

(b) t = 4 mm

(c) t = 6 mm

(d) t = 8 mm

99

100

3 Steel Plate Shear Walls with Considering the Gravity Load …

Fig. 3.11 Effect of wall thickness t on stress distribution of the walls with height a = 1.2 m and width b = 2.4 m: a t = 2 mm, b t = 4 mm, c t = 6 mm and d t = 8 mm

(a) t = 2 mm

(b) t = 4 mm

(c) t = 6 mm

(d) t = 8 mm

3.2 The Three-Segment Distribution Under Uniform Compression Fig. 3.12 The deviation of the stress distribution of the walls 1.2 m height and 1.2 m width

(a) t = 2 mm

(b) t = 4 mm

(c) t = 6 mm

(d) t = 8 mm

101

102

3 Steel Plate Shear Walls with Considering the Gravity Load …

Fig. 3.13 The deviation of the stress distribution of the walls 1.2 m height and 1.8 m width

(a) t = 2 mm

(b) t = 4 mm

In the case of a large slenderness ratio, the effective width model can track the stress distribution better than the cosine distribution. The reason is the thin wall has very small effective width and buckling stress (see Figs. 3.9a, 3.10a and 3.11a) and is more likely to act as the effective width model. In the case of a relatively small slenderness ratio, the cosine distribution is better than the effective width model. While in the case of medium slenderness ratio, e.g. the 4 mm thickness wall, when the compression load is smaller than about 7 times of the buckling load, the cosine distribution is better than the effective width model. In contrary, the effective width model is better than the cosine distribution. With an increase of the compression, the deviation increases significantly for both the effective width model and cosine distribution. For the proposed distribution, except the wall with a width of 1.2 m and thicknesses of 2 mm and 4 mm, the deviations of the predicted stress distribution change little. When the edge stress approaches the yield stress, the deviations suddenly increase. The reason is that the buckling mode changes from one-half sine to wrinkling. All three models are not suitable for describing this buckling mechanism. From the deviation of the cosine distribution, Fig. 3.12b shows that when the edge stress is smaller than 3 times of the buckling stress, it can properly predict the stress distribution. Exceeding it, a flattened portion is formed in the middle portion. The proposed distribution is then more accurate than the cosine distribution. Figure 3.12c

3.2 The Three-Segment Distribution Under Uniform Compression

103

Fig. 3.13 (continued)

(c) t = 6 mm

(d) t = 8 mm

and d show the results of the thicknesses of 6 mm and 8 mm, respectively. The deviation of both the cosine distribution and the proposed distribution is between 5 to 10 MPa. However, from Figs. 3.12c, d and 3.13c, d, although the wall has the same slenderness ratio as that in Fig. 3.12c, d, the cosine distribution cannot predict the stress distribution properly. The reason is the walls are rectangular. Once it buckled, a flattened portion is then formed in the middle portion. The cosine distribution cannot consider this phenomenon. It should be noted that the deviation of the stress distributions are not smooth, the reason is the resistance load may fluctuate during the buckling.

3.2.3.6

Load-Bearing Capacity

The resistance load of the wall can be obtained by integrating the compressive stress through the width. Table 3.2 shows the load obtained from the proposed distribution, cosine distribution and effective width model, respectively. The edge stress σ e is 200 MPa. All the walls have a constant height of 1.2 m. The prediction error is calculated by the following equation

104

3 Steel Plate Shear Walls with Considering the Gravity Load …

Fig. 3.14 The deviation of the stress distribution of the walls 1.2 m height and 2.4 m width

(a) t = 2 mm

(b) t = 4 mm

rpred =

Fpred − FFE × 100% FFE

(3.15)

where F FE and F pred are the loads obtained from FE analysis and the three approaches, respectively. In Table 3.2, r p , r c , and r w are the prediction errors of the proposed distribution, the cosine distribution and the effective width model, respectively. Both the proposed distribution and the effective width model show good predictions except slightly underestimate the large slenderness walls and slightly overestimate the small slenderness walls. The cosine distribution cannot be used to predict the load of a slender wall. The best prediction of the cosine distribution is the 1.2 m height and 1.2 m width wall with a thickness of 8 mm. However, it is not accurate enough to predict the load of a rectangular wall, even the slenderness ratio is the same, because of a flattened portion formed in the middle.

3.2 The Three-Segment Distribution Under Uniform Compression

105

Fig. 3.14 (continued)

(c) t = 6 mm

(d) t = 8 mm Table 3.2 Load prediction of the approaches b (m)

t (mm)

Proposed distribution (kN)

Cosine distribution (kN)

Effective width model (kN)

FE analysis (kN)

r p (%)

r c (%)

r w (%)

1.2

2

85.34

222.26

85.93

99.61

−14.33

123.13

−13.73

1.2

4

317.70

532.03

335.57

346.91

−8.42

53.36

−3.27

1.2

6

663.57

797.31

681.62

677.12

−2.00

17.75

0.66

1.2

8

1203.00

1280.27

1202.84

1195.22

0.65

7.12

0.64

1.8

2

86.86

316.83

89.65

101.75

−14.63

211.38

−11.90

1.8

4

373.48

766.89

383.12

373.25

0.06

105.46

2.64

1.8

6

823.54

1198.76

827.94

804.65

2.35

48.98

2.89

1.8

8

1442.27

1727.47

1453.33

1335.10

8.03

29.39

8.86

2.4

2

106.70

426.75

107.24

108.25

−1.43

294.24

−0.93

2.4

4

456.23

1013.92

479.85

392.67

16.19

158.21

22.20

2.4

6

1008.46

1572.37

1042.04

855.98

17.81

83.69

21.74

2.4

8

1763.87

2226.93

1793.18

1487.91

18.55

49.67

20.52

106

3 Steel Plate Shear Walls with Considering the Gravity Load …

3.3 The Three-Segment Distribution Under Compression and In-plane Bending 3.3.1 The Cosine Stress Distribution Figure 3.15a shows a plan view of the Jinta tower. In the case the gravity loads of the boundary columns are different or the bending moment caused by a wind load is considered, a typical steel shear wall will experience uniform compression and inplane bending. The deformation of the shear wall after applying the loads are shown in solid lines in Fig. 3.15b. Because of the distribution of the boundary beams, some gravity load is resisted by the wall which conversely makes it experience linearly changed compression along the width (see Fig. 3.15c). Therefore, it is the classic post-buckling problem of a thin wall under linear distribution compression. The compression force σ (x) along the loaded edge can be expressed as x σ (x) = σ1 [λ + (1 − λ) ] b

(3.16)

where λ is the stress gradient coefficient. When λ = 1, it is a uniform compression, and when λ = 0, it is a triangular distribution compression. σ 1 is the compressive stress at the heavier loaded edge. By solving the compatibility and equilibrium equations simultaneously [26], the vertical stress along the width b can be obtained by    1 x 2π x (σ1 − σcr ) cos σ (x) = σ1 λ + (1 − λ) + (1 + λ) b 1 + β4 b

(3.17)

where σ (x) is the vertical stress at the position x along the horizontal direction, σ cr is the compression buckling stress of a four-edge clamped wall. Equation (3.17) consists two parts, i.e. complementary solution and particular solution. In the first solution, the stress is the same as the compression load changing linearly with a stress gradient λ. In the second part, the stress is a cosine function, which is the same as the solution under uniform compression except the average stress is 0. To avoid the difficulty of obtaining the maximum compressive stress σ 1 , it is better to express σ 1 in the form of the edge stress σ e σ1 =

2σe + (1 + λ)σcr 3+λ

(3.18)

where σ e is the edge stress at the heavier loaded edge (see Fig. 3.15c), which is the same as the stress of the edge column. Equation (3.17) is accurate enough for a wall with slenderness less than 100 or the edge stress is less than 3 times of the buckling stress. Above that, the stress distribution obtained from Eq. (3.17) is not correct.

3.3 The Three-Segment Distribution Under Compression and In-plane Bending

107

Fig. 3.15 The rectangular wall under compression and in-plane bending

(a) A plan view of the Jinta tower

(b) A typical steel shear wall under gravity load

(c) A rectangular wall under linear distribution compression

108

3 Steel Plate Shear Walls with Considering the Gravity Load …

For example, it is assumed that the buckling stress is 20 MPa (compression stress assumed to be positive). For the aspect ratios β = 1.0 and 2.0, the stress gradients λ = 1.0 and 0, and the edge stress equal to 3 times and 10 times of the buckling stress, the stress distributions are shown in Fig. 3.16. When both β and λ equal to 1.0, i.e. square wall under uniform compression, the minimum stress is exactly the same as the buckling stress. In the case β = 2.0 and λ = 1.0, the minimum stress is 7.7 MPa when the edge stress σ e = 60 MPa, while the minimum stress decreases to −35.1 MPa when the edge stress σ e = 200 MPa. When the compression stress is nonuniform, a similar tendency can be observed (see Fig. 3.16). That means the vertical stress at the middle portion changes from compression to tension with an increase of the edge stress. This is caused by the width to height ratio β, which means this theoretical solution can only be used in square walls. Hereafter, the aspect ratio β in Eq. (3.17) is assumed to be 1.0 to avoid this nonphysical phenomenon and the following equation can be obtained    x  (1 + λ)(σ1 − σcr ) 2π x cos σ (x) = σ1 λ + (1 − λ) + b 2 b

Fig. 3.16 Stress distribution of the wall with different aspect ratios and stress gradients

(a) σe=60 MPa

(b) σe=200 MPa

(3.19)

3.3 The Three-Segment Distribution Under Compression and In-plane Bending

109

3.3.2 The Proposed Three Segment Stress Distribution 3.3.2.1

Model Formulation

The stress distribution of Eq. (3.19) is a cosine function (see Fig. 3.17d, hereafter it is called cosine distribution). This equation is suitable for a non-heavily loaded square wall with the slenderness ratio below 100. In the case of a heavily loaded thin wall [21], a constant distribution close to buckling stress appeared in the middle portion, then a three-segment stress distribution is proposed (see Fig. 3.17). For each segment, the distribution is expressed by the following equations     ⎧  x σ1 − σcr x ⎪ ⎪ + (1 + λ) , 0 ≤ x ≤ b1 cos 2π σ1 λ + (1 − λ) ⎪ ⎪ b1 + b2 2 b1 + b2 ⎪ ⎨ σmin , b1 < x ≤ b − b2 σ (x) = ⎪      ⎪ ⎪ ⎪ x −b σ1 − σcr x −b ⎪ ⎩ σ1 λ + (1 − λ) + (1 + λ) , b − b2 < x ≤ b cos 2π b1 + b2 2 b1 + b2

(3.20) where b1 and b2 are the edge widths of the wall (see Fig. 3.17a, b), σ 1 is the compressive stress at the heavier loaded edge (see Fig. 3.15c), σ cr is the compression buckling stress of a four-edge clamped wall, σ min is the minimum stress along the width (see Fig. 3.17). Each function in Eq. (3.20) represents a segment of the stress distribution. The two edge portions can be combined to form a new wall with a width b1 + b2 and an unchanged height a. The stress distribution of this new wall is then a full cosine function that is the same as Eq. (3.19). The stress of the middle portion is assumed to be constant and equals to the minimum stress σ min . If the compressive load is

Fig. 3.17 Vertical stress distributions

110

3 Steel Plate Shear Walls with Considering the Gravity Load …

uniform, i.e. the stress gradient λ equals to 1.0, the minimum stress σ min equals to the compression buckling stress σ cr . The cosine distribution and the effective width model are also shown in Fig. 3.17d and e respectively. It is obvious that the cosine distribution is the upper bound of the proposed distribution when the effective width b1 + b2 equals to b, i.e. no flattened middle portion. In this case, Eq. (3.20) is the same as Eq. (3.6).

3.3.2.2

Determination of the Edge Widths

The effective width model is adopted to define the width b1 + b2 of the edge portions. The effective width of the lighter loaded edge and the heavier loaded edge of the original wall are respectively defined as b1e and b2e (see Fig. 3.17e). By equating the load calculated by the proposed distribution with that obtained from the effective width model, it is b1 σe1 be1 + σe2 be2 =

b σ (x)dx + (b − b1 − b2 )σmin +

0

σ (x)dx

(3.21)

b−b2

Substituting Eq. (3.20) into Eq. (3.21), the width b1 + b2 is b1 + b2 =

σe1 be1 + σe2 be2 − bσmin 1+λ σ1 − σmin 2

(3.22)

where σ e1 and σ e2 are the edge stresses at x = 0 and x = b, respectively. The edge stress can be assumed to equal to the axial stress of the corresponding boundary column under the gravity load. σ min is the minimum stress along the width (see Fig. 3.17). It is clear that when b1 + b2 equals to b, i.e. no flattened portion occurred in the middle portion, the proposed distribution is identical to the cosine distribution. For a thin wall with a small buckling stress σ cr , we have b1 + b2 = 2(be1 + be2 ). Let the derivative of the stress distribution in Eq. (3.19) equal to 0, then the position of the minimum value along the width can be found by the following equation   σ1 b −1 1 − λ sin x0 = 2π 1 + λ π(σ1 − σcr )

(3.23)

Substituting x 0 into Eq. (3.19), the minimum stress σ min in Fig. 3.17a–d can be obtained. Obviously, be1 and be2 in Eq. (3.22) need to be defined. In this study, two strategies, i.e. the Von Karman’s equation [35] and the Bedair’s equation [39], are adopted to determine the effective widths be1 and be2 . Through Karman’s equation

3.3 The Three-Segment Distribution Under Compression and In-plane Bending

be1 =

b 2

b be2 = 2

 

111

σcr σe1

(3.24)

σcr σe2

(3.25)

where b is the wall width. σ cr is the buckling stress of a clamped wall under compression. It is calculated by σcr = kcr

π2E 12 (1 − μ2 )(b/t)2

(3.26)

where E and μ are the elastic modulus and Poisson’s ratio, respectively. t is the wall thickness. k cr is the buckling coefficient [26]  2  a 2 b kcr = 3.6 + 4.3 + 2.5 a b

(3.27)

For Bedair’s equation, it is assumed that the load is separately carried by the two   strips with the widths equal to be1 and be2 and each strip can carry the load from the centerline to its edge. The widths can be expressed by the following equations [39] 

(β 4 + 1)(3 + λ)σ1 b 1 8 β 4 (2 + λ)σ1 + σ1 − β 4 (1 + λ)σcr

(3.28)

(β 4 + 1)(1 + 3λ)σ1 b 1 8 β 4 (1 + 2λ)σ1 + λσ1 − β 4 (1 + λ)σcr

(3.29)

be1 = 

be2 =

The above two equations should satisfy the following condition σ1 2(1 + λ) > σcr 1 + 3λ

(3.30)

After calculating b1 + b2 , by letting the derivative of the stress function in Eq. (3.20) equals to 0, the width of each edge portion of the proposed distribution is then determined by the following equation   σ1 b1 + b2 −1 1 − λ sin +π b1 = x0 = 2π 1 + λ π(σcr − σ1 ) 

(3.31)

From the above equation, the minimum compressive stress along the width b is equal to the stress at x = b1 or x = b − b1 from Eq. (3.20).

112

3.3.2.3

3 Steel Plate Shear Walls with Considering the Gravity Load …

Discussion of the Effective Width

Although it is no physical meaning to compare the effective widths obtained from the models of Bedair and Karman, it is useful to have an insight into determining the proposed distribution. For Bedair’s method, it is suitable for the lightly loaded walls because the flattened portion is not considered. For Karman’s method, it is reliable when the wall is significantly buckled because it was originally proposed to predict the ultimate strength of a thin wall. Therefore, by equating Eqs. (3.28) and (3.24) or Eqs. (3.29) and (3.25), the ratio of the edge stress to the buckling stress that the two effective widths are equal can be obtained. After rearranged and simplified, the following cubic equation can be obtained  a1

σ1 σcr

3

 + a2

σ1 σcr

2

 + a3

σ1 σcr

 + a4 = 0

(3.32)

where a1 , a2 , a3 and a4 are coefficients. They can be calculated by the following equations ⎧ a1 ⎪ ⎪ ⎪ ⎪ ⎨ a2 ⎪ ⎪ a3 ⎪ ⎪ ⎩ a4

= (3λ + 1)2 λ(β 4 + 1)2 + (3λ + 1)2 (λ + 1)(β 4 + 1) = −(3λ + 1)2 (λ + 1)(β 4 + 1) − 16[(2λ + 1)β 4 + λ] = 32[(2λ + 1)β 4 + λ](λ + 1)β 4

(3.33)

= −16(λ + 1)2 β 8

where λ and β are the stress gradient and the width to height ratio, respectively. In the case of a square wall under uniform compression, i.e. λ = 1.0 and β = 1.0, the solution of Eq. (3.32) is 1.0. That means when the applied compression stress σ 1 equals to the buckling stress σ cr , these two effective methods are the same. Otherwise, the effective width obtained by Bedair is always larger than that of Karman. Three more cases with different aspect ratios and stress gradients are calculated and shown in Fig. 3.18. The red lines are the effective width obtained through Bedair’s model and the black lines are through Karman’s. It is shown that the effective widths have an intersection point at σσcr1 = 2.4, 6.0 and 14.5 for case 1, case 2 and case 3, respectively. Before the intersection points, the effective width obtained by Bedair is smaller. While after these points, Karman’s model is smaller. Also, the critical value decreases with an increase of the aspect ratio and stress gradient. This can be seen from Fig. 3.18 that the critical value σσcr1 of case 1 is the smallest and case 3 is the largest. The minimum value calculated by the models of Bedair and Karman is used to determine the edge width of the proposed distribution. Based on Eqs. (3.22), (3.28) and (3.29), the following equation is used 

σe1 be1 + σe2 be2 − bσmin   , be1 + be2 b1 + b2 = min 1+λ σ1 − σmin 2

 (3.34)

3.3 The Three-Segment Distribution Under Compression and In-plane Bending

113

Fig. 3.18 Effective widths obtained from models of Bedair and Von Karman

3.3.3 Finite Element (FE) Analysis 3.3.3.1

FE Model

A FE model was developed using the software LS-DYNA [43]. The FE model was firstly verified by using the results obtained from experiments carried out by our own and by Zara´s et al. [45]. This model is the same as that used in the previous section.

3.3.3.2

Parametric Analysis

Steel walls, with a height of 1200 mm and a width of 1800 mm were used to verify the proposed distribution. The width to height ratio β is 1.5. The thicknesses considered are 2, 4, 6, and 8 mm. The height to thickness ratio is between 150 and 600. The considered stress gradients are 0.25, 0.50, 0.75 and 1.0. The walls were fully constrained for all edges except for the vertical degree of freedom of the top and side edges (see Fig. 3.19). Linear distribution compression with different stress gradients was added at the top edge. The behavior of the steel was described using an elastic– plastic material considering kinematic hardening. The yield stress, elastic modulus, and tangent modulus are 235 MPa, 210 GPa, and 2.1 GPa, respectively. The analysis cases are listed in Table 3.3. The maximum edge stress σ e is assumed to be smaller than 200 MPa because in practical design, the axial stress of columns under the gravity load is normally much less than the yield stress 235 MPa.

114

3 Steel Plate Shear Walls with Considering the Gravity Load …

Fig. 3.19 Finite element model

Table 3.3 Configurations of the parametric analysis

3.3.3.3

Configuration

b × a (m)

Stress gradient λ

t (mm)

1

1.8 × 1.2

0.25

2/4/6/8

2

1.8 × 1.2

0.5

2/4/6/8

3

1.8 × 1.2

0.75

2/4/6/8

4

1.8 × 1.2

1.0

2/4/6/8

Vertical Stress Distribution

The vertical stress of each element at the middle height is chosen to check the stress distribution (see Fig. 3.19). Because the element size is 30 mm × 30 mm, 60 elements are selected. The relative coordinates, i.e. x* = x/b and y* = y/a in the horizontal and vertical directions respectively are used (see Fig. 3.19). The wall thickness is respectively 2 mm, 4 mm, 6 mm, and 8 mm and the corresponding slenderness is respectively 600, 300, 200 and 150. The considered stress gradients are 1, 0.75, 0.50 and 0.25, in which λ = 1 means uniform compression. All the stresses are displayed after normalized by the buckling stress σ cr (see Eq. (3.13)). The stress distributions under increasing edge stresses are shown in Figs. 3.20, 3.21, 3.22 and 3.23. The proposed distribution can predict the vertical stress along the width well, especially when the edge stress is below half of the maximum edge stress (the maximum edge stress is about 200 MPa). For all the considered cases, no tension stress observed

3.3 The Three-Segment Distribution Under Compression and In-plane Bending

115

Fig. 3.20 The stress distribution of the walls with λ = 1.0

(a) t = 2 mm

(b) t = 4 mm

in the middle portion. The minimum stress is close to the compression buckling stress σ cr . When the edge stress is larger than about 100 MPa, the edge portion is severely buckled. The wall has the tendency to form a second or third sines in the edge. Meanwhile, the maximum stress along the width transferred toward the nearby element. When the edge stress is close to the ultimate stress around 200 MPa, the buckling mode changed from one-half sine to two-half sines or three-half sines or even wrinkling. However, this kind of post-buckling is out of the scope of this study. The predicted results of the wall with a thickness of 8 mm by the cosine distribution are shown in Figs. 3.20d, 3.21d, 3.22d and 3.23d. Because the 8 mm wall has a small slenderness ratio, the stress distributions obtained from the cosine distribution and the proposed distribution are similar. However, for most of the cases, the cosine distribution always overestimates the vertical stress and cannot consider the flattened portion in the middle. From Figs. 3.20, 3.21, 3.22 and 3.23, it shows that stress gradients change the stress distribution significantly. First, the flattened portion will move from the middle to the heavier loaded edge with an increase of the stress gradient. The reason is the effective width of the heavier loaded portion is smaller than the other edge. Second, the flattened portion decreases with an increase of the stress gradient. The reason is the heavier loaded edge has the same effective width as that of the corresponding uniform

116

3 Steel Plate Shear Walls with Considering the Gravity Load …

Fig. 3.20 (continued)

(c) t = 6 mm

(d) t = 8 mm

compression wall, while the lighter loaded edge has smaller edge stress and larger effective width. Third, the cosine distribution can predict the stress distribution of a wall with larger stress gradient better (e.g. λ = 0.25), the reason is also the flattened portion in the middle decreased.

3.3.3.4

Effectiveness of the Proposed Distribution

The standard deviation shown in Eq. (3.14) is used to examine the application scope of the effective width model, cosine distribution and the proposed distribution. Figures 3.24, 3.25, 3.26 and 3.27 show the deviations obtained from different stress distributions. For all the considered cases, the proposed distribution has the smallest deviation, i.e. smaller than 10 MPa. The prediction changes little with the slenderness ratios and stress gradients. Before the wall buckled, the deviations of the cosine distribution and the proposed distribution are identical and below 3 MPa. For a wall of a large slenderness ratio like the 2 mm thickness, the effective width method can predict the stress distribution better than the cosine distribution (see Figs. 3.24a, 3.25a, 3.26a and 3.27a). For a wall with a small slenderness ratio like the 6 mm and 8 mm thicknesses, the cosine distribution is better than the effective

3.3 The Three-Segment Distribution Under Compression and In-plane Bending

117

Fig. 3.21 The stress distribution of the walls with λ = 0.75

(a) t = 2 mm

(b) t = 4 mm

width method (see Figs. 3.24c, d, 3.25c, d, 3.26c, d and 3.27c, d). For a wall of a medium slenderness ratio like the 4 mm thickness, the effectiveness depends on the loading level. When the wall was lightly loaded, the cosine distribution is better. When the wall was heavily loaded, the effective width model is better than the cosine distribution. The prediction errors of the effective width model and the cosine distribution increase with an increase of the loading. The proposed distribution is not very sensitive to the loading levels. However, the deviations obtained from all the three distributions suddenly increase when the edge stress approaches the yield stress. The reason is that the walls buckling patterns changed from one-half sine to two-half sines or three-half sines or even wrinkling. The proposed distribution can predict the vertical stress distribution better than that of the cosine distribution, because the flattened portion in the middle of the wall is considered by the proposed distribution. The deviation of the cosine distribution also decreases with an increase of the stress gradient. Taking the 8 mm thickness wall for example, for the stress gradients equal to 1, 0.75, 0.5 and 0.25, the deviation is 34 MPa, 22 MPa, 20 MPa, and 16 MPa, respectively. The reason is the width of the flattened portion decreases with the stress gradient.

118

3 Steel Plate Shear Walls with Considering the Gravity Load …

Fig. 3.21 (continued)

(c) t = 6 mm

(d) t = 8 mm

Because of the fluctuating of the resistance load during the buckling process, the deviations obtained from all three stress distributions are not smooth.

3.3.3.5

Load-Bearing Capacity

The load of the wall is obtained by integrating the compressive stress along the width b. Table 3.4 shows the load obtained from the effective width model, cosine distribution and the proposed distribution. The prediction error is calculated by Eq. (3.15). Although, the proposed distribution will slightly underestimate the strength of the large slenderness walls and slightly overestimate that of the small slenderness walls, it is the best among these three distributions. The load-bearing capacity increases with the stress gradient because the lighter loaded portion of the wall with a small stress gradient is not fully used. For most of the simulated cases, the cosine distribution always significantly overestimates the load-bearing capacity, especially the heavily loaded thin walls. Among all the studied cases, the cosine distribution predicted the 8 mm thickness wall with the stress gradient equals to 0.25 best. Choosing the wall of 8 mm thickness and different stress gradients as examples, the prediction error by the cosine distribution is respectively 14.0%, 17.8%, 21.9%, and 29.4% for the gradient

3.3 The Three-Segment Distribution Under Compression and In-plane Bending

119

Fig. 3.22 The stress distribution of the walls with λ = 0.5

(a) t = 2 mm

(b) t = 4 mm

of 0.25, 0.50, 0.75 and 1.0, while the corresponding prediction error by the proposed distribution is respectively 2.6%, 2.2%, 3.6% and 8.0%. The prediction error of the cosine distribution increases with a decrease of the stress gradient. The reason is that the width of the flattened portion decreases with the stress gradient. In the case λ = 0.25, the prediction error by the cosine distribution is respectively 201.4%, 77.5%, 39.0% and 14.0% for the slenderness ratios equal to 600, 300, 200 and 150, while respectively −14.2%, −4.4%, 5.0% and 2.6% by the proposed distribution. The prediction accuracy of the effective width model is between that of the cosine distribution and the proposed distribution. Generally, the effective width model will underestimate the strength of walls with small stress gradients. The prediction accuracy of the effective width model increases with the stress gradients and decrease with the wall thicknesses.

120

3 Steel Plate Shear Walls with Considering the Gravity Load …

Fig. 3.22 (continued)

(c) t = 6 mm

(d) t = 8 mm

3.4 Shear Strength Considering the Gravity Load 3.4.1 Stresses Under Compression and Shear The stress analysis is the same as that in Chap. 2. For the readers convenience, the main steps of the stress analysis are descripted here again. Assuming the steel plate is elastic, under compression and shear a linear superposition is still applicable. The stress of the shear wall can be divided into two stages, i.e. at buckling and after buckling. The stresses at buckling are ⎧ ⎪ ⎨ σx x = 0 σ yy = σ (x) ⎪ ⎩ σx y = τcr

(3.35)

where σ (x) is the vertical stress along the width. It can be determined by the cosine distribution, the effective width model and the three-segment distribution. τ cr is the elastic shear buckling stress. The stresses after buckling are

3.4 Shear Strength Considering the Gravity Load

121

Fig. 3.23 The stress distribution of the walls with λ = 0.25

(a) t = 2 mm

(b) t = 4 mm

⎧ 2 ⎪ ⎨ σx x (x) = σty (x) sin α σ yy (x) = σ (x) + σty (x) cos2 α ⎪ ⎩ σx y (x) = τcr + 0.5σty (x) sin 2α

(3.36)

where σ ty (x), σ xx (x), σ yy (x) and σ xy (x) are respectively the tensile stress, horizontal stress, vertical stress and shear stress along the width. Hereafter, σ ty (x), σ xx (x), σ yy (x) and σ xy (x) are respectively expressed as σ ty , σ xx , σ yy and σ xy . α is the angle between the horizontal axis and the tensile strip. For a thin wall, the principal stress in z direction, the shear stresses in plane yz and xz are assumed to be zero. The von Mises yield criterion is obtained by the following equation: 2 + σx2x + 6σx2y − 2 f y2 = 0 J (σ ) = (σx x − σ yy )2 + σ yy

(3.37)

Substituting Eq. (3.36) for σ xx , σ yy , and σ xy into Eq. (3.37), the value of σ ty , at which yielding of the steel wall occurs, is calculated by the following equation: σty2 + [σ (x)(3 cos2 α − 1) + 3τcr sin(2α)]σty + σ (x)2 + 3τcr2 − f y2 = 0

(3.38)

122

3 Steel Plate Shear Walls with Considering the Gravity Load …

Fig. 3.23 (continued)

(c) t = 6 mm

(d) t = 8 mm

1 σty = − [σ (x)(3 cos2 α − 1) + 3τcr sin(2α)] 2 1 + [σ (x)(3 cos2 α − 1) + 3τcr sin(2α)]2 − 4[σ (x)2 + 3τcr2 − f y2 ] 2 (3.39) For the commonly used steel walls, the shear buckling stress τ cr is very small. It can be ignored in the calculation.

3.4.2 Shear Strength According to Thorburn et al. [1], the shear walls that experience early buckling can be modeled by a series of pin-ended tensile strips with an inclined angle of θ. For rigidly connected walls, the angle of the tensile field to the vertical axis according to the research by Timler and Kulak [46] is

3.4 Shear Strength Considering the Gravity Load

123

Fig. 3.24 The deviation the walls with λ = 1.0

(a) t = 2 mm

(b) t = 4 mm

 θ = tan

−1

4

   −1 tL 1 H3 1+ 1 + tH + 2 Ac Ab 360Ic L 2

(3.40)

where Ac and Ab are the cross-section areas of the boundary column and beam. Obviously, α + θ = π/2. I c is the moment of inertia of the boundary column. The shear strength of the shear wall can be calculated by integrating the stresses along the width of the plate: L Vp =

L tσx y dx =

0

tσty sin α cos αdx

(3.40)

0

In this work, the shear strength of the boundary frame is not investigated. However, it can be easily determined by the plate-frame interaction model.

124

3 Steel Plate Shear Walls with Considering the Gravity Load …

Fig. 3.24 (continued)

(c) t = 6 mm

(d) t = 8 mm

3.4.3 Discussion of the Approach The relative edge width (b1 /b) of a steel plate with a height 1.2 m and a width 1.8 m was shown in Fig. 3.28. For the thickness of 1 mm, 2 mm, 4 mm, 6 mm, 8 mm, and 10 mm, the slenderness ratio is 1200, 600, 300, 200, 150, and 120, respectively. As anticipated, the buckling stress σ cr , i.e. the stress at b1 /b = 0.5, increases with the wall thickness. In this condition, the proposed distribution is the same as the cosine distribution. After buckling, the width decreases following a square root function of the edge stress. The width of the edge portion increases rapidly with the thickness. When the edge stress is 200 MPa, the relative edge widths are 0.061, 0.114, 0.205, 0.278, 0.340 and 0.391 for the thicknesses of 1 mm, 2 mm, 4 mm, 6 mm, 8 mm and 10 mm, respectively. The corresponding relative widths be /b obtained from the effective width model are respectively 0.064, 0.129, 0.258, 0.387, 0.516 and 0.645. The widths b1 and be are similar when the plate thickness is very small. Ignoring the shear buckling stress, for a shear wall with an inclined angle of 42° and the yield stress of 235 MPa, Fig. 3.29 shows the tensile stress σ ty , horizontal stress σ xx , vertical stress σ yy and shear stress σ xy under increasing edge stress σ e . The tensile stress decreases with an increase of the edge stress. When the edge

3.4 Shear Strength Considering the Gravity Load

125

Fig. 3.25 The deviation the walls with λ = 0.75

(a) t = 2 mm

(b) t = 4 mm

stress increased to the yield stress of the steel, i.e. −235 MPa, the tensile stress σ ty , horizontal stress σ xx and shear stress σ xy become zero. Figure 3.30 shows the shear strength of different plate thicknesses under an increasing edge stress. The shear strength is normalized by the thickness t. Prior to the buckling stress (the dots in Fig. 3.30), the normalized shear strength is the same. After buckling, a thicker wall has a smaller shear strength because of the effect of the gravity load.

3.4.4 Parametric Analysis Steel walls with different widths and thicknesses were chosen. The wall height of 1200 mm was constant. A varying width of 1200, 1800 and 2400 mm were considered. The respective width to height ratio β was then 1, 1.5 and 2, which were the most used ratio in practice. The considered thicknesses were 2, 4, 6 and 8 mm. Consequently, the slenderness ratio was between 150 and 600. The walls were fully constrained at the bottom. For the vertical and top boundaries, all the degrees of freedom are constrained except the vertical one. A horizontal load was applied at the top of the

126

3 Steel Plate Shear Walls with Considering the Gravity Load …

Fig. 3.25 (continued)

(c) t = 6 mm

(d) t = 8 mm

boundary columns and pushover analysis was conducted subsequently. An elastic– plastic material with a kinematic hardening was used to describe the steel of the boundary frame behaviour. The yield stress, Poisson’s ratio, elastic modulus, and tangent modulus were 335 MPa, 0.3, 210 GPa, and 2.1 GPa, respectively. An elastic perfectly plastic material model with a yield stress of 235 MPa and elastic modulus of 210 GPa was used to describe the behavior of the shear wall. Five levels of axial stress, i.e. 0, 50, 100, 150 and 200 MPa, representing different gravity loads were considered. Table 3.5 lists the configuration of the analyzed cases.

3.4.5 Shear Strength By considering the effect of gravity load with different assumed vertical stress distributions, Table 3.6 summarizes the predicted shear capacities. The capacity of the shear walls obtained from FE analysies, V FEM , is determined using the drift angle according to ANSI-AISC 341-05 (Seismic provisions for structural steel buildings [47]). In this specification the capacity is defined at an inter-storey drift angle of 0.02 radians. V stan is shear strength calculated according to the Canadian standard [40]

3.4 Shear Strength Considering the Gravity Load

127

Fig. 3.26 The deviation the walls with λ = 0.5

(a) t = 2 mm

(b) t = 4 mm

Vstan = 0.5Ry f y t Lsin(2α)

(3.41)

where Ry is the ratio of the mean steel yield stress to the specified minimum yield stress. In this work, a ratio Ry of 1.0 is selected. V cos , V w and V are the shear strengths considering the gravity load effect using the cosine model, the effective width model and the three-segment model, respectively. r, r stan , r cos , and r w are respectively the prediction errors of the shear strength in comparison with the FE result. Table 3.6 shows that the three-segment model slightly underestimates the shear strength of the shear walls. For most of the cases, the prediction error is less than 5%. In the case σ e = 0, all of the three models and the standard predict the similar strengths that are very close to the FE results. This proves that the value according to the standard [40] is accurate if the axial stress is not considered. However, with an increase of the axial stress, the standard will significantly overestimate the shear strength. Choosing L12t2 specimens (see the first column of Table 3.6) as an example, the standard overestimates about 0.27%, 1.68%, 4.24%, and 11.53% of the shear strength when the axial stress equals to 50 MPa, 100 MPa, 150 MPa, and 200 MPa, respectively. In the case of L12t8 specimens, the standard model will overestimate

128

3 Steel Plate Shear Walls with Considering the Gravity Load …

Fig. 3.26 (continued)

(c) t = 6 mm

(d) t = 8 mm

about 6.56%, 11.05%, 21.90, and 34.27% when the axial stress equals to 50 MPa, 100 MPa, 150 MPa, and 200 MPa, respectively. In the case of the cosine model, it will significantly underestimate the shear strength of the steel shear wall, especially when the thickness is very small. Like the cases L12t2s200, L18t2s200 and L24t2s200, the cosine model will underestimate correspondingly about 18.21%, 21.25%, and 21.45% of the shear strength compared to the FE results. The predicted error increases with an increase of the width (see the results of the walls with the same thickness t and axial stress σ e but a varying width b). The reason is that the cosine model cannot consider the flattened portion of the shear wall under compression. Consequently, the compression stress is over estimated, especially for a wall with a large aspect ratio and slenderness ratio. The effective width model can predict the shear strength of a wall with a large slenderness ratio while it will underestimate that with a small slenderness ratio. For the specimens of L12t8s200, L18t8s200, and L24t8s200, the effective width model underestimate respectively about 18.92%, 10.31%, and 14.49% of the shear strength in comparison with the FE results. The mean and coefficient of variation (COV) of the predicted errors are also shown in the last two rows. It indicates that the mean value of the predicted error is not very large, i.e. 6.16%, −8.43%, −4.67% and −2.59% for the results according to the standard, the cosine model, the effective width model

3.4 Shear Strength Considering the Gravity Load

129

Fig. 3.27 The deviation the walls with λ = 0.25

(a) t = 2 mm

(b) t = 4 mm

and the three-segment model, respectively. However, the standard will overestimate the shear strength. The COV of the results according to the standard is the largest, followed by that of the cosine model, effective width model and the three-segment model.

3.4.6 Aspect Ratio and Slenderness Ratio The influence of the slenderness ratio and aspect ratio under varying axial stress was discussed. Figure 3.31 shows that the predicted error according to the standard increases with a decrease of the slenderness ratio. That means the axial stress significantly influences the shear strength of a stocky wall. The slenderness ratio used in Jinta Tower is between 170 to 230. The shear strength is overestimated by more than 20% if the axial stress due to the gravity load of the boundary columns is about 200 MPa. Figure 3.31 shows the predicted errors of the results according to the cosine model and effective width model. As discussed in the previous section, the cosine model can predict the shear strength better when the slenderness ratio is small. The effective width model is accurate for walls with a large slenderness ratio.

130

3 Steel Plate Shear Walls with Considering the Gravity Load …

Fig. 3.27 (continued)

(c) t = 6 mm

(d) t = 8 mm

In the case of different aspect ratios, as shown in Fig. 3.32, the predicted error of the results according to the standard decreases with an increasing aspect ratio. The predicted error due to the cosine model increases with an increasing aspect ratio, while that of the effective width model changes little. For almost all the cases except L12t8s100 and L12t8s150, the predicted error of the three-segment model is within 5%.

3.5 Shear Strength Considering the Gravity Load and In-plane Bending Using the same method in Sect. 3.4, the shear strength of the SPSWs with considering the gravity load and in-plane bending can be determined.

3.5 Shear Strength Considering the Gravity Load and In-plane Bending

131

Table 3.4 The load-bearing capacity of the approaches t (mm) λ (mm) Proposed Cosine Effective FE r p (%) distribution distribution width analysis (kN) (kN) model (kN) (kN)

r c (%)

r w (%)

81.77 −14.21 204.27 −29.24

2

0.25

70.15

248.79

57.86

4

0.25

293.70

545.63

231.45

307.38 −4.45

77.51 −24.70

6

0.25

659.87

873.43

520.76

628.35 5.02

39.00 −17.12

8

0.25

1088.51

1209.38

925.80

1060.77 2.62

14.01 −12.72

2

0.5

76.01

258.62

69.44

4

0.5

338.65

657.79

277.74

6

0.5

735.30

1002.92

624.92

720.76 2.02

39.15 −13.30

8

0.5

1222.42

1408.96

1110.96

1195.53 2.25

17.85 −7.07

2

0.75

82.92

289.95

81.01

4

0.75

369.88

762.91

324.03

368.59 0.35

6

0.75

809.60

1139.48

729.07

789.30 2.57

44.37 −7.63

8

0.75

1351.43

1589.19

1296.13

1303.82 3.65

21.89 −0.59

2

1.0

86.88

317.13

92.58

4

1.0

373.54

767.32

370.32

373.25 0.08

105.58 −0.78

6

1.0

823.74

1199.77

833.22

804.65 2.37

49.10 3.55

8

1.0

1442.15

1727.04

1481.29

1335.10 8.02

29.36 10.95

90.88 −16.36 184.58 −23.60 358.94 −5.65

83.26 −22.62

98.18 −15.54 195.34 −17.49 106.98 −12.09

101.75 −14.61 211.67 −9.01

Fig. 3.28 Relative edge width b1 /b of the proposed distribution under an increasing edge stress

3.5.1 Parametric Analysis Steel walls with different thicknesses are chosen to verify the proposed approaches. The height and width of the walls are constant and equal to 1200 mm and 1800 mm, respectively. The width to height ratio β is 1.5. The considered thicknesses are 2,

132

3 Steel Plate Shear Walls with Considering the Gravity Load …

Fig. 3.29 Stress of the wall under increasing edge stress

Fig. 3.30 Shear strength per unit thickness of plates under an increasing edge stress

Table 3.5 Simulated cases Width b (m)

Height a (m)

Thickness t (mm)

Axial stress σ e (MPa)

1.2

1.2

2/4/6/8

0/50/100/150/200

1.8

1.2

2/4/6/8

0/50/100/150/200

2.4

1.2

2/4/6/8

0/50/100/150/200

4, 6 and 8 mm. Consequently, the slenderness ratio is between 150 and 600. The walls were fully constrained at the bottom. For the vertical and top boundaries, all the degrees of freedom are constrained except for the vertical and horizontal ones. An elastic–plastic material with kinematic hardening is used to describe the steel of the boundary frame. The yield stress, Poisson’s ratio, elastic modulus, and tangent modulus are 335 MPa, 0.3, 210 GPa, and 2.1 GPa, respectively. An elastic perfectly plastic model with yield stress 235 MPa and elastic modulus 210 GPa is used to

29.98

L12t6s200

3.31

29.98

L12t6s150

L18t2s0

29.98

L12t6s100

59.99

29.98

L12t6s50

59.99

29.98

L12t6s0

L12t8s200

21.51

L12t4s200

L12t8s150

21.51

L12t4s150

59.99

21.51

L12t4s100

L12t8s100

21.51

L12t4s50

59.99

21.51

L12t4s0

59.99

5.38

L12t2s200

L12t8s50

5.38

L12t2s150

L12t8s0

5.38

5.38

L12t2s100

5.38

L12t2s0

L12t2s50

σ cr (MPa)

Specimen

45.71

46.76

46.76

46.76

46.76

46.76

46.36

46.36

46.36

46.36

46.36

45.94

45.94

45.94

45.94

45.94

45.48

45.48

45.48

45.48

45.48

θ (°)

1.80

0.66

0.76

0.93

1.20

1.20

0.46

0.54

0.66

0.93

1.20

0.39

0.45

0.56

0.79

1.20

0.20

0.23

0.28

0.39

1.20

be (m)

1.80

0.42

0.46

0.52

1.20

1.20

0.33

0.37

0.42

0.52

1.20

0.30

0.33

0.38

0.48

1.20

0.17

0.19

0.23

0.30

1.20

b1 (m)

427

839

924

1014

1057

1066

699

733

784

802

805

473

518

543

549

548

253

271

277

281

282

V FEM (kN)

423

1126

1126

1126

1126

1126

845

845

845

845

845

564

564

564

564

564

282

282

282

282

282

V stan (kN)

Table 3.6 Numerical and predicted shear strength of the shear wall V cos (kN)

423

714

840

935

1031

1126

588

678

743

792

845

401

459

502

534

564

207

235

256

271

282

V w (kN)

423

680

821

926

1031

1126

628

697

748

791

845

442

481

510

534

564

252

262

269

275

282

V (kN)

423

782

867

939

1031

1126

685

727

762

794

845

475

499

519

536

564

260

266

271

276

282

−1.01

−1.01

−1.01

−11.12 −18.92

21.90

−9.00

−8.68

−7.80 −14.88

5.66 −2.38

−15.83 5.66

−10.16

−7.62

−2.38

−4.50 −4.91

−5.16

−1.36

−6.69

−15.37 −1.26

−6.97

−11.26

4.92

−5.97

4.92

−2.62

−0.36

−18.21 −2.74

−3.29

−13.04

−7.51

−3.03

−7.76

2.79

−2.27

−3.70

2.79

0.09

r w (%)

0.09

r cos (%)

34.27

11.05

6.56

5.66

20.91

15.22

7.83

5.31

4.92

19.07

8.93

3.87

2.75

2.79

11.53

4.24

1.68

0.27

0.09

r stan (%)

(continued)

−1.01

−6.73

−6.16

−7.41

−2.38

5.66

−1.95

−0.83

−2.75

−1.01

4.92

0.31

−3.59

−4.44

−2.21

2.79

2.65

−1.71

−2.27

−2.01

0.09

r (%)

3.5 Shear Strength Considering the Gravity Load and In-plane Bending 133

22.01

22.01

22.01

36.00

36.00

36.00

36.00

36.00

2.77

2.77

L18t6s100

L18t6s150

L18t6s200

L18t8s0

L18t8s50

L18t8s100

L18t8s150

L18t8s200

L24t2s0

L24t2s50

13.23

L18t4s200

22.01

13.23

L18t4s150

22.01

13.23

L18t4s100

L18t6s50

13.23

L18t4s50

L18t6s0

3.31

3.31

L18t2s150

13.23

3.31

L18t2s100

L18t4s0

3.31

L18t2s50

L18t2s200

σ cr (MPa)

Specimen

Table 3.6 (continued)

θ (°)

45.94

45.94

47.50

47.50

47.50

47.50

47.50

46.95

46.95

46.95

46.95

46.95

46.36

46.36

46.36

46.36

46.36

45.71

45.71

45.71

45.71

be (m)

0.56

2.40

0.76

0.88

1.08

1.53

1.80

0.60

0.69

0.84

1.19

1.80

0.46

0.53

0.65

0.93

1.80

0.23

0.27

0.33

0.46

b1 (m)

0.46

2.40

0.54

0.59

0.68

0.83

1.80

0.45

0.50

0.57

0.72

1.80

0.37

0.41

0.48

0.61

1.80

0.21

0.23

0.28

0.37

V FEM (kN)

571

571

1403

1537

1596

1643

1653

1083

1168

1218

1247

1251

753

792

830

843

845

394

412

422

427

V stan (kN)

564

564

1686

1686

1686

1686

1686

1266

1266

1266

1266

1266

845

845

845

845

845

423

423

423

423

V cos (kN)

542

564

1210

1365

1459

1562

1686

887

1020

1119

1193

1266

606

693

757

804

845

310

353

384

406

V w (kN)

553

564

1258

1385

1464

1560

1686

984

1073

1139

1195

1266

701

747

781

809

845

387

399

407

414

V (kN)

554

564

1355

1436

1491

1564

1686

1057

1112

1157

1200

1266

738

767

791

813

845

396

404

410

415

−9.90 −10.31

−11.20 −13.75 −1.28 −5.08

20.15 −1.28 −1.23

9.66

−3.07

−1.28

−8.27 5.63

2.62

−5.02

−9.10

−18.07 −4.88

−8.16

−12.63

−8.57

−6.46

−8.11

1.96

1.21 −4.16

1.21 −4.30

−6.94

−19.58

1.96

1.96

16.91

8.41

3.98

1.56

1.21

12.19

6.70

1.78

−5.93

−4.02

−4.64

0.20

−5.69

−0.01

−0.01 −8.85

−1.72

−21.25

7.35 −0.01

−12.51

−3.39 −3.14

−8.96

2.71

0.30 −14.29

−2.91

−4.76

r w (%)

r cos (%)

r stan (%) −0.92

r (%)

(continued)

−2.90

−1.28

−3.41

−6.58

−6.58

−4.76

1.96

−2.40

−4.80

−4.95

−3.77

1.21

−1.97

−3.11

−4.72

−3.65

−0.01

0.52

−1.98

−2.82

−2.72

134 3 Steel Plate Shear Walls with Considering the Gravity Load …

11.08

11.08

11.08

11.08

20.00

20.00

20.00

20.00

20.00

30.02

30.02

30.02

30.02

30.02

–

–

L24t4s100

L24t4s150

L24t4s200

L24t6s0

L24t6s50

L24t6s100

L24t6s150

L24t6s200

L24t8s0

L24t8s50

L24t8s100

L24t8s150

L24t8s200

Mean

COV

2.77

L24t2s200

11.08

2.77

L24t2s150

L24t4s50

2.77

L24t2s100

L24t4s0

σ cr (MPa)

Specimen

Table 3.6 (continued)

θ (°)

–

–

48.17

48.17

48.17

48.17

48.17

47.50

47.50

47.50

47.50

47.50

46.76

46.76

46.76

46.76

46.76

45.94

45.94

45.94

be (m)

–

–

0.93

1.07

1.31

1.86

2.40

0.76

0.88

1.07

1.52

2.40

0.56

0.65

0.80

1.13

2.40

0.28

0.33

0.40

b1 (m)

–

–

0.67

0.74

0.85

1.05

2.40

0.58

0.64

0.74

0.93

2.40

0.46

0.51

0.60

0.77

2.40

0.25

0.29

0.34

V FEM (kN)

–

–

1925

2038

2128

2231

2241

1483

1552

1653

1693

1693

995

1079

1119

1135

1136

526

551

565

V stan (kN)

–

–

2242

2242

2242

2242

2242

1686

1686

1686

1686

1686

1126

1126

1126

1126

1126

564

564

564

V cos (kN)

–

–

1523

1764

1946

2085

2242

1177

1355

1487

1588

1686

806

922

1008

1072

1126

413

470

511

V w (kN)

–

–

1646

1827

1965

2084

2242

1324

1436

1520

1592

1686

949

1005

1046

1081

1126

520

534

545

V (kN)

–

–

1791

1904

2000

2091

2242

1415

1485

1544

1598

1686

994

1029

1059

1085

1126

530

540

547 −1.04 −4.70 −6.51

−21.45 −0.91 −5.52 −9.94

−0.91 −0.78

−4.61

−18.97 −0.46

−0.46 −0.46

0.524

6.16

16.51

10.02

5.36

0.49

0.06

13.65

8.62

1.94

−4.67

−8.43

0.198

−14.49

−20.85 0.484

−7.66 −10.35

−8.57 −13.44

−6.61

−6.55

−10.71

−20.62

0.06

−7.46

−12.69 0.06

−5.98 −8.06

−6.23 −10.04

−0.46

−6.87

−14.51

4.36 13.20

0.59

−0.91

−3.07

−14.76

2.29 7.27

−3.54

−9.44

r w (%)

r cos (%)

r stan (%) −0.14

r (%)

0.087

−2.59

−6.91

−6.57

−6.03

−6.29

0.06

−4.59

−4.30

−6.63

−5.60

−0.46

−0.11

−4.60

−5.43

−4.35

−0.91

0.94

−2.05

−3.03

3.5 Shear Strength Considering the Gravity Load and In-plane Bending 135

136 Fig. 3.31 Predicted errors for different slenderness ratios

3 Steel Plate Shear Walls with Considering the Gravity Load …

3.5 Shear Strength Considering the Gravity Load and In-plane Bending

137

Fig. 3.32 Predicted errors for different aspect ratios

describe the steel wall. A vertical load on the top of the right column results in 200 MPa vertical stress, representing gravity load, is applied. The vertical load on the left column is 25, 50, 75 and 100% of that on the right column. The corresponding stress gradient is then 0.25, 0.50, 0.75 and 1.0. After applying the vertical load, the pushover analysis is conducted. The configuration of the analyzed cases is listed in Table 3.7.

138

3 Steel Plate Shear Walls with Considering the Gravity Load …

Table 3.7 Parametric analysis cases

Thickness (mm)

Stress gradient

Axial stress (MPa)

2/4/6/8

0.25

200

2/4/6/8

0.50

200

2/4/6/8

0.75

200

2/4/6/8

1.00

200

3.5.2 Shear Capacity By considering the effect of gravity load through different vertical stress distributions, the shear capacities are summarized in Table 3.8. Table 3.8 shows that for most of the cases the prediction errors by the three-segment model are less than 5%. For cases t8r025, t8r050 and t8r075, the shear strength is underestimated by −10.19%, −10.09% and −8.40%, respectively. t2r100 is the only case that the predicted shear capacity is larger than the that obtained from FE analysis. That means the threesegment model slightly underestimates the shear capacity of the steel walls. The reason is that the buckling shear stress τ cr is not considered, which will underestimate the shear capacity, especially for a thick wall (e.g. t8r025, t8r050 and t8r075). In the case of different stress gradients, the shear capacity decreases with an increase of the stress gradient, i.e. the triangular stress compression (λ = 0) case has the largest shear capacity, while that of the uniform compression (λ = 1.0) case is the smallest. The reason is for the uniform compression case larger gravity load goes into the steel wall and results in smaller shear strength. The code model will significantly overestimate the shear strength, especially the walls with a small slenderness ratio and large stress gradient. For example, the code model overestimates about 7.33%, 12.22%, 16.90%, and 20.14% of the shear strength for cases t2r100, t4r100, t6r100 and t8r100, respectively. For the cases of t8r25, t8r50, t8r75 and t8r100, the prediction errors are 8.43%, 10.53%, 14.67% and 20.14%, respectively. For most of the cases, the cosine model and effective width model will underestimate the shear strength more than 15%.

3.5.3 Stress Gradient The influence of the compression stress gradient λ is further discussed. The width and height of the steel wall are respectively 1800 mm and 1200 mm. As shown in Fig. 3.33, the shear strength decreases with an increase of the stress gradient. In the case that the heavier loaded edge stress is100 MPa, when the thickness is 10 mm, all the three models have the same effects on the shear strength, but significantly smaller than that of the code model. When the thicknesses are 6 and 2 mm, with an increase of the stress gradient the shear strengths obtained from cosine and effective

3.31

13.23

29.76

52.91

3.31

13.23

29.76

52.91

3.31

13.23

29.76

52.91

3.31

13.23

29.76

52.91

t2r025

t4r025

t6r025

t8r025

t2r050

t4r050

t6r050

t8r050

t2r075

t4r075

t6r075

t8r075

t2r100

t4r100

t6r100

t8r100

52.91

29.76

13.23

3.31

46.10

25.90

11.46

2.79

38.74

21.62

9.35

1.98

30.46

16.62

6.63

0.61

47.50 0.417

46.95 0.385

46.36 0.365

45.71 0.344

47.50 0.410

46.95 0.378

46.36 0.358

45.71 0.337

47.50 0.400

46.95 0.368

46.36 0.348

45.71 0.328

47.50 0.386

46.95 0.352

46.36 0.332

0.417

0.385

0.365

0.344

0.423

0.393

0.375

0.355

0.432

0.404

0.388

0.369

0.444

0.419

0.403

0.386

0.536

0.448

0.368

0.205

0.576

0.475

0.386

0.212

0.616

0.503

0.407

0.223

0.653

0.532

0.429

0.237

0.536

0.448

0.368

0.205

0.600

0.492

0.399

0.219

0.677

0.547

0.439

0.239

0.772

0.617

0.492

0.269

1403

1083

753

394

1470

1129

769

403

1525

1159

787

408

1555

1182

808

413

1686

1266

845

423

1686

1266

845

423

1686

1266

845

423

1686

1266

845

423

1142

887

606

310

1211

933

634

323

1275

976

660

335

1334

1016

685

346

1155

902

617

317

1226

948

645

329

1282

985

667

339

1327

1014

685

347

1321

1057

738

396

1346

1071

746

398

1371

1085

753

400

1396

1099

760

401

20.14

16.90

12.22

7.33

14.67

12.18

9.95

4.96

10.53

9.21

7.40

3.59

8.43

7.08

4.65

2.46

−16.59 −8.40 −19.58 0.49 −18.02 −1.94 −16.70 −2.41 −17.64 −5.83

−19.56 −18.07 −18.64

−16.01 −5.09

−17.31 −21.27

−16.10 −2.98

−17.53 −17.59

−15.91 −10.09 −18.30 −1.25

−15.05 −6.41

−15.83 −19.85

−15.23 −4.33

−16.10 −16.39

−14.63 −10.19 −16.94 −2.10

−17.94

−14.24 −7.07

−14.10 −14.20

−15.17 −5.94

−15.19

r (%) −15.89 −2.79

−16.07

b’ e1 (m) b’ e2 (m) b1 (m) b2 (m) V FEM (kN) V code (kN) V cos (kN) V w (kN) V (kN) r code (%) r cos (%) r w (%)

45.71 0.311

specimen σ cr (MPa) σ min (MPa) A (°)

Table 3.8 Numerical and predicted shear capacity of the steel wall

3.5 Shear Strength Considering the Gravity Load and In-plane Bending 139

140

3 Steel Plate Shear Walls with Considering the Gravity Load …

(a) σ1=100 MPa

(a) σ1=200 MPa Fig. 3.33 Shear strength versus the stress gradient

width models decrease faster than that obtained from the three-segment model. In the case that the heavier loaded edge stress is 200 MPa, the difference of shear strengths obtained from different models is larger than that of 100 MPa cases. That means the gravity load has a higher effect for a stocky heavily loaded wall on the shear strength. Normally, the shear strength obtained from the code model is the largest, followed by the three-segment model, effective width model, and the cosine distribution model.

3.5 Shear Strength Considering the Gravity Load and In-plane Bending

141

The reason is that the code model does not consider the effect of the gravity load, while the cosine model and effective width model overestimate the influence of the gravity load. It should be noted that, for the 10 mm wall under 100 MPa compression, the shear strengths obtained from the cosine model and the three-segment model are almost the same (see Fig. 3.33a). In the case that the wall thickness is the same but the edge stress is 200 MPa, the cosine model and the three-segment model are still the same when the stress gradient is smaller than 0.5 (see Fig. 3.33b). When the stress gradient is larger than 0.6, a flattened portion occurred in the middle portion of the steel wall, and the shear capacity obtained from the three-segment model becomes larger in comparison with that obtained from the cosine model and effective width model.

3.6 Summary A stress distribution of thin steel walls under uniform compression and in-plane bending is proposed. This approach used a three-segment distribution in the two edges and the middle portion. In the edge portions, a cosine function is assumed. In the middle, the stress is assumed to be constant and equal to the elastic buckling stress. Together with the cosine distribution and the effective width model, these three vertical stress distributions are adopted to consider the effect of the gravity load and in-plane bending on the shear capacity of steel walls. By comparing the predicted results with that obtained from the code and FE analysis results, the following conclusions can be drawn: (1)

(2)

(3)

(4)

(5) (6)

The proposed distribution can predict the vertical stress along the width of both the square and rectangular steel walls properly. When the in-plane bending is considered, the proposed model can properly predict the vertical stress distribution of the walls with different stress gradients and slendernesses. For a wall with small slenderness ratio (8 mm) and small stress gradient (λ = 0.25), the proposed distribution and the cosine distribution are similar and much better than that of the effective width model. In the case of a large slenderness ratio or rectangular wall, the proposed distribution is much better than the cosine distribution because the former considers a flattened portion in the middle. The gravity load will significantly reduce the shear strength of the unstiffened shear walls, especially in the case of a heavily loaded stocky wall. This effect is not considered in the current standard and may lead to potential danger. The cosine model will underestimate the shear strength of the shear wall. The predicted accuracy decreases with increasing slenderness ratio and aspect ratio. The effective width model can predict the shear strength of a slender shear wall better than that of a stocky plate. The predicted accuracy changes little with the aspect ratio.

142

3 Steel Plate Shear Walls with Considering the Gravity Load …

(7)

The three-segment model, considering the effect of the gravity load and inplane bending, can accurately predict the shear strength of steel walls of different stress gradients and slenderness ratios. The shear capacity of the steel wall decreases with an increase of the stress gradient of the applied compression, i.e. shear capacity reduction under a triangular distribution compression (λ = 0) is the smallest, while that under a uniform compression (λ = 1.0) is the largest.

(8)

References 1. Thorburn LJ, Kulak GL, Montgomery CJ (1983) Analysis of steel plate shear walls. Report no. 107. Department of Civil Engineering, University of Alberta, Edmonton 2. Driver RG, KulakG L, Kennedy DJL, Elwi AE (1998) Cyclic test of a four-story steel plate shear wall. J Struct Eng-ASCE 124(2):112–120 3. Behbahanifard MR (2003) Cyclic behavior of unstiffened steel plate shear walls. Ph.D. dissertation. Department of Civil Engineering, University of Alberta, Edmonton, Alberta, Canada 4. Moghimi H, Driver RG (2013) Economical steel plate shear walls for low-seismic regions. J Struct Eng-ASCE 139(3):379–388 5. Qu B, Bruneau M, Lin CH, Tsai KC (2008) Testing of full-scale two-story steel plate shear wall with reduced beam section connections and composite floors. J Struct Eng-ASCE 134(3):364– 373 6. Caccese V, Elgaaly M, Chen R (1993) Experimental study of thin steel-plate shear walls under cyclic load. J Struct Eng-ASCE 119(2):573–587 7. Xu L-H, Zhang G, Xiao S-J, Li Z-X (2019) Development and experimental verification of damage controllable energy dissipation beam to column connection. Eng Struct 199:109660 8. Xu, Longhe, Xiao S, Li Z (2018) Hysteretic behavior and parametric studies of a self-centering RC wall with disc spring devices. Soil Dyn Earthq Eng 115:476–488 9. Xu, Longhe, Fan X, Li Z (2020) Seismic assessment of buildings with pre-pressed spring self-centering energy dissipation braces. J Struct Eng-ASCE 146(2):04019190 10. Sabouri-Ghomi S, Roberts TM (1992) Nonlinear dynamic analysis of steel plate shear walls including shear and bending deformations. Eng Struct 14(3):309–317 11. Bhowmick AK, Grondin GY, Driver RG (2014) Nonlinear seismic analysis of perforated steel plate shear walls. J Constr Steel Res 94(3):103–113 12. Berman JW, Bruneau M (2005) Experimental investigation of light-gauge steel plate shear walls. J Struct Eng-ASCE 131(2):259–267 13. Zirakian T, Zhang J (2015) Structural performance of unstiffened low yield point steel plate shear walls. J Constr Steel Res 112(9):40–53 14. Elgaaly M (1998) Thin steel plate shear walls behavior and analysis. Thin-Wall Struct 32(1– 3):151–180 15. Jin SS, Bai JL, Ou JP (2017) Seismic behavior of a buckling-restrained steel plate shear wall with inclined slots. J Constr Steel Res 129:1–11 16. Xu LH, Liu JL, Li ZX (2019) Cyclic behaviors of steel plate shear wall with self-centering energy dissipation braces. J Constr Steel Res 153:19–30 17. Xu LH, Liu JL, Li ZX (2018) Behavior and design considerations of steel plate shear wall with self-centering energy dissipation braces. Thin-Walled Struct 132:629–641 18. Kang THK, Martin RD, Park HG, Wilkerson R, Youssef N (2013) Tall building with steel plate shear walls subject to load reversal. Struct Des Tall Spec Build 22(6):500–520

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19. Lv Y, Li ZX (2016) Influences of the gravity loads on the cyclic performance of unstiffened steel plate shear wall. Struct Des Tall Spec Build 25(17):988–1008 20. Lv Y, Li ZX, Lu GX (2017) Shear capacity prediction of steel plate shear walls with precompression from columns. Struct Des Tall Spec Build 26(12):1–8. e1375 21. Lv Y, Wu D, Zhu YH, Liang X, Shi YC, Yang Z, Li ZX (2018) Stress state of steel plate shear walls under compression–shear combination load. Struct Des Tall Spec Build 27(6):e1450 22. Zhang XQ, Guo YL (2014) Behavior of steel plate shear walls with pre-compression form adjacent frame columns. Thin-Walled Struct 77(4):17–25 23. Sabelli R, Bruneau M (2007) Design guide 20: steel plate shear walls. American Institute of Steel Construction 24. Nie JG, Fan JS, Liu XG, Huang Y (2012) Comparative study on steel plate shear walls used in a high-rise building. J Struct Eng-ASCE 139(1):85–97 25. Tory RG, Richard RM (1981) Steel plate shear wall design. Struct Eng Rev 1(1):35–39 26. Timoshenko SP, Woinowsky-Krieger S (1959) Theory of plates and shells. McGraw-Hill, New York 27. Rhodes J, Harvey JM (1977) Examination of plate post-buckling behavior. J Eng Mech Div 103(3):461–478 28. Usami T (1982) Post-buckling of plates in compression and bending. J Struct Div 108(3):591– 609 29. Bradford MA (1989) Buckling of longitudinally stiffened plates in bending and compression. Can J Civ Eng 16(5):607–614 30. Jana P, Bhaskar K (2006) Stability analysis of simply-supported rectangular plates under nonuniform uniaxial compression using rigorous and approximate plane stress solutions. ThinWalled Struct 44(5):507–516 31. Bakker MC, Rosmanit M, Hofmeyer H (2007) Elastic post-buckling analysis of compressed plates using a two-strip model. Thin-Walled Struct 45(5):502–516 32. Scheperboer IC, Efthymiou E, Maljaars J (2016) Local buckling of aluminium and steel plates with multiple holes. Thin-Walled Struct 99:132–141 33. Komur MA, Sonmez M (2015) Elastic buckling behavior of rectangular plates with holes subjected to partial edge loading. J Constr Steel Res 112:54–60 34. Byklum E, Steen E, Amdahl J (2004) A semi-analytical model for global buckling and postbuckling analysis of stiffened panels. Thin-Walled Struct 42(5):701–717 35. Von-Karman T, Sechler EE, Donnell LH (1932) The strength of thin plates in compression. Trans ASME 54:53–57 36. Abdel-Sayed G (1969) Effective width of thin plates in compression. J Struct Div 95(10):2193– 2204 37. Narayanan R, Chow FY (1983) Effective widths of plates loaded uniaxially. Thin-Walled Struct 1(2):165–187 38. Vilnay O, Rockey KC (1981) A generalised effective width method for plates loaded in compression. J Constr Steel Res 1(3):3–12 39. Bedair O (2009) Analytical effective width equations for limit state design of thin plates under non-homogeneous in-plane loading. Arch Appl Mech 79(12):1173–1189 40. CSA-2001 (2001) Limit state design of steel structures. Canadian Standard Association CAN/CSA-S16-01, Mississauga, Ontario, Canada 41. AISC (2005) Steel construction manual, 13th edn. American Institute of Steel Construction, Chicago, USA 42. FEMA-450 (2003) NEHRP recommended provisions for seismic regulations for new buildings and other structures. Building Seismic Safety Council for Federal Emergency Management Agency, Washington, DC 43. ANSYS (2013) ANSYS mechanical APDL theory reference, Release 15. ANSYS Inc 44. Alinia MM, Shirazi RS (2009) On the design of stiffeners in steel plate shear walls. J Constr Steel Res 65(10–11):2069–2077 45. Zara´s J, Rhodes J, Królak M (1992) Buckling and post-buckling of rectangular plates under linearly varying compression and shear. Part 2: Experimental investigations. Thin-Walled Struct 14(2):105–126

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3 Steel Plate Shear Walls with Considering the Gravity Load …

46. Timler PA, Kulak GL (1983) Experimental study of steel plate shear walls. Structural engineering report no. 114, Department of Civil Engineering, University of Alberta, Edmonton, Alberta, Canada 47. AISC Committee, ANSI-AISC 341-05 (2005) Seismic provisions for structural steel buildings. American Institute of Steel Construction, Inc., Chicago, Illinois, USA

Chapter 4

Cross Stiffened Steel Plate Shear Walls Considering the Gravity Load

Former investigations have shown that the gravity load will significantly reduce the shear strength of the steel plate shear walls (SPSW), especially in the case of a stocky wall that has high buckling stress under heavy compression load. However, only a few have been carried out on stiffened SPSW, even though it has high compression buckling stress. The present study aims to narrow the knowledge gap by conducting a theoretical analysis and a numerical evaluation. The stiffened SPSW is divided into several sub-walls by vertical and horizontal stiffeners. For each sub-wall, a three-segment vertical stress distribution under the gravity load is proposed. The tension field stress of inclination tension strips of each sub-wall, with considering the effect of the gravity load through the proposed three-segment vertical stress distribution, is determined through the von Mises yield criterion. Shear strength of the wall is calculated as the sum of the shear strength of each sub-wall, while that of the boundary frame is determined according to a wall-frame model. To evaluate the proposed approach, an experimentally verified finite element model using the software ANSYS/LS-DYNA is developed. The results show that the proposed approach can accurately consider the gravity load effect of the stiffened SPSWs of different stiffeners configurations.

4.1 Introduction Steel plate shear wall (SPSW) has been widely used in tall structures and retrofit of reinforced concrete structures. It has high lateral load-bearing capacity and excellent stiffness. Post-buckling stiffened SPSWs with local buckling prevented in design were used in the USA, Japan, and Canada in the early years. After the work by Thorburn et al. [1], the unstiffened thin SPSW has been world-widely examined and used. Driver et al. [2], Behbahanifard [3], Qu et al. [4], Vian et al. [5] did some of the high-impact studies by conducting cyclic loading tests. Verma and Dipti [6] studied the shear contribution of boundary frames in resisting the lateral force of SPSW © Science Press 2022 Y. Lv, Steel Plate Shear Walls with Gravity Load: Theory and Design, https://doi.org/10.1007/978-981-16-8694-8_4

145

146

4 Cross Stiffened Steel Plate Shear Walls Considering …

systems. In recent years, because the buildings using SPSWs as lateral resistance system are much higher than that were built before, uncomfortable noise due to shear buckling under a wind load and compression buckling under the gravity load occurred frequently. Stiffened SPSW is an alternative to avoid the local buckling and attracting the researchers’ attention. Takanashi et al. [7] conducted some of the earliest tests and found that the stiffened SPSWs are very ductile. The drift angle, in some cases, can exceed 0.1. Sabouri-Ghomi and Sajjadi [8] conducted cyclic tests on two one-storey SPSWs, in which one with stiffeners and the other without stiffeners. They concluded that stiffened SPSWs increased 26% in the energy dissipation capacity and 51.1% in the shear stiffness, while the shear strength does not change much in comparison with the unstiffened SPSWs. Other research of stiffened SPSWs includes done by Alinia and Dastfan [9], Alavi and Nateghi [10], Nie et al. [11], etc. However, only a few of the above studies considered the effect of the gravity load on the shear capacity. In reality, the walls, especially the heavily loaded stocky and stiffened walls, always take a large amount of the gravity load. The Tianjin World Financial Center is the highest building using SPSWs [11] all over the world. This building has 75 storeys and a height of 336.9 m. Stiffened SPSWs with high stiffness were used to avoid the unpleasant buckling sound at serviceability condition under a wind load. The steel panels of the walls at the bottom several storeys have a thickness of 25 mm, while that at the top has a thickness of 18 mm, resulting in a slenderness ratio between 170 and 230. Although the engineers installed the infill steel panels 15 storeys slower than the boundary columns and beams, it still might take some gravity load. Other building structures using SPSWs [12] as lateral force resistance system include Shin Nittetsu Building in Tokyo, Olive View Medical Center at Sylmar in California, Hyatt Regency Hotel [13] in Texas, Canam Manac Group headquarters in Quebec and so on. The slenderness ratio of the above buildings is respectively in the range of 230–610, 250–300, 250–295 and 536. The SPSWs with a slenderness ratio in this range always enter the post-buckling stage under compressive load and inversely take a lot of the gravity load, which results in a shear capacity decrease under a wind load or an earthquake load. Zhang and Guo [14] proposed a reduction coefficient to account for the influence of the gravity load on the shear capacity of SPSWs. They concluded that the shear capacity of SPSWs is significantly impaired by the pre-compression from frame columns produced in the construction process, which must be considered in design. Lv et al. [15–18] assumed a vertical stress distribution of the wall under the gravity load. The effect of the gravity load was then considered by reducing the tension field stress of the inclination strips. Their results show that the compression buckling stress of the wall is the key factor to influence the shear strength of a wall with considering the gravity load. The SPSWs stiffened by the horizontal and vertical stiffeners have relatively high compression buckling stress. The shear capacity may decrease more in comparison with that of unstiffened SPSWs. Consequently, the shear capacity without considering the influence of the gravity load maybe overestimated, and it may lead to potential danger. In this work, the stiffened SPSW is treated as sub-walls divided by the stiffeners. A three-segment vertical stress distribution of the SPSWs under the gravity load is

4.1 Introduction

147

adopted to consider the effect of the gravity load on the shear capacity reduction. The tension field stress of the inclined tensile strip, considering the effect of the vertical stress distribution, is determined according to the von Mises yield criterion. By integrating the shear stress in respect to the width of each sub-wall, the shear strength of the wall can be calculated, while that of the boundary frame through a wall-frame interaction model. Results obtained from an experimentally verified finite element model is used to evaluate the proposed approach.

4.2 Shear Strength Model 4.2.1 Stress Under Gravity Load The vertical stress under the gravity load is used to consider the effect of the gravity load on the shear strength of the SPSWs. For a well-designed stiffened SPSW, it can experience local buckling instead of global buckling under shear and compression. The stiffened wall can be divided into several sub-walls by the horizontal and vertical stiffeners. Figure 4.1 shows a typical stiffened SPSW under the gravity load. Figure 4.1a is the stiffened SPSW before and after applying the gravity load on the columns. The columns are shortened under the gravity load. Because of the high stiffness of the boundary beam, the steel panel together with the vertical stiffeners are compressed and shortened with the boundary columns, which makes the panel experience uniform compression (see Fig. 4.1b). For each sub-wall, the vertical stress of the middle plane can be expressed [19] by the following equation:

Fig. 4.1 A typical stiffened SPSW

148

4 Cross Stiffened Steel Plate Shear Walls Considering …

σcos (x) = σ1 +

2m 4 (σ1 − σcr ) 2π x cos β4 + m4 sx

(4.1)

where σ cos (x) is the compression stress at x along the width at the middle height of a sub-wall, σ 1 is the uniform compressive load, σ cr is the compression buckling stress of a four-edge clamped panel, sx and sy is respectively the spacing of the vertical stiffeners and horizontal stiffeners, β is the aspect ratio of each sub-wall defined as sx /sy , m is the number of the half-waves in the longitudinal direction. In the case of a stocky square steel panel, i.e. the slenderness ratio is below 100, Eq. (4.1) is accurate enough to obtain the vertical stress distribution [20, 21]. However, the most widely used slenderness ratio of SPSWs is between 200 and 400 [12]. In this case, before buckling, the compressive stress is homogenously distributed. After buckling, the vertical stress of the edge portions increases with an increase of the compression load, while that of the central portion changes little and kept constant distribution close to the compression buckling stress. The edge portions always experience higher vertical stress than that of the middle portion. Consequently, as shown in Fig. 4.2, a three-segment distribution of each sub-wall, i.e. at each edge segment a cosine distribution changing from edge stress to compression buckling stress and in the middle segment a constant distribution of the buckling stress, is proposed [20, 22, 23] and expressed by the following equation ⎧σ +σ σe − σcr x e cr ⎪ + cos(π ) , 0 ≤ x ≤ b1i ⎪ ⎪ 2 2 b ⎪ 1i ⎨ σ (x) = σcr , b1i < x ≤ sx − b1i ⎪ ⎪ ⎪ ⎪ ⎩ σe + σcr + σe − σcr cos(π x − b ) , sx − b1i < x ≤ sx 2 2 b1i

Fig. 4.2 Schematic diagram of the three-segment vertical stress distribution

(4.2)

4.2 Shear Strength Model

149

where b1i is the width of the edge portion of the ith sub-wall (see Fig. 4.2). It can be determined by assuming the load computed from the effective width model [24] is the same as that of the proposed model, the width b1i can be expressed as b1i =

σe bei − sx σcr σe − σcr

(4.3)

where bei is the effective width of a sub-wall determined by the model proposed by von Karman [24]. σ cr is the buckling stress of a four-edge clamped plate under compression [19]. bei and σ cr can be determined by Eqs. (4.4) and (4.5), respectively. 

σcr × sx σe

(4.4)

π2E 12 (1 − μ2 )(sx /t)2

(4.5)

bei = σcr = kcr

where E and μ is respectively the modulus of elasticity and Poisson’s ratio. t is the wall thickness. k cr is the buckling coefficient [19]. σ e is the stress at the edge and at the vertical stiffeners (see Fig. 4.2), which is equal to the axial stress of the boundary columns. If the uniform compressive load σ 1 is known, it can also be specified by taking x = 0 or x = sx into Eq. (4.1).

4.2.2 Stress Under Compression and Shear Under the gravity load and shear force, assuming that the vertical stiffeners experience the same compression stress as that of the boundary columns, the stresses of a sub-wall can be divided into two stages, i.e. prior to buckling and after buckling. The horizontal direction, vertical direction and the direction along the thickness of the wall was defined as x, y and z axis, respectively. Ignoring the stresses in z direction, i.e. σ zz , σ yz and σ xz , the stresses of a sub-wall at shear buckling are determined by the following equations: ⎧ ⎪ ⎨ σx x = 0 σ yy = σ (x) ⎪ ⎩ σx y = τcr

(4.6)

where σ (x) is the vertical stress at position x along the width (see Eq. (4.2)), τ cr is the elastic shear buckling stress, which can be calculated by the equation bellow [19]: τcr = ks

π2E 12(1 − μ2 )(sx /t)2

(4.7)

150

4 Cross Stiffened Steel Plate Shear Walls Considering …

where k s is the elastic shear buckling coefficient. It is determined by [19] the following equation: ⎧  2 ⎨ 8.89 + 5.6 s y , for sx ≥ s y s  x ks = ⎩ 5.6 + 8.89 s y 2 , for sx ≤ s y sx

(4.8)

After buckling, the stresses of a sub-wall can be determined by the following equations: ⎧ 2 ⎪ ⎨ σx x (x) = σty (x) sin α σ yy (x) = σ (x) + σty (x) cos2 α ⎪ ⎩ σx y (x) = τcr + 0.5σty (x) sin 2α

(4.9)

where α is the angle of the inclination tension strip along the horizontal axis, σ xy (x), σ xx (x), σ yy (x) and σ ty (x) are respectively the shear stress, horizontal stress, vertical stress and tension field stress of the sub-wall along the width. Hereafter, σ ty (x), σ xx (x), σ yy (x) and σ xy (x) are written in short as σ ty , σ xx , σ yy and σ xy , respectively. According to the von Mises yield criterion, the following equation is obtained: 2 2 + σx2x + 6σx2y − 2 f y,p =0 J (σ ) = (σx x − σ yy )2 + σ yy

(4.10)

where f y,p is the yield strength of the infill steel panel. Substitute the values of σ xx , σ yy , and σ xy in Eq. (4.9) into Eq. (4.10), the yield field stress of the inclination tension strips σ ty is calculated by the following two equations: 2 =0 σty2 + [σ (x)(3 cos2 α − 1) + 3τcr sin(2α)]σty + σ (x)2 + 3τcr2 − f y,p

(4.11)

1 2 ] [σ (x)(3 cos2 α − 1) + 3τcr sin(2α)]2 − 4[σ (x)2 + 3τcr2 − f y,p 2 1 − [σ (x)(3 cos2 α − 1) + 3τcr sin(2α)] (4.12) 2

σty =

The inclination angle θ of the tension field strips to the vertical axis of a rigidly connected wall according to the research of Timler and Kulak [25] is

θ = tan

−1

4

  −1 tb 1 a3 1+ 1 + tH + 2 Ac Ab 360Ic b2

(4.13)

where a and b are respectively the height of a wall from centerlines of the boundary beams and the span from the centerlines of the boundary columns. Ac and Ab are respectively the cross-section area of the columns and beams. I c is the moment of inertia of the columns. Here, α = 90 − θ.

4.2 Shear Strength Model

151

4.2.3 Shear Capacity Generally, the SPSWs have two types of failure modes under in-plane shear loading, i.e. shear failure mode and flexure failure mode [8]. From the results by Park et al. [26], the shear strength is defined as the smaller of the shear strength obtained from shear failure mode and flexure failure mode. The shear capacity of the infill steel panel can be calculated by the following equation: b Vsp =

b t σxy dx = t

0

b (τcr + σty sin α cos α) dx = τcr bt + t

0

σty sin α cos α dx 0

(4.14) In the case of a shear failure mode [26], the shear strength V s is the sum of the shear strength of the boundary frame V sf and the steel panel V sp . V sf is determined by the following equation: Vsf = 2(Mpc1 + Mpc2 + Mpb1 + Mpb2 )/H

(4.15)

where H is the height of the wall, M pc and M pb are respectively the plastic moment of the boundary column and the boundary beam. In the case of a flexure failure mode, the shear strength V f is the sum of the shear strength of the boundary frame V ff and the steel panel V sp . V ff is determined by the following equation: Vff = Ac ( f y,c − σg,c )L/H − Pg δ/H

(4.16)

where L is the span of the wall, Ac is the area of the cross section of the column, f y,c is the yield stress of the steel of the column, σ g,c is the axial stress of the column under the gravity load Pg , and δ is the lateral displacement of the wall at the maximum shear strength [26]. The shear strengths of the wall experiencing a flexure failure mode and a shear failure mode are respectively determined by Eqs. (4.17) and (4.18): Vf = Vff + Vsp

(4.17)

Vs = Vsf + Vsp

(4.18)

The shear capacity of an SPSW is determined by the following equation: V = min ( Vf , Vs )

(4.19)

The calculation process of the proposed approach can be illustrated as in Fig. 4.3.

152

4 Cross Stiffened Steel Plate Shear Walls Considering …

Fig. 4.3 Illustration of the calculation process

4.2.4 Discussions of the Proposed Model It is assumed that the unstiffened SPSW has a height of 1.2 m, a width of 1.8 m and a thickness of 4 mm. Different stiffeners configurations are considered and the relative edge width (b1 /b) is shown in Fig. 4.4. In Fig. 4.4, HiLj means the wall has i horizontal and j vertical stiffeners. For example, H0L2 represents no horizontal stiffener and 2 vertical stiffeners. All the stiffeners have the same section of 6 mm × 60 mm on both sides. It shows that prior to buckling, the relative edge width is 0.5. After buckling, the relative width decreases rapidly following a square root function of the edge stress. The walls H0L2 and H1L2 have much larger edge width in comparison with walls H0L0, H1L1, H0L1 and H1L0. When the edge stress is 200 MPa, the relative edge widths are 0.20, 0.22, 0.21, 0.23, 0.28 and 0.37 for the wall H0L0, H1L0, H0L1, H1L1, H0L2 and H1L2, respectively. Assuming the inclination angle is 45°, the tension field stress obtained from Eq. (4.9) is normalized by the yield stress of the steel panel, σty∗ =σty / f y,p (see Fig. 4.5). The normalized tension field stress σty∗ decreases with an increase of the compression stress σ e . For different stiffener configurations, the normalized tension field stresses of walls H0L0 and H0L2 are almost the same. Walls H0L0, H0L1 and H0L2 have larger tension field stress than that of walls H1L0, H1L1 and H1L2. The reason is that the later ones have larger compression buckling stress and edge portion width, i.e. more of the gravity load is sustained by the wall, which decreased the tension field stress. In the case of different inclination angles (see Fig. 4.6), a larger inclination

4.2 Shear Strength Model

153

Relative effective width

0.5

0.4

0.3 H0L0 H0L1 H0L2

0.2 0

50

H1L0 H1L1 H1L2

100 150 Edge stress σe (MPa)

200

Fig. 4.4 Relative effective width of walls under different axial stress

Fig. 4.5 Normalized tension field stress of walls under different axial stresses

angle results in a smaller tension field stress except the cases that no vertical stress or a maximum edge stress (216 MPa for H1L1 and 233 MPa for H0L0) is considered. Figure 4.7 shows the shear capacity of the walls of different stiffener configurations under increasing edge stress. The shear strength is normalized by the shear

154

4 Cross Stiffened Steel Plate Shear Walls Considering …

Fig. 4.6 Normalized tension field stress of walls with different inclination angles

Fig. 4.7 Normalized shear strength of walls with different axial stresses

strength of the wall H0L0 without considering the gravity load. The horizontal stiffeners are more effective than the vertical stiffeners on increasing the shear strength. Prior to buckling, the normalized shear strength decreases linearly with an increase of the edge stress. After buckling, the shear strength of walls H0L1, H0L2, H1L1 and H1L2 decreases faster than that of walls H0L0 and H1L0. The reason is that walls H0L0 and H1L0 have smaller compression buckling stress and sustained less of the gravity load than that of walls H0L0, H0L2, H1L1 and H1L2, resulting in less effect of the gravity load on the shear strength reduction.

4.2 Shear Strength Model

155

Fig. 4.8 Shear of walls with different axial stress due to tension field action

Equation (4.11) indicates that the shear strength of a wall is comprised of the shear strength from the shear buckling stress and the tension field stress. Figure 4.8 shows the shear strength of the wall and the shear strength due to the tension field stress under increasing edge stress. The shear strength due to the shear buckling strength of wall H1L1 is much larger in comparison with that of wall H0L0, because the stiffeners significantly increased the shear buckling stress.

4.3 Numerical Analysis 4.3.1 Finite Element Model A finite element model (FEM) was built in the software ANSYS/LS-DYNA [27]. The FEM was firstly verified by using the existing experiment results, and then used to conduct a parameter analysis. In the FEM (see Fig. 4.9), the panel and stiffeners are described by layered shell elements SHELL163. The thickness of the layers through the thickness of the shell is determined by the locations of each layer. If the plane section assumption is adopted, the strain and curvature of each layer can be calculated by the strain and curvature of the neutral layer. With the strain and stress, the internal force of the element is then calculated by integrating the stress through the thickness in respect to the area of each layer. The boundary frame is modeled by the fibre beam elements BEAM161. The columns and beams are divided into dozens of fibres. The internal force of the beam elements then can be determined by integrating the stress

156

4 Cross Stiffened Steel Plate Shear Walls Considering …

Fig. 4.9 Finite element model

through the beam section in respect to the area of each fibre. Alinia and Shirazi [28] examined the initial imperfection between 1/100,000 and1/500 of the width of the panel, only a little difference was observed among the results. The reason is that a slender wall buckled very early, which does not have much influence on its postbuckling behaviour. Consequently, the initial imperfection is not considered in this study. The FEM verification was conducted by using the test results obtained from Driver et al. [2, 29] and Behbahanifard [3], which can be found in reference [15] that reliable results can be gotten by this FEM.

4.3 Numerical Analysis

157

Table 4.1 Configurations of the considered walls Wall

a (m)

b (m)

t w (mm)

sx (m)

sy (m)

t s (mm)

ws (mm)

Axial stress (MPa)

H0L0

1.2

1.8

4

1.8

1.2

6

120

50/100/150/200

H1L0

1.2

1.8

4

1.8

0.6

6

120

50/100/150/200

H0L1

1.2

1.8

4

0.9

1.2

6

120

50/100/150/200

H1L1

1.2

1.8

4

0.9

0.6

6

120

50/100/150/200

H0L2

1.2

1.8

4

0.6

1.2

6

120

50/100/150/200

H1L2

1.2

1.8

4

0.6

0.6

6

120

50/100/150/200

4.3.2 Parametric Analysis Parameter analysis is conducted on a wall with a height of 1.2 m, width of 1.8 m and thickness of 4 mm. The aspect ratio β defined as width to height is 1.5. The slenderness ratio, i.e. height to thickness, of the unstiffened SPSW is 300. The column section, i.e. depth × flange width × web thickness × flange thickness, is 300 mm × 300 mm × 20 mm × 20 mm. The top beam has the corresponding dimensions of 500 mm × 250 mm × 50 mm × 50 mm. This beam is designed very rigid to ensure that the tension field can form below the beam. All the beam to column joints adopt moment connections. Stiffeners with a section of 6 mm × 60 mm are welded vertically and horizontally on both sides of the wall. The wall is fully connected to the boundary elements. The bottom edge of the steel panel and bottom of the boundary columns are fully constrained. Out of plane, translational degree of freedom of the boundary frame is constrained. Uniform compression acting as the gravity load from the storeys above is added at the top edge of the wall. The behaviour of the boundary frame is described by an elastic plastic material model with considering a kinematic hardening criterion. The yield strength, Young’s modulus, and hardening modulus are 335 MPa, 210 GPa, and 2.1 GPa, respectively. The infill steel panel is described by an elastic-perfectly plastic model with a yield strength of 235 MPa and Young’s modulus of 210 GPa. Table 4.1 shows the cases considered in this study, in which t w and t s are the thickness of the wall and stiffener, respectively. sx and sy are the spacing of the vertical stiffeners and the horizontal stiffeners, respectively.ws is the width of the stiffeners.

4.3.3 Evaluation of the Vertical Stress Distribution Figure 4.10 shows the vertical stress distribution of the walls under the gravity load. The edge stress is 200 MPa. The black lines and red lines in Fig. 4.10 are stresses obtained from the FE analysis and the proposed model, respectively. Although the stress at the vertical stiffeners is slightly smaller than that at the boundary columns, the assumption that the vertical stress at the stiffeners is the same as that of the

158

4 Cross Stiffened Steel Plate Shear Walls Considering …

Fig. 4.10 Stress distribution obtained from FEM and proposed model

column is still acceptable. The proposed stress distribution is accurate enough to predict the vertical stress of the walls. Except for walls H0L0 and H1L0, no unstressed portion experienced in the middle portion because the compression buckling stress is significantly enhanced by the stiffeners. In such a case, the vertical stress distribution is the same as the cosine distribution of Eq. (4.1).

4.3.4 Evaluation of the Shear Strength The shear strength of the stiffened SPSW can be determined by integrating the shear stress through the width of each sub-wall using Eq. (4.11). In Table 4.2, V sp and Vsp,FEM are respectively the shear capacity obtained from the proposed model and that from FE analysis. r sp is the prediction error in comparison with the corresponding FE analysis result. In this work, the shear capacity of the boundary frame obtained by the frame-plane interaction model is the same and equals to 5545 kN, i.e. all the shear walls experienced a shear-dominated deformation mode. The results in the last two columns in Table 4.2 show that the frame-plane interaction model can accurately predict the shear strength. Hereafter, only the shear capacity of the steel panel is discussed. The inclination angle is assumed to be 45°. The edge stress obtained from the FE analysis is a little different from the original planed values of 50, 100, 150 and 200 MPa. Therefore, the actual edge stress is used in the prediction of the shear strength. Results in Table 4.2 show that the proposed model can accurately consider

H1L2

H1L1

H1L0

H0L2

0.9

188

0.6

0.9

141

45

0.9

1.8

227

94

1.8

170

0.9

1.8

47

1.8

0.6

190

57

0.6

114

0.6

95

143

0.9

221

0.6

0.9

166

48

0.9

111

1.8

195

0.9

1.8

55

1.8

98

146

H0L1

1.8

49

H0L0

sx (m)

σ e (MPa)

Wall

0.6

0.6

0.6

0.6

0.6

0.6

0.6

0.6

0.6

1.2

1.2

1.2

1.2

1.2

1.2

1.2

1.2

1.2

1.2

1.2

1.2

sy (m)

Table 4.2 Shear strength prediction

k

9.35

6.38

6.38

6.38

6.38

4.59

4.59

4.59

4.59

6.35

6.35

6.35

6.35

7.60

7.60

7.60

7.60

6.38

6.38

6.38

6.38

τ cr (MPa)

77.37

52.78

52.78

52.78

52.78

38.02

38.02

38.02

38.02

13.14

13.14

13.14

13.14

15.72

15.72

15.72

15.72

13.19

13.19

13.19

13.19

θ (degree)

45

45

45

45

45

45

45

45

45

45

45

45

45

45

45

45

45

45

45

45

45

V sp,FEM (kN)

940.86

727.17

791.38

821.51

872.41

749.42

803.82

856.21

886.53

703.94

750.78

817.84

830.84

714.45

767.8

802.29

820.77

727.98

817.45

803.56

834.97

Reduction (%)

–

16.65

9.29

5.83

–

15.47

9.33

3.42

–

15.27

9.64

1.56

–

12.95

6.45

2.25

–

12.81

2.10

3.76

–

V sp (kN)

909.59

751.29

815.65

863.08

895.16

790.27

826.91

854.34

877.21

653.25

728.53

784.28

823.52

655.59

737.09

795.93

838.35

772.16

801.49

822.91

841.90

r sp (%)

V sf,FEM (kN)

5440 5440

−1.05 −0.22

5460

−3.32

(continued)

5460

5460

5460

5460

5440

3.32

3.07

5.06

2.61

5.45

5440

5500

−7.20

2.87

5500 5500

−4.10

5490

−0.88 −2.96

5500 5500

−4.00 −8.24

5500 5500

−0.79

5500

2.14

6.07

5500 5500

2.41

5490

−1.95

0.83

4.3 Numerical Analysis 159

Wall

sx (m)

0.6

0.6

0.6

σ e (MPa)

90

134

179

Table 4.2 (continued)

0.6

0.6

0.6

sy (m)

k

9.35

9.35

9.35

τ cr (MPa)

77.37

77.37

77.37

θ (degree)

45

45

45

V sp,FEM (kN)

742.3

775.92

820.23

Reduction (%)

21.10

17.53

12.82

V sp (kN)

744.97

812.32

862.17

r sp (%)

0.36

4.69

5.11

V sf,FEM (kN)

5460

5460

5460

160 4 Cross Stiffened Steel Plate Shear Walls Considering …

4.3 Numerical Analysis

161

the reduction effect of the gravity load. The prediction error is less than 5% for most of the cases except that of H0L0, H0L1 and H0L2 under 200 MPa edge stress, their corresponding prediction errors are 6.07%, −8.24% and −7.20%, respectively. The vertical stiffener has less influence on the shear capacity of the wall, while the horizontal stiffener can significantly increase the shear strength. Such as the walls of H1L0, H1L1, and H1L2 under 50 MPa edge stress, the shear capacities are respectively 886.35 kN, 872.41 kN and 940. 86 kN. The corresponding strength of the walls of H0L0, H0L1, and H0L2 are 834.97 kN, 820.77 kN and 830.84 kN, respectively. The reason is that the horizontal stiffener can increase the shear buckling stress (see H1L0, H1L1 and H1L2). The reduction ratio of the shear capacity due to the gravity load, is defined as shear capacity of the wall under different edge stresses to that of the corresponding wall under 50 MPa edge stress. Obviously, the reduction ratio increases significantly with the edge stress, especially when the edge stress increases from 150 to 200 MPa. For the walls under the same edge stress, the reduction ratios of H0L0, H0L1 and H0L2 are similar and smaller than that of H1L0, H1L1 and H1L2. That means the horizonal stiffener has more unfavorable effect than that of the vertical stiffener in reducing the shear capacity. The reason is that a horizontal stiffener halves the height of the wall and significantly increases the compression buckling stress of the sub-wall and more gravity load is sustained by the infill steel panel, resulting in significant shear strength reduction.

4.4 Summary With considering the influence of the gravity load, the shear strength of the stiffened SPSW was discussed in this study. By dividing the wall into sub-walls by the stiffeners, a three-segment vertical stress distribution under the gravity load is proposed to consider a reduction of the tension field stress of each sub-wall. The results obtained from the proposed model and FE analysis were compared and the following conclusions can be drawn: (1)

(2)

(3)

The proposed approach, with considering the effect of the gravity load through the three-segment distribution, can reasonably predict the shear strength of the stiffened SPSWs. In most of the cases studied in this paper, the prediction errors are within 5%. The shear strength of a wall with horizontal stiffeners decreases more than that with vertical stiffeners due to the effect of the gravity load. The reason is that the walls with horizontal stiffeners have higher compression buckling stress and larger edge portion width, which make the wall sustain more of the gravity load. The horizontal stiffeners are more effective than the vertical stiffeners to increase the shear capacity of the walls due to the increase of the shear buckling stress.

162

4 Cross Stiffened Steel Plate Shear Walls Considering …

(4)

It should be noted that the rigidity and arrangement of stiffeners will influence the buckling modes of a steel plate shear wall, which could result in different shear strength of the stiffened steel plate shear walls.

References 1. Thorburn LJ, Kulak GL, Montgomery CJ (1983) Analysis of steel plate shear walls. Report no. 107. Department of Civil Engineering, University of Alberta, Edmonton 2. Driver RG, Kulak GL, Kennedy DJL, Elwi AE (1998) Cyclic test of a four-storey steel plate shear wall. ASCE J Struct Eng-ASCE 124(2):112–120 3. Behbahanifard MR (2003) Cyclic behavior of unstiffened steel plate shear walls. PhD dissertation, Department of Civil Engineering, University of Alberta, Edmonton, Alberta, Canada 4. Qu B, Bruneau M, Lin CH, Tsai KC (2008) Testing of full-scale two-storey steel plate shear wall with reduced beam section connections and composite floors. J Struct Eng-ASCE 134(3):364– 373 5. Vian D, Bruneau M, Tsai KC, Lin YC (2009) Special perforated steel plate shear walls with reduced beam section anchor beams. I: Experimental investigation. J Struct Eng-ASCE 135(3):211–220 6. Verma A, Dipti RS (2017) Estimation of lateral force contribution of boundary elements in steel plate shear wall systems. Earthq Eng Struct Dyn 46(7):1081–1098 7. Takahashi Y, Takeda T, Takemoto Y, Takagai M (1973) Experimental study on thin steel shear walls and particular steel bracing under alternating horizontal loadings. In: IABSE Symposium, resistance and ultimate deformability of structures acted on by well defined repeated loads, preliminary report, Lisbon, IABSE, Zurich, Switzerland 8. Sabouri-Ghomi S, Sajjadi SR (2012) Experimental and theoretical studies of steel shear walls with and without stiffeners. J Constr Steel Res 75:152–159 9. Alinia MM, Dastfan M (2007) Cyclic behaviour, deformability and rigidity of stiffened steel shear panels. J Constr Steel Res 63(4):554–563 10. Alavi E, Nateghi F (2012) Experimental study of diagonally stiffened steel plate shear walls. J Struct Eng-ASCE 139(11):1795–1811 11. Nie JG, Zhu L, Fan JS, Mo YL (2013) Lateral resistance capacity of stiffened steel plate shear walls. Thin-Walled Struct 67:155–167 12. Sabelli R, Bruneau M (2007) Design guide 20: steel plate shear walls. American Institute of Steel Construction 13. Tory RG, Richard RM (1981) Steel plate shear wall design. Struct Eng Rev 1(1):35–39 14. Zhang XQ, Guo YL (2014) Behavior of steel plate shear walls with pre-compression from adjacent frame columns. Thin-Walled Struct 77(4):17–25 15. Lv Y, Li ZX (2016) Influences of the gravity loads on the cyclic performance of unstiffened steel plate shear wall. Struct Des Tall Spec Build 25(17):988–1008 16. Lv Y, Li L, Wu D, Chen Y, Li ZX, Chouw N (2019) Shear-displacement diagram of steel plate shear walls with pre-compression from adjacent frame columns. Struct Des Tall Spec Build 28(5):e1585 17. Lv Y, Li ZX, Lu GX (2017) Shear capacity prediction of steel plate shear walls with precompression from columns. Struct Des Tall Spec Build 26(12):1–8. e1375 18. Lv Y, Wu D, Zhu YH, Liang X, Shi YC, Yang Z, Li ZX (2018) Stress state of steel plate shear walls under compression-shear combination load. Struct Des Tall Spec Build 27(6):e1450 19. Timoshenko SP, Woinowsky-Krieger S (1959) Theory of plates and shells. McGraw-Hill, New York

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20. Lv Y, Zhao Z, Lv JQ, Chouw N, Li ZX (2020) A stress distribution of thin rectangular steel wall under a uniform compression. Int J Struct Stab Dyn 20(3):2050037 21. Lv Y, Lv JQ, Zhao Z (2020) Vertical stress distributions of a thin rectangular steel wall under compression and in-plane bending. Int J Struct Stab Dyn 20(8):2050037 22. Lv Y, Zhao Z, Lv JQ, Li ZX, Chouw N (2020) Determination of shear strength of steel shear walls with three different vertical stress distributions for considering the gravity load effect. J Constr Steel Res 170:106113 23. Zhao Z, Lv Y (2021) Shear strength of steel plate shear walls considering the gravity load and in-plane bending moment effect by vertical stress distributions. J Build Eng 44(12):103012 24. von-Karman T, Sechler EE, Donnell LH (1932) The strength of thin plates in compression. ASME Trans 54:53–57 25. Timler PA, Kulak GL (1983) Experimental study of steel plate shear walls. Structural engineering report no. 114. Department of Civil Engineering, University of Alberta, Edmonton, Alberta, Canada 26. Park HG, Kwack JH, Jeon SW, Kim WK, Choi IR (2007) Framed steel plate wall behavior under cyclic lateral loading. J Struct Eng-ASCE 133(3):378–388 27. ANSYS (2013) ANSYS mechanical APDL theory reference, Release 15. ANSYS Inc 28. Alinia MM, Shirazi RS (2009) On the design of stiffeners in steel plate shear walls. J Constr Steel Res 65(10–11):2069–2077 29. Driver RG (1997) Seismic behavior of steel plate shear walls. PhD dissertation, Department of Civil Engineering, University of Alberta, Edmonton, Alberta, Canada

Chapter 5

Corrugated Steel Plate Shear Wall Considering the Gravity Load

Four scaled one-storey single-bay steel plate shear wall (SPSW) specimens with 2 flat panel and 2 corrugated panel were tested to determine their behavior under cyclic loadings. The shear walls had moment-resisting beam-to-column connections. One flat panel steel wall and one corrugated steel wall were tested with a vertical load about 486 kN at the top of the boundary columns through a force distribution beam to simulate the gravity load from the upper storeys. A horizontal cyclic load was then applied at the top of the specimens. The specimen behaviour, envelope curves, hysteresis curves were analyzed. A finite element model adopting layered shell element model was built and verified by the test results. Then parametric study was conducted to examine the influences of the gravity load on the shear capacity of the corrugated steel shear walls.

5.1 Introduction The corrugated steel plate shear wall (CSPSW) is an alternative to traditional shear walls with flat plates. Researchers conducted extensive studies on its mechanical properties through numerical simulation, theoretical analysis, and experiment. Roudsari et al. [1] simulated two different thicknesses of trapezoidal and sinusoidal corrugated steel plate shear walls with different geometrical openings under seismic load conditions. The results show that increasing the thickness of the corrugated steel plate can increase the load-bearing capacity. In steel plate shear walls without openings, trapezoidal plates can withstand greater lateral loads than sinusoidal plates. Farzampour and Laman [2] and Jeffrey studied the performance of unstiffened corrugated steel plate shear walls with and without openings. They also proposed an ultimate strength prediction method for corrugated steel plate shear walls optimized for rectangular opening positions. The results found that the initial stiffness and ductility of non-opening corrugated shear walls are generally higher than those of unstiffened shear walls, especially when the thickness is small. Zhao et al. [3] mainly conducted © Science Press 2022 Y. Lv, Steel Plate Shear Walls with Gravity Load: Theory and Design, https://doi.org/10.1007/978-981-16-8694-8_5

165

166

5 Corrugated Steel Plate Shear Wall Considering the Gravity …

nonlinear pushover and cyclic analysis on CSPSW and SPSW models, and conducted parameter studies on different panel and frame structures and gravity load effects. The results show that CSPSWs with deeper corrugations have higher lateral stiffness, lateral strength, and energy dissipation capacity than CSPSWs with shallower corrugations, while CSPSWs with shallower corrugations have higher lateral stiffness and ductility, but have higher lateral strength. The direction or structure of the corrugation has little effect on the cycle performance of CSPSWs, especially when no gravity load is applied. For all the cases studied, compared with SPSWs, CSPSWs have a stable hysteresis curve, almost no shrinkage, and are less sensitive to the effects of gravity loads or weaker boundary frames. Yadollahi et al. [4] used the finite element method to analyze and study the nonlinear behavior of the corrugated steel plate shear wall under the horizontal load. The research results concluded that in the equal-sized wall, the trapezoidal slab has higher energy dissipation capacity, ductility and ultimate bearing capacity than sinusoidal corrugated steel plate shear walls. Comparing with the trapezoidal corrugated board, the corrugation depth has a greater impact on the stiffness, ultimate bearing capacity and energy dissipation of the sinusoidal board. Broujerdian et al. [5] proposed a combination of corrugated and flat plates to form an effective composite section of seismic steel shear wall. Numerical studies show that the wall, in comparison with the traditional steel plate shear wall, has higher the elastic stiffness, ultimate strength and energy absorption capacity. The proposed wall also reduces the stress of the main frame, thereby reducing the ductility requirements and achieving a more economical system. Ghodratian-Kashan and Maleki [6] mainly studied the cyclic performance of the double corrugated steel plate shear wall. A single-compartment test piece was designed and simulated, and parameters including the direction of infill plates, the connection between infill plates and the column, the connection between the infill plates, the thickness of infill plates and the aspect ratio of the panel were studied. Responses of interest are force–deformation relationship, initial stiffness, ultimate strength and energy dissipation capacity. It is concluded that increasing the aspect ratio of the plate can effectively improve the cyclic performance of the double corrugated steel plate shear wall. Farzampour et al. [7] investigated the seismic performance of corrugated steel plate shear walls subjected to monotonic and cyclic loads. Numerical simulation results show that when there are no openings inside the steel plates, compared with the corresponding simple steel plate shear wall, the corrugated shear wall will develop the same damping capacity under seismic load. However, compared with the corresponding simple SPSWs, the corrugated shear wall with openings has quite good performance. Deng et al. [8] proposed a high efficiency analysis model (HEAM) of the modular steel structure corrugated steel shear wall. The skeleton curve and hysteretic regulation of CSPSWs are proposed. The general formula for determining the hysteresis model of CSPSW is deduced. Feng et al. [9] established a theoretical model to predict the overall elastic buckling performance of trapezoidal corrugated steel shear walls reinforced by horizontal steel strips. A finite element model was established to verify the theoretical model, and on this basis, parameter studies were carried out to discuss the influence of design parameters such as rigidity constants, width-to-height ratio, and thickness of the trapezoidal corrugated steel plate on elastic buckling. Dou et al. [10]

5.1 Introduction

167

analyzed the elastic shear buckling performance of the infill plate in the sinusoidal corrugated steel plate shear wall. A correction formula for the bending stiffness of sinusoidal corrugated plates is proposed, and a fitting equation for predicting global and local shear buckling loads is established based on finite element analysis. It is found that for the sinusoidal corrugated-filled plate, only global buckling and local buckling can be observed in the lowest buckling mode of the characteristic buckling analysis, and the interaction buckling behavior is not significant. For corrugated plates, the repetition number of corrugations or the ratio of plate width to corrugation wavelength is an important factor affecting the buckling load. Zhao et al. [11] proposed an improved empirical formula for the basic period of corrugated steel plate shear walls. Farzampour et al. [12] introduced a promising type of lateral-loadresisting system-corrugated steel shear wall with reduced boundary beam section (RBS-CSSW), and proposed a step-by-step design procedure based on corrugated plate-frame interaction (CPFI). Qiu et al. [13] used a quasi-static loading to conduct a cyclic quasi-static test on three CSPSW specimens of 1/3-scaled, two-story, singlecompartment, different corrugation orientations and structures, and a traditional flat steel plate to study the performance of corrugated steel plate shear wall. The test shows that, compared with the SPSW specimen, the CSPSW specimen has higher lateral stiffness and elastic buckling ability, and the pinch on the hysteresis curve is smaller. Wang et al. [14] tested six shear wall specimens (flat, vertical and horizontal corrugated steel plate shear walls and corresponding steel plate reinforced concrete composite shear walls) with a ratio of 1:2. The deformation capacity and failure mode of the specimen under cyclic loading are studied. The failure modes, deformation and energy dissipation capacity, stiffness and bearing capacity degradation characteristics of the corrugated steel plate shear wall are analyzed, and the design formulas for the shear capacity of the corrugated steel plate shear wall are proposed. The test results found that the lateral stiffness, ductility and energy dissipation capacity of the composite shear wall are better than those of the steel plate shear wall. Emami et al. [15] conducted hysteresis experiments on three 1/2-scale single-layer singlespan non-stiffened steel plate shear walls and corrugated transversely arranged, and corrugated longitudinally arranged trapezoidal corrugated steel plate shear walls. Cao and Huang [16] conducted cyclic tests on two CSPSWs with inelastic buckling and single-span and two-story, and established a numerical model to simulate the experiment. The results show that through the reasonable design of corrugated parameters, the corrugated steel plate shear wall can avoid elastic buckling, and the shear wall has high initial stiffness, buckling, strength, energy consumption and ductility. Wang et al. [17] used experiment and simulation methods to study the relationship between out-of-plane stiffness and in-plane stiffness. The failure modes of the three specimens under cyclic loading were tested. Nie et al. [18] conducted a shear performance experiment on 8 trapezoidal corrugated steel web I-beam specimens, and mainly discussed their influence on load–deflection response to study the shear performance of the plate. Based on linear elastic buckling analysis, a simplified calculation formula for the elastic shear buckling strength of trapezoidal corrugated steel plates considering three different shear buckling modes was derived. Further, a

168

5 Corrugated Steel Plate Shear Wall Considering the Gravity …

nonlinear buckling analysis was carried out to study the shear strength related to the initial geometric imperfections. However, little research considered the influence of the gravity load on the performance of the corrugated steel shear wall. To evaluate the adequacy of an analytical model available in the literature in predicting the influences of gravity loads on the cyclic performance of the corrugated SPSW physical experiments were performed in this chapter. Four scaled corrugated SPSWs under compression-shear interaction were designed and tested. The envelope curves, the hysteresis curve and the maximum shear capacity obtained from the experiments were then used in the evaluation of the numerical model.

5.2 Experiment Design The test parameter was the vertical load applied at the top boundary columns and the infill steel plate types. Test specimens are one-storey walls. The height of the specimen was 0.75 m, and the width was 1.1 m. The columns were 1 m apart from center to center. Figure 5.1 shows the size and configuration of the specimens. Totally four specimens with the same dimensions were constructed. The specimens can be divided into two groups, one with a corrugated steel plate (specimen B and D) and the other with a flat steel plate (specimen A and C). The plate thickness was 2.1 mm Q235 steel with the yield strength of 232 MPa. The size of the infill plate was 600 mm × 900 mm. The frame members are built-up sections made of Q345 steel with the yield strength of 456 MPa. The boundary columns, i.e. H-overall depth (d) × flange width (bf ) × web thickness (t w ) × flange thickness (t f ), have respectively the dimension of 100 mm × 100 mm × 6 mm × 8 mm. The top beam, connected to the actuator, has the corresponding dimensions of 150 mm × 100 mm × 6 mm × 9 mm. This beam was stiff to ensure a smooth transfer of the load to the tension field occurred below the beam. Moment connections were used at all beam-to-column joints. Connection of the beam flanges to the columns was constructed using complete penetration groove welds. The beam webs were welded to the column flange by two-sided fillet. The infill steel plate was welded directly to the boundary beams and columns with continuous fillet welds on both sides. Figure 5.1 shows a sketch of the test specimen with the vertical load and the lateral cyclic load. The constant vertical load of specimens A and B was 486 kN, which results in an axial stress of the boundary columns about 100 MPa. The vertical load was kept steady during the whole test. The beam, that distributed the vertical load, was hinged to the top of the two boundary columns. A hydraulic jack generated the vertical load at the top of the load distribution beam. It was supported by a stiff steel frame. Specimens C and D, for comparison, were only loaded in the horizontal direction without considering the effect of the vertical load. Horizontal cyclic load acted at the centerline of the top beam. A hydraulic jack, supported by a laboratory reaction wall, generated the horizontal load. Up to the first yield in the steel plate a force-controlled load was applied. Depending on the

5.2 Experiment Design

169

Fig. 5.1 Dimensions and set-up of the specimens SPSW1 and SPSW3 (units: mm)

Axial load

Load distribution beam 1075

100

25

30

100

50 150

600×950×2.0mm Steel plate

200

50

500

H100×6×100×8

HM 150×6×100×9

H100×6×100×8

20

25

HM200×6×150×9 20

160

20

870

(a) Dimensions of the specimens

(b) Set-up of the test

20

160

20

170

5 Corrugated Steel Plate Shear Wall Considering the Gravity …

Table 5.1 Properties of steel Steel

Thickness (mm)

Yield stress (MPa)

Ultimate stress (MPa)

Elongation

Q345

5/10

456

562

21.4

Q235

2

232

376

18.3

load combination the first yielding stage was achieved by increasing the horizontal load from ±50 kN to ±200 kN with an increment of 50 kN. In the subsequent load cycles, the loading protocol is different from any codes or standard and the existing research that a displacement-controlled loading was performed in the push direction until the specimen failure by increasing the displacement after each three cycles from 4 mm, while in the pull direction, the specimen was always pulled back with a maximum load of 200 kN. This kind of loading protocol was used because most of the structures under earthquake can not return back to the centerline because of the random property of the earthquake motions. Two linear variable differential transformers (LVDTs) were installed at the base of the boundary column and at the center line of the top beam. Table 5.1 lists the results of the coupon tests of the two materials Q345 and Q235. Three coupons were tested for each material, and the average value were used for the subsequent analyses (Fig. 5.2).

5.3 Test Results 5.3.1 Hysteretic Curve The hysteretic curves of the specimens are shown in Fig. 5.3. The curves were constructed with the lateral force and the corresponding top displacement. The force was recorded by the load cell and the top displacement was obtained by the LVDT installed at the centerline of the top beam. It demonstrates that all the steel walls experienced excellent deformation and energy dissipation capacity. No noticeable pinching phenomena can be seen in the hysteretic loops for all the specimens. The vertical load applied at the top of the boundary columns has little influence on the hysteretic performance of the walls. All the tests were stopped when the deformation was too large which may cause instability of the specimen. Figure 5.3a shows that the maximum deformation of the flat wall with and without considering the vertical load are about 62.77 mm and 44.4 mm, respectively, and the corresponding storey drifts are 9.66% and 6.83%. It should be noted that even at the end of the test, the force resistance is still increasing. The hysteretic curves were almost linear when the load was smaller than 300 kN. The yielding point is not very obvious. After yielded of the specimen, the unloading stiffness and reloading stiffness were slightly lower than the elastic yield stiffness. Because the reverse load was only 100 kN, significant unrecoverable deformation of the wall existed. The unrecoverable deformation increased

5.3 Test Results

171

Axial load

Load distribution beam 100

1075

100

25

30

50 150

600×950×2.0mm Steel plate

200

50

500

H100×6×100×8

HM 150×6×100×9

H100×6×100×8

20

25

HM200×6×150×9 20

160

160

20

870

20

15

30 °

(a) Dimensions of the specimens

30°

10

26

48 120

26

10

(b) Corrugated steel plate Fig. 5.2 Dimensions and set-up of the specimens SPSW2 and SPSW4 (units: mm)

20

172

5 Corrugated Steel Plate Shear Wall Considering the Gravity …

(c) Set-up of the test Fig. 5.2 (continued)

with the displacement at the top of the specimen. The loops of the repeated three cycles for each loading stage are almost coincident, that means the shear strength and stiffness changed little under the same loading stage. The reason may due to the loading protocol that the revered load is only 100 kN. Figure 5.3b shows that the hysteric curves of the corrugated steel walls with and without considering the vertical load do not like the flat shear wall.

5.3 Test Results

173

800

Shear (kN)

600 400 200 0 -10

10

30

50

Specimen A: with considering the vertical load Specimen C: without considering the vertical load

-200 -400

Displacement (mm)

(a) flat wall

800

Shear (kN)

600

400 200 0

-10

0 -200 -400

10

20

30

40

50

60

Specimen B: with considering the vertical load Specimen D: without considering the vertical load Displacement (mm)

(b) corrugated wall Fig. 5.3 Hysteretic curves of specimens with and without considering the vertical load

174

5 Corrugated Steel Plate Shear Wall Considering the Gravity …

5.3.2 Envelope Curves Figure 5.4 obviously shows that the corrugated wall exhibited a very clear yielding point. Table 5.2 shows the stiffness, strength, deformation at the yielding point and the ultimate point. The yielding displacement and yield force for specimen C are 3.51 mm and 412.37 kN, respectively, and those for specimen D are 4.79 mm and 453.06 kN. The corresponding yield drifts are 0.52 and 0.71%. The maximum deformation of the corrugated wall with and without considering the vertical load are about 62.32 mm and 44.79 mm, respectively, and the corresponding storey drifts are 9.82% and 6.63%. Before buckling, the shear strength and stiffness of the corrugated wall and the flat wall are almost the same. The shear-deformation curves coincident. After buckling, the shear strength of the corrugated walls is smaller than that of the flat shear walls. It should be noted that a small yield platform exhibited for the corrugated walls. The formation of the platform is due to the gradually flatten of the corrugation. With the increase of the displacement, tension field the corrugated wall was developed until the failure of the specimen, which results in an increase of the shear force.

900 800

700 Shear (kN)

600 500 Specimen A: Flat+gravity Specimen B: Corrugated+gravity Specimen C: Flat Specimen D: Corrugated

400 300 200 100 0 0

Fig. 5.4 Envelope curve

20

40 Displacement (mm)

60

80

5.3 Test Results

175

Table 5.2 Summary of test results Specimen K 0 (kN/mm) Py (kN) d y (mm) θy

Pmax (kN) d max (mm) θ max

μ

A

164

507.8

6.75

1.038 813.5

63.33

9.743

9.38

B

157

453.1

4.79

0.737 756.3

62.34

9.591 13.01

C

152

520.5

6.58

1.012 817.0

44.38

6.828

D

149

412.4

3.51

0.540 726.0

44.32

6.818 12.63

6.74

Note K 0 denotes the initial stiffness; Py denotes the yield load; d y denotes yield displacement; θ y denotes yield interstory drift; Pmax denotes the peak load; d max denotes the displacement at peak load; θ max denotes the interstory drift at peak load; μ denotes the ductility ratio

5.3.3 Specimens’ Behaviour Specimens A and C were firstly loaded by a vertical load at the top of the loading beam. The load is 480 kN and kept constant in the following test. The vertical load results in a vertical stress of the boundary columns of 100 MPa. Under pure gravity loads, no buckling occurred in specimens A and C. Then the horizontal cyclic load was added along the centerline of the to beam. All the four specimens experienced excellent deformation and energy dissipation capacity. The existing of the axial load at the top of the boundary columns has little influence on the failure mode of the steel plate shear walls. The horizontal corrugated steel wall has larger deformation capacity in comparison with that of the flat steel wall. The corrugated infill steel plate was firstly flattened before the tension field strips formed. The positions of the flattened corrugated plate are along the diagonal line of the upper joint to the opposite bottom joint. However, even at the end of the test, the corrugated plate was not flattened at the middle part along the beams and the middle height along the columns. The shear stiffness of the corrugated steel plate shear wall is smaller than that of the flat steel wall before the tension field strips were fully formed. Choosing specimen A as an example, until the top displacement reached 4 mm, there was no buckling in the infill steel plate. Tension field strips, formed from the lower left corner to the upper right corner, have an inclination angle near to 45°. During the first load of 300 kN, the first loud bangs occurred. In the subsequent cycles, these noises continued to occur. With an increase of the top displacement, various parts of the infill steel plate progressed to yield. A large residual deformation formed at the end of each pull or push loading with a further increase of residual deformation in the subsequent cycles. No tears occurred in the whole test. The ultimate deformation at the top of the specimen reached more than 62 mm. The force resistance only dropped by about 15%. At the end of the test, the compression column experienced local buckling in the flange, and only a slight out-of-plane deformation was observed (Fig. 5.5).

176

5 Corrugated Steel Plate Shear Wall Considering the Gravity …

(a) Specimen A

(b) Specimen B Fig. 5.5 Failure mode of specimens

5.3 Test Results

177

(c) Specimen C

(d) Specimen D Fig. 5.5 (continued)

178

5 Corrugated Steel Plate Shear Wall Considering the Gravity …

5.4 Finite Element Simulation 5.4.1 Finite Element Model Verification A finite element model was built in ANSYS/LS-DYNA. The SPSWs were simulated by shell elements. A plastic kinematic material model was used to describe the behavior of the steel. The yield strength, ultimate strength and fracture strain of different steels were shown in Table 5.1. The hardening modulus is chosen as 0.01 of the elastic modulus. The material model was shown in Fig. 5.6. Except the tested steel walls, the wall with a vertical corrugated infill steel plate was also considered. The walls with a flat infill steel plate, a horizontal corrugated infill steel plate and a vertical corrugated infill steel plate was show in Fig. 5.7. The corresponding finite element model was shown in Fig. 5.8. The element size is about 25 mm that was determined by a size sensitivity analysis of the model. Specimen A was simulated and the simulation results are shown in Fig. 5.9. Figure 5.9 shows that good agreement is obtained by comparing the hysteresis curves derived from the numerical model and the tests results. The pushover curve indicates a little higher shear strength and stiffness in comparison with the hysteresis curves. The failure mode of specimen A obtained from the simulation is shown in Fig. 5.10. It is indicated that the finite element model proposed in the chapter can

σ 0.01Es

fy

0 Fig. 5.6 Material model of steel

ε

5.4 Finite Element Simulation

179

(a) Flat infill steel plate

(b) Horizontal corrugated infill steel plate

(c) Vertical corrugated infill steel plate

Fig. 5.7 The simulated walls

180

5 Corrugated Steel Plate Shear Wall Considering the Gravity …

Fig. 5.8 The finite element model of the walls

(a) Flat infill steel plate

(b) Horizontal corrugated infill steel plate

(c) Vertical corrugated infill steel plate

5.4 Finite Element Simulation

900

181

Test Pushover FEM

Sjear (kN)

600

300

0 -5

5

-300

15

25

35

45

Displacement (mm)

Fig. 5.9 Comparison of the simulation and test results for specimen A

accurately describe the behavior of the steel plate shear walls and the can be used to analyze the performance of the walls of different configurations.

5.4.2 Parametric Analysis In this chapter, the height to thickness ratio, the axial stress of the boundary columns is analyzed for different steel walls. The parameters considered are shown in Table 5.3. The considered axial stresses are 50, 100, 150 and 200 MPa. The wall thicknesses are 2 mm, 4 mm and 6 mm, which result in slenderness ratios of 300, 150 and 100 respectively. The geometric sizes of the wall are the same to the tests, while the yield strength of the steel is chosen to be 235 MPa. About 36 pushover analyses are conducted.

5.4.3 Infill Steel Plate Types The shear-displacement relationship of the walls with a thickness of 4 mm and axial stresses of 50 MPa and 200 MPa is shown in Fig. 5.11. Figure 5.11 shows that the shear-displacement relationship curves of different infill steel plates are different.

182

5 Corrugated Steel Plate Shear Wall Considering the Gravity …

(a) Simulation result

(b) Test result Fig. 5.10 Failure mode obtained from the FEM and test Table 5.3 Parametric analysis Notation

Plate type

Thickness t (mm)

Axial load p (MPa)

FTtLp

Flat

2/4/6

50/100/150/200

VTtLp

Vertical corrugated

2/4/6

50/100/150/200

HTtLp

Horizontal corrugated

2/4/6

50/100/150/200

5.4 Finite Element Simulation

183

1400

Shear (kN)

1200 1000 800

FT4L50 FT4L200 HT4L50 HT4L200 VT4L50 VT4L200

600 400 200 0 0

10

20 30 Displacement (mm)

40

50

Fig. 5.11 Shear-displacement relationship of different infill plate types

The shear of the flat walls always increases with an increase of the displacement, while those of the corrugated walls experience elastic deformation until gradually buckle of the infill plate. The shear strength of the corrugated wall at buckling is larger than that of the flat wall. After buckling occurred, the shear of the corrugated walls began to decrease with the displacement. The buckling strength of the horizontal corrugated wall is smaller than that of the vertical corrugated walls. The shear force of the horizontal corrugated wall also decreases faster than that of the vertical corrugated wall. With the increase of the displacement, the corrugated walls are gradually flattened and the shear force begin to increase. The horizontal corrugated wall begins to increase at a displacement of 12 mm, while that of the vertical corrugated wall at a displacement of 26 mm. When the displacement is about 30 mm in this study, the shear force of the two corrugated walls is almost equal. After 30 mm, the shear force of the horizontal corrugated wall increases faster than that of the vertical corrugated wall. In all the cases, the flat wall has the largest shear strength in all the three types of walls. The axial stress due to the gravity load of the boundary columns has some influences on the shear force of the wall. Before buckling, the gravity load influences the flat wall significantly, but after buckling, the gravity load has larger influence on the horizontal corrugated wall. It seems that the gravity load has little influence on the vertical corrugated walls in this study.

184

5 Corrugated Steel Plate Shear Wall Considering the Gravity …

5.4.4 Vertical Load

1800 1600 1400 1200 1000 800 600 400 200 0

FT6L100 FT6L150 FT6L200 FT6L50

0

10

20 30 Displacement (mm)

40

50

Shear (kN)

(a) Flat infill steel plate 1800 1600 1400 1200 1000 800 600 400 200 0

HT6L100 HT6L150 HT6L200 HT6L50

0

10

20 30 Displacement (mm)

40

50

(b) Horizontal corrugated infill steel plate

Shear (kN)

Fig. 5.12 Shear-displacement relationship of the walls with a slenderness ratio of 100

Shear (kN)

Figures 5.12, 5.13 and 5.14 show the shear displacement relationship of the wall of different slenderness and wall types. It indicated that the gravity load has some effects

1800 1600 1400 1200 1000 800 600 400 200 0

VT6L100 VT6L150 VT6L200 VT6L50

0

10

20 30 Displacement (mm)

40

(c) Vertical corrugated infill steel plate

50

5.4 Finite Element Simulation 1400 1200 Shear (kN)

1000

800

FT4L100 FT4L150 FT4L200 FT4L50

600 400 200 0 0

10

20 30 Displacement (mm)

40

50

(a) Flat infill steel plate 1400 1200 Shear (kN)

1000 800

HT4L100 HT4L150 HT4L200 HT4L50

600 400 200 0 0

10

20 30 Displacement (mm)

40

50

(b) Horizontal corrugated infill steel plate 1200 1000 Shear (kN)

Fig. 5.13 Shear-displacement relationship of the walls with a slenderness ratio of 150

185

800 600

VT4L100 VT4L150 VT4L200 VT4L50

400 200 0

0

10

20 30 Displacement (mm)

40

(c) Vertical corrugated infill steel plate

50

Shear (kN)

900 800 700 600 500 400 300 200 100 0

FT2L50 FT2L100 FT2L150 FT2L200

0

10

20 30 Displacement (mm)

40

50

(a) Flat infill steel plate 700 600 500 Shear (kN)

Fig. 5.14 Shear-displacement relationship of the walls with a slenderness ratio of 200

5 Corrugated Steel Plate Shear Wall Considering the Gravity …

400

HT2L100 HT2L150 HT2L200 HT2L50

300 200 100 0 0

10

20 30 Displacement (mm)

40

50

(b) Horizontal corrugated infill steel plate

700 600 500 Shear (kN)

186

400 300

VT2L100 VT2L150 VT2L200 VT2L50

200 100 0 0

10

20 30 Displacement (mm)

40

(c) Vertical corrugated infill steel plate

50

5.4 Finite Element Simulation

187

Table 5.4 Shear strength decrease due to the gravity load Walls

50 MPa

100 MPa

FT6

1109.35

1103.58

FT2

389.48

HT4

834.33

HT2

296.58

280.8

VT6

1257.83

VT4

1004.25

150 MPa

200 MPa

−0.5%

1091.88

−1.6%

408.13

4.7%

394.83

1.4%

378.7

841.83

0.9%

841.96

0.9%

858.94

2.9%

−5.3%

271.27

−8.5%

267.23

−8.9%

1262.97

0.4%

1263.67

0.5%

1255.4

−0.2%

1005.75

0.1%

1004.86

0.06%

1001.07

−0.3%

1076.53

−3% −2.8%

on the shear strength of the walls. The influence on the wall with a small slenderness, thick wall, is larger than that of a thin wall. For the three types of walls, the influence on the horizontal corrugated wall is largest, followed by the flat wall and the vertical corrugated wall. As shown in Table 5.4, for example, the shear strength of wall FT6 decreases about 0.5%, 1.6% and 3% when the axial stress of the boundary columns increases from 50 to 100 MPa, 150 MPa and 200 MPa respectively. In the case of HT2, the shear strength decreases about 5.3%, 8.5% and 8.9% when the axial stress of the boundary columns increases from 50 to 100 MPa, 150 MPa and 200 MPa respectively. In the case of VT4, the shear strength almost has no difference with the increase of the axial stress of the columns.

5.5 Summary Experimental studies have been conducted on three types of steel shear walls, i.e., the flat shear wall, horizontal corrugated shear wall and vertical corrugated shear wall. The experimental results verified that the proposed numerical simulation model is effective and accurate. Then the parametric study was carried out on walls with different slenderness and gravity loads. The following conclusions can be obtained: (1) (2) (3)

(4)

The proposed numerical simulation method can be used to simulate the corrugated steel plate shear wall. The gravity load has largest influence on the horizontal corrugated wall, followed by the flat wall and the vertical corrugated wall. For the three types of walls, the vertical corrugated wall has the best performance regarding the shear strength, deformation capacity and the effect of the gravity load. The shear strength of the corrugated steel plate shear wall experienced elastic stage, decrease stage and increase stage with the increase of the displacement. For the vertical corrugated wall with a small slenderness, i.e., VT6, a stable post buckling stage exists.

188

5 Corrugated Steel Plate Shear Wall Considering the Gravity …

References 1. Roudsari SS, Soleimani SM, Hamoush SA (2021) Analytical study of the effects of opening characteristics and plate thickness on the performance of sinusoidal and trapezoidal corrugated steel plate shear walls. J Construct Steel Res 182(7):106660 2. Farzampour A, Laman JA (2015) Behavior prediction of corrugated steel plate shear walls with openings. J Constr Steel Res 114(11):258–268 3. Zhao Q, Sun J, Li Y et al (2017) Cyclic analyses of corrugated steel plate shear walls. Struct Des Tall Spec Build 26(16):e1351 4. Yadollahi Y, Pakar et al (2015) Evaluation and comparison of behavior of corrugated steel plate shear walls. Latin Am J Solids Struct 12(4):763–786 5. Broujerdian V, Ghamari A, Abbaszadeh A (2021) Introducing an efficient compound section for steel shear wall using flat and corrugated plates. Structures 33(1):2855–2871 6. Ghodratian-Kashan SM, Maleki S (2021) Numerical investigation of double corrugated steel plate shear walls. J Civ Eng Construct 10(1):44–58 7. Farzampour A, Mansouri I, Hu JW (2018) Seismic behavior investigation of the corrugated steel shear walls considering variations of corrugation geometrical characteristics. Int J Steel Struct 18(4):1–9 8. Deng EF, Zong L, Wang HP et al (2020) High efficiency analysis model for corrugated steel plate shear walls in modular steel construction. Thin-Walled Struct 156:106963 9. Feng L, Sun T, Ou J (2021) Elastic buckling analysis of steel-strip-stiffened trapezoidal corrugated steel plate shear walls. J Construct Steel Res 184:106833 10. Dou C, Jiang ZQ, Pi YL et al (2016) Elastic shear buckling of sinusoidally corrugated steel plate shear wall. Eng Struct 121(8):136–146 11. Zhao Q, Qiu J, Zhao Y et al (2020) Estimating fundamental period of corrugated steel plate shear walls. KSCE J Civ Eng 24(2):3023–3033 12. Farzampour A, Mansouri I, Lee CH et al (2018) Analysis and design recommendations for corrugated steel plate shear walls with a reduced beam section. Thin-Walled Struct 132:658–666 13. Qiu J, Zhao Q, Cheng Y et al (2018) Experimental studies on cyclic behavior of corrugated steel plate shear walls. J Struct Eng-ASCE 144(11):04018200 14. Wang W, Ren Y, Lu Z et al (2019) Experimental study of the hysteretic behaviour of corrugated steel plate shear walls and steel plate reinforced concrete composite shear walls. J Constr Steel Res 160(9):136–152 15. Emami F, Mofid M, Vafai A (2013) Experimental study on cyclic behavior of trapezoidally corrugated steel shear walls. Eng Struct 48(48):750–762 16. Cao Q, Huang J (2018) Experimental study and numerical simulation of corrugated steel plate shear walls subjected to cyclic loads. Thin-Walled Struct 127:306–317 17. Wang W, Luo Q, Sun Z et al (2021) Relation analysis between out-of-plane and in-plane failure of corrugated steel plate shear wall. Structures 29:1522–1536 18. Nie JG, Zhu L, Tao MX et al (2013) Shear strength of trapezoidal corrugated steel webs. J Constr Steel Res 85(6):105–115

Chapter 6

Summary and Future Work

The research presented in this book focused on development of an improved seismic design method of SPSWs with considering the effects of the gravity load. This book contains a detailed description of the main development efforts for the new design method. The following sub-sections highlight the most important findings, followed by recommendations for future research.

6.1 Main Conclusions (1)

(2)

The proposed distribution can predict the vertical stress along the width of both the square and rectangular steel walls properly. When the in-plane bending is considered, the proposed model can properly predict the vertical stress distribution of the walls with different stress gradients and slendernesses. For a wall with small slenderness ratio (8 mm) and small stress gradient (λ = 0.25), the proposed distribution and the cosine distribution are similar and much better than that of the effective width model. In the case of a large slenderness ratio or rectangular wall, the proposed distribution is much better than the cosine distribution because the former considers a flattened portion in the middle. The load-bearing capacity decreases with a decrease of the stress gradient because the lighter loaded portion of the wall is not fully used. The proposed distribution can be adopted to consider the negative effect of the gravity load and in-plane bending on the shear strength of steel walls. Based on the post plastic buckling strength of the infill steel plate and effective length concept, the infill steel plate is partitioned into three zones, for zone I and zone III, the compression stress of the infill steel plate is assumed to equal to the axial stress of the boundary column, and for zone II, the compression stress of the infill steel plate is elastic buckling stress. Then the compression stress distribution of the infill steel plate is proposed. Simulation results match well with the theoretical analysis results. The infill steel plate is partitioned

© Science Press 2022 Y. Lv, Steel Plate Shear Walls with Gravity Load: Theory and Design, https://doi.org/10.1007/978-981-16-8694-8_6

189

190

(3)

(4)

6 Summary and Future Work

into three zones, for zone I and zone III, the compression stress of the infill steel plate is assumed to equal to the axial stress of the boundary column, and for zone II, the compression stress of the infill steel plate is elastic buckling stress. Both the yielding stress and shear capacity of the infill steel plate will decrease due to the gravity loads applied at the top of the boundary columns. The inclination angle of zone I and III measured from the vertical direction is larger than that of zone II, because the compression stress in zone I and III is greater than that of zone II. Shear-displacement diagram of SPSW under compression-shear interaction is obtained, and the load-carrying capacity and deformation according to the buckling of infill steel plate, yield of Zone I and Zone III, yield of Zone II and yield of the boundary frame are presented and verified. The gravity load will significantly reduce the shear strength of the unstiffened shear walls, especially in the case of a heavily loaded stocky wall. This effect is not considered in the current standard and may lead to potential danger. The cosine model will underestimate the shear strength of the shear wall. The predicted accuracy decreases with increasing slenderness ratio and aspect ratio. The effective width model can predict the shear strength of a slender shear wall better than that of a stocky plate. The predicted accuracy changes little with the aspect ratio. The three-segment model, considering the effect of the gravity load and in-plane bending, can accurately predict the shear strength of steel walls of different stress gradients and slenderness ratios. The shear capacity of the steel wall decreases with an increase of the stress gradient of the applied compression, i.e. shear capacity reduction under a triangular distribution compression (λ = 0) is the smallest, while that under a uniform compression (λ = 1.0) is the largest. The shear strength of a wall with horizontal stiffeners decreases more than that with vertical stiffeners due to the effect of the gravity load. The reason is that the walls with horizontal stiffeners have higher compression buckling stress and larger edge portion width, which make the wall sustain more of the gravity load. The horizontal stiffeners are more effective than the vertical stiffeners to increase the shear capacity of the walls due to the increase of the shear buckling stress. It should be noted that the rigidity and arrangement of stiffeners will influence the buckling modes of a steel shear wall, which could result in different shear strength of the stiffened steel shear walls.

In practical project, it is strongly recommended that the shear capacity and stiffness should be reduced by a factor. This factor mainly influenced by the depth-thickness ratio of the infill steel plate and the magnitude of the gravity loads on the infill steel plate.

6.2 Future Work

191

6.2 Future Work SPSWs have been studied for more than half a century. However, some issues are still not clear and need to be further investigated. (1)

(2)

(3) (4)

Experimental testing of thin wall under uniform compression and linear changed compression should be a major research effort. The tests are essential to verify the effectiveness of the proposed three-segment distribution model. Seismic performance of SPSW under earthquake excitation should be conducted. The existing research mainly focused on the static properties of SPSW, however, as the main lateral loads resisting system, the stiffness, strength, energy dissipation capacity under earthquake excitations are more reasonable to reflect the behaviour of the walls. Shaking table tests should be carried out on SPSWs to check the availability of the existing analytical models. New SPSW like the corrugated SPSW is an alternative to traditional shear walls with flat plates.

A Short Description of the Book This book is written by subject experts based on the recent research results in steel plate shear walls with considering the gravity load effect. It establishes a vertical stress distribution of the walls under compression and in-plane bending load and an inclination angle of the tensile field strips. The stress throughout the inclined tensile strip, considering the effect of the vertical stress distribution, is determined using the von Mises yield criterion. The shear strength is calculated by integrating the shear stress along the width. The proposed theoretical model is verified by tests and numerical simulations. Researchers, scientists and engineers in the field of structural engineering can benefit from the book. As such, this book provides valuable knowledge, useful methods and practical algorithms that can be considered in practical design of building structures adopting a steel plate shear wall system.