Behavior and Design of High-Strength Constructional Steel 0081029314, 9780081029312

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Behavior and Design of HighStrength Constructional Steel

Woodhead Publishing Series in Civil and Structural Engineering

Behavior and Design of High-Strength Constructional Steel Edited by

Guo-Qiang Li Yan-Bo Wang

Woodhead Publishing is an imprint of Elsevier The Officers’ Mess Business Centre, Royston Road, Duxford, CB22 4QH, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, OX5 1GB, United Kingdom Copyright © 2021 Elsevier Ltd. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-08-102931-2 (print) ISBN: 978-0-08-102932-9 (online) For information on all Woodhead Publishing publications visit our website at https://www.elsevier.com/books-and-journals

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Contents

List of Contributors Preface

xiii xv

Part I General

1

1

3

Introduction Yan-Bo Wang and Guo-Qiang Li 1.1 Introduction 1.1.1 Advantages and limits of high-strength constructional steel 1.1.2 Current situation of steel production and consumption 1.1.3 Applications of high-strength constructional steel References

Part II 2

Materials

Material properties and statistical analysis of high-strength steels Yan-Bo Wang, Guo-Qiang Li and Yuan-Zuo Wang 2.1 Introduction 2.2 Mechanical properties of high-strength steels 2.3 Three-dimensional constitutive model of high-strength steels 2.3.1 Fundamental definition 2.3.2 Applicability of von Mises yield criterion for high-strength steel 2.3.3 Effect of stress triaxiality and Lode angle 2.3.4 Proposed plasticity model of high-strength steel 2.3.5 Validation of the proposed plasticity model for HSS 2.4 Statistical analysis of Q690 high-strength steel 2.4.1 Data retrieval and statistics of material properties 2.4.2 Structural reliabilities using Q690 steel 2.4.3 Target reliability 2.4.4 Partial coefficients of resistance and design strength 2.5 Summary References

3 3 6 6 14

15 17 17 18 18 18 27 40 43 45 49 49 51 56 56 59 59

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Contents

Hysteretic behavior of high strength steels under cyclic loading Guo-Qiang Li, Yan-Bo Wang and Fei-Fei Sun 3.1 Introduction 3.2 Cyclic behavior of high strength steels 3.2.1 Experimental program 3.2.2 Hysteretic behavior 3.3 Hysteretic model and verification 3.3.1 Hysteretic model 3.3.2 Verification 3.4 Summary References Uniform material model for constructional steel Le-Tian Hai, Guo-Qiang Li and Yan-Bo Wang 4.1 Introduction 4.2 Experimental observations 4.2.1 Data source 4.2.2 Cyclic hardening/softening behavior of the yielding surface and bounding surface 4.2.3 Kinematic hardening rule 4.2.4 Degradation of the elastic modulus 4.3 Theoretical modeling 4.3.1 Framework of constitutive modeling 4.3.2 Two-step hardening and three-step softening hardening/ softening model 4.3.3 Kinematic hardening model 4.3.4 Elastic stiffness degradation 4.3.5 Tension compression asymmetry 4.4 Capability of the constitutive model 4.4.1 Isotropic hardening/softening evolution surface 4.4.2 Kinematic hardening evolution surface 4.4.3 Cyclic behaviors described in the constitutive model of various steels 4.5 Simplification of cyclic hardening/softening constitutive model 4.6 Cyclic parameter calibration 4.6.1 Calibration method 4.6.2 Calibration of model parameters 4.6.3 Verification of calibration result 4.7 The evolution laws of constitutive model parameters 4.7.1 Evolution laws of monotonic parameters 4.7.2 Evolution laws of cyclic model parameters 4.8 Simplified evaluation approach for cyclic model parameter 4.9 Comparison of prediction results on different structural steels 4.10 Summary References

63 63 64 64 67 74 74 86 91 92 93 93 95 95 96 103 105 107 107 108 115 116 117 117 117 119 120 121 128 128 130 130 130 130 141 145 146 146 150

Contents

5

Properties of high-strength steels at and after elevated temperature Guo-Qiang Li, Huibao Lyu, Lei Huang and Xin-Xin Wang 5.1 Introduction 5.2 Mechanical properties of high-strength steels at elevated temperatures 5.2.1 Methodology 5.2.2 Behaviors of high-strength steels at elevated temperature 5.2.3 Temperature-dependent elastic modulus and yield strength of high-strength steels 5.2.4 Comparative study 5.3 Creep behavior of high-strength steels at elevated temperatures 5.3.1 Creep phenomenon and curves 5.3.2 Setup and specimens in creep test 5.3.3 Creep test procedure 5.3.4 Creep behavior of high-strength steels 5.3.5 Numerical creep models 5.4 Mechanical properties of high-strength steels after fire 5.4.1 Behavior of high-strength steel after fire 5.4.2 Mechanical properties of high-strength steels after fire 5.4.3 Mechanical properties of high-strength steel bolts after Fire 5.5 Summary References

Part III 6

vii

Members

Behavior and design of high-strength steel members under compression Yan-Bo Wang, Guo-Qiang Li and Tian-Ji Li 6.1 Introduction 6.2 Material properties 6.3 Residual stresses in welded high-strength steel box sections and H-sections 6.3.1 Sectioning method 6.3.2 Assessment of residual stresses in welded Q460 steel sections 6.3.3 Assessment of residual stresses in welded Q690 steel sections 6.3.4 Simplified residual stress model for welded Q460 steel sections 6.3.5 Simplified residual stress model for welded Q690 steel sections 6.4 Behavior of high-strength steel columns 6.4.1 Experiment program

153 153 153 153 154 157 159 161 161 163 163 165 173 183 183 186 192 201 202

205 207 207 208 209 209 211 214 219 223 224 224

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Contents

6.4.2 Overall buckling behavior of Q460 columns 6.4.3 Overall buckling behavior of Q690 columns 6.5 Parametric analysis and design recommendation 6.5.1 Parametric analysis of Q460 columns 6.5.2 Design of welded Q460 steel columns 6.5.3 Parametric analysis of Q690 columns 6.5.4 Design of welded Q690 steel columns 6.6 Summary References 7

8

Behavior and design of high-strength steel members under bending moment Xiao-Lei Yan, Guo-Qiang Li and Yan-Bo Wang 7.1 Introduction 7.2 Experimental investigation 7.2.1 Material properties 7.2.2 Specimens 7.2.3 Initial geometric imperfections 7.2.4 Experimental setup and instrumentation 7.2.5 Failure mode and experimental procedures 7.2.6 Load deflection curves 7.2.7 Load strain curves 7.2.8 The investigation of effective length 7.2.9 Finite element modeling 7.3 Parametric study and analysis 7.3.1 Initial geometric imperfections and residual stresses 7.3.2 Effect of residual stress 7.3.3 Effect of width-to-thickness ratio and height-to-thickness ratio 7.4 Comparison with current design codes 7.4.1 Prediction of current codes 7.4.2 Comparison with current design codes 7.5 Summary References Behavior and design of high-strength steel columns under combined compression and bending Kwok-Fai Chung, Tian-Yu Ma, Guo-Qiang Li and Xiao-Lei Yan 8.1 Introduction 8.2 Experimental investigation 8.2.1 H-sections 8.2.2 Box sections 8.3 Numerical investigation 8.3.1 H-sections 8.3.2 Box sections

232 240 249 249 256 258 263 267 269

271 271 272 272 273 274 276 277 280 281 283 285 289 289 292 295 297 297 299 302 303

305 305 306 306 324 334 334 343

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9

10

ix

8.4

Design recommendation 8.4.1 H-sections 8.4.2 Box sections 8.5 Summary References

346 346 353 354 355

Hysteretic behavior of high-strength steel columns Guo-Qiang Li, Yan-Bo Wang and Suwen Chen 9.1 Introduction 9.2 Experimental program 9.2.1 Specimens 9.2.2 Experimental setup 9.2.3 Measurement arrangement 9.2.4 Loading protocols 9.3 Experimental results 9.3.1 Experimental phenomenon 9.3.2 Hysteretic response 9.4 Numerical simulation 9.4.1 Material model and mesh 9.4.2 Geometric and boundary conditions 9.4.3 Initial imperfection 9.4.4 Verification of finite element model 9.5 Parametric analyses and discussions 9.5.1 Parameter design 9.5.2 Influence of width-to-thickness ratio of flange 9.5.3 Influence of web height thickness ratio 9.5.4 Influence of axial force ratio 9.6 Hysteretic model 9.6.1 Hysteretic model incorporated with damage behavior 9.6.2 Simplified hysteretic model 9.7 Summary References

357

Behavior of high-strength steel columns under and after fire Wei-Yong Wang, Guo-Qiang Li and Wen-Yu Cai 10.1 Introduction 10.2 Behavior of restrained high-strength steel columns under fire 10.2.1 Specimen preparation 10.2.2 Test set-up and measurements 10.2.3 Test procedure 10.2.4 Test results 10.2.5 Comparison with restrained mild steel columns 10.3 Postfire residual capacity of high-strength steel columns with axial restraint 10.3.1 Test setup and specimens

357 358 358 360 362 364 368 368 374 381 381 382 383 384 384 384 387 394 396 397 397 405 411 411 413 413 413 413 415 417 417 423 424 424

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Contents

10.3.2 Instrumentation 10.3.3 Test procedure and results 10.3.4 Numerical simulation 10.3.5 Parametric studies 10.3.6 Simplified formulation 10.4 Creep buckling experiments of high-strength steel columns at elevated temperatures 10.4.1 Specimen preparation 10.4.2 Test setup 10.4.3 Instrumentation 10.4.4 Test procedures 10.4.5 Experimental results 10.5 Creep buckling prediction of high-strength steel columns at elevated temperatures 10.5.1 Numerical prediction of creep buckling test on Q690 high-strength steel column 10.5.2 Numerical prediction of creep buckling test on ASTM A992 steel column 10.5.3 Theoretical study on creep buckling behavior of steel columns 10.5.4 Parametric study for creep buckling behavior of high-strength steel columns 10.5.5 Creep buckling load factor for current codes of the practices 10.6 Summary References

Part IV 11

Connections

Bolted connections Yan-Bo Wang, Guo-Qiang Li, Yi-Fan Lyu and Kun Chen 11.1 Introduction 11.2 Bearing-type bolted connections for high-strength steels 11.2.1 Behavior of single-bolt connection 11.2.2 Behavior of two-bolt connection in parallel 11.2.3 Behavior of multibolt connection in tandem 11.2.4 Comparison with current design codes 11.3 Slip critical type bolted connections for high-strength steels 11.3.1 Introduction 11.3.2 Experimental programs 11.3.3 Experimental results 11.3.4 Discussion 11.3.5 Design recommendation 11.3.6 Summary

428 429 432 433 442 444 445 447 448 448 449 458 459 465 469 476 479 489 489

491 493 493 493 494 514 523 525 529 529 533 538 548 552 554

Contents

12

xi

11.4

Experimental study on slip factor of hybrid connections 11.4.1 Introduction 11.4.2 Specimen of slip critical test 11.4.3 Experimental results 11.4.4 Discussion 11.4.5 Design recommendation 11.4.6 Summary References

554 554 555 555 558 562 562 563

Welded connections Fei-Fei Sun, Ming-Ming Ran, Guo-Qiang Li and Yan-Bo Wang 12.1 Introduction 12.2 Experimental investigation 12.2.1 Material information 12.2.2 Gas metal arc welding 12.2.3 Digital image correlation measurement and calibration 12.2.4 Measured load-carrying capacity and deformation capacity of butt joints 12.2.5 Linear correlation between strength and hardness 12.2.6 Measured hardness distribution curves of butt joints 12.3 Summarization of experimental results 12.3.1 Three hardness distribution patterns 12.3.2 Strain distribution for each hardness distribution pattern 12.3.3 Strength loss for specimens with different softened heataffected zone width 12.3.4 Strength increase due to the constraint 12.3.5 Ductility loss due to mismatched connections 12.4 Applicability of Eurocode 3 Part 1 12 12.5 Strength model of butt welds 12.5.1 Theoretical model 12.5.2 Finite element model 12.5.3 Formula modification for butt joints with rigid constraint under axisymmetric condition 12.5.4 Unified formula for butt joints with rigid constraint under all conditions 12.5.5 Unified formula for butt joint with nonrigid constraint under all conditions 12.5.6 Undermatched cases without softened heat-affected zone 12.5.7 Softened heat-affected zone cases 12.5.8 Interpretation of Eq. (12.15) 12.5.9 Verification of strength model 12.6 Design proposal 12.6.1 Design formula 12.6.2 Design strength References

565 565 568 568 569 569 571 576 578 578 578 582 584 584 585 588 589 589 591 595 597 598 599 602 602 603 607 607 608 610

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Contents

Part V 13

Structural systems and economic analysis

Application of high-strength steels in seismic zones and case studies Guo-Qiang Li and Yan-Bo Wang 13.1 Introduction 13.2 Limits related to application of high-strength steel in seismic structures 13.2.1 Effect of material properties on the ductility of structural members 13.2.2 Limits of current design codes 13.3 Proposed methods for application of high-strength steel in seismic structures 13.3.1 Determination of design earthquake action 13.3.2 Selection of structural systems 13.3.3 Adjustment of reliability index 13.4 Information of the case study 13.5 Comparison of structural performance between normal strength steel solution and high-strength steel solution 13.5.1 Period and period ratio 13.5.2 Performance of structures under frequent earthquakes and wind 13.5.3 Performance of structures under rare earthquakes 13.6 Economic evaluation of high-strength steel structures 13.6.1 Evaluation of the prices of structures using different steels 13.6.2 Evaluation of steel consumption 13.6.3 Occupied area of structural members 13.6.4 Foundation construction cost 13.7 Summary References

index

613 615 615 615 615 616 617 618 618 619 620 623 623 623 625 626 627 627 628 628 629 629 631

List of Contributors

Wen-Yu Cai Tongji University, Shanghai, P.R. China Kun Chen Tongji University, Shanghai, P.R. China Suwen Chen Tongji University, Shanghai, P.R. China Kwok-Fai Chung Hong Kong Polytechnic University, Hong Kong, P.R. China Le-Tian Hai Tongji University, Shanghai, P.R. China Lei Huang Tongji University, Shanghai, P.R. China Guo-Qiang Li Tongji University, Shanghai, P.R. China Tian-Ji Li Tongji University, Shanghai, P.R. China Huibao Lyu Tongji University, Shanghai, P.R. China Yi-Fan Lyu Tongji University, Shanghai, P.R. China Tian-Yu Ma Hong Kong Polytechnic University, Hong Kong, P.R. China; Tongji University, Shanghai, P.R. China Ming-Ming Ran Tongji University, Shanghai, P.R. China; Sichuan University, Chengdu, P.R. China Fei-Fei Sun Tongji University, Shanghai, P.R. China Wei-Yong Wang Chongqing University, Chongqing, P.R. China Xin-Xin Wang Tongji University, Shanghai, P.R. China Yan-Bo Wang Tongji University, Shanghai, P.R. China Yuan-Zuo Wang Tongji University, Shanghai, P.R. China Xiao-Lei Yan Tongji University, Shanghai, P.R. China; Chengdu University of Technology, Chengdu, P.R. China

Preface

High-strength constructional steel with a minimum yield strength no less than 460 MPa has been used in some pilot projects of high-rise buildings, large-span buildings, bridges, transmission towers, and offshore structures in recent years, due to the advantages in structural and architecture efficiency, cost competency as well as resource saving and pollution reducing. Although the demand for increasing the strength of steel in construction section has attracted sufficient concerns from the iron and steel industry to provide high-strength constructional steel with competitive quality and price, the regular and widespread application of high-strength steel structures is still limited because the lack of systematic research. With the increase in strength, other mechanical properties of high-strength steel show a considerable difference compared with normal strength steel, such as the decrease in strain hardening, elongation ratio, ductility, and capacity of accumulated plastic deformation for energy dissipation. This may influence the buckling behavior of steel members subjected to compression and bending, the capacity of connections to overcome stress concentration, and redistribute internal forces in structural members and the seismic performance of steel members and structures. Moreover, the specific chemical composition and metallurgical and rolling technologies of high-strength steel may result in different mechanical properties at and after exposed to elevated temperatures caused by welding or fire, which may influence the welding residual stresses and hence the ultimate bearing capacity of highstrength steel members at ambient temperature, and the resistance of high-strength steel members and connections in and after fire. This book is based on the systematic research on the behavior and design of high-strength constructional steel inspired by the above issues to provide researchers and engineers with extensive information and confidence to use high-strength constructional steels in a safe and economic manner. This book is basically divided into five parts. Part I introduces the general information of high-strength steel, including its structural and architecture advantages, social and environmental advantages, an overview of application of high-strength steel in construction projects in China and overseas, and the current situation of world steel production and consumption, as well as the challenge of using high-strength steel for construction especially in seismic areas. Part II provides a detailed discussion of the mechanical properties of high-strength steel at ambient temperature subjected to monotonic and cyclic loads and the degradation of mechanical properties at and after exposed to elevated temperatures. Part III explains the behavior of elementary members subjected to different types of loads and load combinations. The buckling behaviors of high-strength steel box- and H-section columns with imperfections of geometry and

xvi

Preface

residual stress induced by welding are included in terms of experimental and numerical investigations. The hysteretic behavior of high-strength steel members is also discussed. Moreover, the analysis and design of bolted and welded connections for integration of high-strength steel members are included in Part IV. Finally, the methodology for the application of high-strength steel structure in seismic zone is proposed with a case study in Part V. We would like to express our grateful appreciation to the chapter contributors to this book and many other researchers who contributed their achievements we referred. We would also like to express our heartfelt thanks to the research students who helped the research we conducted at Tongji University. Moreover, the support to our research from the National Natural Science Foundation of China and Ministry of Science and Technology of China is gratefully acknowledged. Without their support, this book would not have become a reality. Guo-Qiang Li and Yan-Bo Wang

Introduction

1

Yan-Bo Wang and Guo-Qiang Li Tongji University, Shanghai, P.R. China

1.1

Introduction

High-strength constructional steel refers to a family of constructional steels with minimum yield strength of 460 MPa. The yield strength is defined as the stress related to the initiation of plastic strain, which will cause the permanent deformation in the designed steel members. Compared to conventional mild steels with yield strength ranging from 235 to 355 MPa and carbon content lower than 0.3%, high-strength constructional steels are higher in yield strength and lighter in weight when bearing the same load. Moreover, currently emerging high-strength constructional steel (also known as high-performance steel) can further achieve an optimized balance of strength, weldability, ductility, toughness, corrosion resistance, and formability. Such steels have promising applications in worldwide major projects [1-3].

1.1.1 Advantages and limits of high-strength constructional steel 1.1.1.1 Cost-efficiency The price of high-strength constructional steel per ton is generally higher than that of normal strength constructional steel. However, according to the survey results from Chinese constructional steel market, the price of steel per million Pascal (MPa) reduces with the increase of steel yield strength, as shown in Fig. 1.1. Thus with the same designed strength, high-strength constructional steel is generally more cost-effective than that of the normal strength constructional steel. This advantage is more remarkable for nonbuckling steel members. For example, up to 64% of weight will be reduced for tensile steel members when yield strength of the adopted steel changes from 355 to 960 MPa. This reduced weight will lead to an effective 32% decrease in cost for corresponding materials. Similar achievements can also be found in long-span bridges and stocky columns near the base of highrise buildings. The reduction in self-weight and material cost can approach 40% and 20%, respectively.

1.1.1.2 Environmental friendly World steel production is indirectly dependent on coal. According to the World Coal Association [4], 70% of steel is produced in basic oxygen furnaces, where Behavior and Design of High-Strength Constructional Steel. DOI: https://doi.org/10.1016/B978-0-08-102931-2.00001-2 © 2021 Elsevier Ltd. All rights reserved.

4

Behavior and Design of High-Strength Constructional Steel

Figure 1.1 Comparison of normalized price among different steel grades: (A) steel plate and (B) H shaped steel.

coking coal is a vital ingredient in this steel-making process. A further 29% of steel is produced in electric arc furnaces. Much of the electricity used in this process is generated from coal-fired power stations. World crude steel production was 1.5 billion tons in 2011 and about 761 million tons of coking coal was used in the production of steel. The use of coal for steel production has been inevitably associated with a number of environmental challenges such as greenhouse air and particulate emissions. Because of global warming, regional climate change, as well as the effect of particle pollution on public health and safety, there is currently growing requirement that technology developments have to be part of the solution to climate change and environmental pollution. This is particularly true for steel because its use is significantly growing in many major economies, including the largest and fastest growing countries such as China. By using high-strength constructional steel, fewer resources are needed for fulfilling a given structural function, which will alleviate the negative impact on the environmental.

Introduction

5

1.1.1.3 Architectural and structural advantages The application of high-strength constructional steel in construction can effectively reduce the member size and structure weight. Due to such advantage, a more rational structural design for large and complicated structures may be achieved. The advantages for the application of high-strength constructional steel are summarized as follows: G

G

G

G

G

As an alternate option for conventional mild carbon steels, adopting high-strength constructional steel can reduce plate thickness and member size because of its increased material strengths. Consequently, the usable floor area of the structures could be increased and the overall weight of the structures can be reduced. Light-weight and slim members will provide convenient solutions for special architectural form in design and construction of complicate structures. Due to the reduction in plate thickness and member size, problems in welding thick plates can be effectively avoided and the amount of welding are reduced. Transportation and erection, and foundations are more economic due to the lighter weight structures. Reduction in mass of the structures can effectively improve the seismic action of the whole structure.

1.1.1.4 Limits With the recent development of high-strength constructional steel and advances in metallurgical technology, high-strength constructional steel can be produced with a reasonable cost and preferable performance, including Chinese steel grades of Q460 Q690 (yield strength 460 690 MPa), European steel grades of S690 S960 (yield strength 690 960 MPa), and US high-performance steel HPS70 (yield strength 485 MPa). However, the application of such high-strength steels is limited by current design codes. The Chinese code for design of steel structures GB 500172017 limits steel grades up to Q460. European and US specifications for steel structures allow the use of high-strength constructional steel up to grades of S700 (700 MPa) and ASTM A514 (690 MPa), but the current specifications are established on the test results and analyses of mild carbon steels. These steels usually have the nominal yield strength ranging from 235 to 345 MPa. Consequently, more work are needed to evaluate whether the members fabricated from high-strength constructional steels can be designed according to the existing codes or whether the code terms need modifications for high-strength constructional steels. In addition, seismic design of buildings requires high ductility of steel due to the expectation of inelastic behavior of structural elements and connections under rare earthquakes. However, with the increase in strength, yield to tensile strength ratio for high-strength constructional steel is increased and elongation ratio is decreased, which is unfavorable for structural seismic behaviors. For high-strength steel, it is difficult to meet the seismic design requirement on the ductility for normal strength of steel. Therefore it is important to study the suitability of structural elements fabricated with high-strength constructional steel for their applications in seismic zones.

6

Behavior and Design of High-Strength Constructional Steel

1.1.2 Current situation of steel production and consumption There was an increase of 84% in steel production worldwide between 2000 and 2013, with China experiencing a 540% increase during the same period (Fig. 1.2). During the same time, global use of steel had increased 81% from 2000 to 2013, while the use of steel in China experienced an increase over 440% during the same period. Because of the rapid industrialization and urbanization, China accounted for approximately 47.3% of the global steel production in 2013 (Fig. 1.3), with more than half of which were applied in the construction. Even for developed countries, constructional steel possesses the largest market share. For example, 42% of total US steel was used in construction in 2010, as shown in Fig. 1.4. Because of architectural and structural merits, high-strength constructional steel has been extensively used in high-rise buildings, large span buildings and bridges in the past two decades. However, high-strength constructional steel is only a small fraction of all kinds of constructional steel. According to China Steel Construction Society annual report of 2016, the most used steel in construction is Q345 (nominal yield strength is 345 MPa), which accounts for 62% of the total steel consumption, but Q460 high-strength steel occupies only 4% of the total consumption, as shown in Fig. 1.5.

1.1.3 Applications of high-strength constructional steel High-strength constructional steels have been available in the United States and Japan since 1960s. However, high-strength constructional steel of early times did not find extensive applications in building and bridge structures until the 1990s due to encountered poor weldability and inadequate ductility. With recent advances in techniques of quenching and tempering processes, significantly improved ductility, weldability and fracture toughness can achieved for high-strength constructional steels. Based on such progress, a worldwide application of high-strength steel in the design and construction of building and bridge structures has been witnessed.

Figure 1.2 Annual crude steel production. Data from: World Steel Association [5].

Introduction

7

Figure 1.3 Global use of steel in 2013. Data from: World Coal Association [4].

Figure 1.4 US steel shipments by market classification in 2010. Data from: American Iron and Steel Institute [6].

1.1.3.1 Application in China High-strength constructional steel Q460E (yield strength 460 MPa) was applied in China Central Television Headquarters. This building is a 234 m, 44-story skyscraper located in the Beijing central business district. Without using Q460E the construction of the building was e a structural challenge, due to its striking style and the fact that it is located in a seismic zone. Two L-shaped high-rise tower linked at the top causes considerably high stresses in the connection location between mega-cantilevered steel truss and the perimeter steel frames as shown in

8

Behavior and Design of High-Strength Constructional Steel

Figure 1.5 The percentage of each steel grade consumed in 2016 of mainland China. Data from: China Steel Construction Society [7].

Figure 1.6 China Central Television Headquarters. Courtesy: China Construction Steel Corp. Ltd.

Fig. 1.6. By using a total of 750 tons of high-strength constructional steel Q460E, the above structural challenge receives a satisfactory solution. High-strength constructional steel Q460E was also used in Chinese National Stadium (known as Bird’s Nest), which is shown in Fig. 1.7. This building was famous landmark for China and the city of Beijing. The 2008 Olympic Games were successfully held in this building which offered gross floor area of 254,600 m2 and seating capacity of 91,000, including 11,000 temporary seats. If normal strength steel Q345D was applied the plate thickness would be 220 mm, which would cause

Introduction

9

Figure 1.7 The Chinese National Stadium.

Figure 1.8 The Green Tower Central Plaza in Zhengzhou.

difficulties in plate welding, transportation, and erection. By application of Q460E the plate thickness of the columns was reduced from 220 to 110 mm. Convenient and efficient welding, transportation, and erection were achieved. High-strength steels Q460, Q550, and Q690 steels were used in the Green Tower Central Plaza, which is shown in Fig. 1.8. This building is located in Zhengzhou Comprehensive Transportation Hub, which is a super high-rise building with 66 floors above ground. The height of the building is 300 m with a total floor area of

10

Behavior and Design of High-Strength Constructional Steel

Figure 1.9 The T2 tower and T3 tower of Pudong Financial Plaza in Shanghai.

approximately 250,000 m2. The structural towers adopt braced frame core tube belt truss system, wherein the outer frame is composed of encased steel columns, steel beams, and steel bracings. Those high-strength steels (total 568.7 tons) were mainly adopted in the south tower. Due to such application, 20% saving of steel consumption was achieved. High-strength steel Q460C was applied in the Pudong Financial Plaza (Fig. 1.9), which is located at Century Avenue in Shanghai. The project comprised three office buildings and a commercial building. The height of the T1 tower is 194 m. For the T2 tower and T3 tower, 20 floors above ground and 4 floors underground with the height of 100 m were designed. The overall structural system was a steel frameconcrete core system. For this project the size of frame column was initially 900 3 900 3 50 with the application of Q345 steel. By application of Q460C the size of frame column was reduced to 700 3 700 3 55. More floor area was spared while 500 tons of steel were saved.

1.1.3.2 Applications outside of China High-strength steel has also been applied in many major projects outside of China. The typical projects include Landmark Tower (600 MPa steel columns) in Yokohama (Fig. 1.10), NTV Tower in Tokyo (Fig. 1.11), Sony Centre (S460 and

Introduction

Figure 1.10 Landmark Tower in Yokohama. Source: en.wikipedia.org/wiki/Yokohama_Landmark_Tower.

Figure 1.11 NTV Tower in Tokyo. Source: en.wikipedia.org/wiki/Nittele_Tower.

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Behavior and Design of High-Strength Constructional Steel

Figure 1.12 Sony Centre in Berlin. Source: pixabay.com/photos/sony-center-berlin-architecture-4616105/.

Figure 1.13 Star City in Sydney. Source: en.wikipedia.org/wiki/The_Star,_Sydney.

S960 members of roof truss) in Berlin (Fig. 1.12), and Star City (650 and 690 MPa columns, Fig. 1.13), Latitude Tower (Bisplate80 steel members in transfer truss, Fig. 1.14) in Sydney and One World Trade Center in New York (A913 steel columns, Fig. 1.15). Application of high-strength steels into bridges can also be found. For example, high-performance steel (HPS345, HPS485, and HPS690) were used in over 400 bridges in 43 states of United States.

Introduction

Figure 1.14 Latitude Tower in Sydney. Source: en.wikipedia.org/wiki/Latitude_(building).

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Behavior and Design of High-Strength Constructional Steel

Figure 1.15 One World Trade Center in New York. Source: www.pexels.com/zh-cn/photo/53212.

References [1] Bjorhovde R. Development and use of high performance steel. J Constr Steel Res 2004;60 (3 5):393 400. [2] Pocock G. High strength steel use in Australia, Japan and the US. Struct Eng 2006;84 (21):27 30. [3] Wright W. High-performance steel: research to practice. Public Roads 1997;60(4):34. [4] http://www.worldcoal.org [5] http://www.worldsteel.org [6] http://www.steel.org [7] http://www.cncscs.org.cn/

Material properties and statistical analysis of high-strength steels

2

Yan-Bo Wang, Guo-Qiang Li and Yuan-Zuo Wang Tongji University, Shanghai, P.R. China

2.1

Introduction

The mechanical properties of high-strength steel (HSS) is different from mild steel. In practical construction, some critical steel members and connections are generally working under the complex multiaxis stress states. Accordingly the elasto-plastic behavior of HSS under various stress states should be well investigated, which is a foundation for the analysis and application of HSS structures [1]. The plastic behavior of structural metals is characterized by the constitutive model which is composed of yield criterion, flow rule, and hardening rule. The importance of yield function for actual engineering computations has been well noted. The von Mises yield criterion is generally adopted to describe the plasticity of typical ductile metals. There is one basic tenet of von Mises yield model: the influences of stress triaxiality and Lode angle on metal plasticity are negligible. It has been found that with lower ductility of materials, such as concrete and geomaterials, stress triaxiality, and Lode angle dependences of yield strength of these materials become more significant [2
14]. Compared with mild steel, HSS has higher strength but lower ductility, which implies that the brittleness of HSS is more significant than mild steel. Therefore the influences of stress triaxiality and Lode angle on the plasticity model of HSSs need to be reexamined. Moreover, for practical application, the design values of strength of HSSs need to be specified. Therefore it is necessary to carry out statistical analysis on the uncertainty of the properties of HSSs and determine the partial coefficient of material resistance in the limit state design method through reliability analysis, so as to determine the design strength value of HSSs (Figs. 2.1 and 2.2).

40

20 110 290

5

76

R1

t =10

76

Figure 2.1 Details of tensile specimens of Q550, Q690, and Q890 HSS (t 5 10 mm). Behavior and Design of High-Strength Constructional Steel. DOI: https://doi.org/10.1016/B978-0-08-102931-2.00002-4 © 2021 Elsevier Ltd. All rights reserved.

18

Behavior and Design of High-Strength Constructional Steel

R15

130 170 380

25 45

45

25

t =20

20

Figure 2.2 Details of tensile specimens of Q550, Q690, and Q890 HSS (t 5 20 mm).

2.2

Mechanical properties of high-strength steels

In order to obtain the statistical parameters of material properties, such as yield strength, tensile strength, and elongation of HSSs (Q460, Q550, Q690, and Q890), the standard specimens of HSSs produced by Wuyang Iron and Steel Liability Co., Ltd., Nanjing Iron and Steel Group Co., Ltd., and Jiangyin Xingchen Special Steel Co., Ltd. were taken to carry out the monotonic tensile tests. The chemical composition of different steel plates is shown in Tables 2.1
2.4. The specimens manufactured from steel plates with thickness ranging from 6 to 40 mm were tested. The dimensions of the representative standard specimens are shown in Table 2.1 and Fig. 2.3. The tests were carried out according to the method for tensile testing of metal materials at room temperature (GB/T 228-2002) [15]. The results are summarized in Tables 2.5
2.8. In order to facilitate comparison, the typical stress
strain curves of steels produced from three different steel and iron companies are plotted in Figs. 2.4
2.7. It is observed from Fig. 2.6 that the yield platform of Q690D HSS is obvious, which is similar to mild steels. However, the length of the yield platform section is relatively short with the strain range from 0.003 to 0.02. Subsequently the strengthening stage is reached until the steel reaches the ultimate strength with strain about 0.06. It can be found in Fig. 2.6 that the Q690 HSS with 10 mm thickness of Nanjing Iron and Steel Group Co., Ltd. has no obvious yield platform during tensile tests, while the Q690 HSS specimen taken from 16 mm thick steel plate has obvious yield platform. In addition, the strain hardening of Q690 HSS is not as significant as that of mild steels, which means the larger yield strength ratio (mean value 5 0.94) of Q690 HSS.

2.3

Three-dimensional constitutive model of high-strength steels

2.3.1 Fundamental definition The stress state can be described in terms of three stress tensor invariants, and principal stresses (σ1 , σ2 , σ3 ) are often adopted. Mathematically three principal stresses are roots of the characteristic equation, given by

Table 2.1 Chemical composition of Q460 high-strength steel. Factory Shanghai Hengtai Iron and Steel Group Co., Ltd. Wuyang Iron and Steel Liability Co., Ltd

Thickness (mm)

C

Si

Mn

P

S

Al

Nb

Ti

Cr

Mo

V

Ni 1 Cu

CEV

10 20

0.15 0.104

0.272 0.267

1.384 1.311

0.025 0.022

0.008 0.006

0.010 0.030

0.012 0.022

0.007 0.017

0.024 0.020

0.003 0.003

0.064 0.004

0.041 0.014

0.402 0.329

10 20

0.170 0.160

0.290 0.330

1.480 1.560

0.018 0.021

0.006 0.008

0.029 0.040

0.009 0.037

0.001 0.002

0.010 0.040

0.001 0.027

0.002 0.052

Table 2.2 Chemical composition of Q550 high-strength steel. Factory Shanghai Hengtai Iron and Steel Group Co., Ltd. Wuyang Iron and Steel Liability Co., Ltd Nanjing Iron and Steel Group Co., Ltd.

Thickness (mm)

C

Si

Mn

P

S

Al

Nb

Ti

Cr

Mo

V

Ni 1 Cu

CEV

10 20

0.052 0.087

0.239 0.242

1.232 1.215

0.026 0.029

0.006 0.006

0.034 0.021

0.002 0.008

0.020 0.017

0.303 0.226

0.003 0.080

0.006 0.005

0.015 0.015

0.320 0.352

10 20 20

0.170 0.160 0.13

0.270 0.280 0.26

1.280 1.240 0.134

0.019 0.015 0.012

0.001 0.001 0.001

0.035 0.025


0.001 0.015 0.024

0.024 0.019 0.015

0.280 0.180 0.24

0.002
0.08

0.001
0.002

0.06

0.42

Table 2.3 Chemical composition of Q690D high-strength steel. Factory

Thickness (mm)

C

Si

Mn

P

S

Cr

Al

Mo

Ti

Cu

Nb

Ni

V

10 16 30 40 6 8 10 16 20 20

0.17 0.15 0.17 0.12 0.133 0.133 0.133 0.143 0.130 0.06

0.19 0.3 0.28 0.27 0.26 0.25 0.26 0.24 0.25 0.2

1.41 1.34 1.22 1.00 1.38 1.43 1.39 1.39 0.13 1.6

0.009 0.012 0.02 0.012 0.012 0.011 0.011 0.013 0.013 0.1

0.003 0.004 0.002 0.004 0.001 0.001 0.001 0.001 0.001 0.02

0.03 0.22 0.18 0.48 0.27 0.42 0.27 0.37 0.33 0.32

0.036 0.045 0.04 0.045 0.032 0.031 0.038 0.027
0.029

0.01 0.17 0.122 0.35 0.14 0.29 0.14 0.24 0.15 0.18

0.017 0.02 0.017 0.014 0.013 0.015 0.015 0.015 0.016 0.015

0.02
0.01
0.02 0.02 0.02 0.03
0.03

0.02 0.017 0.021 0.027 0.027 0.026 0.027 0.025 0.024 0.018

0.02
0.01
0.03 0.04 0.04 0.04
0.01

0.002 0.0027 0.001 0.006 0.001 0.003 0.002 0.002
0.036

Wuyang Iron and Steel Liability Co., Ltd

Nanjing Iron and Steel Group Co., Ltd.

Jiangyin Xingchen Special Steel Co., Ltd.

Table 2.4 Chemical composition of Q890D high-strength steel. Factory Shanghai Hengtai Iron and Steel Group Co., Ltd. Wuyang Iron and Steel Liability Co., Ltd Nanjing Iron and Steel Group Co., Ltd.

Thickness (mm)

C

Si

Mn

P

S

Al

Nb

Ti

Cr

Mo

V

Ni 1 Cu

CEV

10 20 10 20 20

0.114 0.160 0.170 0.160 0.160

0.284 0.330 0.350 0.330 0.26

1.015 1.250 1.080 1.250 1.02

0.019 0.011 0.010 0.011 0.008

0.008 0.002 0.001 0.002 0.001

0.032 0.041 0.036 0.041


0.015 0.021 0.020 0.021 0.026

0.017 0.017 0.010 0.017


0.231 0.270 0.230 0.270 0.440

0.509 0.560 0.560 0.560 0.531

0.048 0.048 0.047 0.048 0.043

0.472 0.130

0.472 0.553

0.150

Material properties and statistical analysis of high-strength steels

21

Figure 2.3 Details of tensile specimens of Q690D HSS (t 5 16 and 30 mm).

Table 2.5 Mechanical properties of Q460 high-strength steel. No.

1 2 3 4 5 6 7 8 Mean value

Thickness (mm)

Yield strength (MPa)

Tensile strength (MPa)

Elongation (%)

Yield strength ratio

10 10 10 20 20 20 30 30

418 489 479 511 507 493 463 468 479

591 612 607 645 642 634 590 593 614

24 27 24 26 24 27 28 25 26

0.70 0.80 0.79 0.79 0.79 0.78 0.78 0.79 0.78

σ3 2 I1 σ3 2 I2 σ 2 I3 5 0 where I1 , I2 ; and I3 are functions of principal stresses that are defined as I1 5 σ 1 1 σ 2 1 σ 3 I2 5 σ 1 σ 2 1 σ 2 σ 3 1 σ 3 σ 1

(2.1)

22

Behavior and Design of High-Strength Constructional Steel

Table 2.6 Mechanical properties of Q550 high-strength steel. No.

1 2 3 4 5 6 7 8 9 10 11 12 Mean value

Thickness (mm)

Yield strength (MPa)

Tensile strength (MPa)

Elongation (%)

Yield strength ratio

10 10 10 10 20 20 20 20 20 20 30 30

700 705 696 824 672 635 638 697 696 702 700 694 697

773 773 766 890 746 707 706 745 748 751 764 760 761

18 17 17 20 17 19 17 19 21 19 24 28 20

0.91 0.91 0.91 0.91 0.90 0.90 0.90 0.94 0.93 0.94 0.92 0.91 0.92

Table 2.7 Mechanical properties of Q690D high-strength steel. No.

Thickness (mm)

Yield strength (MPa)

Tensile strength (MPa)

Elongation (%)

Yield strength ratio

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

6 6 6 6 6 6 6 8 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

781 784 777 779 775 781 788 798 755 751 749 845 848 855 821 827 824 816 818 784 786 726 791

822 819 812 816 810 817 824 848 814 815 807 878 880 884 863 868 866 858 853 831 837 794 844

14 16 16 15 16 16 16 18 17 17 17 14 15 15 14 14 14 16 16 16
16 14

0.95 0.96 0.96 0.95 0.96 0.96 0.96 0.94 0.93 0.92 0.93 0.96 0.96 0.97 0.95 0.95 0.95 0.95 0.96 0.94 0.94 0.91 0.94 (Continued)

Material properties and statistical analysis of high-strength steels

23

Table 2.7 (Continued) No.

Thickness (mm)

Yield strength (MPa)

Tensile strength (MPa)

Elongation (%)

Yield strength ratio

24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

10 10 10 10 10 10 10 10 15 15 15 16 16 16 16 16 16 16 16 16

754 773 773 761 764 759 774 754 778 762 761 788 772 779 779 772 756 780 772 782

808 812 809 805 808 805 811 807 820 815 810 847 827 833 834 827 810 834 827 837

17 16 16 15 16 16 16 16 17 19 19 20 22 23 20 21 21 18 20 20

0.93 0.95 0.96 0.95 0.95 0.94 0.95 0.93 0.95 0.93 0.94 0.93 0.93 0.94 0.93 0.93 0.93 0.94 0.93 0.93

44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

16 16 16 16 16 16 16 16 20 20 20 20 20 20 20 20 20 25 30 30 30 30 30

781 780 776 777 775 773 776 800 809 808 813 801 801 802 793 739 770 772 734 736 739 736 740

836 835 831 837 828 832 831 859 831 833 836 826 826 826 822 784 821 832 784 786 789 792 790

19 19 22 21 21 21 20 20 17 16 17 19 19 18
18 18 16 18 20 18 19 18

0.93 0.93 0.93 0.93 0.94 0.93 0.93 0.93 0.97 0.97 0.97 0.97 0.97 0.97 0.96 0.94 0.94 0.93 0.94 0.94 0.94 0.93 0.94 (Continued)

24

Behavior and Design of High-Strength Constructional Steel

Table 2.7 (Continued) No.

67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 Mean value

Thickness (mm)

Yield strength (MPa)

Tensile strength (MPa)

Elongation (%)

Yield strength ratio

30 40 40 40 40 40 40 40 40 40 40 40 40 40 40

738 759 761 761 756 758 763 757 766 761 758 756 776 774 777 777

790 810 812 813 807 809 809 810 811 809 808 808 824 825 824 823

18
18 19 17 18 18 18 18 19 20 19 17 17 16 18

0.93 0.94 0.94 0.94 0.94 0.94 0.94 0.93 0.94 0.94 0.94 0.94 0.94 0.94 0.94 0.94

Table 2.8 Mechanical properties of Q890 high-strength steel. No.

1 2 3 4 4 5 6 7 8 9 10 11 12 Mean value

Thickness (mm)

Yield strength (MPa)

Tensile strength (MPa)

Elongation (%)

Yield strength ratio

10 10 10 10 20 20 20 20 20 20 20 30 30

1073 1079 1070 1067 1025 998 1015 1032 954 953 951 980 978 1013

1111 1114 1098 1106 1063 1038 1052 1095 995 995 993 1028 1027 1055

14 14 13 15 19 19 19 17 17 17 17 22 20 17

0.97 0.97 0.97 0.97 0.96 0.96 0.96 0.94 0.96 0.96 0.96 0.95 0.95 0.96

Material properties and statistical analysis of high-strength steels

25

700 600

Stress (MPa)

500 400 300 Shanghai Hengtai Iron and Steel Group Co., Ltd. 10 mm Shanghai Hengtai Iron and Steel Group Co., Ltd. 20 mm Shanghai Hengtai Iron and Steel Group Co., Ltd. 30 mm

200 100 0 0

0.05

0.1

0.15

0.2

0.25

0.3

Strain (m/m) Figure 2.4 Typical stress
strain curves of Q460 HSS. HSS, High-strength steel.

900 800 700

Stress (MPa)

600 500 400

Nanjing Iron and Steel Group Co., Ltd. 20 mm Shanghai Hengtai Iron and Steel Group Co., Ltd. 10 mm Shanghai Hengtai Iron and Steel Group Co., Ltd. 20 mm Shanghai Hengtai Iron and Steel Group Co., Ltd. 30 mm

300 200 100 0 0

0.05

0.1

0.15

0.2

0.25

0.3

Strain (m/m) Figure 2.5 Typical stress
strain curves of Q550 HSS. HSS, High-strength steel.

I3 5 σ 1 σ 2 σ 3

(2.2)

These three functions are totally independent of each other, and I1 , I2 , and I3 can be also taken as stress tensor invariants. An arbitrary stress state can be represented geometrically in the Cartesian coordinates which take principal stresses as the axis,

26

Behavior and Design of High-Strength Constructional Steel

1,000

(A)

Stress (MPa)

800

600

400 Wuyang Iron and Steel Liability Co., Ltd. 10 mm 200

Wuyang Iron and Steel Liability Co., Ltd. 16 mm Wuyang Iron and Steel Liability Co., Ltd. 30 mm

0 0

0.05

0.1

0.15

Strain (m/m) 1,000 (B)

Stress (MPa)

800

600

400 Nanjing Iron and Steel Group Co., Ltd. 10 mm 200

Nanjing Iron and Steel Group Co., Ltd. 16 mm Xingchen Special Steel Co., Ltd. 20 mm

0 0

0.05

0.1

0.15

Strain (m/m) Figure 2.6 Typical stress
strain curves of Q690D HSS.

as shown in Fig. 2.8. The Cartesian coordinates can also be transferred to cylindrical coordinates (σm ,θ,σeq ) which are defined as 1 1 I1 5 ðσ 1 1 σ 2 1 σ 3 Þ 3 3 
  1 r 3 θ 5 arccos 3 σ

σm 5

(2.3) (2.4)

Material properties and statistical analysis of high-strength steels

27

1,200

Stress (MPa)

1,000 800 600 Nanjing Iron and Steel Group Co., Ltd. 20 mm Shanghai Hengtai Iron and Steel Group Co., Ltd. 10 mm Shanghai Hengtai Iron and Steel Group Co., Ltd. 20 mm

400 200 0 0

0.05

0.1

0.15

0.2

0.25

Strain (m/m) Figure 2.7 Typical stress
strain curves of Q890 HSS.

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1 σ5 ðσ1 2σ2 Þ2 1 ðσ2 2σ3 Þ2 1 ðσ3 2σ1 Þ2 2

(2.5)

where  r5

27 ðσ1 2σm Þðσ2 2σm Þðσ3 2σm Þ 2

1=3 (2.6)

The parameter θ (0 # θ # π3) is often referred to as the Lode angle, and σ is the von Mises equivalent stress. The Cartesian coordinates can also be transferred to spherical coordinates which adopt (η, θ, σ) as axis. The parameter η is often adopted to describe triaxiality of stress state, defined by η5

σm I1 =3 5 σ σ

(2.7)

2.3.2 Applicability of von Mises yield criterion for high-strength steel 2.3.2.1 Specimens Three strength grades of HSS are investigated in the present study, respectively named as Q550, Q690, and Q890, and the chemical compositions of these steels are summarized in Table 2.9. The experimental program is summarized in Table 2.10, and the corresponding initial values of stress triaxiality and Lode angle at the crack

28

Behavior and Design of High-Strength Constructional Steel

2

| ' |=

'

2 3

P

Deviatoric plane

(Lode angle)

O

1

Hydrostatic pressure line

' =

3

m

3

Figure 2.8 Geometry representation of Cartesian coordinates, cylindrical coordinates, and spherical coordinates.

Table 2.9 Chemical composition and hardness value of steels (%). Steel grade Q550 Q690 Q890

C

Mn

P

S

Si

Ni

Cr

0.17 0.17 0.17

1.28 1.41 1.08

0.019 0.009 0.01

0.001 0.003 0.002

0.27 0.19 0.35

0.01 0.02 0.42

0.28 0.03 0.23

ID

Repeat

η0

θ0

Table 2.10 Overview of experimental specimens. Loading type Monotonic uniaxial tension

Specimen type Smooth round bar Notched round bar

Flat grooved plate

Monotonic uniaxial compression

Pure shear specimen Cylinder specimen Notched cylinder specimen

SRB NRB-1 NRB-2 NRB-3 FGP-1 FGP-2 FGP-3 PS CS NCS-1 NCS-2 NCS-3

3 3 3 3 3 3 3 3 3 3 3 3

1 3

1.026 0.739 0.556 1.045 0.713 0.624 0 2 13 2 1.314 2 0.907 2 0.649

0 0 0 0 π 6 π 6 π 6 π 6 π 3 π 3 π 3 π 3

Material properties and statistical analysis of high-strength steels

29

initiation set are calculated using Bridgman’s formulas and listed in Table 2.10. The photographs of representative specimens are shown in Fig. 2.9.

2.3.2.2 Smooth round bar tensile test For the von Mises yield function, there is only hardening function σ ðε Þ needed to be calibrated. Because two different thicknesses (10 and 30 mm) of steels are used to manufacture specimens, we design two types of uniaxial tensile tests (flat specimen with rectangular cross section for 10-mm steel plate and smooth round bar tensile test

Figure 2.9 Photographs of representative specimens.

30

Behavior and Design of High-Strength Constructional Steel

for 30-mm steel plate) to obtain the hardening functions. The dimensions of the flat specimen and smooth round bar are shown in Fig. 2.10. All uniaxial tensile tests are performed on an MTS machine with the loading speed of 0.3 mm/min to achieve the quasi-static loading condition. The extensometer with length of 20 mm is assembled on the specimen to measure the elongation δ. The typical load-elongation curves of three different strength grades of HSS are shown in Fig. 2.11. The engineering stress
strain curve is given by σE 5

P A0

(2.8)

εE 5

δ δ0

(2.9)

where A0 is the initial area of the minimum cross section and δ0 is the initial gauge section length. Based on engineering stress
strain curve, the true stress
strain can be calculated by σT 5 σE ð1 1 εE Þ

(2.10)

εT 5 lnð1 1 εE Þ

(2.11)

Because these two transformation equations are based on incompressible assumption, Eqs. (2.10) and (2.11) are no longer valid in the postnecking phase in which severe nonuniform deformation occurs in the necking region. In order to solve this problem, a number of correction methods are proposed in previous publications [16
19]. In the present study, a power function is used to fit the true stress
strain curve obtained from prenecking phase, and then the approximate true stress
strain curve after necking can be obtained by extrapolating the fitting curve.

20 40

R=25

120 350 d=10

R10

15

(A)

10

120 300 (B)

Figure 2.10 Dimensions of flat specimen and smooth round bar. (A) Flat specimen with rectangular cross section (thickness of steel plate 5 10 mm). (B) Smooth round bar (thickness of steel plate 5 30 mm).

Material properties and statistical analysis of high-strength steels

31

Figure 2.11 Load-elongation curves of flat specimen and smooth round bar tensile tests. (A) Flat specimen tensile test (thickness of steel plate 5 10 mm). (B) Smooth round bar tensile test (thickness of steel plate 5 10 mm).

Figure 2.12 True stress
strain curves of HSS.

32

Behavior and Design of High-Strength Constructional Steel

The true stress
strain curves for two different thickness steel plates are plotted in Fig. 2.12. The calibrated values of Young’s modulus E, Poisson’s ratio υ; and yield strength for three grades HSS are summarized in Tables 2.11 and 2.12. The finite element analyses are carried out to verify the hardening function by examining the discrepancy between experimental data and simulation results of tensile tests. The FE models of flat specimen and smooth round bar are shown in Table 2.11 Mechanical properties of HSS (thickness of steel plate 5 10 mm). Steel

Young’s modulus E (GPa)

Poisson’s ratio υ

Yield strength (MPa)

Q550 Q690 Q890

198 216 206

0.3 0.3 0.3

537 713 1146

HSS, High-strength steel.

Table 2.12 Mechanical properties of HSS (thickness of steel plate 5 30 mm). Steel

Young’s modulus E (GPa)

Poisson’s ratio υ

Yield strength (MPa)

Q550 Q690 Q890

185.5 204 208

0.3 0.3 0.3

662 745 924

HSS, High-strength steel.

Figure 2.13 FE model of flat specimen and smooth round bar. (A) Flat specimen with rectangular cross section (thickness of steel plate 5 10 mm) and (B) smooth round bar (thickness of steel plate 5 30 mm).

Material properties and statistical analysis of high-strength steels

33

Figure 2.14 A comparison of load-elongation curves between experiments and FE analyses (FS with rectangular cross section, SRB): (A) Q550, (B) Q690, and (C) Q890. FS, Flat specimen; SRB, smooth round bar.

34

Behavior and Design of High-Strength Constructional Steel

Fig. 2.13. Due to the symmetrical shape of flat specimen and axisymmetric shape of smooth round bar, only 1/8 of the full flat specimen is modeled using solid elements (C3D8R), and a quarter of full smooth round bar is modeled using axisymmetric elements (CAX8R). After comparing different mesh densities, a fine mesh size is selected, and the necking region is further refined with the minimum mesh size of 0.2 and 0.05 mm, respectively. As illustrated in Fig. 2.14, the load-elongation curves of smooth round bar simulation agree with the corresponding experimental results well, which implies that the calibrated mechanical properties of HSSs are valid.

2.3.2.3 Notched round bar tensile test A group of notched round bar tensile tests are conducted to investigate the plastic behavior of HSS in different triaxiality stress states. It is worth mentioned that the Table 2.13 Dimensions of notched round bar (Q690). ID

R0 (mm)

D0 (mm)

L0 (mm)

NRB-1 NRB-2 NRB-3

2.5 5.0 10.0

10.03 10.04 10.03

5.02 8.61 13.27

Figure 2.15 Dimensions of notched round bar.

Figure 2.16 FE model of notched round bar.

Material properties and statistical analysis of high-strength steels

35

Figure 2.17 Load-elongation curves of notched round bar tensile tests. (A) Q550, (B) Q690, and (C) Q890.

36

Behavior and Design of High-Strength Constructional Steel

Lode angle θ of notched round bar is identical with that of smooth round bar, but the corresponding ranges of triaxiality are different. The range of triaxiality depends on the radii of the notch of the notched round bar. In present study, three values of radii of the notches are designed: 2.5, 5.0, and 10.0 mm. Actual dimensions of notched round bars are listed in Table 2.13, and the other dimensions are identical to Fig. 2.15. These notched round bars are all manufactured from steel plates with the thickness of 30 mm. Due to the symmetrical shape of the smooth round bar, only a quarter is modeled by axisymmetric elements (CAX8R) with finer mesh in the notched region with the minimum mesh size of 0.05 mm, as shown in Fig. 2.16. The extensometer is assembled on the specimen to measure the elongation δ of the gauge section with length of 20 mm. The typical load-elongation curves of three HSSs are shown in Fig. 2.17. Based on numerical simulation results, the evolution of the stress triaxiality at the center of the minimum cross section is obtained, as shown in Fig. 2.18. It is found that in the whole loading process, the value of stress triaxiality on the center of the minimum cross section is higher than 1/3 which represents the uniaxial tensile stress state. For comparison, the evolution of stress triaxiality on the center of the minimum cross section of smooth round bar under tensile loading is also plotted. As mentioned before, the geometry notch to a smooth round bar increases the stress triaxiality inside the neck. It can be observed in Fig. 2.19 that the analysis results by using von Mises criterion gives a well prediction of the actual load-elongation response of the notched round bar tensile tests. The stress triaxiality on the minimum cross section of the notched round bar tends to be higher than that of the smooth round bar, but there is no distinct discrepancy (less than 3.5%) appears between the analysis results by using von Mises criterion and the actual responses. Moreover, the same observation appears in the other notched round bar tensile tests with Q550 and Q890 HSSs (the discrepancies are less than 4% and 3%, respectively). It indirectly implies the influence of stress triaxiality on the plasticity model of HSSs is negligible.

Figure 2.18 Evolution of the stress triaxiality at the center of the minimum cross section.

Material properties and statistical analysis of high-strength steels

Figure 2.19 Notched round bar load-elongation curves for Q690. (A) R 5 2.5 mm, (B) R 5 5.0 mm, and (C) R 5 10.0 mm.

37

38

Behavior and Design of High-Strength Constructional Steel

2.3.2.4 Cylinder specimen compression test The von Mises yield criterion assumes that there is no difference between compressive and tensile yield strength of the HSS at the given strain. In order to check whether the HSS shows strength differential effect, another group of cylinder specimen compression tests is conducted. The cylinder specimen compression test corresponds to stress state of θ 5 π=3 and η , 0. In order to prevent buckling failure of the specimen, the ratio of height H to diameter D of the specimen is set as 2.0 as shown in Fig. 2.20. Lubricant is applied to the machine-specimen interfaces to reduce the friction effect. The extensometer with length of 10 mm is assembled on the specimen to measure the elongation δ. The typical load-elongation curves of three HSS are shown in Fig. 2.21. The FE model of the cylinder, as shown in Fig. 2.22, is discretized using 4-node axisymmetric elements with the minimum mesh size of 0.05 mm. The numerical simulation results are compared to the experimental results in Fig. 2.23 (only the results of Q690 are plotted). It is observed that the FE results using von Mises yield function underestimates the experimental response with about 7% error for Q690 HSS (3% for Q550 and 5% for Q890), which means the strength differential effect exists for HSSs, but von Mises yield function fails to characterize it. The strength differential can arise from

Figure 2.20 Dimension of cylinder specimen.

Figure 2.21 Load-elongation curves of cylinder specimen compression tests.

Material properties and statistical analysis of high-strength steels

39

Figure 2.22 FE model of cylinder specimen.

Figure 2.23 Cylinder specimen load-elongation curves for Q690.

the dependence of yield condition on stress triaxiality or Lode angle. Because it has been demonstrated the stress triaxiality independence of HSS based on notched round bar tensile tests, the discrepancy between von Mises result and experimental result implies that the yield strength of the HSS is greater in uniaxial compression with θ 5 π=3 than in uniaxial tension with θ 5 0 at the given strain.

2.3.2.5 Pure shear test The specimen configuration for pure shear loading is shown in Fig. 2.24. During the loading process, an upward boundary condition is applied at the top and the bottom is fixed. It is observed that severe deformation occurs in the gauge section and the crack initiates at the pure shear section. The extensometer is assembled on the specimen to measure the elongation δ of the gauge section with the length of 20 mm. The typical load-elongation curves of three HSS are shown in Fig. 2.25. Corresponding numerical simulations, the pure shear specimen is modeled using

40

Behavior and Design of High-Strength Constructional Steel

Figure 2.24 Dimension of pure shear specimen.

Figure 2.25 Load-elongation curves of pure shear tests.

8-node solid elements, and the whole model consists of 71280 elements with finer mesh at the pure shear section, as shown in Fig. 2.26. As shown in Fig. 2.27, it can be observed that for Q690 HSS, the simulation results by using von Mises criterion overpredicts the actual load-elongation response of the pure shear test with maximum error equal to 16% (31% for Q550 and 15% for Q890). It is worth mentioning that the Lode angle and stress triaxiality on the minimum cross section of the pure shear specimen (θ 5 π=6, η . 1=3) are different from that of the smooth round bar (θ 5 0, η 5 1=3). Because it has been demonstrated that the influence of the stress triaxiality on plasticity model is negligible, the reason for this discrepancy is differences between plane strain strength (θ 5 π=6) and tensile strength (θ 5 0), which means that the influences of Lode angle on the plasticity model of HSS is not negligible.

2.3.3 Effect of stress triaxiality and Lode angle The experimental study has shown that the von Mises yield criterion calibrated from smooth round bar tensile tests can predict the notched round bar tensile tests

Material properties and statistical analysis of high-strength steels

41

Figure 2.26 FE model of pure shear specimen.

Figure 2.27 Pure shear specimen load-elongation curves for Q690.

but fails to prediction the real responses of compressive and shear tests, which indirectly implied that the Lode angle effect should be considered in plasticity model of HSS, and stress triaxiality influence is negligible. In order to investigate the influence of the stress triaxiality on the yield criterion of HSS specifically, the initial yield strength of HSS under various stress triaxiality state are calculated. Because the stress state on the minimum cross section of the notched round bar is triaxial, the average true stress σaverage , which is calculated using load and current area of the minimum cross section, is not equal to the equivalent stress σ. In the present study, the Bridgman modified function is adopted to derive the equivalent stress, which is given by σ 5 σaverage 

11

2R a

1

ln 1 1



a 2R

(2.12)

42

Behavior and Design of High-Strength Constructional Steel

where R and a are curvature of the notched region and radius of minimum cross section, respectively. According to the results of four types specimen tensile tests (SRB, NRB-1, NRB-2, and NRB-3), corresponding initial yield strength data is calculated. The stress triaxiality at the center of the minimum cross section is various for different notched radii, and Bridgman’s function is generally used to calculate the stress triaxiality at the center of the minimum cross section. However, based on numerical simulation, a corrected formula was proposed by Wierzbicki and Bao, which is given by η5

1 pffiffiffi
a 1 2ln 1 1 3 2R

(2.13)

The initial yield strength and corresponding stress triaxiality data points are plotted in Fig. 2.28, and the dashed-line represents the average value. It can be found that there is little discrepancy between values of the initial yield stress (deviation is less than 0.5%). Therefore the influence of stress triaxiality on the plasticity model of HSS is negligible. In order to evaluate the influence of the Lode angle on the yield criterion of HSS specifically, the initial yield strength of HSS under various Lode angle states (SRB, for θ 5 0; PS, for θ 5 π=6; CS, for θ 5 π=3) are calculated and compared. Eqs. (2.14) and (2.15) are used to calculate the yield stress based on experimental results of ST-1-1 and CC-1-1. σ θ5π=6 5

pffiffiffi F 3 A

(2.14)

σ θ5π=3 5

F h  A0 h 0

(2.15)

Because it has been proved that the stress triaxiality has little effect on plasticity model of HSS, the shape of yield surface does not change along the hydrostatic

Figure 2.28 Relationship between initial yield strength and stress triaxiality for Q690.

Material properties and statistical analysis of high-strength steels

43

Figure 2.29 Initial yield strength data point on deviatoric plane. Table 2.14 An overview of parameters in the new function of HSS. Steel grade Q550 Q690 Q890

Cπ6

Cπ3

0.88 0.92 0.89

1.02 1.06 1.04

HSS, High-strength steel.

pressure line. Therefore the initial yield stress points can be compared on the same deviatoric plane, as plotted in Fig. 2.29. It can be observed that the classical yield criterions, von Mises yield criterion, fails to characterize the differences between compressive, shear, and tensile strength of HSS.

2.3.4 Proposed plasticity model of high-strength steel According to the experimental results, it has been demonstrated that the Lode angle effect should be considered in the plasticity model of HSS, and influence of stress triaxiality is negligible. Therefore, a new yield function, which considers the Lode angle effect, is adopted to predict plastic behavior of HSS and is defined as h
 π πi  f 5 σ 2 σðεÞ Cπ6 1 Cθ 2 Cπ6 U θ 2 = 6 6 8 π > 1 for 0 # θ # > < 6 Cθ 5 π π > > : Cπ3 for 6 , θ # 3

(2.16)

(2.17)

44

Behavior and Design of High-Strength Constructional Steel

There are two parameters Cπ6 and Cπ3 in the new proposed yield function. According to results of four types experiments mentioned in Section 2.3, these parameters are calibrated using the inverse method to minimize discrepancy between experimental and simulated load-elongation curves, as listed in Table 2.14. The numerical results using the new proposed plasticity model are plotted in Figs. 2.30 and 2.31 (only the results of Q690 are plotted). By examining the discrepancy between experimental data and simulation results, the new plasticity model gives better prediction of plastic behavior of HSS with an error less than 1%.

Figure 2.30 Cylinder specimen load-elongation curves for Q690.

Figure 2.31 Pure shear specimen load-elongation curves for Q690.

Material properties and statistical analysis of high-strength steels

45

0

N

1

Figure 2.32 Dimension of notched cylinder specimen. Table 2.15 Dimensions of notched cylinder specimen (Q690). ID NCS-1 NCS-2 NCS-2

D0 (mm)

D1 (mm)

L (mm)

10.07 10.07 9.97

15.02 15.03 14.95

4.98 8.69 13.27

NCS, Notched cylinder specimen.

Figure 2.33 FE model of notched cylinder specimen.

2.3.5 Validation of the proposed plasticity model for HSS In this section, based on experimental results of two new types of specimens: notched cylinder compression tests and pure shear tests, the validation work of the new plasticity model of HSS is conducted.

2.3.5.1 Notched cylinder compression test The geometry of notched cylinder is shown in Fig. 2.32 and details are listed in Table 2.15. There are three radius of notch assigned, equal to 5.0 and 10.0 mm, respectively. The FE model of the notched cylinder, which is also modeled using

46

Behavior and Design of High-Strength Constructional Steel

Figure 2.34 Load-elongation curves of notched cylinder compression tests (A) Q550, (B) Q690, (C) Q890.

Material properties and statistical analysis of high-strength steels

47

4-node asymmetrical elements (CAX4R), is similar to the conventional cylinder as shown in Fig. 2.33. The extensometer is assembled on the specimen to measure the elongation δ of the gauge section with the length of 15 mm. The typical load-elongation curves of three HSS are shown in Fig. 2.34. Under compression condition, the notched cylinder has the same value of Lode angle, but the corresponding values of stress triaxiality are different. A comparison of the evolution of stress triaxiality on the center of the middle height cross section is done, as shown in Fig. 2.35. The simulation results by using von Mises yield function and new yield function are plotted in Fig. 2.36 (only the results of Q690 are plotted). Compared with the simulation with von Mises yield criterion (errors are about 5%), the numerical simulation using new plasticity model gives a better prediction of the load-elongation curve with error less than 1% which implies that the new plasticity model is applicable for HSS under compression condition.

2.3.5.2 Flat grooved plate tensile test The flat grooved plate has the same range of triaxiality with that of the notched round bar, but because flat grooved plate tensile test is a plane strain case, which means that the value of Lode angle is around π=6. The dimensions of the flat grooved plate are illustrated in Fig. 2.37, and three values of radius of the grooves are machined: 1.25, 5.00, and 15.00 mm. The extensometer is assembled on the specimen to measure the elongation δ of the gauge section with the length of 20 mm. The typical loadelongation curves of three HSS are shown in Fig. 2.38. As for the FE model of flat grooved plate, 8-node solid element (C3D8R) is adopted, and due to symmetric geometry of the specimen, 1/8 of the full specimen is modeled, as shown in Fig. 2.39. As shown in Fig. 2.40, the FE analyses results using von Mises and new proposed yield function are compared to the experimental results (only the results of

Figure 2.35 Evolution of the stress triaxiality on the center of the minimum cross section.

48

Behavior and Design of High-Strength Constructional Steel

Figure 2.36 Notched cylinder load-elongation curves for Q690. (A) R 5 2.50 mm, (B) R 5 5.00 mm, and (C) R 5 15.00 mm.

Material properties and statistical analysis of high-strength steels

49

Figure 2.37 Dimension of flat grooved plate.

Q690 are plotted). It can be observed that the simulation with von Mises yield criterion overpredicts the experimental load-elongation curve up to 25%. The numerical simulation using new plasticity model gives a better prediction of the loadelongation curve with error less than 6% which implies that the new plasticity model is applicable for HSS under plane strain condition. In summary, notched cylinder compression tests and flat grooved plate tensile tests are conducted to demonstrate the validation of the new plasticity model of HSS. According to experimental and numerical results, the new plasticity model gives better prediction of the real response than von Mises yield function.

2.4

Statistical analysis of Q690 high-strength steel

2.4.1 Data retrieval and statistics of material properties In addition to the results mentioned above, 319 test samples of HSS with 690 MPa strength grade were collected from other publications [20
26]. The thickness of steel plate ranges from 4 to 40 mm. According to the origin, these steel samples were mainly manufactured from China, Europe, the United States, and Australia, and respective proportion is shown in Fig. 2.41. According to the production time of steels, 14% of steel samples were produced before 2000 and 86% after 2000. The yield strength of 14 samples in the statistical data cannot meet the requirements of material specifications, which is eliminated in the process of statistical analysis. The measured yield strengths of two other samples were 887 and 920 MPa, which were much higher than the yield strength of other samples, and were also excluded. Combining 202 valid samples collected from literatures with 81 monotonic tensile tests data, the statistical values of yield strength of Q690 steel are calculated, as shown in Table 2.16. With the assumption that 384 samples are in normal distribution, the chi-square test is used to verify the hypothesis, and the test significance level is 0.05. The actual distribution of statistical samples is compared with the hypothetical normal distribution as shown in Fig. 2.42. At the 0.05 significance level, the critical chi-square value is 14.07, and the cumulative chi-square value is 6.45, less than the critical value, which means the assumption of normal distribution is acceptable.

50

Behavior and Design of High-Strength Constructional Steel

Figure 2.38 Load-elongation curves of flat grooved plate tensile tests (A) Q550 (B) Q690 (C) Q890.

Material properties and statistical analysis of high-strength steels

51

Figure 2.39 FE model of flat grooved plate.

2.4.2 Structural reliabilities using Q690 steel 2.4.2.1 Determination of structural reliability The safety of a structure is governed by R$S

(2.18)

where R is the structural resistance and S is the effects of combined loads on the structure. Since both the structural resistance and combined load effects are random variables, the structural reliability can be determined by Ps 5 PðR $ SÞ

(2.19)

2.4.2.2 Statistical analysis of structural resistance The resistance of structural members is many influenced by factors and these uncertainties can be treated as random variables. The resistance of structural members is a function of multiple random variables, which can be expressed as follows: R 5 χ0 χf χA χP

(2.20)

where χ0 is the random variable of differences of material properties between structural components and specimens; χf is the random variable of material property

52

Behavior and Design of High-Strength Constructional Steel

Figure 2.40 Flat grooved plate load-elongation curves for Q690. (A) R 5 1.25 mm, (B) R 5 5.00 mm, and (C) R 5 15.00 mm.

Material properties and statistical analysis of high-strength steels

53

Figure 2.41 The proportion of origin. Table 2.16 Statistic parameters of yield strength for 690 MPa grade of steel. Nominal yield strength (MPa) 690

Mean value μ (MPa)

Standard deviation σ (MPa)

Nominal value χ

Variation coefficient δ

776.1

33.7

1.125

0.0434

Figure 2.42 Histogram and PDF of 690 MPa steel.

uncertainty of specimen; χA is the random variable of geometry uncertainty of specimen; and χP is the random variable of resistance computing model uncertainty.

54

Behavior and Design of High-Strength Constructional Steel

Table 2.17 Statistic parameters of resistance uncertainties for Q690 steel members. Random variables

χXi

δXi

χf χ0

1.13 0.92

0.043 0.032

χA χP

0.98 1.05 1.07 1.12 1.07 1.09 1.14

0.049 0.07 0.10 0.10 0.10 0.12 0.12

Uniaxial tension Uniaxial compression Eccentric compression Uniaxial tension Uniaxial compression Eccentric compression

χR

Note: χXi is the ratio of variable χi to nominal; δXi is the variation coefficient of χi .

The random variable of material property uncertainty of specimen is taken as the values listed in Table 2.16, the random variable of resistance computing model uncertainty is taken as the value given by Shi Gang et al. The random variable of geometry uncertainty of specimen is taken as the statistical value in Steel Structural Design Code GB20017-2017. The other random variables refer to the Unified Standard for Reliability Design of Building Structures GB50068-2001. These statistic variables of resistance uncertainties for Q690 steel members are summarized in Table 2.17.

2.4.2.3 Statistical analysis of load effect The loads imposed on the structures can be divided into permanent load and variable load. In the analysis, the variable load only considers the variable load on the residential floor, office floor, and the wind load. The permanent load conforms to the normal distribution, while the variable on the floor and wind load conform to the extreme I distribution. The Unified Standard for Reliability Design of Building Structures GB50068-2001 gives the values of the above statistical parameters of load effect. G

G

G

G

Statistical parameters of permanent load. Because the variation of statistic characteristics of permanent load with time can be neglected, according to the Unified Standard for Building Structural Design, the statistical parameter of permanent load can be calculated by χG 5 μG =SGK 5 1:060

(2.21)

δG 5 0:070

(2.22)

Statistical parameter of variable load on floor. The Unified Standard for Design of Building Structures GB50068-2001 stipulates that the standard value of variable load is SQK 5 1:5 kN=m2 , while in Lode Code for the Design

Material properties and statistical analysis of high-strength steels

G

G

G

G

55

of Building Structures (GB 50009-2012) [27], the standard value of variable load increases to 2:0 kN=m2 . Accordingly it is necessary to revise the calculation results as follows: For offices: χQ 5 μQ =SQK 5 0:524

(2.23)

δ 5 0:288

(2.24)

For residential buildings: χQ 5 μQ =SQK 5 0:644

(2.25)

δ 5 0:233

(2.26)

Statistical parameter of wind load. In Lode Code for the Design of Building Structures (GB50009-2001), the recurrence period of basic wind pressure is raised from 30 to 50 years. The current design code of steel structures follows this revision, so it is necessary to revise the statistical parameters of wind load when carrying out reliability analysis. Because generally the revised basic wind pressure increases by 10%, with the assumption that the distribution of wind load does not change, the statistical parameters of wind load should be divided by the correction factor of 1.1 (Table 2.18).

Not in the direction of wind: χ 5 1:109=1:1 5 1:008

(2.27)

δ 5 0:193

(2.28)

In the direction of wind: χ 5 0:999=1:1 5 0:908

(2.29)

δ 5 0:193

(2.30)

Table 2.18 Updated statistic parameters of loads. Load type Permanent load Variable load on office floor Variable load on residential floor Wind load (not in the direction of wind) Wind load (in the direction of wind)

χ

δG

Distribution form

1.060 0.524 0.644 1.008

0.070 0.288 0.233 0.193

Normal distribution Extreme value type I distribution Extreme value type I distribution Extreme value type I distribution

0.908

0.193

Extreme value type I distribution

56

Behavior and Design of High-Strength Constructional Steel

2.4.3 Target reliability According to the Unified Standard for Reliability Design of Building Structures (GB 50068-2001), the reliability index of the ultimate bearing capacity of structural members should not be less than the limit value in Table 2.19.

2.4.4 Partial coefficients of resistance and design strength According to Load Code for the Design of Building Structures (GB 50009-2012) [27], the following load combinations should be considered when calculating the resistance partial coefficients: G

G

G

Permanent action 1 floor variability (G 1 L) RK 5 γ R ð1:2SGK 1 1:4SLK Þ

(2.31)

RK 5 γ R ð1:35SGK 1 0:7 3 1:4SLK Þ

(2.32)

Permanent action 1 wind load action (G 1 W) RK 5 γ R ð1:2SGK 1 1:4SWK Þ

(2.33)

RK 5 γ R ð1:35SGK 1 0:7 3 1:4SWK Þ

(2.34)

where γ R is the resistance partial coefficient; SGK is the standard value of permanent load effect; SLK is the standard value of floor load effect, which should be calculated for office and residential building; SWK is the standard value of wind load effect, which should be calculated in the wind direction and not in the wind direction.

The ratio of variable load to permanent load ρ should be considered: when ρ 5 0.25, the permanent load plays a controlling role, which is calculated using Eqs. (2.32) and (2.34); when ρ 5 0.5, 1.0, or 2.0, the variable plays a controlling role, which is calculated using Eqs. (2.31) and (2.33). According to Eqs. (2.31)
(2.34), the resistance partial coefficients of all kinds of components are calculated. Accordingly the discrepancy between the standard value of bearing capacity of all kinds of components and the standard value of bearing capacity calculated directly according to probability reliability is minimized.

Table 2.19 Targeted reliability index under ultimate limit states. Failure type Ductile failure Brittle failure

Safety grade 1 3.7 4.2

2 3.2 3.7

3 2.7 3.2

Material properties and statistical analysis of high-strength steels

Hi 5

2 X
2 X
RKij 2RKij 5 RKij 2γ Ri Sj j

57

(2.35)

j

Sj 5 γ G SGKj 1 γ Q SQKj

(2.36)

where RKij is the resistance standard value of the type i structure under the j load effect ratio, which is calculated by the target reliability index using probability method; RKij is the resistance standard value of the type i structure under the j load effect ratio, which is calculated from Eqs. (2.31) to (2.34). In order to minimized the value of Hi P γ Ri 5

 j RKij Sj P 2 j Sj

(2.37)

where RKij 5

μRij χRi

(2.38)

Therefore the resistance partial coefficients of the type i structure μRij can be calculated using Eqs. (2.37) and (2.38) based on the type i structure under the j load effect ratio according to the target reliability index calculated using the probability method. In the calculation method of structural reliability, the central point method and checking point method are practical method. The checking point method is adopted in the calculation of structural reliability, and the following formulas are used to calculate the ur .

g μX1 1 α1 βσX1 ; ?; μXn 1 αn βσXn 5 0    @g  σ  n
X Xi αi 5 

2 i1=2 @Xi  P @g  σ X X  i @Xi

(2.39)

(2.40)

i51

Xi 5 μXi 1 αi βσXi

(2.41)

where gðÞ is the functional function of structure; Xi is random variables in limit state equation; and αi is directional cosine from origin to checking point. In Eq. (2.39), only ur is unknown and can be calculated using iteration method. Hence the partial coefficients of resistance γ Ri of different structural members and design values of steel strength can be obtained using Eqs. (2.37) and (2.38).

58

Behavior and Design of High-Strength Constructional Steel

The partial coefficients obtained for various Q690 steel structural members are listed in Table 2.20. In the Code for Design of Steel Structures GB50017-2017 [28], the partial coefficients of resistance of Q235 steel under various combinations of action effects are unified as 1.087. By comparison, under the action of “G 1 L,” the resistance partial coefficient of Q690 steel is less than that of Q235 steel (except for the axially compressed members under the variable load combination of residential floors), while in the combination of “G 1 W,” the resistance partial coefficient of Q690 steel is larger than that of Q235 steel. This is because: (1) The statistical parameter of Q690 HSS (χm 5 χf χ0 5 1:035) is smaller than that of Q235 steel (χm 5 1:08); (2) In the Load Code for the Design of Building Structures (GB 50009-2001) (2006 edition), the standard value of variable load of office floors and residential floors increases from LK 5 1:5kN=m2 to SQK 5 2:0kN=m2 ; (3) In the unified standard GBJ 68
84 [29], the β is allowed to be adjusted as the specified value 6 0.25. After several trial calculations, the value of γ R is taken as 1.106 and the corresponding reliability indexes β under various combinations of action effects are listed in Table 2.8. It can be found that the reliability index of Q690 HSS is higher than that of Q235 steel under “G 1 L” combination, and the reliability index level of Q690 HSS under “G 1 W” combination is equal to that of Q235 steel. Therefore, it is suggested that the resistance partial coefficient of Q690 HSS be taken as γ R 5 1:106. According to the Specification for Classification of Thickness in High Strength Low Alloy Structural Steels (GB/T 1591
2008) [30], for Q690 HSS, the minimum Table 2.20 Partial factors of resistance (β 5 3:2). Load combination

Axial tension

axial compression

Eccentric compression

1.031 1.053 1.264 1.177

1.081 1.125 1.318 1.244

1.030 1.054 1.234 1.152

G 1 L office G 1 L residence G 1 W in the wind direction G 1 W not in the wind direction

Table 2.21 Expected reliability index under different load combinations if γR 5 1:106. Load combination G1L G 1 W not in the wind direction

Q690 3.67 2.67

Q235 3.20 2.69

Q690 3.52 2.54

Axial tension

Axial compression

Eccentric compression

Q235 3.07 2.55

Q690 3.74 2.78

Q235 3.16 2.77

Material properties and statistical analysis of high-strength steels

59

yield strength of steel plate is 690 and 670 MPa when the thickness of plate t # 16 mm and 16 mm , t # 40 mm, respectively. Considering the design value of material strength is f 5 fy =γ R , the design value of Q690 steel strength is determined as:f 5 625MPa for t # 16 mm; f 5 605MPa for 16 mm , t # 40 mm (Table 2.21).

2.5

Summary

In order to obtain the statistical parameters of material properties, such as yield strength, tensile strength and elongation of HSSs (Q460, Q550, Q690, and Q890), the standard specimens of HSSs produced by Wuyang Iron and Steel Liability Co., Ltd., Nanjing Iron and Steel Group Co., Ltd., and Jiangyin Xingchen Special Steel Co., Ltd. were taken to carry out the monotonic tensile tests. Moreover, based on a series of experiments with six specimen geometries, the plasticity model has been reexamined in case of three grades of HSS. The following conclusions are obtained: 1. High-strength structural steels exhibit appreciable strength but reduced deformability compared to normal-strength steels. The yield-to-tensile ratios for Q550D, Q690D, and Q890D are more than 0.90, while the elongations range from 12% to 20%, indicating that these three structural steels are prohibited according to current Chinese seismic code. However, the yield-to-tensile ratio for Q460D steel is 0.78, and the elongation is larger than 20%; thus Q460 steel can be directly used in steel buildings in seismic zones as normal-strength steel. 2. The von Mises yield function, a classical metal plasticity model which assumes the Lode angle have no effect on yield strength, is no longer applicable well for HSS, especially in shear and compressive condition, and the maximum error reaches 25%. 3. The proposed yield function with consideration of Lode angle effect can give more accurate prediction of the actual response. According to experimental and numerical results, all material parameters of plasticity models of three types of HSS are calibrated. 4. With the target reliability index, the resistance partial coefficient varies significantly under different combinations of action effects, and the value of under “G 1 W” combinations is obviously larger. When the resistance partial coefficient is taken as 3.2, the reliability index of Q690 HSS member is equivalent to that of Q235 steel member. Therefore it is suggested that the resistance partial coefficient of Q690 HSS member γR 5 1:106. The design value of Q690 steel strength is taken as follows: f 5 625MPa for t # 16 mm; f 5 605MPa for 16 mm , t # 40 mm.

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Behavior and Design of High-Strength Constructional Steel

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506. [20] Beg D, Hladnik L. Slenderness limit of Class 3 I cross-sections made of high strength steel. J Constr Steel Res 1996;38(3):201
17. [21] da Silva LS, et al. Statistical evaluation of the lateral-torsional buckling resistance of steel I-beams, Part 2: Variability of steel properties. J Constr Steel Res 2009;65 (4):832
49. [22] Rasmussen KJR, Hancock GJ. Plate slenderness limits for high-strength steel sections. J Constr Steel Res 1992;23(1
3):73
96. [23] Rasmussen KJR, Hancock GJ. Tests of high-strength steel columns. J Constr Steel Res 1995;34(1):27
52. [24] Shi G, Zhu X. Research on resistance model uncertainty of high-strength steel axial compression members. Build Struct 2014;44(16):36
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42.

Material properties and statistical analysis of high-strength steels

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[26] Usami T, Fukumoto Y. Welded box compression members. J Struct Eng-ASCE 1984;110(10):2457
70. [27] Load code for the design of building structures (GB 50009-2012). Beijing, China: Standards Press of China; 2012. [28] Code for seismic design of buildings (GB 50017-2017). Beijing, China: China Planning Press; 2017. [29] Unified standard for reliability design of building structures (GB 50068-2001). Beijing, China: Standards Press of China; 2001. [30] High strength low alloy structural steels (GB/T 1591-2008). Beijing, China: Standards Press of China; 2008

Hysteretic behavior of high strength steels under cyclic loading

3

Guo-Qiang Li, Yan-Bo Wang and Fei-Fei Sun Tongji University, Shanghai, P.R. China

3.1

Introduction

For seismic design of structures, it is not economical to restrict structures in elastic state under severe earthquake actions. Instead, it is expected that structures should be able to endure rational inelastic deformations in order to dissipate earthquake energy. Therefore the seismic performance of high-strength steel (HSS) should be examined if it is related to the application of HSS in structural design with required earthquake-resistance. With the increase in yield strength, the yield to tensile strength ratio gets closer to 1.0 and the elongation ratio decreases. Ductility requirements specified in Eurocode3 [1] and Eurocode8 [2] are in terms of the tensile to yield strength ratio (fu/fy $ 1.10), the elongation ratio (not less than 15%) and the ultimate strain (not less than 15 times of the yield strain). According to the Chinese Code for Seismic Design of Buildings GB 50011-2010 [3], the ductility requirements of steel (fy/fu # 0.85, elongation ratio $ 20%) are more stringent than those of Eurocode3 due to the expectation of inelastic behavior of structural elements and connections under rare earthquake actions. The important mechanical properties of HSS can hardly meet the requirements specified in GB 50011-2010 for earthquake resistance. Currently, the application of HSS in seismic design is restricted by existing seismic design codes and constructional practice, which are established on the study of conventional steel. A number of experimental observations [4,5] have been made on the hysteretic behavior of metal materials subjected to uniaxial cyclic loading. However, due to lack of cyclic test data of HSS structural members, the verification of hysteretic model on member level for HSS was limited in the previous researches. Therefore better understanding of the inelastic cyclic behavior of HSS, as well as the reliable model for predicting the seismic behavior of HSS members, is important for applying HSS to seismic resistance structures. Moreover, the necessity of developing and calibrating a cyclic model for HSS is highlighted. In the experiments reported in this chapter [6], cyclic behaviors of four types of HSS—Q460D, Q550D, Q690D, and Q890D—were evaluated. The cyclic stress evolution law, cyclic backbone, and energy dissipation behavior were analyzed through the measured stress
strain hysteretic curves. The cyclic backbones were then calibrated using Ramberg
Osgood model. The behavior differences between the four steel grades were compared and discussed. Based on the test results, three Behavior and Design of High-Strength Constructional Steel. DOI: https://doi.org/10.1016/B978-0-08-102931-2.00003-6 © 2021 Elsevier Ltd. All rights reserved.

64

Behavior and Design of High-Strength Constructional Steel

widely used cyclic constitutive models were calibrated, including Chaboche model (combined isotropic/kinematic hardening model) [7], Dong
Shen model (cumulative damage
based cyclic constitutive model) [8], and Giuffre
Menegotto
Pinto (GMP) model [9]. To compare the applicability of the three constitutive model, the calibrated key parameters were used to simulate the aforementioned cyclic coupon tests and their simulation results were compared with the test results. The comparison reveals that Dong
Shen model is the most favorable constitutive model for high strength steels with cyclic softening.

3.2

Cyclic behavior of high strength steels

3.2.1 Experimental program 3.2.1.1 Test materials The structural steel plates studied in this chapter were manufactured by Wuyang Iron and Steel Co. Ltd and consisted of four different steel grades—Q460D, Q550D, Q690D, and Q890D.

3.2.1.2 Cyclic test specimens The cyclic test specimens were cut in the rolled direction and their geometric dimensions are illustrated in Fig. 3.1. In order to achieve considered uniaxial compressive strain without overall buckling under fully reversed cyclic strains, a R3

(A)

6 .2

80

25 2

10

30

5 t=10

(C)

80

20 25 230

(B)

80

8 28

25 241

28

R35 R

44

16

t=16

80

Figure 3.1 Typical cyclic experimental specimen details: (A) dimensions of specimens fabricated of 10 mm steel plate, (B) dimensions of specimens fabricated of 16 mm steel plate, and (C) specimens for cyclic experiments.

Hysteretic behavior of high strength steels under cyclic loading

65

relatively small length-to-width (1.5
2.0) was adopted in the design of cyclic specimen with thicknesses of 10 and 16 mm, as shown in Fig. 3.1. The 16 mm specimens were machined from 20 mm steel plate since there exists visible rustiness on the front surface of 20 mm steel plate. All steel coupons were mounted and loaded in hydraulic grips during the testing. Monotonic and cyclic loadings were applied by a servo-hydraulic universal testing machine (MTS 880) as shown in Fig. 3.2. All of the tests were strain-controlled using the measured displacement from the extensometer over the gage length divided by the extensometer gage length.

3.2.1.3 Loading protocols The cycles were fully reversed with constant strain rate 0.2%s21 for purpose of minimizing the influence of temperature increase caused by rapid working of inelastic strain. A total of three triangular waveform load histories of constant amplitude and multi-steps amplitude were applied and started with a tensile excursion, as shown in Fig. 3.3. H-1 is a strain history with constant amplitude of 4.0%. For H-2 loading case, the strain amplitude increases from 0.75% to 1.5% with amplitude increment of 0.25%. Each strain amplitude was repeated for 3 cycles to achieve a fair saturation. The form of H-3 loading case is identical to that of H-2,

Figure 3.2 Experimental setup and instrumentation.

66

Behavior and Design of High-Strength Constructional Steel

however, the strain amplitude increases from 0.50% to 2.0% with amplitude increment of 0.50%. The selection of load pattern refers to the previous similar cyclic loading tests. Constant amplitude loading history is a basic load pattern to clearly understand the cyclic behavior since the maximum strain remains constant in each loop. Besides, 6 4% is a representative amplitude that steels may encounter in earthquake actions and be therefore commonly selected in cyclic loading coupon tests. The authors make the ultimate amplitude of step-wise loading H-2 and H-3 respectively equal to 6 1.5% and 6 2%, which are also typical low-fatigue strain amplitude. These two loading histories both present four monotonically increasing strain levels of which the maximum level was continuously loaded until final fracture took place. H-1 and H-2 (H-3) can form a counterpart to study the difference between constant and step-wise amplitude, while H-2 and H-3 produce a counterpart to discuss the influence of different step-wise evolution path. Moreover, the H2 and H-3 can be treated as constant amplitude loading history when strain level reaches its maximum state. Thus these three constant amplitude loading histories can be used to study the influence of amplitude on cyclic behavior. These strain pattern and magnitudes ensured significant plastic behavior and were representative of the strain measured during cyclic tests on built-up beam-columns. All specimens were tested to failure, which was defined as a complete fracture or a reversal stress that did not achieve 20% of the maximum recorded cyclic stress. Detailed cyclic loading histories for all four types of structural steels with a total of 13 individual tests were illustrated in Table 3.1.

Figure 3.3 Cyclic loading histories: (A) Constant amplitude history H-1, (B) stepwise history H-2, and (C) stepwise history H-3.

Table 3.1 Cyclic loading histories of specimens. Loading histories H-1 H-2 H-3

Specimens 4R10-1, 5R10-1, 5R16-1, 6R10-1, 6R10-3, 8R10-1 4R10-2, 5R10-2, 5R16-2, 6R10-2, 8R10-2 5R16-3, 6R10-3

Note: The initial number of the specimen identification denotes the steel grade, the letter “R” denotes the initial letter of “reversal,” the number prior to the hyphen denotes the steel plate thickness, the number after the hyphen denotes the cyclic loading number. For instance, the identification of “4R10-2” indicates the specimen is fabricated of Q460 steel plate whose thickness is 10 mm and the specimen is loaded under the H-2 cyclic reversal loading.

Hysteretic behavior of high strength steels under cyclic loading

67

3.2.2 Hysteretic behavior 3.2.2.1 Stress
strain hysteretic curve Constant amplitude and stepwise cyclic loading history were chosen to characterize cyclic behavior of all types of steel plates quasistatically. Moreover, one additional stepwise cyclic loading history was applied on specimens with thickness equals to 16 mm. It should be noted that nearly all coupons were tested to fracture (Fig. 3.4A) without buckling in compression except for 6R10 whose instability is schematically depicted in Fig. 3.4B. Hysteresis curves for all tests are presented in Fig. 3.5. Generally, all stress
strain curves are full and stable in the preliminary cycles but endure remarkable decrease in size and considerable change in shape subsequently. Even in the constant amplitude tests, this phenomenon is evident for all studied steel grades. The stress value corresponding to the maximum strain in each cyclic loop is defined as cyclic peak stress. To thoroughly understand the influence of plasticity accumulation, the cyclic peak stress normalized by yield strength fy is plotted against the cycles normalized by failure cycle Nf for each steel grade in Fig. 3.6. Besides, comparison between different steel grades with identical cyclic loading history is shown in Fig. 3.7. It can be seen from Fig. 3.6 that for all steel grades the cyclic stress softening effect is obvious and its development shows considerable dependence on strain history and steel grades. As shown in Fig. 3.6A, constant amplitude specimen 4R10-1 whose strain amplitude is 4% presents larger hardening amount than that of stepwise amplitude specimen 4R10-2 whose largest amplitude is 1.5% during the whole

Figure 3.4 Failure model of specimens: (A) typical low-cycle fatigue fracture of specimens and (B) instability observed in 6R10-1.

68

Behavior and Design of High-Strength Constructional Steel

Figure 3.5 Cyclic stress
strain curves: (A) 4R10-1, (B) 4R10-2, (C) 5R10-1, (D) 5R10-2, (E) 5R16-1, (F) 5R16-2, (G) 5R16-3, (H) 6R10-1, (I) 6R10-2, (J) 6R10-3, (K) 8R10-1, and (L) 8R10-2.

process, revealing the truth that stress hardening is dominated by strain amplitude although plastic strain accumulation dose also produce hardening to some extent. As the cycle numbers reach to around 0.7Nf for 4R10-1 and 0.9Nf for 4R10-2, respectively, the stress stepped into softening phase whose evolutionary rate for 4R10-2 is considerably larger than that of 4R10-1, indicating the constant amplitude loading history may cause earlier but milder softening. For the test results from specimens fabricated of Q550D, Q690D, and Q890D, a slight hardening in the initial cycles took place before subsequent cyclic softening, which sustained for more than 95% of the ultimate cycle number. Similar with Q460D steel, softening speeds of specimens with constant amplitude are considerably slower than those of specimens with stepwise amplitude except for 6R10-1 which was terminated due to unexpected buckling. The comparison of different structural steels under identical strain

Hysteretic behavior of high strength steels under cyclic loading

69

Figure 3.6 Relationship of normalized cyclic peak stress and normalized cycles: (A) Q460D, (B) Q550D (t 5 10 mm), (C) Q550D (t 5 16 mm), (D) Q690D, and (E) 890D.

history shown in Fig. 3.7 indicates that the increase in yield strength generally synchronizes the decrease in cyclic hardening capacity and increase in cyclic softening capacity. Moreover, the increase in yield strength amplifies the evolutionary rate of cyclic softening behavior apparently. Table 3.2 summarizes the primary mechanical characteristics of the cyclic tested curves, including the elastic modulus E, the yield strength fy , the yield strain εy , the overall maximum stress during the whole cycles fcyc;u , the corresponding monotonic ultimate strength fu , the ratio of cyclic ultimate strength and monotonic ultimate

70

Behavior and Design of High-Strength Constructional Steel

Figure 3.7 Comparison of evolution law of cyclic peak stress between different steel grades: (A) Constant amplitude loading specimens and (B) Stepwise amplitude loading specimens.

P strength fcyc;u =fu , the total cyclePnumber N, the total cumulative P plastic strain εp , the total cumulative ductility ε =ε , the total strain energy E , and the total p y s P hysteretic energy E. It is clear that cyclic ultimate strengths are noticeably smaller than those observed in monotonic tests for steels Q550D, Q690D, and Q890D, further indicating these steels possess significant softening. The total cycles of constant amplitude tests are apparently smaller than those of stepwise amplitude tests, resulting in considerable different in the values of the total cumulative plastic strain, the total cumulative ductility, the total strain energy, and the total hysteretic energy. This is because the maximum amplitude level of stepwise amplitude tests is lower than the constant amplitude. On the other hand, the feature of step-by-step increment of stepwise amplitude test provides a milder accumulation of material damage compared to the constant amplitude test.

3.2.2.2 Cyclic skeleton curves Ramberg
Osgood model is a commonly used expression to simulate strain
stress relations and backbone curves of steels without a distinct yield point. This model is an exponential-form function containing three material-dependent parameters, as expressed in the following equation: ε 5 εe 1 εp 5

 σ 1=n0 σ 1 0 Es K

(3.1)

where Es is elastic modulus, K 0 is hardening modulus, n0 denotes hardening rate. εe represents elastic strain, and εp is plastic strain. In this study, Ramberg
Osgood parameters are calibrated for all types of tested steels and tabulated in Table 3.3 along with corresponding values of two additional mild steels from reference. It can be seen that the improvement of yield strength synchronizes the increase of K 0 and the decrease of n0 .

Table 3.2 Summary of mechanical characteristics from cyclic loading tests. Specimen

E (GPa)

fy (MPa)

εy (%)

fcu (MPa)

fu (MPa)

fcu/ fu

N

P εp

P εp/εy

P Es (MPa)

591.2 673.3 645.1 801.5 1048.5

0.42 0.49 0.52 0.56 0.67

761.6 729.8 736.7 858.8 1079.3

610.6 764.9 753.1 890.4 1105.9

1.25 0.95 0.98 0.96 0.98

20 34 25 11 22

2.9 4.4 3.6 1.5 2.9

690.3 899.1 682.4 274.7 427.8

1930.0 2055.1 2254.8 1145.8 2212.3

1.93 2.06 5.77 1.15 2.21

595.2 686.9 628.1 679.4 824.3 784.9 1049.3

0.42 0.51 0.52 0.47 0.58 0.56 0.68

637.8 677.8 717.6 726.7 816.0 837.6 1042.4

610.6 764.9 753.1 753.1 890.4 890.4 1105.9

1.04 0.89 0.95 0.96 0.92 0.94 0.94

281 145 285 201 283 35 126

13.5 6.7 13.2 12.8 12.8 5.2 4.9

3227.2 1314.7 2550.4 2727.2 2214.4 907.0 716.4

3948.8 3466.3 6586.3 6509.0 5963.5 3112.9 3730.8

3.95 3.47 16.86 16.66 5.96 3.11 3.73

P E (MNmm)

Constant amplitude 4R10-1 5R10-1 5R16-1 6R10-1 8R10-1

273.8 222.3 201.4 224.8 225.6

Stepwise amplitude 4R10-2 5R10-2 5R16-2 5R16-3 6R10-2 6R10-3 8R10-2

267.8 224.5 200.3 255.1 218.8 219.8 217.9

72

Behavior and Design of High-Strength Constructional Steel

Table 3.3 Ramberg
Osgood parameters for HSS. Specimen 4R10-2 5R10-2 5R16-2 6R10-2 8R10-2 LYP100 LYP160 Q235B Q345B

Es (GPa)

K 0 (MPa)

268 224 200 219 218 195 205 200 205

750 950 900 950 1250 161 181 808 828

n0 0.046 0.048 0.052 0.043 0.037 0.450 0.350 0.098 0.097

To clearly figure out the main differences between monotonic response and cyclic response, Fig. 3.8 compares cyclic backbones developed using calibrated Ramberg
Osgood model and monotonic and cyclic test data of each specimen. As indicated in these figures, the simulation of cyclic backbones based on Ramberg
Osgood model presents good agreement with the cyclic test results. On the other hand, the distinction between monotonic and cyclic backbones for different steel grades varies. In detail, the stress predicted by cyclic backbone of Q460D steel coupon is larger than its monotonic curve, while the two curves are fairly closed for Q550D steel coupons. However, the cyclic backbone of Q690D and Q890D is lower than their monotonic curves. The truth reveals that monotonic curves can be used conservatively and representatively as the cyclic backbone for Q460D and Q550D steels, however, this approach may produce unsafe prediction for Q690D and Q890D steels.

3.2.2.3 Energy dissipation behavior The equivalent viscous damping ξ eq is a commonly used index to quantify and compare cyclic energy dissipation ability of steel coupons and specimens. As shown in Fig. 3.9, ξeq is a function of two adjacent half-cycle energy EABC and ECDA normalized by elastic strain energy quantity EOBE and EODF , being expressed as Eq. (3.2). With a hypothesis where the cyclic excursion is rectangular, ξeq equals to its maximum value 2=π. In other words, the increase of ξeq represents the enlargement in the fullness of cyclic loops. Fig. 3.10 compares the energy dissipation index ξeq of the tested HSS specimens under different loading histories, while Fig. 3.11 depicts the ξeq of different steel grades under identical loading histories. The following characteristics can be observed: It appears that for all steel grades the overall equivalent viscous damping of constant amplitude test (4R10-1, 5R10-1, 5R16-1, 6R10-1, 8R10-1) is larger than that of stepwise amplitude test (4R10-2, 5R10-2, 5R16-2, 6R10-2, 6R10-3, 8R10-2) except for Q690D. The reason is that the constant amplitude tests possess two times of the maximum amplitude compared to the stepwise amplitude tests. Besides, the values of ξ eq increase gradually along with the strain level increment in stepwise

Hysteretic behavior of high strength steels under cyclic loading

73

Figure 3.8 Comparison of Ramberg
Osgood model with cyclic experimental backbone and monotonic curve. (A) Cyclic backbones of different specimens. (B) Cyclic backbones of Q460D steel specimens. Cyclic backbones of Q550D (10 mm) steel specimens. (D) Cyclic backbones of Q550D (16 mm) steel specimens. (E) Cyclic backbones of Q690D steel specimens. (F) Cyclic backbones of Q890D steel specimens.

amplitude tests, which once again emphases the dominant influence of the absolute value of strain amplitude. In addition, the values of ξeq obviously decrease with the accumulation of cycles due to the cyclic softening of stress. As shown in Fig. 3.11A, in the preliminary 40% cycles the ξeq of different steel grades remain in

74

Behavior and Design of High-Strength Constructional Steel

Figure 3.9 Definition of energy dissipation capacity.

ξ eq 5

1 2ðEABC 1 ECDA Þ 4π EOBE 1 EODF

(3.2)

a stable phase and present little distinction. Subsequently, the ξ eq step into softening phase one after another and their difference grow rapidly since the softening rates of structural steels vary. However, the ξeq shown in Fig. 3.11B presents a larger difference than that in Fig. 3.11A even in the stable phase. For instance, the specimen 4R10-2 and 8R10-2 both reach strain amplitude equals to 1.5% in the stable phase but the ξeq of 4R10-2 is around 30% larger than that of 8R10-2. Generally, the ξeq decreases with the increase of steel grades under stepwise amplitude cyclic strain, indicating deterioration of fullness in cyclic loop. Therefore the evolution law of energy dissipation indexes with respect to cycles strongly depends on steel grades, strain amplitude, and loading history. Furthermore, the stable value of ξeq is within 0.18
0.25 for structural steels with nominal yield strength larger than Q460MPa. According to the data provided in references, the value of ξeq is in the range of 0.47
0.60 for LYP100 and LYP160 and slightly greater than 0.4 for Q235 and Q345. This implies the truth that the cyclic loops of the LYP and conventional steels are fuller than HSS.

3.3

Hysteretic model and verification

3.3.1 Hysteretic model 3.3.1.1 Chaboche model In previous research conducted by Chaboche, cyclic hardening was decomposed into isotropic part, which corresponds to the expansion of the yield surface, and kinematic part, representing the offset of the yield surface. For uniaxial case, the stress amplitude of elastic region and strain
stress relation of plastic region are two keys to define isotropic and kinematic hardening respectively in a stable cyclic loop as shown in Fig. 3.12. The evolution law of isotropic part given by Chaboche

Hysteretic behavior of high strength steels under cyclic loading

75

Figure 3.10 Relationship of ξ eq and normalized cycles: (A) energy dissipation capacities of Q460D steel specimens, (B) energy dissipation capacities of Q550D steel specimens, (C) energy dissipation capacities of Q550D steel specimens, (D) energy dissipation capacities of Q690D steel specimens, and (E) Energy dissipation capacities of Q550D steel specimens.

defines the relationship of the size of the yield surface and the cumulative plastic strain, the expression of which is shown as in the following equation:   σ0 5 σj0 1 QN 1 2 e2b~εp

(3.3)

where σ0 denotes the current size of the yield surface (half of the stress amplitude in elastic range), σ|0 is the initial value of σ0 , QN is the maximum change of σ0 , b is the evolution rate of σ0 with respect to ε~ p , and ε~ p is the equivalent plastic strain. As indicated by Eq. (3.4), the isotropic part is a close-form exponential function

76

Behavior and Design of High-Strength Constructional Steel

Figure 3.11 Comparison of evolution law of ξ eq between different steel grades: (A) energy dissipation capacities of constant amplitude loading specimens and (B) energy dissipation capacities of stepwise amplitude loading specimens.

Figure 3.12 The isotropic and kinematic components in Chaboche model.

which tends to gradually reach a saturation state with the increase of cumulative plastic strain. Clearly the hardening process in early cycles is obviously quicker than that in later cycles. Besides, the signs of the parameter QN are of great importance since they represent the expanding (cyclic isotropic hardening) and shrinking of the yield surface (cyclic isotropic softening). To calibrate material-dependent parameters QN and b, the stress amplitude of elastic region in the ith loop σ0i and its corresponding equivalent plastic strain ε~ pi were extracted from cyclic test curves. Then QN and b were identified by nonlinear fitting using the processed test data groups σ0i ; ε~ pi as shown in Fig. 3.13. In previous research conducted by Chaboche, cyclic hardening was decomposed into isotropic part, which corresponds to the expansion of the yield surface, and kinematic part, representing the offset of the yield surface. For uniaxial case, the stress amplitude of elastic region and strain
stress relation of plastic region are two keys to define isotropic and kinematic hardening respectively in a stable cyclic loop as shown in Fig. 3.12. The evolution law of isotropic part given by Chaboche defines the relationship of the size of the yield surface and the cumulative plastic strain, the expression of which is shown as in the following equation:

Hysteretic behavior of high strength steels under cyclic loading

77

Figure 3.13 Curve fitting of isotropic components in Chaboche model: (A) isotropic components of Q460D steel specimens, (B) isotropic components of Q550D (10 mm) steel specimens, (C) isotropic components of Q550D (16 mm) steel specimens, (D) isotropic components of Q690D steel specimens, and (E) isotropic components of Q890D steel specimens.

  σ0 5 σj0 1 QN 1 2 e2b~εp

(3.4)

where σ0 denotes the current size of the yield surface (half of the stress amplitude in elastic range), σ|0 is the initial value of σ0 , QN is the maximum change of σ0 , b is the evolution rate of σ0 with respect to ε~ p , and ε~ p is the equivalent plastic strain. As indicated by Eq. (3.3), the isotropic part is a close-form exponential function which tends to gradually reach a saturation state with the increase of cumulative plastic strain. Clearly the hardening process in early cycles is obviously quicker than that in later cycles. Besides, the signs of the parameter QN are of great important since it determines the prediction size of yield surface is either expansion or shrinkage, corresponding respectively to cyclic isotropic hardening and softening. To calibrate material-dependent parameters QN and b, the stress amplitude of elastic region in the ith loop σ0i and its corresponding equivalent plastic strain ε~ pi were extracted from cyclic test curves. Then QN andb were identified by nonlinear fitting using the processed test data groups σ0i ; ε~ pi as shown in Fig. 3.13. Calibration results are summarized in Table 3.7. It should be noted that the test data used for calibration is constant amplitude test since the influence of strain amplitude is without consideration. To consider the tension
compression asymmetry, the sizes of yield surface in tension and compression zone are plotted against the cumulative plastic strain separately in Fig. 3.13. Besides, the prediction curves based on the calibration Chaboche model are also depicted in Fig. 3.13. It can be seen from the table that most calibrated values of QN are negative, indicating the size of the yield surface decreases exponentially as the plastic strain cumulates and saturates to a certain value of σ0 5 σj0 2 jQN j. On the contrast, the

78

Behavior and Design of High-Strength Constructional Steel

values of QN for mild steels are positive with reference to the calibration results conducted by Wang, which further implies the cyclic softening is more prominent in HSS rather than cyclic hardening. In addition, the evolution law of σ0 provided by Chaboche can be used to properly simulate the previous half phase of the overall cyclic process. However, the prediction fails to achieve good agreement with test result in the subsequent cyclic softening phase, especially for the specimen 5R10-1 and 8R10-1 shown in Fig. 3.16. Moreover, the compression curves are beyond the tension curves for all steel grades, resulting in the size of yield surface in compression zone is larger than that in tension zone. The evolution law of kinematic part is described by Eq. (3.4), in which αk is backstress, Ck =γ k defines the maximum change in back stress, γ k denotes the rate at which the backstress develops as the plastic strain εp increases. Since the evolution law of backstress varies with different strain ranges, superposition of several backstresses is a general approach to achieve more accurate simulation result, as shown in Eq. (3.5). In this study, strain
stress relations of plastic region in a full and stable loop were extracted from constant amplitude cyclic curves and calibrated via nonlinear fitting. The calibration results of kinematic parameters are summarized in Table 3.4 and the comparison between the calibration curves and test curves is plotted in Fig. 3.14. αk 5

αi 5

Ck ð1 2 e2γk εp Þ 1 αk;1 e2γk εp γk N X

αk

(3.5)

(3.6)

k51

3.3.1.2 Giuffre
Menegotto
Pinto model GMP model is a commonly used constitutive model for steel reinforcement and has already been implemented in OpenSees as the constitutive model for STEEL02 material. This model is proposed by Menegotto and Pinto and further refined by Filippou et al. [10]. Unlike the decomposition of cyclic hardening in Chaboche model, this model defines the unloading
reloading strain
stress path directly using a gradual transition from an elastic part with elastic modulus E0 to a hardening part with strain-hardening modulus E1 as shown in Fig. 3.15. Besides, the point Aðε0 ; σ0 Þ presented in the figure is where the elastic line and strain-hardening line meet, while point Bðεr ; σr Þ is the place at which the last strain reversal ends. The transition process is described as Eq. (3.6), where the parameter b denotes the strain-hardening ratio of the E1 and the E0 , the parameter Rðξ Þ is a factor to control the shape and Bauschinger effect of the reloading process, and updates with the strain amplitude between point A and B. The expression of RðξÞ is described in Eq. (3.9). It should be noted that Eq. (3.9) is a function of normalized strain and

Table 3.4 Calibration results of Chaboche model for HSS. Specimen

4R10-1 5R10-1 5R16-1 6R10-1 8R10-1

σj1 0

Q1 N

(MPa)

(MPa)

502.2 518.9 426.0 540.5 756.9

2 10 2 50 10 2 30 2 120

b1 iso 3 10 10 3 8

σj2 0

Q2 N

(MPa)

(MPa)

675.7 642.8 708.5 739.3 969.4

2 80 2 80 2 80 2 50 2 80

b2 iso 3 10 3 3 3

Ckin;1

Ckin;2

Ckin;3

(MPa)

(MPa)

(MPa)

22,100 18,300 102,500 38,100 9800

23,000 7700 34,000 85,600 25,900

1600 3800 1400 3900 13,800

γ1

γ2

γ3

115 345 643 153 883

465 98 134 670 235

4 26 27 2 52

80

Behavior and Design of High-Strength Constructional Steel

Figure 3.14 Curve fitting of kinematic components in Chaboche model: (A) kinematic components of Q460D steel specimens, (B) kinematic components of Q550D (10 mm) steel specimens, (C) kinematic components of Q550D (16 mm) steel specimens, (D) kinematic components of Q690D steel specimens, and (E) kinematic components of Q890D steel specimens.

Figure 3.15 Giuffre
Menegotto
Pinto model.

Hysteretic behavior of high strength steels under cyclic loading

81

stress indexes which are related to the coordinate of point A and B, as indicated in Eqs. (3.7) and (3.8). σ 5 bUε 1

ð1 2 bÞUε

(3.7)

ð11ε RÞ1=R

σ 5

σ 2 σr σ0 2 σr

(3.8)

ε 5

ε 2 εr ε0 2 εr

(3.9)

Rðξ Þ 5 R0 2

CR1 ξ CR2 1 ξ

(3.10)

where ξ updates at each strain reversal. R0 is the initial value of R. CR1 and CR2 are material-dependent parameters and should be calibrated through test results. To further refine the constitutive model described above, an isotropic hardening is added into the Menegotto and Pinto model by Filippou et al. With consideration of elastic region expansion, yield stress is defined as a linear function of the maximum plastic strain, as shown in the following equation:   σst εmax 5 a1U 2 a2 σy εy

(3.11)

where σst is the current yield stress, εmax is the absolute value of maximum plastic strain in current reversal. a1 and a2 are material-dependent parameters that respectively control the expansion rate and threshold value of initiation. All material parameters are calibrated and summarized in Table 3.5.

3.3.1.3 Dong
Shen model In order to properly consider the cyclic deterioration of the elasticity modulus and the yielding stress, Shen and Dong proposed a cumulative damage mechanics model

Table 3.5 Calibration results of Giuffre
Menegotto
Pinto model. Specimen

fy ðMPaÞ

4R10-2 5R10-2 5R16-2 6R10-2 8R10-2

595 687 628 824 1049

b0

R0

0.010 0.056 0.088 0.055 0.105

1.00 1.25 1.01 1.27 1.05

CR1

CR2

a1

a2

1 1 1 1 1

0.031 0.009 0.004 0.003 0.003

0.325 0.315 0.550 0.475 0.565

1 1 1 1 1

82

Behavior and Design of High-Strength Constructional Steel

using both strain and cumulative damage index as basic variables. The definition of the damage model is described as in the following equation: N X εpm εp β ip (3.12) p 1 εu εu i51 where N is the number of half cycles, εpm is the maximum experienced plastic strain, εpi is the plastic strain at the ith half cycle, εpu is the ultimate monotonic plastic strain, and β i is the weighted factor of the cumulative plastic strain. The definition of the typical cyclic loop is depicted in Fig. 3.16 where the points An and An11 respectively denote the starting point of the nth and (n 1 1)th halfcycle, the points Bn and Bn11 are the place at which the elastic range ends in the nth and (n 1 1)th half-cycle and the points Cn and Cn11 are the end points of the transition range from elasticity to strain-hardening range in the nth and (n 1 1)th half-cycle. The points group in each half-cycle divided the unloading
reloading process into three phases—elastic range, transition range (elastic
plastic range), and plastic range. The stress
strain relationships are defined in Eq. (3.13), where σD and ED , respectively, represent the reduced yield stress and elasticity modulus with consideration of damage accumulation; the parameters a, b, and c are the parabola factors that control the shape of the transition range; kD is the damage-

D 5 ð1 2 β Þ

Figure 3.16 Dong
Shen model.

8 jσAn 2 σj # γσD < σ 5 σAn 1 ED ðε 2 εAn Þ σ 5 aε2 1 bε 1 c γσD # jσAn 2 σj # ð2 1 ηÞσD : c 5 σCn 1 kd ED ðε 2 εCn Þ jσCn 2 σj $ ð2 1 ηÞσD

(3.13)

Hysteretic behavior of high strength steels under cyclic loading

83

based strain-hardening coefficient, and γ and η are the material-dependent parameters. A reasonable approach to correlate the damage index with σD and ED is to use three successive straight lines to achieve the phenomenal three-stage evolution law. Thus the expressions of σD and ED are defined as Eq. (3.13) where the ki , ξi , si , and ζ i ði 5 1; 2; 3Þ are material-dependent constants. Di and Di ði 5 1; 2Þ are threshold damage values for the three phases. Moreover, the kD given by Dong
Shen model is an increasingly increasing function, which is unsuitable to be implemented in the HSS since the decrease in kD is observed apparently from experimental data as shown in Fig. 3.18. In this chapter, the expression of kD is modified as an exponential Eq. (3.15), where k0 denotes the initial value of kD , km denotes the maximum change in kD while kb is the evolutionary rate of kD as damage index increases. The determination of the transition parameters a, b and c is based on the value of kD , σD and ED , and the coordinates of Bn ðε1 ; σ1 Þ and Cn ðε2 ; σ2 Þ. The expression is described as in Eq. (3.16). 8 < σ D 5 k1 D 1 ξ 1 σ D 5 k2 D 1 ξ 2 : σ D 5 k3 D 1 ξ 3

D # D1 D1 # D # D2 D $ D2

(3.14)

8 < ED 5 s 1 D 1 ζ 1 ED 5 s 2 D 1 ζ 2 : ED 5 s 3 D 1 ζ 3

D # D1 D1 # D # D2 D $ D2

(3.15)

  kD 5 k0 2 km 1 2 e2kb D 8 σ 1 2 σ 2 1 k d ED ð ε 2 2 ε 1 Þ > > >a5 > > ðε2 2ε1 Þ2 > >   > > < kd ED ε21 2 ε22 2 2ε2 ðσ1 2 σ2 Þ b5 ðε2 2ε1 Þ2 > > > > > > > 2 kd ED ðε2 2 ε1 Þ 1 σ1 2 σ 2 > > : c 5 σ2 2 kd ED ε2 1 ε2 ðε2 2ε1 Þ2

(3.16)

(3.17)

Since the values of εpi and εpm in each half cycle can be obtained based on the above definition under certain cyclic loading, the constant parameter β can be derived as Eq. (3.18) through setting the value of damage index D at failure as 1.0.   1 2 εpm =εpu β5 N P  p p  p p εi =εu 2 εm =εu i51

(3.18)

Table 3.6 Calibration results of SOY parameters for Dong
Shen model. Specimen

k11 ðMPaÞ

s1 1 ðMPaÞ

k21 ðMPaÞ

s1 2 ðMPaÞ

k31 ðMPaÞ

s1 3 ðMPaÞ

D1 c1

D1 c2

4R10-2 5R10-2 5R16-2 6R10-2 8R10-2 Specimen 4R10-2 5R10-2 5R16-2 6R10-2 8R10-2

2 50 2 250 2 500 2 300 2 200 k12 ðMPaÞ 2 50 2 250 2 500 2 300 2 200

510 610 700 690 910 s2 ð 1 MPaÞ 510 610 700 690 910

2 10 2 60 2 100 2 150 2 100 k22 ðMPaÞ 2 10 2 60 2 100 2 150 2 100

510 540 580 640 880 s2 ð 2 MPaÞ 510 540 580 640 880

2 1200 2 1200 2 1200 2 1200 2 1200 k32 ðMPaÞ 2 1200 2 1200 2 1200 2 1200 2 1200

1530 1630 1510 1440 1860 s2 3 ðMPaÞ 1530 1630 1510 1440 1860

0.13 0.34 0.31 0.33 0.19 D2 c1 0.13 0.34 0.31 0.33 0.19

0.85 0.95 0.91 0.80 0.97 D2 c2 0.85 0.95 0.91 0.80 0.97

Table 3.7 Calibration results of stiffness parameters for Dong
Shen model. Specimen

k11 ðMPaÞ

s1 1 ðMPaÞ

k21 ðMPaÞ

s1 2 ðMPaÞ

k31 ðMPaÞ

s1 3 ðMPaÞ

D1 c1

D1 c2

4R10-2 5R10-2 5R16-2 6R10-2 8R10-2 Specimen 4R10-2 5R10-2 5R16-2 6R10-2 8R10-2

2 10 2 220 2 10 2 200 2 200 k12 ðMPaÞ 2 10 2 20 2 10 2 20 2 20

230 240 170 260 230 s2 1 ðMPaÞ 240 210 210 210 220

2 10 2 20 2 10 2 20 2 20 k22 ðMPaÞ 2 10 2 10 2 10 2 20 2 20

230 200 170 200 200 s2 2 ðMPaÞ 240 210 210 210 220

2 1500 2 800 2 700 2 800 2 1000 k32 ðMPaÞ 2 700 2 800 2 700 2 500 2 500

1510 940 800 820 1090 s2 3 ðMPaÞ 830 960 820 590 650

0.13 0.20 0.31 0.33 0.19 D2 c1 0.13 0.20 0.31 0.31 0.15

0.85 0.95 0.91 0.80 0.91 D2 c2 0.85 0.95 0.89 0.78 0.89

Table 3.8 Calibration results of hardening parameters for Dong
Shen Model. Specimen 4R10-2 5R10-2 5R16-2 6R10-2 8R10-2

β

k0

km

kb

aðMPaÞ

bðMPaÞ

cðMPaÞ

γ

η

0.0090 0.0088 0.0025 0.0020 0.0115

0.69 0.60 0.68 0.74 0.90

2 0.22 2 0.35 2 0.44 2 0.63 2 0.49

14.81 7.66 4.74 2.49 8.12

126,880 1,644,010 372,360 2,134,370 2,736,860

24,180 4720 20,360 7100 29,680

560 530 530 540 800

1.33 1.34 1.38 1.32 1.38

0.2 0.2 0.2 0.2 0.2

86

Behavior and Design of High-Strength Constructional Steel

Similar with the calibration of Chaboche model, the calibration of Dong
Shen model also takes the tension
compression asymmetry into consideration. The calibrated parameters including elastic module parameters, yield stress parameters, and strain-hardening parameters are separately listed in Tables 3.6
3.8. Besides, the experimental size of the yield surface (half stress amplitude of elastic range) is plotted against the damage index as SOY-D curves for all tested steel grades. To compare the calibration results, the predicted SOY-D curves based on the calibration parameters are also depicted simultaneously, as shown in Fig. 3.17. Similarly, the relationships of the damaged elastic modulus and damage index (ED-D curves) and that of strain-hardening coefficient (kD-D curves) are plotted in Fig. 3.18.

3.3.2 Verification To further verify the adaptability of the above three constitutive models for HSS, the calibrated models are used to simulate the cyclic loading tests presented previously. The simulation results and test data are compared in Fig. 3.19. It can be seen that the simulations derived from Dong
Shen model present good agreement with test data. The prominent phenomenon during damage accumulation including cyclic softening, deterioration of elastic modulus, and strain-hardening coefficient can be properly presented. On the other hand, the Chaboche model can produce good accuracy in the several front cycles; however, the ability of this model to simulate the subsequent cyclic softening in the latter cycles for HSS is obviously of inadequacy since the cyclic loop does not show any shrinkage. This leads to considerable overestimation of the practical cyclic strength and energy dissipation ability. Among these three constitutive models, GMP model is proved to be the most unsuitable one. As shown in Fig. 3.19, the cyclic loop expands once the strain amplitude increases and shows no change if the strain amplitude remains constant. This is because the GMP model ignores the influence of cumulative plastic strain on the strain
stress relationship and presents no definition on any softening behavior. In other words, GMP is a good choice to simulate the cyclic loops for low yield point steels whose isotropic hardening is much more prominent than other behaviors. In summary, Dong
Shen model is suitable to be incorporated in the simulation of seismic performance on HSS structures. Chaboche model can also be used to achieve rational results when the cycle number is within half failure cycle number. The GMP model is not recommended to be applied in the numerical investigation of HSS structures.

Hysteretic behavior of high strength steels under cyclic loading

87

Figure 3.17 SOY-D and Ed-D curves: (A) Q460D SOY-D curves, (B) Q460D Ed-D curves, (C) Q550D (10 mm) SOY-D curves, (D) Q550D (10 mm) Ed-D curves, (E) Q550D (16 mm) SOY-D curves, (F) Q550D (16 mm) Ed-D curves, (G) Q690D SOY-D curves, (H) Q690D Ed-D curves, (I) Q890D SOY-D curves, and (J) Q890D Ed-D curves.

88

Behavior and Design of High-Strength Constructional Steel

Figure 3.18 Relationship of strain-hardening coefficients with damage index: (A) 4R10-2, (B) 5R10-2, (C) 5R16-2, (D) 6R10-2, and (E) 8R10-2.

Hysteretic behavior of high strength steels under cyclic loading

Figure 3.19a Comparison between experimental results and simulations.

89

90

Figure 3.19b (Continued)

Behavior and Design of High-Strength Constructional Steel

Hysteretic behavior of high strength steels under cyclic loading

3.4

91

Summary

In this chapter the cyclic behaviors of high strength structural steels including Q460D, Q550D, Q690D, and Q890D were thoroughly investigated through experimental study. The cyclic mechanical indexes and energy dissipation capacity were analyzed and compared. Based on Ramberg
Osgood model, the cyclic backbones were calibrated for all tested steels. Moreover, three widely used cyclic constitutive models including Dong
Shen model, Chaboche model, and GMP model were identified through cyclic test data. To demonstrate their applicability to HSS, the simulation results derived from the three calibrated models were compared to the test curves and their differences were discussed. According to these experimental and theoretical investigations, the following conclusions can be drawn: 1. The cyclic behavior greatly alters its monotonic behavior for all tested steel grades. The ratio between cyclic tensile strength and monotonic tensile strength for Q460D is greater than 1.0, while their values for three other HSSs are smaller than 1.0, indicating that cyclic softening rather than hardening becomes prominent behavior for Q550D, Q690D, and Q890D steels. This phenomenon shows considerable dependence on strain history and steel grades. Generally, the increase in yield strength synchronizes the decrease in cyclic hardening and increase in cyclic softening. 2. The cyclic backbones for all tested steels are calibrated using Ramberg
Osgood model. The distinction between monotonic and cyclic backbones for different steel grades varies. In detail, the stress predicted by cyclic backbone of Q460D steel coupon is larger than its monotonic curve, while the two curves are fairly closed for Q550D steel coupons. However, the cyclic backbone of Q690D and Q890D is lower than their monotonic curves. The increase in steel grade generally improves the value of K but reduces the value of n for Ramberg
Osgood model. 3. The evolution law of energy dissipation indexes for HSS is different from mild steels due to obvious cyclic stress softening and strongly depends on steel grades, strain amplitude, and loading history. The energy dissipation index ξ eq of Q460 steel specimen is around 30% larger than that of Q890 steel specimen under identical strain amplitude. The stable value of ξ eq for HSS is within 0.18
0.25 which is considerably lower than that of mild steels (0.47
0.60). This implies the truth that the cyclic loops of the mild steels are plump than HSS and the increase in steel grade deteriorates the fullness of cyclic loops. 4. Three widely applied cyclic constitutive models are calibrated and examined, including Chaboche mode, GMP model, and Dong
Shen model. The calibration parameters are incorporated into simulations of the aforementioned cyclic coupon tests. Based on the comparison of experimental results and simulation results, it can be concluded that Dong
Shen model is the most suitable model to be implemented in the simulation of seismic performance on HSS structures. Chaboche model can also be used to achieve rational results when the cycle number is within half failure cycle number. The GMP model is not recommended to be applied in the numerical investigation of HSS structures.

92

Behavior and Design of High-Strength Constructional Steel

References [1] Eurocode CEN. 3: Design of steel structures-Part 1-1: General rules and rules for buildings. CEN, Brussels: European Committee for Standardization, 2005. [2] Code P. Eurocode 8: Design of structures for earthquake resistance-part 1: general rules, seismic actions and rules for buildings. Brussels: European Committee for Standardization, 2005. [3] Code for seismic design of buildings (GB 50017-2017). Beijing: China Planning Press; 2017. [4] Wang YB, Li GQ, Cui W, Chen SW, Sun FF. Experimental investigation and modeling of cyclic behavior of high strength steel, Journal of Constructional Steel Research 2015;104:37
48. [5] Wang YB, Li GQ, Sun X, Chen SW, Hai LT. Evaluation and prediction of cyclic response of Q690D steel, Structures and Buildings 2017;170(110):788
803. [6] Hai LT, Sun FF, Zhao C, Li GQ, Wang YB. Experimental cyclic behavior and constitutive modeling of high strength structural steels. Construction and Building Materials 2018;189:1264
1285. [7] Chaboche JL. A review of some plasticity and viscoplasticity constitutive theories. Int J Plast 2008;24(10):1642
93. [8] Shen ZY, Dong B. An experiment-based cumulative damage mechanics model of steel under cyclic loading. Adv Struct Eng 1997;1(1):39
46. [9] Menegotto M. Method of analysis for cyclically loaded R. C. plane frames including changes in geometry and non-elastic behavior of elements under combined normal force and bending. In: Proc. of IABSE symposium on resistance and ultimate deformability of structures acted on by well defined repeated loads; 1973. 15
22. [10] Filippou FC, Popov EP, Bertero VV. Effects of bond deterioration on hysteretic behavior of reinforced concrete joints. Report No. UCB/EERC-83/19, Berkeley, California, 1983.

Uniform material model for constructional steel

4

Le-Tian Hai, Guo-Qiang Li and Yan-Bo Wang Tongji University, Shanghai, P.R. China

4.1

Introduction

The reliability of the structural design for earthquake resistance is challenged both on the demand and capacity sides due to the extreme uncertainties of earthquake motions, as well as the lack of well understanding of the cyclic behaviors of structural material and members. Moreover, the demand for advanced building material and devices such as high-strength steel (HSS) [1
6] and metallic dampers [7
11] is increasing based on the balance between economy and safety. Accordingly, steel and composite structures fabricated from various grades of steels, including lowyield-point (LYP) steel, normal-strength steel, and HSS, will certainly gain increasing engineering application in seismic zones. Therefore the seismic performance of those materials needs to be thoroughly investigated. To obtain more distinct insights on the hysteretic behavior and seismic performance of members and systems with various structural steels, full-range cyclic plasticity behavior under arbitrary load path must be well recognized for the commonly and potentially used structural steels, including LYP, mild steel, and HSS. Numerical simulation is becoming the most popular way in dynamic analysis since the considerable cost of large-scale testing is far beyond reach of most research funds. To this end, an appropriate description of hardening and softening evolution is the foundation of characterizing the strength and stiffness deterioration in steel members under cyclic deformation, which plays a key role in the damage evaluation in the numerical simulation of structures. Therefore a constitutive model that is able to accurately describe the cyclic hardening, cyclic softening, strain range effect [12] with consideration of the variation between different steel grades is in urgent need. Cyclic plasticity constitutive models for structural steels have gained prominent development in the past decades. A comprehensive review of the related researches was conducted by Chaboche [13]. A typical approach to generalize the cyclic nonlinear plasticity is the so-called multisurface model that traces the nonlinear plasticity in stress space through a series of intermediate surfaces. Lee et al. [14] describe the hysteretic characteristics of sheet metals using such piecewise multisurface approach. Despite such remarkable simulation accuracy, the implementation of the multisurface model is considerably burdensome since numerous surfaces must be provided along with enormous storage requirement in numerical simulations. In Behavior and Design of High-Strength Constructional Steel. DOI: https://doi.org/10.1016/B978-0-08-102931-2.00004-8 © 2021 Elsevier Ltd. All rights reserved.

94

Behavior and Design of High-Strength Constructional Steel

order to reduce this complexity, two-surface model was developed by Dafalias and Popov [15], Lee et al. [16], which consists of an inner yield surface and an enclosing bounding surface (BS) as shown in Fig. 4.1. Besides, the cyclic hardening and softening as well as mean-relaxation phenomena of structural steels were described by Elnashai and Izzudin [17]. Through torsion
compression experimental analysis, the yield plateau was implemented into the two-surface model by Shen [18]. Jia and Kuwamura [19] improved the two-surface model to cover large plastic strain ranges. The other typical approach widely used in simulations is the so-called single-surface model that depends on several tensorial variables governed by a series of independent differential equations [20]. The parameters of the single-surface model are convenient to be calibrated, and the required storage space in numerical implementation is considerably smaller than that of multisurface models, because the constitutive formulations can be explicitly integrated under proportional loading. The current isotropic and kinematic constitutive models are assumed to evolve separately and approach a series of asymptotic values, resulting in the fact that cyclic softening is solely contributed by the decrease in stress amplitude rather than the change in plastic modulus. This may lead to insufficient ability to rationally simulate the cyclic hardening and softening behaviors. To improve the performance of isotropic and kinematic model, Zhu et al. [21] developed a finite cyclic elastoplastic constitutive model that contains a new kinematic rule and an exponential isotropic rule. Recently, more sophisticated hardening models for sheet metals have been further developed by Yoshida et al. [22]. They proposed an anisotropic hardening model with consideration of the Bauschinger effect. Zhu et al. [23] developed a logarithmic stress rate
based constitutive model.

nij

Bounding surface

2

δ

3

nij

2κ 3

2κ 3

σ1

Oy O

σ2

Ob

σ3

Figure 4.1 Dafalias
Popov two-surface constitutive model.

Yielding surface

Uniform material model for constructional steel

95

Modified versions of the combined isotropic and kinematic hardening model were developed; however, the evolution of the full-range loops was not fully addressed. To consider the damage and deterioration effect in steel structures, several equivalent cyclic models were investigated. Representatively, an equivalent constitutive model was proposed by Wang [24] to consider the deterioration caused by local buckling through modifications on compression part of cyclic backbones. The model parameters are correlated to the width
thickness ratios of component plates, which play a key role in the initiation and evolution of local buckling. Dong and Shen [25] developed a damage-based cyclic constitutive model, the damage index of which is a function of maximum and cumulative deformation. The stress concentration and local buckling can be considered in this model through multiplying the damage index by a magnification coefficient. However, such constitutive models have been facing the difficulty of parameter calibration, especially for those multisurface models. At least, three categories of cyclic loading histories are needed to calibrate the 18 parameters in the two-surface model proposed by Shen [26]. Cyclic loading test turns out to be an expensive way to investigate the cyclic behavior and calibrate the existing constitutive models. To reduce this cost and simplify the implementation of these modeling work, Hu et al. [27] proposed a new constitutive model for full-range elastoplastic behavior of structural steels with yield plateau and developed a very concise calibration procedure via some empirical assumptions, which enable that the cyclic parameters can be evaluated through monotonic coupon test. The aforementioned models were mostly based on the experimental observations from mild steels, although a series of experimental investigations has been carried out on LYP [28] and HSS [28
32] due to the requirements of applications in construction. Wang et al. [33] revealed that HSS presents more remarkable cyclic softening behavior than that of mild steels [28]. Therefore the further study on LYP steel and HSS is necessary to understand the influences of steel grades on cyclic response and update the current constitutive models to include different grades of steel. Moreover, a uniform constitutive model that can be used for LYP, mild steel, and HSS is necessary. This chapter presents a uniform constitutive model that is applicable for structural steels with nominal yield stress ranging from 100 to 890 MPa based on the framework of two-surface model and the available test data. Moreover, since monotonic tension experiments can be easily conducted, a group of formulations for evaluating the cyclic hardening and softening behaviors of steels with various grades based on common monotonic properties are proposed.

4.2

Experimental observations

4.2.1 Data source Cyclic loading coupon tests aiming to study cyclic behavior of different structural steels have been conducted by a number of researchers. As for LYP, coupons made of LYP100 and LYP160 with nominal yield stress, respectively, equal to 100 and

96

Behavior and Design of High-Strength Constructional Steel

160 MPa were tested to investigate their cyclic response by Wang et al. [34]. With the identical research methodologies, LYP225 steel was studied by Wang [35], Shi et al. [36], and Ge et al. [37]. Besides, low fatigue tests of LYP100 and LYP225 under seven constant amplitudes were conducted by Saeki et al. and Dusicka et al. [28]. The investigation on cyclic behavior of mild steels should refer to the current constitutive models. Several detailed cyclic loading tests on mild steels, including Q235, Q345, GR345, and HT440, were, respectively, conducted by Shi et al. [38], Wu [39], Dusicka et al. [28], and Luo et al. [40]. Regarding structural steels with nominal stress greater than 460 MPa, monotonic and cyclic behaviors of several HSS and HPS potentially used in steel structures were experimentally studied by Shi (Q460C and Q460D) [29,30], Cui et al. (Q460D) [32], Dusicka et al. [28] (HPS485), Wang et al. (Q690D) [33], Lu co-workers (Q690GJ) [41,42], Wang (Q800) [43], and Zhao (Q460D, Q550D, Q690D, and Q890D) [44], respectively. Moreover, low fatigue tests and cyclic loading tests on welding materials for Q235, Q345, Q460, and Q690GJ were, respectively, performed by Wu [39], Shi et al. [45], and Lu [41]. The basic material properties of the steels with the yield strength ranging from 76.5 to 1067.2 MPa are summarized in Table 4.1, where fy is the yield strength, fu and εu are the ultimate strength and its corresponding strain, and Δ is fracture elongation.

4.2.2 Cyclic hardening/softening behavior of the yielding surface and bounding surface As for metallic materials that obey von Mises yielding criterion in threedimensional stress space, cyclic hardening behavior is contributed from the expansion of yielding surface (YS) and BS, as well as the pure motion of YS as depicted in Fig. 4.2. On the other hand, cyclic softening behavior consists of the shrinkage of YS and BS. Regarding single-dimensional condition, the size of BS and YS as well as the pure motion amount of YS, respectively, corresponds to the peak stress, flow stress, and back stress, as shown in Fig. 4.3 Thus the evolution laws of BS and YS are able to be studied through the peak stress, flow stress, and back stress with consideration of influence by steel grade. Previous cyclic constitutive models such as the Chaboche model generally consider the influence of instantaneous strain and cumulative plastic strain (CPS). In this chapter the influence of steel grade is investigated in order to propose a uniform model that can be utilized to simulate possible structural steels. From uniaxial cyclic coupon tests the values of stress indexes, including peak stress, flow stress, and back stress, can be decomposed from each individual uniaxial cyclic loop as schematically depicted in Fig. 4.4. To estimate the influence of plasticity accumulation, the obtained stress values should be, respectively, correlated with the CPS to form their relationship curves, which is called CS
CP (cyclic stress
CPS) curves herein. The processed CS
CP curves are shown in Figs. 4.5 and 4.6, respectively, for the constant amplitude cyclic coupon tests cited and variable amplitude cyclic coupon tests. Fig. 4.5 shows that the hardening and softening

Uniform material model for constructional steel

97

Table 4.1 Summary of basic mechanical properties of target structural steels. Steel grade

fy (MPa)

fu (MPa)

εu (%)

Δ (%)

fy/fu

Researchers

LYP100 LYP100 LYP160 LYP225 LYP225 LYP225 LYP225 Q235B Q345B GR345 Q235 Q345 Q235a Q235L Q235T W235L W235T Q345L Q345T W345L W345R HT440 Q460C Q460D Q460-W Q460D-11 Q460D-21 Q460D HPS485 Q550D-10 Q550D-16 Q690GJ Q690GJ-W Q690D16A Q690D-16B Q690D-16C Q690D-40L Q690D-40R Q690D Q800 Q890D

85.0 76.5 129.0 212.0 242.0 199.0 242.5 407.0 429.0 353.0 300.0 385.0 316.7 233.5 224.3 444.8 456.0 301.3 301.1 365.5 388.8 501.0 470.0 466.0 481.0 505.8 464.0 479.1 503.0 695.7 645.1 737.0 560.0 770.0

261.0 257.0 273.0 295.0 324.0 325.0 303.9 588.0 589.0 534.0 415.0 540.0 515.6 383.0 386.0 575.0 591.7 466.3 463.2 482.1 508.5 688.0 620.0 568.0 670.0 597.5 585.9 610.6 590.0 764.9 753.1 825.0 790.0 826.2

25.00
28.60 25.86
24.83 8.17 11.80 10.10


15.74 16.48 18.99 12.28 12.59 15.54 15.72 11.90 11.49
12.43 12.37 8.54 6.48 14.00 13.63
7.43 4.93 7.00 11.81 6.12

55.00
54.00 49.11
50.00 44.03 46.10 45.65
36.00 25.50 27.50 36.00 36.00 18.00 24.00 20.00 30.00 28.00 24.00
45.50 46.00 20.37 23.70 30.40 22.00
20.00 14.00 21.00 20.00 21.00

0.33 0.30 0.47 0.72 0.75 0.61 0.80 0.69 0.73 0.66 0.72 0.71 0.61 0.61 0.58 0.77 0.77 0.65 0.65 0.76 0.76 0.73 0.76 0.82 0.72 0.85 0.79 0.78 0.85 0.91 0.89 0.89 0.71 0.93

Wang et al. [34] Dusicka et al. [28] Wang et al. [34] Ge et al. [37] Dusicka et al. [28] Shi et al. [36] Wang [35] Shi [38]

779.0 775.0 760.0 776.0 823.7 893.0 1067.2

834.0 832.0 810.0 824.0 890.4 914.0 1105.9

5.90 6.29 5.88 6.02 4.19
6.84

19.00 21.00 18.00 17.00 12.00 15.30 15.00

0.93 0.93 0.94 0.94 0.93 0.98 0.97

Dusicka et al. [28] Luo et al. [40] Wu [39]

Dusicka et al. [28] Shi [29,30]

Cui et al. [32] Zhao [44] Dusicka et al. [28] Zhao [44] Lu co-workers [41,42] Wang et al. [33]

Zhao [44] Wang [43] Zhao [44]

98

Behavior and Design of High-Strength Constructional Steel

YS softening YS hardening YS Back stress

Influenced by strain, cumulative plastic strain, steel grade

BS softening BS hardening BS

Influenced by strain, cumulative plastic strain, steel grade

σ1 Oy

O

σ3

σ2

Initial YS

Figure 4.2 Evolution of yielding surface and bounding surface.

σ Distance between α *(δ) BS and YS G

H

A

I C

B

Yielding surface (YS) 2 f y (2κ)v

O Two times flow stress Peak stress

Back stress

Shrinkage and expansion of BS

D

ε

E F

Bounding surface (BS)

Figure 4.3 Definition of peak stress, flow stress, and back stress.

behaviors have a significant dependency on strain amplitudes, loading history, and steel grades. Fig. 4.5A consists of four CS
CP curves obtained from constant amplitude cyclic tests of LYP225 coupons. It can be observed that experiment 8#0.01 with the strain amplitude of Δε 5 6 1% initially presents a hardening region and then reaches a plateau region followed by a steady softening develops in the latest 15% region until the low-cycle fatigue fracture occurs. With the increase in the applied constant strain amplitude, the magnitude of the plateau region is increased, while the length of the plateau region is reduced. Fig. 4.5B depicts the CS
CP curves obtained from constant amplitude cyclic tests of Q460D coupons.

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99

Figure 4.4 Decomposition of individual cyclic loop.

The cyclic hardening of Q460D is decreased compared to that of LYP225. Fig. 4.5C
G depicts the CS
CP curves obtained from constant amplitude cyclic tests of Q550, Q690, and Q890 coupons. Different from those steels with lower yield points, HSSs step into softening region after an ignorable initial hardening then reaches the plateau region followed with a rapidly softening region in the final 10% cycles. The peak stress (the size of BS) and flow stress (the size of yield surface) develop with similar tendency in LYP and HSS. The back stress (the motion of yield surface) nearly keeps constant during the accumulation process except the last 10%
15% cycles. On the other hand, considerable tension
compression asymmetry is observed in the CS
CP curves as shown in Fig. 4.5A, especially for the LYP steels under small strain amplitude. The stress magnitude in compression zone is around 10% higher than that in tension zone for LYP steels. This difference declines to about 5% for Q690 steel. Compared to cyclic softening under tension, the softening under compression is less significant. This phenomenon is more notable for LYP steel than for HSS. Fig. 4.6 shows the CS
CP curves obtained from variable amplitude cyclic tests of LYP steels, mild steels, and HSS coupons. It can be seen that the back stress has experienced an increase up to a certain stress value with the increase in the absolute value of strain rather than the CPS. This indicates the applied absolute value of strain plays a key role in kinematic hardening, which is discussed in Section 4.2.3. To further understand the influence of steel grades on cyclic hardening and softening behaviors, constant amplitude cyclic coupon tests with the strain amplitude ranging from 0.5% to 7.0% of different structural steels are analyzed and compared. The cyclic stress amplitude to yield strength ratio, which is denoted as cyclic stress ratio, is plotted against CPS in Fig. 4.7. Generally, the increase in yield-to-tensile ratio (YTR) is accompanied with the decrease in hardening capability and the increase in softening capability, leading to the different loading surface evolution

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Behavior and Design of High-Strength Constructional Steel

Figure 4.5 CS
CP curves of the cyclic coupon tests with constant amplitude: (A) LYP225 by Wang [35], (B) Q460D by Zhao [44], (C) Q550D-1 by Zhao [44], (D) Q550D-2 by Zhao [44], (E) Q690D by Wang et al. [33], (F) Q690D by Zhao [44], and (G) Q890D by Zhao [44]. CS
CP, Cyclic stress
cumulative plastic strain.

Uniform material model for constructional steel

Figure 4.6 CS
CP curves of the cyclic coupon tests with variable amplitude: (A) LYP100 by Wang et al. [34], (B) LYP160 by Wang et al. [34], (C) LYP225 by Shi et al. [36], (D) Q235 by Shi et al. [38], (E) Q345 by Shi et al. [38], (F) Q460C by Shi [29], (G) Q460D-2 by Cui et al. [32], and (H) Q690D-1 by Wang et al. [33]. CS
CP, Cyclic stress
cumulative plastic strain.

101

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Behavior and Design of High-Strength Constructional Steel

Figure 4.7 Cyclic stress ratios against cumulative plastic strain for different grades of steel: (A) strain amplitude Δε 5 6 0.5%, (B) strain amplitude Δε 5 6 0.75%, (C) strain amplitude Δε 5 6 1.0%, (D) strain amplitude Δε 5 6 1.5%, (E) strain amplitude Δε 5 6 2.0%, (F) strain amplitude Δε 5 6 3.0%, (G) strain amplitude Δε 5 6 4.0%, (H) strain amplitude Δε 5 6 5.0%, (I) strain amplitude Δε 5 6 6.0%, and (J) strain amplitude Δε 5 6 7.0%.

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103

among LYP, mild steels, and HSS. As shown in Fig. 4.7C, the hardening capacity of LYP100 (YTR 5 0.30) is much higher than those of mild steel and HSS. The maximum cyclic stress ratio of LYP100 is 4.5, while the maximum cyclic stress ratio of mild steels (YTR 5 0.66
0.78) ranges from 1.2 to 1.5. Nevertheless, HSS with nominal yield strength greater than 550 MPa (YTR 5 0.89
0.97) shows a cyclic stress ratio less than 1.0 during the overall cycling process soften, indicating a pure softening process with the plastic strain accumulation. As shown in Fig. 4.7B, both curve 3 and curve 5 are derived from the cyclic tests on Q235 steel but with different yield
tensile ratios that are 0.61 and 0.72. The maximum cyclic stress ratios are around 1.4 and 1.2, respectively. Similarly, this observation can also be obtained in comparison between the curve 6 and curve 7 as shown in Fig. 4.7C. On the other hand, the increase in steel grade from mild steel to HSS remarkably changes the shape of the evolution curve, as illustrated in the comparison between curve 6 and curve 7 in Fig. 4.7B. The evolution curves of the mild steels and LYP exhibit a similar trend. At the end of softening region, a sharp softening behavior is evident for all structural steels before final fracture. For instance, the cyclic stress ratios of LYP100 drop from 3.8, 4.5, and 4.8 to 2.2, 3.5, and 4.3, respectively, under cyclic strain amplitudes of 2%, 4%, and 6%, respectively. It is noted that the increase in cyclic strain amplitude may reduce the total softening amount.

4.2.3 Kinematic hardening rule To figure out the relationship between the back stress and the full-range CPS, the back stress was extracted from the hysteresis curves and translated from the starting point to the coordinate origin. Then the translated back stress was plotted against the CPS as the kinematic hardening evolution surfaces (KHESs) in Fig. 4.8. It can be seen that with the increase in the CPS, considerable decrease in amplitude and change in shape of the back stress curve are observed, which lead to prominent tension
compression asymmetry before fracture. Far from the fracture point, CPS does not cause much difference on KHES. However, this should be further specifically examined through proper quantitative indexes. Since the Chaboche model is evident to gain sufficient accuracy for exponential stress
strain relationship, its kinematic parameters are chosen as the quantitative indexes for the back stress herein. Normally, superposition of several components consisted of various amplitude and saturation rates is adopted in the Chaboche model to consider the influence of different strain ranges. However, the calibration of material parameters may encounter excessive difficulties when more than two components are utilized. In this chapter the back stress is assumed to consist of two components with identical amplitude but different saturation rate as 
 
 ak 5 a 1 2 exp 2χ1 εp 1 a 1 2 exp 2χ2 εp

(4.1)

where αk denotes the overall back stress, α is the saturation magnitude of each back stress component, χ1 and χ2 are two saturation rates of back stress components.

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Behavior and Design of High-Strength Constructional Steel

Figure 4.8 Evolution surfaces of back stress for various steels: (A) LYP225 steel [35], (B) Q460D steel. [44], (C) Q550D steel [44], (D) Q690D steel [33], and (E) Q890D steel [44].

With the development of the CPS, the cyclic loop becomes plump and these two indexes increase. There are three parameters that need to be calibrated. The calibrated results are plotted against CPS in Fig. 4.9 that describes the evolution of amplitude and saturation rate indexes with respect to CPS. It can be seen that the index α of LYP225 generally experiences an increasestable-decrease process, as shown in Fig. 4.9A. The first saturation rate index χ1 ranges from 800 to 100 in tension zone and 800
20 in compression zone, while the second saturation rate index χ2 ranges from 50 to 5 in tension zone and 50
10 in compression zone. As shown in Fig. 4.9B, χ1 decrease rapidly in the initial cycles and step into a stable stage until a second rapid decrease occurs, while χ2 continuously stay in a stable stage after an initial slight decrease then sharply increase in the final several cycles, as depicted in Fig. 4.9C. Regarding HSS such as Q460D, Q550D, Q690D, and Q890D, similar three-stage evolution characteristics are

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105

Figure 4.9 Evolution laws of kinematic parameters for various steels: (A) α of LYP225 steel [35], (B) χ1 of LYP225 steel [35], and (C) χ2 of LYP225 steel [35].

observed for index χ1. However, the final stage of χ1 for most HSS in tension zone exhibits slight increase or stable status rather than decrease except for Q460D steel. As for index χ2, the three-stage evolution characteristic is also evident. The evolution laws of these two indexes indicate that the fullness of cyclic loop evolves with a three-stage characteristic as plastic strain accumulates for either LYP or HSS. It can be observed that the values of χ1 within the second stable stage are, respectively, around 400
500 for LYP225 steels, 200
250 for Q460 and Q550 steels, and 150
200 for Q690 and Q890 steels. The values of χ2 within the second stable stage are, respectively, around 10
30 for LYP225 steels, 30
40 for Q460 and Q550 steels, 20
30 for Q690, and 30
40 Q890 steels. Generally, the value range of χ2 is small and similar to different structural steels compared to that of χ1. Besides, the values of χ1 for HSS are obviously less than that of LYP steel. This further implies that the cyclic loop of LYP steel is more plumpy than that of HSS.

4.2.4 Degradation of the elastic modulus The evolution of the elastic modulus is shown in Fig. 4.10. The elastic modulus decreases rapidly from the nondamage elastic modulus E0 to a plateau region of Es after the initial cycles, followed by the final degradation stage with much higher deterioration rate than that of the plateau region. The elastic modulus corresponds to the final fracture is denoted as Ef. The difference between the

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Behavior and Design of High-Strength Constructional Steel

Figure 4.10 Evolution laws of elastic modulus for various steels: (A) LYP225 steel [35], (B) Q460 steel [44], (C) Q550 steel [44], (D) Q550 steel [44], (E) Q690 steel [33], (F) Q690 steel [44], and (G) Q890 steel [44].

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107

nondamaged elastic modulus E0 and fracture elastic modulus Ef is denoted as Ed. The ratio of Ef/E0 ranges from 0.2 to 0.6, while the ratio of Ed/E0 that represents the maximum degradation ranges from 0.4 to 0.8. The ratio of Ef/E0 is affected by cyclic loading. Besides, tension
compression asymmetry also exists and acts as a difference in the value of Ed/E0. Generally, the Ed/E0 under compression is smaller than that under tension. All structural steels exhibit such three-stage characteristics. The evolution law of elastic modulus does not show obvious dependency on steel grades.

4.3

Theoretical modeling

The constitutive modeling investigated herein is within the scope of rate-independent plasticity. Time and temperature effects are ignored. The classical plasticity concept such as von Mises criteria and flow rule is adopted. The main concern is focused on the mathematical description of the effect of cumulative plasticity on the postyielding behavior of various structural steels under cyclic loading.

4.3.1 Framework of constitutive modeling With the hypothesis of small strain the total strain is commonly decomposed into the elastic and plastic parts as ε 5 εe 1 εp

(4.2)

where εe and εp represent the elastic and plastic part, respectively. The evolution law of the postyield surface in the stress space is defined with respect to its position and size based on the classical combined hardening model, given as f 5 J ðσ 2 X Þ 2 R 2 k

(4.3)

where X denotes the back stress that stands for the center motion of the yield surface, R is the isotropic hardening parameter that describe the change in size of the yield surface, k is the initial size of the yield surface, and J is the distance in stress space as for von Mises material and expressed as 

1=2 2 0 0 0 0 J ðσ 2 X Þ 5 ðσ 2X Þ:ðσ 2X Þ 3

(4.4)

The flow rule complies with the normality rule dεp 5 dλ

@f @σ

(4.5)

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Behavior and Design of High-Strength Constructional Steel

As both the isotropic and kinematic hardening parameters approach their saturation size, the yield surface reaches its so-called BS, which is assumed as f b 5 J ð σ 2 X m Þ 2 Rm 2 k

(4.6)

Based on the analysis of the collected test data, it is well recognized that the peak stress (BS) and flow stress (YS) are two primary stress indexes that show complex evolution. Thus the concept of two-surface model incorporated with the observed characteristics seems to be a suitable selection to lay the foundation of the constitutive modeling in this study. To this end, several features must be defined to include evolution rules of both peak stress and flow stress (isotropic hardening/softening) and evolution rules of back stress (kinematic hardening) with respect CPS. To further improve the simulation performance, another three features, including elastic modulus degradation, strain range dependence, and steel grade dependence, are also considered. These described characteristics and their corresponding methodologies are summarized in Table 4.2, which set up the framework of the uniform constitutive modeling for various steels in this study.

4.3.2 Two-step hardening and three-step softening hardening/softening model The aforementioned experimental observations reveal that the prominent features of LYP and HSS are cyclic hardening and cyclic softening, respectively. As shown in Fig. 4.5
4.7, the evolution laws of cyclic hardening and softening present remarkable dependency on strain amplitude and CPS range. Thus superposition of several components is suggested to provide a precise simulation result according to the approach given by Chaboche [46]. The cyclic hardening behavior of steel can be decomposed into two stages, the first of which describes the initial rapid hardening achieved shortly in the earliest 20% of the total CPS, while the second defines the later gradual hardening with a relatively low saturation rate. It is known that the material degradation develops in three stages, including microcrack incubation, microcrack growth, and microcrack penetration. The first cyclic softening stage defines the rapid decrease in peak stress and flow stress caused by microcrack incubation, which occurs more noticeably among HSS than mild and LYP. In addition, the second stage describes the gradual decrease in aforementioned cyclic stress indexes as the sizes of microcracks proceed to increase, followed by a sharp stress drop, which is the third stage of microcracks penetration. Since the two stages of cyclic hardening and three stages of cyclic softening are considered, this model can be named 2H3S as expressed in the following equation. σc 5 σm 1 σRc 2 σQc 5 σm 1

2 X i21

iÞ σðRc 2

3 X i21

iÞ σðQc

(4.7)

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109

Table 4.2 Modeling methodology based upon experimental observations. Concept

Representative experimental observation

Modeling methodology

Yielding criteria Flow rule Hardening/softening rule 1 Isotropic hardening of flow stress (yield surface)





von Mises yield criteria Consistency condition

Two-step hardening, including initial rapid hardening and subsequent steady hardening

2

Isotropic softening of flow stress (yield surface)

3

Evolutionary law of peak stress (bounding surface) Kinematic rule

Three-stage softening, including initial rapid softening, subsequent steady softening, and final sharp softening Peak stress flow stress present similar trends but different quantity with flow stress

To define two cyclic hardening components, the first of which should be endowed with a considerably large evolutionary rate, while the second should exhibit a relatively low speed To define three cyclic softening components whose evolutionary rates should successively be large, small then large To use the preceding 2H3S model of the flow stress with different component quantity distribution To define the relationship between the back stress and cumulative plastic strain

4

5

Elastic modulus degradation

6

Strain range dependence

7

Steel grade dependence

The amplitude and speed indexes of back stress of each cycle varies with respect to the accumulation of cyclic cycles Three-stage degradation (Fig. 4.11) Each component quantity show significant dependence on strain range The increase in monotonic yield stress or yield ratio considerably affect the evolution law of cyclic stress

To utilize the framework of the aforementioned three-step softening To correlate the 2H3S model parameters with strain range To correlate the 2H3S model parameters with monotonic properties

2H3S, Two-step hardening and three-step softening.

where σc and σm represent the cyclic and monotonic stress, respectively, σRc and σQc denote the total amount of cyclic hardening and softening severally at the loadðiÞ ing point, and σðiÞ Rc and σ Qc denote the subcomponents of hardening and softening parts, respectively. It must be noted that there are two key parameters describing the evolution law of each component. The first is the reserved capacity parameter (RCP), which represents the maximum hardening/softening quantity (saturation size) of a cyclic

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Behavior and Design of High-Strength Constructional Steel

Figure 4.11 Three-stage damage evolution law.

component, and the second is the shape parameter that determines the saturation rate that cyclic hardening/softening amount saturates to its maximum amount with respect to CPS. These two types of parameters both perform dependency on steel grade and strain amplitude, which are exquisitely considered in the following constitutive mathematic modeling. Since the two cyclic hardening components and the first cyclic softening component tend to saturate to their RCPs exponentially, the refreshing progress of these components is evaluated on the basis of the closedform exponential function as ðiÞ

iÞ σðRc 5 RðciÞ U

1 2 e2bc

p=pm

i 5 1; 2

ðiÞ

1 2 e2bc ðiÞ

1Þ σðQc 5 Qðc1Þ U

1 2 e2rc

(4.8)

p=pm

(4.9)

ð1Þ

1 2 e2rc

Based on the experimental features, the second and third softening components are expressed as " jÞ σðQc

5 QðcjÞ U

ðjÞ

e2rc

ðpm 2pÞ=pm

ðjÞ

2 e2rc ðjÞ

1 2 e2rc

# (4.10)

ðiÞ where RðiÞ c and Qc are the aforementioned RCPs of the ith cyclic hardening and the ðiÞ first cyclic softening component; bðiÞ c and rc are the shape parameters of cyclic hardening and softening components, respectively. While p denotes the CPS, and

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111

pm is the ultimate CPS with certain strain amplitude, which can be determined through Manson
coffin equation. Apparently, all components equal to zero when ðiÞ p 5 0 and, respectively, saturate to RðiÞ c or Qc as p asymptotically approaches its ultimate value of pm. To consider the strain amplitude effect on the maximum value of the ith cyclic hardening components, RðiÞ c is assumed as the product of a material-dependent constant RðiÞ and a piecewise proportional factor ξðiÞ Rc ðεÞ as shown in Eq. (4.11), and the cm ðiÞ three segments of ξRc ðεÞ are, respectively, represented by a constant, a monotone increasing, and monotone decreasing function as shown in the following equation. iÞ iÞ ðε Þ RðciÞ ðεÞ 5 Rðcm Uξ ðRc

iÞ ξ ðRc 5

8 > > > > > > > < ai 1 > > > > > > > :

(4.11)

ai ð1 2 ai Þ:ðε 2 εst Þ=ðεc 2 εst Þ  2 β i ðε2εst Þ=ðεc 2εst Þ21 1 ðεc 2 εst Þ=ðεc 2 εst Þ ε=εc 2 γ i ε=εc 21 1 ε=εc


  εA εy ; εst εA½εst ; εc Þ   εA εc ; εf (4.12)

The definition of ξ ðiÞ Rc ðεÞ is relied on the following assumptions: Assumption 1: In the plateau region the RCP remains constant. Assumption 2: The strain amplitude could enlarge the RCP considerably in the hardening region until a critical strain amplitude εc is reached, where the RCP enlargement achieves its saturation value. Assumption 3: As the εc is exceeded, the RCP nonlinearly decreases since the potential cyclic hardening is exhausted by the initial monotonic strain hardening. Typical distributions of ξðiÞ Rc with respect to strain amplitude based on different material constants are plotted in Fig. 4.12. It can be seen that the value of ξðiÞ Rc becomes unity when the strain amplitude reaches its critical value εc, otherwise, it ranges from zero to unity. The parameter αi describes the proportion of RCP among plateau region in the overall maximum RCP, while β i and γ i, respectively, determine the evolutionary rates of RCP with respect to strain amplitude below and above the critical amplitude εc. Along with the increase in the values of β i and γ i, the shape of ξðiÞ Rc presents remarkable concave feature as shown in Fig. 4.12. As experimentally observed, the cyclic stress indexes normally experience a steady period then endure a rapid decrease, namely, the second and third softening components. Empirically, the total softening quantity of these two stages inversely decreases along with the increase in strain amplitude. In other words, the enlargement of strain amplitude would increase the potential of cyclic brittle failure that

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Behavior and Design of High-Strength Constructional Steel

Figure 4.12 Distribution of RðiÞ Rc with respect to strain amplitude.

generally occurs with limited softening. Thus the total softening quantity can be assumed as a monotone increase function as given by 0Þ 0Þ Qðc23Þ 5 Qðcm U ξðQc

0Þ ξ ðQc


 arctan½ωðε 2 εc Þ 2 arctan ω εf 2 εc 
 5 arctanðωεc Þ 1 arctan ω εf 2 εc

(4.13) (4.14)

where ξcð23Þ denotes the total reserved capacity of the second and third softening components, which is defined as the product of a material-dependent parameter Qð0Þ cm and a strain-dependent factor ξ ð0Þ Qc to consider the influence of steel grade and strain amplitude. The evolution law of ξð0Þ Qc with respect to strain amplitude is plotted in Fig. 4.13, where parameter ω is used to define the softening rate, an increase of which would result in severer drop near the critical strain amplitude εc. It has been found experimentally that the softening quantities of the second softð3Þ ening component Qð2Þ c and the third softening component Qc as well as their proð23Þ portions in the total value of Qc significantly depend on strain amplitude. To be specific, in case of fairly small cyclic strain amplitude, the softening quantity of the stress indexes is quite large and mainly contributed by component Qð3Þ c , since the quite slow damage rate accumulation limits the total softening amount during the second softening stage corresponding to component Qð2Þ c . Conversely, the component Qð2Þ dominates the cyclic behavior under considerable strain amplitude c because of the much higher damage evolutionary rate than those under small amplið3Þ tude, while the total amount of Qð2Þ c and Qc encounters a considerable drop. In other words, the increasing strain amplitude synchronizes the increase in proportion ð3Þ of Qð2Þ c with the decrease in proportion of Qc . Accordingly, a strain-based partition

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113

coefficient ξð2Þ Qc is introduced to determine the percentage of the second and third ð3Þ softening components, the reserved capacities of which are Qð2Þ c and Qc , respectively, expressed as ð2Þ ð23Þ Qð2Þ c 5 Qc U ξ Qc

(4.15)

 ð2Þ ð23Þ Qð3Þ c 5 Qc U 1 2 ξ Qc

(4.16)

2ηε Þ ξ ð2Þ Qc 5 κU ð1 2 e

(4.17)

where κ, η, and ω are model constants. The relationships between the strain amplitude and the partition coefficient as well as softening components are plotted in Figs. 4.13 and 4.14 with a set of assumptive model constants. As illustrated by Eqs. (4.8) and (4.9), the shape parameters and respectively describe the change rate of hardening and degradation components with respect to CPS. The superscript (2) in denotes the hardening rate of the second hardening component, while the superscript (2) in describes the damage accumulaton rate corresponding to “the middle stable evolutionary state” as shown in Figure 4.11. It has been well recognized from experimental summarizations that the increasing strain amplitude increases the shape parameters and. However, such increase tends to reach critical saturation vaules. Thus and are assumed as piecewise functions with a saturation value of and respectively as Eqs(4.18) and (4.19). On the other hand, the shape parameters in Eq.(4.8), and in Eq.(4.10) are regarded as much larger constants than and . Such arrangment is to match the truth that the corresponding hardening and softening components saturate with much quicker speed than “the middle stable evolutionary state”.

ð0Þ Figure 4.13 Distribution of ξ Qc with respect to strain amplitude.

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Behavior and Design of High-Strength Constructional Steel

Figure 4.14 Distribution of QðiÞ c with respect to strain amplitude.

Figure 4.15 Influence of ζ on cyclic stress evolution law.


 8 2Þ   bðcm ε=εc > <
 εA εy ; εc 2
ð2Þ bc 5 ζ ε=εc 21 1 ε=εc >   : 2Þ εA εc ; εf bðcm

(4.18)


 8 ð2 Þ   rcm ε=εc > <
εA εy ; εc 
 2 rcð2Þ 5 ζ ε=εc 21 1 ε=εc >   : ð2Þ εA εc ; εf rcm

(4.19)

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115

Figure 4.16 Distribution of cyclic stress envelop with respect to strain amplitude.

ð2Þ ð3Þ ð2Þ where bð1Þ are model constants, and rcm and bð2Þ c ,ζ,rc , and rc cm are materialdependent parameters. The influences of a series of assumed ζ on the cyclic stress evolution law are plotted in Fig. 4.15. The cyclic stress envelope that describes the boundary considering cyclic hardening and softening behaviors is shown in Fig. 4.16.

4.3.3 Kinematic hardening model As summarized in the existing experimental investigation, the cyclic softening phenomenon is observed once the CPS reaches certain amount and behaves as the flattening tendency of the cyclic loops. In the constitutive modeling of BLY160 conducted by Xu et al. [12], it is assumed that the motion region of the yield surface expands along with the shrinkage of the yield surface radius, describing the increase in back stress and decrease in flow stress during the strain accumulation process. Simultaneously, the quantity difference between the expansion of motion region and shrinkage of yield surface leads to a reduction in the size of BS, indicating a decrease in peak stress that is called stress deterioration. However, this settlement is necessary to be further discussed since test observations of other steel grades show the motion region may undergo either increase or decrease during different stages of a certain cyclic process, producing a more sophisticated development in BS. Nevertheless, the identification of the kinematic saturation magnitude parameter α and saturation rate parameters χ1 and χ2 given in Eq. (4.1) must be further conducted since the distinction between the BS and yield surface (the difference value of peak stress and flow stress in uniaxial case) can solely define the amplitude of kinematic component.

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Behavior and Design of High-Strength Constructional Steel

Traditionally, researchers get used to simulate the kinematic parts using superposition of more than three kinematic subcomponents with constant evolutionary rates and magnitudes, the values of which are calibrated with the stabilized loop after subtracting the elastic region. This approach is proved to be a good choice to consider strain range effect under early cycles with relatively small strain amplitude, synchronously limiting its availability for other cases. Furthermore, this approach cannot well recognize the influence of CPS that successively differs the kinematic rule in every cyclic half cycle. To briefly extend its usage, via trial and error, the kinematic hardening is divided into two parts with identical hardening quantity but different evolutionary rate. The action of accumulation plasticity is incorporated through correlating the evolutionary rates and CPS as depicted in Eqs. (4.20) and (4.21). To this end, the kinematic part extracted from each cyclic half cycle is calibrated through the preceding approach to achieve the basic evolution law of the two saturation rate parameters that are schematically depicted Fig. 4.9. It can be noted that the first-rate parameter behaves as the similar trend of previous three-stage softening, while the second almost remains constant except the abrupt improvement in the last several cycles. Since larger rate parameters signify earlier achievement of kinematic saturation, the fullness of cyclic loop would successively decrease along with the evolution of the two rate parameters. ð0Þ Among those parameters, χð0Þ 1 and χ2 , respectively, denote the initial saturation ð2Þ rapidity of the two kinematic components, while the χð1Þ 1 and χ1 describe the early and subsequent decrease amount of χ1 in the first kinematic component, and χð1Þ 1 and χð2Þ 2 present the initial slight decrease amount and the subsequent increase amount of χ2 in the second kinematic component. On the other hand, d1 and d2 ð1Þ define the evolutionary rates, respectively, corresponding to χð1Þ 1 and χ2 , as well ð2Þ ð2Þ as χ1 and χ2 . χ1 5 χð10Þ 1 χð11Þ U

 2d2 ðpm 2pÞ=pm  1 2 e2d1 p=pm e 2 e2d2 ð2 Þ 1 χ U 1 1 2 e2d1 1 2 e2d2

(4.20)

χ2 5 χð20Þ

 2d2 ðpm 2pÞ=pm  1 2 e2d1 p=pm e 2 e2d2 ð2 Þ 1 χ2 U 1 2 e2d1 1 2 e2d2

(4.21)

1 χð21Þ U

4.3.4 Elastic stiffness degradation It has been well recognized that deteriorations of both strength and stiffness in cyclic loading tests of various steels are caused by microcrack development and behave as a three-stage softening form. The aforementioned sections have decomposed cyclic softening process into three components corresponding to the three distinct stages. Thus a typical evolutional law of elastic stiffness is

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117

introduced based on the above softening system as given in the following equation. ξKc 5 ξKc1 1 ξ Kc2 1 ξKc3 5 λU

2rc1 p=m

12e 1 1 2 e2rc

8 2 3 < 2rc2 ðpm 2pÞ=pm 2rc2 e 2e 5 1 ð1 2 λÞU κ U ð1 2 e2ηε ÞU 4 2 : 1 2 e2rc 2 3) 2rc3 ðpm 2pÞ=pm 2rc3 e 2e 5 1 ½1 2 κU ð1 2 e2ηε ÞU 4 3 1 2 e2rc (4.22)

Em 5 E1 Ufy 1 E2

(4.23)


 Ed 5 1 2 ζ Kc UEm E0

(4.24)

where ξKc1, ξKc2, and ξ Kc3 denote the three partition coefficients corresponding to the three distinct softening stages, λ represents the proportionality coefficient of ξKc1 among all three components, Em is the maximum reduction proportion of elastic stiffness, and E0 is the initial elastic stiffness. Besides, test observations show that Em presents considerable dependency on steel grades, which will be discussed in later sections.

4.3.5 Tension
compression asymmetry Two sets of parameters in the preceding models are utilized for the purpose of defining the asymmetry, including the stress amplitude, shape of cyclic loop, and elastic stiffness degradation. To be specific, the hardening RCPs in compression zone should be greater than those in tension zone, while the softening RCPs in the compression zone are suggested to be less than those in tension zone. Moreover, the elastic stiffness and kinematic rule also show slight difference for compression and tension zones. This tension
compression asymmetry can be attained using different sets of constant parameters.

4.4

Capability of the constitutive model

4.4.1 Isotropic hardening/softening evolution surface Since the proposed constitutive model provides two independent variables, including strain amplitude and CPS, an isotropic hardening/softening evolution surface (IHSES) can be obtained by superposing the five subcomponents under different combination of strain and CPS. Conceptually different from the existing strain
stress constitutive models, the IHSES depicts distinctly the complex evolution law

118

Behavior and Design of High-Strength Constructional Steel

Figure 4.17 IHSES of LYP steel: (A) IHSES, (B) XOY plane, (C) YOZ plane, and (D) XOZ plane. IHSES, Isotropic hardening/softening evolution surface; LYP, low-yield-point.

and provides a convenient approach to capture either the flow stress (YS) or peak stress (BS) under arbitrary cyclic loading. Representative IHSES and its projections on the three coordinate planes are illustrated in Fig. 4.17 for LYP steel. It can be seen that the evolution surface shown in Fig. 4.17 denotes the influence of both strain and CPS on the cyclic stress indexes, either flow stress or peak stress. The shape and amplitude of this surface can be changed through using different material constant to satisfy the accuracy demand for different grades of steel. The projection plane XOY describes the low-cycle fatigue criterion that presents the combination of strain and CPS at fracture point. Besides, the cyclic stress amplitude can as well be depicted using contour lines as shown in Fig. 4.17B. The projection plane XOZ illustrates a distinct stress
strain relationship that presents not only the monotonic curve but also the envelop of the cyclic hardening and softening behaviors. The projection YOZ exhibits the CS
CP curves (relationship of cyclic stress indexes and CPS) under a variety of strain amplitude. Obviously, the characteristics of considerable hardening behavior existing in LYP are suitably simulated in Fig. 4.17. A surface can also be properly used for HSS as plotted in Fig. 4.18 along with its projection planes. Clearly, the surface shown in Fig. 4.18 is concave rather than protruding compared in that shown in Fig. 4.17, indicating a softening effect driven by CPS.

Uniform material model for constructional steel

119

Figure 4.18 IHSES of HSS: (A) IHSES, (B) XOY plane, (C) YOZ plane, and (D) XOZ plane. IHSES, Isotropic hardening/softening evolution surface; HSS, high-strength steel.

4.4.2 Kinematic hardening evolution surface The aforementioned constitutive model considers the influence of plasticity accumulation on kinematic hardening rule via correlating the kinematic saturation rate parameters χ1 and χ2 with CPS. Similar with IHSES, the KHES, which is defined as the back stress surface related with strain and CPS, can be produced by incorporating the accumulation-based magnitude parameter α and saturation rate parameters χ1 and χ2 into back stress. It should be noted that the IHSES of BS includes the contribution of kinematic hardening, and thus the magnitude parameter α of kinematic hardening can be obtained through the difference of the IHSES of BS and YS. Besides, the saturation rate parameters χ1 and χ2 upgrade as plastic strain accumulates. Typical KHES for LYP225 and Q690 steels are depicted in Fig. 4.19. It should be noted that the strain and CPS are both normalized by their maximum values in order to compare the stress magnitude and surface shape of KHES. Generally, the KHES exhibits inconspicuous change within almost 90% CPS range for either Q690 steel or LYP225 steel. However, the shape and magnitude of back stress encounter obvious deterioration in Q690 steel in the last several cycles. The magnitude deterioration in LYP225 steel is not as much as that of Q690 steel. However, the surface still present slight shape change in the final cycles closed to low-cycle fatigue fracture. This behavior is caused by the decrease of parameter χ1 in the final several cycles.

120

Behavior and Design of High-Strength Constructional Steel

Figure 4.19 KHES for LYP225 and Q690 steels: (A) viewport I for KHES of LYP225 steel, (B) viewport II for KHES of LYP225 steel, (C) viewport I for KHES of Q690D steel, and (D) viewport II for KHES of Q690D steel. KHES, Kinematic hardening evolution surface.

4.4.3 Cyclic behaviors described in the constitutive model of various steels The characteristics described in the constitutive model for the cyclic behavior of steel are elaborately illustrated in Fig. 4.20. The cyclic hardening and softening amount can be directly calculated through Eqs. (4.8)
(4.21) at certain combination of strain and CPS. The unloading
reloading process can be divided into the elastic region and plastic region. Besides, the elastic modulus in the unloading
reloading process upgrades using Eqs. (4.22)
(4.24) The asymmetry in kinematic hardening, isotropic hardening, and elastic modulus degradation can be achieved through different sets of material constant in compression and tension zone, respectively. The amplitude of elastic region can be determined with IHSES, and the amplitude and shape of plastic region can be determined with KHES. What distinguishes the proposed constitutive model from traditional models is that cyclic hardening/softening behavior and elastic modulus degradation are coupled nonlinearly with full-range interactive strain and CPS. In addition, the described behaviors are able to consider tension
compression and steel grade dependency.

Uniform material model for constructional steel

121

Figure 4.20 Functions of the proposed constitutive model.

4.5

Simplification of cyclic hardening/softening constitutive model

The proposed uniform full-range isotropic and kinematic hardening and softening cyclic plasticity two-surface constitutive model included eight indispensable parts, including the basic monotonic material properties, low-cycle fatigue properties, isotropic hardening for yield surface, isotropic hardening for BS, isotropic softening for yield surface, isotropic softening for BS, kinematic components, and elastic modulus degradation. Moreover, different types of test results and calibration methods are, respectively, required in each part as illustrated in Part I
Part VIII. Part I: The basic mechanical parameters, including yield stress σy, ultimate tensile stress σu, and its corresponding strain εu, strain of yield plateau end point εst, fracture strain εf, and Young’s modulus E, can be easily identified through typical strain
stress curve derived from monotonic coupon test. Part II: The low-cycle fatigue plasticity parameters include fatigue ductility coefficient ε0f and fatigue ductility exponent C of Manson
Coffin equation, the calibration of which demands at least four low-cycle fatigue tests to be conducted. Part III
VI: The calibration of isotropic hardening and softening parameters requires a range of constant amplitude cyclic loading tests (Fig. 4.21), from which the CPS
stress (CPS
S) curves for flow stress (yield surface) and peak stress (BS) can be obtained. The shape parameters α, β, and γ, which are utilized to describe the dependence of RCPs on strain range, can be calibrated through nonlinear fitting of correlation between the maximum values of cyclic stress indexes (flow stress and peak stress) and strain range, as shown in Fig. 4.22. Since the cyclic stress indexes harden in an exponent manner during the earlier cycles, the so-called rate 1Þ 2Þ parameters bðcm , bðcm , and ζ that describe such exponentially hardening rate with

122

Behavior and Design of High-Strength Constructional Steel

Figure 4.21 Constant amplitude cyclic loading coupon tests and their hysteretic stress
strain relationships.

Figure 4.22 Definition of CPS
S curve and calibration procedure for parameters of isotropic components. CPS
S, Cumulative plastic strain
stress.

respect to CPS and their dependency on strain range can be obtained via correlating the saturation rate identified from the CPS
S curves with the strain range, as shown in the CPS
S curve shown in Fig. 4.22. Similarly, the shape parameters for both RCP and shape parameters of isotropic softening components can also be achieved using cyclic test results and strain range. The isotropic hardening and softening in flow stress (YS) behave as the identical tendency of that in peak stress (BS) along with the plasticity accumulation. However, it has been noted that their hardening and softening magnitudes exist considerable difference. Thus identical sets of shape parameters and rate parameters but two different sets of RCP parameters are calibrated and utilized. Part VII: As for the calibration of the kinematic hardening rule with flattening effect, the plastic part of cyclic loops should be extracted through translating the yield point of each half cycle to the coordinate origin after removing the elastic part, as shown in Fig. 4.23A. According to the developed modeling, the evolution law of back stress can be expressed as superposition of two exponential components with identical saturation value that equals to half stress amplitude (marked as α in Fig. 4.23B) of the extracted plastic part as Eq. (4.25) expressed. The flattening effect is considered through correlating the two rate parameters χ1 and χ2 of saturation in each component with CPS. Since the saturation value of each back stress component (half stress amplitude of the extracted plastic part) can be determined

Uniform material model for constructional steel

123

Figure 4.23 Calibration procedure for parameters of kinematic components: (A) extraction of elastic parts and (B) calibration of kinematic parameters.

using the difference between the peak stress and flow stress, the identification of kinematic hardening rule is to solely calibrate the parameters in the correlation formula Eqs. (4.26) and (4.27) that define the evolution law of saturation rate in each kinematic component with respect to CPS by using nonlinear curve-fitting as shown in the χ
p curve depicted in Fig. 4.23B. σb 5 σb1 1 σb2 5


 α 
 α  1 2 exp 2χ1 ε 1 1 2 exp 2χ2 ε 2 2

(4.25)

χ1 5 χð10Þ 2 χð11Þ

 2d2 ðpm 2pÞ=pm  1 2 e2d1 p=pm 2 e2d2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2 Þ e 2 χ b2 2 4ac 1 1 2 e2d1 1 2 e2d2

(4.26)

χ2 5 χð20Þ 2 χð21Þ

 2d2 ðpm 2pÞ=pm  1 2 e2d1 p=pm 2 e2d2 ð2 Þ e 2 χ 2 1 2 e2d1 1 2 e2d2

(4.27)

where σb1 and σb2, respectively, denote the first and second back stress components; χ1 and χ2, respectively, denote the saturation rate of the two components; α ð1Þ ð2Þ ð0Þ ð1Þ ð2Þ denotes the back stress saturation value; and χð0Þ 1 ,χ1 ,χ1 ,χ2 ,χ2 ,χ2 , d1 and d2 denote material constant parameters. Part VIII: With regard to the elastic stiffness degradation, only two parameters need to be calibrated, including the partition coefficient λ and parameter Em. These two parameters can be identified through correlation between the CPS and elastic modulus. The parameters that need to be calibrated are summarized by the aforementioned categories in Table 4.3 that classifies all parameter categories as steel grade dependent and steel grade independent. Accordingly, there exist 44 parameters in total to be determined through different measurements derived from monotonic tests, lowcycle fatigue tests, and cyclic loading tests, resulting in considerable complexity for application. Thus a simplification should be made to reduce the number of constant parameters based on appropriate assumptions. As the evolution law of ξiRc that describes the dependence of RCP on strain range behaves as concave form, when β i and γ i are larger than 1.0, and switches to a

Table 4.3 Cyclic constitutive parameters need to be calibrated.

1 2

Categories

Steel grade dependent

Steel grade independent

Parameters quantity

Basis mechanical properties Low-cycle fatigue properties based on Manson
coffin formula Isotropic components

σy σu εst εu εf ε0f C

ν 5 0:3 E 5 206; 000


7 2

Reserved capacity parameters




3 4 5 6 7

Isotropic hardening for yield surface Isotropic hardening for bounding surface Isotropic softening for yield surface Isotropic softening for bounding surface Kinematic components

1Þ ð2Þ Rym Rðym

εcR

α1 β 1 γ 1 α2 β 2 γ 2

Rate parameters 1Þ ð2Þ bðcm bcm ζ

1Þ ð2Þ Rðbm Rbm 1Þ ð0Þ Qym Qðym 1Þ ð0Þ Qðbm Qbm

εcS

ωκη

ð2Þ ð3Þ νrcm rcm

11

d1 d2

8

8

Elastic modulus degradation Parameters quantity

Em 18

χð10Þ χð11Þ χð12Þ χð20Þ χð21Þ χð22Þ λ 26




2 44




14

Uniform material model for constructional steel

125

convex shape when β i and γ i are smaller than 1.0. To simplify the model with consideration of such two features, the first assumption regarding the shape parameters of hardening components is employed and extrapolated herein as follows: Assumption 1: α1 5 α2 , β 1 5 γ 1 . 1:0, and β 2 5 γ 2 5 1=β 1 , 1:0 As for the evolution law of isotropic softening components, empirical value of the rate parameter ω is fairly close to that of magnitude parameter η. Besides, the parameter κ that defines the ultimate proportion of the second softening quantity in the total quantity of the second and third softening components can also be reasonably assumed. Thus the second assumption is adopted as follows: Assumption 2: ω 5 η; κ 5 0.5 The five hardening/softening components are separately applied to different hardening/softening stages with various behaviors. Specifically, the first hardening component corresponds to the initial sharp hardening, while the second is utilized to define the subsequent gradual hardening. Thus the saturation rate of the first component should be considerably larger than that of the second component. Similarly, the evolutionary rates of the first and third softening components should be far greater than that of the second softening component. To this end, the third assumption is carried out as follows: 2 1 3 1 Assumption 3: b2cm 5 0:1U b1cm ; rcm 5 0:1Urcm ; rcm 5 2 Urcm

Tension
compression asymmetry expressed as a more significant hardening characteristic subjected to compression stress as well as a more remarkable softening behavior in the mid-to-end stage under tension stress. Thus this model adopts the fourth assumption to consider this effect as following: 1 2 2 Assumption 4: R2 i 5 1:1Ri ; Q23 5 0:8UQ23

Regarding the parameters in Eqs. (4.26) and (4.27) that describe the dependence of kinematic rule on CPS, a series of assumptions can also be empirically applied based on the overall curve shape of the experimental observation. In detail, parameter χ1 and χ2 both present a three-stage evolution characteristic as plasticity accumulates. The main difference between them is that χ1 monotonically decreases while χ2 experiences a decrease-stable-increase three-stage process. The superscripts (0), (1), and (2), respectively, denote the initial value, the saturated amount of the first stage, and third stage for parameter χ1 or χ2. Generally, the parameter ð0Þ ð2Þ ð0Þ ð1Þ ð0Þ ð0Þ ð0Þ ratios, including χð1Þ 1 = χ1 ; χ1 = χ1 ; χ2 = χ1 , and χ2 = χ1 , can be treated as constant based on experimental observation. Moreover, the evolutionary rate of the third stage d2 should be fairly larger than that of the first stage that are determined

126

Behavior and Design of High-Strength Constructional Steel

by d1. Thus parameter ratio of d2/d1 can also been set as proper constant. In general, the fifth assumption is presented as follows: ð0Þ ð2Þ ð0Þ ð0Þ ð0Þ ð1Þ ð0Þ Assumption 5: χð1Þ 1 5 0:25χ1 ; χ1 5 0:4χ1 ; χ2 5 0:25χ1 ; χ2 5 0:25χ2 ; d2 5 5d1

Assuming the cyclic stress reaches its ultimate value when strain and CPS attain a combination of (εcR,pc), the cyclic global ultimate stress can be expressed as

 fcyc ðεcR ; pc Þ 5 fmono ðεcR Þ 1 Rm1 1 2 e2b1 pc 1 Rm2 1 2 e2b2 pc ð3Þ 2 Qm1 ð1 2 e2r1 pc Þ 2 σð2Þ Qc 2 σ Qc

(4.28)

where fcyc(εcRpc) denotes the current cyclic stress value, and fmono(εcR) denotes the monotonic stress corresponding to the current strain εcR. Generally, the two cyclic hardening components and the first softening component reach their saturation state when ultimate cyclic stress is attained. Thus the following expression should be satisfied.


 1 2 e2b1 pc  1:0

(4.29)




 1 2 e2b2 pc  1:0

(4.30)

ð1 2 e2r1 pc Þ  1:0

(4.31)

Besides, the cycles are still out of the mid-to-end stage where the second and third softening components play a key role on cyclic behavior. Thus the following expression should be gained. 2Þ σðQc 0

(4.32)

3Þ 0 σðQc

(4.33)

By substituting Eqs. (4.29)
(4.33) into Eq. (4.28), the expression is obtained in the following equation. fcyc 5 fmono 1 Rm1 1 Rm2 2 Qm1

(4.34)

Regarding LYP steels and mild steels, experimental observations indicate that their initial softening amount is negligible. Thus Eq. (4.34) can be reformed as ðRm1 1Rm2 Þ 5 fcyc 2 fmono ðεcR Þ

(4.35)

As for HSSs, the maximum cyclic hardening amount is unable to exceed the maximum monotonic hardening amount, thus Eq. (4.34) can be reformed as ðRm1 1Rm2 Þ 5 fu 2 fmono ðεcR Þ

(4.36)

Uniform material model for constructional steel

127

The known ðRm1 1Rm2 Þ can be multiplied by a distribution factor ρ0 to predict values of the two components as Rm1 5 ρ0 ðRm1 1Rm2 Þ

(4.37)


 Rm2 5 1 2 ρ0 ðRm1 1Rm2 Þ

(4.38)

We take the assumption that ρ0 5 0.5 according to several tentative calibration. Thus the sixth assumption is herein obtained as follows: Assumption 6: Eqs. (4.35
4.38); ρ0 5 0.5 The material will attain its global maximum stress with cumulative strain accumulation under constant amplitude cyclic strain loading, an amplitude of which equals to the critical strain εcR. Regarding extremely LYP steels, even fairly small strain amplitude will result in considerable and speedy cyclic hardening as cycles accumulate. Besides, the strain corresponding to the ultimate stress εu is relatively large. Thus the ratio of εcR/εu should be a small value. As yield strength increases, the cyclic hardening effect weakens, and the strain ratio εcR/εu needed to reach its ultimate hardening amount must be increased. When YTR is up to 0.85, the cyclic hardening is fairly small, and the critical strain is εcR closed to εu. Therefore the strain ratios εcR/εu can be treated as 1.0 in case that YTR is greater than 0.85. As for the critical strain εcS that defines the strain corresponding to the attainment of ultimate softening amount, the evolution law of strain ratio εcS/εu should be opposite to that for cyclic hardening case. Specifically, the strain ratio εcS/εu is supposed to be fairly small when yield-to-tensile is extraordinary large since extremely small strain will cause considerable cyclic softening for ultra-HSSs. Based on previous qualitative analysis, two piecewise-assumed formulations were developed to describe the correlations between the critical strain ratios and the YTR, respectively, as Eqs. (4.39) and (4.40) described. The relationship curves are shown in Fig. 4.24. Assumption 7: Eqs. (4.39
4.40) 8 0:25
 εcR < 5 0:25 1 1:36 U fy =fu 2 0:3 : εu 1:0

fy =fu , 0:3 0:3 , fy =fu , 0:85 fy =fu . 0:85

(4.39)

8 0:5
 fy =fu , 0:3 εcS < 5 0:5 2 0:45U fy =fu 2 0:3 0:3 , fy =fu , 0:85 : εu 0:25 fy =fu . 0:85

(4.40)

Consequently, the number of the constant parameters need to be calibrated reduces from 44 to 29 based on the preceding assumptions, leading to considerable

128

Behavior and Design of High-Strength Constructional Steel

Figure 4.24 Assumed relationships between critical strain and yield-to-tensile ratio.

simplification on subsequent parameter calibration. The simplified constitutive model parameters are summarized in Table 4.4 by categories. However, the dependence of these parameters on the steel grade needs to be further investigated to develop a uniform approach on evaluating the cyclic response. To this end, the simplified constitutive model was calibrated through available cyclic coupon test results on structural steel with nominal yield stress ranged from 100 to 890 MPa. Since the existing cyclic stress
strain data are derived from different loading histories, including constant amplitude tests and variable amplitude tests, and not all coupon tests are loaded to the ultimate low-cycle fatigue failure condition, the calibrations have to be completed through trial and error to make the best fit.

4.6

Cyclic parameter calibration

4.6.1 Calibration method The monotonic and cyclic coupon tests can provide several key relationship curves for calibration, including monotonic stress
strain curve, cyclic stress
strain hysteretic curve, cyclic CPS
peak stress curve, cyclic CPS
flow stress curve (CPS
S), and cyclic CPS
elastic modulus curve (CPS
E). The calibration of model parameters demands the previous experimental curves. The basic calibration procedure is summarized in the following sections. 1. Monotonic properties, including fy, fu, εu, εf, E, and YTR are obtained through monotonic experimental stress
strain relationship. 2. Critical strains εcR and εcS, respectively, for cyclic hardening and softening are calculated by substituting the YTR into Eqs. (4.39) and (4.40). Then the calculated εcR is substituted into Eqs. (4.35) and (4.36) to obtain the hardening reserve capacities of flow stress and peak stress, especially denoted as ðRym1 1Rym2 Þ and ðRbm1 1Rbm2 Þ . Then the assumed

Uniform material model for constructional steel

129

Table 4.4 Parameters need to be calibrated for simplified constitutive model. Categories

Parameters

Quantity

Monotonic parameters Low-cycle fatigue properties

σy σu εst εu εf v E 0 εf C b v y fcyc;u Qb m1 Qm23 Qm1 Qm23 αβγω pR pb0 py0 1 ζ b1cm rcm  λEm χ1ð0Þ χð0Þ 2 d1

7 2

Stress parameters Shape parameters Proportional parameters Rate parameters Elastic modulus parameter Kinematic parameter In total

5 4 3 3 2 3 29

partition factor ρ0 is substituted into Eqs. (4.37) and (4.38) to calculate the initial calibray y b tion results for parameters Rb m1 , Rm2 , Rm1 , and Rm2 . 3. Based on the experimental CPS
S curve for flow stress and peak stress, initial calibration y y b results for cyclic softening parameters Qb m1 , Qm1 , Qm23 , and Qm23 can be attained though Eqs. (4.41)
(4.44). b b Qb m1 5 f0 2 f0:3N $ 0

(4.41)

y y Qy m1 5 f0 2 f0:3N $ 0

(4.42)

b b Qb m23 5 f0:5N 2 fN

(4.43)

y y Qy m23 5 f0:5N 2 fN

(4.44)

where f0b and f0y , respectively, denote the peak stress and flow stress in the first cycle; y b f0:5N and f0:5N , respectively, denote the peak stress and flow stress corresponding to the 0.5Nf in the CPS
S curve derived from constant amplitude cyclic coupon tests; fNb and fNy , respectively, denote the peak stress and flow stress corresponding to the final fracture point; and Nf denotes the cycles corresponding to the low-cycle fatigue fracture. 4. Successively adjust the values of distribution factors ρy0 ,ρb0 , and ρR; shape factors α, β, γ, 1Þ ð2Þ and ω; and rate parameters bðcm ,rcm , and ζ until the simulated CPS
S curves match well with their experimental versions. 5. Back stress parameters for each cycle are calibrated using Eq. (4.25). Then the calibrated parameters are correlated to plastic cumulative strain to form the relationship curve (CPS
KRP). Successively adjust the value of related constant parameters until the simulated CPS
KRP curve match well with their experimental versions. 6. Initial calibration results of elastic modulus parameters λ and Em are calculated according to Eqs. (4.45) and (4.46) based on the experimental CPS
E curve. λ 5

E0 2 E0:2N E0 2 EN

(4.45)

130

Behavior and Design of High-Strength Constructional Steel

Em 5

E0 2 EN E0

(4.46)

where E0.2N denotes the elastic modulus corresponding to 0.2Nf in the CPS
E curve derived from constant amplitude cyclic coupon tests, and EN denotes the elastic modulus corresponding to final fracture. 7. Successively adjust the values of distribution factor ρ; shape factors α, β, γ, and ω; speed ð0Þ 1Þ ð1Þ ,rcm , and ζ; kinematic parameters χð0Þ parameters bðcm 1 ,χ2 , and d1; elastic modulus para meters λ and Em until the CPS
S curve and stress
strain hysteretic curve match well with their experimental versions. The corresponding values are treated as the calibration results.

The initial values and value ranges for each parameter are tabulated in Tables 4.5 and 4.6.

4.6.2 Calibration of model parameters Cyclic model parameters were calibrated for 22 batches of structural steels with 9 steel grades using their cyclic coupon test data. The nominal yield strength of these steels ranges from 100 to 890 MPa. The calibrated hardening parameters, softening parameters, and shape parameters were summarized in Table 4.7. The calibrated elastic modulus parameters, evolutionary speed parameters, and kinematic parameters were tabulated in Table 4.8.

4.6.3 Verification of calibration result To verify the accuracy of cyclic constitutive model and its calibration results, the calibrated parameters were incorporated into a self-programmed software used to predict cyclic response. Then each cyclic coupon test used for parameter calibration was simulated using corresponding cyclic loading protocol. The simulated CPS
S curves and stress
strain hysteretic curve were then extracted and compared with the corresponding experimental curves as shown in Figs. 4.25
4.35. It can be seen that all simulated curves provide good agreement with the experimental curves for each structural steel, indicating that the proposed constitutive model and their parameters are capable of giving accurate description.

4.7

The evolution laws of constitutive model parameters

4.7.1 Evolution laws of monotonic parameters In previous studies, monotonic curve of structural steel is generally modeled as a piecewise function. To attain further simplification a new monotonic model with single expression is proposed in the following equation.

Table 4.5 Initial values and value ranges of cyclic model parameters: I. Stress parameters Parameter Initial value Range

Qb m1 0.2fy (0
0.5)fy

fcyc,u 2fy (1
5)fy

Shape parameters Qy m1 0.25fy (0
0.5)fy

Qb m23 0.2fy (0.1
0.8)fy

Qy m23 0.5fy (0.1
0.8)fy

α 0.5 0.1
0.5

β 0.3 0.3
0.7

γ 0.5 0.2
0.8

ω 50 20
80

Table 4.6 Initial values and value ranges of cyclic model parameters: II. Proportional parameters Parameter Initial value Range

p0 0.5 0.3
0.8

pR 1.0 0.5
1.5

Rate parameters b1cm 300 100
500

1 rcm 80 50
150

Elastic modulus parameter ζ 0.1 0
0.3

λ 0.1 0
0.5

Em 0.5 0.1
0.8

Kinematic parameter χð0Þ 1 400 200
1000

χð0Þ 2 50 10
100

d1 30 10
50

Table 4.7 Calibration results for hardening, softening, and shape parameters. Category

Hardening parameters

Softening parameters

Shape parameters

Steel grade

fcyc,u

py0

pb0

pR

Qb m1

Qb m23

Qy m1

Qy m23

α

β

γ

ω

LYP100 LYP160 LYP225 LYP225 Q235 Q345 Q460C Q460D Q460D-11 Q690D-40L Q690D-16A Q690D-16B Q690D-16C Q690D-40R Q690GJ Q690GJW Q460WJ Q460D Q550D-10 Q550D-16 Q690D Q890D

268.3 265.3 309.1 415.2 583.1 590.3 637.4 581.7 570.5 874.2 803.9 804.0 874.2 816.3 811.2 753.5 653.8 768.9 733.1 751.3 866.0 1095.0

0.47 0.50 0.50 0.56 0.51 0.44 0.48 0.45 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.52 0.49 0.64 0.50 0.50 0.50 0.50

0.39 0.36 0.50 0.55 0.43 0.47 0.58 0.62 0.60 0.50 0.50 0.50 0.50 0.50 0.50 0.38 0.44 0.65 0.50 0.50 0.50 0.50

0.70 0.63 1.00 0.69 0.92 0.85 0.91 0.71 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.73 1.00 0.91 1.00 1.00 1.00

0.0 0.0 0.0 0.0 0.0 0.0 2.0 3.0 0.0 70.8 31.0 54.5 91.0 100.6 22.4 4.2 22.0 4.3 51.1 129.6 89.4 50.1

152.6 28.9 96.2 34.9 185.1 47.1 292.5 365.1 411.9 443.9 600.0

0.0 0.0 0.0 0.0 0.0 0.0 2.0 2.0 2.0 61.6 37.3 40.4 57.1 90.8 38.7 19.6 9.9 0.0 83.6 112.5 90.7 67.9

129.0 26.9 26.0 0.0 214.6 61.9 227.5 247.4 276.0 450.6 468.0

0.52 0.55 0.34 0.50 0.43 0.52 0.58 0.55 0.55 0.58 0.42 0.65 0.58 0.56 0.45 0.50 0.62 0.44 0.46 0.40 0.40 0.40

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 60 60 60 60 60

Uniform material model for constructional steel

133

Table 4.8 Calibration results for elastic modulus, evolutionary speed, and kinematic parameters. Rate parameters

Kinematic parameters

Category

Elastic modulus parameters

Steel grade

λ

Em

ζ

b1cm

1 rcm

χð0Þ 1

χð0Þ 2

d1

LYP100 LYP160 LYP225 LYP225 Q235 Q345 Q460C Q460D Q460D-11 Q690D-40L Q690D-16A Q690D-16B Q690D-16C Q690D-40R Q690GJ Q690GJW Q460WJ Q460D Q550D-10 Q550D-16 Q690D Q890D

0.17 0.10 0.10 0.10 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.30 0.30 0.30 0.30 0.30

0.45 0.45 0.45 0.58 0.45 0.45 0.45 0.45 0.45 0.62 0.48 0.55 0.58 0.65 0.35 0.35 0.45 0.35 0.35 0.35 0.35 0.35

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

200 150 150 180 150 150 150 150 150 150 150 150 150 150 150 150 150 100 100 100 100 100

30 30 50 25 30 30 30 30 30 30 30 30 30 30 30 30 30 25 40 30 25 25

1000 800 600 800 600 500 440 500 550 250 350 450 250 200 250 200 360 400 400 500 400 400

181 150 132 113 81 83 55 60 83 10 10 10 10 10 13 30 54 48 48 60 48 48

10 8 15 10 10 10 10 10 10 10 10 10 10 10 10 10 10 12 12 12 12 12



 f εp 5 fy 1 R 1 2 e2bεp 2 Q e2rðεfp 2εp Þ=εfp 2 e2r

(4.47)

where the second term in the right is monotonic hardening component while the third is the monotonic softening component; εst denotes the strain corresponding to the yield plateau end point, and εp 5 ε 2 εst denotes the plastic strain, and εfp 5 εf 2 εst denotes monotonic fracture plastic strain; parameters R and Q, respectively, denote the maximum amount of hardening and softening components; and parameters b and r denote the evolutionary speed of hardening and softening components with respect to plastic strain. In this chapter, in total 480 monotonic experimental curves were collected for calibration of the proposed monotonic constitutive model. The calibrated parameters were then correlated with yield strength or YTR as shown in Fig. 4.36. The following relationships can be fitted as Eqs. (4.48)
(4.55).

134

Behavior and Design of High-Strength Constructional Steel

Figure 4.25 Comparison between experimental and simulation results for LYP100 steel: (A) Hysteretic curves of coupon, (B) Hysteretic curves of coupon, (C) Hysteretic curves of coupon, (D) CPS-S curves of coupon A1, (E) CPS-S curves of coupon A2, and (F) CPS-S curves of coupon A4 [34].

Figure 4.26 Comparison between experimental and simulation results for LYP160 steel: (A) Hysteretic curves of coupon B2, (B) Hysteretic curves of coupon B4, (C) CPS-S curves of coupon B2, and (D) CPS-S curves of coupon B4 [34].

Uniform material model for constructional steel

135

Figure 4.27 Comparison between experimental and simulation results for LYP225 steel: (A) Hysteretic curves of coupon S3-1, (B) Hysteretic curves of coupon S4-1, (C) Hysteretic curves of coupon S7, (D) CPS-S curves of coupon S3-1, (E) CPS-S curves of coupon S4-1, and (F) CPS-S curves of coupon S7 [35].

Figure 4.28 Comparison between experimental and simulation results for Q235 steel: (A) Hysteretic curves of coupon L6-1, (B) Hysteretic curves of coupon L6-2, (C) CPS-S curves of coupon L6-1, and (D) CPS-S curves of coupon L6-2 [38].

136

Behavior and Design of High-Strength Constructional Steel

Figure 4.29 Comparison between experimental and simulation results for Q345 steel: (A) Hysteretic curves of coupon H3-1(1), (B) Hysteretic curves of coupon H3-3, (C) Hysteretic curves of coupon H7-3, (D) CPS-S curves of coupon H3-1(1), (E) CPS-S curves of coupon H3-3, and (F) CPS-S curves of coupon H7-3 [38].

Figure 4.30 Comparison between experimental and simulation results for Q460C steel: Hysteretic curves of coupon BM3-1, (B) Hysteretic curves of coupon BM3-3, (C) Hysteretic curves of coupon BM3-4, (D) CPS-S curves of coupon BM3-1, (E) CPS-S curves of coupon BM3-3, and (F) CPS-S curves of coupon BM3-4 [30].

Uniform material model for constructional steel

137

Figure 4.31 Comparison between experimental and simulation results for Q460D steel: (A) Hysteretic curves of coupon H3-2, (B) Hysteretic curves of coupon H3-3, (C) Hysteretic curves of coupon H6-1, (D) CPS-S curves of coupon H3-2, (E) CPS-S curves of coupon H33, and (F) CPS-S curves of coupon H6-1 [29].

Figure 4.32 Comparison between experimental and simulation results for Q550D: (A) Hysteretic curves of coupon 5R10-1, (B) Hysteretic curves of coupon 5R10-2, (C) Hysteretic curves of coupon 5R16-1, (D) CPS-S curves of coupon 5R10-1, (E) CPS-S curves of coupon 5R10-2, and (F) CPS-S curves of coupon 5R16-1 [44].

138

Behavior and Design of High-Strength Constructional Steel

Figure 4.33 Comparison between experimental and simulation results for Q690D steel: (A) Hysteretic curves of coupon C16-A1, (B) Hysteretic curves of coupon C16-A2, (C) Hysteretic curves of coupon C16-A3m (D) CPS-S curves of coupon C16-A1, (E) CPS-S curves of coupon C16-A2, and (F) CPS-S curves of coupon C16-A3 [33].

Figure 4.34 Comparison between experimental and simulation results for Q690GJ steel: (A) Hysteretic curves of coupon CL1-1, (B) Hysteretic curves of coupon CL1-2, (C) Hysteretic curves of coupon CL1-3, (D) CPS-S curves of coupon CL1-1, (E) CPS-S curves of coupon CL1-2, and (F) CPS-S curves of coupon CL1-3 [42].

Uniform material model for constructional steel

139

Figure 4.35 Comparison between experimental and simulation results for Q890D: (A) Hysteretic curves of coupon 8R10-1, (B) Hysteretic curves of coupon 8R10-2, (C) CPS-S curves of coupon 8R10-1, and (D) CPS-S curves of coupon 8R10-2 [44]. 1. Parameters b and r can be expressed using monotonic yield strength as the following equations.  fy 2 1 0:05fy 1 13:78 b 5 2 0:48 100

(4.48)

 2 fy 2 0:03fy 1 15:78 r 5 0:16 100

(4.49)

2. Parameters R and Q can be formulated using both yield strength and YTR as the following equations. R 5 7:19fy e23:92fy =fu

(4.50)

Q 5 3:44fy e22:47fy =fu

(4.51)

3. The strain corresponding to yield plateau end point, the ultimate strength, and fracture point can be linearly fitted as the following equations. εst 5 15:62 2 0:014fy εy

(4.52)

εu 5 2 0:37fy =fu 1 0:41

(4.53)

140

Behavior and Design of High-Strength Constructional Steel

(Continued)

Uniform material model for constructional steel

141

Figure 4.37 Predicting monotonic curves of different yield-to-tensile ratio: (A) monotonic curves with yield plateau and (B) monotonic curves without yield plateau. εf 5 2 0:65fy =fu 1 0:79

(4.54)

4. The correlation between the yield strength and YTR can be expressed as the following equations.

h i fy 5 0:97 1 2 e23:82ðfy =1000Þ fu

(4.55)

By using the previous expressions, the monotonic curve can be predicted solely according to the yield strength. Fig. 4.37A and B, respectively, provides the predicted monotonic curves with and without yield plateau.

4.7.2 Evolution laws of cyclic model parameters The aim of this section is to develop an evaluation approach for cyclic constitutive parameters relying on monotonic parameters. Since YTR is a simple but clear index for classifying the structural steels, it was selected as the monotonic parameter with which the cyclic constitutive parameters were correlated herein. In this section, low-fatigue fracture criterion and cyclic parameters were correlated with YTR. The relationship between cyclic global ultimate tensile-to-yield ratio and monotonic YTR can be fitted using exponential equation (Fig. 4.38A) expressed as (4.56)

L

fcu 5 0:9977 UYTR21:21 fy

Figure 4.36 Evolution laws of monotonic model parameters: (A) correlation between b and fy, (B) correlation between r and fy, (C) correlation between R/fy and fy/ fu, (D) correlation between Q/fy and fy/fu, (E) correlation between εst/εy and fy/fu, (F) correlation between εu and fy/ fu., (G) correlation between εf and fy/fu, and (H) correlation between fy/fu and fy.

142

Behavior and Design of High-Strength Constructional Steel

(Continued)

Uniform material model for constructional steel

143

As depicted in Fig. 4.38B, parameters ρy0 and ρu0 can be treated as constant value of 0.5 due to its small variation. Partition parameter ρR denotes the ratio of maximum cyclic hardening amount of flow stress and that of peak stress. As for HSSs, the cyclic hardening behavior is fairly small that the influence of partition parameter is negligible. Thus the value of parameter ρR can be taken as 0.7 according to the corresponding value of LYP steels conservatively. The proportional parameter fiso/fy can be predicted using a linear expression (Fig. 4.38E) as fiso 5 0:14 UYTR 1 0:67 fy

(4.57)

The structural steels, YTR of which is smaller than 0.8, are generally LYP steels and normal-strength steels. Experimental observation showed that the initial cyclic softening in LYP steel and normal-strength steel is negligible but that in HSS is within 0.20fy. In this chapter, formulations expressed as Eqs. (4.58) and Eq. (4.59) are proposed regarding the prediction of softening parameters Qbm1 and Qym1 . It can be seen that two critical YTRs YTR1 and YTR2 were incorporated and, respectively, valued as 0.85 and 0.4. Qbm1 5 Qym1 5 0:2

arctan½50ðYTR 2 YTR1 Þ 1 arctanð50YTR1 Þ arctanð50YTR1 Þ 1 arctan½50ð1 2 YTR1 Þ

Qbm23 5 Qym23 5 0:5 1 1:2

(4.58)

arctan½50ð1 2 YTR2 Þ 2 arctan½50ðYTR 2 YTR2 Þ arctanð50U YTR2 Þ 1 arctan½50ð1 2 YTR2 Þ (4.59)

The relationships between kinematic parameters and YTR are provided in Fig. 4.38F and G, including calibration data and fitting curves, expressions of which are, respectively, in the following equations. χð10Þ 5 2 649 UlnðYTRÞ 1 274

(4.60)

χð20Þ 5 2 254 UYTR 1 262

(4.61)

As for elastic modulus, this chapter suggests to predict their value piecewise as expressed in the following equations.

0:45 0:35

fy , 460 MPa fy $ 460 MPa

(4.62)

L

Em 5

Figure 4.38 Evolution law of cyclic constitutive parameters: (A) correlation between TYR and YTR, (B) correlation between ρ0 and YTR, (C) correlation between Q1/fy and YTR, (D) correlation between Q23/fy and YTR, (E) correlation between fiso/fy and YTR, (F) correlation between χ1(0) and YTR, and (G) correlation between χ2(0) and YTR. HSS, High strength steels; LYP: low yield point steels; MS, mild steels; TYR, tensile-to-yield ratio; YTR, yield-totensile ratio

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Behavior and Design of High-Strength Constructional Steel

λm 5

0:10 0:25

fy , 345 MPa fy $ 345 MPa

(4.63)

The values of distribution parameters ρy0 ,ρb0 , and ρR ; shape parameters α,β,γ, and 1Þ ð1Þ ω; and rate parameters bðcm ,rcm ,ζ, and d1ð1Þ can be treated as global constant due to its small variations. In other words, their values are identical as given in Table 4.9 for any steels the nominal yield strength of which ranges from 100 to 890 MPa. The exact cyclic softening amount is related to the low-fatigue fracture criterion. Thus the relationship between low-fatigue fracture criterion and YTR should simultaneously be developed. Generally, the cycles corresponding to low-fatigue fracture can be calculated using Manson
Coffin equation. In the constant amplitude cyclic coupon tests on five kinds of steels conducted by Peter [28], it was found that the influence of strain rate and steel grades on Manson
Coffin parameter is fairly small. Thus low-fatigue failure curves on 13 steel grades were collected and plotted in Fig. 4.39 without consideration of strain rate effect. Since the contribution of elastic term in Manson
Coffin equation is small when strain steps into plastic range, the plastic term is adopted to fit the data in Fig. 4.39B. The fitting equation is shown in Eq. (4.64). The CPS corresponding to low-fatigue fracture is the product of half cycle number and plastic strain amplitude, which can be further integrated as Eq. (4.65). Thus the formulation of ultimate CPS and yield strength is established.

Table 4.9 Global model constant parameters. ρy0 0.5 ω 50

ρb0 0.5 b1cm 140

ρR 0.7 1 rcm 30

α 0.5 ζ 0.1

β 0.5 d1ð1Þ 10

γ 0.5

Figure 4.39 Manson
Coffin curve for 13 structural steels: (A) S-N curve derived from 13 steels and (B) Manson-Coffin fitting curve.

Uniform material model for constructional steel


2c
20:521 Δεp 5 εf 2Nf 5 0:353 2Nf 2
20:92
ðc21Þ=c
1=c 2εf 5 0:27 ε2fy =E pm 5 2Δεp Nf 5 Δεp

4.8

145

(4.64) (4.65)

Simplified evaluation approach for cyclic model parameter

Most practical construction projects are unable to provide the experiment condition for cyclic parameter calibration. Thus the calibration of cyclic model parameter needs to be simplified using a new evaluation approach. Thereby two tasks must be solved by this evaluation approach. 1. The monotonic curves can be directly predicted using yield strength and ultimate strength given by metallic report. 2. Cyclic model parameters, including isotropic cyclic hardening/softening parameters, kinematic parameters, elastic modulus parameters, and low-fatigue failure parameters, can be directly evaluated using the previous monotonic properties predicted.

The metallic report generally provides the yield strength, ultimate strength, and elongation of the structural steel. The yield-to-tensile can also be calculated in case that the ultimate strength is unable to be provided. Based on the known yield-totensile strength and yield strength, strain corresponding to ultimate strength, fracture strain, monotonic hardening parameter and softening parameter can be predicted using Eqs. (4.48)
(4.55). The cyclic parameters are to be predicted using the following procedure. 1. Predict the cyclic tensile-to-yield ratio and cyclic global ultimate tensile strength based on YTR through Eq. (4.56). 2. Predict the critical strain corresponding to cyclic hardening and cyclic softening, respectively, using Eqs. (4.39) and (4.40). Then predict their corresponding monotonic stress (so-called critical stress) by substituting the critical strain into Eq. (4.47). 3. The difference of cyclic global ultimate stress and critical stress is the total cyclic hardening reserved capacity. Each reserved capacity can be obtained through multiplying partition factor 0.5 with the total cyclic hardening reserved capacity. 4. Predict the two cyclic softening reserved capacities by substituting the critical strain and YTR into Eqs. (4.58) and (4.59). 5. Predict the two kinematic hardening parameters by substituting YTR into Eqs. (4.60) and (4.61). 6. Predict the current two cyclic hardening amount and three cyclic softening amount by substituting the model constant parameters, current strain, and current CPS into Eq. (4.8), Eq. (4.9), and Eq. (4.10), respectively. 7. Predict the BS size by summing the monotonic stress with the previous five hardening/ softening components in procedure (6). 8. Predict the back stress using Eqs. (4.1)
(4.21). 9. Predict the elastic modulus using Eqs. (4.62) and (4.63).

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Behavior and Design of High-Strength Constructional Steel

Figure 4.40 Cyclic response prediction procedure using proposed evaluation method.

This procedure is schematically depicted in Fig. 4.40 to provide a more clear description. It can be seen that even only yield strength is provided, the cyclic constitutive parameters can be predicted conveniently.

4.9

Comparison of prediction results on different structural steels

To further verify the prediction results and understand the characteristics difference of steel grades, several comparison analyzes were conducted in this section. The concepts of IHSES and KHES were proposed to provide a general view on the cyclic response characteristics. The IHSES and KHES were produced through the evaluation approach given in the previous sections herein and plotted in Figs. 4.41 and 4.42, respectively. It can be seen that yield strength greatly alters the IHSES since the IHSES transforms from outer convex surface to inner concave surface with increase in yield strength. The difference of KHES is relatively small except for the sectors closed to low-fatigue failure. The surface shown in Fig. 4.43 is elastic modulus evolutionary surface. It can be seen both strain and CPS may deteriorate the elastic modulus in a different manner. Besides, this effect can be described conveniently though the proposed evaluation method.

4.10

Summary

Based on the experimental observations on cyclic behavior of structural steels with nine steel grades with yield strength ranges from 100 to 890 MPa, a uniform

Uniform material model for constructional steel

147

Figure 4.41 Predicted IHSES of different structural steel using proposed evaluation method: (A) LYP100 steel, (B) LYP160 steel, (C) Q235 steel, (D) Q345 steel, (E) Q460 steel, (F) Q550 steel, (G) Q690 steel, and (H) Q890 steel. IHSES, Isotropic hardening/softening evolution surface.

148

Behavior and Design of High-Strength Constructional Steel

Figure 4.42 Predicted KHES of different structural steel using proposed evaluation method: (A) LYP100 steel, (B) LYP160 steel, (C) Q235 steel, (D) Q345 steel, (E) Q460 steel, (F) Q550 steel, (G) Q690 steel, and (H) Q890 steel. KHES, Kinematic hardening evolution surface.

Uniform material model for constructional steel

149

Figure 4.43 Predicted EMES using proposed evaluation method. EMES, Elastic modulus evolution surface.

full-range constitutive model considering cyclic hardening and softening was studied. A two-surface mixed isotropic and kinematic hardening/softening constitutive model was developed, and the following concluding remarks can be summarized: 1. The stress indexes, including flow stress, peak stress, and back stress, present considerable dependency on strain amplitude, CPS as well as steel grades. 2. The evolution law of isotropic components with respect to CPS for LYP steels generally exhibits a three-stage characteristic that consists of a hardening, a stable, and a final softening stage. On the other hand, those of HSS present a three-stage softening characteristics. The degradation of elastic modulus exhibits similar three-stage softening behavior as presented in the cyclic stress softening behavior of HSS. 3. The developed two-surface constitutive model provides a rigorous approach to describe the strain
CPS
stress relationship under arbitrary loading history. Simultaneously, the degradation of elastic modulus, tension
compression asymmetry, and steel grade dependency is considered in this constitutive model. 4. According to the proposed constitutive model, the novel concept of IHSES and KHES is introduced as a sophisticated description of the evolution law of isotropic part and kinematic part. These two concepts can be utilized to view the general cyclic characteristics of various structural steels. 5. The proposed constitutive model is able to descript of the cyclic hardening and softening behaviors with respect to full-range instantaneous strain and CPS, as well as the kinematic hardening and elastic modulus affected by plasticity accumulation. The proposed constitutive model is applicable for LYP, mild steels, and HSS. 6. The proposed cyclic isotropic and kinematic hardening and softening two-surface constitutive model was simplified with several basic assumptions, leading to considerable reduction in cyclic parameter quantity. 7. A simplified evaluation procedure was developed, leading to considerable convenience for model application since the cyclic model parameters can be directly predicted relying on yield strength. The proposed model and parameter evaluation procedure were suitable for structural steels, nominal yield strength of which ranges from 100 to 890 MPa.

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Behavior and Design of High-Strength Constructional Steel

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88.

Properties of high-strength steels at and after elevated temperature

5

Guo-Qiang Li, Huibao Lyu, Lei Huang and Xin-Xin Wang Tongji University, Shanghai, P.R. China

5.1

Introduction

High-strength steel (HSS) is quite vulnerable to the fire and high temperature due to the high thermal conductivity and low specific heat. To ensure the safety of the HSS structures in an entire fire event, it is required to better understand the behavior of HSS in fire. Therefore essential properties of HSS subjected to fire is introduced in this chapter. In addition, the fire damaged steel structures may be reused after structural inspection, safety appraisal and necessary repair. However, effects of heating and cooling of steel in fire are similar to tempering and annealing, thus mechanical properties of steel after fire will be different from the initial properties. Therefore it is necessary to obtain mechanical properties of steel after fire for evaluating the postfire behavior and load bearing capacity and l of steel structures. Previous studies on HSSs shows that the steel grade and cooling way had significant effects on the postfire mechanical properties of HSS, higher strength steel lost more yield strength and elastic modulus than the lower strength steels after fire [1
4] and the effect of natural cooling is very different from that of water cooling [5
7]. It has been also found that the mechanical properties of high-strength steel bolts after fire and ultimate tensile strength can lose up to 33% after fire. Therefore to ensure the safety of high-strength steel structures after fire events, it is the fundamental to investigate into the mechanical properties of high-strength steel plates and bolts after fire.

5.2

Mechanical properties of high-strength steels at elevated temperatures

5.2.1 Methodology The high temperature mechanical properties for steel can be usually obtained through two different types of tests, transient test and steady-state test. In a transient-state test, the test specimen is first loaded to a target stress level and then exposed to an increasing temperature. Strain is measured as a function of temperature during the test. In a steady-state test, the test specimen is first heated to Behavior and Design of High-Strength Constructional Steel. DOI: https://doi.org/10.1016/B978-0-08-102931-2.00005-X © 2021 Elsevier Ltd. All rights reserved.

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Behavior and Design of High-Strength Constructional Steel

a specific temperature level and then loaded to failure. Strain is measured as a function of applied engineering stress, as the case for a conventional room temperature tension test. One of the key issues discussed in the literature [8
10] in regard to the test methods is the role of creep in the final stress
strain curves. For transient-state tests, the degree to which creep is represented in the final stress
strain curves depends on the heating rate, with the lower heating rates giving a greater amount of creep. For steady-state tests, the degree to which creep is represented in the final stress
strain curves depends on the loading rate, with lower load rates giving a greater amount of creep. For either of the test method, the effects of creep are included in the final stress
strain curves in an approximate manner, and cannot be compared directly to real fire conditions. Although both methods can be considered an approximate representation of the heating and stress conditions in an actual structure in an actual fire, both methods can still provide useful information on high temperature properties of steel for structural fire engineering design.

5.2.2 Behaviors of high-strength steels at elevated temperature 5.2.2.1 Tests and specimens The mechanical properties of Chinese high-strength steel Q550, Q690, and Q890 were derived herein by using the steady-state test. The tests were carried out in accordance with GB/T4338-2006 [11]. A total of nine elevated temperature levels were considered, which were 200 C, 300 C, 400 C, 450 C, 500 C, 550 C, 600 C, 700 C, and 800 C. Fig. 5.1 shows the shape and dimensions of the test specimens. The test specimens were cut from quenched and tempered Q550, Q690, Q890 structural steel plates with a nominal thickness of 20 mm.

5.2.2.2 Fracture modes of high-strength steel at elevated temperatures Fig. 5.2 shows the coupons after completion of testing. The coupons exhibit different colors depending on the test temperature. The relation between the color and

Figure 5.1 Dimensions of specimen for steady-state tests (unit in mm) [12].

Properties of high-strength steels at and after elevated temperature

155

Figure 5.2 Failure modes of the specimens at various temperatures. (A) Q550. (B) Q690. (C) Q890.

surface appearance of the steel with maximum temperature exposure can be useful for the postfire investigations of HSS structures. Also, as seen in these photos, the length of the necked region of the coupon increases at higher temperatures.

5.2.2.3 Engineering stress
strain curves of high-strength steel at elevated temperatures Fig. 5.3 shows the measured engineering strain
stress curves for different steels at elevated temperatures. At high temperatures, the stress
strain curves of Q690 and Q890 exhibit no yield plateau. Q550 steel exhibited no yield plateau at both room and high temperatures. Furthermore, as the temperature increases, the linear portion of the stress
strain curve shortens, the strain corresponding to ultimate strength decreases, and the descending stage of the curve shows a gentler slope. The strain at fracture tends to decrease with increasing temperature, up to approximately 400 C
500 C. For higher temperatures, the strain at fracture increases significantly compared to that at room temperature, indicating the much larger ductility for HSSs at high temperatures.

5.2.2.4 Definition of yield strength There is no general agreement on the definition of yield strength of steel at high temperature. Various definitions for the yield strength have been reported in the literature such as f0.2, T, f0.5, T, f1.0, T, f1.5, T and f2.0, T. As shown in Fig. 5.4, the 0.2% offset yield strength (f0.2,T) is the intersection point of the stress
strain curve and the proportional line offset by 0.2% strain, while the yield strengths of f0.5,T, f1.0,T, f1.5,T and f2.0,T are those corresponding to the stress at total strain levels of 0.5%, 1.0%, 1.5%, and 2.0%. In the current design codes and standards, different yield strengths are recommended for structural fire engineering design. For instance, EC3 [13] specifies that the yield strength at high temperature is f0.2,T. ECCS [14] specifies that, above 400 C, the yield strength is taken as f0.5,T; and below 400 C, the yield strength is corresponding to an interpolation between f0.2,T and f0.5,T. BS5950 [15] specifies that the yield strength at high temperature can be taken as f0.5,T, f1.5,T or f2.0,T,

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Behavior and Design of High-Strength Constructional Steel

Figure 5.3 Stress
strain curves for Q550, Q690, and Q890 for temperatures ranging from 20 C to 800 C. (A) Q550, (B) Q690, and (C) Q890. [12].

Figure 5.4 Definition of nominal yield strengths [12].

Properties of high-strength steels at and after elevated temperature

157

depending on the type of structural members. CECS200 [16] recommends the use of f1.0, T as yield strength for structural fire safety design.

5.2.2.5 Elastic modulus, yield strengths, ultimate strength, and ultimate strain of high-strength steel at elevated temperature Tables 5.1
5.3 show the measured elastic modulus, yield strengths with different definitions, ultimate strength and ultimate strain at elevated temperatures up to 800 C for Q550, Q690, and Q890 steel, respectively. In general, the measured elastic modulus, yield strength, and ultimate tensile strength are decreased with the increase of the temperatures. There is no explicit correlation between the ultimate strain and the temperature, but for high temperature the ultimate strain is relatively larger than that at most of low temperatures.

5.2.3 Temperature-dependent elastic modulus and yield strength of high-strength steels To analyze and simulate the impact, fires and resulting collapse, the US National Institute of Standards and Technology (NIST), in the part of its report on the collapse of the World Trade Center [17], set up the true stress
strain models of structural steels by characterizing many grades of steel recovered from the buildings, namely the NIST steel stress
strain constitutive model [18]. The general NIST steel stress
strain model accounts for several factors, including temperature dependence of elastic modulus, yield strength, strain-hardening behavior, and strain rate sensitivity. The NIST steel stress
strain model described for temperature dependence of elastic modulus and yield strength herein, without considering the strainhardening behavior and strain rate effects, was used in curve fitting of the test data, which has the expression (5.1) and (5.2) as follows [19,20]: Table 5.1 Elastic modulus, yield strengths, ultimate strength, and ultimate strain for Q550 at elevated temperatures. T ( C)

ET (MPa)

f0.2,T (MPa)

f0.5,T (MPa)

f1.0,T (MPa)

f1.5,T (MPa)

f2.0,T (MPa)

fu,T (MPa)

εu,T (%)

20 200 300 400 450 500 550 600 700 800

214,459 211,190 206,166 195,955 178,243 177,007 147,701 127,236 87,430 32,782

651 631 587 537 503 430 367 282 93 27

651 631 592 545 509 441 373 283 95 28

692 720 675 623 573 471 382 271 91 29

706 747 704 642 586 472 377 258 88 29

715 760 719 651 586 469 369 248 86 28

747 778 763 656 588 473 383 285 96 29

17.36 14.74 17.65 16.45 16.76 20.37 33.37 44.25 87.06 172.67

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Behavior and Design of High-Strength Constructional Steel

Table 5.2 Elastic modulus, yield strengths, ultimate strength, and ultimate strain for Q690 at elevated temperatures. T ( C)

ET (MPa)

f0.2,T (MPa)

f0.5,T (MPa)

f1.0,T (MPa)

f1.5,T (MPa)

f2.0,T (MPa)

fu,T (MPa)

εu,T (%)

20 200 300 400 450 500 550 600 700 800

196,779 201,351 196,923 189,423 178,032 174,203 171,271 152,784 79,979 22,606

771 706 690 641 607 543 446 319 93 19

773 700 681 634 599 542 451 329 98 21

770 729 723 676 637 563 461 335 98 22

771 740 736 687 643 560 452 326 97 23

772 748 745 692 642 552 442 318 95 24

795 773 759 694 644 564 463 337 100 24

19.84 17.89 17.61 18.43 17.46 19.41 25.02 27.39 70.37 132.66

Table 5.3 Elastic modulus, yield strengths, ultimate strength, and ultimate strain for Q890 at elevated temperatures. T ( C)

ET (MPa)

f0.2,T (MPa)

f0.5,T (MPa)

f1.0,T (MPa)

f1.5,T (MPa)

f2.0,T (MPa)

fu,T (MPa)

εu,T (%)

20 200 300 400 450 500 550 600 700 800

204,976 204,400 201,032 183,367 175,592 174,246 150,692 132,509 60,189 23,209

1003 893 847 798 761 698 593 441 104 37

1002 855 803 753 712 668 563 435 110 39

1010 938 904 849 810 738 633 472 117 43

1021 965 933 875 831 750 638 473 118 44

1027 981 948 889 840 752 638 467 119 44

1057 1025 990 906 845 754 640 475 120 45

16.64 13.87 16.97 16.97 18.68 22.17 26.91 27.14 70.80 162.02



  ET 1 T 220 e1 1 T 220 e2 5 exp 2 2 2 e3 2 e4 E20

(5.1)



  fy;T 1 T 220 r1 1 T 220 r2 5 r5 1 ð1 2 r5 Þexp 2 2 2 r3 2 r4 fy;20

(5.2)

where e1 to e4 are parameters, determined by regression analysis of the test data on high temperature elastic modulus; r1 to r5 are parameters, determined by regression analysis of the test data on high temperature yield strength; and k1, k2, k3, k4, and n

Properties of high-strength steels at and after elevated temperature

159

Table 5.4 Values of the parameters in NIST model for curve fitting (elastic modulus and 0.2% offset strength).

e1 e2 e3 e4 r1 r2 r3 r4 r5

Q550

Q690

Q890

All

1.421 3.803 782 C 580 C 6.239 (4.963) 1.815 (6.866) 576 C (512 C) 729 C (1910 C) 0

8.308 2.684 656 C 1035 C 6.245 (5.953) 0.461 (0.3328) 547 C (541 C) 7921 C (184,000 C) 0

2.076 6.442 1044 C 620 C 8.031 (7.734) 0.758 (0.9486) 575 C (574 C) 1207 C (1359 C) 0

7.026 2.723 657 C 810 C 6.700 (5.98) 0.945 (1.168) 563 C (546 C) 1256 C (1932 C) 0

are parameters considering the strain-hardening behavior, determined by regression analysis of the measured stress
strain curves. The yield strength in the NIST model, fy,T is based on the 0.2% offset definition (f0.2). The reduction factor as shown in expression (5.2) and (5.3) is defined as the ratio of the high temperature property to the corresponding room temperature property. According to curve-fitting approach, the calibrated parameters for the NIST model (e1 to e4 for elastic modulus and r1 to r5 for f0.2) for different steels are shown in Table 5.4. The values for parameters r1 to r5 for 1.0% proof strength (f1.0) are also provided in the table for reference purpose. Fig. 5.5 shows the reduction factors for all three grades of steel tested in this program. Curve fitting of the dataset including all steels was also performed and the values of the curve-fitting parameters are reported in Table 5.4. As shown in Fig. 5.5A, the reduction factors of elastic modulus for the three steels were very similar at both low and very high temperatures. However, from 550 C to 700 C, somewhat larger differences among the three steels can be observed. For example, at 600 C, the difference between the reduction factors of elastic modulus for Q690 and Q550 was 0.18 (30%). As shown in Fig. 5.5B
F, the reduction factors of yield strength for different steels were similar. At low temperatures, the reduction factors of yield strength for different steels shows greater variation. For example, at 200 C, the difference of f2.0, T/ f 2.0,20 for Q550 and Q890 was 0.11 (11%). The calibrated NIST model fit well with the dataset of all steels for predicting reduction factors for elastic modulus and yield strength with temperature.

5.2.4 Comparative study Fig. 5.6 compares the results of reduction factors for elastic modulus from this study with the results from other researchers and from codes. Test data considered herein include those reported by Chen et al. [21] (marked as “BISPLATE80” in Fig. 5.6), Qiang et al. [22,23] (“S460N”and “S690”), Choi et al. [24] (“HSA800”), and Chiew et al. [4] (“SQ690L”). The values of reduction factors predicted by Maraveas et al. [25]. The nominal yield strength for S690, SQ690L, BISPLATE80,

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Behavior and Design of High-Strength Constructional Steel

Figure 5.5 Elevated temperature reduction factors for Q550, Q690, and Q890 steels for modulus of elasticity and for yield strength (f0.2, f0.5, f1.0, f1.5 and f2.0). (A) ET/E20. (B) f0.2,T/ f0.2,20. (C) f0.5,T/f0.5,20. (D) f1.0,T/f1.0,20. (E) f1.5,T/f1.5,20. (F) f2.0,T/f2.0,20.

and Q690 steels is 690 MPa, and the nominal yield strength for HSA800 is 650 MPa. Fig. 5.6A shows significant differences in reduction factors for elastic modulus among these different steels that have similar nominal yield strength. Fig. 5.6B shows that the various models for predicting elastic modulus reduction factors differ significantly from one another and from the test data in this program.

Properties of high-strength steels at and after elevated temperature

161

Figure 5.6 Comparison of reduction factors of elastic modulus predicted by others (A) and by codes (B) with the test data.

Figure 5.7 Comparison of reduction factors of yield strength (f2.0) predicted by others (A) and by codes (B) with the test data [12].

This observation is consistent with previous studies on other HSSs. The current models tend to underestimate the elastic modulus, which is also consistent with previous studies. Fig. 5.7 compares the results for reduction factors of yield strength (f2.0) from this study with the results from other researchers and from codes. Fig. 5.7A again shows considerable variation in reduction factor for yield strength among HSSs with similar nominal yield strength. Fig. 5.7B shows that there is also variations among various models, with the calibrated NIST model showing the best prediction of the experimental data.

5.3

Creep behavior of high-strength steels at elevated temperatures

5.3.1 Creep phenomenon and curves When exposed to elevated temperatures, steel structures undergo increasing permanent deformations with time even when the applied stress level is below that of

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Behavior and Design of High-Strength Constructional Steel

Figure 5.8 Creep response of steel at elevated temperature [26].

yield stress and this time-dependent deformation is referred to as creep. Thus creep can be defined as an increase in strain in a solid material under constant stress over a period. According to the previous studies, once the exposed temperature is beyond the 30% of the material melting temperature, the creep deformation is quite marked [26]. Dislocation glide, dislocation climb, or diffusional-flow mechanisms may generate creep deformation [27]. Typical strain
time curves obtained from creep tests at constant loading, applied over sufficiently long periods of time, often exhibit three characteristic stages as shown in Fig. 5.8. In stage 1: referred to as primary or transient creep; following setting-in of instantaneous elastic strain, ε0, the material deforms rapidly but at a decreasing rate. The duration of this stage is typically relatively short in relation to the total creep curve. The formation of this stage may be due to a process analogous to work hardening at lower temperatures [28]. In stage 2: referred to as secondary or steady-state creep, the creep strain rate reaches a minimum value and remains approximately constant over a relatively longer period. No material strength is lost during these first two stages of creep. The characterized “creep strain rate” typically refers to the constant rate in this second stage [29]. In stage 3: referred to as tertiary creep, the creep strain rate accelerates rapidly and attains ruptures when the material is unable to withstand the applied loading. In tertiary creep, the strain rate exponentially increases with stress because of the necking phenomena or internal cracks or voids decrease the effective area of the specimen. Strength is quickly lost in this stage while the shape of the material is permanently changed. The acceleration of creep deformation in the tertiary stage eventually leads to material fracture. In structural fire engineering, significance of creep deformations under fire conditions has been highlighted by various researchers [30-35]. In addition, recent studies have clearly shown that the extent of creep is influenced by the chemical composition of steel and this varies for different grades of steel [36], that is, different types of steel exhibit different creep response. Therefore it is very necessary to investigate into the creep effects on the performance of HSS in fire.

Properties of high-strength steels at and after elevated temperature

163

Figure 5.9 Test machine and instruments for steady-state test. (A) Test machine; (B) furnace; (C) extensometer; and (D) specimen with thermocouple [12,26,36].

Figure 5.10 Dimensions of steel coupon for high temperature creep test [26]. (A) Q460 steel specimen [38,39]. (B) Q550, Q690, and Q890 steel specimens.

5.3.2 Setup and specimens in creep test For the creep tests of HSSs at elevated temperature, steel coupons are generally installed into the grips of the MTS machine with furnace to create high temperature environment, as shown in Fig. 5.9. The shape and size of the coupons for creep tests are shown in Fig. 5.10, which were prepared in accordance with GB/T 228.1
2010 [37].

5.3.3 Creep test procedure In the creep tests, the electric furnace is tuned on to heat to the predetermined temperature at a heating rate of 15 C/min and then kept constant during the subsequent test. When the temperature reaches the predetermined temperature for 15 minutes

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Behavior and Design of High-Strength Constructional Steel

(to ensure the uniform temperature of the whole specimen), the specimen is loaded. After loading to the predetermined stress level, the retaining force remains constant (engineering stress remains unchanged) until the specimen fractures or the loading time exceeds 6 hours [38,39]. The stress level is defined as α5

σ f1;T

(5.3)

where, α is stress level (stress ratio), σ is actual stress applied on the specimen, f1;T is stress corresponding to the material strain of 1%. The measuring range of the extensometer is (15.5 mm) and therefore it is removed when the displacement exceeds 5 mm. It is considered that the subsequent deformation occurs at the cross-beam position located at the parallel direction of the specimen. The displacement of cross-beam section after the extensometer removing together with the displacement of extensometer can be obtained to determine the strain of the specimens. The details are as follow: εn1 5

Ln Lb

(5.4)

εn2 5

Lt Hn 2 Ht 1 Lb Lp

(5.5)

where εn1 is strain of the specimen before the extensometer removing, Ln is displacement of the extensometer at room temperature, Lb is gauge length of the specimen (50 mm), εn2 is strain of the specimen after the extensometer removing, Lt is displacement of the extensometer when the extensometer removing, Hn is displacement of the cross-beam after the extensometer removing, Ht is displacement of the cross-beam when the extensometer removing, Hp is total displacement of the parallel direction of the specimen, and Lp - original gauge length of the parallel direction of the specimen. However, the above measured strain includes three parts εmec, εth, and εcp which are contributed by the stress, high temperature, and creep effects, respectively. In the test, during the heating process and the constant temperature stage before loading, by controlling the experimental instrument, the stress of the specimen can be removed and therefore εth can be taken as 0. The stress-induced strain is taken as the strain at the completion of loading and therefore εmec 5 ε0 (ε0 is defined as the strain of the specimen at the end of loading process). Creep strain can be obtained by eliminating the strain caused by thermal strain and stress from the total strain as follows: εcp ðσ; T; tÞ 5 ε 2 εmec ðσ; TÞ 2 εth ðTÞ

(5.6)

Therefore εcp ðσ; T; tÞ 5 ε 2 ε0 ðσ; TÞ

(5.7)

Properties of high-strength steels at and after elevated temperature

165

Then the creep rate can be calibrated as follows: ε_ cp 5

dεcp dt

(5.8)

where ε_ cp is the creep rate at high temperature and εcp is the creep strain, t is time of period

5.3.4 Creep behavior of high-strength steels 5.3.4.1 Creep-time curves at various stress levels a. Creep-time curves—Q460 Steel

The measured typical creep strains of Q460 steel at various stress levels and at different temperatures, from 300 C to 900 C, are plotted in Fig. 5.11. Generally at a given temperature, the creep strain at a higher stress level is larger than that at a lower stress level. The general trend of creep response in lower temperatures (300 C
400 C), shown in Fig. 5.11A,B, can be grouped into two stages, namely primary (stage I) and secondary (stage II). The increase in secondary creep strain with time during stage II is dependent both on temperature and stress levels. At low stress levels, there is little secondary creep generated at 300 C and 400 C. However, at temperatures of 400 C and higher stress level of 476 MPa, the tertiary creep occurs. When the temperature is very high (500 C
900 C), the trend of creep response can be also generally grouped into two stages (Fig. 5.11C
G), namely secondary (stage II) and tertiary (stage III). The primary creep becomes very short and the deformation fully developed in very few minutes. The quick rise in creep strain with time in stage II is due to thermal softening. Secondary creep is highly important under fire exposure conditions because it dominates the creep response, occur in a constant rate, and over short period of time. In stage III, the tertiary creep increases at a faster rate due to reduced cross section of the specimen resulting from necking phenomenon which results in higher stresses for the same level of applied load. With the increase of temperature, the ductility of steel increases and maximum creep values become higher. The effect of temperature on creep response of Q460 steel is illustrated in Fig. 5.12. It can be seen from Fig. 5.12A
C, for temperatures range of 300 C
700 C, creep at a certain stress level increased significantly with temperature, even the elevation of temperature is only 50 C. For example, at stress level of 406 MPa, the creep rate at temperature of 450 C is much larger than that at temperature of 400 C. However, at temperature range of 800 C
900 C, the general trend observed above is not applicable, as is shown in Fig. 5.12D. The creep strain at 900 C is similar with that at 800 C under the same stress level. The trends in the figure indicate that creep deformation increases substantially with increasing temperature up to 800 C and then keep constant with the increase of temperature.

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Behavior and Design of High-Strength Constructional Steel

Figure 5.11 Creep strain response in Q460 steel at various stress levels [38,39]. (A) Creep strain at 300 C at various stress levels. (B) Creep strain at 400 C at various stress levels. (C) Creep strain at 500 C at various stress levels. (D) Creep strain at 600 C at various stress levels. (E) Creep strain at 700 C at various stress levels. (F) Creep strain at 800 C at various stress levels. (G) Creep strain at 900 C at various stress levels.

Properties of high-strength steels at and after elevated temperature

167

Figure 5.12 Creep strain response in Q460 steel at various temperatures [38,39]. (A) Stress levels of 406 and 457 MPa. (B) Stress levels of 38 and 178 MP. (C) Stress levels of 25.5 MPa. (D) Stress levels of 13, 19, and 32 MPa.

b. Creep-time curves—Q550 Steel

The measured typical creep strains of Q550 steel at various stress levels and at different temperatures, from 400 C to 800 C, are shown in Fig. 5.13 and Table 5.5, the development of creep strain of Q550 steel varies greatly under different temperature and stress levels. For example, the total creep strain can reach 103% when T 5 800 C and α 5 0.8, while the total creep strain is only 0.039% when T 5 400 C and α 5 0.4. At the same temperature and different stress levels, the creep strain is quite different. At 400 C, the total creep deformation at α 5 0.8 is 0.45%, which is 10 times of the total creep strain at α 5 0.4. At the same temperature, the total creep strain increases nonlinearly with the increase of stress ratio. Under the same stress ratio, the total creep will increase with the temperature in the range of 400 C
700  C. However, when the stress ratio is 0.4 and 0.6, the total creep strain at 800 C is smaller than that at 700 C. c. Creep-time curves—Q690 Steel

The measured typical creep strains of Q690 steel at various stress levels and at different temperatures, from 400 C to 800 C, are shown in Table 5.6 and Fig. 5.14. The development of creep strain of Q690 steel varies greatly under different

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Behavior and Design of High-Strength Constructional Steel

Figure 5.13 Creep strain curves of Q550 Steel [26]. (A) 400 C. (B) 550 C. (C)700 C. (D) 800 C.

Table 5.5 Steel Q550—creep strain at different temperature and stress level. α/T

400 C

550 C

700 C

800 C

0.4 0.6 0.8

0.039 0.116 0.451

1.406 7.500 42.353

2.964 9.271 60.121

2.115 8.600 103.232

Table 5.6 Steel Q690—creep strain at different temperature and stress level. α/T

400 C

550 C

700 C

800 C

0.4 0.6 0.8

0.021 0.051 0.219

0.624 3.698 27.520

6.138 22.486 83.359

4.468 20.553 119.015

Properties of high-strength steels at and after elevated temperature

169

Figure 5.14 Creep strain curves of Q690 Steel [26]. (A) T 5 400 C. (B) T 5 550 C. (C) T 5 700 C. (D) T 5 800 C.

temperature and stress levels. For example, the total creep strain can reach 120% when T 5 800 C and α 5 0.8, while the total creep strain is only 0.021% when T 5 400 C and α 5 0.4. At the same temperature and different stress levels, the creep strain is quite different. At 400 C, the total creep deformation at α 5 0.8 is 0.291%, which is 10 times of the total creep strain at α 5 0.4. At the same temperature, the total creep strain increases nonlinearly with the increase of stress ratio. Under the same stress ratio, the total creep will increase with the increase of temperature in the range of 400 C
700 C. However, when the stress ratio is 0.4 and 0.6, the total creep strain at 800 C is smaller than that at 700 C. d. Creep-time curves—Q890 Steel

The measured typical creep strains of Q890 steel at various stress levels and at different temperatures, from 400 C to 800 C, are shown in Fig. 5.15 and Table 5.7. The development of creep strain of Q890 steel varies greatly under different temperature and stress levels. For example, the total creep strain can reach 76% when T 5 800 C and α 5 0.8, while the total creep strain is only 0.032% when T 5 400 C and α 5 0.4. At the same temperature and different stress levels, the creep strain is quite different. At 400 C, the total creep deformation at α 5 0.8 is 0.229%, which

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Behavior and Design of High-Strength Constructional Steel

Figure 5.15 Creep strain curves of Q890 Steel [26]. (A) T 5 400 C. (B) T 5 550 C. (C) T 5 700 C. (D) T 5 800 C. Table 5.7 Steel Q890—creep strain at different temperature and stress level. α/T

400 C

550 C

700 C

800 C

0.4 0.6 0.8

0.0316 0.0844 0.229

0.686 2.904 25.335

3.957 15.693 78.471

3.080 18.688 75.785

is 10 times of the total creep strain at α 5 0.4. At the same temperature, the total creep strain increases nonlinearly with the increase of stress ratio. Under stress ratio of 0.4 and 0.8, the total creep will increase with the increase of temperature in the range of 400 C
700 C. However, when the stress ratio is 0.6, the total creep strain at 800 C is smaller than that at 700 C.

5.3.4.2 Creep rate curves and demarcation point for three stages of high-strength steels Fig. 5.16 shows the creep rate curves, with comparison of creep curves, of Q550, Q690, and Q890 steels at various elevated temperatures and stress levels. It can be

Properties of high-strength steels at and after elevated temperature

171

Figure 5.16 Creep rate curves and three demarcation point of high-strength steels [26]. (A) Q550 T 5 550 C, α 5 0.6. (B) Q550 T 5 800 C, α 5 0.8. (C) Q690 T 5 550 C, α 5 0.6. (D) Q690 T 5 550 C, α 5 0.8. (E) Q 890 T 5 550 C, α 5 0.6. (F) Q890 T 5 550 C, α 5 0.8.

seen that the creep rate in the intermediate stage is relatively constant, the creep rate in this stage can be approximately defined as the creep rate of the steels. The creep rate of Q550, Q690, and Q890 steels are listed in Tables 5.8
5.10 at various elevated temperatures and stress levels respectively. Obviously at the same temperature, with the increase of stress ratio, the creep rate increases nonlinearly. Under the same stress ratio, the creep rate also increases with the increase of temperature in the range of 400 C
700 C, but that at 800 C is closed to or even lower than that at 700 C.

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Behavior and Design of High-Strength Constructional Steel

Table 5.8 Steel Q550—creep rate (min21). α/T

400 C

550 C

700 C

800 C

0.4 0.6 0.8

7.07E-07 2.26E-06 9.21E-06

3.46E-05 3.03E-04 1.03E-03

7.44E-05 2.39E-04 9.30E-04

5.37E-05 2.28E-04 9.95E-04

Table 5.9 Steel Q690—creep rate (min21). α/T

400 C

550 C

700 C

800 C

0.4 0.6 0.8

4.33E-7 1.24E-6 3.92E-6

1.38E-5 9.04E-5 6.35E-4

1.78E-4 5.51E-4 1.41E-3

1.03E-4 3.93E-4 1.34E-3

Table 5.10 Steel Q890—creep rate (min21). α/T

400 C

550 C

700 C

800 C

0.4 0.6 0.8

7.26E-07 1.55E-06 4.43E-06

1.24E-05 6.66E-05 6.99E-04

1.04E-04 4.33E-04 1.38E-03

9.20E-05 4.93E-04 1.14E-03

Table 5.11 Demarcation point for three creep stages of Q550 steel (min). α/T

0.4 0.6 0.8

400 C

550 C

700 C

800 C

t1

t2

t1

t2

t1

t2

t1

t2

37 29 28






31 34 4



74

15 10 7



271

14 13 5



213

Notes: t1 demarcation point of stage 1 and 2, t2 demarcation point of stage 2 and 3.

According to the characteristics of creep rate curves, the demarcation point for three different creep stages of can be calibrated. The demarcation times of three creep stages of various HSSs at various elevated temperatures and stress levels are summarized in Tables 5.11
5.13.

5.3.4.3 Comparison of creep strain of Q550, Q690, and Q890 The comparison of the creep strains of three high-strength grades of Q550, Q690 and Q890 steels are shown in Fig. 5.17. It can be found that at 400 C, the creep phenomenon of the three steels is not obvious, and the maximum creep strain is less than 0.5%. Under different stress ratios, the creep development of Q550 steel is

Properties of high-strength steels at and after elevated temperature

173

Table 5.12 Demarcation point for three creep stages of Q690 steel (min). α/T

0.4 0.6 0.8

400 C

550 C

700 C

800 C

t1

t2

t1

t2

t1

t2

t1

t2

40 36.9 43






43.3 29.7 6



93.8

10 7.7 7.2


320 113

16.7 13 7.5


263 213

Table 5.13 Demarcation point for three creep stages of Q890 steel (min). α/T

0.4 0.6 0.8

400 C

550 C

700 C

800 C

t1

t2

t1

t2

t1

t2

t1

t2

42 37 36






45.5 41 6



89

9 9 5



142

19 13 4



154

faster than that of Q890 steel, and the creep development of Q690 steel is the slowest. At 550 C, the creep development process of Q690 steel is similar to that of Q890 steel at all stress levels, and the creep development rate is slower than that of Q550 steel. At high stress level, the creep tests of all three steels was terminated due to the fracture of the specimens. The fracture time of Q550 steel is approximately 125 minutes, and that of the other two steels is about 175 minutes. The total creep strain of Q550 steel is about 50%, and that of the other two steels is about 25%. At 700 C, the creep strain rate of Q690 steel is the fastest, higher than that of Q890 steel, and the creep rate of Q550 steel is the slowest at the same stress ratio. When the stress ratio is 0.8, among the three steels, the time of Q690 steel to reach failure is shortest, which is about 180 minutes, and that of Q550 steel is longest, which is about 330 minutes. In general, the creep strain rate of Q550 steel is fastest at lower temperature (550 C and below), and that of Q690 steel is quite similar to that of Q890 steel. At higher temperatures (700  C and above), the creep strain rate of Q690 steel is fastest, followed by that of Q890 steel, and that of Q550 steel is slowest.

5.3.5 Numerical creep models At present, there are a number of creep models proposed by previous researchers to evaluate the creep properties of steel at high temperatures according to their own research results. In this section, based on reviewing the existing creep models,

Figure 5.17 Comparison of creep strains of high-strength steels [26]. (A) T 5 400 C, α 5 0.4. (B) T 5 400 C, α 5 0.6. (C) T 5 400 C, α 5 0.8. (D) T 5 550 C, α 5 0.4. (E) T 5 550 C, α 5 0.6. (F) T 5 550 C, α 5 0.8. (G) T 5 700 C, α 5 0.4. (H) T 5 700 C, α 5 0.6. (I) T 5 700 C, α 5 0.8.

Properties of high-strength steels at and after elevated temperature

175

Figure 5.18 Burger’s model schematic diagram [26].

including Burger’s model [40
42], Norton model [43], Field & Field model [44], and ANSYS [45] built-in combined time hardening model with the creep test results of Q460, Q550, Q690, and Q890 steels, a new creep model, defined as three-stage high temperature creep model for HSS, is proposed as well.

5.3.5.1 Burger’s model Burger’s [40] model, also known as Rheological model, can be used to describe instantaneous creep and steady-state creep. The model takes the elastic strain ε1, viscoelastic strain ε2 and viscous flow ε3 into account. The schematic diagram is shown in Fig. 5.18. The model function is shown as Eq. (5.9).where, E2,η1,η2 are creep properties parameters required to be calibrated from test results. Burger’s model is used to fit the creep test results of Chinese Q550, Q690, and Q890 steels. Fig. 5.19 shows the fitting results of Burger’s model for Q550, Q690, and Q890 steels.

5.3.5.2 Norton’s model Norton [43] model was derived from the classical power law of creep (Norton
Bailey law). The classical power law of creep embodies the first and second creep periods (transient and steady-state creep) in a formula. The third creep period (tertiary creep) is not considered. The form of Norton
Bailey’s law is shown in Eq. (5.10).  εcp 5 σ

 1  t 1 2 eð2E2 =η1 Þt 1 E2 η2



εccp 5 c0 σc1 tc2 eð2cT =TÞ c1 . 1and0 , c2 # 1 where, T ( C) is temperature, c1,c2,c3 creep property parameters

(5.9) (5.10)

176

Behavior and Design of High-Strength Constructional Steel

Figure 5.19 Burger’s creep model curves [26]. (A) Q550 steel (B) Q690 steel. (C) Q890 steel.

Norton further develops Bailey
Norton’s law and puts forward Norton’s theory, which assumed that the intermediate stage (steady-state creep) rate of creep can be described by formula. The Norton model is shown in Formula (5.11). ε_cp 5 b1 σb2 expðb3 =TÞ

(5.11)

where, b1, b2, b3 is creep parameters need to be calibrated, b1 . 0 The set of Norton creep model curves for Q550, Q690, and Q890 steels according to the creep test results are shown in Figs. 5.20
5.22, respectively.

5.3.5.3 Field & field model In structural fire engineering, a commonly creep model is a power law model proposed by Field and Field [44]. Its basic form is shown in Formula (5.12). εcp 5 atb σc

(5.12)

where, t is time, σ is stress, a,b,c are creep property parameters depends on the temperatures

Figure 5.20 Norton’s creep model curves for Q550 steel [26]. (A) T 5 400 C. (B) T 5 550 C. (C) T 5 700 C. (D) T 5 800 C.

Figure 5.21 Norton’s creep model curves for Q690 steel [26]. (A) T 5 400 C. (B) T 5 550 C. (C) T 5 700 C (D) T 5 800 C.

178

Behavior and Design of High-Strength Constructional Steel

Figure 5.22 Norton’s creep model curves for Q890 steel [26]. (A) T 5 400 C. (B) T 5 550 C. (C) T 5 700 C. (D) T 5 800 C.

Figs. 5.23
5.26 show the set of Field & Field creep model curves for Q460, Q550, Q690, and Q890 steels according to the creep test results.

5.3.5.4 Combined time hardening model in ANSYS There are 13 creep models built in finite element computer program ANSYS, including Strain Harding, Time Harding and Combined Time Hardening models. The combined time hardening model is suitable for the first and intermediate stages of creep (transient and steady-state creep). This model is based on Zienkiewicz and Cormeau’s unified theory of plasticity and creep strain [45], without considering the coupling effect between time, stress or temperature. The expression of the composite time hardening model is shown in formula (5.13). εcp 5

c1 σcs 2 tc3 11 e2c4 =Ts 1 c5 σcs 6 te2c7 =Ts c3 1 1

(5.13)

where, t is creep time, Ts ( C) is temperature, c1Bc7 are creep parameters Employing the creep test results, the set of Combined Time Hardening creep model curves for Q460, Q550, Q690, and Q890 steels are calibrated in Figs. 5.27
5.30, respectively.

Figure 5.23 Field & Field creep model curves for Q460 steel [38,39]. (A) At temperature of 400 C. (B) At temperature of 550 C. (C) At temperature of 700 C. (D) At temperature of 800 C.

Figure 5.24 Field & Field creep model curves for Q550 steel [26]. (A) T 5 400 C. (B) T 5 550 C. (C) T 5 700 C. (D) T 5 8000 C.

180

Behavior and Design of High-Strength Constructional Steel

Figure 5.25 Field & Field creep model curves for Q690 steel [26]. (A) T 5 400 C. (B) T 5 550 C. (C) T 5 700 C. (D) T 5 800 C.

5.3.5.5 Three-stage creep model Based on previous creep studies, a three-stage creep model of steel at high temperature is proposed according to the test results of HSSs and creep behavior of the three stages (transient, steady-state and tertiary creep stages) [26]. This model fits for the creep behavior of steels at temperatures ranging from 400 C to 800 C. The effect of phase changes in microscopic structure of steels at 800 C on creep is considered as well. It is expressed by the following formulas. 1

Stage 1:εcp 5c1Uαc2UðT-Ta Þc3U t2 ð0 # t # t1 Þ Stage 2:εcp 5

ðt 


c4Uαc5U exp 2

t1

c6 T 2 Tb

(5.14)

 dt 1 ε1 ðt1 , t # t2 Þ

 c9 c10 Stage3:εcp 5 c7U α Uexp 2 2 1 ε2 ðt2 , tÞ T t 2 t2 1 c11

(5.15)




c8

(5.16)

where εcp is creep strain, α is stress ratio [at 800 C and should be adjusted due to the phase change in steel; m 5 0:9α 1 0:02 (Q550); m 5 0:75α 1 0:0833 (Q690);

Figure 5.26 Field & Field creep model curves for Q890 steel [26]. (A) T 5 400 C. (B) T 5 550 C. (C) T 5 700 C. (D) T 5 800 C.

Figure 5.27 Combined time hardening creep model curves for Q460 steel [38,39]. (A) At temperature of 400 C. (B) At temperature of 550 C. (C) At temperature of 700 C. (D) At temperature of 800 C.

Figure 5.28 Combined time hardening creep model curves for Q550 steel [26]. (A) T 5 400 C. (B) T 5 550 C. (C) T 5 700 C. (D) T 5 800 C.

Figure 5.29 Combined time hardening creep model curves for Q690 steel [26]. (A) T 5 400 C. (B) T 5 550 C. (C) T 5 700 C. (D) T 5 800 C.

Properties of high-strength steels at and after elevated temperature

183

Figure 5.30 Combined time hardening creep model curves for Q890 steel [26]. (A) T 5 400 C. (B) T 5 550 C. (C) T 5 700 C. (D) T 5 800 C.

m 5 0:625α 1 0:1117 (Q890)], T is temperature ( C), t is creep time (min), t1 is demarcation time between stage 1 and 2, t2 is demarcation time between stage 2 and 3, ε1,ε2 are creep strains at the end of stage 1 and stage 2 respectively, and Ta, Tb, c1Bc11 are creep parameters needed to be calibrated. The set of three-stage creep model curves for Q550, Q690, and Q890 steels calibrated with the creep test results are shown in Figs. 5.31
5.33, respectively. The calibrated three-stage creep model parameters at various elevated temperatures for Q550, Q690, and Q890 steels are listed in Tables 5.14
5.16, respectively.

5.4

Mechanical properties of high-strength steels after fire

5.4.1 Behavior of high-strength steel after fire If the peak temperature in fire exceeding the tempering temperature of steel when it was produced, the softening phenomenon may occur. So, the mechanical properties of steel, especially for HSS, may reduce significantly after fire, which has essential impact on the structural member and even the whole structure with HSS. The

Figure 5.31 Three-stage creep model curves for Q550 steel [26]. (A) T 5 400 C. (B) T 5 550 C. (C) T 5 700 C. (D) T 5 800 C.

Figure 5.32 Three-stage creep model curves for Q690 steel [26]. (A) T 5 400 C. (B) T 5 550 C. (C) T 5 700 C. (D) T 5 800 C.

Properties of high-strength steels at and after elevated temperature

185

Figure 5.33 Three-stage creep model curves for Q890 steel [26]. (A) T 5 400 C. (B) T 5 550 C. (C) T 5 700 C. (D) T 5 800 C.

Table 5.14 Three-stage creep model creep parameters for Q550 steel. c1 ( C21 min20.5)

c2

c3

c4 (min21)

c5

c6 ( C)

c7

1.31E-3 c8

3.053 c9 ( C)

2.46E-3 c11 (min)

3.400

25.52E2

4.191 Ta ( C) 399.97

22.527 Tb ( C)

25.474

3.28E-1 c10 (min) 8.50E2

1.05E2

400.542

Table 5.15 Steel Q690
three-stage creep model creep parameters for Q690 steel. c1 ( C21 min20.5)

c2

c3

c4 (min21)

c5

c6 ( C）

c7

1.1E-6 c8 4.81

2.64 c9 ( C) 3545.6

1.52 c10 (min) 10,064

0.03 c11 (min) 466.23

4.34 Ta ( C) 362.32

832.41 Tb ( C) 295.51

1.96E10

186

Behavior and Design of High-Strength Constructional Steel

Table 5.16 Three-stage creep model creep parameters for Q890 steel. c1 ( C21 min20.5)

c2

c3

c4 (min21)

c5

c6 ( C）

c7

2.225E-5 c8 3.546

2.571 c9 ( C) 1.996E3

1.025 c10 (min) 1.078E3

2.286 c11 (min) 117.849

3.973 Ta ( C) 391.078

4.287E3 Tb ( C) 43.226

5.149E3

reduction of the postfire mechanical properties of steel may be affected by cooling way, either cooling in air or in water. If steel subjected to fire cools in water, quenching effect may occur. So, postfire mechanical properties of steel will be different if cooling ways are different. The postfire stress
strain curves of Q550, Q690, and Q890 steels experienced various peak temperatures and cooled in air (CIA) or in water (CIW) are presented in Fig. 5.34. It can be seen the behavior of HSSs after fire is different with steel grades, peak temperatures experienced in fire and cooling ways. The postfire strength of HSSs may be significantly reduced, while the ductility may be, in general, increased. It can also be found that there is a yield plateau in the stress
strain curves of Q550 and Q690 steels cooled in air, while for that cooled in water the stress
strain curves exhibit yield plateau during the temperature range below 700 C, but no yield plateau above 700 C. As for the stress
strain curves of Q890 steel, either cooled in air or in water, they exhibit yield plateau below 700 C, but no yield plateau above 700 C. When the Q890 steel is heated below 600 C, the stress
strain curves obtain either from CIA case or from CIW case are quite similar to those at ambient, indicating that the effect of heating within 600 C followed by cooling process on the mechanical properties of Q890 steel is relatively negligible.

5.4.2 Mechanical properties of high-strength steels after fire Postfire mechanical properties of high-strength steel Q550, Q690, and Q890, including yield strength, ultimate strength, elastic modulus, ultimate elongation, and the reduction factors, calculated as the ratio of the test value of specimens after experiencing an elevated temperature T to the original value of specimens at ambient temperature, are presented in Tables 5.17
5.19, respectively. In these tables, fy,T (fy), fu,T (fu), ET (E) and εT (ε) are respectively the postfire yield strength, ultimate strength, elastic modulus and ultimate elongation of HSS after experiencing temperature T (at ambient temperature). So, fy,T/fy, fu,T/fu, ET/E, and εT/ε can be defined as the reduction factors for the postfire yield strength, ultimate strength, elastic modulus and ultimate elongation of HSS, respectively. The reduction factors of postfire yield strength, ultimate strength, elastic modulus and ultimate elongation of high strength Q550, Q690, and Q890 steels cooled in air and cooled in water are shown in Figs. 5.35
5.37, respectively. The yield

Properties of high-strength steels at and after elevated temperature

187

Figure 5.34 Postfire stress
strain curves of Q550, Q690, and Q890 structural steel experiencing various temperature levels [7]. (A) Q550—Cooled in air. (B) Q550—Cooled in water. (C) Q690—Cooled in air. (D) Q690—Cooled in water. (E) Q890—Cooled in air. (F) Q890—Cooled in water.

strength is defined as either lower boundary of yield plateau, or determined by the 0.2% offset method for no yield plateau case. If the reduction factor is greater than 1, for example, for the postfire ultimate elongation, it means the value after fire is increased to be greater than its original one. It can be seen that, whether cooled down in air or in water, HSS can both regain most of its nominal yield strength if exposed temperatures is below 600 C. Furthermore, the yield strength in the two cooling modes both follows a similar reduction trend when the experienced temperatures ranging from 600 C to roughly

Table 5.17 Values and reduction factors of postfire mechanical properties of Q550 steel. Parameter

fy,T/MPa

fy,T/fy

Cooling method

CIA

CIW

CIA

713 707 714 710 707 556 458 432 346

713 702 705 707 711 578 470 504 945

1.00 0.99 1.00 1.00 0.99 0.78 0.64 0.61 0.48

T/  C

20 200 400 500 600 700 750 800 900

fu,T/MPa

fu,T/fu

ET/Gpa

ET/E

εT/%

εT/ε

CIW

CIA

CIW

CIA

CIW

CIA

CIW

CIA

CIW

CIA

CIW

CIA

CIW

1.00 0.99 0.99 0.99 1.00 0.81 0.66 0.71 1.33

781 775 784 776 777 647 659 610 544

781 774 774 780 782 677 907 1030 1222

1.00 0.99 1.00 0.99 0.99 0.83 0.84 0.78 0.70

1.00 0.99 0.99 1.00 1.00 0.87 1.16 1.32 1.56

206 207 207 206 207 205 200 200 202

206 210 206 209 205 204 207 203 207

1.00 1.00 1.00 1.00 1.00 0.99 0.97 0.97 0.98

1.00 1.02 1.00 1.02 0.99 0.99 1.01 0.99 1.01

16.3 17.3 16.6 16.9 16.9 21.1 26.5 29.6 30.9

16.3 16.9 16.9 16.3 16.8 20.1 16.5 14.6 8.0

1.00 1.06 1.02 1.03 1.04 1.29 1.62 1.81 1.89

1.00 1.04 1.04 1.00 1.03 1.23 1.01 0.89 0.49

Table 5.18 Values and reduction factors of postfire mechanical properties of Q690 steel. Parameter

fy,T/MPa

fy,T/fy

Cooling method

CIA

CIW

CIA

803 865 852 788 658 518 476 370

803 841 856 782 672 528 473 966

1.00 1.08 1.06 0.98 0.82 0.64 0.59 0.46

T/  C

20 300 400 500 600 700 800 900

fu,T/MPa

fu,T/fu

ET/Gpa

ET/E

εT/%

εT/ε

CIW

CIA

CIW

CIA

CIW

CIA

CIW

CIA

CIW

CIA

CIW

CIA

CIW

1.00 1.05 1.07 0.97 0.84 0.66 0.59 1.20

869 886 884 821 727 585 577 928

869 874 884 825 738 603 913 1363

1.00 1.02 1.02 0.94 0.84 0.67 0.66 0.60

1.00 1.01 1.02 0.95 0.85 0.69 1.05 1.57

206 205 201 206 207 207 216 205

206 210 207 207 205 195 217 206

1.00 1.00 0.98 1.00 1.00 1.01 1.05 0.99

1.00 1.02 1.01 1.00 0.99 0.95 1.05 1.00

13.1 11.8 12.1 13.5 15.2 24.2 30.0 31.7

13.1 13.1 13.0 14.5 16.6 23.8 14.1 8.8

1.00 0.91 0.92 1.03 1.17 1.86 2.30 2.43

1.00 1.00 1.00 1.11 1.27 1.82 1.08 0.67

Table 5.19 Values and reduction factors of postfire mechanical properties of Q890 steel. Parameter

fy, T/MPa

fy, T/fy

Cooling method

CIA

CIW

CIA

1029 1040 1034 1035 1043 799 861 922

1029 1040 1035 1046 1045 801 1209 1204

1.00 1.01 1.00 1.01 1.01 0.78 0.84 0.90

T/  C

20 300 400 500 600 700 800 900

fu,T/MPa

fu,T/fu

ET/Gpa

ET/E

εT/%

εT/ε

CIW

CIA

CIW

CIA

CIW

CIA

CIW

CIA

CIW

CIA

CIW

CIA

CIW

1.00 1.01 1.01 1.02 1.02 0.78 1.18 1.17

1065 1075 1067 1071 1074 860 889 928

1065 1073 1071 1081 1077 863 1217 1362

1.00 1.01 1.00 1.01 1.01 0.81 0.84 0.87

1.00 1.01 1.01 1.01 1.01 0.81 1.14 1.28

210 211 212 213 214 210 210 209

210 212 212 214 208 213 212 212

1.00 1.01 1.01 1.02 1.02 1.00 1.00 1.00

1.00 1.01 1.01 1.02 0.99 1.02 1.01 1.01

12.8 13.0 12.9 13.0 13.3 16.8 19.2 11.9

12.8 12.7 13.2 13.7 13.7 17.4 11.2 9.8

1.00 1.02 1.01 1.02 1.04 1.32 1.50 0.94

1.00 0.99 1.04 1.07 1.08 1.36 0.88 0.76

190

Behavior and Design of High-Strength Constructional Steel

Figure 5.35 Reduction factors of postfire yield strength of high-strength steels [7]. (A) Q550 steel (B) Q690 steel. (C) Q890 steel.

Figure 5.36 Reduction factors of postfire ultimate strength of high-strength steels [7]. (A) Q550 steel (B) Q690 steel. (C) Q890 steel.

Properties of high-strength steels at and after elevated temperature

191

Figure 5.37 Reduction factors of postfire elastic modulus of high-strength steels [7]. (A) Q550 steel. (B) Q690 steel. (C) Q890 steel.

750 C, with approximately 2/3 residual yield strength retained at around 750 C. However, the postfire yield strength differentiate very much for cooled in air with cooled in water when the experienced temperatures exceeding about 750 C due to quenching effect. The postfire yield strength cooled in air keep decreasing for Q550 and Q690 steels, but regain for Q890 steel. However, the postfire yield strength cooled in water increases rapidly when in the experienced temperatures over 750 C, even greater than 1.2 times of the original yield strength if the exposed peak temperature up to 900 C. The reduction regulation of postfire ultimate strength of HSS is similar to that of postfire yield strength. The ultimate strength after fire is retained if exposed fire temperatures below 600 C but reduced when exposed fire temperatures ranging from 600 C to 700 C, whether cooled in air or in water. It is the same that the postfire ultimate strength cooled in air keep decreasing for Q550 and Q690 steels, but regain for Q890 steel when the exposed temperatures exceeding 700 C. However, the postfire ultimate strength cooled in water increases also rapidly after experiencing 700 C of fire, even greater than more than 1.3 times of the original ultimate strength if the exposed peak temperature up to 900 C.

192

Behavior and Design of High-Strength Constructional Steel

Obviously the postfire elastic modulus of high-strength steel keeps approximately unchanged with its original value, within 5% fluctuation, for both cooling modes. However, although the postfire ultimate elongation maintains approximately its original value when the heated temperature is below 600 C, the ultimate elongation of Q550, Q690, and Q890 steels after fire for CIA case keeps increasing up to more than 1.5 times of its original value when the heated temperature exceeds 600 C, except for Q890 steel to increase until 800 C and then decrease even to the value less than its original when the heated peak temperature exceeds 800 C. For CIW case, the postfire ultimate elongation of various grades of HSSs all increases in the range of the heated temperatures between 600 C and 700 C and then keep decreasing in the range of the heated temperatures between 700 C and 900 C, to at most 0.75 times of its original value. By fitting the measured results obtained in the tests, two sets of predictive equations were proposed for the reduction factors of postfire mechanical properties of high strength Q550, Q690, and Q890 steel cooled in air and cooled in water, as shown in Tables 5.20
5.22, respectively Fig. 5.38.

5.4.3 Mechanical properties of high-strength steel bolts after Fire 5.4.3.1 Specimens of high-strength bolts High-strength bolts are widely used for connection in steel structure construction. The mechanical properties of the typical 10.9 and 8.8 Grade bolts and raw materials at normal temperature are presented in Table 5.23. The mechanical properties of high-strength bolts after fire including stress
strain curves, strength and elasticity modulus of 10.9 and 8.8 Grade bolts can be Table 5.20 Proposed reduction factors of postfire mechanical properties of Q550 steel.  CIA 1; 20 C # T # 600 C fy;T 5 26 2 fy 2:905e T 2 0:006051T 1 3:575; 600 C # T # 900 C  CIW 1; 20 C # T # 600 C fy;T fy 5 1:065e27 T 3 2 0:0002186T 2 1 0:1469T 2 31:44; 600 C # T # 900 C  CIA 1; 20 C # T # 600 C fu;T 5 fu 1:453e26 T 2 2 0:003149T 1 2:361; 600 C # T # 900 C  CIW 1; 20 C # T # 600 C fu;T 5 26 2 fu 8:815e T 2 0:01122T 1 4:557; 600 C # T # 900 C CIA

ET E

51

CIW

ET E

5 1

CIA

εT/ε 5

CIW

εT/ε 5

1; 20 C # T # 600 C 2 4:564e T 1 0:00981T 2 3:205; 600 C # T # 900 C 26 2



1; 20 C # T # 600 C 8:201e28 T 3 2 0:0001981T 2 2 0:1551T 2 38:44; 600 C # T # 900 C

Table 5.21 Proposed reduction factors of postfire mechanical properties of Q690 steel.  CIA 1; T # 500 C fy;T 5 fy 1:693e26 T 2 2 0:003687T 1 2:42; 500 C , T # 900 C  CIW 1; T # 500 C fy;T 28 3 2 fy 5 5:78e T 2 0:0001095T 1 0:06651T 2 12:11; 500 C , T # 900 C  CIA 1; T # 400 C fu;T 27 2 fu 5 4:102e T 2 0:001356T 1 1:477; 400 C , T # 900 C  CIW 1; T # 500 C fu;T 28 3 25 2 fu 5 1:524e T 2 1:89e T 1 0:004909T 1 1:366; 500 C , T # 900 C CIA

ET E

51

CIW

ET E

5 1

CIA

εT/ε 5

CIW

εT/ε 5



1; T # 500 C 2 7:002e28 T 3 1 0:0001448T 2 2 0:09349T 1 20:29; 500 C , T # 900 C 1; T # 400 C 2 3:489e T 1 5:719e T 2 0:02874T 1 5:5796; 400 C , T # 900 C 28 3

25 2

Table 5.22 Proposed reduction factors of postfire mechanical properties of Q890 steel. CIA fy;T fy

CIW fy;T fy

CIA fu;T fu

CIW fu;T fu

8 > >
> : 0:0005912T 1 0:3637; 800 C , T # 900 C 8 1; T # 600 C > > < 2:328 2 0:002213T; 600 C , T # 700 C 5 0:003964T 2 1:996; 700 C , T # 800 C > > : 1:215 2 4:937e25 T; 800 C , T # 900 C 8 1; T # 600 C > > < 2:154 2 0:001920T; 600 C , T # 700 C 5 0:0002750T 1 0:6154; 700 C , T # 800 C > > : 0:0003670T 1 0:5418; 800 C , T # 900 C 8 1; T # 600 C > > < 2:140 2 0:001900T; 600 C , T # 700 C 5 0:003324T 2 1:516; 700 C , T # 800 C > > : 0:001367T 1 0:04856; 800 C , T # 900 C

CIA

ET E

51

CIW

ET E

51 8 1; T # 600 C > > < 0:003173T 2 0:9035; 600 C , T # 700 C 5 εT=ε 0:001871T 1 0:1947; 700 C , T # 800 C > > :   4:9136 2 0:005680T; 800 C , T # 900 C 8 1; T # 600 C > > < 0:003630T 2 1:178; 600 C , T # 700 C 5 4:743 2 0:004830T; 700 C , T # 800 C > > : 1:802 2 0:001150T; 800 C , T # 900 C

CIA εT ε

CIW εT ε

194

Behavior and Design of High-Strength Constructional Steel

Figure 5.38 Reduction factors of postfire ultimate elongation of high-strength steels [7]. (A) Q550 steel. (B) Q690 steel. (C) Q890 steel.

obtained through uniaxial tension experiments. The effects of fire exposure on high-strength bolts can be obtained through comparing the experimental results to the case of normal temperature. The engineering stress
strain curves and reduction factor of the elasticity modulus and strength of high-strength bolts after fire can be used to analyze the postfire load bearing capacity of connections. In order to simulate the situations in real fire, both natural cooling and water cooling can be considered in the experimental study. Both heating and cooling has a great effect on the mechanical properties of high-strength bolts after fire, and the effect of natural cooling is very different from that of water cooling.

5.4.3.2 Tests proceedure The following procedure has been adopted: G

G

Heat the bolt specimens up to the desired temperature at 20 C/min furnace temperature in electrical furnace and maintain the temperature for 60 minutes. The temperature is recorded by K type thermal couple. Cool the specimens to the ambient temperature using either natural cooling method which relies on the air only or water cooling method which puts the specimens in the water to

Table 5.23 Mechanical properties of high-strength bolts and raw materials. Grade and size of bolts

10.9 s, M20 3 120 8.8 s, M20 3 120

Raw material

20MnTiB 45# steel

Surface color

Black Black

Mechanical properties of raw material

Mechanical properties of bolts

Nominal tensile strength σ b(MPa)

Nominal yield strength σ 0.2(MPa)

Elongation δ 5 (%)

Shrinkage Ψ (%)

Capacity (kN)

Elongation δ 5 (%)

1190 655

990 385

13.0 24.0

58.0 46.0

281.5 $ 203

$ 12

G

G

provide forced cooling. Water cooling will cool specimens rapidly, so quenching in this procedure has a greater effect on the mechanical properties on high-strength bolts than what happened in practice when water is sprayed on the hot steel structures. Conduct uniaxial tension tests on the cooled specimens until failure occurs. The rate of loading is kept at 25 με/s. The test device of tension test is shown in Fig. 5.39. Two displacement meters were symmetrically fixed on two sides of bolt clamp. Obtain the stress
strain curve, strength and elasticity modulus using the methods given in GB/T228-2010 [37].

Totally 11 temperature points have been chosen which are 100 C, 200 C, 300 C, 400 C, 500 C, 600 C, 700 C, 750 C, 800 C, 850 C, and 900 C. Three specimens were tested at each temperature point. The high-strength bolts which have not exposed to fire were also tested for comparison.

5.4.3.3 Failure modes of high-strength bolts after fire The failure mode of some bolts is presented in Figs. 5.40
5.43. After heating and cooling the surface color of the bolts differs from the original color and is affected by the attained temperature and cooling method. This may be used during the postfire inspection to determine the highest temperature attained in the bolts.

Figure 5.39 Device of tension test of bolts [5].

Figure 5.40 Failure of 10.9 s high-strength bolts: natural cooling [5]. (A) Attained temp. 300 C. (B) Attained temp. 70 C. (C) Attained temp. 750 C, (D) Attained temp. 900 C.

Properties of high-strength steels at and after elevated temperature

197

Figure 5.41 Failure of 10.9 s high-strength bolts: water cooling [5]. (A) Attained temp. 200 C. (B) Attained temp. 300 C. (C) Attained temp. 750 C. (D) Attained temp. 900 C.

Figure 5.42 Failure of 8.8 s high-strength bolts: natural cooling [5]. (A) No fire exposure. (B) Attained temp. 400 C. (C) Attained temp. 700 C. (D) Attained temp. 800 C.

Figure 5.43 Failure of 8.8 Grade high-strength bolts: water cooling [5]. (A) Attained temp. 300 C. (B) Attained temp. 600 C. (C) Attained temp. 750 C. (D) Attained temp. 800 C.

5.4.3.4 Engineering stress
strain relationship of high-strength bolts after fire Stress
strain curves of 8.8 and 10.9 Grade bolts after fire are shown in Figs. 5.44 and 5.45, respectively. The stress
strain curves of bolts attained above 700 C have a softening effect at the beginning of loading. The reasons for this may be due to the shape phase change of the microstructures of steel at about 725 C. The lattice structure of steel is rearranged in process of loading.

198

Behavior and Design of High-Strength Constructional Steel

Figure 5.44 Stress
strain curves of 8.8 s high-strength bolts after fire. (A) 8.8 Grade boltsNatural cooling. (B) 8.8 Grade bolts-Water cooling.

Figure 5.45 Stress
strain curves of 10.9 Grade high-strength bolts after fire [5]. (A) 10.9 Grade bolts-Natural cooling. (B) 10.9 Grade bolts-Water cooling.

5.4.3.5 Reduction factors of high-strength bolts after fire Because there is no definite point on the stress
strain curves of bolts where elastic strain ends and plastic strain begins, the yield strength is defined as the point at which 0.2% plastic strain has taken place. The reduction factors of strength and elasticity modulus for 8.8 and 10.9 s high-strength bolts after exposed to different temperatures are given in Tables 5.24 and 5.25, respectively. The yield strength of grade 8.8 and 10.9 s bolt at elevated temperatures reported by Kirby [46] and Li [47] is also given in Fig. 5.46.

Table 5.24 Reduction factors of strength and elasticity modulus of 8.8 s bolts after fire. Attained temp. Tm ( C)

20 100 200 300 400 500 600 700 750 800 850 900

Yield strength σ0.2, 2 Tm (N/mm )

Tensile strength σb,Tm (N/mm2)

Reduction factor of yield strength σ0.2, Tm/σ 0.2

Reduction factor of tensile strength σ b,Tm/σb

Reduction factor of elasticity modulus ETm/E

Natural cooling

Water cooling

Natural cooling

Water cooling

Natural cooling

Water cooling

Natural cooling

Water cooling

Natural cooling

Water cooling

837.0 812.0 821.5 805.0 843.5 824.0 731.5 530.0 447.0 258.7 312.5 425.5

837.0 781.5 794.7 844.5 831.0 843.7 761.0 620.5 720.0 257.5 785.5 790.0

910.5 913.3 925.5 903.0 951.0 930.0 836.0 634.0 594.5 513.0 598.5 697.5

910.5 901.0 900.7 952.5 936.0 951.7 860.5 704.5 849.0 257.5 928.5 948.5

1.00 0.97 0.98 0.96 1.01 0.98 0.87 0.63 0.53 0.31 0.37 0.51

1.00 0.93 0.95 1.01 0.99 1.01 0.91 0.74 0.86 0.31 0.94 0.94

1.00 1.00 1.02 0.99 1.04 1.02 0.92 0.70 0.65 0.56 0.66 0.77

1.00 0.99 0.99 1.05 1.03 1.05 0.95 0.77 0.93 0.28 1.02 1.04

1.00 0.97
0.94 0.86 0.93 0.53 0.50 0.32 0.38 0.38 0.26

1.00 0.94 0.95 1.06 0.71 0.78 0.88 0.51 0.74 0.34 0.57 0.72

Table 5.25 Reduction factors of strength and elasticity modulus of 10.9 s bolts after fire. Attained temp. Tm ( C)

20 100 200 300 400 500 600 700 750 800 850 900

Yield strength σ 0.2, 2 Tm (N/mm )

Tensile strength σb,Tm (N/mm2)

Reduction factor of yield strength σ 0.2, Tm/σ 0.2

Reduction factor of tensile strength σ b,Tm/σb

Reduction factor of elasticity modulus ETm/E

Natural cooling

Water cooling

Natural cooling

Water cooling

Natural cooling

Water cooling

Natural cooling

Water cooling

Natural cooling

Water cooling

1129.5 1136.5 1162.7 999.0 1163.0 943.0 850.0 659.0 539.5 564.0 332.0 425.0

1122.5 1136.5 1115.0 1088.0 1096.0 953.0 812.0 647.3 869.5 972.0 1235.5 1120.3

1152.0 1157.5 1183.0 1016.5 1175.0 979.5 886.5 722.0 621.0 664.5 499.0 605.0

1152.0 1157.5 1191.0 1171.0 1168.0 1004.5 866.0 800.3 1233.5 1372.0 1409.0 1330.0

1.00 1.02 1.03 0.88 1.03 0.83 0.75 0.58 0.48 0.50 0.29 0.38

1.00 1.01 0.99 0.97 0.98 0.85 0.72 0.58 0.77 0.87 1.10 1.00

1.00 1.00 1.03 0.88 1.02 0.85 0.77 0.63 0.54 0.58 0.43 0.53

1.00 1.00 1.03 1.02 1.01 0.87 0.75 0.69 1.07 1.19 1.22 1.15

1.00 1.00 1.03 0.88 1.02 0.85 0.77 0.63 0.54 0.58 0.43 0.53

1.00 0.92 0.86 1.02 0.80 1.00 0.92 0.67 0.28 0.51 0.39 0.38

Properties of high-strength steels at and after elevated temperature

201

Figure 5.46 Reduction factors of strength and elasticity modulus of high-strength bolts after fire [5]. (A) 8.8 s high-strength bolts. (B) 10.9 s high-strength bolts.

5.4.4

Summary

The following conclusions can be summarized based on the test results: 1. When high-strength bolts have achieved temperatures not in excess of 400 C, the mechanical properties of the bolts cooling after fire are not significantly affected regardless of the cooling method. Almost 100% strength of the bolts is regained and the ductility of them is changed slightly. 2. When the temperature of high-strength bolts attained is between 400 C and 700 C, the stress
strain curves become softer and more ductile. With the attained temperature increasing, the strength and elastic modulus of the bolts cooling after fire reduce significantly. Furthermore, the different effects of natural cooling and water cooling on mechanical properties of high-strength bolts begin to appear, especially on the postyield stress
strain curves at the attained temperature of 700 C. 3. When the temperature of high-strength bolts attained is above 700 C, the cooling method appears to have a great effect on the mechanical properties of high-strength bolts after fire. After natural cooling, the stress
strain curves become softer and more ductile, and the strength is regained slightly. Whereas, after water cooling, the ductility reduces significantly, and with the attained temperature increasing the elasticity modulus remains decreasing with reduced rate, whereas the strength is regained to a great extent. For 8.8 s high-strength bolts subjected to water cooling, the postfire strength recovers to almost the same as the normal strength (but the postfire strength decreases sharply at the attained temperature of 800 C), while for 10.9 s high-strength bolts subjected to water cooling the postfire strength even exceeds the normal strength approximately 20% when the attained temperature is in excess of 800 C. In order to find out the reason, further observation of metallurgical structure should be carried out. 4. 400 C and 700 C are the key attained temperatures to the mechanical properties of highstrength bolts after fire.

202

Behavior and Design of High-Strength Constructional Steel

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9 [in Chinese]. [7] Li GQ, Lyu H, Zhang C. Post-fire mechanical properties of high strength Q690 structural steel. J Constr Steel Res 2017;132:108
16 (Complete). [8] Kirby BR, Preston RR. High temperature properties of hot-rolled, structural steels for use in fire engineering design studies. Fire Safety J 1988;13(1):27
37. [9] Qutinen J. Mechanical properties of structural steels at high temperatures and after cooling down. PhD thesis, Helsinki University of Technology, 2007. [10] Lee J, Morovat MA, Hu G, et al. Experimental investigation of mechanical properties of ASTM A992 steel at elevated temperatures. Eng J 2013;50(4):249
72. [11] GB/T 4338-2006. Metallic materials—tensile testing at elevated temperature. Beijing: China Standard Press; 2006. [12] Huang L, Li GQ, Wang XX, et al. Correction to: high temperature mechanical properties of high strength structural steels Q550, Q690 and Q890. Fire Technol 2018;. [13] EC3-1-2 Eurocode 3: design of steel structures—Part 1
2: general rules—structural fire design. British Standards Institution, 2005. [14] European recommendation for the fire safety of steel structure—calculation of the fire resistance of loadbearing element and structural assemblies exposed to the standard fire. European Convention for Constructional Steelwork (ECCS), Technical Committee —Fire Safety of Steel Structures, 1983. [15] BS 59508:1990 Structural use of steelwork in building part 8: code of practice for fire resistant design. British Standards Institution, 1990. [16] CECS200:2006. Chinese technical code for fire safety of steel structure in buildings. Beijing: China Planning Press; 2006. [17] Luecke WE, McColskey JD, McCowan CN, Banovic SW, Fields RJ, Foecke T, et al. Federal building and fire safety investigation of the world trade center disaster: mechanical properties of structural steel. Technical report NCSTAR 1-3D. Gaithersburg, Maryland: National Institute of Standards and Technology (NIST); 2013. [18] Luecke WE, Banovic SW, McColskey JD. High-temperature tensile constitutive data and models for structural steels in fire. NIST technical note 1714. Gaithersburg, Maryland: National Institute of Standards and Technology (NIST); 2011. [19] Choe L, Zhang C, Luecke WE, et al. Influence of material models on predicting the fire behavior of steel columns. Fire Technol 2017;53(1):375
400. [20] NIST, Temperature-dependent material modeling for structural steels: formulation and application. NIST technical note 1907, 2016, National Institute of Standards and Technology, Gaithersburg, Maryland.

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[21] Chen J, Young B, Uy B. Behavior of high strength structural steel at elevated temperatures. J Struct Eng 2006;132(12):1948
54. [22] Qiang XH, Bijlaard FSK, Kolstein H. Elevated-temperature mechanical properties of high strength structural steel S460N: experimental study and recommendations for fireresistance design. Fire Safety J 2013;55:15
21. [23] Qiang XH, Bijlaard F, Kolstein H. Dependence of mechanical properties of high strength steel S690 on elevated temperatures. Steel Constr 2012;30(30):73
9. [24] Choi IR, Chung KS, Kim DH. Thermal and mechanical properties of high-strength structural steel HSA800 at elevated temperatures. Mater Design 2014;63:544
51. [25] Maraveas C, Fasoulakis ZC, Tsavdaridis KD. Mechanical properties of high and very high steel at elevated temperatures and after cooling down. Fire Sci Rev 2017;6(1):3. [26] Wang XX. Research on Creep Behavior of High Strength Structural Steels over 500 MPa. Master’s Thesis. Tongji University, 2019 [In Chinese]. [27] Mu XY. Creep mechanics [in Chinese]. China: Xian JiaoTong University Press; 1989. [28] Courtney TH. Mechanical behavior of materials. Waveland Press; 2005. [29] Thermal Creep. NADCA Design. Retrieved 2017-03-29. https://www.diecastingdesign. org/thermal-creep [30] Morovat MA, Engelhardt MD, Helwig TA, Taleff EM. High-temperature creep buckling phenomenon of steel columns subjected to fire. J Fire Struct Eng 2014;5 (3):189
202. [31] Harmathy TZ. A comprehensive creep model. J Basic Eng 1967;89(3):496
502. [32] Fields BA, Fields RJ. Elevated temperature deformation of structural steel. Gaithersburg, MD: Report NISTIR 88-3899, NIST; 1989. [33] Kodur VKR, Dwaikat M. Effect of high temperature creep on the fire response of restrained steel beams. Mater Struct 2010;43(10):1327
41. [34] Kodur VKR, Dwaikat M. Response of steel beam-columns exposed to fire. Eng Struct 2009;31(2):369
79. [35] Kodur VKR, Aziz EM. Effect of temperature on creep in ASTM A572 high-strength low-alloy steels. Mater Struct 2014. Available from: https://doi.org/10.1617/s11527014-0262-2. [36] Li GQ, Wang XX, Zhang C. Experimental research on creep properties of domestic high strength steel Q690 at elevated temperatures. In: Proceedings of graduate student thesis of Civil Engineering Department at Tongji University, 2019 [In Chinese]. [37] Standardization Administration of the People’s Republic of China (SAC), Metallic materials—tensile testing—part 1: method of test at room temperature. GB/T 228.12010, 2010, China Standard Press, Beijing. [38] Wang W, Yan S. Creep behavior in high strength Q460 steel. J Tongji Univ Nat Sci 2016;44(6) [in Chinese]. [39] Wang W, Yan S, Liu J. Studies on temperature induced creep in high strength Q460 steel, Mater Structu 50(1), 2017,68. [40] Brnic J, Canadija M, Turkalj G, et al. Behaviour of S 355JO steel subjected to uniaxial stress at lowered and elevated temperatures and creep. Bull Mater Sci 2010;33 (4):475
81. [41] Findley WN, Lai JS, Onaran K. Creep and relaxation of nonlinear viscoelastic materials. New York: Dover Publications; 1989. [42] Brnic J, Canadija M, Turkalj G, et al. Structural steel ASTM A709-behavior at uniaxial tests conducted at lowered and elevated temperatures, short-time creep response, and fracture toughness calculation. J Eng Mech Asce 2010;136(9):1083
9.

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[43] Norton FH. The creep of steel at high temperatures. New York: McGraw-Hill Inc; 1929. [44] Field BA, Field RJ. The prediction of elevated temperature deformation of structural steel under anisothermal conditions. Gaithersburg: Departement of Commerce National Institute of Standards and Technology; 1989. [45] Zienkiewicz OC, Cormeau IC. Visco-plasticity-plasticity and creep in elastic solids—a unified numerical solution approach. Int J Numer Method Eng 1974;8:821
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4 [in Chinese].

Behavior and design of highstrength steel members under compression

6

Yan-Bo Wang, Guo-Qiang Li and Tian-Ji Li Tongji University, Shanghai, P.R. China

6.1

Introduction

High-strength steel (HSS) is especially useful for heavily loaded members such as columns in lower storeys of high-rise buildings. However, the current Chinese steel structures design code GB 50017-2003 [1] limits steel strength grade up to Q420 (nominal yield strength 420 MPa). HSS has been located in the research scope of the introduction of multiple column curves to the design codes, and column curves were selected for different combinations of shape, steel grade, bending axis, and manufacturing method [2
4]. European and American specifications for steel structures allow the use of HSS up to steel grades of S700 (700 MPa) and A514 (690 MPa) [5,6], but the current column curves for predicting the maximum strengths of centrally loaded columns were selected based on the available experimental and analytical studies of mild carbon steels usually with nominal yield strength from 235 to 345 MPa [4,7]. Bjorhovde and coworkers [2,3] have generated a total of 112 maximum strength column curves, which form the theoretic basis for the American column curves. Nevertheless, some combinations of shape and HSS were limited due to lack of data for residual stress distributions, since actually measured rather than assumed residual stresses were taken into account. Therefore tentative choices of columns curve for the steel grade A572 (Grade 65) with a nominal yield strength of 460 MPa have been made without an experimental confirmation. The determination of European column curves was based on the theoretical and experimental studies of Ref. [4]. Unlike Ref. [3], European Convention for Constructional Steelwork (ECCS) [4] assumed a compressive residual stress level of 10% of the yield strength for both box and H welded shapes of HSS in the absence of experimental confirmation. As a major factor influencing the buckling behavior of columns, emphases should be imposed on the effect of residual stresses on the maximum strength of HSS welded columns. Moreover, the assumption of the residual stress distributions for different shapes of HSS should be validated by actual measurements. Consequently, further work has been conducted to investigate whether the members fabricated from HSS can be designed according to the existing codes or whether the codes need to be modified to include HSS columns. Behavior and Design of High-Strength Constructional Steel. DOI: https://doi.org/10.1016/B978-0-08-102931-2.00006-1 © 2021 Elsevier Ltd. All rights reserved.

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Behavior and Design of High-Strength Constructional Steel

A lot of valuable studies have been done regarding the application of HSS, but studies on the overall buckling behavior and maximum strength of centrally loaded columns fabricated from HSS are not sufficient. The measurement of residual stresses in welded built-up A514 steel (nominal yield strength 690 MPa) shapes and the corresponding short and long columns tests were first carried out at Lehigh University [8,9]. Although the measured compressive residual stresses on A514 steel shapes were slightly higher than those on A7 (nominal yield strength 250 MPa) steel shapes, it was found that the residual stresses were less detrimental to the maximum strength of HSS columns than that of conventional carbon steel
fabricated columns compared on a nondimensionalized basis [8,9]. Rasmussen and Hancock [10] conducted a series of tests on the ultimate bearing capacity of HSS box and I-section columns (nominal yield strength 690 MPa). They compared the tests with column design strengths of the Australian steel structures standard AS4100. It was concluded that the higher curve (αb 5 20.5) was the appropriate curve for columns fabricated from flame-cut HSS plates with a nominal yield strength of 690 MPa. Shi and Bijlaard [11] carried out a series of finite element analyses on the buckling behavior of HSS welded H-section columns based on the experiments achieved by Rasmussen and Hancock [10]. It was found that the verified finite element model can predict the overall buckling behavior very well. The concern of this chapter is to select a curve for HSS welded box- and Hsection columns with a nominal yield strength of 460 and 690 MPa by experimental and parametrical investigations. Twenty-four specimens comprising box- and Hcolumns with slenderness ratios in the range of 30
80 were fabricated by flame-cut Q460 and Q690 steel plates and monitored until failure under axial compression via adopting ideally pin-ended supports. The ultimate bearing capacities and loaddeformation curves were obtained from the test and compared with the design columns strength of GB 50017-2003 and Eurocode3. Moreover, an extensively parametric analysis was carried out based on the experimentally verified nonlinear finite element model for welded flame-cut box- and H-section columns.

6.2

Material properties

Material tests were carried out to determine the material stress
strain characteristics of Q460 and Q690 steel columns. Tension coupons were cut from the 11 and 21 mm Q460 steel plates as well as the 16 mm Q690 steel plates. The cutting direction was perpendicular to the rolling direction according to the specifications of GB/T 29751998 [12]. The tension coupons were tested in accordance with the specifications of GB/T 228-2002 [13], and the results are summarized in Table 6.1. The mean values of the test specimens are given in Table 6.1 respectively, where Rp0.2 is the 0.2% proof stress, which is adopted as the yield strength of steel plates, Rm is ultimate tensile stress, E is Young’s modulus and Δ% is the percentage of elongation after fracture. Since the material properties of two 21 mm parent plates were different, the corresponding test specimens were marked with labels (1) and (2).

Behavior and design of high-strength steel members under compression

209

Table 6.1 Measured mechanical properties. Thickness Q460 11 mm Q460 21 mm (1) Q460 21 mm (2) Q690 16 mm

E (GPa)

Rp0.2 (MPa)

Rm (MPa)

207.8 217.6 217.6 233.5

505.8 540.9 464 772

597.5 617.6 585.9 826.2

Rp0.2/Rm 0.846 0.876 0.792 0.930

Δ (%) 23.7 29.0 30.4 21.0

Note: The material properties of two 21 mm parent plates were different, so the test results were denoted by 21 (1) and 21 mm (2), respectively.

6.3

Residual stresses in welded high-strength steel box sections and H-sections

It is well known that residual stresses exist in most structural steel members induced by welding, flame cutting, uneven cooling, or cold forming during processes of manufacture and fabrication. Although the internal equilibrium residual stresses are not detrimental to the resistances of cross sections of steel members, the presence of residual stress will significantly jeopardize the stiffness of compression members and shorten the fatigue life of steel members under periodical load or dynamic load. An effective and efficient pattern of residual stress is necessary for numerical modeling and practical design of steel structures, especially when using high-strength steel.

6.3.1 Sectioning method The sectioning method was adopted by lots of researchers after presented by Kalakoutsky in 1888 and is very popular for measuring residual stresses in structure steel members. It has proven itself adequate, accurate, and economical if proper care is taken in the preparation of the specimen and the procedure of measurement [14]. As most of the previous experimental results of residual stresses in box sections were measured by sectioning method, this method was also employed in this assessment (Fig. 6.1). The main steps of the sectioning method for the measurement of residual stress over the cross section of steel specimens can be described as follows: 1. Determination of sectioning locations and drilling holes The cutting lines and centers of gage holes were marked in the middle of specimens. Based on the marks, gage holes penetrating the full-thickness of the specimens were created by a punch. Then, the fillet angle of the holes was enlarged to 45 degrees, which is to guarantee the accuracy of testing results. All strips in the sectioning area were distinguished via a different serial number. 2. Measurement before sectioning After preparation of gage holes and before succeeding measurements, an air pump was used to clean the holes, which is essential for the subsequent measurements. As shown in Fig. 6.2, the gage lengths were initially measured and recorded by the Whittemore strain gage. The external surface of box-section specimens was measured, whereas the internal and external surfaces of H-section specimens were measured. Three sets of measurement

210

Behavior and Design of High-Strength Constructional Steel

Figure 6.1 Process of sectioning method.

Figure 6.2 Initial measurement. data for every gage length were recorded if the errors among them were less than 0.005 mm. To eliminate the deviations due to temperature variation, the gage length of a reference bar was gauged at the start and end of every measurement. 3. Sectioning specimens A sawing machine was used to cut the sectioning zones from the middle of the specimens, as shown in Fig. 6.3. Then, in line with the marked lines, the zones were sliced into strips by utilizing a wire cutting machine (see Fig. 6.4). To eliminate the influence of released heat due to the gasification of steel, a cooling fluid was applied in the sectioning process. The width of the strips on the weld was 14 (specimen R-B-7), 15 (specimen R-B10), and 12 mm (specimen R-B-13), while that of the other strips for the box-section specimens and the strips of the H-section specimens was approximately 10 mm. 4. Measurement after sectioning

Behavior and design of high-strength steel members under compression

211

Figure 6.3 Sectioning of measuring zone.

Figure 6.4 Sectioning with a wire cutting machine: (A) partial and (B) complete.

Completely sectioned strips are shown in Fig. 6.5. Before implementing a new round of measurements, an important step to clean the iron dust and grease around the gage holes was carried out. Three sets of readings were recorded again according to the measurement procedure conducted before sectioning. The calculation of the released strains needs to consider the exclusion of the temperature strains from the measured results. If bending deformation occurs in the sectioned strips, the deflection measurement is essential as well. Some strips curved after sectioning.

6.3.2 Assessment of residual stresses in welded Q460 steel sections 6.3.2.1 Specimens The test specimens were fabricated from Q460 steel (nominal yield strength 460 MPa) plates of 11 mm in thickness which were made in China. The original plate was flame cut into small component plates. Four component plates were welded together to form a box-section specimen by manual gas metal arc welding

212

Behavior and Design of High-Strength Constructional Steel

Figure 6.5 Sectioned strips.

Figure 6.6 Definition of symbols: (A) box section and (B) H-section.

(GMAW), as given in Fig. 6.6A. Two flange component plates with a nominal thickness of 21 mm and one web component pate with a nominal thickness of 11 mm were welded together to form an H-section, as shown in Fig. 6.6B. As current practice does not employ complete penetration welding for columns except in beam-to-column connection region, the component plates were connected by incomplete penetration welding. The electrode was ER55-D2 with the same nominal yield strength of Q460 steel. In order to reduce the shrinkage deformations caused by welding heat, an optimized welding sequence was adopted. The welding parameters used in the fabrication were presented in Table 6.2. The measured geometric dimensions of three box-section specimens and three H-section specimens were shown in Tables 6.3 and 6.4, respectively. The symbols in Tables 6.3 and 6.4 are illustrated in Fig. 6.6. In order to reduce end effects, the specimens must be

Behavior and design of high-strength steel members under compression

213

Table 6.2 Welding parameters. Current (A) 190
195

Voltage (V)

Velocity (mm/s)

28
30

2.3

Table 6.3 Actual dimensions. Specimen R-B-8 R-B-12 R-B-18

D (mm)

b (mm)

t (mm)

b/t

L (mm)

110.9 156.5 219.8

88.2 133.6 196.9

11.40 11.44 11.42

7.7 11.7 17.3

840 1020 1280

Table 6.4 Actual dimensions. Specimen

D (mm)

H (mm)

tf (mm)

tw (mm)

d/tf

h/tw

L (mm)

R-H-3 R-H-5 R-H-7

156.00 225.25 314.00

168.00 243.75 319.50

21.39 21.23 21.20

11.49 11.33 11.63

3.4 5.0 7.1

10.9 17.8 23.8

1550 1300 1060

long enough to provide a distance of two times the lateral dimension from the ends to the test regions. Hence, the sum of four times D and 400 mm was adopted for the length of each specimen, as shown in Table 6.3. It is reasonable to begin with a semidestructive method to avoid the interaction of two test regions. So the sectioning method was conducted after the hole-drilling method had been finished.

6.3.2.2 Measured residual stress Based on the difference in the length of each strip measured before and after sectioning, the residual stresses were calculated by multiplying the released strains by Young’s modulus. Figs. 6.7
6.12 show the residual stress distributions of different sections. The solid symbols in Figs. 6.7
6.9 present the residual stresses in component plates I and III. The hollow symbols in Figs. 6.7
6.9 present the residual stresses in component plates II and IV. Since the inappropriate preparation of original gage lengths, the gray symbol in Fig. 6.8 was estimated value according to the distributions of plates I and III. The other two gray symbols in Fig. 6.9 were the linear interpolations of neighbor strips instead of invalid value. Using the measured residual stresses, the equilibrium condition for the whole sections was checked. Theoretically, since no external forces exist, equilibrium requires that the integration of the stresses over the whole section must be zero. But in fact, a difference of 3.3 MPa in compression is computed for R-B-8, 4.9 and 2.8 MPa in compression for R-B-12 and R-B-18, respectively. These small differences may be attributed to the effect of saw cutting and accumulated experimental errors [14].

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Figure 6.7 Residual stress distribution in R-B-8 by sectioning method.

Figs. 6.10
6.12 show the residual stress distributions of sections R-H-3, R-H-5, and R-H-7. The solid dots in Figs. 6.10
6.12 represent the residual stresses in the outside surface of flange plates or in the right surface of web plates, while the hollow dots represent residual stresses in the inside surface of flange plates or in the left surface of web plates. Because of the inappropriate preparation of gage holes in some strips of section R-H-7, the original gage lengths exceeded the full-scale value of the Whittemore strain gage. The linear interpolation results of neighbor strips are shown in Fig. 6.12 as gray dots instead of invalid values.

6.3.3 Assessment of residual stresses in welded Q690 steel sections 6.3.3.1 Specimens Six specimens including box and H-sections (Fig. 6.13) were fabricated from flame-cut steel plates (nominal yield strength, 690 MPa) via GMAW. The gauge

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215

Figure 6.8 Residual stress distribution in R-B-12 by sectioning method.

length was 200 mm on the test specimens. To eliminate the end effects, the length of specimens was the sum of 4B and 200, in which B is the lateral dimension of cross section, although the theoretical value is 2.0 times the dimension. Full penetration welding was used to connect component plates. The filler wire used was ER120S-G with the same nominal yield strength of the base material. To diminish shrinkage deformations, an optimal welding sequence (i.e., antisymmetric welding sequence) was adopted. The welding parameters used in the welding process are listed in Table 6.5, and the measured geometrical dimensions of the specimens are shown in Tables 6.6 and 6.7.

6.3.3.2 Measured residual stresses Magnitudes of residual stress on box and H-sections can be determined by the application of Hooke’s law and measured strain data. For the bowed strips the measurement data needs to be modified. The specific procedure refers to the research presented by Cruise and Gardner [15]. Fig. 6.14 indicates the residual stress distribution on the exterior surface of the box-section specimens. Fig. 6.15 shows the residual stress distribution on the exterior and internal surfaces of H-section specimens. In the figure the solid block describes the stress distribution on the external

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Behavior and Design of High-Strength Constructional Steel

Figure 6.9 Residual stress distribution in R-B-18 by sectioning method.

Figure 6.10 Residual stress distribution in R-H-3 by sectioning method.

Behavior and design of high-strength steel members under compression

Figure 6.11 Residual stress distribution in R-H-5 by sectioning method.

Figure 6.12 Residual stress distribution in R-H-7 by sectioning method.

217

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Behavior and Design of High-Strength Constructional Steel

Figure 6.13 Sections of specimens: (A) box section and (B) H-section.

Table 6.5 Welding parameters. Diameter (mm) ϕ1.2

Type

Current type

Gas composition

Flow rate (L/min)

Electric current (A)

Volts (V)

Travel speed (cm/min)

Semiauto

DCEP

80%Ar 1 20%CO2

15
20

260
290

28
30

25
35

Table 6.6 Measured dimensions of box section. Specimen label

B (mm)

h (mm)

t (mm)

h/t

L (mm)

R-B-7 R-B-10 R-B-13

141.21 192.04 235.64

108.83 159.78 203.04

16.19 16.13 16.30

6.7 9.9 12.5

802 1009 1187

Table 6.7 Measured dimensions of H-section. Specimen

B

bf

tf (tw)

H

h

bf/

h/

L

label

(mm)

(mm)

(mm)

(mm)

(mm)

tf

tw

(mm)

R-H-6

209.03

96.40

16.22

206.30

173.86

5.9

10.7

1075

R-H-7

240.25

112.07

16.11

239.63

207.42

7.0

12.9

1202

R-H-8

261.60

122.67

16.27

258.40

225.87

7.5

13.9

1283

(left) surface, and the internal (right) surface. Whittemore strain gage flange and web because

hollow block represents the stress distribution on the For the H-section specimens, measurement by using a cannot be conducted near the conjunction between the this conjunction space is too narrow. However, the stress

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219

Figure 6.14 Experimental results of box section: (A) R-B-13, (B) R-B-10, and (C) R-B-7.

levels in these locations can be calculated via sectional self-equilibrium conditions under residual stresses.

6.3.4 Simplified residual stress model for welded Q460 steel sections The residual stresses on welded sections are caused by the nonuniform thermal strains during welding. The width of the tensile zone is almost unaltered for boxand H-section specimens with different plate slenderness ratios. This phenomenon can be explained via a thermal envelope model proposed by Barroso et al. [16]. rffiffiffiffiffi 2 qw T 2 T0 5 πe 2hcρx

(6.1)

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Behavior and Design of High-Strength Constructional Steel

Figure 6.15 Experimental results of H-section: (A) R-H-8, (B) R-H-7, and (C) R-H-6.

where qw denotes the heat input per unit welded length, c is the specific heat, ρ represents the density of steel products, T0 denotes the initial temperature, h is the thickness, and x is the distance to the weld bead. Based on Eq. (6.1), the deduction that various sections possess the same heat envelope curve around weld beads can be obtained if the identical heat input per unit welded length, specific heat, density, and thickness of plates are adopted. The conclusion indicates that the width of the residual tension zones is approximately identical for each section with diverse plate slenderness ratios.

6.3.4.1 Box sections A simplified residual stress distribution was given based on the results of sectioning method, as plotted in Fig. 6.16, where w is the width of tensile residual stress in plates I and III, α and β are already given in Table 6.8. So w can be obtained by applying equilibrium condition.

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221

Figure 6.16 Simplified residual stress pattern (unit: mm).

Table 6.8 Ratios of residual stresses. Specimen

α 5 σrt =Rp0:2

β 5 σrc =Rp0:2

R-B-8 R-B-12 R-B-18

0.555 (0.874) 0.678 0.787

2 0.255 (20.253) 2 0.195 2 0.142

6.3.4.2 H-sections Unlike the compressive residual stresses presented in the flange tips of H-sections fabricated from guillotined plates, H-sections manufactured from flame-cut plates are characterized by tensile residual stress at the flange tips. This favorable residual stress distribution could result in higher ultimate capacity for welded H-section columns, especially for those under bending about weak axis. The measured residual stresses by sectioning method have the same pattern as the typical residual stress pattern of flame-cut welded H-sections. The large differences in residual stresses between outside (right) and inside (left) surfaces of the flanges (webs) indicate that the sectioned strips were bent in the longitudinal direction. These bending stresses are usually induced by the sectioning procedures [15]. In order to obtain the membrane stresses, residual strains on opposite sides were averaged to calculate the effective residual stress, based on the assumption that the residual stresses are

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uniformly distributed through thickness. If the thicknesses of all component plates are less than 25 mm, the variation of residual stresses across thickness is recognized as negligible [17]. Therefore the assumption is applicable. The mean values of residual stresses are shown in Table 6.9 and a corresponding simplified residual stress pattern for H-sections fabricated from flame-cut plates is proposed based on the results of sectioning method as shown in Fig. 6.17. In Table 6.9 and Fig. 6.17, αi and β i are the ratios of tensile and compressive residual stresses over yield strength of base metal, respectively, and the subscribe letter i indicates the location of residual stress block. Since the measurement in blocks α1 were not available, the tensile residual stress ratios α1 were obtained by applying equilibrium condition in the measured residual stress distributions. It is founded by Young and Robinson [18] that the rectangular shape of tension block will result in more conservative column strength than the triangular and trapezoidal shapes. In addition, for the convenience of introducing residual stresses in numerical models, the rectangular shape was adopted in this simplified residual stress pattern.

Table 6.9 Ratios of residual stresses. Specimen R-H-3 R-H-5 R-H-7

α1

α2

β1

β2

1.039 0.900 0.731

0.080 0.243 0.488

2 0.408 2 0.271 2 0.195

2 0.152 2 0.235 2 0.131

Figure 6.17 Simplified residual stress pattern.

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223

6.3.5 Simplified residual stress model for welded Q690 steel sections 6.3.5.1 Box sections A simplified residual stress model for the box sections is illustrated in Fig. 6.18, which is based on measurement results. In the model, α and β are tensile and compressive residual stress ratios listed in Table 6.10. And the width of the tensile zone, w, can be determined by equilibrium conditions.

6.3.5.2 H-sections Fig. 6.19 shows the residual stress model of the H-sections fabricated by flame-cut plates. In Table 6.11 and Fig. 6.19, αi and β i denote the ratios of residual tensile

Figure 6.18 Residual stress model of box section. Table 6.10 Residual stress ratios of box-section specimens. Specimen label R-B-7 R-B-10 R-B-13

α

β

0.394 0.445 0.496

2 0.137 2 0.126 2 0.119

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Behavior and Design of High-Strength Constructional Steel

Figure 6.19 Residual stress model of H-section. Table 6.11 Residual stress ratios of H-section specimens. Specimen label R-H-6 R-H-7 R-H-8

α1

α2

β1

β2

0.432 0.311 0.286

0.06 0.101 0.07

2 0.136 2 0.078 2 0.103

2 0.027 2 0.063 2 0.012

and compressive stresses about the realistic yield stress, where the subscript i represents the different locations of residual stress blocks.

6.4

Behavior of high-strength steel columns

6.4.1 Experiment program In order to study the behavior of axially compressed HSS columns with steel grades of Q460 and Q690, 24 specimens with various width-to-thickness ratios and different column slenderness were manufactured from flame-cut Q460 and Q690 steel plates. Q460 and Q690 steel with the nominal yield strength of 460 and 690 MPa are equivalent to S460 and S690 in EN 10025. The columns were loaded concentrically between two pinended supports with a hydraulic jack with a capacity of 10,000 kN. Due to the thick component plates and large cross sections, the maximum strength of the specimens was over 9000 kN. Two curved surface supports, as shown in Fig. 6.20, were fabricated and used at both ends of each test specimen. For the H-section columns, the buckling axis was set

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225

as the weak axis. The curved surface supports are free to rotate about the weak axis as the perfect hinged connection while are fixed about strong axis of cross section.

6.4.1.1 Test specimen data and fabrication procedure Flame-cut component plates were welded together to form box- and H-section specimens by manual GMAW, as shown in Fig. 6.21. The electrode ER55-D2 and ER120S-G were adopted for the welding of Q460 and Q690 steel column specimens. As current practice does not employ complete penetration welding for columns except in beam-to-column connection zone, the component plates were connected by incomplete penetration welding except the 500 mm length from each end. In order to reduce the shrinkage deformations caused by welding heat, an optimized welding sequence was adopted. The measured geometric dimensions of six test specimens are shown in Tables 6.12 and 6.13. Symbols in tables are illustrated

Figure 6.20 Curved surface support.

Figure 6.21 Definition of symbols.

Table 6.12 Measured dimensions of specimens. Steel grade

Specimen

D (mm)

t (mm)

Q460

B-8-80-1 B-8-80-2 B-12-55-1 B-12-55-2 B-18-38-1 B-18-38-2 B-30-1

110.3 112.0 156.5 156.3 220.2 220.8 236.23

11.40 11.49 11.43 11.42 11.46 11.46 16.20

B-30-2 B-50-1 B-50-2 B-70-1 B-70-2

236.47 192.37 192.52 140.88 140.48

16.10 16.02 16.02 16.07 16.08

Q690

d/t

L (mm)

Lt (mm)

A (mm2)

7.7 7.8 11.7 11.7 17.2 17.3 12.6

3000 2940 2940 2940 2940 2940 2501

3320 3260 3260 3260 3260 3260 2811

12.7 10.0 10.0 6.8 6.7

2502 3300 3302 3300 3299

2812 3610 3612 3610 3609

λ

λn

4505 4618 6633 6617 9565 9594 14,258

81.7 78.9 54.9 55.0 38.2 38.1 31.2

1.283 1.240 0.862 0.863 0.600 0.598 0.571

14,192 11,301 11,310 8023 8001

31.2 50 49.9 70.3 70.5

0.571 0.915 0.913 1.287 1.29

Table 6.13 Measured dimensions of specimens. Steel grade

Specimen

B (mm)

H (mm)

tw (mm)

tf (mm)

L (mm)

Lt (mm)

A (mm)

λy

λyn

Q460

H-3-80-1 (1) H-3-80-2 (1) H-5-55-1 (2) H-5-55-2 (1,2) H-7-40-1 (1) H-7-40-2 (1)

154.5 154.7 227.75 229.0 308.75 308.25

171.25 171.25 245.75 245.5 317.25 318.5

11.52 11.36 11.54 11.62 11.47 11.46

20.99 20.98 21.33 21.15 21.03 21.20

3000 2984 3000 3000 3000 3000

3320 3304 3320 3320 3320 3320

7976 7959 12,058 12,046 16,140 16,230

82.5 81.9 56.2 56.0 41.5 41.6

1.301 1.291 0.834 0.857 0.655 0.656

Q690

H-30-1 H-30-2 H-50-1 H-50-2 H-70-1 H-70-2

260.85 260.82 241.75 240.47 209.21 209.38

259.19 260.35 236.3 238.15 204.78 205.24

16.08 16.25 16.03 16.16 16.26 16.24

16.08 16.25 16.03 16.16 16.26 16.24

1701 1700 2602 2601 3201 3202

2011 2010 2912 2911 3511 3512

12,040 12,179 11,024 11,098 9605 9606

32 32 49.7 50 69 69

0.586 0.586 0.91 0.915 1.263 1.263

Note: L is the clear column length; Lt is the total length of the specimen between the two pinned-supports; A is the area of cross section; λy 5 Lt/ry is column slenderness about the weak axis; λyn is nondimensional slenderness about the weak axis. Because the material properties of two 21 mm parent plates were different, the corresponding test specimens were marked with (1) and (2), as shown in Table 6.13. For example, the two flange plates of specimen H-3-80-1 (1) are both fabricated from type (1) 21 mm plates; the top flange plate of specimen H-5-55-2 (1,2) is fabricated from type (1) 21 mm plates while the bottom flange plate is fabricated from type (2) 21 mm plates.

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Behavior and Design of High-Strength Constructional Steel

in Fig. 6.21. The specimens were designed to investigate the behavior of welded HSS columns under axially loading. Thus plate slenderness ratios of all specimens were designed to prevent the occurrence of local buckling. In Chinese code for design of steel structures GB 50017-2003 [1]: For plates supported along one longitudinal edges

sffiffiffiffiffiffiffiffi b 235 ; # ð10 1 0:1λÞ tf fy

30 # λ # 100

(6.2) For plates supported along both longitudinal edges

h # ð25 1 0:5λÞ tw

sffiffiffiffiffiffiffiffi 235 ; fy

30 # λ # 100

(6.3) where λ is the column slenderness ratio, fy is the nominal yield strength, which is equal to 460 or 690 MPa herein. According to λ, the plate slenderness limits are varying from 9.3 to 14.2 for Q460 steel outstand flanges (7.6
11.7 for Q690 steel flanges) and that are ranging from 28.6 to 53.6 for Q460 steel-encased webs (23.3
43.8 for Q690 steel webs). In the European code for design of steel structures Eurocode3 [19]: sffiffiffiffiffiffiffiffi b 235 For plates supported along one longitudinal edges (6.4) # 14 tf fy For plates supported along both longitudinal edges

h # 42 tw

sffiffiffiffiffiffiffiffi 235 fy

(6.5)

If the nominal yield strength fy is assumed equal to 460 MPa, the plate slenderness limits of Class 3 section will be 10.0 for Q460 steel outstand flanges (8.2 for Q690 steel flanges) and 30.0 for Q460 steel webs (24.5 for Q690 steel webs). The value ranges of width-to-thickness ratio and height-to-thickness ratio selected in Tables 6.12 and 6.13 aim to cover generally and extreme cases.

6.4.1.2 Test setup and test procedures All the specimens were tested under concentric loading with a 10,000 kN universal testing machine at Tongji University. The photographs of the test setup are illustrated in Fig. 6.22. Both the bottom and top supports were set to fixed about strong axis and pin-supported about weak axis. The arrangement of linear varying displacement transducers (LVDTs) and strain gauges is shown in Fig. 6.23. The axial deformations of the specimens were measured by V1 and V2 LVDTs. LVDT H01
H03 were placed at the mid-length of the column to record the in-plane lateral deflection. The out-ofplane lateral deflections were captured by LVDT H06
H08. Strain gauges were attached at the mid-height cross section of the specimens to monitor the loading force and verify the measured geometric imperfections. The real-time load
deflection and load
shortening curves displayed in monitor during the entire loading processes were used to adjust and govern the experiments.

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229

Figure 6.22 Specimens test set-up.

Before the actual test a preload with 10% of the predicted maximum column strength was used to check the test instrumentations and then unloaded. In the actual test, the axial load was applied on the column at a rate of 1 mm/min until the peak load was reached. Then the load rate was increased in the postpeak range. Finally, the test specimen was unloaded to finish the test procedure when the test load decrease to 60% of the peak load. The observed failure modes of the specimens, with member slenderness ratios from 40 to 80, are all identified as the overall buckling of the whole columns, as shown in Fig. 6.24. Local buckling is not observed before the peak load. The out-of-plane lateral deflections were found very small as expected that the columns can be considered as in-plane bending.

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Figure 6.23 Test arrangement of (A) LVDTs and (B) strain gauges. LVDT, Linear varying displacement transducer.

Behavior and design of high-strength steel members under compression

Figure 6.24 Specimens after failure.

231

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Behavior and Design of High-Strength Constructional Steel

6.4.2 Overall buckling behavior of Q460 columns 6.4.2.1 Test results 1. Box sections

The obtained ultimate bearing capacities of columns are summarized in Table 6.14, where Pcr is the measured ultimate strength of columns, A is the cross section area, fy is the measured yield strength. The axial load versus mid-height deflection curves of all specimens are presented in Fig. 6.25. Because the readings obtained from H01 to H03 differ slightly from each other for the specimens B-8-80 and B-12-55, the average deflection of H01
H03 was used in Fig. 6.25A and B. It can be seen that the specimens show a stable load
deflection relationship in loading process and all curves trend to drop gradually and exhibit a very ductile behavior. It should be noted that, for the specimen B-8-80-2, the maximum strength was very close to Table 6.14 Measured ultimate strength of specimens. Specimen

Pcr (kN)

Pcr/Afy

B-8-80-1 B-8-80-2 B-12-55-1 B-12-55-2 B-18-38-1 B-18-38-2

1122.5 1473.5 2591.0 2436.5 3774 4010.0

0.493 0.631 0.772 0.728 0.780 0.826

Figure 6.25 Load
deflection curves: (A) specimens B-8-80-1 and B-8-80-2, (B) specimens B-12-55-1 and B-12-55-2, (C) specimen B-18-38-1, and (D) specimens B-18-38-2.

Behavior and design of high-strength steel members under compression

233

Figure 6.26 Local buckling of the specimens: (A) specimen B-18-38-1 and (B) specimen B18-38-2.

the Euler critical load and it behaved in a similar way as bifurcation buckling (Fig. 6.25A), which is not normal in practice. The initial geometric imperfection of the specimen B-8-80-2, which is 0.19m of the column effective length, was much smaller than the other specimens. However, it is not enough to yield bifurcation buckling. The appearance of the buckling plateau in the specimen B-8-80-2 may be attributed to dynamic effect. The mid-height deflection at peak load was 6.4 mm and rapidly increased to 24.2 mm within two seconds, although the loading rate was set at 1 mm/min. Consequently, a singular point was caused in the softening range by the dynamic effect. The buckling behavior of the specimen B-8-80-2 was further investigated and discussed in Section 4.4. In addition, it can be seen from Fig. 6.25C and D that the displacement recorded by the mid-width placed LVDT H02 exceeds those recorded by the corner LVDTs H01 and H03 when the test load decreased to about 80% of the maximum strength. The local buckling found on the specimens with the larger plate slenderness ratios is the good explanation for the difference among the measurements of LVDTs H01
H03, as shown in Fig. 6.26. Fig. 6.27A
C shows the typical axial load-strain curves measured at the midlength of the specimens B-8-80-1, B-12-55-1, and B-8-38-1, respectively. As shown in Fig. 6.23, the strain gages S010 and S12 were attached on the left side of the box section while the strain gages S04 and S06 were attached on the right side. The strain gages S02 and S08 were attached on the mid-width of the box section. Therefore the bending effect can be identified from the difference in the axial strains between the two component plates parallel the bending axis at mid-height cross section. Since initial eccentricity and out of straightness is inevitable in practice, 1/1000 of the column length has been considered as the acceptable initial geometric imperfection. It can be seen from Fig. 6.27 that there is almost no difference in the recorded axial strain between the two component plates when the test load remains low. However, with the increase of the test load, the second-order effect becomes appreciable and the difference in the recorded strains between the extreme compressive and tensile fibers becomes significant. The strain of the middle fibers, such as that recorded by strain gages S02 and S08, can be recognized as the average

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Behavior and Design of High-Strength Constructional Steel

Figure 6.27 Load
strain curves: (A) specimen B-8-80-1, (B) specimen B-12-55-1, and (C) specimen B-18-38-1.

strain of all fibers which highlights the utilization level of the cross-sectional resistances. The middle fiber strains of the specimens B-2-80-1, B-12-55-1, and B-1838-1 are 51%, 83%, and 89% of fy/E. The same as the buckling factor, the middle fiber strain to fy/E ratio is increasing with the decrease of the column slenderness ratio. For the specimen B-8-80-1 the recorded strain of the extreme compressive fibers at the peak load point is 86% of fy/E, which seems an elastic-buckling behavior. However, if considerable residual stress is considered, some extreme fibers in compressive residual blocks may have yielded. The extreme compressive fiber strains at the peak load point are 128% and 174% of fy/E for the elastic
plastic buckled specimens B-12-55-1 and B-18-38-1, respectively. 2. H-sections

The axial load versus mid-height deflection curves of all specimens are presented in Fig. 6.28. It can be seen that each specimen shows a stable load
deflection relationship in loading process and all curves trend to drop gradually which exhibit a very ductile behavior. However, the load
deflection curves differ from each other. By comparing the two curves with the same cross section and slenderness, as shown in Fig. 6.28A for the specimens H-3-80-1 and H-3-80-2, it can be observed that the specimen H-3-80-2 with a smaller initial geometric imperfection than the specimen H-3-80-1 shows a higher bending stiffness and a higher ultimate strength, although in the unloading branch they trend to converge. This phenomenon could also be observed in Fig. 6.28B and C for the series H-5-55 and H-7-40, respectively.

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235

Figure 6.28 Load
deflection curves: (A) specimens H-3-80-1 and H-3-80-2, (B) specimens H-5-55-1 and H-5-55-2, and (C) specimens H-7-40-1 and H-7-40-2.

Moreover, by comparing A
C in Fig. 6.28, it can be found that the specimens with larger slenderness are more sensitive to initial geometric imperfections. The specimen H-3-80-1 starts bending at the beginning of loading, while the specimens H-7-40-1 and H-7-40-2 start bending at about 50%
60% of peak load. It can also be seen that the effect of initial geometric imperfections in the specimens H-7-40-1 and H-7-40-2 on bending stiffness and ultimate strength is insignificant when compared with the specimens H-3-80-1 and H-3-80-2. The obtained ultimate bearing capacities of columns are summarized in Table 6.15, where Pcr is the measured ultimate strength of columns, Aw and Af are the area of flanges and webs, respectively, fyf and fyw are the measured yield strength of flanges and webs, respectively. The typical axial load
strain curves measured at the mid-height of the columns are shown in Fig. 6.29A
C for specimens H-3-8-1, H-5-55-1 and H-7-40-2, respectively. As shown in Fig. 6.23B, strain gauges S01 and S09 were attached on the left flange tips of the mid-height cross section, while strain gauges S03 and S07 were attached on the right flange tips. Strain gauges S02 and S08 were attached on the mid-width of flanges. It is easy to identify the beginning of bending and convenient to determine the yield status of the mid-height cross section by using the load
strain curves. Although all specimens were designed as axial concentrically loaded columns with the initial geometric imperfection less than 1/1000 of the column length, the bending of specimen H-3-80-1 occurred at the beginning of loading. It can be seen that, when the load reached about 65% of the peak load, the strains of

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Behavior and Design of High-Strength Constructional Steel

Table 6.15 Measured ultimate strength of specimens. Specimen H-3-80-1 (1) H-3-80-2 (1) H-5-55-1 (2) H-5-55-2 (1,2) H-7-40-1 (1) H-7-40-2 (1)

Pcr (kN)

Pcr/(Awfyw 1 Affyf)

1913 2107.5 4357.5 4290 7596.5 7534.5

0.449 0.496 0.765 0.708 0.881 0.869

Figure 6.29 Load
strain curves: (A) specimen H-3-80-1, (B) specimen H-5-55-1, and (C) specimen H-7-40-2.

S03 and S07 started decreasing and turned into tensile strains before peak load point. In contrast, the uniform loading was achieved for specimen H-7-40-2 until 96% of peak load was reached and the extreme tensile fibers still kept in compressive status when peak load achieved. For extreme compressive fibers, extreme compressive strain gauges (S01 and S09) of specimen H-3-80-1 yielded before reaching Rp0.2/E. With the increase in slenderness, extreme compressive strain gauges (S03 and S07) of specimen H-7-40-2 yielded at compressive strain near Rp0.2/E. The strains of middle fibers, such as that recorded by strain gauges S02 and S08, can be recognized as the average strain of all fibers which highlight the stress status

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237

of the entire mid-height cross section. For specimen H-3-80-1 the recorded strains of the middle fibers corresponding to peak load are only 48.8% of Rp0.2/E. The middle fiber strain corresponding to peak load is 86.8% of Rp0.2/E for specimen H5-55-1 and 109.2% of Rp0.2/E for specimen H-7-40-2. The same as reduction factor, the middle fiber strain to Rp0.2/E ratio is increasing with the decrease of the column slenderness ratio.

6.4.2.2 Comparison of test results with design codes The Chinese specifications Code for design of steel structures GB 50017-2003 [1] and Eurocode3 [19] of the European Committee for Standardization were used to predict the ultimate bearing capacities of columns. The measured values, such as geometric dimensions and material properties, were employed in the evaluation. The predicted values were compared with the test results. GB 50017-2003 According to GB 50017-2003 [1], the capacity of a column is calculated as: N5

ϕAfy γR

(6.6)

where γ R is the partial factor, ϕ is the column slenderness reduction factor. For

λn # 0:215:

ϕ 5 1 2 a1 λ2n

For

λn . 0:215:

ϕ5

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2 1  2 2 a2 1a3 λn 1λ2n 2 4λ2n a 1 a λ 1 λ 2 3 n n 2λ2n

(6.7) (6.8)

where λn is the normalized column slenderness, imperfection coefficients a1 , a2 , and a3 should be employed according to corresponding buckling curves (a, b, and c). Eurocode3

According to Eurocode3 [19], the capacity of a column is obtained as: Nb;Rd 5

χAfy γ M1

(6.9)

where γ M1 is the partial factor, χ is the column slenderness reduction factor. For λn # 0:2:χ 5 1 for

λn . 0:2:χ 5

1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Φ 1 Φ2 2 λ2n

(6.10) (6.11)

where

  Φ 5 0:5 1 1 αðλn 2 0:2Þ 1 λ2n

(6.12)

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Behavior and Design of High-Strength Constructional Steel

Imperfection factor α should be determined by the corresponding buckling curve (a0, a, b, c, and d). 1. Box sections

The ultimate strengths obtained from the test and the multiple buckling curves of GB 50017-2003 and Eurocode3 were compared on the nondimensional basis, as shown in Figs. 6.30 and 6.31. According to GB 50017-2003, the buckling curve c is the appropriate design curve for the box columns with d/t ratio no more than 20.

Figure 6.30 Comparison of test results and column curves of GB 50017-2003 using measured values.

Figure 6.31 Comparison of test results and column curves of Eurocode3 using measured values.

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239

The comparison in Fig. 6.30 shows that the measured ultimate strengths are higher than the buckling curve c of the GB 50017-2003 by 27% on average. Moreover, except for the specimen B-18-38-1, the test results are even higher than the buckling curve b of GB 50017-2003. For Eurocode3, the buckling curve c is the appropriate design curve for the box columns with d/t ratio less than 30. The comparison in Fig. 6.31 shows that the maximum strengths of the specimens are underestimated by the buckling curve c about 20% in average. The main reason of the underestimation on 460 MPa welded box columns is the less detrimental effect of the residual stresses in HSS columns than those in conventional steel columns. The ratios of the measured compressive residual stress to the yield strength of the test specimens are lower than those adopted for conventional steel in the current codes [20]. However, the limited test results are not sufficient to find an appropriate design curve for fabricated Q460 HSS box columns. Therefore a further and extensive numerical study is necessary. 2. H-sections

The ultimate strengths obtained from test and the multiple buckling curves of GB 50017-2003 and Eurocode3 are compared on the nondimensional basis, as shown in Figs. 6.32 and 6.33. The comparison shows that the buckling curve b of GB 50017-2003 is higher than the measured ultimate strengths by 7.9% and the buckling curve a of GB 50017-2003 cannot safely predict the ultimate strength with a difference of 3.7% in average. For Eurocode3, curve c significantly underestimated the ultimate strength of the specimens by 17.8%. The main reason for this is the ignoring of the beneficial effect of the tensile residual stresses induced by flame-cut. The overall buckling will be delay by the tensile residual stresses presented in the flange tips. The ultimate strengths obtained from test are higher than

Figure 6.32 Comparison of test results and column curves of GB 50017-2003 using measured values.

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Behavior and Design of High-Strength Constructional Steel

Figure 6.33 Comparison of test results and column curves of Eurocode3 using measured values.

curve b of Eurocode3 by 8.1% on average and slightly lower than curve a. However, the limited test results are not sufficient to evaluate the maximum strength of Q460 HSS columns. Therefore a further numerical study is necessary to extend the database.

6.4.3 Overall buckling behavior of Q690 columns 6.4.3.1 Test results The obtained experimental results of the specimens are listed in Table 6.16, in which Pu is the measured ultimate bearing capacity of the specimens; fy is the measured yield strength; and A denotes the sectional area of the specimens. The load
deflection relations at the mid-height of the specimens of columns with box section and H-section are shown in Fig. 6.34 and Fig. 6.35. Except specimens B-30 and H-30 series, the average readings from LVDTs H01-H03 of the other specimens were adopted to represent the mid-height deflection because of the slight deviations between such LVDTs. The failure mode of all the 12 test columns is the overall buckling as expected. Slight crookedness of the specimens is gradually expanded under increasing axial load, when overall buckling occurs, the curved surface supports can reflect the free rotation at both ends, as shown in Fig. 6.36. When the monitored load approaches the ultimate capacity of the columns, the slope of the load
deflection curves gradually decreases. Once the specimens reach the limit state (i.e., the occurrence of overall buckling), the lateral deflection considerably increases with the slow decrease of the monitored load. When the loading force reaches 60% of the ultimate strength of the specimens, the tested column is treated as a complete failure and unloaded.

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241

Table 6.16 Ultimate strength of columns. Specimen label B-30-1 B-30-2 B-50-1 B-50-2 B-70-1 B-70-2



H-30-1 H-30-2 H-50-1 H-50-2 H-70-1 H-70-2

Pu (kN)

Pu/Afy

5771.5 9751.5 6444.5 7180.0 3258.5 2897.0

0.649 0.890 0.739 0.822 0.526 0.469

8493.0 8994.0 7207.0 7124.5 3039.0 3690.0

0.914 0.957 0.847 0.832 0.421 0.498

Note:  represents the specimen was straightened by flame heating method.

Figure 6.34 Load
deflection curves of box columns: (A) B-30-1, (B) B-30-2, (C) B-50-1 and B-50-2, and (D) B-70-1 and B-70-2.

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Behavior and Design of High-Strength Constructional Steel

Figure 6.35 Load
deflection curves of H columns: (A) H-30-1, (B) H-30-2, (C) H-50-1 and H-50-2, and (D) H-70-1 and H-70-2.

Figure 6.36 Support rotation.

The initial geometric imperfections of all the specimens are listed in Table 6.17, in which v0 and e0 are the initial deflection and loading eccentricity, respectively. As depicted in Fig. 6.35, the slope of the load
deflection curves and ultimate strength of the specimens decreases with increasing initial geometric imperfections. The columns are increasingly sensitive to initial imperfections with the increase of the slenderness ratios. Note that the ultimate strength of B-30-1 is less than that of B-30-2. One factor was that the former has much larger initial crookedness against the latter. Another factor was that the flame was used for straightening of the specimen in the process of fabrication and the flame heating decreased the strength of steel [21].

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243

Table 6.17 Initial geometric imperfections. Specimen label

e0 (mm)

B-30-1 B-30-2 B-50-1 B-50-2 B-70-1 B-70-2 H-30-1 H-30-2 H-50-1 H-50-2 H-70-1 H-70-2

0.8 2.4 0.1 2 0.8 0.9 2 0.5 1.0 0 2 1.5 2 0.5 2 0.8 0

v0 (mm)

ðe0 1 v0 Þ=Le 3 1023

27.0 2.5 2 1.0 2 1.5 2 1.0 2 1.0 1.0 0.5 1.0 2 0.5 2 2.0 2 1.5

9.89 1.74 0.25 0.64 0.03 0.42 0.99 0.25 0.17 0.34 0.80 0.43

Table 6.18 Material properties of specimens B-30-1. Specimen B-1 B-2 B-3 B-4

fy0 (MPa)

fu0 (MPa)

fy (MPa)

fy0 /fy

617 630 780 625

720 753 836 743

772 772 772 772

0.80 0.82 1.01 0.81

The measured yielding strength and ultimate strength of the steel exposed to flame are listed in Table 6.18, where B-1 to B-4 are the steel coupons taken from specimen B-30-1 after test as illustrated in Fig. 6.37, fy0 and fu0 are the measured yielding strength and tensile strength of the steel exposed to flame, respectively, and fy is the measured yielding strength of the steel unexposed to flame. The stress
strain curves of coupons B-1 to B-4 is shown in Fig. 6.38. Given the breakdown of the extensometer attached onto B-4, a hardening stage of this specimen was undetected and not depicted in stress
strain curve. It can be found that the yield strength of coupon B-3 unexposed to flame was almost unchanged. By contrast, the yield strength of coupons B-1, B-2, and B-4 exposed to flame decline to 80%, 82%, and 81% of its original yielding strength, respectively. The aforementioned results indicate that the straightening method by flame heating is unfavorable to the strength of columns made of Q690 high-strength steel. Twelve and thirteen strain gages were, respectively, attached to the exterior at mid-height of box and H columns. For the critical cross-section at the mid-height of the box columns, strain gages S02 and S08 monitored the strains at the mid-width of the webs, which can be acknowledged as the average strain of this section, whereas S04, S06, S10, and S12 presented the strains at the edges of the flanges of the box sections. For the H columns, S02 and S08 can express the average strain of

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Behavior and Design of High-Strength Constructional Steel

Figure 6.37 Areas of specimen B-30-1 exposed to flame and location of coupons used for material test.

Figure 6.38 Stress
strain relation of coupons B-1 to B-4.

the section at the mid-width of the flanges, whereas S03, S07, S01, and S09 provided the strains at the tips of the H section flanges. Fig. 6.39 indicates that when reaching the failure load, the compressive strain at mid-width of the webs for specimen B-30-2 accounts for 101.8% of the yield strain, fy/E, which can be recognized as strength failure. For specimen B-70-1 with a large slenderness, this strain occupies 57.5% of the yield strain under the failure load. This demonstrates that B-70-1 buckled in elastic state. It is found that the average strain at the failure of specimen B-50-1 constitutes 85.9% of the yield strain, and the compressive flange completely yields. So, the instability mode of specimen B50-1 can be categorized under elastic
plastic type. It can be seen from Fig. 6.40 that the compressive strains located in the middle of the sectional flanges of the specimens H-30-1, H-50-2, and H-70-2 account for 159.2%, 95.1%, and 56% of the yield strain, fy/E, respectively. These demonstrate the failure of the specimens H-30-1 and H-70-2 can be regarded as the sectional yielding mode, elastic
plastic instability mode, and elastic instability mode respectively.

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245

Figure 6.39 Load
strain curves of box columns: (A) B-30-2, (B) B-50-1, and (C) B-70-1.

6.4.3.2 Test results The codes for design of steel structures GB 50017-2003 [1], Eurocode 3 [19], and ANSI/AISC 360-10 [22] were adopted to estimate the ultimate strength of the specimens. The measured geometric sizes and material properties were applied in the estimation. The estimated load-bearing capacities of the specimens were compared against the experimental results. In GB 50017-2003 [1] the design buckling resistance of an axial compression member is expressed as: Nd 5 ϕAfy

(6.13)

where ϕ is the reduction factor for overall buckling. When λ # 0:215;

ϕ 5 1 2 α1 λ

2

(6.14) 2

When λ . 0:215;

ϕ5

½ðα2 1 α3 λ 1 λ Þ 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ðα2 1α3 λ1λ Þ2 2 4λ  2λ

2

(6.15)

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Behavior and Design of High-Strength Constructional Steel

Figure 6.40 Load
strain curves of box columns: (A) B-30-2, (B) B-50-1, and (C) B-70-1.

in which λð

Þ is the nondimensional slenderness; and α1, α2 and α3 are the factors corresponding to the appropriate types of cross sections (class a, b, and c sections). The design resistance against axial compressive buckling of a column is taken in Eurocode 3 [19] as: Nd 5 χAfy

(6.16)

where χ is the reduction factor for overall buckling, which can be expressed as follows: When λ # 0:2;

χ51

When λ . 0:2;

χ5

(6.17)

1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Φ 1 Φ2 2 λ

(6.18)

2

in which Φ 5 0:5½1 1 αðλ 2 0:2Þ 1 λ , and α is the imperfection factor for the corresponding buckling curve. In ANSI/AISC 360-10 [22] the design buckling resistance of an axially compressed column is calculated as: Nd 5 φc fcr A

(6.19)

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247

in which φc is the resistance factor, for which a value of 0.9 is recommended. fcr represents the critical stress, written as follows: sffiffiffiffi h i E ; fcr 5 0:658fy =fe fy (6.20) When λ # 4:71 fy sffiffiffiffi E ; When λ . 4:71 fy

fcr 5 0:877fe

(6.21)

where λ is the slenderness ratio and fe 5 π2 E=λ2 . If nondimensional slenderness λ substitutes for the slenderness ratio λ in Eqs. (6.20) and (6.21), these two equations can be obtained as follows: When λ # 1:5;

χ5

When λ . 1:5;

χ5

2i fcr h 5 0:658λ fy

fcr 0:877 5 2 fy λ

(6.22)

(6.23)

The comparison between the measured ultimate bearing capacity of the specimens and the relevant buckling curves of GB 50017-2003, Eurocode 3, and ANSI/ AISC 360-10 were conducted on the basis of nondimensionalization, as shown in Figs. 6.41
6.43. In accordance with the related provisions of GB 50017-2003 (2003), the buckling curve c is suited for welded box columns with a maximum width-to-thickness ratio of 20. The specimens B-30, B-50, and B-70 satisfy this condition. Fig. 6.41A illustrates that the reduction factors of all the specimens of the box columns are higher than buckling curves “c” and “b,” except that of specimen B-30-1, which was impaired by flame heating and needs to be rejected. This finding suggests that the provisions for welded box columns with Q690 steel are conservative. If the average data in Fig. 6.41A are adopted, the buckling curve “a” could be appropriate for this kind of columns. GB 50017-2003 specifies that for flame-cut welded H columns with plates of less than 40 mm, buckling curve “b” is appropriate. The specimens H-30, H-50, and H-70 are suitable for classification under this provision. However, as shown in Fig. 6.41B, the reduction factors of all the columns are higher than buckling curve “a,” except H-70-1, which was also straightened by flame heating. These results indicate that curve “a” may be suited for Q690 flame-cut welded H columns. As specified in Eurocode 3, buckling curve “c” is appropriate for welded box columns with a less than 30 width-to-thickness ratio. The specimens B-30, B-50, and B-70 are suitable for classification under this provision. Fig. 6.42A shows that the reduction factors of the specimens are larger than that depicted by curve “a,” except the unusable results of specimen B-30-1. The result indicates that buckling curve “a” is more appropriate than curve “c.” For the welded H columns

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Behavior and Design of High-Strength Constructional Steel

Figure 6.41 Comparison of test results and GB 50017-2003: (A) box and (B) H columns.

with plates of less than 40 mm, buckling curve “c” is suited when buckling occurs about the minor axis. However, Fig. 6.42B depicts that except for the reduction factor of specimen H-70-1, those of the other columns are higher than curve “a.” The obtained experimental data suggest that curve “a” is appropriate for this type of H columns. For the design of steel columns under axial compression a single buckling curve is employed as stipulated in ANSI/AISC 360-10. The curve is suitable for compression columns without slender elements. It can be seen from Fig. 6.43 that the buckling curve is in general conservative for reduction factors of both welded Q690 steel box- and H-columns if ignoring the unusable results of specimens B-30-1 and H-70-1 which were impaired by flame heating, which concludes that the provisions of the American code for welded Q690 box and H columns without slender elements are conservative.

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249

Figure 6.42 Comparison of test results and Eurocode 3: (A) box and (B) H columns.

6.5

Parametric analysis and design recommendation

6.5.1 Parametric analysis of Q460 columns Based on the verified numerical model [23
26], parametric analysis was carried out to supplement extra data to the limited test results. Initial geometric imperfections and residual stresses are the major factors influencing the maximum column strength. The verified model was employed to conduct the parametric analysis to investigate the effect of initial geometric imperfections and residual stresses on the ultimate bearing capacities of axially compressed HSS columns.

250

Behavior and Design of High-Strength Constructional Steel

Figure 6.43 Comparison of test results and ANSI/AISC 360-10: (A) box and (B) H columns.

Box sections

A total of 120 pin-ended columns buckling about the x
x axis and y
y axis were analyzed in the parametric analysis. The specimens were divided into five groups with various d/t ratios of 8
18. For each section, there were 12 columns with slenderness ranging from 20 to 130. The dimensions and material properties of the specimens are shown in Table 6.19. H-sections

A total of 72 pin-ended columns buckling about strong and weak axes were analyzed in the parametric analysis. The specimens were divided into three groups with different section slenderness ratios. For each section, there were 12 columns

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251

Table 6.19 Dimensions and material properties of box-section specimens for parametric study. Section B-8 B-10 B-12 B-15 B-18

D (mm)

t (mm)

d (mm)

d/t

A (mm2)

E (GPa)

fy (MPa)

112 136 156 194 220

11.44 11.44 11.44 11.44 11.44

89.12 113.12 133.12 171.12 197.12

7.8 9.9 11.6 15.0 17.2

4602 5700 6615 8354 9544

207.8 207.8 207.8 207.8 207.8

505.8 505.8 505.8 505.8 505.8

Table 6.20 Dimensions and material properties of H-section specimens for parametric study. Section

B (mm)

H (mm)

H-3 H-5 H-7

154.7 229.0 308.3

171.3 245.5 318.5

tw (mm) 11.36 11.62 11.46

tf (mm)

b/tf

h/tw

E (GPa)

fy (MPa)

20.98 21.15 21.20

3.4 5.1 7.0

11.4 17.5 24.1

217.6 217.6 217.6

540.9 540.9 540.9

with slenderness ranging from 20 to 130. The dimensions and material properties of specimens are shown in Table 6.20.

6.5.1.1 Geometric imperfection sensitivity In order to investigate the sensitivity of columns of different steel grades to the geometric imperfection, two series of columns with identical cross-sectional sizes but different yield strengths were carried out, as shown in Fig. 6.44. Rather than residual stresses, only a central bow of Lt/1000 was considered in the models. The elastic modulus of the series Q235 is 206 GPa and the yield strength is 235 MPa. Because the calculated buckling curves with different flange slenderness ratios differ slightly from each other, the buckling curve for section H-7 was adopted to represent the whole series. It can be seen from Fig. 6.44 that the columns fabricated from Q460 HSS is less sensitive to the geometric imperfection than those fabricated from Q235 carbon steel. The buckling curve of Q460 steel is higher than the Q235 curve by 4.5% for buckling about weak axis and 6.6% for buckling about strong axis at λyn (λxn) 5 0.97.

6.5.1.2 Effect of residual stresses 1. Box sections

In addition to the initial geometric imperfection, the residual stress is the other initial imperfection, which influences the buckling behavior of the welded box columns. In order to evaluate the effect of the residual stresses on the maximum

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Behavior and Design of High-Strength Constructional Steel

Figure 6.44 Sensitivity of different steel grades to geometric imperfection: (A) weak and (B) strong axis buckling.

Figure 6.45 Effect of the residual stresses on the buckling curves: (A) comparison of the buckling curves and (B) relative difference of the reduction factors.

strength of the columns, the buckling curves of sections B-8, B-12, and B-18 with the consideration of the initial geometric imperfections and the simplified residual stresses were compared with the buckling curve which only took the geometric imperfection of 1mLt in to account, as shown in Fig. 6.45A. And the relative difference of the reduction factors between curves B-8 and B-12 was introduced for the discussion, as shown in Fig. 6.45B. It can be observed that the buckling curves B-8, B-12, and B-18 are very close to the curve Lt/1000 for the stub and very slender columns. However, the effect of residual stresses is obvious for the columns with intermediate slenderness. Meanwhile, it is can be seen that the buckling curves B-8 and B-18 intersect at λn 5 0.79. With the decrease in the nondimensional slenderness, the highest curve B-18 becomes the lowest curve, which means the smaller compressive residual stress becomes more harmful to the axial loaded columns. This is not in conformity with the generally recognized regular that the bigger value of compressive residual stress is more harmful to the buckling strength of the columns. According to this phenomenon, the buckling curves were divided into four

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253

parts and discussed individually. Part 1 represents the case of strength failure. Part 4 represents the case of elastic buckling. Parts 2 and 3 represent the case of elastic
plastic buckling. For Part 2, χ 1 jβ j $ 1. For Part 3, χ 1 jβ j , 1. The boundary between Parts 2 and 3 is the line χ 5 1 1 β. Part 1 (stub columns): Rather than overall buckling, the failure mode of the stub columns in Part 1 is strength failure. Owing to the sufficiently good ductility of steel material, residual stresses are not detrimental to the plastic resistances of cross sections. The whole cross section is in the plastic status under the maximum load. Therefore the reduction factors of the columns with varied cross-sectional dimensions and thus different residual stresses are all close to 1.0 for ideal elastic
plastic materials. Part 4 (very slender columns): Due to the large slenderness, the very slender columns in Part 4 tend to buckle elastically. Because the whole mid-length cross section is still in the elastic status, residual stresses make no difference to the buckling curves. All reduction factors in Part 4 are very close to the Euler curves. Part 3 (slender columns): Because the reduction factor χ in Part 3 is less than 1 1 β, the sum of the axial force induced compressive residual stress   stress  andthe is less than the yield strength, P=Afy 1 σrc =fy 5 χ 1 jβ j , 1. However, the appearance of the initial bow of Lt/1000 will cause a moment at the mid-length cross section. If the corresponding curvature is φ, the moment induced compressive stress will be 2yφE in elastic   range.  For the extreme compressive fibers, if we have P=Afy 2 yφE=fy 1 σrc =fy . 1, the inelasticity will be developed in the compressive residual stress blocks. The continuing development of the inelastic zones will result in a gradual deterioration of the flexural stiffness and a final overall buckling of the column. In this case the increase in the magnitudes of compressive residual stresses will accelerate the development of the inelastic zones, which is detrimental to the ultimate bearing capacities of the columns. Therefore the buckling curve B-8 with the compressive residual stress ratio of 20.255 is lower than the buckling curve B-18 with the compressive residual stress ratio of 20.142 by up to 9.4% in Part 3. Fig. 6.46A shows the stress distribution and status of the critical cross section of B-8 at peak load.

Figure 6.46 Stress distribution status of mid-height cross section at peak load: (A) B-8, λn 5 1.283 and (B) B-18, λn 5 0.600.

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Behavior and Design of High-Strength Constructional Steel

Part 2 (intermediate columns): The reduction factor χ in Part 2 is not less than 1 1 β, so the superimposing of the axial force induced resid  stress  and  the compressive  ual stress is not less than the yield strength, P=Afy 1 σrc =fy 5 χ 1 jβ j $ 1. However, due to the appearance of the tensile residual stresses, the tensile residual blocks  in the  four  corners of the box section are still in the elastic status, P=Afy 1 σrt =fy 5 χ 2 α , 1. If a perturbation of Lt/1000 is imposed on the axial loaded column, the bending resistance can only be provided by the elastic blocks. The larger area of the elastic blocks is, the higher bending stiffness is. Therefore the residual stress pattern with a large area of tensile residual stresses is preferred in Part 2. According to the static equilibrium of the cross section: β 3 Ac 1 α 3 At 5 0 (6.24) where Ac is the total area of the compressive residual stress blocks, At is the total area of the tensile residual stress blocks, and: A c 1 At 5 A

(6.25)

Substitution for Ac in Eq. (6.24) gives: At β 5 β2α A

(6.26)

It shows the ratios of the tensile residual stress area to the cross-sectional area for welded box sections with different width-to-thickness ratios. It can be seen from Fig. 6.47B that, not the same as in Part 3, the buckling curve B-8 is higher than curve B-18 by about 3.2%
4.6% in Part 2. The stress distribution of the critical cross section of B-18 at peak load is shown in Fig. 6.19B. 2. H-sections

The ultimate bearing capacities of 72 columns obtained from the parametric analysis were compared with the design strengths predicted by Eurocode3 [19] and GB 50017-2003 [1] on the nondimensional basis. A initial bow of 1/1000 of the column length and 3 residual stress distributions for H-sections in different sizes were considered in the parametric study. Fig. 6.47 shows the comparison of theoretical curves with Eurocode3 buckling about weak and strong axes. The more compact section is the greater magnitudes of compressive residual stresses are. It can be observed that the residual stresses are more detrimental to the maximum strength of columns with more compact sections. The limits of the width-to-thickness ratio b/tf of outstand flanges for Q460 steel range from 8.1 to 14.2 according to different design codes [1,6,19]. Because section H-3 with a too small b/tf can hardly be adopted in structural members and section H-5 cannot represent the commonly used columns, the buckling curve for section H-7 was used to select an appropriate curve for welded H-section columns fabricated from flame-cut 460 MPa HSS plates. ECCS [4] considered the beneficial effect of high yield strength and assumed a compressive residual stress level of 10% of the yield strength for HSS (430 MPa) welded H-section with rolled flanges. The theoretical curves M39m and M40m

Behavior and design of high-strength steel members under compression

255

Figure 6.47 Comparison of numerical results with Eurocode3: (A) y
y axis, (B) x
x axis, and (C) details of the comparison of curve H-7.

were generated for buckling about weak and strong axes, as shown in Fig. 6.47A and B. It is expected that tensile residual stresses at the flange toes induced by flame-cut will be beneficial for bending about weak axis. Therefore curve a, which is close to curve M39m and lower than curve M40m, was recommended for HSS welded flame-cut H-section columns buckling about two principle axes. It can be found from Fig. 6.47A and B that curves H-7 (buckling about y
y and x
x axes) are much lower than curves a, M39m, and M40m for short and intermediate columns. On the other hand, the suggested curve a has not been adopted in Eurocode3 to design HSS H-section columns in the absence of experimental confirmation. In Eurocode3, H-section columns have to be designed according to curves b (x
x axis) and c (y
y axis). Fig. 6.47C shows the details of the comparison between curves H-7 and curves b and c of Eurocode3. It can be seen that the curves for section H-7 are higher than their design curves c (y
y axis) and b (x
x axis) by 16.1% and 9.0% in average. Although curve b is considered appropriate to design the welded flame-cut 460 MPa columns buckling about the y
y axis, the specified curve c should not be modified due to the desire to keep the same curve for columns manufactured by different methods (welded from flame-cut/rolled/sheared plates).

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Behavior and Design of High-Strength Constructional Steel

6.5.2 Design of welded Q460 steel columns 6.5.2.1 Welded box-section columns In order to select an appropriate buckling curve for the design of Q460 HSS welded columns, a series of FEM models in the same dimensions as specified in Table 6.19 but with a yield strength of 460 MPa was developed. The 1mL t initial geometric imperfections and the simplified residual stresses as shown in Figs. 6.46 and 6.47 were considered in the analysis. It was found that the adopted residual stress pattern was slightly more harmful to the maximum column strength with buckling about the y
y axis than buckling about the x
x axis (Fig. 6.6). Thus the y
y axis series was used to compare with the prediction of Eurocode3 [19] and GB 50017-2003 [1], as shown in Figs. 6.48 and 6.49. The limit of the width-to-thickness ratio d/t of the component pates for welded 460 MPa HSS box section ranges from 28.4 to 30 according to the different design codes [1,6,19]. Because section B-8 can represent the extreme case and section B-18 can represent the most commonly used columns, the buckling curve for the both sections were used to select an appropriate curve for the welded box columns fabricated from 460 MPa HSS plates. Fig. 6.48A shows the nondimensional comparison between the theoretical curves and the design curves of Eurocode3. The effect of the residual stresses on the buckling curves shows the same regular as the four parts discussed previously. The decreased detrimental effect of residual stresses has been evaluated by ECCS [4] by assuming a compressive residual stress level of 10% of the yield strength for HSS (430 MPa) welded box sections. The theoretical curve M1011m was generated for box sections with moderate sized welds, as shown in Fig. 6.48A. Because curve a is the lower boundary of the theoretical curve M1011m, it has been suggested for the design of welded HSS box

Figure 6.48 Comparison of numerical results with Eurocode3: (A) y
y axis and (B) details of the comparison.

Behavior and design of high-strength steel members under compression

257

Figure 6.49 Comparison of numerical results with GB 50017-2003. (A) y
y axis and (B) details of the comparison.

columns with moderate size welds buckling about both principle axes. However, it can be found from Fig. 6.48A and B that curves B-8 and B-18 are much lower than curves a and M1011m for short and intermediate columns. To be on the safe side, the suggested curve a has not been adopted in Eurocode3 to design HSS box columns in the absence of the experimental confirmation. In Eurocode3, welded box columns have to be designed according to curve c. The selection of curve c is based on the numerical analysis considering the compressive residual stress of 29%
82% of the yield strength for normal strength steel. Fig. 6.48B shows the details of the comparison between the computed curves and curves a, b, and c of Eurocode3. It can be seen that the generated theoretical curves for sections B-8 and B-18 are higher than the design curve b by 9.1% and 10.3% in average, respectively. Therefore curve b is considered as appropriate as the design curve of the welded 460 MPa box columns buckling about both principle axes. Fig. 6.49 shows the comparison of the theoretical curves with curves a, b, and c of GB 50017-2003 for welded 460 MPa box columns buckling about both principle axes. According to GB 50017-2003, curve c was adopted to design the welded box columns with d/t ratio no more than 20. It is found that the theoretical curves for sections B-8 and B-18 are higher than curve b by 8.8% and 9.7% in average, respectively. Therefore curve b is recommended for the design of Q460 HSS welded box columns.

6.5.2.2 Welded H-section columns Fig. 6.50 shows the comparison of the theoretical curves with curves a, b and c of GB 50017-2003 for buckling about weak and strong axes. In GB 50017-2003 [1], curve b was adopted to design the welded H-section columns fabricated from flame-cut plates buckling about two principle axes. It is found that curves H-7 are higher than curve b by 7.3% and 8.6% in average for buckling about y
y and x
x axes, respectively. Therefore curve b is recommended to extend to design the Q460 steel columns.

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Behavior and Design of High-Strength Constructional Steel

Figure 6.50 Comparison of numerical results with GB 50017-2003: (A) y
y axis, (B) x
x axis, and (C) details of the comparison of curve H-7. Table 6.21 Dimensions of box columns. Specimen label

B (mm)

b (mm)

t (mm)

h/t

A (mm2)

Iu (cm4)

B-7 B-10 B-13

140.88 192.37 236.47

108.74 160.33 204.27

16.07 16.02 16.10

6.8 10.0 12.7

8023 11,301 14,192

2118 5906 11,548

6.5.3 Parametric analysis of Q690 columns Parametric analysis is conducted to investigate the effects of initial geometric imperfections and residual stresses on the ultimate bearing capacities of the Q690 high-strength steel welded columns. Thirty-six welded box columns and 36 welded H-columns under axial compression are adopted in parametric study. The box columns comprise three sectional width-to-thickness ratios (7
13) and 12 slenderness ratios (20
130). The H-columns also contain three width-to-thickness ratios ranging from 6 to 8 and 12 slenderness ratios with a range from 20 to 130. Tables 6.21 and 6.22 list the sectional dimensions of box- and H-columns. Fig. 6.51A and B represents the cross sections of

Behavior and design of high-strength steel members under compression

259

Table 6.22 Dimensions of H-columns. Specimen label

B (mm)

bf (mm)

H (mm)

h (mm)

H-6 H-7 H-8

209.38 240.47 260.82

96.57 112.16 122.29

205.24 238.15 260.35

172.76 205.83 227.85

tf, tw (mm)

bf/ tf

h/ tw

A (mm2)

5.9 6.9 7.5

10.6 12.7 14.0

9606 11,098 12,179

16.24 16.16 16.25

Figure 6.51 Sectional symbols of columns: (A) box section and (B) H-section.

Figure 6.52 Elastic-ideally plastic stress
strain relation of Q690 steel.

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Behavior and Design of High-Strength Constructional Steel

Figure 6.53 The effect of residual stresses on box columns.

the box- and H-columns, respectively. An elastic-ideally plastic stress
strain relation based on the material measurement, as shown in Fig. 6.52, is applied in analysis.

6.5.3.1 Effects of residual stresses To study the effect of residual stresses on the ultimate bearing capacities of Q690 welded box- and H-columns, the models include box columns (section types: B-7, B-10, and B-13) and H columns (section types: H-6, H-7, and H-8) bent about the minor and major axes with 12 slenderness ratios. The models are divided into two groups: one that takes into account initial curvature (Le/1000) and another that considers combined initial curvature and residual stresses. Fig. 6.53 displays the simulation results of the box columns, which indicate that the effect of residual stresses on the box columns is concentrated on the nondimensional slenderness ratios with a range from 0.5 to 1.2. The buckling reduction factor with consideration of residual stresses is lower than that without consideration of the stresses by a maximum of 3.9%. Fig. 6.54A and B demonstrates that the analysis results of the Hcolumns bent about the minor and major axes. For the minor axis, the reduction factor of the H-columns allowing for residual stresses decreases by a maximum of 2.0%. For the major axis, the factor decreases by a maximum of 2.8%.

6.5.3.2 Effects of width-to-thickness of sections without residual stress 1. Box columns

Three series of box-column models with only initial geometric imperfection are built and analyzed for studying the effects of sectional width-to-thickness on buckling curves. The initial deflection is 1m of the corresponding column length. The

Behavior and design of high-strength steel members under compression

261

Figure 6.54 The effect of residual stresses on H-columns: buckling about (A) minor and (B) major axes.

Figure 6.55 Effects of width-to-thickness of box sections without residual stress.

material properties of the Q690 box columns are presented in Fig. 6.52. The analysis results shown in Fig. 6.55 indicate that the three buckling curves for sections B7, B-10, and B-13 are very close, which shows the effects of width-to-thickness of sections without residual stress is negligibly small. 2. H-columns

Three series of H-column models bending about the minor and major axes are established, in which only initial geometric imperfections are considered. The initial curvature applied in the models is set as 1m of the related column length. The material properties of the H-columns are identical with the corresponding box columns. The analysis results for the minor and major axes are shown in Fig. 6.56A and B, respectively. It can be concluded that the width-to-thickness of sections without residual stress has very small influence in the buckling curves for sections H-6, H-7, and H-8 buckling about the minor and major axes.

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Behavior and Design of High-Strength Constructional Steel

Figure 6.56 Effect of width-to-thickness of H-sections without residual stress: (A) buckling about minor and (B) major axes.

6.5.3.3 Response of various steel grades to initial deflections 1. Box columns

Three series of box-column models with three steel grades (Q690, Q460, and Q235) are built and analyzed, in which section B-10 with only initial geometric imperfections are selected. The initial deflection is 1m of the corresponding column length. The material properties of the Q690 box columns are presented in Fig. 6.52. The Young’s modulus of the Q460 and Q235 columns is 206 GPa. The yield strengths of the Q460 and Q235 columns are 460 and 235 MPa, respectively. It can be observed from Fig. 6.57 that the buckling curve of the Q690 box columns is higher than the Q460 columns, which is higher than that of the Q235

Behavior and design of high-strength steel members under compression

263

Figure 6.57 Box-column response of various steel grades to initial deflections.

columns. A large difference occurs among the buckling reduction factors of the Q690, Q460, and Q235 box columns for the nondimensional slenderness ratios ranging from 0.5 to 1.5. The buckling curves of the Q690 and Q460 columns are higher than that of the Q235 columns by 6.9% and 4.4%, respectively. These findings suggest that the sensitivity of the box columns to the initial geometric imperfections tends to decrease with increasing yield strength. 2. H-columns

Three series of H-column models with three steel grades (Q690, Q460, and Q235) are built, in which section H-7 bending about the minor and major axes are adopted and only initial geometric imperfections are allowed for. The initial curvature applied in the models is set as 1m of the related column length. The material properties of the Q690, Q460, and Q235 H-columns are identical with the corresponding box columns. As indicated in Fig. 6.58A and B, the buckling curve of the Q690 H-columns is higher than the Q460 column curve. Likewise, the Q460 column curve is higher than the Q235 column curve. When the nondimensional slenderness ratios range from 0.5 to 1.5, a large difference exists among the buckling reduction factors of the Q690, Q460, and Q235 H-columns. For the reduction factor about the minor axis, the largest ratio between the factor of the Q460 and Q235 columns reaches 1.041 and that between the Q690 and Q235 columns reaches 1.072. For the columns buckling about the major axis, the maximum ratios of the reduction factors between the Q460, Q690, and Q235 columns amount to 1.047 and 1.073, respectively. These findings reveal that the columns are less sensitive to the initial geometric imperfections with increasing yield strength.

6.5.4 Design of welded Q690 steel columns To determine an applicable buckling curve for designing Q690 high-strength steel welded columns, a series of numerical models with the same sizes listed in

264

Behavior and Design of High-Strength Constructional Steel

Figure 6.58 Effect of width-to-thickness of H-sections without residual stress: buckling about (A) minor and (B) major axes.

Tables 6.23 and 6.24 was created and analyzed. The initial deflection with 1m column length and the residual stress patterns were applied in the models.

6.5.4.1 Comparison with GB 50017-2003 code As specified in GB 50017-2003, buckling curve “c” is suggested to design welded box columns with a width-to-thickness ratio less than 20. In order to evaluate the feasibility of the item for Q690 welded box columns, the analysis results that involve sections B-7, B-10, and B-13 are compared with buckling curves “a,” “b,” and “c” in the code (Fig. 6.59). It can be seen from Fig. 6.59

Behavior and design of high-strength steel members under compression

265

Table 6.23 Dimensions of box columns. Specimen label

B (mm)

h (mm)

t (mm)

h/t

B-7 B-10 B-13

140.88 192.37 236.47

108.74 160.33 204.27

16.07 16.02 16.10

6.8 10.0 12.7

A (mm2)

Iu (cm4)

8023 11,301 14,192

2118 5906 11,548

Table 6.24 Dimensions of H-columns. Specimen label

B (mm)

bf (mm)

H (mm)

h (mm)

H-6 H-7 H-8

209.38 240.47 260.82

96.57 112.16 122.29

205.24 238.15 260.35

172.76 205.83 227.85

tf, tw (mm) 16.24 16.16 16.25

b f/ tf

h/ tw

A (mm2)

5.9 6.9 7.5

10.6 12.7 14.0

9606 11,098 12,179

Figure 6.59 Comparison between test results of box columns and GB 50017-2003.

that the buckling reduction factors based on the numerical simulation are higher than the buckling curves “a,” “b,” and “c” by an average of 5%, 15%, and 28%, respectively. Thus, for design of Q690 welded box columns in GB 50017-2003, buckling curve “a” is appropriate. For flame-cut welded H-column with plates of lower than 40 mm in GB 500172003, buckling curve “b” is recommended. H-6, H-7, and H-8 series bent about the minor and major axes are adopted in the numerical analysis. The comparison between the analysis results and buckling curves “a” and “b” are carried out as shown in Fig. 6.60A and B. As shown in the figures, the numerical results of the Hcolumns bent about the minor and major axes are higher than the buckling curves “a” and “b.” On average the numerical results for the minor axis yield percentages

266

Behavior and Design of High-Strength Constructional Steel

Figure 6.60 Comparison between test results of H-columns and GB 50017-2003: buckling about (A) minor and (B) major axes.

of 4% and 14% and those for the major axis yield percentages of 6% and 16%. Consequently, buckling curve “a” is suitable for design of Q690 welded H-columns with reference to GB 50017-2003 [1].

6.5.4.2 Comparison with Eurocode 3 Eurocode 3 suggests that buckling curve “c” is appropriate for welded box columns with a width-to-thickness ratio lower than 30. However, the provision for box columns welded by Q690 steel needs to be validated. The columns with sections B-7, B-10, and B-13 are analyzed, after which the analysis data are compared to

Behavior and design of high-strength steel members under compression

267

Figure 6.61 Comparison between test results of box columns and Eurocode 3.

buckling curves “a0,” “a,” “b,” and “c” (Fig. 6.61). The figure shows that the analysis results of Q690 welded box columns are higher than buckling curves “a0,” “a,” “b,” and “c” by 2%, 8%, 16%, and 25% in average, respectively. For the use of curve “a0,” a potential risk exists in overestimating the ultimate capacities of the box columns with slenderness ratios ranging from 20 to 40. Accordingly, curve “a” should be selected as a reasonable design curve for Q690 welded box columns. As indicated in Eurocode 3, buckling curves “c” and “b” are respectively recommended to design flame-cut welded H-columns (plate thickness ,40 mm) bent about the minor and major axes. To verify the item, Fig. 6.62 plots the numerical results for H-6, H-7, and H-8 series buckling about the minor and major axes against curves “a0,” “a,” “b,” and “c.” As shown in the figure, the buckling reduction factors of the H-columns bent about the minor axis are higher than buckling curves “a0,” “a,” “b,” and “c” by an average of 1%, 7%, 15%, and 24%, respectively. The factors for the major axis are higher than buckling curves “a0,” “a,” and “b” by 3%, 8%, and 16% in average, respectively. Considering a potential risk of overestimating the ultimate capacities of Q690 flame-cut welded H-columns in slenderness ratios with a range from 20 to 30, curve “a” is an advisable choice for design of such columns.

6.6

Summary

1. The residual stress distributions of welded box section and H-section fabricated from Q460 and Q690 high-strength steel plates were measured. The corresponding simplified residual stress distributions were proposed. The comparison of test result with those of mild carbon steel shows that the residual stress ratios of high-strength steel welded box

268

Behavior and Design of High-Strength Constructional Steel

Figure 6.62 Comparison between test results of H-columns and GB 50017-2003: buckling about (A) minor and (B) major axes.

section and H-section tend to be less detrimental to the column strength than the ordinary steel H-sections. 2. Base on the experimental and numerical results, the buckling curves b specified in both Eurocode3 and GB 50017-2003 are recommended as the design curve for the welded boxsection and H-section columns with the nominal yield strength of 460 MPa. For the welded box- and H-section columns with the nominal yield strength no less than 690 MPa, the buckling curves a specified in both Eurocode3 and GB 50017-2003 are recommended. 3. Compared to normal strength steels welded columns, higher buckling curves are applicable for high-strength steel welded columns. This is mainly attributed to the reduced residual stress ratios of high-strength steel welded sections and the lower sensitivity to initial geometric imperfection of high-strength steel columns.

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269

References [1] GB 50017-2003. Code for design of steel structures. Beijing: China Architecture & Building Press; 2003 [in Chinese]. [2] Bjorhovde R, Tall L. Maximum column strength and the multiple column curve concept, Fritz Laboratory report. Lehigh University; 1971. [3] Bjorhovde R. Deterministic and probabilistic approaches to the strength of steel columns. Bethlehem, PA: Lehigh University; 1972. [4] European Convention for Constructional Steelwork. Manual on stability of steel structures. ECCS Publication; 1976. [5] CEN. Eurocode 3: Design of steel structures, Part 1
12: Additional rules for the extension of EN 1993 up to steel grades S700, EN 1993-1-12. Brussels: European Committee for Standardization; 2007. [6] American Institute of Steel Construction. Specification for structural steel buildings, ANSI/AISC 360-10. Chicago, Illinois: AISC; 2010. [7] Ziemian RD. Guide to stability design criteria for metal structures. 6th ed. Hoboken, NJ: John Wiley & Sons, Inc; 2010. [8] Odar E, Nishino F, Tall L. Residual stresses in welded built-up T-1 shapes, Fritz Laboratory report. Lehigh University; 1965. [9] Nishino F, Tall L. Experimental investigation of the strength of T-1 steel columns, Fritz Laboratory report. Lehigh University; 1970. [10] Rasmussen KJR, Hancock GJ. Tests of high strength steel columns. J Constr Steel Res 1995;34(1):27
52. [11] Shi G, Bijlaard, FSK. Finite element analysis on the buckling behaviour of high strength steel columns. In: Proceedings of the 5th international conference on advances in steel structures. Singapore; 2007. [12] GB/T 2975-1998. Steel and steel products: location and preparation of test pieces for mechanical testing. Beijing: China Standard Press; 1998 [in Chinese]. [13] National Standardization Technical Committees. GB/T 228-2002 metallic materials: tensile testing at ambient temperature. Beijing: China Standard Press; 2002 [in Chinese]. [14] Tebedge N, Alpsten G, Tall L. Residual-stress measurement by the sectioning method. Exp Mech 1973;13(2):88
96. [15] Cruise RB, Gardner L. Residual stress analysis of structural stainless steel sections. J Constr Steel Res 2008;64(3):352
66. [16] Barroso A, Can˜as J, Pico´n R, Parı´s F, Me´ndez C, Unanue I. Prediction of welding residual stresses and displacements by simplified models. Experimental validation. Mater Des 2010;31(3):1338
49. [17] Alpsten G, Tall L. Residual stresses in heavy welded shapes, Fritz Laboratory report. Lehigh University; 1969. [18] Young BW, Robinson KW. Buckling of axially loaded welded steel columns. Struct Eng 1975;53(5):203
7. [19] CEN. Eurocode 3: Design of steel structures, Part 1-1: General rules and rules for buildings, EN 1993-1-1. Brussels: European Committee for Standardization; 2005. [20] Wang YB, Li GQ, Chen SW. The assessment of residual stresses in welded high strength steel box sections. J Constr Steel Res 2012;76(0):93
9. [21] Qiang XH, Bijlaard FSK, Kolstein H. Post-fire mechanical properties of high strength structural steels S460 and S690. Eng Struct 2012;35(0):1
10.

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[22] American Institute of Steel Construction. Specification for structural steel buildings, ANSI/AISC 360-16. Chicago, IL: AISC; 2016. [23] Li TJ, Li GQ, Chan SL, Wang YB. Behavior of Q690 high-strength steel columns: Part 1: Experimental investigation. J Constr Steel Res 2016;123:18
30. [24] Li TJ, Liu SW, Li GQ, Chan S-L, Wang YB. Behavior of Q690 high-strength steel columns: Part 2: Parametric study and design recommendations. J Constr Steel Res 2016;122:379
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91.

Behavior and design of highstrength steel members under bending moment

7

Xiao-Lei Yan1,2, Guo-Qiang Li1 and Yan-Bo Wang1 1 Tongji University, Shanghai, P.R. China, 2Chengdu University of Technology, Chengdu, P.R. China

7.1

Introduction

Although a lot of valuable research work has been carried out to improve the application of high-strength steel (HSS) [1
9], the study on the flexural behavior and ultimate bending resistance of HSS beams is limited. The measurements of ultimate resistance to lateral torsional buckling of hot rolled I-section beams in Grade 55 steel and the residual stress distributions were performed by Dibley [10]. A total of 30 specimens of experimental investigation were performed under uniform bending to verify the proposed design curves, which is the basis of the bending design in BS 449 [11] for plate girders. The experimental investigation of the local stability of welded HSS with the yield stress around 800 MPa, was presented by Beg and Hladnik [12]. They tested 10 beams with different slenderness ratios of flange plates up to ultimate load and carried out numerical analysis on the effect of slenderness ratios of flange and web plate. In addition, a flexural
torsional experimental study on full-scale I-section beams fabricated from 800 MPa HSS were carried out by Lee et al. [13]. They estimated the effect of the flange slenderness on the flexural strength and rotation capacity. Bradford et al. [14] developed a nonlinear finite element (FE) model to investigate the flexural
torsional buckling ultimate capacity of various grade HSS beams. It was concluded through numerical simulation and parameter analysis that the buckling strength of the 960 MPa HSS beam was higher than that of the steel beam with a yield stress not higher than 690 MPa under the same normalized slenderness ratio. The main achievements of previous research focused on the local buckling, rotation capacity, and numerical simulation of HSS beams. However, if the beam does not have sufficient lateral stiffness and support, the beam may buckle around the minor bending axis and result in reduced bending moment resistance, as shown in Fig. 7.1. In China, welded I-section beams are widely used in the construction of steel structure. Compared to hot rolled I-section, the welded I-section shows a different pattern of residual stress distribution, especially for flame-cut flanges [5]. To address these concerns, in this research, an investigation on the overall buckling behavior of four-welded I-section beams with slenderness ratios of 95 and 105 were Behavior and Design of High-Strength Constructional Steel. DOI: https://doi.org/10.1016/B978-0-08-102931-2.00007-3 © 2021 Elsevier Ltd. All rights reserved.

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Behavior and Design of High-Strength Constructional Steel

Figure 7.1 I-section of flexural
torsional buckling.

carried out. The ultimate strength of flexural
torsional buckling of HSS beams were obtained from the tests and compared with steel structure design specifications. Moreover, an extensive parametric analysis was performed based on the experimentally verified nonlinear FE model with consideration of the effect of the measured welded residual stress distribution on the flexural
torsional buckling behavior of HSS welded I-section beams.

7.2

Experimental investigation

7.2.1 Material properties The Q460C HSS plates used in this experimental investigation was a high-strength low alloy structural steel of with the nominal yield strength is 460 MPa according to GB/T 1591-2008 [15]. The approximately equivalent steel to Q460C is S460 in EN 10025-6 [16], but the latter requires an additional verification of impact energy at 220 C. According to GB/T 228-2002 [17], four tension coupons of the 10 mm steel plate were cut from steel plates and tested on material mechanical properties. Two strain gages and an extensometer were arranged at the mid-height of each coupons to measure the longitudinal strains and the longitudinal deformation. The typical stress
strain characteristic of Q460C HSS was measured, and the curves are shown in Fig. 7.2. The values of the tension tested results are summarized in Table 7.1, where fy is the 0.2% proof stress, which was adopted as the yield strength of Q460C steel, fu is the ultimate tensile stress, E is Young’s modulus, and Δ is the percentage of elongation after fracture. The yield strength of the tension coupon is determined by the proof strength with 0.2% plastic strain since there is no obvious yield plateau observed for the tested Q460 steel.

Behavior and design of high-strength steel members under bending moment

273

Figure 7.2 Stress
strain curves. Table 7.1 Measured mechanical properties. E (GPa)

fy (MPa)

fu (MPa)

fy/fu

Δ (%)

505.8

597.5

0.486

23.7

207.8

7.2.2 Specimens To evaluate the flexural
torsional buckling behavior of Q460 HSS-welded I-section beams, four specimens with slenderness ratio between 95 and 155 about minor axis were fabricated from flame-cut Q460 steel plates. Three strips of 10 mm thickness steel plates were welded together to form doubly symmetric I-section beams by fillet welds of manual gas metal arc welding, as shown in Fig. 7.3. The electrode ER55-D2 of equivalent matching strength with Q460 steel plates was applied to achieve weld, and the size of the fillet weld was 10 mm. The measured dimensions of all the sections are listed in Table 7.2. The BAVG is the average value of BA and BC. Similarly the HAVG is the average value of HB and HD. L is the total length of the specimen, L1 and L2 are the distance from the beam end to the inner support, as shown in Fig. 7.9. The radius of gyration (ry) of cross-section for all specimens are also summarized in Table 7.2. The specimens were designed to investigate the overall buckling behavior of welded HSS beams subjected bend moment. In the European code for design of steel structures Eurocode 3, the width-tothickness ratios of one-side supported plate is restricted to b # 14ε tf

(7.1)

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Behavior and Design of High-Strength Constructional Steel

Figure 7.3 I-section of specimen.

And, the width-to-thickness ratios of two-side supported plate is restricted to h # 124ε (7.2) tw qffiffiffiffiffiffi where ε 5 235 460 5 0:71 in this chapter. The slenderness ratio limits of class 3 are 9.94 for flange plates and 88.04 for web plates. The specimen I-95-9-33 was designed with the minor axis slenderness ratio of 95, the width-to-thickness ratio (tbf ) of 9, and the height-to-thickness (thw ) of 33, which represents the commonly used Q460 I-section beams. The specimen I-155-518 with tbf of 5, thw of 18 and the minor axis slenderness ratio of 95 was designed to achieve the elastic flexural
torsional buckling of beam. The slenderness ratios of flange and web plates are all lower than the limits of slenderness ratio, either specified in the Eurocode 3 or in the GB50017-2017. Thus the local buckling of plate will designed to prevent the occurrence before reaching the peak load point, and the failure mode of overall buckling will dominate the ultimate bearing capacities of the beams.

7.2.3 Initial geometric imperfections In practice, initial imperfections were introduced during manufacturing process. The initial geometric imperfection in this study is comprised of out of straightness, out of flatness, and initial distortion, as shown in Fig. 7.4. A wire was attached on the surface of the I-section flange and web, and it go through the center line of the plate from one end of the beam to the other. Any deviation of the web and flange at mid-length was regarded as the initial out of straightness, and the values of Δ in Table 7.3 were the average of multiple measurements. The out of flatness and initial distortion were measured by rectangular steel ruler. The maximum values of out of flatness and initial distortion are summarized in Table 7.3.

Table 7.2 Measured dimensions of test specimens. Specimens

BA (mm)

BC (mm)

BAVG (mm)

tf (mm)

tW (mm)

HB (mm)

HD (mm)

HAVG (mm)

L (mm)

L1 (mm)

L2 (mm)

ry (mm)

Classification

I-155-5-18-1 I-155-5-18-2 I-95-9-33-1 I-95-9-33-2

101.6 99.9 179.9 180.2

102.1 100.1 179.1 180.1

101.9 100.0 179.5 180.15

10.08 10.08 10.08 10.08

10.9 10.0 10.6 10.2

198.1 200.9 350.2 350.8

205.8 199.1 350.8 349.2

202.0 200.0 350.5 350.0

7921 8005 8996 9010

985 988 1491 1492

986 985 1495 1490

21.9 21.0 38.0 37.8

Class 1 Class 1 Class 3 Class 3

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Behavior and Design of High-Strength Constructional Steel

Figure 7.4 Geometrical imperfection. (A) Out of straightness. (B) Out of flatness. (C) Initial distortion. Table 7.3 Result of geometrical initial imperfection. Specimens I-155-5-18-1 I-155-5-18-2 I-95-9-33-1 I-95-9-33-2

Δ (mm)

δ1 (mm)

δ2 (mm)

δ3 (mm)

δ4 (mm)

HB (mm)

HD (mm)

1.8 2.2 3.8 4.1

0.5 0.6 1.2 1.0

0.8 0.6 1.0 0.9

0.9 1.2 2 3.5 2.2

1.6 0.8 2 4.0 2.5

198.1 200.9 350.2 350.8

205.8 199.1 350.8 349.2

7.2.4 Experimental setup and instrumentation The experiment was carried out at the State Key Laboratory for Disaster Reduction in Civil Engineering in Tongji University. Generally loads can be easily applied to beams by actuators provided with primary deflections in the direction of that load. However, I-section beams subjected to failure of flexural
torsional buckling and Isection beams are susceptible to a deflected and twisted equilibrium position, such as that shown in Fig. 7.1. Under the circumstances, the request for the following up to the side-sway of the jacks would be a challenge for test setup. This problem has received a variety of potential solutions, which can be divided into three categories: gravity loading method [18], gravity-load simulator [19], and effective length method [10,20]. In view of the high target load and the complexity of the gravityload simulator, the effective length method is used in this study, as shown in

Behavior and design of high-strength steel members under bending moment

277

Fig. 7.5. Four-point loading was used so that the center unsupported span carried a uniform bending moment. At both ends of the beam, vertical loads were applied by actuators, while the torsional deformation is constrained by the cylindrical steel rod attached to both left and right sides of the I-section, as shown in Figs. 7.6(A) and 7.7. On other hand, the quarter points of the beam were constrained by vertical spherical supports and lateral cylindrical supports, as shown in Figs. 7.6(B) and 7.8. Consequently the effective length of Le of simply supported beam under equal end moments was equivalently simulated, as shown in Fig. 7.9. To identify the effective length of the simulated simply supported beam with equal end moments, a series of strain gauges (labeled as “S-xx”) and the linear varying displacement transducers (LVDTs, labeled as “D-xx”) were arranged, as shown in Fig. 7.10. Measurement type I was consisted of two axial strain gages arranged symmetrically at the two edges of the compression flange and two lateral LVDTs. The measurement type I (the Sections 1
5 and the Sections 6
10 in Fig. 7.11) was set to identify the position of the point of inflection with an interval of 200 mm. The measurement type II was set at the mid-length Section 11 to measure the maximum strain and deformation in the beam, which was consisted of nine axial strain gauges and six LVDTs, as shown in Fig. 7.10(B).

7.2.5 Failure mode and experimental procedures As expected, both the specimens I-155-5-18-1 and I-95-9-33-1 failed due to lateral torsional buckling, and the local buckling did not occur before the peak load, as

Figure 7.5 Test load devices.

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Behavior and Design of High-Strength Constructional Steel

Figure 7.6 Sectional plane of test load devices. (A) Cross-section of 1-1. (B) Cross-section of 2-2.

shown in Figs. 7.12 and 7.13. Local buckling was not observed before the peak load. The maximum lateral deformation appeared at the mid-length section and the deformed shape was the same as the assumed shape in Fig. 7.9. At the initial stage of loading, the in-plane deflection of the specimen developed slightly, and increased linearly with the increase of load. The flexural specimen arched slowly with the increase of load. The out-of-plane deflection and twisting of the specimen could not be observed when the applied load was less than 70% of the ultimate load. After that, the in-plane deflection increased faster and the outplane deflection became obvious. After the achieved ultimate resistance, it was difficult to stabilize the load. Finally unloading of the actuator and the failure of the beam were observed.

Behavior and design of high-strength steel members under bending moment

Figure 7.7 Loading end.

Figure 7.8 Lateral support and pin-ended.

279

280

Behavior and Design of High-Strength Constructional Steel

Figure 7.9 Buckling mode of a lateral unsupported beam under four-point loading. (A) Elevation view. (B) Plan view. (C) Idealized buckling mode of a lateral unsupported beam.

7.2.6 Load
deflection curves The measured in-plane and out-plane load
deflection curves of the four specimens were plotted in Figs. 7.14
7.17. The in-plane flexure load
deflection curve consisted of three parts: elastic stage, inelastic hardening stage, and inelastic softening stage. The curve showed a linear relationship at the initial loading stage. Then, the slope of the load
lateral deflection curve decreased at about 70% of the ultimate load. After reached the ultimate bearing capacity, the deflection developed very fast until failure. For the out-plane load
deflection curve, there was little deflection and torsion can be observed during the elastic stage. As the load approaching the ultimate bearing capacity, the compression flange rotated outwards in the direction of the initial bow. The onset time of torsional deformation was closely related to the initial geometrical imperfection of the specimen. The bigger the initial geometrical defects, the larger torsional deformation of the beam, and vice versa. Similarly after reached the ultimate bearing capacity, the lateral deflection and torsion of the beam increased rapidly until failure.

Behavior and design of high-strength steel members under bending moment

281

Figure 7.10 Layout of strain gauges and LVDTs. (A) Measurement type I. (B) Measurement type II. LVDTs, Linear varying displacement transducers.  Note: The Sections 7.1
7.10 were measurement of type I-section, which were arranged near the predicted point of inflection with an interval of 200 mm; The Section 7.11 was measurement of type II-section, which was the mid-length section.

Figure 7.11 Diagram of test position.

7.2.7 Load
strain curves The typical load
strain curves measured at the mid-length section of the specimen I-95-9-33-2 are shown in Fig. 7.18, including the strains of compression flange, tension flange, and web. At the initial loading stage with N # 185 kN, the compressive flange strain gauge S11, S12, and S13 showed compressive strains, while the tensile flange strain gauge S17, S18, and S19 showed tensile strains. With the increase of load, the curves measured from different strain gauges are almost coincident to each other. When the loading reaches 185 kN m, the Fig. 7.19 shows the linear relationship of mid

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Behavior and Design of High-Strength Constructional Steel

Figure 7.12 Initial loaded specimens and specimens near failure. (A) I-155-5-18-1. (B) I-959-33-1.

cross-section strain of I-95-9-33-2, where S11 and S18 are strain gauges in the middle of the flange plate, and S14, S15, and S16 are strain gauges on the web. With the increase in load and 185 kN m , M , 293.4 kN m, the specimen showed obvious out-of-plane deflection and rotation at the mid-length section of the beam. As approaching to the ultimate bearing capacity, the strain increment was

Behavior and design of high-strength steel members under bending moment

283

Figure 7.13 Partial enlarged detail of lateral support.

Figure 7.14 Load
deflection curves of I-155-5-18-1. (A) Out-of-plane load
deflection curves. (B) In plane load
deflection curves.

intensified. Some strain gauges were even damaged due to excessive deformation. Due to the lateral deflection, the specimen subjected to biaxial bending. About the minor axis of the I-section, the left side (S11 and S17) subjected to compression and the right side (S13 and S19) subjected to tension, as shown in Fig. 7.18.

7.2.8 The investigation of effective length The effective length of the specimen about minor axis could not be considered as ideally simply supported beam because of the arrangement of restraints and the inevitable friction between the support and the specimen. In the test, the uniform

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Behavior and Design of High-Strength Constructional Steel

Figure 7.15 Load
deflection curves of I-155-5-18-2. (A) Out-of-plane load
deflection curves. (B) In plane load
deflection curves.

Figure 7.16 Load
deflection curves of I-95-9-33-1. (A) Out-of-plane load
deflection curves. (B) In plane load
deflection curves.

Figure 7.17 Load
deflection curves of I-95-9-33-2. (A) Out-of-plane load
deflection curves. (B) In plane load
deflection curves.

Behavior and design of high-strength steel members under bending moment

285

Figure 7.18 N-ε Curves of I-95-9-33-2. (A) Compression flange. (B) Tension flange. (C) Web.

section specimen with equal end moments must have inflection points in the middle span, as shown in Fig. 7.9. The position of inflection points can be identified by analyzing the measured strain. The strain increments caused by the lateral deflection of middle span and side spans have different sign since a hogging segment must have followed with a sagging segment. Therefore the inflection points can be determined by measuring the adjacent sections with opposite signs of the strain increment. Taking the specimen, I-95-9-33-2 as an example, the specific method to determine the effective length was shown in Table 7.4. The effective lengths of the four specimens and the ultimate bearing capacity to bending moment were summarized in Table 7.5.

7.2.9 Finite element modeling To simulate the flexural
torsional buckling behavior of HSS beams, a threedimensional (3D) FE model was developed by using the general FE software ANSYS [14].

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Behavior and Design of High-Strength Constructional Steel

Figure 7.19 Linear relationship of mid cross strain of I-95-9-33-2.

Table 7.4 Measurement of points of inflection. Section no. 1 2 3 4 5 6 7 8 9 10

Strain gauge

ε180

ε190

ε180 2 ε190

Deflection

S01 S02 S01 S02 S01 S02 S01 S02 S01 S02 S01 S02 S01 S02 S01 S02 S01 S02 S01 S02

21355 21700 21346 21494 21202 21439 21762 21268 21736 21203 21853 21401 21901 21370 21454 21653 21406 21569 21498 21886

21091 22258 21283 21787 21274 21548 22066 21210 22152 2998 22348 21121 22137 21308 21529 21761 21327 21822 21151 22341

2264 558 263 293 72 109 304 258 416 2205 495 2280 236 262 75 108 279 253 2347 455

Sagging Sagging Sagging Hogging Hogging Hogging Hogging Sagging Sagging Sagging

Notes: The section no. was the location of cross-section, which was illustrated in Fig. 7.11; Strain gauge was illustrated in Fig. 7.10(B); ε180 was strain, when loading reached 180 kN; ε190 was strain, when loading reached 190 kN.

Behavior and design of high-strength steel members under bending moment

287

7.2.9.1 Initial geometric imperfections and residual stresses The initial imperfections include both the initial geometric imperfections and residual stresses, which are important factors that impair the flexural
torsional buckling strength of steel beams. With the consideration of longitudinal geometric imperfections in the analysis, the first order of eigenvalue buckling mode were used to simulate the initial out of straightness. The Δ is the measured maximum deflection, as shown in Fig. 7.4(A). In addition, the out of flatness and initial distortion were simulated through the locations of element node, as summarized in Table 7.3. The simplified residual stress model for Q460welded I-section proposed by [5] was adopted and implemented as initial stresses of the numerical model, as shown in Fig. 7.20. The signal αi and β i are the ratios of tensile and compressive residual stresses over yield strength of base metal, the subscript “i” indicates the location of the residual stress block. It can be obtained by quadratic polynomial fitting calculation [7].

Table 7.5 Ultimate bearing moment and effective length. Specimens I-155-5-18-1 I-155-5-18-2 I-95-9-33-1 I-95-9-33-2

Figure 7.20 Simplified residual stress.

Mu (kN m)

Le (mm)

68.5 73.6 303.45 285.75

3310 3220 3550 3620

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Behavior and Design of High-Strength Constructional Steel

7.2.9.2 Boundary conditions and mesh The boundary conditions of simply supported beam, which allowed to rotate about major and minor axis and warping displacements while restraining in-plane and out-of-plane deflections and twists, were established. The twist rotation of all end sections about longitudinal axis was prevented, meanwhile the horizontal and vertical displacement of the centroid of the web and the lateral displacements of all nodes of the section were constrained. The SHELL181 element was adopted for the simulation of flexural specimen, which is a high-order 3D element with 4-node, as shown in Fig. 7.21. The length of the meshed element is 20 cm in the longitudinal direction.

7.2.9.3 Material modeling The material behavior was modeled as an elastic
linear relationship with Von Mises’ yield function. The uniaxial stress
strain relationship for the HSS material was established by the bilinear stress
strain curve, as illuminated in Fig. 7.22. The measured Young’s modulus E, yield strength fy, ultimate strength fu, yield strain εy, and ultimate strain limit εu were used in the numerical simulation. The Poisson’s ratio of HSS was taken as 0.3.

7.2.9.4 Verification of finite element model To verify the accuracy of the FE model, the numerical results of the moment resistance to flexural
torsional buckling were compared with the experimental results,

Figure 7.21 Mesh section.

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289

Figure 7.22 Stress
strain relationship. Table 7.6 Comparison of experimental and predicted strengths. Specimens I-155-5-18-1 I-155-5-18-2 I-95-9-33-1 I-95-9-33-2

Experimental (kN)

Computed (kN)

Computed / Experimental

68.5 73.6 202.3 190.5

66.3 67.9 195.6 185.8 Mean value Standard deviation

0.968 0.923 0.968 0.975 0.959 0.024

as summarized in Table 7.6. The mean value of experimental to numerical ratio is 0.959 with standard deviation of 0.024. Based on the comparisons, the FE model can provide a reliable simulation to the lateral torsional buckling behavior of 460 MPa HSS-welded I-section beams. Consequently a parametric study was performed based on the FE analysis.

7.3

Parametric study and analysis

In the parametric study, a total of 150 specimens subjected to flexural
torsional buckling were investigated. The specimens were divided into six groups with different heights of I-sections. For each group, there were 25 beams with slenderness ranging from 60 to 300, which fall within the range of flexural
torsional buckling failure according GB50017-2017. The commonly used cross-sectional sizes in construction are adopted in the parametric analysis, as shown in Table 7.7 and Fig. 7.3. The initial deflection with 1m beam length and the simplified residual stress described in Fig. 7.20 were simulated in the simple residual stress model [5].

7.3.1 Initial geometric imperfections and residual stresses To compare the effect of geometric imperfection on the flexural
torsional buckling behavior bewteen Q460 HSS and Q235 NSS welded I-section beams, two groups of

Table 7.7 Dimensions of specimens for parametric study. Specimens I200 I320 I500 I580 I730 I850

B (mm)

tf (mm)

tw (mm)

h0 (mm)

H (mm)

b/tf

h0/tw

Classification

100 130 180 240 360 450

11 15 20 22 24 25

7 9 12 14 18 18

178 290 460 536 684 800

200 320 500 580 732 850

3.2 3.0 3.2 4.1 6.1 7.6

22.3 28.9 35.0 35.1 35.3 41.7

Class 1 Class 1 Class 1 Class 1 Class 1 Class 3

Behavior and design of high-strength steel members under bending moment

291

beams with identical cross-section sizes but different yield strengths (235 and 460 MPa) were undertaken. A central bow of Le/1000 was established in the numerical model. The material properties of the groups Q460 are given in Table 7.1. The elastic modulus of the groups Q235 steel is 206 GPa and the yield strength is 235 MPa. The comparison of the results of the groups of Q235 and Q460 are illuminated in Fig. 7.23, where the X-axis is defined as the normalized slenderness ratio. λn 5

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Wp f y =Mcr

π Mcr 5 l

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi π2 EIy ðGIt 1 EIω 2 Þ l

(7.3)

(7.4)

The Y-axis is defined as reduction factor ϕb 5

Mb Wp fy

(7.5)

where Mb is the predicted buckling moment resistance of I-beam and Wp is the plastic section modulus. Fig. 7.23 depicts the numerical analysis of the groups I500 of Q235 and Q460 steel beams, arranged by nondimensional slenderness ratio λn. It was demonstrated that the beams fabricated from Q460 HSS is less sensitive to the geometric imperfection than those fabricated from Q235 steel. The flexural
torsional buckling curve of Q460 steel is higher than the Q235 steel curve by 6.19%.

Figure 7.23 Sensitivity of different steel grades to geometric imperfection.

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Behavior and Design of High-Strength Constructional Steel

7.3.2 Effect of residual stress Residual stress is an important factor affecting the ultimate bearing capacity of beams in terms of the onset time of buckling. There is a close relationship between the peak tensile and compressive residual stress and width to thickness ratio and height to thickness ratio of the I-section. Therefore the variation of width or height to thickness ratios is accompanied by the variation of residual stress. The numerical results of six section groups with and without considering residual stress were shown in Figs. 7.24
7.28. It can be observed that the presence of residual stress impairs the ultimate bearing capacity of the beams. The effect of residual stress is small but cannot be ignored if the normalized slenderness ratio is 0.5 , λn # 1.3. To quantify the influence of residual stress on the ultimate bearing capacity to bending moment, the influence index of residual stress is defined as: Δϕ% 5

ϕ0 2 ϕi 3 100% ϕ0

(7.6)

where ϕ0 is the stability coefficient of the beam without considering residual stress, and ϕi is the stability coefficient of the beam with consideration of residual stress. The typical influence index of the section I200 is shown in Fig. 7.29. When λn $ 1:3, the flexural member tends to elastic flexural and torsional buckling. The predicted buckling coefficients are close to the critical value determined by elastic

Figure 7.24 Comparison of group I200.

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293

Figure 7.25 Comparison of group I320.

Figure 7.26 Comparison of group I500.

flexural and torsional buckling. The influence index of residual stress is less than 1%, which can be ignored in practice. When 0:5 # λn , 1:3, the flexural member trends to inelastic- lateral torsional buckling. The influence index increases with the

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Behavior and Design of High-Strength Constructional Steel

Figure 7.27 Comparison of group I580.

Figure 7.28 Comparison of group I730.

decrease in beam slenderness. When λn $ 0.85, the influence factor of residual stress is up to 18%. When λn , 0.5, the failure of the beam is mainly depended on the sectional strength rather than buckling resistance.

Behavior and design of high-strength steel members under bending moment

295

Figure 7.29 Comparison of group I850.

7.3.3 Effect of width-to-thickness ratio and height-to-thickness ratio To investigate the effect of width-to-thickness and height-to-thickness ratio, six groups of specimens with different cross-section sizes and slenderness ratios were carried out, as shown in Table 7.7 and Fig. 7.3. The width-to-thickness ratio b/tf varies from 3.2 to 7.6 and the height-to-thickness ratio h0/tw varies from 22.3 to 41.7. The groups of I200, I320, I500, and I580 are belong to class 1, in the EN1993-1-1. And the groups of I730 and I850 are classified into class 2 and class 3, respectively. According to the current design code, a nominal initial geometric imperfection of L/1000 of the span or 3 mm (whichever is the greater) were taken into account in the parametric study. The cross-sectional residual stress was applied as an initial stress at the element integration points, and the magnitude was determined based on the relevant experimental results as described in Ref. [5]. The residual stress distribution in I-shaped cross-section is closely related to width-tothickness ratio and height-to-thickness ratio. Fig. 7.30 compares the flexural
torsional stability coefficient between the groups of I200, I320, and I500. The groups of I200, I320, and I500 are designed with similar width-to-thickness ratios but different height-to-thickness ratios. The comparison shows that the flexural
torsional stability coefficient curves almost coincide to each other, which indicates the flexural
torsional stability coefficient is not affected by the height-to-thickness ratio of the web for compact I-sections. Fig. 7.31 compares the flexural
torsional stability coefficient between the groups of I500, I580, and I730. As can be seen from Table 7.7, the groups of I500, I580, and I730 are designed with the same height-to-thickness ratio but different

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Behavior and Design of High-Strength Constructional Steel

Figure 7.30 Relative difference parameters of residual stress of group I200.

Figure 7.31 Comparison of group I200, I320, and I500.

width-to-thickness. Fig. 7.31 shows that the flexural
torsional stability coefficient of groups I730 members is slightly higher than that of groups I500 and I580 with intermediate slenderness ratios. While, the difference between group I730 and I500 are 4.2%, as shown in Fig. 7.32. With the increase of section width and thickness

Behavior and design of high-strength steel members under bending moment

297

Figure 7.32 Comparison of group I500, I580, and I730.

ratio, the residual compressive stress ratio β 1 decreases. Therefore the influence of residual stress on the critical bending moment of beams decreases.

7.4

Comparison with current design codes

In design codes for steel structures, the design method for the member capacity to resist flexure-torsional buckling is based on an empirical strength curve. To evaluate the applicability of current GB50017-2017 [21], ANSI/AISC 360
16 [22], and EN1993-1-1 [23] for Q460 HSS-welded I-section beams, the buckling resistances obtained from numerical simulation and experimental results of beams subjected to uniform bending are compared with the prediction of codes.

7.4.1 Prediction of current codes According to stability theory of steel structure, the overall buckling failure of beams in bending can be divide into three types, elastic bucking, elastic-plastic buckling, and plastic bending. For the slender beams, the ultimate bending moment is close to the elastic flexural
torsional buckling moment. The design codes are basically consistence with the elastic flexural
torsional curve. For stock beams, the failure is only determined by fully plastic moment of cross-section. For the beams with intermediate slenderness, the collapse of beam is observed with the bending moment smaller than either fully plastic moment resistance of cross-section or elastic critical moment. The current design codes have different design curves due to the

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Behavior and Design of High-Strength Constructional Steel

differences in affected factors such as steel grade, cross-section, residual stress, and geometric initial imperfections. As specified in GB50017-2017, the beam resistance to flexural
torsional buckling is expressed as, Mx 5 ϕb Wx f

(7.7)

where ϕb is the stability factor, and it is equal to the reduction factor for lateral torsional buckling. When ϕb # 0.6, 2sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3   4320 Ah 4 λy t1 2 11 1 ηb 5ε2k ϕ b 5 βb 2 U 4:4h λy Wx

(7.8)

When ϕb . 0.6, the factor ϕb0 takes place of ϕb, and 0

ϕb 5 1:07 2

0:282 ϕb

(7.9)

where ϕb0 is the stability factor of beams of intermediate slenderness. Wx is the elastic moduli about the major X-axis. In the ANSI/AISC 360-16, the buckling design resistance for doubly symmetric compact I-shaped member bent about their major axis should be verified against as follow equations. When Lb # Lp, the limit sate of lateral torsional buckling does not apply. When Lp , Lb # Lr,      Lb 2 Lp Mn 5 Cb Mp 2 Mp 2 0:7Fy Sx # Mp Lr 2 Lp

(7.10)

When Lb . Lr Mn 5 Fcr Sx # Mp

(7.11)

where Lb is the length between points that are either braced against lateral displacement of the compression flange or braced against twist of the cross-section. Lr is the minimum length of elastic lateral torsional buckling. Lp is the minimum length of bended yielding. And Cb π2 E Fcr 5 2 Lb rts

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi Jc Lb 2 1 1 0:078 Sx h0 rts

(7.12)

Behavior and design of high-strength steel members under bending moment

299

The design curve of ANSI/AISC 360-16 are divided into three parts by limiting the unsupported length of the beam. According to EN1993-1-1, the laterally unrestrained member subject to major axis bending can be calculated by the following equations. Mb;Rd 5 χLT Wy

fy γ M1

(7.13)

where Wy is the appropriate section modulus as follows: Wy 5 Wpl,y for Class 1 or 2 cross-sections; Wy 5 Wel,y for Class 3 cross-section; Wy 5 Weff,y for Class 4 cross-section. In this research, it involves class 1, class 2, and class 3 sections. χLT is the reduction factor for lateral torsional buckling. 8 > < χLT # 1:0 1 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi butχLT # 1:0; or χLT # χLT 5 2 > 2 : λ LT ΦLT 1 Φ2LT 2 βλ LT

(7.14)

where h i   2 ΦLT 5 0:5 1 1 αLT λ LT 2 λ LT;0 1 βλ LT λ LT 5

rffiffiffiffiffiffiffiffiffi Wy fy Mcr

(7.15)

(7.16)

In EN 1993-1-1, the reduction factor χLT determined by two design methods of flexural
torsional beams, a simple but conservative general case method and a less conservative strict method for rolled sections or equivalent welded sections. According to the code, there are four design curves involving a and b curves for roll I-section and c and d curves for welded I-section. In the general case of EN 1993-1-1, β 5 1.0, λ LT;0 5 0:2, αLT 5 0.49 for c curve, and αLT 5 0.76 for d curve. For the strict method for rolled sections or equivalent welded sections, β 5 0.75 and λ LT;0 5 0:4, correspondingly. The cross-section and limits of h/b determined the imperfection factor αLT . In the experimental research of Section 7.2, the four specimens is applicable for curve c. In the numerical analysis of Section 7.3, the group I200 is applicable for curve c, and other groups are applicable for curve d.

7.4.2 Comparison with current design codes The test results and numerically simulated resistances of beams subjected to uniform bending are compared with the design curves of GB50017-2017, BS EN19931-1:2005, and ANSI/AISC360-16, as shown in Figs. 7.33 and 7.34. The design

300

Behavior and Design of High-Strength Constructional Steel

Figure 7.33 Comparisons of test results with current design codes.

Figure 7.34 Comparisons of numerical analysis with current design codes.

formulations of ANSI/AISC 360
16 and the GB50017-2017 for identifying the flexural
torsional buckling strength are according to the lateral unbraced length, therefore the unbraced length is expressed in terms of the normalized slenderness ratio according to Eq. (7.3).

Behavior and design of high-strength steel members under bending moment

301

The test results of four specimens in Section 7.2 is higher than all related curve of three national codes. To further quantify the comparison results, Eq. (7.17) is adopted. C5

ϕTest 2 ϕCode 3 100% ϕCode

(7.17)

where ϕTest is reduction factor of test value defined in Eq. (7.5), and ϕCode is reduction factor of three national codes. These comparison results are listed in Table 7.8. The ultimate bear capacity of specimens agree well with GB50017-2017 and ANSI/ AISC360-16, but EN1993-1-1 is relatively conservative. Fig. 7.34 depicts the normalized numerical analysis of FE models in comparison with design results. For slender beams, most of analysis results are close to the elastic buckling curve. The Eurocode 3 Strict (c) curve and the GB50017-2017 curve are the lower and upper limits of the corresponding numerical results, respectively. For the beams with intermediate slenderness (which 0.5 , λn # 1.3), it can be seen from Fig. 7.34 that the GB 50017-2017 curve is the lower limit of the analysis results while the ANSI/AISC360-16 curve is close to the average of numerical results. The comparison of numerical results with design codes is expressed by Eq. (7.18), C5

ϕNA 2 ϕCode 3 100% ϕCode

(7.18)

where ϕNA is reduction factor of numerical analysis. According to the parameter analysis in Section 7.2, the group of I200 and I850 are the lower and upper limits of the corresponding numerical results, respectively. The design codes are compared with the group I200 and I850 in Table 7.9. The differences between the calculated stability coefficient and those of the ANSI/AISC 360
10 and GB 500172017 are within 7.37%. This indicates that the current ANSI/AISC 360
16 and GB 50017-2017 can provide an accurate prediction of the buckling resistance of Q460 HSS-welded I-section beams. For slender beams, the predictions by ANSI/AISC 360
16 and the GB50017-2017 are close to each other. For beams with intermediate slenderness, ANSI/AISC 360
16 provides relatively most accurate predictions Table 7.8 Comparison between test results and current design codes. Specimens I-155-5-18-1 I-155-5-18-2 I-95-9-33-1 I-95-9-33-2

CGB50017 (%)

CANSI/AISC360 (%)

CEC3 General (c) (%)

CEC3 Strict (c) (%)

1.97 7.31 6.62 3.25

4.70 9.95 3.53 0.60

58.75 66.91 57.31 53.12

29.17 35.88 30.33 26.69

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Behavior and Design of High-Strength Constructional Steel

Table 7.9 Comparison between numerical analysis results and current design codes. Codes CGB50017 CANSI/AISC360 CEC3 General (c) CEC3 General (d) CEC3 Strict (c) CEC3 Strict (d)

Groups

Mean value (%)

Standard devation (%)

I200 I850 I200 I850 I200 I850 I200 I850 I200 I850 I200 I850

-0.75 7.37 3.33 5.25 26.45 31.36 41.46 48.18 8.51 13.03 19.12 23.51

9.05 6.18 2.34 3.28 8.63 11.22 9.13 17.32 6.96 9.71 9.25 15.32

compared with other standards, although it slightly overestimates the buckling resistance. The Chinese standards is reliable to beams of intermediate slender with slightly overestimation for beams with relatively small slenderness. For stock beams, the curve of GB50017-2017 does not consider the plastic development capacity. The main reason is that the buckling resistance is determined by the yield criterion of edge fiber. The buckling resistance curve predicted by the general method of Eurocode 3 appears to be the most conservative one.

7.5

Summary

Four beams of experimental research on the flexural
torsional buckling behavior of Q460 steel welded I-section beams subjected to uniform bending is presented. Considering the affection of geometrical imperfections and residual stresses, a nonlinear FE model was established and verified against experimental results. With the verified numerical model, parametric analysis was carried out to evaluate the effect of steel strength, initial imperfections, beam slenderness, and width-to-thickness ratio on the buckling resistance of Q460 HSS double symmetric I-section beams. The experimental and numerical results are compared with the current design codes. Based on this research, the following conclusions have been made: 1. The validated nonlinear FE model with consideration of geometrical imperfection and residual stress can give accurate prediction of the flexural
torsional buckling behavior of Q460 steel welded I-section beams subjected to uniform bending. 2. It is demonstrated that the presence of residual stresses would impair the buckling resistances of steel beams. The buckling resistance for Q460 HSS beams with welded I-section are higher than that of Q235 NSS beams based on the normalized strength. The main reason is that the higher grade of steel, the less sensitive to geometric imperfection and welding residual stresses.

Behavior and design of high-strength steel members under bending moment

303

3. The assessment of current design codes shows that American National Standard (AISC 360-16) can provide an accurate prediction of the moment capacities of beams fabricated from Q460 HSS. EN1993-1-1 is more conservative than Chinese Code (GB50017-2017) and American National Standard (AISC 360-16).

References [1] Fukumoto Y. New constructional steels and structural stability. Eng Struct 1996;18:786
91. [2] Ricles JM, Sause R, Green PS. High-strength steel: implications of material and geometric characteristics on inelastic flexural behavior. Eng Struct 1998;20:323
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65. [6] Wang YB, Li GQ, Chen SW, Sun FF. Experimental and numerical study on the behavior of axially compressed high strength steel box-columns. Eng Struct 2014;58 (2014):79
91. [7] Wang YB, Li GQ, Chen SW, Sun FF. Experimental and numerical study on the behavior of axially compressed high strength steel columns with H-section. Eng Struct 2012;43(2012):149
59. [8] Yan XL, Wang YB, Li GQ. Q460C welded box-section columns under eccentric compression. Proceedings of the Institution of Civil Engineers 2018;171(8):611
24. [9] Shi G, Wang M, Bai Y, Wang F, Shi Y, Wang Y. Experimental and modeling study of high-strength structural steel under cyclic loading. Eng Struct. 2012;37:1
13. [10] Dibley JE. Lateral torsional buckling of I-sections in grade 55 steel. ICE Proceedings 1969;43(4):599
627. [11] Specification for the use of structural steel in building. Metric units. BS 449-2 1969. [12] Beg D, Hladnik L. Slenderness limit of class 3 I cross-sections made of high strength steel. J. Constr. Steel Res. 1996;38(3):201
17. [13] Lee C-H, Han K-H, Uang C-M, Kim D-K, Park C-H, Kim J-H. Flexural strength and rotation capacity of I-shaped beams fabricated from 800-MPa steel. J. Struct. Eng. ASCE 2013;139(6):1043
58. [14] Bradford Mark A, Liu Xinpei. Flexural
torsional buckling of high-strength steel beams. J Construct Steel Res 2016;124(2016):122
31. [15] GB/T 1591-2008, High strength low alloy structural steels. China Standard Press, Beijing, China (in Chinese). [16] EN 10025-6: Hot rolled products of structural steels, Part 6: technical delivery conditions for fiat products of high yield strength structural steels in the quenched and tempered condition. CEN (2014), Brussels, Belgium. [17] GB/T 228-2002. Metallic materials-tensile testing at ambient temperature[S]. Beijing: China Standards Press; 2002 (in Chinese). [18] Tong Genshu. Out plane stability of steel structures. China Construction Industry Press, Beijing. (in Chinese).

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[19] Yarimci E, Yura JA, Lu LW. Techniques for testing structures permitted to sway. SESA Spring meeting held in ottawa, Ontario, Canada. [20] Bangfei Han. Experimental study on overall stability of steel beam. J Shijiazhuang Railway Institute 1992;5(3):57
64 (in Chinese). [21] GB 50017-2017. Code for design of steel structures. Beijing: China Architecture & Building Press; 2017 [in Chinese]. [22] American Institute of Steel Construction. Specification for structural steel buildings, ANSI/AISC 360-16. Chicago, Illinois: AISC;; 2016. [23] British Standards Institution (BSI), BS EN 1993-1-1: 2005 Eurocode 3: Design of Steel Structures Part 1-1: General Rules and Rules for Buildings, BSI, London, 2005

Behavior and design of highstrength steel columns under combined compression and bending

8

Kwok-Fai Chung1, Tian-Yu Ma1,2, Guo-Qiang Li2 and Xiao-Lei Yan2 1 Hong Kong Polytechnic University, Hong Kong, P.R. China, 2Tongji University, Shanghai, P.R. China

8.1

Introduction

This chapter presents both experimental [1,2] and numerical investigations [2,3] on high-strength steel (HSS) columns under combined compression and bending. Two types of sections, including welded H-sections and welded box-sections, are considered for the columns. In the experimental investigation, a total of eight columns of welded H-sections made of Q690 steel and a total of seven columns of welded boxsections made of Q460 steel were tested under eccentric loads. Initial out-ofstraightness of these columns was measured during the preparation of the test, and initial loading eccentricity was acquired by different methods. In the test process, applied load, local strain, axial deformation, and lateral deflection were obtained. All the columns failed by overall buckling. The buckling resistances measured in these tests were compared to their corresponding design results according to different design codes. Then, numerical investigations were performed by using finite element models. Geometrical and material nonlinearities were incorporated in these finite element (FE) models, and these models showed excellent capability of replicating the key test data. Upon validation of the FE models, parametric studies were conducted to evaluate different effects influencing buckling resistances of HSS columns of welded sections. Afterward, applicability of design rules on steel columns under combined compression and bending given in different design codes is assessed by means of the ratios of the FE to design buckling resistances. And design proposals on HSS columns under combined compression and bending were made.

Behavior and Design of High-Strength Constructional Steel. DOI: https://doi.org/10.1016/B978-0-08-102931-2.00008-5 © 2021 Elsevier Ltd. All rights reserved.

306

Behavior and Design of High-Strength Constructional Steel

8.2

Experimental investigation

8.2.1 H-sections 8.2.1.1 Test program A total of eight slender columns of welded H-sections were tested under combined compression and bending about minor axis of their cross sections. These columns were made from high-strength Q690 steel plates with nominal thicknesses of 6, 10, and 16 mm. Four sections of different cross-sectional dimensions, Sections H1, H2, H3, and H4, were involved. Their nominal dimensions and section classifications according to EN 1993-1-1 and ANSI/AISC 360-16 are shown in Fig. 8.1, while their measured dimensions and section properties are summarized in Table 8.1.

8.2.1.2 Specimen fabrication All the specimens were fabricated under the following steps: G

G

G

Cut the steel plates into strips and assemble them to form H-sections with tack welds. A preheating of 120
C was applied to web-to-flange junctions to facilitate good quality welding. For each section the web was connected to the flanges with fillet welds on both sides of the web. Gas metal arc welding (GMAW) with a fillet size of 6 mm was used for Sections H1 and H2, while submerged arc welding (SAW) with a fillet size of 10 mm was used for Sections H3 and H4. The fillet sizes were assigned to be the same as the web thicknesses to ensure structural adequacy. Each fillet was formed in a single run that was staggered with a length of 500600 mm along the column length to minimize distortion due to welding. Technical information on electrodes and welding parameters are shown in Table 8.2. Since the electrical parameters fluctuated during welding, average values were taken.

z 10

z 10

y y

6 120

y 140

6

z 16

z 16

170

150

y 232

282

10

10

200

Class1 Compact

Class3 Noncom pact

Class2 Compact

(A)

(B)

(C)

250

Class3 Noncompact (D)

Figure 8.1 Nominal cross-sectional dimensions of welded H-sections: (A) Section H1, (B) Section H2, (C) Section H3, and (D) Section H4.

Table 8.1 Measured dimensions and section properties of welded H-sections. Test

EH1P EH1Q EH2P EH2Q EH3P EH3Q EH4P EH4Q

Section depth

Section width

Flange thickness

Web thickness

Specimen length

Effective length

Area

Second moment of area

Radius of gyration

H

b

tf

tw

Ls

Leff

A

Iz

iz

(mm)

(mm)

(mm)

(mm)

(mm)

(mm)

(mm2)

( 3 106 mm4)

(mm)

140.0 141.2 170.0 170.0 231.8 231.7 284.2 282.0

119.6 119.8 149.3 149.7 201.5 200.7 250.1 249.9

9.90 9.91 9.90 9.92 15.98 15.97 15.97 15.93

5.83 5.85 5.81 5.85 9.92 9.95 9.92 9.93

1612 2410 1613 2410 1613 2412 1611 2410

1992 2790 1993 2790 1993 2792 1991 2790

3070 3085 3827 3847 8422 8397 10490 10448

2.83 2.84 5.49 5.54 21.81 21.54 41.66 41.47

30.4 30.3 37.9 38.0 50.9 50.6 63.0 63.0

Table 8.2 Information on welding electrodes and welding parameters. Section

Welding method

Welding electrodes Product designation

H1 and H2 H3 and H4

GMAW SAW

CHW-80C1 CHW-S80

GMAW, Gas metal arc welding; SAW, submerged arc welding.

Welding parameters

Diameter (mm)

Yield strength (N/mm2)

Tensile strength (N/mm2)

Voltage (V)

Current (A)

Speed (mm/s)

Fillet size (mm)

1.2 4.0

660 680

760 760

30 36

240 450

4.1 6.1

6 10

308

G

Behavior and Design of High-Strength Constructional Steel

After the H-sections were assembled, a pair of Q345 steel 30 mm thick end plates was welded onto both ends of the H-sections. Triangular stiffeners were welded to strengthen the connections between end plates and flanges.

8.2.1.3 Material properties To obtain the material properties of Q690 steel plates, a total of nine tensile tests were carried out. The stressstrain curves for all the tensile tests are plotted in Fig. 8.2. It is found that for all the Q690 steel plates with different thickness, no definite yielding plateau is observed, and strain hardening started shortly after occurrence of yielding. The measured material properties are summarized in Table 8.3. It should be noted that EN 1993-1-12 specifies the following ductility criteria for steel materials with steel grades from S460 up to S700: (1) fu/fy $ 1.05, (2) elongation at failure not less than 10%, and (3) εu $ 15 fy/E. It is shown that all the steel plates satisfy these ductility criteria, and they are readily qualified to be HSS materials to EN 1993-1-12.

8.2.1.4 Test setup All the tests were conducted with a 1000 t universal servo-controlled testing machine, and the general test setup is shown in Fig. 8.3. A pair of attachments was connected to both ends of the H-sections through bolts. These attachments provided an eccentricity of about 100 mm along the minor axis of the H-sections for the applied compressive loads. And they also enabled the H-sections to rotate freely at both ends about the minor axis of the H-sections. Therefore these columns were tested under combined compression and bending. 1000

Stress, σ (N/mm2)

800

600

400

200

0 0.00

T06 - A/B/C T10 - A/B/C T16 - A/B/C 0.05

0.10

0.15

Strain, ε Figure 8.2 Stressstrain curves of Q690 steel plates.

0.20

0.25

Table 8.3 Mechanical properties of Q690 steel plates. Nominal thickness t (mm)

Coupon

6

T06-A T06-B T06-C Average T10-A T10-B T10-C Average T16-A T16-B T16-C Average

10

16

Young’s modulus E (kN/mm2) 210 210 209 210 212 214 211 212 208 206 212 209

Yield strength fy (N/mm2) 771 764 763 766 753 758 756 756 800 797 804 800

Tensile strength fu (N/mm2)

Ratio fu/fy

Strain at fu εu

819 810 817 815 788 796 794 793 855 833 843 844

1.06 1.06 1.07 1.06 1.05 1.05 1.05 1.05 1.07 1.05 1.05 1.05

0.059 0.060 0.058 0.059 0.065 0.078 0.067 0.070 0.064 0.065 0.068 0.066

Elongation at fracture A (%) 15.5 15.3 16.0 15.6 18.2 18.9 18.7 18.6 19.7 17.9 19.3 19.0

310

Behavior and Design of High-Strength Constructional Steel

e = 100 mm

N

SG 2

SG 1 A

DT 6 A

Specimen length, Lcr

Effective length, Leff

DT 1

SG 3

A-A DT 5

SG 5

SG 4

SG 6 DT 2

B

B DT 2/3 DT 3 SG 7

C

C

SG 8 B-B

SG 9

SG 10

DT 4 SG 11

C-C

SG 12 LVDT Strain gauge

Figure 8.3 Test setup for welded H-sections under eccentric loads.

A total of 12 strain gauges were mounted onto the outer surfaces of the flanges of the H-sections at three cross sections, namely, Sections AA, BB, and CC. At each section, four strain gauges were installed 10 mm away from the edges of the flanges. Displacement transducers DT1, DT2, DT3, and DT4 were used to measure lateral deflections of the H-sections at these three sections along the direction of the major axis of their cross sections. It should be noted that any difference in the measurements between Transducers DT2 and DT3 would give a twisting of the H-sections along their longitudinal axes. Transducer DT5 was used to capture any lateral deflection of the H-section along the minor axis at Section BB for a monitoring purpose while Transducer DT6 was used to measure axial deformations of the H-sections.

8.2.1.5 Initial out-of-straightness Before testing, initial out-of-straightness of each H-section was measured. A steel wire was attached to the surface of one flange of the H-section, and it ran through the centerline of the flange from Sections AA to CC. Any deviation of the flange at the midheight of the H-section was regarded as the initial out-of-straightness of this

Behavior and design of high-strength steel columns under combined compression and bending

311

flange, denoted as v1. Measurement was repeated on the other flange to obtain v2. The average value of v1 and v2 was considered to be the initial out-of-straightness of the H-section, denoted as v. The initial out-of-straightness for all the H-sections is summarized in Table 8.4. Due to limitations of this measuring method, any initial out-of-straightness smaller than the radius of the steel wire, that is, 0.25 mm, could not be recognized, and this situation is denoted with “.” It is shown that the absolute values of the measured out-of-straightness of all the H-sections are smaller than 1.0 mm, also far away from 1/1000 of their effective lengths, Leff. Thus these initial out-of-straightnesses would have very little effects on buckling behavior of the H-sections. In general, quality of the workmanship in fabricating these high-strength Q690 steel welded H-sections was considered to be high and readily achieved in modern fabrication shops.

8.2.1.6 Test procedures In the initial stage of testing a load was applied at a loading rate of 30105 kN/min onto the H-sections, depending on their cross-sectional areas. Under this loading condition the average stress rate for each H-section was kept to be smaller than 10 N/ mm2/min. This loading rate was maintained until 80% of the predicted (or designed) buckling resistances of each H-section were attained. The calculation of the predicted buckling resistances will be discussed later. Then, a displacement control was adopted with a deformation rate of 0.5 mm/min. Under this displacement rate the average strain rate for each H-section was smaller than 0.00025/min, and hence, these tests should be regarded as static tests. The tests terminated after the applied load attained its maximum value and dropped down to 85% of the maximum value.

8.2.1.7 Test results Failure modes and failure loads All the H-sections failed in overall flexural buckling about minor axis of their cross sections, as shown in Fig. 8.4. The maximum applied load observed in each test Table 8.4 Initial out-of-straightness at midheight of welded H-sections. Test EH1P EH1Q EH2P EH2Q E3HP E3HQ E4HP E4HQ

v1 (mm)  2 0.5 1 0.2 2 0.5  2 1.5 2 0.5 

v2 (mm) 1 0.3 2 0.5  2 0.5  2 0.5 2 0.5 

v 5 (v1 1 v2)/2 (mm) 1 0.2 2 0.5 1 0.1 2 0.5 0 2 1.0 2 0.5 0

Leff (mm) 1992.0 2790.1 1993.3 2790.3 1993.3 2791.6 1990.5 2790.1

|v|/Leff ( 3 1023) 0.1 0.2 0.1 0.2 0.0 0.4 0.3 0.0

Notes: (1) “” represents a value smaller than 0.25, and it may be taken to be 0. (2) The signs of these values indicate the positions of the deviations.

312

Behavior and Design of High-Strength Constructional Steel

Figure 8.4 Overall flexural buckling of welded H-sections under eccentric loads: (A) test EH1P, (B) test EH1Q, (C) test EH2P, (D) test EH2Q, (E) test EH3P, (F) test EH3Q, (G) test EH4P, and (H) test EH4Q.

Behavior and design of high-strength steel columns under combined compression and bending

313

was regarded as the measured buckling resistance of the corresponding column, as summarized in Table 8.5.

Loaddeformation relationships For all welded H-sections, lateral deflections of the flanges at midheight were recorded by Transducers DT2 and DT3. Hence, the average values of these two transducer readings were regarded as the lateral deflections, Δy, of the welded H-sections. The relationships between the applied load, N, and the lateral deflection, Δy, of all welded H-sections are shown in Fig. 8.5. It should be noted that the maximum differences between these two transducer readings were found to be smaller than 0.2 mm throughout the testing, and hence, twisting of the H-sections at midheight of the welded H-sections was considered to be insignificant in the present tests. Moreover, axial deformations of the welded H-sections, Δx, were measured with Transducer DT6. The relationships between the applied load, N, and the axial deformation, Δx, of all the welded H-sections are shown in Fig. 8.6. It is shown that both the lateral deflections, Δy, and the axial deformations, Δx, increase almost linearly with an increase of the applied load, N, up to failure in all tests. After the failure loads, Ntest, were attained, unloading took place gradually with further deformations in all the welded H-sections. In general, all of these loaddeformation relationships are considered to be similar to those slender columns of welded H-sections made of conventional steel materials.

Loadstrain relationships For each H-section, there were a total of 12 strain gauges mounted onto the outer surfaces of their two flanges, and these strain gauges were divided into a group of 4 strain gauges at three different cross sections, Sections AA, BB, and CC. It is interesting to plot development of these axial strains measured at these three cross sections during load application. Fig. 8.7 plots relationships between the applied load, N, and the axial strains, εx, at various cross sections of test EH3P for easy comparison. It should be noted that under the presence of combined compression and bending at midheight of the welded H-section, that is, Section B-B, measured resultant axial stresses from strain gauges SG 6 and 8 are found to be in compression while those from strain gauges SG 5 and 7 are in tension. As the applied load increases, the strain readings of Table 8.5 Measured buckling resistances of welded H-sections. Test EH1P EH1Q EH2P EH2Q EH3P EH3Q EH4P EH4Q

Ntest (kN)

λz

λz

328 250 527 418 1698 1376 2662 2276

1.26 1.77 1.01 1.42 0.77 1.08 0.62 0.87

66 92 53 74 39 55 32 44

Behavior and Design of High-Strength Constructional Steel

600 400 200 0

0

50

100

150

Lateral deflection, Δy (mm)

Applied load, N (kN)

Applied load, N (kN)

314

N

600 400

Δy

200 0

0

50

600 400 200

0

50

100

150

Lateral deflection, Δy (mm)

400 200 0

0

50

Applied load, N (kN)

Applied load, N (kN)

150

(D)

1000

0

50

100

Lateral deflection, Δy (mm)

2000

1000

0

150

0

50

100

150

100

150

Lateral deflection, Δy (mm)

(E)

(F) 3000

Applied load, N (kN)

3000

Applied load, N (kN)

100

Lateral deflection, Δy (mm) 3000

2000

2000

1000

0

N

600

(C) 3000

0

150

(B) Applied load, N (kN)

Applied load, N (kN)

(A)

0

100

Lateral deflection, Δy (mm)

0

50

100

Lateral deflection, Δy (mm)

(G)

150

2000

1000

0

0

50

Lateral deflection, Δy (mm)

(H)

Figure 8.5 Relationships between applied loads and lateral deflections for welded H-sections under combined compression and bending: (A) test EH1P, (B) test EH1Q, (C) test EH2P, (D) test EH2Q, (E) test EH3P, (F) test EH3Q, (G) test EH4P, and (H) test EH4Q.

200 0

0

40

20

60

Axial deformation, Δx (mm)

Applied load, N (kN)

600

400

600

Applied load, N (kN)

Applied load, N (kN)

600

Applied load, N (kN)

Behavior and design of high-strength steel columns under combined compression and bending

600

N Δx

400 200 0

0

20

0

20

40

60

Axial deformation, Δx (mm)

200 0

0

20

Applied load, N (kN)

Applied load, N (kN)

60

(D)

1000

0

20

40

Axial deformation, Δx (mm)

2000

1000

0

60

0

20

40

60

40

60

Axial deformation, Δx (mm)

(F)

(E) 3000

Applied load, N (kN)

3000

Applied load, N (kN)

40

Axial deformation, Δx (mm) 3000

2000

2000

1000

0

N

400

(C) 3000

0

60

(B)

200 0

40

Axial deformation, Δx (mm)

(A) 400

315

0

20

40

Axial deformation, Δx (mm)

(G)

60

2000

1000

0

0

20

Axial deformation, Δx (mm)

(H)

Figure 8.6 Relationships between applied loads and axial deformations for welded H-sections under combined compression and bending: (A) test EH1P, (B) test EH1Q, (C) test EH2P, (D) test EH2Q, (E) test EH3P, (F) test EH3Q, (G) test EH4P, and (H) test EH4Q.

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Behavior and Design of High-Strength Constructional Steel

Applied load, N (kN)

2000

-fy/E

fy/E

z SG 1

1500

SG 2 y

1000 SG 1 SG 2 SG 3 SG 4

500 0 –0.06

–0.04

–0.02

0.00

Strain, εx

0.02

0.04

SG 3

SG 4

A –A

0.06

(A)

Applied load, N (kN)

2000

-fy/E

z

fy/E

1500

SG 5

SG 6

1000 SG 5 SG 6 SG 7 SG 8

500 0 –0.06

–0.04

–0.02

0.00

Strain, εx

0.02

0.04

y SG 7

SG 8

B–B

0.06

(B)

Applied load, N (kN)

2000

-fy/E

z

fy/E SG 9

1500

SG 10 y

1000 SG 9 SG 10 SG 11 SG 12

500 0 –0.06

–0.04

–0.02

0.00

Strain, εx

0.02

0.04

SG 11

SG 12

C –C

0.06

(C)

Figure 8.7 Relationships between applied load and axial strains of welded H-sections—test EH3P: (A) Section AA, (B) Section BB, and (S) Section CC.

these four strain gauges exceed the value of the yield strain, εy(or fy /E). Hence, yielding occurs, leading the H-section to fail in an elastoplastic manner.

Initial loading eccentricity The initial loading eccentricities at Section AA, eA, and Section CC, eC, are defined as the eccentricities of the rotation centers at the H-section ends with respect to the line passing through the centerline of the flange at Sections AA and CC before loading,

Behavior and design of high-strength steel columns under combined compression and bending

317

respectively, as shown in Fig. 8.8. The measured initial loading eccentricities, eA and eC, for the welded H-sections in an elastic stage were obtained by using the applied load readings, and the corresponding strain readings and lateral deflections under the applied load at Sections AA and CC, respectively. For Section AA the strain distribution of the whole cross section was obtained using strain readings measured by strain gauges SG5, SG6, SG7, and SG 8. The stress distribution at this cross section was then obtained by using the stressstrain curves obtained from the coupon tests, and hence, the corresponding internal moment under the applied load, MA,SG, was computed accordingly. Consequently, the initial loading eccentricity at Section AA, eA, was readily obtained through equilibrium consideration as follows: eA 5

MA;SG 2 dA N

(8.1)

N A line passing through the centerline of the flange at Sections A–A and C–C

dA

Under the applied load, N

eA

vB

Before loading dC

(eA+eC)/2 eC

N Figure 8.8 Measurement of loading eccentricity. Table 8.6 Loading Eccentricities of all the welded H-sections. Test

Section AA eA (mm)

Section CC eC (mm)

Average (eA 1 eC)/2 (mm)

EH1P EH1Q EH2P EH2Q E3HP E3HQ E4HP E4HQ

1 101.5 1 96.4 1 106.0 1 98.4 1 100.1 1 101.9 1 97.8 1 99.1

1 101.4 1 95.8 1 101.6 1 101.0 1 96.2 1 102.9 1 102.3 1 98.0

1 101.5 1 96.1 1 103.8 1 99.7 1 98.2 1 102.4 1 100.1 1 98.6

318

Behavior and Design of High-Strength Constructional Steel

where N is the applied load in the initial loading stage, dA is the corresponding lateral deflection at Section AA under the applied load, N. The initial loading eccentricity at Section CC, eC, was obtained from a similar process. Table 8.6 summarizes the initial loading eccentricities of all the welded H-sections. The average value of the initial loading eccentricities of each welded H-section was employed to calculate the first order applied moment in subsequent analyses and calibration.

8.2.1.8 Applicability of design rules Applicability of design rules given in EN 1993-1-1, ANSI/AISC 360-16, and GB 50017-2003 for welded H-sections under combined compression and bending is assessed through calibration against the test results. In these design rules the effects of axial compression and bending moments are summed up linearly, while nonlinear effects of applied bending moments are accounted for by interaction factors. In general, two formulae should be satisfied, each of which corresponds to member buckling about a principal plane. However, as there was no applied moment about the major axis of the cross sections of the H-sections, lateral-torsional buckling did not occur in the tests. Hence, the critical formula is the one corresponding to flexural buckling of the H-sections under bending about minor axis.

EN 1993-1-1 According to EN 1993-1-1, members who are subjected to combined compression and bending should satisfy the following equations: NEd My;Ed 1ΔMy;Ed M 1ΔMz;Ed
 1kyz z;Ed 1kyy #1 χy NRk =γM1 Mz;Rk =γM1 χLT My;Rk =γM1

(8.2)

NEd My;Ed 1ΔMy;Ed M 1ΔMz;Ed
 1kzz z;Ed 1kzy #1 χz NRk =γM1 Mz;Rk =γM1 χLT My;Rk =γM1

(8.3)

where NEd, My,Ed, and Mz,Ed are the design values of the compression force and the moments about the major (y) and the minor (z) axes along the member, respectively; NRk, My,Rk, and Mz,Rk are the characteristic values of resistances to compression force and the bending moments about the major (y) and the minor (z) axes, respectively; ΔMy;Ed and ΔMz;Ed are the moments due to the shift of the centroidal axes for Class 4 sections; χy and χz are the reduction factors due to flexural buckling about the major (y) and the minor (z) axes, respectively; χLT is the reduction factor due to lateral-torsional buckling; kyy, kyz, kzy, and kzz are the interaction factors; and γ M1 is the partial factor for resistance of members to instability assessed by member checks. The reduction factors χy and χz in the first terms in Eqs. (8.2) and (8.3) are determined by a suitable selection of flexural buckling curves. According to the complementary rules in EN 1993-1-12 for high-strength Q690 steel welded H-sections, a curve “c” is recommended to calculate the flexural buckling resistances of

Behavior and design of high-strength steel columns under combined compression and bending

319

the H-sections for buckling about the minor axis of the cross sections. The interaction factors kyy, kyz, kzy, and kzz in the second and the third terms may be obtained from two different approaches given in Annexes A and B, respectively. It should be noted that the main difference between these two approaches is the way of presenting different structural effects. As Annex A emphasizes transparency, each structural effect is accounted for by an individual factor. However, Annex B works with simplicity and allows some structural effects to be combined into a global factor. Based on these two approaches, the design resistances NEC3,c for all the H-sections were calculated through iterations. In the calculations, measured dimensions and mechanical properties as well as total initial geometrical imperfections were adopted. All the moment resistances of the H-sections are given by their plastic moduli even though Sections H2 and H4 are considered to be merely Class 3 sections. Table 8.7 summarizes both the failure loads Ntest and the design resistances NEC3,c of the H-sections. It should be noted that: G

G

According to the approach given in Annex A, the values of Ntest/NEC3,c are found to range from 1.06 to 1.11 with an average value of 1.09. According to the approach given in Annex B, the values of Ntest/NEC3,c are found to range from 1.10 to 1.24 with an average value of 1.20.

Comparison between the test and the design resistances may be illustrated through plotting test values onto the graphs of normalized interaction curves according to the approaches in Annexes A and B for each of the four H-sections in Fig. 8.9. As shown in the graphs, Annex B tends to give more conservative results when compared with Annex A. It should be noted that in order to improve structural efficiency of the design rules, curve “a” is suggested to be used in the flexural buckling design of the welded H-sections to give the axial buckling resistances NEC3,a of the sections under combined compression and bending. The values of NEC3,a are also summarized in Table 8.7 for direct comparison with those of NEC3,c. It should be noted that: G

G

According to the approach given in Annex A, the values of Ntest/NEC3,a are found to range from 1.02 to 1.07 with an average value of 1.05. According to the approach given in Annex B, the values of Ntest/NEC3,a are found to range from 1.03 to 1.16 with an average value of 1.11.

Hence, by selecting a proper parameter in designing flexural resistances of the H-sections, the approaches in both Annexes A and B are shown to be significantly improved in giving conservative and yet efficient resistances for high-strength Q690 steel welded H-sections under combined compression and bending about minor axis. The use of curve “a” in designing flexural resistances of welded H-sections is also supported by other researchers as structural effects of residual stresses are proportionally less pronounced in these sections, when compared with those of conventional steel materials.

Table 8.7 Calibration of EN 1993-1-1 for Q690 steel welded H-sections under combined compression and bending. Test

Ntest (kN)

λ

λ

EN 1993-1-1: Annex A NEC3;c (kN)

EH1P EH1Q EH2P EH2Q EH3P EH3Q EH4P EH4Q Average

328 250 527 418 1698 1376 2662 2276 

66 92 53 74 39 55 32 44 

1.26 1.77 1.01 1.42 0.77 1.08 0.62 0.87 

295 231 477 386 1599 1250 2481 2075 

EN 1993-1-1: Annex B

Ntest =NEC3;c

NEC3;a (kN)

Ntest =NEC3;a

NEC3;c (kN)

Ntest =NEC3;c

NEC3;a (kN)

Ntest =N

1.11 1.08 1.10 1.08 1.06 1.10 1.07 1.10 1.09

306 240 495 402 1666 1310 2579 2179 

1.07 1.04 1.07 1.04 1.02 1.05 1.03 1.04 1.05

272 227 425 362 1424 1129 2185 1848 

1.21 1.10 1.24 1.15 1.19 1.22 1.22 1.23 1.20

293 238 461 393 1523 1238 2303 2018 

1.12 1.05 1.14 1.06 1.12 1.11 1.16 1.13 1.11

Behavior and design of high-strength steel columns under combined compression and bending

321

1.0

1.0 Test EH1P, λz = 1.26

Test EH2P, λz = 1.01 Section H2, Annex A, λz = 1.01

Section H1, Annex A, λz = 1.26 Section H1, Annex B, λz = 1.26

0.8

Section H2, Annex B, λz = 1.01

0.8

Test EH2Q, λz = 1.42

Test EH1Q, λz = 1.77 Section H1, Annex A, λz = 1.77 0.6

Section H2, Annex A, λz = 1.42 0.6

Section H2, Annex B, λz = 1.42

N/Npl

N/Npl

Section H1, Annex B, λz = 1.77

0.4

0.4

0.2

0.2

0.0 0.0

0.2

0.4

0.6

0.8

0.0 0.0

1.0

0.2

0.4

0.6

M/Mpl,z,Rd

M/Mpl,z,Rd

(A)

(B)

1.0

0.8

1.0

1.0 Test EH3P, λz = 0.77

Test EH4P, λz = 0.62

Section H3, Annex A, λz = 0.77

Section H4, Annex A, λz = 0.62

Section H3, Annex B, λz = 0.77

0.8

Section H4, Annex B, λz = 0.62

0.8

Test EH3Q, λz = 1.08

Test EH4Q, λz = 0.87

Section H3, Annex A, λz = 1.08 0.6

Section H4, Annex A, λz = 0.87 0.6

Section H4, Annex B, λz = 0.87

N/Npl

N/Npl

Section H3, Annex B, λz = 1.08

0.4

0.4

0.2

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.0

0.2

0.4

0.6

M/Mpl,z,Rd

M/Mpl,z,Rd

(C)

(D)

0.8

1.0

Figure 8.9 Normalized interaction curves to EN 1993-1-1: (A) Section H1, (B) Section H2, (C) Section H3, and (D) Section H4.

ANSI/AISC 360-16 ANSI/AISC 360-16 is applicable to steel grades up to 690 N/mm2 (ASTM A514 and A709 steel). For doubly and singly symmetric members subject to combined compression and bending, the following equations in ANSI/AISC 360-16 should be satisfied: Pr When $ 0:2 Pc When

Pr , 0:2 Pc

  Mry Pr 8 Mrx 1 1 # 1:0 9 Mcx Pc Mcy

(8.4)

  Mry Pr Mrx 1 1 # 1:0 2Pc Mcx Mcy

(8.5)

322

Behavior and Design of High-Strength Constructional Steel

where Pr is the design axial force, Pc is the axial buckling resistance, Mrx and Mry are the design moments about the major (x) and the minor (y) axes, respectively, and Mcx and Mcy are the moment resistances about the major (x) and the minor (y) axes, respectively. It should be noted that the design axial forces, Pr,ANSI, for all the H-sections are calculated through iterations. Comparison between the test resistances Ntest and the design resistances Pr,ANSI is shown in Table 8.8, and corresponding normalized interaction curves are plotted in Fig. 8.10. It is found that the design rules in ANSI/ AISC 360-16 tend to provide close but slightly unconservative predictions to the failure loads of high-strength Q690 steel welded H-sections under combined compression and bending about minor axis.

GB 50017-2003 It should be noted that Q690 steel is beyond the scope of GB 50017-2003, and hence, it is necessary to verify its applicability to design Q690 steel materials. For members under combined compression and bending, the following equations in GB 50017-2003 should be satisfied: βty My N βmx Mx 1 #f 1η 0 ϕx A γx Wx ð1 2 0:8ðN=NEx ÞÞ ϕby Wy

(8.6)

βmy My N β Mx 1 η tx 1 #f 0 ϕy A ϕbx Wx γy Wy ð1 2 0:8ðN=NEy ÞÞ

(8.7)

where N is the design value of the compression force, Mx and My are the design moments about the major (x) and the minor (y) axes, respectively, ϕx and ϕy are the reduction factors for flexural buckling about the major (x) and the minor (y) axes, respectively, ϕbx is the reduction factor for lateral-torsional buckling, Table 8.8 Calibration of ANSI/AISC 360-16 for Q690 steel welded H-sections under combined compression and bending. Test

EH1P EH1Q EH2P EH2Q EH3P EH3Q EH4P EH4Q Average

Ntest (kN)

328 250 527 418 1698 1376 2662 2276 

λ

66 92 53 74 39 55 32 44 

λ

1.26 1.77 1.01 1.42 0.77 1.08 0.62 0.87 

ANSI/AISC 360-16 Pr;ANSI (kN)

Ntest =Pr;ANSI

335 256 515 426 1713 1409 2449 2182 

0.98 0.98 1.02 0.98 0.99 0.98 1.09 1.04 1.01

Behavior and design of high-strength steel columns under combined compression and bending 1.0

323

1.0 Test EH1P, λz = 1.26

Test EH2P, λz = 1.01

Section H1, λz = 1.26 0.8

Section H2, λz = 1.01 0.8

Test EH1Q, λz = 1.77

Test EH2Q, λz = 1.42

Section H1, λz = 1.77

Section H2, λz = 1.42 0.6

N/Npl

N/Npl

0.6

0.4

0.4

0.2

0.2

0.0 0.0

0.2

0.4

0.6

0.8

0.0 0.0

1.0

0.2

M/Mpl,z,Rd

0.4

0.6

0.8

1.0

M/Mpl,z,Rd

(A)

(B)

1.0

1.0 Test EH3P, λz = 0.77

Test EH4P, λz = 0.62

Section H3, λz = 0.77 0.8

Section H4, λz = 0.62 0.8

Test EH3Q, λz = 1.08

Test EH4Q, λz = 0.87

Section H3, λz = 1.08

Section H4, λz = 0.87

N/Npl

0.6

N/Npl

0.6

0.4

0.4

0.2

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.0

0.2

0.4

0.6

M/Mpl,z,Rd

M/Mpl,z,Rd

(C)

(D)

0.8

1.0

Figure 8.10 Normalized interaction curves to ANSI/AISC 360-16: (A) Section H1, (B) Section H2, (C) Section H3, and (D) Section H4.

 
 0 0 ϕby 5 1:0, NEx 5π2 EA= 1:1λ2x , NEy 5π2 EA= 1:1λ2y , λx ; λy are the slendernesses for flexural buckling about the major (x) and the minor (y) axes, respectively, A is the cross sectional area, Wx and Wy are the elastic moduli about the major (x) and the minor (y) axes, respectively, βmx and βmy are equivalent uniform in-plane moment factors about the major (x) and the minor (y) axes, respectively, βtx and βty are equivalent uniform out-of-plane moment factors about the major (x) and the minor (y) axes, respectively, γx and γy are factors considering material plasticity when bending about the major (x) and the minor (y) axes, respectively, η 5 1:0 for members susceptible to torsional deformation, η 5 0:7 for members not susceptible to torsional deformation, f design yield strength of the steel material, and E is Young’s modulus of the steel material.

324

Behavior and Design of High-Strength Constructional Steel

A curve “b” is recommended to calculate flexural resistances of welded H-sections made of Q420 steel, the highest steel grade incorporated in GB50017-2003. The corresponding design resistances NGB,b are adopted for calibration against the test results. In addition, a curve “a” is also permitted in the code, and the corresponding design resistances NGB,a are also adopted for calibration. It should be noted that both the design resistances NGB,a and NGB,b are obtained through iterations. Comparison between the failure loads and the design resistances are shown in Table 8.9, while normalized interaction curves are plotted in Fig. 8.11. It is found that the test results are significantly higher than the design resistances obtained with either curve “b” or curve “a.”

8.2.2 Box sections 8.2.2.1 Material properties The Q460C steel used in this study is high-strength low alloy structural steel with the nominal yield strength of 460 MPa in GB/T 1591-2008 [4]. The nearest equivalent steel of Q460C according to EN 10025-6 [5] is S460N but the latter requires an additional verification of impact energy at 220
C. Tensile coupon tests were carried out to measure the mechanical properties of Q460C steel. A total of nine coupons were cut from the 11 mm parent plates. The cutting direction was perpendicular to the rolling direction according to GB/T 2975-1998 [6]. The tensile coupons were tested in accordance with GB/T 228-2002 [7]. Fig. 8.12 shows that, unlike normal-strength steel, no significant strain hardening appears in HSS Q460C steel. For some tensile coupons, there is no well-defined yield plateau. The average values of the test results are summarized in Fig. 8.12, where fy is the 0.2% proof stress, which is adopted as the yield strength of Q460C steel, fu is the ultimate tensile stress, E is Young’s modulus, and Δ is the percentage of elongation after fracture. Table 8.9 Calibration of GB 50017-2003 for Q690 steel welded H-sections under combined compression and bending. Test

EH1P EH1Q EH2P EH2Q EH3P EH3Q EH4P EH4Q Average

Ntest (kN)

328 250 527 418 1698 1376 2662 2276 

λ

66 92 53 74 39 55 32 44 

λ

1.26 1.77 1.01 1.42 0.77 1.08 0.62 0.87 

GB 50017-2003 NGB;b (kN)

Ntest =NGB;b

NGB;a (kN)

Ntest =NGB;a

268 218 425 361 1380 1149 2062 1860 

1.23 1.15 1.24 1.16 1.23 1.20 1.29 1.22 1.21

275 223 437 371 1419 1191 2110 1930 

1.19 1.12 1.21 1.13 1.20 1.16 1.26 1.18 1.18

Behavior and design of high-strength steel columns under combined compression and bending 1.0

325

1.0 Test EH1P, λz = 1.26

Test EH2P, λz = 1.01

Section H1, λz = 1.26 0.8

Section H2, λz = 1.01 0.8

Test EH1Q, λz = 1.77

Test EH2Q, λz = 1.42

Section H1, λz = 1.77

Section H2, λz = 1.42 0.6

N/Npl

N/Npl

0.6

0.4

0.4

0.2

0.2

0.0 0.0

0.2

0.4

0.6

0.8

0.0 0.0

1.0

0.2

0.4

0.6

M/Mpl,z,Rd

M/Mpl,z,Rd

(A)

(B)

1.0

0.8

1.0

1.0 Test EH4P, λz = 0.62

Test EH3P, λz = 0.77

Section H4, λz = 0.62

Section H3, λz = 0.77 0.8

0.8

Test EH3Q, λz = 1.08

Test EH4Q, λz = 0.87 Section H4, λz = 0.87

Section H3, λz = 1.08 0.6

N/Npl

N/Npl

0.6

0.4

0.4

0.2

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.0

M/Mpl,z,Rd

(C)

0.2

0.4

0.6

0.8

1.0

M/Mpl,z,Rd

(D)

Figure 8.11 Normalized interaction curves to GB 50017-2003: (A) Section H1, (B) Section H2, (C) Section H3, and (D) Section H4.

8.2.2.2 Specimen design and fabrication To evaluate the behavior of Q460C box columns under eccentric compression, seven specimens with sectional width to thickness ratios from 7.6 to 17.5 were fabricated from flame-cut Q460C steel plate. Four 11 mm component plates were welded together to form a box-section specimen by manual GMAW, as shown in Fig. 8.13. The electrode ER55-D2 was used to achieve equivalent-matching weld with Q460C steel. As current practice does not employ complete penetration welding for columns except in the beam-to-column connection zone, the component plates were connected by incomplete penetration welding except the 500 mm length from each end. In order to reduce the effect of shrinkage deformation caused by

326

Behavior and Design of High-Strength Constructional Steel

700

Stress (MPa)

600 500 400

E=207.8 GPa fy=505.8 MPa fu=597.5 Mpa Δ =23.7%

300 200 100 0 0

0.05

Strain ε

0.1

0.15

Figure 8.12 Stressstrain relationship of Q460C steel.

D y W3

z

z

t

d

D

W1

W2

W4 y

Figure 8.13 Definition of symbols.

welding heating and cooling, the optimized welding sequence (W1!W2!W3!W4) was adopted, as shown in Fig. 8.13. The measured geometric dimensions of the seven test specimens are shown in Table 8.10. The specimens were named in terms of B-d/t-λ-X-n, where B is for box-section and n is the specimen number. The tests were designed to investigate the overall buckling behavior of the welded HSS box columns under eccentric load. Thus premature local buckling is prevented by limiting the plate slenderness ratio. In the Chinese Code for design of steel structures GB 50017-2003 [8]: sffiffiffiffiffiffiffiffi 235 d=t # ð25 1 0:5λÞ ; fy

30 # λ # 100

(8.8)

The plate slenderness limits are ranging from 28.6 to 53.6 depending on the λ varying from 30 to 100.

Behavior and design of high-strength steel columns under combined compression and bending

327

In the European code for design of steel structures Eurocode 3 [9]: sffiffiffiffiffiffiffiffi 235 d=t # 42 fy

(8.9)

Thus the plate slenderness limit of Class 3 section is 30.0. The d/t of 17.5 (section B-18) was selected to represent the commonly used aspect ratio for Q460C steel; the d/t of 11.5 (section B-12) was considered as the lower boundary of the columns fabricated from Q460C steel in industry practice; and the section B-8 with d/t of 7.7 was designed to investigate the extreme case. The plate slenderness ratios are all lower than the plate slenderness limits, specified either in the GB 50017-2003 or in the Eurocode 3. Thus premature local buckling is not expected to occur before the peak load, and the overall buckling will dominate the ultimate bearing capacities of the test columns.

8.2.2.3 Out-of-straightness and loading eccentricity Initial geometric imperfection consists of initial out-of-straightness and initial loading eccentricity, as shown in Fig. 8.14. The initial out-of-straightness was unavoidably caused by the nonuniform weld shrinkage during the manufacture process. A prestressed string was attached tightly to the two ends of the component plate to provide a straight reference line, and the deviations at seven points with equal interval were measured. The measurement process was repeated after rotating the Table 8.10 Measured dimensions of test specimens. Specimen

B-8-80-X-1 B-8-80-X-2 B-8-80-X-3 B-12-55-X-1 B-12-55-X-2 B-18-38-X-1 B-18-38-X-2

D

t

mm

mm

110.0 110.8 112.5 155.2 153.3 222.0 219.8

11.5 11.5 11.3 11.5 11.5 11.4 11.5

d/t

7.6 7.6 8.0 11.5 11.3 17.5 17.1

L

Le

A

I

r

mm

mm

mm2

cm4

mm

3000 2940 3000 2940 2940 2940 2940

3320 3260 3320 3260 3260 3260 3260

4531 4581 4574 6610 6523 9603 9582

743 767 791 2290 2200 7120 6950

40.5 40.9 41.6 58.9 58.1 86.1 85.2

λ

82.0 79.6 79.8 55.4 56.1 37.9 38.3

Note: L is the net length of the column not accounting end plates, Le is the effective length of column between two pinned supports, A is the area of box-section, I is the moment of inertia of box-section, r is the gyration radius, and λ is the slenderness where λ 5 Le/r.

Figure 8.14 Geometric imperfections.

328

Behavior and Design of High-Strength Constructional Steel

specimens by 90 degrees. To reduce measurement errors the average value of initial out-of-straightness v0 obtained on both sides was adopted. The values of v0 at midheight cross section are summarized in Table 8.11. The test specimens were set under eccentric loading. The intentional loading eccentricity was introduced by the deviation between the central axis of column cross section and the central axis of end plate, as shown in Fig. 8.15. The measured loading eccentricities of both ends were averaged and summarized in Table 8.11.

8.2.2.4 Test setup and loading procedure The specimens were tested under eccentric loading with a 10,000 kN universal testing machine at Tongji University. The schematic diagram of the test setup is shown in Fig. 8.16. Two curved surface supports were used at both ends of each specimen, which can be recognized as ideal hinged connection in the bending plane, while they are fixed about the other principal axis of the cross section. All specimens were set to pin-supported about the yy axis and fixed the about zz axis. The arrangement of the linear varying displacement transducers (LVDTs) and the strain gages is shown in Fig. 8.16. The axial deformation of the specimens was measured by the LVDTs V1 and V2. The LVDTs H01H03 were placed at the midlength of the column to record the in-plane lateral deflections. The out-of-plane lateral Table 8.11 Initial geometric imperfections. Specimen B-8-80-X-1 B-8-80-X-2 B-8-80-X-3 B-12-55-X-1 B-12-55-X-2 B-18-38-X-1 B-18-38-X-2 Centraal axis of end d palte

e0 mm

v0 mm

48.1 54.6 53.4 55.4 53.4 66.0 65.5

2 4.5 2 6.0 2 6.1 2 2.8 3.5 2 1.0 2 1.7

Ceentral axis e of column 0

Centraal axiss

Figure 8.15 Loading eccentricity.

Behavior and design of high-strength steel columns under combined compression and bending

329

Actuator P≤10,000 kN Horizontal displacement constraint

V2

1

V4

V3

V1 1-1 Top cross section

H04

2

H01 2

H06

H02

H05 H03 V6

V5 H8

S09-S07 S12-S10

L

V1 V2 Le

H01~03

S04-S06

1

V4 V6

V3 V5

H07

S01-S03 2-2 Midheight cross section In-plane LVDT Out-of-plane LVDT Strain gauge

Figure 8.16 Test setup.

deflection was captured by the LVDT H06. The strain gages were attached to the midlength cross section of the specimens to monitor the loading force. The realtime loaddeflection curves and load-shortening curves displayed in the monitor during the entire loading processes were used to adjust and govern the experiments. Before the actual test a preload with 10% of the predicted maximum column strength was used to check the test instrumentations and then unloaded. In the actual test the axial load was applied on the column at a rate of 1 mm/min until the peak load was reached. Then the load rate was increased in the postpeak range. Finally, the test specimen was unloaded to finish the test procedure when the test load decreased to 80% of the peak load.

8.2.2.5 Assessment of test result Overall buckling behavior The measured axial load versus midheight deflection curves are shown in Fig. 8.17. The average deflection of H01H03 is used, since the readings obtained from

330

Behavior and Design of High-Strength Constructional Steel

700 (A) 600

Load (kN)

500 400 300 B-8-80-X-1 B-8-80-X-2 B-8-80-X-3

200 100 0 0

20

40

60

80

100

120

Deflection (mm) 1500

(B)

Load (kN)

1200 900 600 B-12-55-X-1 B-12-55-X-2

300 0 0

20

40

60

80

100

120

Deflection (mm) 3000 (C)

Load (kN)

2500 2000 1500 1000

B-18-38-X-1 B-18-38-X-2

500 0 0

20

40

60

80

100

120

Deflection (mm) Figure 8.17 Loaddefection curves: (A) B-8-80 series, (B) B-12-55 series, and (C) B-18-38 series.

Behavior and design of high-strength steel columns under combined compression and bending

331

H01H03 differs slightly from each other. The specimens show a stable loaddeflection relationship, including initial elastic loading branch, inelastic hardening branch, and gradual softening branch, which indicates a ductile behavior under large deformation. The measured out-of-plane lateral deflection is so small that can be ignored. The observed failure mode of the specimens, with various member slenderness ratios of 3880, is identified as flexural buckling in the intended direction, as shown in Fig. 8.18. Local buckling was not observed before peak load. For the specimens B-18-38-X-1 and B-18-38-X-2, local buckling was observed at midheight cross section just before the load dropping to 80% of the maximum load. The measured ultimate loads Pu of the specimens subjected to eccentric loading are summarized in Table 8.12. The typical axial loadstrain curves measured at the midheight cross section are shown in Fig. 8.19. As shown in Fig. 8.16, the strain gages S05 and S11 were

Figure 8.18 Initial loaded specimens and specimens near failure: (A) B-8-80-X-1, (B) B-1255-X-1, and (C) B-18-38-X-1. Table 8.12 Comparison of experimental and predicted ultimate resistances. Specimen

B-8-80-X-1 B-8-80-X-2 B-8-80-X-3 B-12-55-X-1 B-12-55-X-2 B-18-38-X-1 B-18-38-X-2

Pu (kN) 598.5 598.0 599.0 1204.5 1264.5 2532.0 2394.0

Pu /Afy

GB 50017 NGB (kN)

0.261 448.6 0.258 443.1 0.259 451.3 0.360 967.2 0.383 954.7 0.521 1846.8 0.494 1821.3 Mean value Standard deviation

GB 50017 NGB/Pu

Eurocode 3 NEC3 (kN)

Eurocode 3 NEC3/Pu

0.75 0.74 0.75 0.80 0.76 0.73 0.76 0.76 0.02

509.8 487.3 532.4 1041.7 1098.1 1417.5 1363.4

0.85 0.81 0.89 0.86 0.87 0.56 0.57 0.77 0.14

332

Behavior and Design of High-Strength Constructional Steel

700 (A) 600

Load (kN)

500 400

S02 S05 S08 S11 Yield strain

300 200 100 0 –15,000

–10,000

–5000

0

5000

10,000

−6

Strain (×10 ) 1400

(B) 1200

Load (kN)

1000 800

S02 S05 S08 S11 Yield strain

600 400 200

0 –15,000 –10,000

–5000

0

5000

10,000

5000

10,000

−6

Strain (×10 ) 2800

(C) 2400

Load (kN)

2000 1600 1200 800

S02 S05 S08 S11 Yield strain

400 0 –15,000 –10,000

–5000

0 −6

Strain (×10 )

Figure 8.19 Loadstrain curves: (A) B-8-80-X-2, (B) B-12-55-X-2, and (C) B-18-38-X-2.

Behavior and design of high-strength steel columns under combined compression and bending

333

attached on the midwidth of right and left flanges of the box-section, respectively. The strain gages S02 and S08 were attached on the middepth of webs. Consequently, the bending effect can be identified from the difference of the axial strain between the component plates. Fig. 8.19 shows that, due to the preset eccentricity, the midheight cross section is subjected to compression and bending at the beginning of loading. Thus the specimens show a limit load instability instead of bifurcation buckling of concentrically loaded columns. With the increase in test load the second-order effect becomes obvious, and the difference of axial strain between the extreme compressive and tensile fibers becomes remarkable. The extreme compressive fibers (S05) yields before peak load for all specimens, while the strains of extreme tensile fibers (S11) are lower than yield strain (fy/E) before peak load. However, it should be noted that RS induced by welding process could advance or postpone yielding of steel, depending on the superimposed compressive or tensile RS. The strain of middle fibers such as that recorded by strain gages S02 and S08, which can be recognized as the average strain of the cross section, reveals the stress ratio of the cross sectional resistances. At peak load the middle fiber strains of the specimens with column slenderness ratio of 80, 55, and 38 are 63%, 88%, and 92% of fy/E, respectively. Similar as buckling factor, the middle fiber strain to fy/E ratio increases with the decrease in column slenderness ratio.

Comparison of test results with design codes Slender steel members subjected to combined bending and axial compression are generally limited by buckling resistance. Thus in addition to cross section resistance, the beamcolumn member should be verified against buckling interaction formula. In the case of uniaxial eccentric loading, the buckling interaction formulae can be reduced as follows: According to GB50017-2003

According to Eurocode 3

βmy My N # fy 1 0 χy A γy W1y ð1 2 η1 ðN=NEy ÞÞ

My 1ΔMy N
 #1 1kyy χy NRk =γM1 χLT My;Rk =γM1

(8.10)

(8.11)

where N and My are the design values of the compression force and the maximum moments about the yy axis along the column, ΔMy is the moment due to the shift of the centroidal axis for Class 4 sections, NRk and My,Rk are the characteristic compressive resistance and characteristic moment resistance about yy axis of the cross section, χy is the reduction factor due to flexural buckling, βmy is the equivalent coefficient of moment distribution, W1x is the elastic section modulus, kyy is the interaction factor, γM1 5 1:00; γy is inelastic development factor and γy 5 1:05; η1 is the compensation coefficient with η1 5 0:8 for normal-strength steel, and 0 NEy 5 π2 EA=ð1:1λ2 Þ. According to both GB 50017-2003 and Eurocode 3, the buckling curve “c” is the design curve for the tested box-section with d/t ratio less than

334

Behavior and Design of High-Strength Constructional Steel

20 and ΔMy is zero. Considering box-section member with twisting restrained supports at both ends, it is not susceptible to lateral-torsional buckling. Thus the reduction factor χLT 5 1 was adopted in the calculation. Table 8.12 compares the design strengths of GB 50017-2003 and Eurocode 3 with the test strengths. All test results are higher than the predicted value, indicating the conservative of the current design formulae. The ultimate load bearing capacity of the tested Q460C box-section columns is underestimated by 24% and 23% in accordance with GB 50017-2003 and Eurocode 3, respectively. The similar underestimation has been found in the concentrically loaded Q460C HSS box columns [10], however, which is less pronounced than that of eccentrically loaded Q460C HSS box columns. The main reason for the underestimation of ultimate resistance is recognized as the less detrimental effect of imperfections on HSS members, such as RS and initial out-of-straightness caused by welding process, than those on normal-strength steel columns. Moreover, the compressive RS to the yield strength ratios of Q460C HSS box-sections are lower than those adopted for normal-strength steel in the current codes [11]. It is noted that Eurocode 3 can give better perdition than GB 50017-2003 for columns with high slenderness. However, for columns with medium slenderness, Eurocode 3 gives more conservative results than GB 50017-2003 by up to 44%.

8.3

Numerical investigation

In this numerical investigation, validated numerical models were initially developed through accurate replication of the test results. Parametric studies were performed subsequently to examine further the influences of member residual stresses and material tensile to yield strength ratios on the structural response of HSS columns of welded sections under combined compression and bending.

8.3.1 H-sections Having validated the numerical models against the experimental results [3], a series of parametric studies was performed, focusing on residual stresses and material tensile to yield strength ratios. The measured residual stress ratios of Q690 steel columns of welded H-sections were found to be significantly smaller than the assumed values by ECCS for conventional steel columns of welded H-sections, and thus the adverse effect from residual stresses on Q690 steel columns of welded H-sections was anticipated to be less than that on conventional steel columns of welded H-sections. The degree of residual stress effect for Q690 steel and Q235 steel columns of welded H-sections will be compared in the following studies. Another factor that may affect the buckling resistances of Q690 welded H-sections is material tensile to yield strength ratios. The tensile to yield strength ratios indicate the abilities of strain hardening of the materials. For conventional steel material, strain hardening develops at the end of a long yield plateau. In the simulation of conventional steel

Behavior and design of high-strength steel columns under combined compression and bending

335

members the resistance of which is dominated by instability, an elastic-perfectly plastic model is practically adopted, and the strain hardening is neglected. However, for Q690 steel, the strain hardening starts once the material is yielding. Therefore for Q690 steel columns of welded H-sections, the neglection of strain hardening may lead to a conservative design. The contribution of strain hardening to the buckling resistances is related to the tensile to yield strength ratios. In the following studies the effect of tensile to yield strength ratios is evaluated for Q690 steel columns of welded H-sections under combined compression and bending. In the generation of FE models, all the steel columns were modeled by using shell elements S4R. Section H3 was selected as a typical cross section size. The nondimensional slendernesses of these FE models ranged from 0.6 to 1.8 with an interval of 0.4. To normalize the initial loading eccentricity with the cross-sectional properties, an initial loading eccentricity ratio is defined by Eq. (8.12), where e is the initial loading eccentricity, A is the area of cross section, and Wel is the elastic modulus of cross section. For each nondimensional slenderness the initial loading eccentricity ratios varied from 0.0 (compression only) to 20.0 (combined compression and bending): ε5

eA Wel

(8.12)

The shape of the initial out-of-straightness was assumed to be a sinusoidal shape, and the amplitude was replaced by Leff/1000. The initial out-of-straightness and the initial loading eccentricity were positioned on different sides of the longitudinal axis of the welded H-sections.

8.3.1.1 Residual stresses To evaluate the effect of residual stresses on Q690 steel columns of welded H-sections, buckling resistances were obtained from two groups of FE models, in which one with residual stresses and the other one without residual stresses. The residual stress amplitudes for Q690 steel welded H-section H3 were shown in Table 8.13. Table 8.13 FE model information for the evaluation of effect of residual stresses. Group 1 2 3 4 5 6 7 8

Bending axis

Steel grade

Residual stresses

Initial out-ofstraightness v

Model numbers

Nondimensional Slenderness

Major

690

No Yes No Yes No Yes No Yes

Leff/1000

60 60 60 60 60 60 60 60

0.6/1.0 1.4/1.8

235 Minor

690 235

336

Behavior and Design of High-Strength Constructional Steel

A total of 240 numerical models were established, and their information is summarized in Table 8.13. In this study the overall flexural buckling was not restricted in minor axis but extended to major axis. The material behavior of Q690 steel was modeled as an elastic-linear hardening relationship, as shown in Fig. 8.20. The mechanical properties of Q690 steel were obtained in standard the tensile tests, and the key parameters in true stresslogarithmic plastic strain behavior are shown in Fig. 8.20. To compare the effect of residual stresses on Q690 steel columns of welded Hsections with that on Q235 steel columns, FE models with Q235 steel were generated, and their model information is also summarized in Table 8.13. The residual stress pattern for Q235 steel columns of welded H-sections is shown in Table 8.14. The material behavior of Q235 steel was modeled as a multilinear relationship, as shown in Fig. 8.20. The yield strength was taken to be its nominal value, 235 MPa, and the tensile strength was 370 MPa according to GB/T 700 and EN 10025-2 [5]. The εst,log and εu,log are the strain at the end of yield plateau and the ultimate strain, respectively. The key parameters in true stresslogarithmic plastic strain behavior are shown in Table 8.14. Figs. 8.21 and 8.22 depict the normalized compression and bending relationships of FE models, arranged by nondimensional slenderness, λ. In these figures, NFEM is the buckling resistance obtained from the finite element models, MFEM 5 NFEMee,

σ fu,true fy,true pl

pl

εst,log o εy,log

εu,log εst,log

εu,log

ε

Figure 8.20 True stresslogarithmic strain relationships for Q235 steel plates. Table 8.14 True stresslogarithmic plastic strain relationship for Q235 steel plates in FE models. Young’s Modulus E (kN/ mm2) 210

True yield strength fy, true (N/ mm2)

True strength onset of strain hardening fst,true (N/mm2)

logarithmic plastic strain onset of strain hardening εpl st;log

True tensile strength fu, true (N/ mm2)

Logarithmic plastic ultimate strain εpl u;log

235

240

0.024

444

0.182

Behavior and design of high-strength steel columns under combined compression and bending

337

1.0 (A)

⎯λy=0.6 Section H3 Section H3, residual stress

0.8

NFEM / Npl

⎯λy=1.0 Section H3 Section H3, residual stress

0.6

⎯λy=1.4

0.4

Section H3 Section H3, residual stress ⎯λy=1.8

0.2

Section H3 Section H3, residual stress 0.0 0.0

0.2

0.4

0.6

0.8

1.0

MFEM / Mpl,y 1.0 ⎯λz=0.6

(B)

Section H3 Section H3, residual stress

0.8

NFEM / Npl

⎯λz=1.0 Section H3 Section H3, residual stress

0.6

⎯λz=1.4

0.4

Section H3 Section H3, residual stress ⎯λz=1.8

0.2

Section H3 Section H3, residual stress 0.0 0.0

0.2

0.4

0.6

0.8

1.0

MFEM / Mpl,y Figure 8.21 Effect of residual stresses on buckling resistances of Q690 and Q235 steel columns of welded H-sections under combined compression and major axis bending: (A) Q690 steel columns of welded H-sections and (B) Q235 steel columns of welded H-sections.

338

Behavior and Design of High-Strength Constructional Steel

1.0 (A)

⎯λz=0.6 Section H3 Section H3, residual stress

0.8

NFEM / Npl

⎯λz=1.0 Section H3 Section H3, residual stress

0.6

⎯λz=1.4

0.4

Section H3 Section H3, residual stress ⎯λz=1.8

0.2

Section H3 Section H3, residual stress 0.0 0.0

0.2

0.4

0.6

0.8

1.0

MFEM / Mpl,z 1.0

⎯λz=0.6

(B)

Section H3 Section H3, residual stress

0.8

NFEM / Npl

⎯λz=1.0 Section H3 Section H3, residual stress

0.6

⎯λz=1.4

0.4

Section H3 Section H3, residual stress

⎯λz=1.8

0.2

Section H3 Section H3, residual stress 0.0 0.0

0.2

0.4

0.6

0.8

1.0

MFEM / Mpl,z Figure 8.22 Effect of residual stresses on buckling resistances of Q690 and Q235 steel columns of welded H-sections under combined compression and minor axis bending: (A) Q690 steel columns of welded H-sections and (B) Q235 steel columns of welded H-sections.

Behavior and design of high-strength steel columns under combined compression and bending

339

Npl 5 Aefy, and Mpl is the sectional plastic resistance to bending moment. It is found that: 1. The presence of residual stresses reduces the buckling resistances of both Q690 and Q235 steel columns of welded H-sections under combined compression and bending. Compared with Q235 steel columns of welded H-sections, the effect degree becomes less on Q690 steel columns of welded H-sections with the same nondimensional slenderness and initial loading eccentricity ratio. 2. For both Q690 and Q235 steel columns of welded H-sections, when the nondimensional slendernesses of the columns are 0.6 and 1.0, the effect of residual stresses decreases with the increase of initial loading eccentricity ratio. However, when the nondimensional slendernesses of the columns are 1.4 and 1.8, the effect of residual stresses initially increases and subsequently decreases with the increase of initial loading eccentricity ratio.

Overall, residual stresses can significantly reduce the buckling resistances of Q690 steel columns of welded H-sections under combined compression and bending. But compared with Q235 steel columns of welded H-sections, the effect of residual stresses is decreased for Q690 steel columns of welded H-sections due to the smaller ratios of residual stresses to yield strength.

8.3.1.2 Tensile to yield strength ratios To evaluate the effect of tensile to yield strength ratios on Q690 steel columns of welded H-sections, buckling resistances were obtained from FE models with two different material stressstrain models, (1) an elastic-linear hardening model (see Fig. 8.23) and (2) an elastic-ideally plastic model, as shown in Fig. 8.23. The mechanical properties of these two models are listed in Table 8.15, and only E and fy,true were included in the elastic-ideally plastic model. A total of 240 numerical models were established, and their information are summarized in Table 8.15. The residual stress pattern for Q690 steel welded H-section H3 is listed in Table 8.15. The overall flexural buckling was not restricted in minor axis but extended to major axis. To compare the effect of tensile to yield strength ratios on Q690 steel columns of welded H-sections with that on Q235 steel columns of welded H-sections, FE models with Q235 steel were generated, and their model information is also σ fy,true

o εy,log

Figure 8.23 Elastic-ideally plastic model.

ε

340

Behavior and Design of High-Strength Constructional Steel

Table 8.15 FE model information for the evaluation of the effect of tensile to yield strength ratios. Group

1 2 3 4 5 6 7 8

Bending axis

Steel grade

fu/fy

Residual stresses

Initial out-ofstraightness v

Model numbers

Nondimensional slenderness

Major

690

1.00 1.05 1.00 1.57 1.00 1.05 1.00 1.57

Yes

Leff/1000

60 60 60 60 60 60 60 60

0.6/1.0 1.4/1.8

235 Minor

690 235

summarized in Table 8.15. Two different material stressstrain models were considered, (1) a multilinear model (see Fig. 8.20) and (2) an elastic-ideally plastic model (see Fig. 8.23). The mechanical properties of these two models are listed in Table 8.14, and only E and fy,true were excluded in the elastic-ideally plastic model. A total of 240 numerical models were established, and their information is summarized in Table 8.15. The overall flexural buckling was not restricted in minor axis but extended to major axis. Figs. 8.24 and 8.25 depict the normalized compression and bending relationships of FE models, arranged by nondimensional slenderness, λ. It is found that: 1. For Q235 steel columns of welded H-sections, even though the strain hardening is considered in the stressstrain model, there is no increase in the buckling resistances of the models. This has been commonplace and lies in the fact that there is a long yield plateau in the stressstrain relationship of Q235 steel, and when the H-sections fail, significant strain hardening does not take place. 2. For Q690 steel columns of welded H-sections, when considering strain hardening, some increases in the buckling resistances of the models are visible only in columns with nondimensional slenderness of 0.6. Because when a relatively short column fails, the strain is able to develop well beyond the yield strain, and thus the buckling resistance can take advantage of strain hardening. While when a long column fails, the strain is not able to develop beyond the yield strain, and the strain hardening cannot be mobilized. 3. The maximum increase in buckling resistance improved by strain hardening is only 2.0%, thus the elastic-ideally plastic model can be used in the modeling of buckling resistances of Q690 steel columns of welded H-sections under combined compression and bending without causing too much conservatism.

Overall, strain hardening can slightly improve the buckling resistances of Q690 steel short columns of welded H-sections, and the neglection of strain hardening will not result in a conservative design. From the parametric studies earlier, it is concluded that the residual stresses can significantly affect the buckling resistances of Q690 steel columns of welded H-sections under combined compression and bending. Compared with the conventional steel

Behavior and design of high-strength steel columns under combined compression and bending

341

1.0 (A)

⎯λy=0.6 Section H3, ideal plastic Section H3, fu=1.05f y

0.8

NFEM / Npl

⎯λy=1.0 Section H3, ideal plastic Section H3, fu=1.05f y

0.6

⎯λy=1.4

0.4

Section H3, ideal plastic Section H3, fu=1.05f y 0.2

0.0 0.0

⎯λy=1.8 Section H3, ideal plastic Section H3, fu=1.05f y 0.2

0.4

0.6

0.8

1.0

MFEM / Mpl,y 1.0 ⎯λy=0.6

(B)

Section H3, ideal plastic Section H3, fu=1.57f y

0.8

NFEM / Npl

⎯λy=1.0 Section H3, ideal plastic Section H3, fu=1.57f y

0.6

⎯λy=1.4

0.4

Section H3, ideal plastic Section H3, fu=1.57f y 0.2

0.0 0.0

⎯λy=1.8 Section H3, ideal plastic Section H3, fu=1.57f y 0.2

0.4

0.6

0.8

1.0

MFEM / Mpl,y Figure 8.24 Effect of tensile to yield strength ratios on buckling resistances of Q690 and Q235 steel columns of welded H-sections under combined compression and major axis bending: (A) Q690 steel columns of welded H-sections and (B) Q235 steel columns of welded H-sections.

342

Behavior and Design of High-Strength Constructional Steel

1.0 (A)

⎯λz=0.6 Section H3, ideal plastic Section H3, fu=1.05f y

0.8

NFEM / Npl

⎯λz=1.0 Section H3, ideal plastic Section H3, fu=1.05f y

0.6

⎯λz=1.4

0.4

Section H3, ideal plastic Section H3, fu=1.05f y 0.2

0.0 0.0

⎯λz=1.8 Section H3, ideal plastic Section H3, fu=1.05f y 0.2

0.4

0.6

0.8

1.0

MFEM / Mpl,z 1.0 ⎯λz=0.6

(B)

Section H3, ideal plastic Section H3, fu=1.57f y

0.8

NFEM / Npl

⎯λz=1.0 Section H3, ideal plastic Section H3, fu=1.57f y

0.6

⎯λz=1.4

0.4

Section H3, ideal plastic Section H3, fu=1.57f y 0.2

0.0 0.0

⎯λz=1.8 Section H3, ideal plastic Section H3, fu=1.57f y 0.2

0.4

0.6

0.8

1.0

MFEM / Mpl,z Figure 8.25 Effect of tensile to yield strength ratios on buckling resistances of Q690 and Q235 steel columns of welded H-sections under combined compression and minor axis bending: (a) Q690 steel columns of welded H-sections and (b) Q235 steel columns of welded H-sections.

Behavior and design of high-strength steel columns under combined compression and bending

343

materials, the effect of residual stresses becomes less pronounced in Q690 steel columns of welded H-sections. It should be noted that strain hardening barely contributes to the buckling resistances of the Q690 steel columns of welded H-sections.

8.3.2 Box sections Since the finite element model can successfully predict the ultimate strength and the loaddeflection behavior of the columns, the developed FE model is used to generate more data rather than the limited test results. The main parameters are width to thickness ratio (d/t in Table 8.16), column slenderness λ (from 10 to 160) and eccentricity ratio, ε0 5 e0A/W1y (from 0 to 20 along yy and zz axes, respectively). The varying of RS is implicitly involved in the adoption of different cross sections with varying width to thickness ratios. The cross section dimensions and material properties are summarized in Table 8.16. A total of 2016 pinpin columns with consideration of simplified RS pattern and 1000/Le initial bow were analyzed. The results show that columns with eccentricity caused bending about yy axis achieve slight lower ultimate strength than those bending about zz axis. Thus on the conservative side, the following discussion is based on the eccentricity settled along zz axis.

8.3.2.1 Effect of slenderness To evaluate the effect of the slenderness on the behavior of eccentrically loaded columns, the calculated interaction curves of B-15 series with varying column slenderness are shown in Fig. 8.26. It can be observed that, with lower slenderness, the columns achieve higher buckling interaction curves. From the comparison of the predicted results with those without consideration of RS, the influence of RS on the interaction curves is revealed. Stocky columns with slenderness no more than 20 trend to be strength failure. Since RS does not impair cross-sectional strength, the interaction curves with and without consideration of RS almost coincide with each other (less than 1%). With increased slenderness higher than 20, columns are susceptible to overall buckling. The presence of compressive RS in compression side of midheight cross section will bring forward the yielding and reduce the moment of inertia. Consequently, for columns with mediate slenderness, the interaction curves with consideration of RS are lower than those ignored RS. Table 8.16 Dimensions and material properties of specimens for parametric study. Section

D (mm)

B-8 B-10 B-12 B-15 B-18 B-20

105.6 127.6 148.5 187.0 211.2 242.0

t (mm)

d

d/t

A (mm2)

E (GPa)

fy (MPa)

11

83.6 105.6 126.5 165.0 189.2 220.0

7.6 9.6 11.5 15.0 17.2 20.0

4162.4 5130.4 6050.0 7744.0 8808.8 10164.0

207.8

505.8

344

Behavior and Design of High-Strength Constructional Steel

For very slender column with slenderness over 120, it tends to buckle elastically. Thus the influence of RS on the ultimate strength of column is reduced.

8.3.2.2 Effect of eccentricity ratio In order to evaluate the influence of eccentricity ratio, the deviations between predictions with and without consideration of RS (NRS and N0) are plotted against ε0 in Fig. 8.27, where the deviation is defined as: ΔN 5

NRS 2 N0 3 100% N0

N /Af y

1.0

λ 20

0.8

40

0.6

60 70

0.4 0.2

(8.13)

Without RS With RS

80 90 100 120 140

0.0 –0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

M /1.05W1y f y Figure 8.26 Interaction curves of the B-15 series with varying column slenderness. 4 2

ΔN (%)

0 –2 –4 λ = 40 λ = 70

–6 –8 –10 0

5

10

15

ε 0 = e0 A / W1y

Figure 8.27 Effect of eccentricity ratio, B-15 series.

20

25

Behavior and design of high-strength steel columns under combined compression and bending

345

Because the ultimate strength of stub column and very slender column is less sensitive to RS, only ΔN 2 ε0 curves of mediate (λ 5 40) and slender (λ 5 70) columns of B-15 series are shown. According to Fig. 8.27, the influence of RS is significant with eccentricity ratio about no more than 5. With the increase in load eccentricity, the moment resistance of cross section becomes more importance than the buckling resistance of columns in compression. Thus the detrimental effect of RS decreases to less 2% with eccentricity ratio more than 5. One the other hand, the two ΔN 2 ε0 curves with slenderness of 40 and 70 perform differently in detail with ε0 ranging from 0 to 5. There is even positive effect of RS on the ultimate strength of columns with slenderness of 40 under small eccentricity, especially for 0.7 , ε0 , 4.5. The different stress states of midheight cross section between the two columns may give a reason for such deviation in ΔN 2 ε0 curves. (1) For columns with slenderness of 70, the middle fiber strains (S02 and S05 in Fig. 8.16) of midheight reach
cross  section
 about 75% of fy/E near peak load. Before peak load, we have N=Af y 2 zΦE=f y , 1, where Φ is
the curvature 
at midheight. 
Thesuperimposition of compressive RS may result in N=Af y 2 zΦE=f y 1 σrc =f y . 1. In this case the presence of RS accelerates the yielding process of compressive side that is detrimental to the ultimate bearing capacities of slender columns. (2) For columns with mediate slenderness of 40, the middle fiber strains of midheight cross section are much higher than those of slender columns that are about 90% of fy/E near peak load. The fibers in compressive
 side
should  have yielded under combined bending and axial compression, N=Af 2 zΦE=f
y 
y .1. However,
 the presence of tensile RS in the corners results in N=Af y 2 zΦE=f y 1 σrt =f y , 1. Thus the corners are still in elastic status and able to provide additional bending resistance rather than the cross section without RS.

8.3.2.3 Effect of width to thickness ratio Fig. 8.28 shows that the calculated interaction buckling curves with different d/t ratios differ from each other. 1.0 0.8

λ 20

B-10 B-15 B-20

40

N /Afy

60

0.6 0.4 0.2 0.0 –0.2

70 80 90 100 120 140

0.0

0.2

0.4

0.6

0.8

N /Af y Figure 8.28 Comparison of interaction curves with varying d/t.

1.0

1.2

346

Behavior and Design of High-Strength Constructional Steel

8 λ =40 λ =70

ΔND/t (%)

6 4 2 0 –2 0

5

10

15

20

25

ε 0 = e0 A / W1y

Figure 8.29 Strength difference between columns with d/t ratios of 10 and 20.

To evaluate the influence of d/t ratio, the strength difference versus eccentricity ratio curves is shown in Fig. 8.29. The strength difference between columns with d/ t ratios of 10 and 20 is defined as: ΔND=t 5

NB-10 2 NB-20 3 100% NB-10

(8.14)

where NB-10 and NB-20 are the predicted ultimate strength of columns with d/t ratios of 10 and 20, respectively. The indirect influence of width to thickness ratio on the buckling behavior is derived from the effect of RS. The magnitude of the compressive residual stress ratio β decreases with the increase in d/t ratio, while the tensile residual stress ratio α is proportional to d/t ratio. Under small load eccentricity with ε0 , 5, the effect of d/t is as same as discussed in Section 5.3. However, under large load eccentricity with ε0 $ 5, it is different from the case of under small load eccentricity. In addition
to the compression the fibers in the tension side have yielded 
side before peak load, N=Afy 1 zΦE=fy . of compressive
1. The presence

 RS delays the yielding of fibers in tension side, N=Af y 1 zΦE=f y 1 σrt =f y , 1. The higher magnitude of compressive RS ratio is, the more beneficial to the ultimate strength of columns will be. This is the reason that the interaction curves of the B-10 series are higher than those of the B-20 series under large load eccentricity.

8.4

Design recommendation

8.4.1 H-sections In this section the applicability of design rules given in EN 1993-1-1, ANSI/ AISC 360-16, and GB 50017-2003 for Q690 steel columns of welded H-sections

Behavior and design of high-strength steel columns under combined compression and bending

347

under combined compression and bending is assessed by means of the ratios of the FE to design buckling resistances. In these design rules, effects of axial compression and bending moments are summed up linearly, while nonlinear effects of applied bending moments are accounted for by interaction factors. Since torsional deformation is beyond the scope of this research, factors related to torsional deformation were taken to be 1.0 in the calculation of design buckling resistances. Large numbers of FE models for Q690 steel columns of welded H-sections under combined compression and uniaxial bending were established with variations in cross section sizes, nondimensional slendernesses, and initial loading eccentricity ratios. Cross section sizes included Sections H1, H2, H3, and H4 and nondimensional slendernesses included 0.6, 1.0, 1.4, and 1.8. Initial loading eccentricity ratios varied from 0.0 to 20.0. A total of 15 initial loading eccentricities were adopted, which were 0.0 to 1.0 with an interval of 0.2 and 1.5, 2.0, 3.0, 4.0, 5.0, 6.0, 8.0, 10.0, and 20.0. Residual stresses were incorporated in FE models. Initial out-ofstraightnesses were taken to be sinusoidal shapes. The amplitudes of initial out-ofstraightnesses complied with the theoretical background of corresponding design rules. In the assessment of both EN 1993-1-1 and GB 50017-2003, the amplitudes were 1/1000 of effective lengths, while for ANSI/AISC 360-16 the amplitudes were 1/1500 of effective lengths.

8.4.1.1 EN 1993-1-1 According to EN 1993-1-1, the buckling resistance for a steel member under combined compression and bending should satisfy the following equations: NEd My;Ed 1ΔMy;Ed M 1ΔMz;Ed
 1kyz z;Ed 1kyy #1 χy NRk =γM1 Mz;Rk =γM1 χLT My;Rk =γM1

(8.15)

NEd My;Ed 1ΔMy;Ed M 1ΔMz;Ed
 1kzz z;Ed 1kzy #1 χz NRk =γM1 Mz;Rk =γM1 χLT My;Rk =γM1

(8.16)

where NEd, My,Ed, and Mz,Ed are the design values of the compression force and the moments about the major (y) and the minor (z) axes along the member, respectively; NRk, My,Rk, and Mz,Rk are the characteristic values of resistances to compression force and the bending moments about the major (y) and the minor (z) axes, respectively; ΔMy;Ed and ΔMz;Ed are the moments due to the shift of the centroidal axes for Class 4 sections; χy and χz are the reduction factors due to flexural buckling about the major (y) and the minor (z) axes, respectively; χLT is the reduction factor due to lateral-torsional buckling; kyy, kyz, kzy, and kzz are the interaction factors; and γ M1 is the partial factor for resistance of members to instability assessed by member checks.

348

Behavior and Design of High-Strength Constructional Steel

In the first terms of Eqs. (8.15) and (8.16), the reduction factors, χy and χz, are related to the in-plane buckling behavior, and they are determined from the studies on columns under axial compression. In EN 1993-1-1, for welded H-sections made of 690 N/mm2 steel, curves “b” and “c” are currently proposed to calculate column buckling resistances about major (y) axis and minor (z) axis, respectively. Based on the residual stress measurement and the test results, higher column curves may be applicable for designing Q690 steel columns of welded H-sections, and the assessment of higher column curves will be conducted later. In the second terms the reduction factor χLT is related to the lateral-torsional buckling behavior, and it was taken to be 1.0 in calculations of design buckling resistances. The interaction factors kyy, kyz, kzy, and kzz in the second and the third terms can be obtained from two different approaches given in Annexes A and B, respectively. It should be noted that the main difference between these two approaches is the way of presenting different structural effects. Annex A emphasizes transparency, and each structural effect is accounted for by an individual factor. However, Annex B works with simplicity and allows some structural effects to be combined into a global factor. In this study the design rules given in Annexes A and B in EN 19931-1 for Q690 steel columns of welded H-sections are assessed. It should be noted that in Annex A the factor CmLT and aLT are also related to torsional deformation, and they were both taken to be 1.0. In Annex B, there are two sets of formulae for the calculation of interaction factors kij, one for members not susceptible to torsional deformations and the other one for members susceptible to torsional deformations. The former was adopted since the effect of torsional deformation was neglected. For H-section about major axis the average ratios of FE to design buckling resistances by using curves “a” and “b” are summarized in Table 8.17. It is found that the average ratios using either curves “a” or “b” are slightly larger than 1.00 with standard variations smaller than 0.04. It means that both curves “a” and “b” could provide conservative predictions to buckling resistances about major axis. However, compared with curve “b,” the ratios using curve “a” are closer to 1.00. Thus the design rules using curve “a” should be recommended owing to its advantage in safety and accuracy in estimating Q690 steel columns of welded H-sections under combined compression and major axis bending. For H-section about minor axis the average ratios of FE to design buckling resistances by using curves “a” and “b” are summarized in Table 8.18. For Annex A, even though the average ratios using curve “b” are larger than 1.00, a few unconservative design results are observed in columns with nondimensional slendernesses of 0.6 and 1.0. The minimum value of all the ratios using curve b is 0.96. To the contrary, only a few unconservative design results are found when using curve “c,” and the minimum value of all the ratios using curve “c” is 0.99. For Annex B a few unconservative design results are found when using either curve “b” or “c”. Overall, the design rules using curve “c” should be proposed due to acceptable accuracy in designing Q690 steel columns of welded H-sections under combined compression and minor axis bending.

Table 8.17 Ratios of FE to design buckling resistances about major axis according to EN 1993-1-1. H-section (major axis)

Curve “a” Annex A

H1 H2 H3 H4

Curve “b” Annex B

Annex A

Annex B

Average

Standard deviation

Average

Standard deviation

Average

Standard deviation

Average

Standard deviation

1.02 1.01 1.04 1.03

0.02 0.03 0.02 0.02

1.04 1.02 1.05 1.04

0.02 0.03 0.02 0.02

1.06 1.05 1.07 1.06

0.03 0.03 0.03 0.04

1.09 1.07 1.10 1.09

0.02 0.02 0.02 0.03

Table 8.18 Ratios of FE to design buckling resistances about minor axis according to EN 1993-1-1. H-section (minor axis)

Design with curve “b” Annex A

H1 H2 H3 H4

Design with curve “c”

Annex B

Annex A

Annex B

Average

Standard deviation

Average

Standard deviation

Average

Standard deviation

Average

Standard deviation

1.02 1.02 1.04 1.04

0.03 0.05 0.04 0.04

1.06 1.07 1.09 1.09

0.04 0.04 0.03 0.03

1.05 1.06 1.08 1.08

0.04 0.06 0.06 0.06

1.11 1.12 1.14 1.14

0.04 0.04 0.04 0.04

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Behavior and Design of High-Strength Constructional Steel

8.4.1.2 ANSI/AISC 360-16 According to ANSI/AISC 360-16, the buckling resistance for a steel member under combined compression and bending should satisfy the following equations: Pr When $ 0:2 Pc

0 1 Mry A Pr 8 @ Mrx # 1:0 1 1 9 Mcx Pc Mcy

Pr When , 0:2 Pc

0 1 M Pr M ry A rx # 1:0 1@ 1 2Pc Mcx Mcy

(8.17)

(8.18)

where Pr is the design axial force, Pc is the axial buckling resistance, Mrx and Mry are the design moments about the major (x) and the minor (y) axes, respectively, and Mcx and Mcy are the moment resistances about the major (x) and the minor (y) axes, respectively. In the first terms of Eqs. (8.17) and (8.18), Pc is related to the in-plane buckling behavior, and it is determined from the studies on columns under axial compression. In this standard, there is only one column curve for the calculation of buckling resistances of all types of steel columns, and the applicability of this column curve will be examined later. In the second and third terms, Mcx is the lower value obtained according to the limit states of yielding, lateral-torsional buckling, and compression flange local buckling. In the calculation of Mcx herein, the limit state of lateral-torsional buckling was neglected. Mcy is the lower value obtained according to the limit states of yielding and flange local buckling. For H-section about major axis the average ratios of FE to design buckling resistances by using curves “a” and “b” are summarized in Table 8.19. It is found that all the ratios are larger than 1.00 with standard deviations no greater than 0.05. The minimum value of all the ratios is 0.97. That means the design rules in ANSI/AISC 360-16 could provide accurate predictions to buckling resistances of Q690 steel columns of welded H-sections under combined compression and major axis bending. For H-section about minor axis the average ratios of FE to design buckling resistances are summarized in Table 8.20. It is found that except Section H4, the average ratios for the rest sections are smaller than 1.00. The minimum value of all the ratios is 0.91. The main reason should be that only one column curve is adopted in Table 8.19 Ratios of FE to design buckling resistances about major axis according to ANSI/AISC 360-16. H-sections H1 H2 H3 H4

Average

Standard deviation

1.04 1.04 1.05 1.06

0.05 0.04 0.05 0.04

Behavior and design of high-strength steel columns under combined compression and bending

351

Table 8.20 Ratios of FE to design buckling resistances about minor axis according to ANSI/AISC 360-16. H-sections H1 H2 H3 H4

Average

Standard deviation

0.97 0.98 0.99 1.01

0.05 0.05 0.05 0.04

this code, and this curve is the mean curve of the band for the group with the largest amount of data in Structural Stability Research Council (SSRC) column categories. Thus overestimation on buckling resistance of eccentrically compressed columns will occur when neglecting the resistance factors. Therefore the design rules in ANSI/AISC 360-16 should be arguably considered applicable for designing Q690 steel columns of welded H-sections under combined compression and minor axis bending.

8.4.1.3 GB 50017-2003 According to GB 50017-2003, the buckling resistance for a steel member under combined compression and bending should satisfy the following equations: β My N β M  mx x  1 η ty 1 #f ϕx A γ Wx 1 2 0:8 N0 ϕby Wy x N

(8.19)

β My  my  #f γy Wy 1 2 0:8 NN0

(8.20)

Ex

N β Mx 1 η tx 1 ϕy A ϕbx Wx

Ey

where N is the design value of the compression force, Mx and My are the design moments about the major (x) and the minor (y) axes, respectively, ϕx and ϕy are the reduction factors for flexural buckling about the major (x) and the minor (y) axes, respectively, ϕbx
is the for   lateral-torsional buckling;  reduction factor ϕby 5 1:0, N0Ex 5 π2 EA= 1:1λ2x , N0Ey 5π2 EA= 1:1λ2y , λx and λy are the slendernesses for flexural buckling about the major (x) and the minor (y) axes, respectively, A is the cross-sectional area, Wx and Wy are the elastic moduli about the major (x) and the minor (y) axes, respectively, βmx and βmy are equivalent moment factors for in-plane stability about the major (x) and the minor (y) axes, respectively, βtx and βty are equivalent moment factors for out-of-plane stability about the major (x) and the minor (y) axes, respectively, γx and γy are plasticity adaptation factors for bending about the major (x) and the minor (y) axes, respectively, η 5 1.0 for members susceptible to torsional deformation η 5 0.7 for members not susceptible to torsional deformation, f design yield strength of the steel material, and E is Young’s modulus of the steel material.

352

Behavior and Design of High-Strength Constructional Steel

In the first terms of Eqs. (8.19) and (8.20), the reduction factors, ϕx and ϕy, are related to the in-plane buckling behavior, and they are determined from the studies on columns under axial compression. Curve “b” is recommended to calculate column buckling resistances of welded H-sections with steel grades no greater than Q420 steel. The assessment of the column curve “b” and the higher column curve “a” for designing Q690 steel columns of welded H-sections will be conducted later. In the second term of Eq. (8.19), the reduction factor ϕbx is related to the lateraltorsional buckling behavior, and it was taken to be 1.0 in calculations of design buckling resistances. In this design code the cross-sectional plastic moduli are evaluated by the product of plasticity adaption factors and cross-sectional elastic moduli. For welded Hsections the plasticity adaption factors for major (x) and minor (y) axes bending are taken to be 1.05 and 1.20, respectively. That may lead to a partial use of plastic bending resistances of compact and noncompact sections. For H-section about major axis the average ratios of FE to design buckling resistances by using curves “a” and “b” are summarized in Table 8.21. It is found that the average ratios using curve “a” are slightly larger than 1.00 with standard variations no greater than 0.04. The minimum value of all the ratios using curve “a” is 0.97. Noting that the selection of column curves in this code is based on the “mean value” criteria, slight unconservatism should be allowed. Therefore the design rules using curve a could provide accurate predictions to buckling resistances of Q690 steel columns of welded H-sections under combined compression and major axis bending. For H-section about minor axis the average ratios of FE to design buckling resistances by using curves “a” and “b” are summarized in Table 8.22. All the average ratios using curve “a” are greater than 1.00 but the standard deviations approach 0.10. This is because some FE results are significantly higher than the design results using curve “a.” All the average ratios using curve “b” are significantly larger than 1.10, and their standard deviations are relatively reduced. The minimum value of all the ratios using curve “b” is 0.98. Therefore the design rules using curve “b” could make predictions with satisfactory accuracy to buckling resistances of Q690 steel columns of welded H-sections under combined compression and minor axis bending.

Table 8.21 Ratios of FE to design buckling resistances about major axis according to GB 50017-2003. H-sections

H1 H2 H3 H4

Design with curve a

Design with curve b

Average

Standard deviation

Average

Standard deviation

1.07 1.05 1.08 1.06

0.04 0.04 0.03 0.03

1.12 1.10 1.13 1.11

0.02 0.02 0.02 0.03

Behavior and design of high-strength steel columns under combined compression and bending

353

Table 8.22 Ratios of FE to design buckling resistances about minor axis according to GB 50017-2003. H-sections

H1 H2 H3 H4

Design with curve a

Design with curve b

Average

Standard deviation

Average

Standard deviation

1.06 1.07 1.08 1.08

0.10 0.09 0.08 0.08

1.11 1.11 1.13 1.13

0.08 0.07 0.06 0.06

8.4.2 Box sections The interaction formula Eq. (8.21) is derived from the yield criteria of extreme edge fiber base on the elasticity assumption, which is in the form of the following equation: My N 1 5 fy 0 χy A W1y ð1 2 χy ðN=NEy ÞÞ

(8.21)

To reflect the inelastic development of beamcolumns in practice, it is rewritten in form of the following equation: βmy My N 1 # fy 0 χy A γy W1y ð1 2 η1 ðN=NEy ÞÞ

(8.22)

by introducing the inelastic development factor γy and the compensation coefficient η1. The compensation coefficient of 0.8 is determined by the regression analysis of the numerical and experimental results of beamcolumns with the nominal yield strength no more than 420 MPa. To this end, it is necessary to check whether it is suitable for HSS such as Q460C. The results of parametric analysis (NFE) are compared with the interaction NGB in GB 50017-2003, as summarized in Table 8.23, where ε0,min and ε0,max are the eccentricity ratios corresponding to the minimum and maximum values of NFE/NGB, respectively. The comparison shows that the current design code underestimates the ultimate strength of Q460C beamcolumn by 10%18%. To determine the compensation coefficient η1 for Q460C welded box columns, a total of 2016 FE results are used in this study. Owing to the less sensitivity to RS and initial geometric imperfection of HSS columns, the recommended buckling curve “b” is adopted here for Q460C welded box columns [10]. The same value of inelastic development of 1.05 is adopted due to the good ductility of Q460C steel beamcolumn [12]. Hence, the compensation coefficient for Q460C welded box columns is obtained by regression analysis, which is 0.67. Table 8.23 shows the proposed modification of interaction formula (NM,GB) agrees better with the FE

354

Behavior and Design of High-Strength Constructional Steel

Table 8.23 Comparison of FE results and design code prediction. Λ

Min (NFE/ NGB)

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 Mean value Standard deviation

1.05 1.08 1.11 1.11 1.12 1.11 1.12 1.11 1.10 1.10 1.10 1.09 1.09 1.10 1.10 1.10 1.10 0.02

ε0, min

0.2 0.2 20 20 20 20 20 20 10 20 20 20 20 20 20 20

Max (NFE/ NGB) 1.12 1.15 1.16 1.17 1.19 1.23 1.26 1.26 1.24 1.23 1.20 1.18 1.15 1.14 1.14 1.13 1.18 0.05

ε0, max

0.6 1 1 2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 1

Min (NFE/ NM,GB) 1.05 1.05 1.04 1.03 1.03 1.03 1.02 1.03 1.03 1.03 1.03 1.03 1.03 1.03 1.03 1.03 1.03 0.01

ε0, min

0.2 0.2 0.2 0.2 0.2 0.6 1 1 2 2 2 2 2 2 4 4

Max (NFE/ NM,GB) 1.12 1.13 1.11 1.10 1.11 1.09 1.08 1.09 1.10 1.10 1.09 1.08 1.07 1.05 1.05 1.04 1.09 0.03

ε0, max

1 1 2 2 4 6 6 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

results than the current design code. The underestimation of ultimate strength is reduced to 3%9%.

8.5

Summary

Based on the studies hereinabove, the following remarks can be summarized as follows: 1. In the determination of design buckling resistances of Q690 steel columns of welded Hsections under combined compression and major axis bending, the design rules in EN 1993-1-1 using curve “a,” and the design rules in ANSI/AISC 360-16 using the single curve, and the design rules in GB 5007-2003 using curve “a” are proposed; 2. In the determination of design buckling resistances of Q690 steel columns of welded Hsections under combined compression and minor axis bending, the design rules in EN 1993-1-1 using curve “c,” the design rules in ANSI/AISC 360-16 using the single curve, and the design rules in GB 5007-2003 using curve “b” are proposed. 3. In the determination of design buckling resistances of Q460 steel columns of welded boxsections under combined compression and bending, the buckling curve “b,” the inelastic development factor of 1.05, and the compensation coefficient of 0.67 are recommended according to the interaction formula in GB 50017-2003.

Behavior and design of high-strength steel columns under combined compression and bending

355

References [1] Ma TY, Hu YF, Liu X, Li GQ, Chung KF. Experimental investigation into high strength Q690 steel welded H-sections under combined compression and bending. J Constr Steel Res 2017;138:44962. [2] Yan XL, Li GQ, Wang YB. Q460C welded box-section columns under eccentric compression. Proc Inst Civil Eng-Struct Build 2018;171(8):61124. [3] Ma TY, Li GQ, Chung KF. Numerical investigation into high strength Q690 steel columns of welded H-sections under combined compression and bending. J Constr Steel Res 2018;144:11934. [4] GB/T 1591-2008. High strength low alloy structural steels. 2008. [5] European Committee for Standardization. CEN. EN 10025-6, Hot rolled products of structural steels-part 6: technical delivery conditions for fiat products of high yield strength structural steels in the quenched and tempered condition. Brussels: European Committee for Standardization; 2004. [6] China Standard Press. GB/T 2975-1998 Steel and steel products: location and preparation of test pieces for mechanical testing. Beijing: China Standard Press; 1998 (in Chinese). [7] National Standardization Technical Committees. GB/T 228-2002 Metallic materials: tensile testing at ambient temperature. Beijing: China Standard Press; 2002 (in Chinese). [8] China Architecture & Building Press. GB 50017-2003. Code for design of steel structures. Beijing: China Architecture & Building Press; 2003 (in Chinese). [9] European Committee for Standardization. CEN. Eurocode 3: Design of steel structures, Part 1-1: General rules and rules for buildings, EN 1993-1-1. Brussels: European Committee for Standardization; 2005. [10] Wang YB, Li GQ, Chen SW, Sun FF. Experimental and numerical study on the behavior of axially compressed high strength steel box-columns. Eng Struct 2014;58 (0):7991. [11] Wang YB, Li GQ, Chen SW. The assessment of residual stresses in welded high strength steel box-sections. J Constr Steel Res 2012;76:939. [12] Wang YB, Li GQ, Cui W, Chen SW. Seismic behavior of high strength steel welded beam-column members. J Constr Steel Res 2014;102(0):24555.

Hysteretic behavior of highstrength steel columns

9

Guo-Qiang Li, Yan-Bo Wang and Suwen Chen Tongji University, Shanghai, P.R. China

9.1

Introduction

Application of high-strength steel (HSS) in the construction of structures and buildings has to consider the problem of endurability under severe earthquakes. Thus it is important to have a good understanding of the hysteretic behavior of structural members made of HSS subjected to seismic action. The seismic behaviors of normal-strength steel members have been extensively investigated, including the influences of geometrical parameters and loading conditions. Ballio and Castiglioni [1] characterized the cyclic behavior of steel beams based on the test and numerical simulation of 29 specimens with I-shaped sections. The cumulative damage models for predicting the seismic behavior of steel members were discussed. Fadden and McCormick [2] carried out cyclic quasistatic test on 11 hollow structural section beam members and verified that hollow structural sections could be used in seismic applications when width-to-thickness ratio and height-to-thickness ratio were carefully determined to ensure a stable plastic hinge behavior. The behavior of columns subjected to changing compression axial load was investigated by Lamarche and Tremblay [3]. The test results show that steel columns with compact I-shaped sections might be able to sustain a certain inelastic buckling without loss in loadcarrying capacity under seismic actions. According to nonlinear time-history analysis of structures subjected to earthquake ground motions, columns are usually subjected to combined axial load and inelastic rotation derived from story drift. The effects of width-to-thickness ratio, height-to-thickness ratio, and axial load ratio on the cyclic behavior of steel beam
columns have been evaluated. Newell and Uang [4] tested nine full-scale wide
flange columns under high axial force ratio of 35%, 55%, and 75% combined with story drift ratio up to 10%. It is concluded that the tested columns can sustain a large inelastic rotation under high axial loads. Similarly, experimental studies on box columns were carried out by Nakashima and Liu [5] and Kurata et al. [6]. It shows that the box column with compact section can sustain medium axial load (30% of the sectional yield strength) under story drift rotation up to 30% after the occurrence of local buckling. However, the prediction of crashing failure behavior by finite element (FE) analysis deviates from the experiments. This indicates the strong need for further experimental investigations. Not only in steel building frames are hollow box-section columns also widely used in steel bridge piers. However, it is typical with larger width-to-thickness ratio but Behavior and Design of High-Strength Constructional Steel. DOI: https://doi.org/10.1016/B978-0-08-102931-2.00009-7 © 2021 Elsevier Ltd. All rights reserved.

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Behavior and Design of High-Strength Constructional Steel

lower in axial force ratio than those columns in buildings. Kumar and Usami [7] tested 10 hollow box piers under low axial force ratios (16.3% and 20% of the yield strength of piers) combined with cyclic lateral load. They developed a damage model to quantify the damage sustained under earthquake actions. Moreover, Aoki and Susantha [8] investigated the effect of cross-sectional aspect ratio on ultimate strength, ductility, and energy dissipation capacity of rectangular-shaped steel piers. The height-to-width ratio of 1.6 was suggested for the optimal design of member ductility and energy dissipation capacity. As pointed out by Fukumoto [9], the increase in strength will result in the increase in yield-to-tensile strength ratio and the decrease in elongation ratio. Due to the differences in mechanical properties, the seismic performance of HSS members may be different from that of the conventional normal-strength steel members [10]. European and American specifications for steel structures allow the use of HSS up to steel grades of S700 (700 MPa) and A514 (690 MPa) [11,12]. However, the existing design codes are established based on the experimental and analytical studies of mild carbon steels usually with nominal yield strengths from 235 to 355 MPa [13,14]. So as to apply HSS for structures located in seismic zones it is necessary to understand the hysteretic behavior of HSS elements subjected to seismic actions. In this chapter, experimental and numerical studies on the cyclic behavior of welded H-section and box-section beam
columns with nominal yield strength of 460 and 690 MPa are presented. Seven welded H-section and two box-section columns were tested under cyclic horizontal load combined with constant axial load. Ultimate strength and energy dissipation capacity were evaluated according to the observation of the obtained lateral load
displacement hysteretic curves. Moreover, FE models were developed and validated through comparison between the simulation and test results. Based on the verified FE model, parametric analyses were conducted to investigate the effects of axial force ratio, component plate slenderness ratio on the hysteretic performance. Through the generalization of the experimental and numerical results, a hysteretic model incorporated with damage behavior and a simplified moment
curvature hysteretic model for Q460 and Q690 welded column were proposed.

9.2

Experimental program

9.2.1 Specimens HSS frame structures generally consist of HSS columns and mild steel beams. Similar with normal steel frame structures, the vertical members are usually subjected to small variation of axial pressure and periodic horizontal force under earthquake motions. Thus based on the assumptions that the axial pressure is constant and inflection point of curvature is at the column height center, the lower half column in the bottom story can be extracted as the studied mechanical model as shown in Fig. 9.1. This model is a cantilever beam
column, the top of which can move

Hysteretic behavior of high-strength steel columns

359

Figure 9.1 Studied mechanical model.

horizontally under constant vertical load. The failure modes include not only overall instability but also sectional fracture and local buckling. H-section steel beam
columns have been frequently applied in steel frame structures. In terms of their susceptibility to local buckling, H-section steel beam
columns are normally categorized into several categories, the component plate slenderness of which is accordingly restricted by design codes such as Chinese seismic design code GB50011-2010 and European code EC3 [15]. The width
thickness ratio restrictions of H-section members in the Code for Seismic Design of Buildings (GB50011-2010) are related to seismic level and yield strength of materials. Nevertheless, this restriction is correlated with material yield strength, axial compression ratio, area ratio of web and flange, and cross-sectional stress distribution in EC3, leading to a more comprehensive consideration. Therefore the specimen design in this chapter refers to the relevant provisions provided by European code EC3. Specifically, a Class 4 cross section (slender cross section) in EC3 denotes a type of cross section, the plate component slenderness of which is so large that local buckling occurs prior to the attainment of the elastic moment resistance (Mu , Mec ). A Class 3 (semicompact) cross section is one in which local buckling occurs after reaching the elastic moment resistance but before the plastic moment resistance (Mec , Mu , Mpc ). A Class 2 cross section corresponds to one which can reach its plastic moment resistance but solely provide limited rotation capacity due to elastic
plastic local buckling (Mu . Mpc ). Finally, for a Class 1 cross section, the plastic moment resistance can be developed with sufficient rotation capacity to

360

Behavior and Design of High-Strength Constructional Steel

allow redistribution of moments in the steel frames (Mu . Mpc and R . R0 ). In the abovementioned definitions, Mu is the ultimate moment resistance of the cross section; Mec and Mpc are the elastic and plastic moment resistance of the cross section, respectively, including the influence of axial load; R 5 θu =θy is the available rotation capacity of the cross section, where θu is the ultimate rotation and θy is the yield rotation; and R0 is the required rotation capacity that is defined differently in each specification. As more recently recognized, Class 1 and Class 2 sections, denoted as compact sections, are considered suitable for resisting seismic loading due to the fact that they are less susceptible to local buckling than Class 3 and Class 4 sections, which are, respectively, denoted as semicompact and slender sections. Regarding Class 1 and Class 2 cross sections, the expressions of width
thickness ratio restriction for flange and web are shown in Table 9.1. In this study, two H-section and two box-section specimens made of Q460C HSS and five compact H-section beam
column specimens made of Q690D HSS were designed, as shown in Fig. 9.2. The details of experimental specimens are summarized in Table 9.2. The main parameters are cross-sectional width-tothickness ratio and height-to-thickness ratio, which are compared with the classification limits of Eurocode 3, as shown in Fig. 9.3 and 9.4. As shown in Table 9.2, each experiment is identified as H/B/HM-X1-X2-X3, in which H(B) indicates H-section (box-section) specimens processed by Q460C; HM indicates H-section specimens processed by Q690D; and X1, X2, and X3, respectively, indicate the flange width
thickness ratio, web height
thickness ratio, and axial compression ratio. The identifier marked with  denotes this specimen possesses larger height (2510 mm) than others (2310 mm). For example, HM-5.3-220.2 indicates that the specimen is Q690D H-section beam
column, flange width
thickness ratio and web height
thickness ratio of which are 5.3 and 22, respectively. Besides, its axial compression ratio is 0.2. H, B, tw ; and tf , respectively, denote the section height, flange width, web thickness, and flange thickness. b denotes the free extension width of flange, hw 5 H 2 2tf denotes the calculated web height, n denotes the nominal axial compression ratio, and N denotes the axial force. The top plate, bottom, plate and stiffeners are all fabricated of Q345 steel. The plate component details and specimen blueprints of those Q690D-fabricated specimens are, respectively, shown in Table 9.3 and Fig. 9.5. The tensile tests were conducted on the Q460C and Q690D high-strength hotrolled steels for the specimens in accordance with the specifications of GB/T 2282002 [16], and the test results are summarized in Table 9.4. The Q460C and Q690D specimens were built up by welding with ER55-D2 and ER120S-G welding wires, respectively. The selected welding wire is able to provide sufficient strength and toughness to ensure the welding quality for high-strength structural steel.

9.2.2 Experimental setup The experiments were carried out in the structural engineering laboratory of Tongji University. MAS-3000-300/2-300-ton servo actuator was chosen as the vertical

Table 9.1 Plate slenderness limitation according to Eurocode 3. Stress distribution

Flange classification

Full-flange compression Edge compression Edge tension

Class 1

Class 2

9ε 9ε=α  pffiffiffiffi 9ε= α α

10ε 10ε=α  pffiffiffiffi 10ε= α α

Relative depth of compression zone

α.1 0:5 , α , 1 α # 0:5

Web classification Class 1

Class 2

33ε 396ε=ð13α 2 1Þ 36ε=α

38ε 456ε=ð13α 2 1Þ 41:5ε=α

pffiffiffiffiffiffiffiffiffiffiffiffiffi   Where α 5 1 1 nA=Aw =2 denotes the relative depth of compression zone in steel plate, related with axial load ratio, section area, and flange
web area ratio. ε 5 235=fy .

Table 9.2 Specimen details. Identifiers

H-3.8-15.6-0.3 H-7.7-27.7-0.3 B-10.7-10.7-0.3 B-21.6-21.6-0.3 HM-5.3-19-0.2 HM-5.8-19-0.2 HM-5.3-22-0.2 HM-5.3-19-0.35 HM-5.3-19-0.35

Steel grade

Q460C Q460C Q460C Q460C Q690D Q690D Q690D Q690D Q690D

H 3 B 3 tw 3 tf (mm) 220 3 175 3 11 3 21 360 3 340 3 11 3 21 140 3 140 3 11 3 11 260 3 260 3 11 3 11 230 3 220 3 10 3 20 230 3 240 3 10 3 20 260 3 220 3 10 3 20 230 3 220 3 10 3 20 230 3 220 3 10 3 20

Le (mm)

1905 1905  1905 1905 2310 2310 2310 2310 2510

b/ tf 3.8 7.7 10.7 21.6 5.3 5.8 5.3 5.3 5.3

hw/ tw 15.6 27.7 10.7 21.6 19.0 19.0 22.0 19.0 19.0

[b/tf]

[hw/tw]

Class 1

Class 2

Class 1

Class 2

6.4 6.4 23.5 23.5 5.3 5.3 5.3 5.3 5.3

7.1 7.1 27.2 27.2 5.8 5.8 5.8 5.8 5.8

23.6 23.6 23.6 23.6 19.3 19.3 19.3 19.3 19.3

27.2 27.2 27.2 27.2 22.2 22.2 22.2 22.2 22.2

n

N (kN)

0.3 0.3 0.3 0.3 0.2 0.2 0.2 0.35 0.35

1323 2523 868 1675 1733 1863 1782 2967 2967

362

Behavior and Design of High-Strength Constructional Steel

tf

H

tf

tw

h0

h0

tw

tw

H

b

B (A)

b B (B)

Figure 9.2 Cross section of Q460C specimens: (A) H section and (B) box section.

loading device. The actuator provided load sensor and high precision ball hinge. The maximum static test pull/pressure load was 1500/3000 kN with error less than 1%, and the maximum piston stroke was 300 mm. On the other hand, 50-t servo actuator was selected as the horizontal loading device. The maximum static test pull/pressure load was 500/500 kN with error less than 1%, and the maximum piston stroke was 500 mm. These two actuators can meet the requirement of vertical and horizontal loading in this experiment. The top plate was connected to an Lsection loading device that can freely rotate along its central shaft, thereby only shear force and axial load are transferred to the specimens. One end of this device was linked to the horizontal actuator. The other end was connected to the ball hinge of the vertical actuator. The specimen bottom plate was fixed onto the basement through high-strength bolts. The out-of-plane displacement was restrained using two steel trusses outside the plane that clamped the L-section loading device. The contact surface was coated with lubricant to reduce friction. The structural testing system for the H-section specimens is shown in Fig. 9.6.

9.2.3 Measurement arrangement For the specimens fabricated by Q460C, the arrangement of linear varying displacement transducers (LVDTs) and strain gauges is shown in Fig. 9.7. The in-plane lateral displacements were measured by LVDTs D1
D3 and D5
D10 that were distributed along the axial direction. LVDTs D9 and D10 were attached to the column end in horizontal alignment with the top surface of stiffeners, and LVDTs D5
D8 were attached next to D9 and D10 with vertical spacing of 100 mm, as shown in Fig. 9.7A. In order to inspect the rigidity of the fabricated support, lateral displacements of the specimen end plate and the top surface of the support were measured by VLDT D3 and D12, while the potentially relative slipping between them was recorded by LVDTs D13 for comparison. The lateral deformation of the

Hysteretic behavior of high-strength steel columns

32

363

H section

Box section

30

Class 3

Class 3

28

Height to thickness ratio

Class 1

Class 2

26 24

Class 1

22

Class 1

20 18

H-3.8-15.6-0.3 H-7.7-27.7-0.3 B-10.7-10.7-0.3 B-21.6-21.6-0.3

16 14 12 10 8 0

2

4

6

8

10 12 14 16 18 20 22 24 26 28 30

Width to thickness ratio

Figure 9.3 Width-to-thickness and height-to-thickness ratios of Q460C specimens.

 H section

Class 2 

Height to thickness ratio



Class 1

 

+0 +0 +0 +0 +0

    











Width to thickness ratio

Figure 9.4 Width-to-thickness and height-to-thickness ratios of Q690D specimens.

364

Behavior and Design of High-Strength Constructional Steel

Table 9.3 Plate component details. Specimen number

Plate dimension

Amount

Steel grade

1 2 3 4 5 6 7

2600 3 560 3 40 2830 3 630 3 40 2305 3 150 3 30 2190 3 150 3 30 2200 3 150 3 10 22150 3 220 3 20 22150 3 190 3 10

1 1 4 2 4 2 1

Q345 Q345 Q345 Q345 Q345 Q690 Q690

stiffened region of specimens was measured by D14. The out-of-plane lateral deflections were measured by LVDTs D4 and D11. Strain gauges were attached at two cross sections with a distance of 50 mm from both ends of each specimen to monitor the loading force and the moment applied on the specimens. The real time load-deflection curves displayed in monitor during the entire loading history were used to adjust and control the experiments. For the specimens fabricated by Q690D, the arrangement of measurement points is shown in Fig. 9.8. The specimen deformations were measured using 14 LVDT sensors. LVDTs 1
2 were used to monitor the displacement of pin shaft in Lsection loading device. LVDTs 3
4 were used to monitor the in-plane displacement at the top plate; LVDTs 5
6 were used to monitor the out-of-plane displacement at the top plate; LVDTs 7
8 were used to measure the in-plane displacement at one cross section that was 500 mm above the stiffeners; LVDTs 9
10 were used to monitor the out-of-plane displacement at that cross section; LVDTs 11
12 were arranged to monitor the vertical displacement of bottom plate to estimate the rigid rotation; LVDTs 13
14 were set to monitor the in-plane and out-of-plane relative slip regarding the bottom plate and the basement, respectively. To further understand local deformation, a total of 40 strain gauges were arranged on the cross sections A
A, B
B, C
C, and D
D which, respectively, possess distances of 2 L/3, L/3, B and B/2 from the bottom plate. Strain gauges arranged at cross sections A
A and B
B were used to measure the strain within elastic region to estimate the actual axial and shear force effectively transferred to the specimen; C
C and D
D were used to judge whether local buckling occurs at the flange or web. Besides, the collected data can also be used to estimate the curvature of bottom region.

9.2.4 Loading protocols The constant axial load was applied to the specimen by load-control loading. After the vertical loading reached its stable condition, cyclic horizontal load was subsequently applied to the pivot (Fig. 9.6B) in terms of increasing lateral displacement. Besides, the structure was pushed up to complete failure, when the specimen completely lost its resistance against the lateral load.

Hysteretic behavior of high-strength steel columns

365

Figure 9.5 Blueprint of Q690D specimens and components: (A) experimental specimen, (B) 2-2 cross section, (C) 4-4 cross section, (D) 5-5 cross section, (E) 1-1 cross section, (F) 3-3 cross section, and (G) 6-6 cross section.

The loading process includes preloading and formal loading. All measurement sensors were then checked and zero cleared before loading. In the preloading stage, 100 kN axial force was first applied to the top plate, and the anchor and bolt was further tightened. Then a one-cycle horizontal displacement was applied on the top plate, while the data collected by LVDTs and strain gauges were checked. Once all sensors were proved to be normal, the horizontal actuator was placed back to the

366

Behavior and Design of High-Strength Constructional Steel

Table 9.4 Basic mechanical properties of high-strength structural steels. Steel grade

Q460C Q460C Q690D Q690D

Thickness

Yielding strength

Elastic modulus

Ultimate strength

Elongation

t (mm)

fy (MPa)

E (MPa)

fu (MPa)

A (%)

Yieldtotensile ratio fy/fu

21 11 20 10

464 506 810 807

217,600 207,800 205,138 190,933

586 598 833 862

30.4 23.7 16.9 14.7

0.79 0.85 0.97 0.94

Figure 9.6 Experimental setup: (A) experimental setup, (B) L-shape link between the specimen top, and (C) HSS-bolted connection between column end and basement. HSS, High-strength steel.

initial position. Regarding formal loading stage, the axial force was first progressively applied using load-controlling approach, after which the LVDTs were zeroed again. Then horizontal cyclic loading was applied using displacement-controlled loading approach. For the specimens fabricated by Q460C, the lateral cyclic displacement was gradually increased from L/300 to L/30 (L/20 for B-10.7-10.7-0.3) and repeated two cycles in each amplitude of drift ratio, as shown in Fig. 9.9. For the specimens fabricated by Q690D, the cyclic loading includes three cyclic cycle at each displacement amplitude level with level increment equals to δy , which

Hysteretic behavior of high-strength steel columns

367

In-plane LVDT Out-of-plane LVDT

Horizontal load

D4 D1/D2

D5/D6





D7/D8 D9/D10

S01

S06

S02

S05

S03

S04

D14

D3

D13 D12

D11

S01

S06

S02

S05

S03

S04

(A)

(B)

Figure 9.7 Measurement points: (A) arrangement of LVDTs and (B) arrangement of strain gauges. LVDTs, Linear varying displacement transducers.

LVDTs 1-2 LVDTs 3-4 LVDTs 5-6

A

LVDTs 9-10 LVDTs 7-8 LVDTs 1 LVDTs 1 LVDTs 1

A

B

B

C

C

D

D

A–A

B–B

C–C

D–D

LVDT11

(A)

(B)

Figure 9.8 Measurement arrangement: (A) arrangement of LVDT and (B) arrangement of strain gauges. LVDTs, Linear varying displacement transducers.

is defined as the yield displacement when monitored edge fiber reaches its yield strain. In addition, Chinese code for seismic design of buildings (GB50011-2010) requires that the interstory angle of multistory and high-rise steel structures should be smaller than 1/250 and 1/50, respectively, under frequently occurred earthquake

368

Behavior and Design of High-Strength Constructional Steel

0.04

Lateral displacemence (L)

0.03 0.02 0.01 0.00 L/300

–0.01

L/150

L/100

L/80

–0.02

L/60

L/50

–0.03

L/40 L/30

–0.04 0

2

4

6

8

10

12

14

16

18

20

Cycles

Figure 9.9 Cyclic loading protocol.

Table 9.5 Cyclic loading amplitudes. Level

L/250

L/50

δy

2δ y

3δ y

3.5δ y

4δ y

4.5δ y

5δ y

HM-5.3-19-0.2 HM-5.8-19-0.2 HM-5.3-22-0.2 HM-5.3-19-0.35 HM-5.3-19-0.35

8.4 8.4 8.4 9.2 9.2

42.2 42.2 42.2 46.2 46.2

62.3 52.4 50.9 40.3 48.8

124.6 104.8 101.8 80.6 97.6

186.9 157.2 152.7 120.9 146.4

218.1 183.4 178.2 141.1 170.8

249.2 209.6 203.6 161.2 195.2

280.4 235.8 229.1 181.4 219.6

311.5 262 254.5 201.5 244

and severe earthquake. Thus two additional amplitude levels of 1/250 and 1/50 were added to the cyclic loading. The displacement amplitude level is shown in Table 9.5. The loading protocol is schematically depicted in Fig. 9.10. All experiments were continuously loaded until the specimens were significantly destroyed or basically lost their bearing capacity.

9.3

Experimental results

9.3.1 Experimental phenomenon For Q460C fabricated specimens the corresponding deformations after testing are shown in Fig. 9.11. In consistent with the appearance of degradation in the moment
rotation hysteretic curve, local buckling is found in the first cycle with

Hysteretic behavior of high-strength steel columns

369

Figure 9.10 Cyclic lateral loading.

Plasc hinge

Plasc hinge (A)

(B) Local buckling

Local buckling (C)

(D)

Figure 9.11 Failure modes of Q460C specimens: (A) H-3.8-15.6-0.3, (B) B-10.7-10.7-0.3, (C) H-7.7-27.7-0.3, and (D) B-21.6-21.6-0.3.

drift ratio of 1/30 for the specimens H-7.7-27.7-0.3 and the second cycle with drift ratio of 1/30 for the specimens B-21.6-21.6-0.3. The minor bulging was found near the specimen foot and became serious with the increase in repeated cycles, as

370

Behavior and Design of High-Strength Constructional Steel

shown in Fig. 9.11C and D. It can be seen that, although the moment resistance of the specimen series H-7.7-27.7-0.3 and B-21.6-21.6-0.3 deteriorates immediately since the appearance of local buckling, there is no obvious pinching for the postbuckling hysteretic curves. Equivalent viscous damping coefficients are calculated to evaluate the earthquake resisting performance [17], as expressed in the following equation: ξ5

1 Shl 4π Se

(9.1)

where Shl is the energy dissipated in each hysteretic loop, which can be represented by the area of the corresponding closed loop in the total restoring force
displacement diagram; Se is the equivalent elastic strain energy; and Se 5 1=2Dmax Fmax , Dmax and Fmax are the maximum displacement and the corresponding force reached in the loop. Fig. 9.12 shows the equivalent viscous damping coefficient against the normalized story drift ratio by the yield story drift ratio θy. For box-section series, very compact cross section B-10.7-10.7-0.3 shows the highest maximum value of equivalent viscous damping, which is 0.28. The equivalent viscous damping coefficient of the section B-21.6-21.6-0.3 is lower than that of the section B-10.7-10.70.3 during the entire loading history, with maximum value of 0.14. For H-section specimens the Class 1 cross section H-3.8-15.6-0.3 shows the higher equivalent viscous damping curve than the Class 2 cross section H-7.7-27.7-0.3 during the

Figure 9.12 Equivalent viscous damping coefficient for tested specimens.

Hysteretic behavior of high-strength steel columns

371

inelastic deformation range. The maximum values of equivalent viscous coefficient are 0.22 and 0.16 for the sections H-3.8-15.6-0.3 and H-7.7-27.7-0.3, respectively. It should be noted that, with respect to the total energy dissipated within each hysteretic loop, the section H-7.7-27.7-0.3 showed the highest energy dissipation capacity of the tested specimens attributed to the high load bearing capacity. The HM-5.3-19-0.2 specimen remained in elastic range when the LVDT on pin shaft reached 8.4 mm (interstory drift angle of 1/250) and 42.2 mm (interstory drift angle of 1/50). As it reached 62.3 mm, the strain gauge arranged on the edge fiber of D
D cross section indicated that its strain stepped into plastic range. Thus 62.3 mm was adopted as the yield displacement and displacement amplitude increment in the subsequent loading. The corresponding yield horizontal load was 272.6 kN. Slight local buckling then took place in the west flange near the bottom plate when horizontal displacement reached the first cycle with amplitude of 2δy . However, this buckling deformation disappeared as the loading amplitude reversed. When the loading stepped into the second cycle with amplitude of 2δy , the horizontal reaction force began to decrease, and maximum horizontal load of 344.4 kN was therefore obtained. Subsequently, local buckling deformation accumulated remarkably and was unrecoverable along with reversal. The web formed a half-wave local buckling when the displacement amplitude reached the first cycle with amplitude of 3δy , with considerable decrease in horizontal load up to 40% of ultimate bearing capacity. As horizontal load reversed, the welding between the west flange and web near the buckling center encountered fracture. The specimen experienced a sudden drop in horizontal reaction force but still provided bearing capacity to some extent. Accompanied by a tremendous sound, the east flange as well as the welding between web and east flange fractured abruptly. The specimen basically lost its bearing capacity with corresponding interstory drift angle closed to 1/10. This was the end of experiment HM-5.3-19-0.2. As for experiment HM-5.8-19-0.2, the measured yield displacement was 52.4 mm that also exceeded the pin shaft displacement corresponding to interstory drift angle 1/ 50. Local buckling behavior initially took place in the second cycle with displacement amplitude level of 2δy but then vanished with reversal. The unrecoverable local buckling plastic deformation rapidly developed as cyclic loading stepped into the third cycle with displacement amplitude level of 2δy . Subsequently, the web plate began to buckle at the first cycle with displacement amplitude level of 3δy . The horizontal load continuously dropped to 30% of ultimate bearing capacity until the appearance of flange
web welding joint fracture. The final interstory drift angle was more than 1/15. Similar with previous two experiments, experiment HM-5.3-22-0.2 was within elastic region even interstory drift angle reached 1/50. The measured yield displacement and yield load according to identical method with previous experiments were, respectively, 50.9 mm and 305 kN. The local buckling occurred in the first cycle with amplitude of 2δy and vanished as that amplitude reversed. Then this local buckling became unrecoverable when cyclic loading entered the third cycle with amplitude of 2δy where the maximum horizontal load of 408.5 kN was obtained. By the first circle with amplitude of 3δy , the web plate began to buckle locally while the horizontal load decreased rapidly to about 60% of the ultimate bearing capacity.

372

Behavior and Design of High-Strength Constructional Steel

As interstory drift angle reached 1/15, welding joint between the web and the west flange near the buckling center was fractured, leading to a sudden drop in horizontal load. The experiment was then stopped. Regarding experiment HM-5.3-19-0.35, the interstory drift angle corresponding to edge fiber yielding was smaller than 1/50. The yield displacement and load were, respectively, 40.3 mm and 205 kN. By the first cycle with amplitude of 2δy , the local buckling emerged then became unrecoverable while reaching the third cycle of that amplitude. Accompanied by considerable strength decrease, the web plate buckled in the first cycle with amplitude of 3δy . Finally, the welding joint between web and flange near bottom plate was torn off as interstory drift angle reached 1/15 after two cycles accumulated. As for experiment HM-5.3-19-0.35 , dimensions of which were identical with HM-5.3-19-0.35 except the specimen height, the yielding interstory drift angle was larger than 1/50, with corresponding displacement and horizontal load, respectively, equaled to 48.8 mm and 192.1 kN. The subsequent phenomenon, including appearance and growth of local buckling, decrease in strength, and fracture, and their corresponding cycles were identical with those of HM-5.3-19-0.35. The interstory drift angle reached 1/15 when this experiment stopped. The key events and their corresponding interstory drift angles during the overall process are summarized in Table 9.6. It can be seen that except for experiment HM5.3-19-0.35, the yielding interstory drift angles for almost all specimens are larger than 1/50. Besides, the interstory drift angles corresponding to maximum horizontal load for all specimens are larger than 1/50. For each specimen the displacement amplitudes corresponding to maximum horizontal force and that corresponding to local buckling initiation are identical, indicating that the decrease in bearing capacity is mainly driven by local buckling. In addition, the displacement amplitudes corresponding to the status when horizontal force decreased to 85% of ultimate bearing capacity and that corresponding to welding fracture are also identical. However, the latter generally took one more cycle than the former. Thus the ductility index can be calculated based on the displacement corresponding to 85% of ultimate bearing capacity. The potential ultralow-fatigue behavior should not be neglected. The failure modes of experimental specimens are shown in Fig. 9.13, including local buckling of component plates and fracture of welding joint. It is found that local buckling concentrating near the bottom is the dominating failure mode while no overall buckling is observed for any of the test specimens. With plastic buckling deformation accumulated, fracture occurs at the weld between the flange and the web where the largest local buckling deformation takes place. On the other hand, experiment HM-5.3-19-0.2 cross section of which belongs to Class 1 category possesses fracture in its flange plate. This is important because generally a compact section with small component plate slenderness is preferred to be utilized in plasticity or seismic design. Thus Class 1 section is the most favorable type when full-section plasticity and considerable deformation ability are demanded. However, the Class 1 section exhibits the potential risk of flange fracture that is fully unfavorable in structure design. This reveals the truth that the decrease in width
thickness ratio for Q690HSS compact beam
columns may produce severer local buckling deformation instead of weakening that effect. Therefore Q690 HSS H-section beam
column needs not only the

Table 9.6 Main observations with corresponding loading amplitude. Specimens

HM-5.3-19-0.2 HM-5.8-19-0.2 HM-5.3-22-0.2 HM-5.3-19-0.35 HM-5.3-19-0.35 HM-5.3-19-0.2

δ 5 δ 1/  250 (mm)

δ 5 δ y (mm)

50

δ 5 δ 1/  (mm)

Local buckling initiation

Attainment of maximum lateral force

Lateral force drops to 85% of ultimate force

Joint fracture

8.4 8.4 8.4 8.4 9.2 8.4

50.3 45.7 49.4 40.3 55.0 50.3

42.2 42.2 42.2 42.2 46.2 42.2

2.0δy 2.0δy 2.0δy 2.0δy 2.0δy 2.0δy (1)

2.0δy (2) 2.0δy (2) 2.0δy (3) 2.0δy (2) 2.0δy (2) 2.0δy (2)

3.0δy (1) 3.0δy (1) 3.0δy (1) 3.0δy (1) 3.0δy (1) 3.0δy (1)

3.0δy (1) 3.0δy (2) 3.0δy (2) 3.0δy (2) 3.0δy (2) 3.0δy (1)

 Where δy denotes the displacement corresponding to the yielding of cross sectional edge fiber, while δ1/250 and δ1/50, respectively, denote the displacement corresponding to the interstory drift angle of 1/250 and 1/50.

374

Behavior and Design of High-Strength Constructional Steel

(A-1)

(B-1)

(C-1)

(D-1)

(E-1)

(A-2)

(B-2)

(C-2)

(D-2)

(E-2)

(A-3)

(B-3)

(C-3)

(D-3)

(E-3)

Figure 9.13 Failure modes of Q690D specimens: Series A for HM-5.3-19-0.2, Series B for HM-5.8-19-0.2, Series C for HM-5.3-22-0.2, Series D for HM-5.3-19-0.35, and Series E for HM-5.3-19-0.35 , and (A-1): flange fracture and local buckling, (B-1), (C-1), (D-1), and (E-1)—flange local buckling; (A-2), (B-2), (C-2), (D-2), and (E-2)—weld joint fracture; (A-3), (B-3), (C-3), (D-3), and (E-3)—web local buckling.

upper bound of component plate slenderness but also a lower bound. In this chapter the lower bound is suggested to be taken as the plate slenderness limitation value of Class 1 cross section according to EC3 (Fig. 9.14).

9.3.2 Hysteretic response The end moment versus rotation hysteretic curves of two H-section and two boxsection specimens were obtained and summarized in Figs. 9.15 and 9.16. The second order effect of axial force on the moment at the specimen end was considered in addition to the lateral force caused moment. The end moment was calculated from the following equation: M 5 V ðL 1 Lc Þ 1 PΔ

(9.2)

Hysteretic behavior of high-strength steel columns

375

1.5

1.0

1.0

0.5

0.5

0.0

0.0

D

–0.5

D

1.5

–0.5

(A)

(B)

–1.0

–1.0

–1.5 –0.04 –0.03 –0.02 –0.01 0.00 0.01 0.02 0.03 0.04 Storey drift ratio T (rad)

–1.5 –0.04 –0.03 –0.02 –0.01 0.00 0.01 0.02 0.03 0.04 Storey drift ratio T (rad)

1.5

1.5

1.0

1.0

0.5

0.5

0.0

0.0

–0.5

D

D

Figure 9.14 Definition of location of buckling and fracture.

(C)

–0.5

(D)

–1.0

–1.0

–1.5 –0.04 –0.03 –0.02 –0.01 0.00 0.01 0.02 0.03 0.04 Storey drift ratio T (rad)

–1.5 –0.04 –0.03 –0.02 –0.01 0.00 0.01 0.02 0.03 0.04 Storey drift ratio T (rad)

Figure 9.15 Hysteretic behavior of H-section Q460 steel specimens: (A) H-4-16-1, (B) H-416-2, (C) H-8-28-1, and (D) H-8-28-2.

Behavior and Design of High-Strength Constructional Steel

1.5

1.5

1.0

1.0

0.5

0.5

0.0

0.0

D

D

376

–0.5

–0.5

(B)

–1.0

–1.0

–1.5 –0.04 –0.03 –0.02 –0.01 0.00 0.01 0.02 0.03 0.04 Story drift ratio T (rad)

–1.5 –0.04 –0.03 –0.02 –0.01 0.00 0.01 0.02 0.03 0.04 Story drift ratio T (rad)

1.5

1.5

1.0

1.0

0.5

0.5

0.0

0.0

D

D

(A)

–0.5

–0.5

(D)

(C) –1.0

–1.0

–1.5 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 Story drift ratio T (rad)

–1.5 –0.04 –0.03 –0.02 –0.01 0.00 0.01 0.02 0.03 0.04 Story drift ratio T (rad)

Figure 9.16 Hysteretic behavior of box-section Q460 steel specimens: (A) B-10-1, (B) B10-2, (C) B-21-1, and (D) B-21-2.

where V is the lateral force, P is the axial force, P 5 0.3Py, Lc is the length from the top of the column to the rotation center of the pin, and Lc 5 305 mm for all these Q460C specimens. The story drift Δ is obtained by averaging the recorded value of VLDTs D1 and D2. The moment is normalized by the experimental yield moment resistance calculated by using the actually measured values of yield strength. It can be seen from the overall hysteretic behavior of the tested specimens that plump hysteretic loop is developed in all the moment
rotation curves at large story drift ratio, as shown in Figs. 9.15 and 9.16. This indicates that the tested HSS H-section and box-section specimens have a good capacity of energy dissipation even under severe earthquake. Experimental plastic moment resistance Mp (and Mc) without (and with) considering the reduction effect of the applied axial force was plotted together with moment
rotation hysteretic curves to compare and evaluate the moment capacity deterioration behavior under cyclic load. The hysteretic curves for very compact sections H-3.8-15.6-0.3 and B-10.7-10.7-0.3 (Figs. 9.15A, B and 9.16A, B) show that the maximum moments are achieved at the applied maximum story drift ratio at the amplitude of 1/30. This reveals that, since Mp is not achieved for the specimens, those specimens may have ability to reach a higher moment resistance. To this end, further story drift ratios up to 1/20 were applied to the specimen B-10.7-10.7-0.3.

Hysteretic behavior of high-strength steel columns

377

As expressed in Eq. (9.1), the specimen end moment that takes P-Δ effect into account is an inherent indicator to represent the hysteretic behavior of beam
columns. Although the lateral force resistance shows a slowly decreasing, attributed to P 2 Δ effect, the moment resistance achieves the maximum value of 170.8 kN m at θ 5 0.050, which is higher than Mp. Even under the cyclic loading level of θ 5 0.050, no decrease in the moment resistance is observed for the specimen B-10.7-10.7-0.3. In other words, a steady plastic hinge formed at the end of the specimen, as shown in Fig. 9.11B. This highlights the excellent ductility and energy dissipation capacity of HSS columns with very compact cross section under server earthquake. It can be observed from Figs. 9.15C, D and 9.16C, D that, unlike the specimens with very compact cross sections as H-3.8-15.6-0.3 and B-10.7-10.7-0.3, the specimen series H-7.7-27.7-0.3 and B-21.6-21.6-0.3 with relative higher component plate slenderness ratios show a decrease in moment resistance under the cyclic load when the story drift ratio is up to 1/30. The moment capacity of the specimen H-7.7-27.70.3 achieves the maximum value of 1284.3 kN m corresponding to the story drift ratio of 0.025 and decreases to 1167.7 kN m at θ 5 0.033. The component plate slenderness ratio of the specimen B-21.6-21.6-0.3 is slightly less than the limit of Class 1. These two specimens achieved maximum moment capacity of 574.7 and 564.0 kN m corresponding to the maximum story rotation during the first cycle of θ 5 0.033 and then show a progressive deterioration in moment capacity for the continued cycles. It can be observed from the tested specimens that the degradation of moment capacity is associated with local buckling phenomena. For Q690D-fabricated specimens the experimental hysteretic force
displacement curves are schematically depicted in Fig. 9.17. The curves are divided into prelocalbuckling curves marked with yielding points and postlocal-buckling curve marked with buckling initiation points. The yielding point is recorded once the strain measured by strain gauge arranged at the edge of flange plate reaches the calculated yielding strain. The buckling point is attained when visible buckling deformation of flange plate is observed. It can be seen that all buckling point corresponds to the point where the ultimate strength is attained, indicating that the ultimate strength is governed by local buckling rather than fracture. On the other hand, there exists considerable difference between the prelocal- and postlocal-buckling curves. The postlocal-buckling curves present a steadily strength-deteriorated region that does not appear in the prelocal-buckling curves. The unloading
reloading strength amplitude and stiffness of postlocal-buckling curve are smaller than those of prelocal-buckling curve. Besides, the cyclic deterioration behavior in postlocal-buckling curve is much serious than that of prelocal-buckling curve. This is because the growth of local buckling deformation may continuously increase even under identical displacement amplitude. It also needs to be noted that a sharp strength decrease is observed in specimen HM-5.3-19-0.2 and HM-5.8-19-0.2 as fracture occurred near the weld joint of flange and web. In addition, the M
θ (moment
interstory drift angle) curves, moment of which is normalized by elastic moment considering axial load ratio My and θ is normalized by its corresponding yielding value θy ; are shown in Fig. 9.18. It can be seen that all cyclic loops are plump without obvious pinching phenomenon prior to

378

Behavior and Design of High-Strength Constructional Steel

Figure 9.17 Experimental hysteretic curves of Q690 steel specimens: (A) specimen HM-5.319-0.2, (B) specimen HM-5.8-19-0.2, (C) specimen HM-5.3-22-0.2, (D) specimen HM-5.319-0.35, and (E) specimen HM-5.3-19-0.35 .

Hysteretic behavior of high-strength steel columns

379

Figure 9.18 Normalized moment
drift angle curves of Q690 steel specimens: (A) specimen HM-5.3-19-0.2, (B) specimen HM-5.8-19-0.2, (C) specimen HM-1-2-18-2, (D) specimen HM-1-1-18-35, and (E) specimen HM-1-1-20-35.

fracture, indicating that Q690D HSS compact H-section columns can provide good seismic performance and energy dissipation capacity. To further understand the influence of flange, web, slenderness, and axial load ratio on hysteretic curves, the four counterparts are compared in Fig. 9.19. The comparison between HM-5.3-19-0.2 and HM-5.8-19-0.2 illustrates that the strength difference between Class 2 flange and Class 1 flange is fairly small for Q690 HSS Hsection beam
column. However, experiment HM-5.3-19-0.2 was stopped at the first cycle of the largest amplitude due to an abrupt flange plate fracture. Thus experiment HM-5.8-19-0.2 presents more cycles than experiment HM-5.3-19-0.2. The counterparts HM-5.3-19-0.2 and HM-5.3-22-0.2 show that the strength and stiffness of specimen with Class 2 web are considerably greater than those with Class 1 web. Besides, the HM-5.3-22-0.2 also attains more cycles in the largest amplitude than HM-5.3-190.2. Therefore the increase in flange width and web height delays the appearance of

380

Behavior and Design of High-Strength Constructional Steel

Figure 9.19 Comparison between different of Q690 steel specimens: (A) HM-5.3-19-0.2 versus HM-5.8-19-0.2, (B) HM-5.3-19-0.2 versus HM-5.3-22-0.2, (C) HM-5.3-19-0.2 versus HM-5.3-19-0.35, and (D) HM-5.3-19-0.35 versus HM-5.3-19-0.35 .

fracture to some extent. The counterparts HM-5.3-19-0.2 and HM-5.3-19-0.35 indicate that axial load ratio greatly alters not only the strength but also the curve shape especially within the postlocal-buckling region. The experiment HM-5.3-19-0.2 is able to develop considerable deformation with smaller strength degradation but fewer

Hysteretic behavior of high-strength steel columns

381

Figure 9.20 Normalized skeleton curves of Q690 steel specimens.

cycles before fracture. However, the experiment HM-5.3-19-0.35 with the increased axial load ratio of 0.35 loses strength quickly but withstands more cycles after attaining the ultimate strength. Thus the increase of axial load ratio may delay the appearance of fracture. The counterparts HM-5.3-19-0.35 and HM-5.3-19-0.35 do not exhibit obvious difference except for the elastic stiffness and strength. The normalized M
θ skeleton curves for specimens are shown in Fig. 9.20. It can be seen that the five experiments exhibit similar strength and skeleton shape. One difference is that the deterioration rates of HM-5.3-22-0.2 and HM-5.3-19-0.2 are smaller than others. This is because HM-5.3-19-0.2 encountered total fracture in flange plate, while HM-5.3-22-0.2 possess a greater web plate slenderness, further illustrating that H section with Class 2 flange and Class 1 web may produce a better deterioration resistance.

9.4

Numerical simulation

9.4.1 Material model and mesh Since C3D20R (quadratic-reduced integration solid element, as shown in Fig. 9.21) is insensitive to hourglass and shear locking behavior [18], it is suitable to analyzing the behavior of seriously distorted element. In order to simulate the large local deformation, element C3D20R is selected herein to mesh the geometric models. The mesh grid size in flange plate is 10 mm 3 10.5 mm while that in web plate is 10 mm 3 11 mm.

382

Behavior and Design of High-Strength Constructional Steel

9.4.2 Geometric and boundary conditions The geometric characteristics of the FE models are in accordance with those in experiments. Thus the developed FE model consists of rigid top plate, end plate, and column specimen. A reference point, on which the cyclic lateral displacement is applied, was set 50 mm above the top plate, shown as RP-1 in Fig. 9.22A. All freedoms of top plate were coupled with the reference point; thus the reaction force

Figure 9.21 C3D20R element.

Figure 9.22 Geometric and finite element models of simulated steel beam
column: (A) geometric model and (B) finite element model.

Hysteretic behavior of high-strength steel columns

383

of this point is treated as the cyclic lateral strength, which can be plotted against cyclic lateral displacement to form the hysteretic curve. Typical geometric and numerical models were depicted in Fig. 9.22. Taking X-direction as the direction of cyclic loading, the transverse freedom of Y-direction is restrained to constrain the out-of-plane deformation. Besides, the rotation freedoms about X-direction and Z-direction of the top plate are simultaneously constrained to prevent its torsion and out-of-plane bending according to the experimental setup. All six freedoms of the bottom plate are constrained to simulate the fix end condition.

9.4.3 Initial imperfection Welded steel structures evidently possess residual stress that may lead to an earlier development plastic range compared to ideal structures without residual stress. In this study, residual stress and initial geometric deformation are simultaneously considered. Besides, the initial geometric deformation includes not only local deformation but also overall deformation. This imperfection is arranged using the buckle analysis modulus of linear analysis in ABAQUS. Specifically, a buckle analysis was conducted on the FE model under vertical load. Subsequently, the first overall buckling mode and the first local buckling model were extracted to obtain the corresponding node displacements. These node displacements were then multiplied by two different amplified factors and applied on the FE model to form the overall initial geometric imperfection. The local buckling modes and FE model with geometric imperfection are, respectively, shown in Fig. 9.23. As specified in Chinese steel structure manufacture code [19], the maximum deformation of the first-order overall buckling mode is valued as 1/1000. The maximum local deformation of web plate is, respectively, taken as 3.0 and 2.0 mm when plate thickness is under and above 14 mm. As for flange plate, the maximum deformation is taken as the minimum value of b/100 and 3.0 mm. The residual stress is applied on the element integration point of the FE model using initial stress modulus provided by ABAQUS. The magnitude and distribution of residual stress refer to the residual stress model proposed by Ban et al. [20], which is suitable for HSSs.

Figure 9.23 Initial geometric imperfection and residual stress: (A) FEM model with initial geometric imperfection and (B) FEM model with residual stress.

384

Behavior and Design of High-Strength Constructional Steel

Figure 9.24 Q690D high-strength steel beam
column specimens for cyclic tests.

9.4.4 Verification of finite element model To verify the rationality of the aforementioned FE model, it was utilized to simulate the cyclic loading test on Q690 HSS beam
column specimens, as shown in Fig. 9.24. The simulated hysteretic curves are compared with those of experiments [21] in Fig. 9.25. Simultaneously, the failure modes predicted by FE model are also compared with those of experiments, as depicted in Figs. 9.26 and 9.27. The experimental bearing capacities are compared to those obtained with FE simulations as shown in Table 9.7. It can be seen that both failure modes and hysteretic curves derived from FEM simulation match well with the corresponding experimental versions. This reveals the truth that the proposed FE model is able to provide satisfactory description for these HSS beam
columns.

9.5

Parametric analyses and discussions

9.5.1 Parameter design The factors probably influencing the cyclic behavior of HSS beam
columns include flange width-to-thickness ratio, web height-to-thickness ratio, and axial load ratio. The loading history follows the regulation specified by Chinese seismic experiments code JGJ101-96 [22]. The story drift angles corresponding to each

Hysteretic behavior of high-strength steel columns

385

Figure 9.25 Comparison between experimental hysteretic curves and FEM hysteretic curves: (A) specimen HM-5.3-19-0.2, (B) specimen HM-5.8-19-0.2, (C) specimen HM-5.3-22-0.2, (D) specimen HM-5.3-19-0.35, and (E) specimen HM-5.3-19-0.35 .

loading amplitude level are tabulated in Table 9.8. The beam
column specimens are treated as failure once any status listed in Table 9.9 is satisfied. The flange and web slenderness of the Q690 steel beam
column specimens for parametric studies are taken as uniformly spaced sequences of 4, 6, 8, 10, and 12 and 20, 25, 30, 35, and 40, respectively. The axial load ratio sequence is taken as 0.2, 0.4, and 0.6. Each specimen is identified as H-x
y
z format, in which H denotes that the section is H shaped while x, y, and z, respectively, represent the flange width
thickness ratio, web height
thickness ratio as well as axial load ratio. For instance, H-4-20-2 represents a welded H-shaped beam
column, flange widthto-thickness ratio, height-to-thickness ratio, and axial load ratio of which are,

386

Behavior and Design of High-Strength Constructional Steel

Figure 9.26 Failure process of FEM simulation: (A) crack initiation, (B) development of crack, (C) local buckling with crack, and (D) development of local buckling.

Figure 9.27 Comparison between failure mode of FEM and experiment: (A) failure mode of FEM simulation and (B) failure mode of experiment.

Table 9.7 Comparison between experimental results and FEM results. Specimen number

Experiments (kN)

FEM (kN)

Error (%)

HM-5.3-19-0.2 HM-5.8-19-0.2 HM-5.3-22-0.2 HM-5.3-19-0.35 HM-5.3-19-0.35

344.4 348.5 408.5 315.4 287.7

353.9 366.3 389.8 278.4 257.1

2.69 4.88 4.78 13.28 11.90

Hysteretic behavior of high-strength steel columns

387

Table 9.8 Displacement amplitude of loading history. Loading level

1

2

3

4

5

6

7

8

9

Story drift angle Cycles

61/150

61/100

61/90

61/80

61/70

61/60

61/50

61/40

61/30

3

3

3

3

3

3

3

3

3

Table 9.9 Failure definition of beam
column specimens. Number

Failure mode

Criterion

1 2

Ductile fracture Strength deteriorate to certain critical line due to local buckling

Element deletion The critical line is defined as 85% of the maximum strength

respectively, 4.0, 20, and 0.2. The overall slenderness of all the specimens is less or closed to the critical limitation of 100 specified by Chinese seismic building design code GB50011-2010 [23]. The dimensions and section categories according to EC3 and GB50011 are listed in Table 9.10. It should be noted that the category 1, 2, 3, and 4 in this table, respectively, refers to the seismic level of one, two, three, and four according to GB50011-2010, the section category of which is related to seismic level and yield strength of materials. Furthermore, the classification of 1, 2, 3, and 4, respectively, denotes the Class 1, Class 2, Class 3, and Class 4 cross section specified in EC3.

9.5.2 Influence of width-to-thickness ratio of flange Representative cyclic curves simulated are schematically plotted in Fig. 9.28. The corresponding flange width-to-thickness ratios are 4, 8, and 12, while web heightto-thickness ratios are 20, 30, and 40. It can be seen that the increase in flange width-to-thickness ratio brought severer cyclic deterioration of hysteretic curves. For instance, the hysteretic loop of specimen H-4-20-2 stayed plump until the appearance of fracture point. However, the hysteretic loop of specimen H-12-20-2 rapidly flattened even cyclic- displacement loading amplitude was still limited. The skeleton curves of various specimens varying with different flange width-to-thickness ratio were schematically depicted in Fig. 9.29. It is clear that the increase in flange width-to-thickness ratio synchronizes the increase in ultimate moment and decrease in ductility. Furthermore, the increasing magnitude from H-10-20-2 to H-12-20-2 is considerably less than that from H4-20-2 to H-6-20-2. This is because that local buckling limits the elaboration of sectional bearing capacity as flange slenderness increases.

Table 9.10 Q690 steel beam
column specimens for parametric study. Identifier

Sectional dimensions (mm)

Plate slenderness

Overall slenderness

Axial load ratio

Sectional category GB

H-4-40-2 H-6-40-2 H-8-40-2 H-10-40-2 H-12-40-2 H-4-25-2 H-6-25-2 H-8-25-2 H-10-25-2 H-12-25-2 H-4-30-2 H-6-30-2 H-8-30-2 H-10-30-2 H-12-30-2 H-4-35-2 H-6-35-2 H-8-35-2 H-10-35-2 H-12-35-2 H-4-40-2 H-6-40-2 H-8-40-2

EC

h

B

tw

tf

Flange

Web

X

y

n

Web

Flange

Web

Flange

440 440 440 440 440 540 540 540 540 540 640 640 640 640 640 740 740 740 740 740 840 840 840

180 260 340 420 500 180 260 340 420 500 180 260 340 420 500 180 260 340 420 500 180 260 340

20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20

20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20

4 6 8 10 12 4 6 8 10 12 4 6 8 10 12 4 6 8 10 12 4 6 8

20 20 20 20 20 25 25 25 25 25 30 30 30 30 30 35 35 35 35 35 40 40 40

18 17 17 16 16 15 14 14 13 13 13 12 12 11 11 11 11 10 10 10 10 10 9

83 53 38 30 25 88 56 40 31 25 93 58 42 32 26 98 61 43 33 27 102 63 45

0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

1 1 1 1 1 1 1 1 1 1 4 4 4 4 4 5 5 5 5 5 5 5 5

1 2 5 5 5 1 2 5 5 5 1 2 5 5 5 1 2 5 5 5 1 2 5

1 1 1 1 1 1 1 1 1 2 1 2 2 2 3 2 3 3 3 3 3 3 3

1 3 3 4 4 1 3 3 4 4 1 3 3 4 4 1 3 3 4 4 1 3 3

H-10-40-2 H-12-40-2 H-4-40-4 H-6-40-4 H-8-40-4 H-10-40-4 H-12-40-4 H-4-25-4 H-6-25-4 H-8-25-4 H-10-25-4 H-12-25-4 H-4-30-4 H-6-30-4 H-8-30-4 H-10-30-4 H-12-30-4 H-4-35-4 H-6-35-4 H-8-35-4 H-10-35-4 H-12-35-4 H-4-40-4 H-6-40-4 H-8-40-4 H-10-40-4 H-12-40-4 H-4-40-6 H-6-40-6 H-8-40-6 H-10-40-6 H-12-40-6

840 840 440 440 440 440 440 540 540 540 540 540 640 640 640 640 640 740 740 740 740 740 840 840 840 840 840 440 440 440 440 440

420 500 180 260 340 420 500 180 260 340 420 500 180 260 340 420 500 180 260 340 420 500 180 260 340 420 500 180 260 340 420 500

20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20

20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20

10 12 4 6 8 10 12 4 6 8 10 12 4 6 8 10 12 4 6 8 10 12 4 6 8 10 12 4 6 8 10 12

40 40 20 20 20 20 20 25 25 25 25 25 30 30 30 30 30 35 35 35 35 35 40 40 40 40 40 20 20 20 20 20

9 9 18 17 17 16 16 15 14 14 13 13 13 12 12 11 11 11 11 10 10 10 10 10 9 9 9 18 17 17 16 16

35 28 83 53 38 30 25 88 56 40 31 25 93 58 42 32 26 98 61 43 33 27 102 63 45 35 28 83 53 38 30 25

0.2 0.2 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.6 0.6 0.6 0.6 0.6

5 5 1 1 1 1 1 1 1 1 1 1 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 1 1 1 1 1

5 5 1 2 5 5 5 1 2 5 5 5 1 2 5 5 5 1 2 5 5 5 1 2 5 5 5 1 2 5 5 5

3 3 1 1 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2

4 4 1 3 3 4 4 1 3 3 4 4 1 3 3 4 4 1 3 3 4 4 1 3 3 4 4 1 3 3 4 4 (Continued)

Table 9.10 (Continued) Identifier

Sectional dimensions (mm)

Plate slenderness

Overall slenderness

Axial load ratio

Sectional category GB

H-4-25-6 H-6-25-6 H-8-25-6 H-10-25-6 H-12-25-6 H-4-30-6 H-6-30-6 H-8-30-6 H-10-30-6 H-12-30-6 H-4-35-6 H-6-35-6 H-8-35-6 H-10-35-6 H-12-35-6 H-4-40-6 H-6-40-6 H-8-40-6 H-10-40-6 H-12-40-6

EC

h

B

tw

tf

Flange

Web

X

y

n

Web

Flange

Web

Flange

540 540 540 540 540 640 640 640 640 640 740 740 740 740 740 840 840 840 840 840

180 260 340 420 500 180 260 340 420 500 180 260 340 420 500 180 260 340 420 500

20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20

20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20

4 6 8 10 12 4 6 8 10 12 4 6 8 10 12 4 6 8 10 12

25 25 25 25 25 30 30 30 30 30 35 35 35 35 35 40 40 40 40 40

15 14 14 13 13 13 12 12 11 11 11 11 10 10 10 10 10 9 9 9

88 56 40 31 25 93 58 42 32 26 98 61 43 33 27 102 63 45 35 28

0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6

1 1 1 1 1 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5

1 2 5 5 5 1 2 5 5 5 1 2 5 5 5 1 2 5 5 5

3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4

1 3 3 4 4 1 3 3 4 4 1 3 3 4 4 1 3 3 4 4

Note: The marker  denotes that the flange or web slenderness of the specimen larger than the limitation corresponding to category 4 specified in GB50010.

Hysteretic behavior of high-strength steel columns

391

Figure 9.28 Representative cyclic curves of specimens with different flange and web slenderness: (A-1) specimen H-4-40-2, (A-2) specimen H-8-40-2, and (A-3) specimen H-840-2; (B-1) specimen H-4-30-2, (B-2) specimen H-8-30-2, and (B-3) specimen H-12-30-2; and (C-1) specimen H-4-20-2, (C-2) specimen H-8-20-2, and (C-3) specimen H-12-20-2.

In addition, the failure mode of specimen is strongly influenced by flange widthto-thickness ratio. In order to thoroughly study such phenomenon, the failure modes of four representative specimens that possess identical web height-to-thickness ratio and axial load ratio, but different flange width-to-thickness ratios, are schematically depicted in Fig. 9.30 along with their corresponding hysteretic curves. It can be seen that the failure of specimen H-4-25-2, which corresponds to category 1 section specified in GB50011, is governed by ductile fracture, as shown in Fig. 9.30A. Since full-flange plasticity is rapidly developed, the strain of steel fiber reaches its damage initiation strain. Subsequently, stress decreases continuously as strain increased, resulting in considerable cyclic strength and stiffness deterioration of steel member. As for specimen H-6-25-2, the hysteretic curve possesses more uniform cyclic deterioration than that of H-4-25-2. Nevertheless, the initial point of cyclic deterioration is closed to that of H-4-25-2. This is because the increase in flange slenderness leads to that local buckling becomes the dominant failure mode. However, since the flange width-to-thickness ratio is still limited, the cyclic deterioration rate and magnitude are relatively moderate. In addition, it should be noted that the

392

Behavior and Design of High-Strength Constructional Steel

Figure 9.29 skeleton curves of various specimens varying with different width-to-thickness ratios: (A) specimens with different flange width
thickness ratios (axial load ratio 0.2, web height
thickness ratio 20), (B) specimens with different flange width
thickness ratios (axial load ratio 0.4, web height
thickness ratio 20), and (C) specimens with different flange width
thickness ratios (axial load ratio 0.6, web height
thickness ratio 20).

fracture point evolves at the flange
web joint rather than extends along the flange width with certain angle such as that of H-4-25-2. With further increasing flange width-to-thickness ratio, specimen H-10-25-2 presents severer cyclic deterioration than the aforementioned ones. The deterioration rate and magnitude are greater than that of H-6-25-2 due to larger local buckling deformation. Furthermore, the failure mode is local buckling of flange plate, along with welding fracture at flange
web joint, which is similar with that of H-6-25-2. It should be noted that the joint fracture zone is larger than that of H-6-25-2. The reason is because the local deformation at the flange
web joint is greater than previous specimens due to considerable buckling deformation of both flange and web plates. As for specimen H-12-25-2, local buckling rapidly evolves prior to the attainment of plasticity at edge-fiber. Since the flange width-to-thickness ratio is greater than that of all aforementioned specimens, the local buckling deformation increases dramatically, as shown in Fig. 9.30D, leading to considerable cyclic deterioration once again. In summary, the increase in flange width-to-thickness ratio will result in different damage mechanisms and consequently present different cyclic deterioration characteristics. Ultimate story drift angle and sectional plasticity development coefficient are two key quantities to describe the deformability and bearing capacity. Ultimate story

Hysteretic behavior of high-strength steel columns

393

Figure 9.30 The influence of flange width
thickness ratio on failure mode: (A) failure mode and cyclic curve of Q690-4-25-2, (B) failure mode and cyclic curve of Q690-6-25-2, (C) failure mode and cyclic curve of Q690-10-25-2, and (D) failure mode and cyclic curve of Q690-12-25-2.

drift angle is defined as the minimum values corresponding to fracture and 85% ultimate strength in the deterioration range. The sectional plasticity development coefficient is defined as the ratio of ultimate moment and elastic moment considering axial load ratio. The correlation between these two indexes and flange width-tothickness ratio is depicted in Fig. 9.31. It can be seen that the sectional plasticity

394

Behavior and Design of High-Strength Constructional Steel

Figure 9.31 Influence of flange width-to-thickness ratio on interstory drift angle and plasticity development coefficient: (A) specimen with axial load ratio of 0.2, (B) specimen with axial load ratio of 0.4, and (C) specimen with axial load ratio of 0.6.

development coefficient experiences an increase
decrease process as flange widthto-thickness ratio increases. On the other hand, increase in flange width-to-thickness ratio results in decrease in ultimate story drift angle. However, this decrease will be weakened once web slenderness and axial load ratios are relatively large.

9.5.3 Influence of web height
thickness ratio Fig. 9.32 presents the skeleton curves comparison of various specimens varying with different height-to-thickness ratios. According to those graphics in Fig. 9.28, the increase in web height-to-thickness ratio results in severer flattening characteristics of hysteretic curve. Furthermore, it can be seen from Fig. 9.33 that the increase

Hysteretic behavior of high-strength steel columns

395

Figure 9.32 skeleton curves of various specimens varying with different width-to-thickness ratio: (A) specimens with different web height
thickness ratios (axial load ratio 0.2, flange width
thickness ratio4), (B) specimens with different web height
thickness ratios (axial load ratio 0.4, flange width
thickness ratio4), (C) specimens with different web height
thickness ratios (axial load ratio 0.6, flange width
thickness ratio 4), (D) specimens with different web height
thickness ratios (axial load ratio 0.2, flange width
thickness ratio 12), (E) specimens with different web height
thickness ratios (axial load ratio 0.4, flange width
thickness ratio 12), and (F) specimens with different web height
thickness ratios (axial load ratio 0.6, flange width
thickness ratio 12).

in web height
thickness ratio synchronizes increase in bearing capacity and decrease in ductility. This reveals the truth that slender web will result in more significant cyclic deterioration. Similar with the comparison conducted in Section 9.5.2, the ultimate interstory drift and sectional plasticity development coefficient of the simulated specimens is

396

Behavior and Design of High-Strength Constructional Steel

Figure 9.33 Influence of web height-to-thickness ratio on interstory drift angle and plasticity development coefficient: (A) specimen with axial load ratio of 0.2, (B) specimen with axial load ratio of 0.4, and (C) specimen with axial load ratio of 0.6.

plotted against web height-to-thickness ratio. It can be seen that the increase in web height
thickness ratio decreases the sectional plasticity development coefficient and ultimate rotational deformability, further indicating the deterioration of deformability caused by local buckling of web.

9.5.4 Influence of axial force ratio The hysteretic curves with different axial load ratio are schematically depicted in Fig. 9.34. It can be concluded that the increase in axial load ratio will dramatically deteriorate the fullness of hysteretic curves, further reducing the bearing capacity, ductility and energy dissipation ability.

Hysteretic behavior of high-strength steel columns

397

Figure 9.34 Cyclic curves of various specimens varying with different axial load ratio: (A-1) H-12-40-6, (A-2) H-12-40-4, and (A-3) H-12-40-2; (B-1) H-8-40-6, (B-2) H-8-40-4, and (B-3) H-8-40-2; and (C-1) H-4-40-6, (C-2) H-4-40-4, and (C-3) H-4-40-2.

According to the skeleton curves in Fig. 9.35, it can be seen that the increase in axial load ratio may delay the appearance of low-fatigue fracture to some extent regarding compact-sectional specimens. Specifically, the increase of axial load ratio from 0.2 to 0.4 transfers the failure mode from low-fatigue fracture to local buckling, as depicted in Fig. 9.36 that shows the failure mode of specimens Q690-4-250.2 and Q690-4-25-0.4. On the contrary, the increase in axial load ratio rapidly deteriorates the lateral strength of specimen prior to welding joint fracture regarding specimens with much slender section, as shown in Fig. 9.37.

9.6

Hysteretic model

9.6.1 Hysteretic model incorporated with damage behavior Based on the numerical analyses results of Section 9.5, the cyclic deterioration of HSS beam
column members can be classified into two categories: local-buckling-driven deterioration (LBD) and low-cycle fatigue
driven deterioration (LFD). Considering that a compact beam
column member is subjected to extremely small lateral and axial loads, the potential for local buckling to occur is quite low, and the failure mode may

398

Behavior and Design of High-Strength Constructional Steel

Figure 9.35 Skeleton curves of various specimens varying with different axial load ratio: (A) specimens with flange width
thickness ratio 4 and web height
thickness ratio 40, (B) specimens with flange width
thickness ratio 12 and web height
thickness ratio 40, (C) specimens with flange width
thickness ratio 4 and web height
thickness ratio 20, and (D) specimens with flange width
thickness ratio 12 and web height
thickness ratio 20.

be practically governed by a low-cycle fatigue fracture. However, an increase in the plate slenderness ratio, axial load ratio, or cyclic loading amplitude may considerably enhance the proportion of LBD but may lower the proportion of LFD. The deterioration amount of the DLCs controlled by LFD was fairly small within a cycle number of 80%
95% but became extremely large in several of the final cycles. The total deterioration of the DLCs controlled by LBD can be divided into three stages: the initial stage of local buckling, stabilization stage, and further development stage of local buckling. To properly describe the characteristics of DLCs with both LFD and LBD, the following equations were established as: ð 1Þ

ð2Þ FD e2bF ð12DÞ 2 1 5 QF 1 ð1 2 QF Þe2bF D 2bðF1Þ Fm e 21

(9.3)

ð 1Þ

ð2Þ KD e2bk ð12DÞ 2 1 5 Qk 1 ð1 2 Qk Þe2bk D 2bðk1Þ Km 21 e

(9.4)

Hysteretic behavior of high-strength steel columns

Figure 9.36 Influence of axial load ratio on failure mode of compact specimens: (A) specimen Q690-4-25-2, (B) specimen Q690-4-25-4, and (C) specimen Q690-4-25-6.

Figure 9.37 Influence of axial load ratio on failure mode of slender specimens: (A) specimen Q690-12-25-2, (B) specimen Q690-12-25-4, and (C) specimen Q690-12-25-6.

399

400

Behavior and Design of High-Strength Constructional Steel

where the first item on the right side of the equations defines the LFD component, and the second denotes the LBD component. FD and KD denote the deteriorated lateral strength and stiffness under a current loading amplitude, respectively, and Fm and Km denote the lateral strength and stiffness of beam
columns under monotonic loading, respectively. D denotes a damage index and will be discussed subsequently. The coefficients QF and Qk were used to adjust the proportion of the LFD and LBD, respectively. The parameters bðF1Þ and bðF2Þ denote the strength deterioration rates corresponding to the LFD and the LBD, respectively. Similarly, bðk1Þ and bðk2Þ denote the unloading stiffness deterioration rates corresponding to the LFD and LBD components, respectively. With an increase in plate slenderness, axial load ratio, and cyclic loading amplitude, specimens were more likely to encounter local buckling. Thus the parameter values of bðF1Þ and QF should be decreased to allow the DLC to present the three-stage characteristics of LBD. Regarding the DLC of unloading stiffness, a similar approach can be taken to adjust the contribution of LBD and LFD to the DLCs. According to studies on the effect of the key factors on the cyclic deterioration response, bðk1Þ , bðF1Þ , QF , and Qk must correlate with these influencing factors that include the cyclic loading amplitude and flange, web slenderness, and axial load ratios. The correlations are depicted as shown in the following equations:      bðk1Þ 5 κk 1 2 R b U 1 2 R h U ð1 2 nÞU 1 2 R δ

(9.5)

 α  α  α Qk 5 12R b bU 12R h hU ð12nÞαnU 12R δ δ

(9.6)

     bðF1Þ 5 κF 1 2 R b U 1 2 R h U ð1 2 nÞU 1 2 R δ

(9.7)

 β  β  β QF 5 12R b bU 12R h hU ð12nÞβnU 12R δ δ

(9.8)

where αb ,αh ,αn , and αδ are the constant parameters that describe the contribution of flange slenderness (b/t), web slenderness (h0/tw), axial load ratio (N/A/fy), and cyclic loading amplitude (Δθ) to the LBD of strength, respectively. It can be assumed that the influence of web plate slenderness on cyclic deterioration is greater than those of the other three factors if αh is much greater than αb , αn , and αδ . β b , β h , β n , and β δ are the parameters that control the contribution of different influencing factors on unloading stiffness deterioration; κk and κF are the constant parameters that define the maximum value of bðk1Þ and bðF1Þ ; Rb and Rh denote the flange slenderness and web slenderness ratios, respectively; and Rδ denotes the cyclic loading amplitude ratio, which is the ratio of cyclic loading displacement amplitude and the displacement leading to the yield of a member (δy). The normalized Rb , Rh , and Rδ are expressed in the following equation: Rb 5

½Rb max 2 Rb ½Rb max 2 ½Rb min

(9.9)

Rh 5

½Rh max 2 Rh ½Rh max 2 ½Rh min

(9.10)

Hysteretic behavior of high-strength steel columns

Rδ 5

½Rδ max 2 Rδ ½Rδ max 2 ½Rδ min

401

(9.11)

where R b , R h , and R δ are the normalized flange slenderness, web slenderness, and cyclic loading amplitude ratio, respectively. ½Rb min 5 4 and ½Rb min 5 12 denote the upper and lower bounds of the flange slenderness, respectively; ½Rh max 5 20 and ½Rh min 5 40 denote the upper and lower bounds of the web slenderness, respectively; and ½Rδ max and ½Rδ min denote the upper and lower bounds of the cyclic loading amplitude, respectively. These can be calibrated from the FE analysis. The damage index D is determined from the Manson
Coffin equation and Miner’s rule as expressed as in the following equation: X nθ D5 (9.12) Nθf where nθ denotes the number of cycles with a cycle amplitude of θ, and Nθf denotes the cycle number at failure corresponding to the applied cyclic amplitude of θ. The calculation of Nθf can be determined [12] in the following equation:  b  d (9.13) θ 5 a 2Nf 1 c 2Nf where θ denotes the story drift amplitude applied to beam
column members; 2Nf denotes the half cycle number at failure; a, b, and d are constant parameters; and c denotes a parameter correlated with the flange slenderness, web slenderness, and axial load ratios. The determination of c requires a failure criterion that is normally considered to be the status when the lateral strength drops to 85% of the ultimate member strength. However, because full-range deterioration was expected, the failure criterion was set as the status when the lateral strength dropped to 15% of the ultimate strength. Consequently, the correlations between parameter c and the axial load, flange, and web slenderness ratios required rebuilding. Similar to the approaches adopted in previous studies, these correlations were nonlinearly fitted based on previous test results. The parameter values of a, b, and d were then obtained and are shown in Table 9.11, and the calculated values of c are listed in Table 9.12. The equation derived from the curve-fitting method is shown in Eq. (9.14), and comparisons of the calculated failure parameter c from Eq. (9.14) and from FE analysis results are illustrated in Figs. 9.38 and 9.39, respectively. "  # h (9.14) c 5 3:41exp 20:04 Uexpð 22:09nÞ tw (9.14) # "  2    3 b b b 1 0:12 2 1:22 1 10:14 U 20:004 tf tf tf

402

Behavior and Design of High-Strength Constructional Steel

Table 9.11 Model parameters. R

a

B

d

0.924

4.5

2 1.5

2 0.4

Table 9.12 Calculation results for parameter c with corresponding influencing factors. h/tw 5 20n 5 0.2

b/tf 5 6h/tw 5 20

b/tf 5 6n 5 0.2

b/tf

cf

n

cn

h/tw

cw

4 6 8 10 12

6.996 6.332 6.263 6.105 6.140

0 0.2 0.4 0.6

5.619 3.934 2.693 1.285

20 25 30 35 40

6.332 5.479 3.934 3.446 2.973

The ductility of hysteretic curves may also encounter deterioration as the cyclic plastic deformation accumulates. In particular, the deformation corresponding to the peak point of the reloading curve will decrease with the accumulation of cyclic plasticity. In this study, it was assumed that the deterioration law of ductility in terms of deformation was in accordance with that of stiffness, depicted in the following equation: δiu Ki 5 D δmu Km

(9.15)

where δiu denotes the deformation corresponding to the peak strength point of the reloading curve in the ith half cycle, δmu denotes the deformation corresponding to the peak strength point of the monotonic curve, KDi denotes the deteriorated unloading
reloading stiffness of the ith half cycle, and Km denotes the monotonic unloading
reloading stiffness. Normally, a hysteretic curve consists of a constant skeleton curve and a hysteretic rule. To incorporate cyclic deterioration behavior into a hysteretic curve, the skeleton curve was refreshed with the accumulation of cyclic damage. The updated strength with deterioration in the reloading curve of the ith half cycle is expressed in the following equations:    2 A δ =δiu 1 ðB 2 1Þ δ =δiu F 5    2 Fui 1 1 ðA 2 2Þ δ =δiu 1 B δ =δiu     δ 5 δ 2 δi21 p 

(9.16)

(9.17)

Hysteretic behavior of high-strength steel columns

403

Figure 9.38 Relationship between c and influencing factors: (A) relationship between c and b/tf , (B) relationship between c and h/tw , (C) relationship between c and n.

F  5 jF j A5

Kui21 Kpi

(9.18)

(9.19)

where Fui denotes the peak strength point of the ith half cycle, F  denotes the deteriorated strength in the reloading curve of the ith half cycle, δiu denotes the deformation corresponding to the peak strength point of the ith half cycle, δi21 denotes the p residual deformation from the i 2 1th half cycle, Kui21 denotes the unloading stiffness of the i 2 1th half cycle with consideration of cyclic deterioration, Kpi 5 Fui =δiu denotes the secant stiffness corresponding to the peak point of the ith half cycle, and B denotes a shape parameter that controls the softening rate. An increase in B will result in more severe softening behavior. The procedure to predict the hysteretic curves using the proposed hysteretic model was as follows: 1. The monotonic lateral bearing capacity (Fm ) and stiffness (δmu ) were calculated based on the axial load ratio and the geometry of the beam
column members.

404

Behavior and Design of High-Strength Constructional Steel

Figure 9.39 Comparison between models predicted curves (solid lines) and finite element analysis results: (A) b/tf varies from 4 to 12, (B) h/tw varies from 20 to 40, (C) n varies from 0 to 0.6. 2. The damage index D was calculated from the deformation accumulated up to the i 2 1th half cycle by substituting the maximum displacement from the i 2 1th half cycle (δi21 m ) and the half cycle number i into Eqs. (9.12) and (9.13). Then, the damaged lateral strength i21 (FDi ) and stiffness (KDi ) at the current cycle were calculated by substituting δi21 m =δy , Ku , n, Rb , and Rh into Eqs. (9.3)
(9.11).   i21    3. Taking δ 5 δi21  skeleton curve  p 1 δm , the strength F corresponding to the δ on the   i21  i21 was calculated by substituting the values of Ku , δ , Fm , δmu , and δ 5 δi21 p 1 δm  into Eq. (9.14). Then, the reduction coefficients of strength and deformation were given by ηF 5 FDi =F  and ηδ 5 δ =δiu . 4. The refreshed skeleton curve corresponding to the ith half cycle was obtained by multiplying the reduction coefficients ηF and ηδ with the peak strength and deformation from the initial monotonic skeleton curve.

Several constant parameters in the proposed hysteretic model shown in Table 9.13 required calibration. By using the FE analysis approach, the parameters in the proposed hysteretic model were obtained by adjusting the values of each parameter until the predicted results from the proposed theoretical hysteretic model of beam
column member with different flange, web slenderness, and axial load ratios matched well with the simulated results by FE analysis. The calibration

Hysteretic behavior of high-strength steel columns

405

Table 9.13 Calibrated parameters of the proposed hysteretic model. Parameter

αb

αh

αn

αD

κK

B

Value Parameter Value

0.1 βb 0.1

0.4 βh 0.4

0.5 βn 0.8

1 βD 1

30 κF 30

1.2

results are shown in Table 9.13. To verify the proposed hysteretic model, its predicted hysteresis curves were further compared against the test results of different Q690 beam
column members with cross sectional dimensions (Hw 3 B 3 tw 3 tf) of 230 3 220 3 10 3 20 mm4 and 240 3 220 3 10 3 20 mm4 under axial load ratios of 0.2 and 0.35, respectively. The test results were not used to calibrate the proposed theoretical hysteretic model. Fig. 9.40 depicts the hysteresis curves, including the moment
story drift ratio (M
θ) curves and strength
lateral displacement (F
δ) curves predicted from the proposed hysteresis model and obtained from the tests. It is clear that the hysteresis curves generated from the proposed theoretical hysteresis model matched well with those obtained directly from the test results, indicating that the proposed model together with the calibrated parameters was capable of describing the hysteresis behavior of Q690 HSS beam
columns with reasonable accuracy. To further indicate the capability of the proposed hysteretic model, a series of comparable theoretical analyses was conducted on Q690 beam
column members to better comprehend the influence of the axial load, flange plate slenderness, and web plate slenderness ratios and the cyclic loading history on the predicted hysteretic response. Fig. 9.41A shows that an increase in flange plate slenderness resulted in a considerable increase in strength but may lead to a deterioration in ductility. As shown in Fig. 9.41B, the increase in web plate slenderness considerably increased the load bearing capacity but decreased the ductility of the beam
columns. Moreover, as shown in Fig. 9.41C, an increase in axial load ratio not only led to a decrease in the strength and stiffness of the beam
columns but also changed the characteristics of the hysteretic loop. In addition, the comparison shown in Fig. 9.41D indicates that the specimen with five-cycle loading possessed more evident deterioration than that with one-cycle loading. In general, the developed hysteretic model can reasonably describe the effect of influencing factors on cyclic deterioration behavior.

9.6.2 Simplified hysteretic model For the Q460C series of test specimens, a simplified hysteretic model is proposed [24]. End moment
curvature curves are developed to describe the hysteretic performance of the HSS beam
column members. The curvature of specimen foot is calculated by the following equation: Φ5

εl 2 εr H

(9.20)

406

Behavior and Design of High-Strength Constructional Steel

Figure 9.40 Comparison between hysteretic curves obtained from different tests and the proposed model: (A) M
θ curves of HM-5.3-19-0.2, (B) F
δ curves of HM-5.3-19-0.2, (C) M
θ curves of HM-5.3-19-0.35 , (D) F
δ curves of HM-5.3-19-0.35 , (E) M
θ curves of HM-5.8-19-0.2, (F) F
δ curves of HM-5.8-19-0.2, (G) M
θ curves of HM-5.3-19-0.35, and (H) F
δ curves of HM-5.3-19-0.35.

where εl is the average value of strain gauges S01, S02, and S03, and εr is the average value of strain gauges S04, S05, and S06, as shown in Fig. 9.7B. Typical moment
curvature hysteretic curves are shown in Fig. 9.42

Hysteretic behavior of high-strength steel columns

407

Figure 9.41 Influence of different factors on hysteretic curves: (A) influence of flange plate slenderness, (B) influence of web plate slenderness, (C) influence of axial load ratio, (D) influence of cycle loading times.

Based on the simplification and generalization of the experimental moment
curvature hysteretic loops, the hysteretic loops can be approximated by a series of linear segments, as shown in Fig. 9.43A. A trilinear hysteretic model with elastic loading and unloading, inelastic hardening, and transition segments was suggested for hysteretic loops with slight pinching, while a bilinear model without transition segment was recognized to be appropriate for representing plump hysteretic loops. After reach the elastic boundary line QiQ0 i (i 5 1, 2), a critical value of slope γ should be established to determine the necessary for the adoption of a transition segment. If the slope of transition segment is less than the critical value, the elastic loading segment should be extended to the modified elastic boundary line PiP0 i. Otherwise, the trilinear model should be used to reproduce the hysteretic loops. Fig.9.43B shows the proposed hysteretic model, where A (1, 1) and A0 (21, 21) are the reference points; P1P0 1 and P2P0 2 are the reference lines with slope of k1, which pass through points A (1, 1) and A0 (21, 21), respectively; and Q1Q0 1 and Q2Q0 2 are the reference lines with slope of k2, which pass through points (0, 0.7) and (0, 20.7), respectively. The points (0, 0.7) and (0, 20.7) were determined according to the axial force ratio of 0.3. Linear unloading with stiffness of k3 is supposed at any point of the hysteretic curve. The segments ①!②!③!④ represent the four major steps of the hysteretic model. The segment ① represents the elastic loading up to A (1, 1). Continuing to load from point A, it will enter into the

408

Behavior and Design of High-Strength Constructional Steel

Figure 9.42 Hysteretic moment
curvature curves: (A) H-3.8-15.6-0.3, (B) H-7.7-27.7-0.3, (C)B-10.7-10.7-0.3, (D) B-21.6-21.6-0.3.

inelastic hardening stage of the segment ② with the hardening modulus of β. If it unloads from an arbitrary point B and follows by a reverse loading, the linear unloading and reverse loading paths will intersect with the reference line Q2Q0 2 at the point C. There are two cases for the forthcoming transition segment such as the transition segment CA0 connected with the segment ③ and the transition segment DG connected with the segment ④. If the slope of CA0 is higher than γ, CA0 will be adopted as the transition segment up to a new hardening segment such as A0 B0 . In contrast, the slope of DA0 may be smaller than γ. In this case a hardening reference line EF with the same hardening modulus of β is introduced. The transition segment starts from point D with the slope of γ and ends at the intersection point with EF will be adopted instead of DA0 and so forth. Parameters of the proposed trilinear hysteretic model, k 1 , k 2 , k 3, β, and γ, are obtained from the regression of the test data, as summarized in Table 9.14. The predicted end moment
curvature hysteretic curves of H-section and boxsection specimens are plotted in Fig. 9.44 with comparison of experimental hysteretic curves. The comparison between the predicted and experimental curves shows that the proposed trilinear hysteretic model with transition segment is able to reproduce the seismic behavior of HSS beam
column member with a reasonable accuracy. It should be noted that the accuracy of the

Hysteretic behavior of high-strength steel columns

409

1.4 1.2 1.0

P' 1

0.6 0.4

M / My

P1

Critical slope: J

0.8

Q1

Q' 1

0.2 0.0 –0.2

Q2

Q' 2

–0.4 –0.6

P' 2

–0.8

P2

Experimental Reference Linearized

–1.0 –1.2 –1.4 –6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

Curvature ) /) y

(A)

M/M y

E'

G'

γ

A ĸ

P1'

Q1'

3

3

B

ĺ

Ĺ

k

k

3

ķ

F'

β

k

D'

k2

3

Q1

k1

k

P1

Ф/ Фy k

3

k2

Q2

F

C

P2 B'

β

A'

D

γ E

G

k1

Q2'

P2 '

(B)

Figure 9.43 Hysteretic model for HSS beam
column member: (A) generalization of hysteretic curve and (B) proposed hysteretic model. HSS, High-strength steel.

Table 9.14 Variables of the proposed hysteretic model. Section

k1

k2

k3

β

γ

H section Box section

2 0.1 2 0.1

0.08 0.08

1 1

0.031 0.042

0.21 0.36

proposed model is validated under the scope of the presented experimental investigation, where steel grade is Q460, the axial force ratio is 0.3, and the story drift ratio is limit to 1/30 and 1/40 for very compact Class 1 up to Class 2 cross sections. Further experimental and numerical analyses are needed to calibrate the model for, including different values of axial load ratio and cross section slenderness ratio.

410

Behavior and Design of High-Strength Constructional Steel M/M y Experimental Predicted 1.0

0

–1.0

–2.0

0

–1.0

1.0

2.0

Ф/ Ф y

(A) M/M y Experimental Predicted 1.0

0

–1.0

–4.0

–3.0

–2.0

0

–1.0

1.0

2.0

3.0

Ф/ Ф y

(B) M/M y

Experimental Predicted

1.0

0

–1.0

–2.0

0

–1.0

1.0

2.0

Ф/ Ф y

(C) M/M y

Experimental Predicted

1.0

0

–1.0

–3.0

–2.0

–1.0

0

1.0

2.0

Ф/ Ф y

(D)

Figure 9.44 Comparison between experimental and predicted hysteretic curves: (A) H-3.815.6-0.3, (B) H-7.7-27.7-0.3, (C)B-10.7-10.7-0.3, (D) B-21.6-21.6-0.3.

Hysteretic behavior of high-strength steel columns

9.7

411

Summary

Based on the experimental and numerical studies on hysteretic behavior of Q460 and Q690 HSS beam
column, the following summaries are drawn: 1. The plastic local buckling dominates the failure mechanism of all the specimens without overall buckling being observed. 2. The hysteretic curves of all specimens are plump, and there is no obvious pinching phenomenon before fracture. 3. The yielding displacement interstory drift angles of almost all specimens are larger than 1/50, while the axial compression ratio of the larger specimens is slightly less than 1/50. The failure displacement angle of all specimens (bending moment bearing capacity reduced to 85%) is greater than 1/20, which can meet the limit of maximum interstory displacement angle specified in seismic design codes. 4. The energy dissipation ability and evolution law for different specimens under identical cyclic loading are similar. 5. The two proposed hysteretic models are able to reproduce the seismic behavior of HSS beam
column members with a reasonable accuracy.

References [1] Ballio G, Castiglioni CA. Seismic behaviour of steel sections. J Constr Steel Res 1994;29(1
3):21
54. [2] Fadden M, McCormick J. Cyclic quasi-static testing of hollow structural section beam members. J Struct Eng 2011;138(5):561
70. [3] Lamarche C-P, Tremblay R. Seismically induced cyclic buckling of steel columns including residual-stress and strain-rate effects. J Constr Steel Res 2011;67 (9):1401
10. [4] Newell J, Uang C. Cyclic behavior of steel wide-flange columns subjected to large drift. J Struct Eng 2008;134(8):1334
42. [5] Nakashima M, Liu D. Instability and complete failure of steel columns subjected to cyclic loading. J Eng Mech 2005;131(6):559
67. [6] Kurata M, Nakashima M, Suita K. Effect of column base behaviour on the seismic response of steel moment frames. J Earthquake Eng 2005;9(sup2):415
38. [7] Kumar S, Usami T. Damage evaluation in steel box columns by cyclic loading tests. J Struct Eng 1996;122(6):626
34. [8] Aoki T, Susantha K. Seismic performance of rectangular-shaped steel piers under cyclic loading. J Struct Eng 2005;131(2):240
9. [9] Fukumoto Y. New constructional steels and structural stability. Eng Struct 1996;18 (10):786
91. [10] Li GQ, Wang YB, Chen SW. The art of application of high-strength steel structures for buildings in seismic zones. Adv Steel Constr 2015;11(4):492
506. [11] European Committee for Standardization. Eurocode 3: design of steel structures, Part 1
12: Additional rules for the extension of EN 1993 up to steel grades S700, EN 1993-1-12: [S]. Brussels: European Committee for Standardization; 2007. [12] AISC. Specification for structural steel buildings, ANSI/AISC 360-10: [S]. Chicago, IL: AISC; 2010.

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Behavior and Design of High-Strength Constructional Steel

[13] European Convention for Constructional SteelWork. Manual on stability of steel structures. ECCS Publication; 1976. [14] Ziemian RD. Guide to stability design criteria for metal structures. 6th edition [M] NJ: John Wiley & Sons, Inc; 2010. [15] European Committee for Standardization. CEN E. 1-1-Eurocode 3: Design of steel structures-Part 1-1: General rules and rules for buildings. Brussels: European Committee for Standardization; 2005. [16] China Standard Press. GB/T 228-2002 Metallic materials: tensile testing at ambient temperature: [S]. Beijing: China Standard Press; 2002 (in Chinese). [17] Jacobsen LS. Steady forced vibrations as influenced by damping. Trans Am Soc Mech Eng 1930;52(1):169
81. [18] Hibbitt H, Karlsson B, Sorensen P. Abaqus analysis user’s manual version 6.10. Providence, RI: Dassault Syste`mes Simulia Corp; 2011. [19] China Planning Press. Code for acceptance of construction quality of steel structures: GB 50205[S]. Beijing: China Planning Press; 2001. [20] Ban H, Shi G, Shi Y, et al. Study on the residual stress distribution of ultra-highstrength-steel welded sections. Eng Mech 2008;25(SII):57
61. [21] Hai LT, Li GQ, Wang YB, et al. Experimental investigation on cyclic behavior of Q690D high strength steel H-section beam-columns about strong axis. Eng Struct 2019;189:157
73. [22] China Academy of Architectural Sciences. Specificating of testing method for earthquake resistant building: JGJ101-96[S]. Beijing: China Academy of Architectural Sciences; 1996. [23] China Construction Industry Press. Code for seismic design of buildings GB50011[S]. Beijing: China Construction Industry Press; 2010. [24] Wang Y.B., Li G.Q., Cui W., et al. Seismic behavior of high strength steel welded beam-column members. J Constr Steel Res, 2014, 102(0): 245-255

Behavior of high-strength steel columns under and after fire

10

Wei-Yong Wang1, Guo-Qiang Li2 and Wen-Yu Cai2 1 Chongqing University, Chongqing, P.R. China, 2Tongji University, Shanghai, P.R. China

10.1

Introduction

Steel structures may be inevitably exposed to fire hazards. In fire events, deterioration of mechanical properties and creep deformation has been considered as the most important factor affecting the behavior of steel structures. Much research has been carried out on fire resistance of mild steel structures and the research results have been considered in the current codes of practice including BSI 2003, ECS 2005, and GB 51249-2017 [1]. However, there are no available design guidelines for the safety of fire-exposed high-strength steel structures. This chapter presents the research investigations on the behavior of high-strength steel elements subjected to fire. The topics include behavior of restrained highstrength steel columns under or after fire, and experimental and analytical study on creep buckling of high-strength steel columns at elevated temperatures in fire.

10.2

Behavior of restrained high-strength steel columns under fire

This section presents the experimental studies on the behavior of restrained Q460 high-strength steel columns under fire. Eight specimens were tested, of which four were axially restrained and the other four were axially and rotationally restrained. The fire resistance and response of restrained high-strength steel columns were compared with those of restrained mild steel columns.

10.2.1 Specimen preparation The steel plates made of Q460 steel were welded to build up eight column specimens with H-shape cross section (H200 3 195 3 8 3 8), in which four specimens with a length of 4.3 m were designed with axial end restraints, whereas the others with a length of 4.48 m are designed with both axial and rotational end restraints. The restraining stiffness at each end was provided by two H-shaped steel beams made of Q235 steel with a length of 3.2 m. Two cross sections, namely, H200 3 150 3 6 3 9 and H300 3 150 3 6.5 3 9, were fabricated for the beam to Behavior and Design of High-Strength Constructional Steel. DOI: https://doi.org/10.1016/B978-0-08-102931-2.00010-3 © 2021 Elsevier Ltd. All rights reserved.

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Table 10.1 Material properties of steels for column specimens and restraining beams. Steel

Thickness (mm)

Yield strength (MPa)

Ultimate strength (MPa)

Elastic modulus (MPa)

Q235 Q460

9 8

285 585

415 660

2.10 3 105 2.12 3 105

generate two different types of restraining stiffness. The mechanical properties of the test specimens and the restraining beams are obtained by the standard tension coupon test according to the ASTM A370 test protocol [2]. The test results are tabulated in Table 10.1. For the specimens with the axial end restraints the end conditions are hinge connected to the restraining beam to ensure the specimen ends can freely rotate about its weak axis. The end conditions of strong axis are fixed and cannot rotate. For the specimens with both axial and rotational end restraints, extended end-plate connections were used to connect the column ends to the restraining beam in such a way both axial and rotational deformation at column ends are prevented. The aforementioned two types of end connections are illustrated in Fig. 10.1. Different magnitudes of the applied load and restraining stiffness were considered in the tests. The axial restraint ratio β a is defined as: βa 5

Kb 48Eb Ib =l3b 5 Kc Ec Ac =lc

(10.1)

where Kb is the flexural stiffness associated with the mid-span deflection of the restraining beam; Kc is the axial stiffness of the column; Ib is the moment of inertia of the beam; Ac is the column cross-sectional area; and lb and lc are the length of the beam and column, respectively. The rotational restraint ratio is defined as: βr 5

Krb 12Eb Ib =lb 4Eb Ib lc 5 5 Krc 3Ec Ic =lc Ec Ic lb

(10.2)

where Krb is the rotational stiffness associated with the mid-span rotation of the restraining beam; Krc is the end rotational stiffness of the column. The load ratio R is expressed as: R5

N Ncr

(10.3)

where N is the applied load placed on the column top end and Ncr is the ultimate load capacity of the column evaluated based on GB 50017-2017 [3] at ambient temperature.

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Figure 10.1 Connection between column specimen and restraining beam: (A) Hinged connection, (B) extended end-plate connection. Table 10.2 Information of the specimens. Specimen no. S-1 S-2 S-3 S-4 S-5 S-6 S-7 S-8

End restraint

Load (ratio)

βa

βr

0.25 0.40 0.25 0.40 0.20 0.20 0.36 0.38

0.45 0.45 0.17 0.17 0.45 0.17 0.45 0.17

0 0 0 0 36 14 36 14

Axial

Axial and rotational

The detail information about the specimens is tabulated in Table 10.2.

10.2.2 Test set-up and measurements The test specimens are heated in a fire furnace. The dimension of the furnace is 3.6 m wide, 4.6 m long, and 3.3 m high. The maximum heat power generated by the furnace is 5 MW. Eight natural gas burners are installed in the furnace, and the furnace temperature was recorded by 10 thermocouples placed in the test chamber over a fire test. The plan view of the furnace is shown in Fig. 10.2. During the fire test, the temperature readings in thermocouples (noted as FT1
FT8) are used to compare with that of ISO-834 heating curve and the control system automatically adjusts corresponding fuel supply to keep the furnace temperature in pace with that of the heating curve. As shown in Fig. 10.3, a horizontal self-reaction loading system, consisting of a steel frame and two steel restraining beams (top beam and bottom beam), was designed to apply loading on the test specimen and to provide desirable boundary conditions for specimen. The frame was horizontally placed on the top of furnace.

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Figure 10.2 Plan view of the fire furnace.

Figure 10.3 Experimental loading system layout and specimen set-up.

Axial load applied on the test specimen by a horizontal hydraulic jack with a capacity of 1000 kN. The top steel restraining beam provided an axial restraint or both axial and rational restraints on the test columns, and the bottom restraining beam supported by the stub column only provides the rotational restraint to the column end. The bottom beam hinged at both ends is connected to frame using highstrength steel bolts prior to the installation of the test column. After the installation of test specimen and top restraining beam the bolts at the ends of top beam keep loose to apply the load on the restrained column. Until finishing applying the load, the top restraining beam was firmly connected to frame by tightening the highstrength bolts. The top beams placed inside the furnace were protected with fire

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insulation in order to maintain the restraining stiffness during the high-temperature tests. Thermocouples, strain gauges, and linear variable differential transformers (LVDTs) were used to record the thermal and structural responses of the test columns during the fire tests. The instrumentation of the test is shown in Fig. 10.3. Temperatures of the specimen were recorded by nine Type-K thermocouples, 2.0 mm in diameter. The thermocouples were located at the 1/3, 1/2, and 2/3 of the specimen length. Three strain gauges were placed in the short stub column to evaluate the axial force in the restrained specimen. At locations of the top and midheight of the specimen, two LVDTs were installed to measure the axial displacement and lateral deflection of the specimen, respectively. The applied load on the test specimen was measured by a load cell mounted on the hydraulic jack.

10.2.3 Test procedure The test process comprises the following steps: G

G

G

The specimen was first placed into the loading frame, and instrumentations including the thermocouples, strain gauges, and LVDTs were subsequently installed. The load was applied to the specimen by the hydraulic jack from zero up to 10% of target load, kept for 5 minutes, and then released to zero. This preloading process was repeated twice to ensure stable variation trends of load
deformation curves and to examine the operations of the instrumentations. After preloading the load was applied to the specimen with a loading rate of 20 kN/min until it reached the target load. At this point the top restraining beam was firmly connected to the frame and the magnitude of the applied load was maintained throughout the rest of the test. The furnace was turned on and the furnace temperature was controlled in consistence with ISO-834 heating curve. If the axial displacement at the top end of specimen or the lateral deflection at the mid-height of the specimen reached the maximum range (lc/100) or failure limit (lb/20), the furnace was turned off, the applied load was released, and the test is terminated.

10.2.4 Test results The test data were utilized to investigate the behavior of restrained Q460 highstrength steel column subjected to elevated temperature. The effects of the applied load, and axial and rotational restraints on the behavior of Q460 high-strength steel columns were evaluated through the comparison with thermal responses, structural responses, and failure patterns of the specimens.

10.2.4.1 Temperature evolution The temperature evolution in the furnace, column specimens, and top beams are plotted in Fig. 10.4. The furnace temperature was unable to match with that of ISO834 curve initially due to the limited heat power of the furnace. However, a few (3
5) minutes later, the furnace temperature matched well with ISO-834 curve. The column specimen was directly exposed to fire; owing to its small thickness and

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Figure 10.4 Temperature distributions of furnace, specimen, and top restraining beam: (A) Specimen S-1, (B) Specimen S-2, (C) Specimen S-3, (D) Specimen S-4, (E) Specimen S-5, (F) Specimen S-6, (G) Specimen S-7, (H) Specimen S-8.

high thermal conductivity, its temperature increased quickly. The dispersion of temperatures recorded by thermocouples was less than 50 C across the column cross section. Therefore the effect of nonuniform distribution of the temperature in the column was negligible and the average temperature of the column was employed to

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analyze structural responses of the columns. The top restraining beam was wrapped with fire insulation; consequently, its maximum temperature only increased up to approximately 220 C.

10.2.4.2 Axial displacement and lateral deflection When the axial displacement induced by thermal expansion approaches the maximum value, the axial force in restrained column also reaches the maximum value. According to the previous research [4], for a restrained mild steel column in fire, the temperature at which the axial force reaches the maximum magnitude is defined as buckling temperature, whereas the temperature associated with that when the axial force returns to its initial magnitude is defined as the failure temperature or critical temperature. Fig. 10.5 shows the variations of the axial displacements versus time and temperature for different specimens. The axial displacements increased gradually, and then dropped abruptly when reaching maximum values. Since all the specimens failed at elevated temperature within 25 minutes, the unprotected high-strength steel columns are quite sensitive to fire. In the axial displacement
temperature curves, the maximum displacement points, the initial points of the descending segment were defined as the buckling temperature and failure temperature (T1), respectively, tabulated in Table 10.3. The restraining stiffness is another critical factor influencing the fire resistance of the restrained column. At the same applied load ratio, taking an example of specimens S-1 and S-3, the higher axial restraining stiffness yields the less

Figure 10.5 Axial displacements of test specimens: (A) relationship between axial displacement and time, (B) relationship between axial displacement and temperature.

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Table 10.3 Critical temperature and failure time of the specimens. Specimen no. 

Buckling temperature ( C) Failure temperature T1 ( C) Failure temperature T2 ( C)

S-1

S-2

S-3

S-4

S-5

S-6

S-7

S-8

606 620 652

493 510 550

582 625 603

530 564 558

607 655 669

617 688 699

512 564 560

412 454 356

Figure 10.6 Lateral deflection of test specimens: (A) relationship between the lateral deflection and time, (B) relationship between the lateral deflection and specimen’s temperature.

axial displacement and produce greater thermal force, resulting in the lower failure temperature for restrained columns. In Fig. 10.6 specimens S-1 and S-2 reached a metastable state in postbuckling phase. In this state the applied load was resisted by the top restraining beam instead of the column. Note that this phenomenon was only observed for the column specimens with high axial restraining stiffness and relatively low load level, such as specimens S-1 and S-2. In fact, this phenomenon was also observed in the research investigations on the restrained mild steel column reported by Franssen [5] and Wang [6]. For all the specimens the lateral deflection was quite stable at initial stage of heating, followed by an abrupt increase prior to reaching the failure state. It only took a few seconds to transit from stable to failure state; the corresponding increase of the lateral flection was approximately about 200 mm. Note that some specimens

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Figure 10.7 P/P0 of test specimens: (A) relationship between the axial force ratio P/P0 and time duration, (B) relationship between the axial force ratio P/P0 and specimen’s temperature.

experienced negative deflections at the initial stage of heating, such as specimen S4. The negative deflection might be attributed to possible flange curling and torsion in the specimen at the initial heating stage. As for the axially restrained columns, specimen S-6 took longer to fail than specimen S-5 although they were subjected to the same applied load. This might be due to that the top restraining beam connected to specimen S-5 was higher than specimen S-6. The higher stiffness of the top restraining beam, the higher axial restraint ratio in specimen will be, potentially resulting in a higher axial force associated with thermal expansion in the specimen, as illustrated in Fig. 10.7. However, as for the columns with both axial and rotational restraints such as specimens S-7 and S8, the opposite was true. The reason is that compared with that in specimen S-8, the higher rotational restraint ratio in specimen S-7 may result in a stronger capacity of resisting flexural torsional buckling.

10.2.4.3 Axial compressive force in the specimen The variations of the axial force ratio, P/P0, with the time and temperature are showed in Fig. 10.7, where P denotes the axial force calculated based on the elastic modulus and the measured strain of the stub column connected to the bottom end of the test specimen; P0 is the applied load on top of the specimen. The axial force induced by thermal expansion in the axially restrained column is greater for the column with the higher axial restraint ratio and the lower applied load ratio, leading to

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a higher P/P0, such as the case of S-1. A similar phenomenon is also observed for the specimen with both axial and rotational restraints such as specimen S-5 in the figure. The axial force in the column declined quickly after large lateral deformation was observed, indicating the failure of the column. Based on the aforementioned discussion, the failure temperature determined based on the axial displacement in the specimen denoted as T1; to distinguish from T1, that determined based on the axial force in the specimen denoted as T2. The failure temperatures T1 and T2 are presented in Table 10.3 for comparison. There are some discrepancies between the failure temperatures T1 and T2, particularly for specimen S-8. The discrepancies can be attributed to that the calculated axial force may not always accurate using strain obtained from the stub at the top of the specimen since the bottom beam could also resist the axial force from the specimen to some degree.

10.2.4.4 Failure mode Fig. 10.8 shows the failure modes of the test specimens. It is clear that the failure modes of the specimens are influenced by the type of end restraints and magnitude of applied load. For specimens S1
S4 with only axial restraint, the failure mode is the flexural buckling about the weak axis of the specimens. For specimens with both axial and rotational restraints, failure modes depend on the magnitude of applied load. For example, specimens subjected to a low magnitude of applied load,

Figure 10.8 Failure modes in test specimens: (A) S-1, (B) S-2, (C) S-3, (D) S-4, (E) S-5, (F) S-6, (G) S-7, (H) S-8.

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such as specimens S-5 and S-6, flexural buckling about the weak axis dominates their failure modes; however, for the specimens subjected to high magnitude of applied load, such as specimen S-7 and S-8, flexural torsional buckling failure was observed as shown in Fig. 10.8G and H. The flexural torsional buckling failure can be attributed to the nonuniform temperature distribution in these two specimens at the failure time. Compared with the specimens subjected to a lower applied load ratio, those subjected to a higher applied load ratio may fail in flexural torsional buckling at an earlier stage in which the temperature increases quickly and the effects of nonuniform distribution can be more significant than that in the later stage. The nonuniform temperature distribution in the specimen may result in nonuniform stiffness distribution in the specimen. However, the flexural buckling failure normally happened in the later stage due to the low applied load ratios and the corresponding temperature distribution at this stage is relatively uniform. It appears that the failure mode of the specimens is more sensitive to the magnitude of applied load other than the magnitudes of axial and rotational restraints. Further investigations on the behavior of the columns with the high applied load ratio and effect of nonuniform temperature distribution on the column failure mode are needed in future research.

10.2.5 Comparison with restrained mild steel columns In order to investigate the difference of the fire responses between the restrained columns made of high-strength steel and mild steel. Test results of specimen RS97_4 in a fire test conducted by Tan et al. [7] was compared with that of specimen S-2 in this investigation since both specimens have a slenderness ratio of 96, applied load ratio of 0.5 and axial restraint stiffness ratio of 0.16. From Fig. 10.9, there are significant differences between the maximum axial displacement and P/P0 ratio because the length of RS97_4 is only 1.5 m, which is considerably shorter than that of S-2. However, the P/P0 ratios of the two specimens follow a close evolution in the early stage of the elevated temperature. Note that with the same test conditions, the Q460 axially restrained column generally exhibits much better fire

Figure 10.9 Fire resistance comparison between mild and high-strength restrained steel columns: (A) Axial displacement, (B) Axial applied load ratio.

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resistance than mild steel axially restrained column. For a column with an applied load ratio of 0.3, the axial restraint stiffness ratio of 0.34, and a slenderness of 51 similar to those of specimen S-7 in this investigation, the failure temperature of column HEA200-K128-L30 was approximately 515 C as reported by Refs. [8,9], which is less than 564 C obtained from this investigation for specimen S-7. This conclusion also can be drawn by comparing the test results of this investigation with that of tests performed by Ali et al. [10].

10.3

Postfire residual capacity of high-strength steel columns with axial restraint

This part presents experimental and theoretical results of fire and postfire behavior of restrained high-strength steel columns made of Q550 steel. Three specimens were tested, of which one was loaded until failure at ambient temperature and the other two specimens were heated in a furnace and loaded until failure after cooling down to determine their residual load-bearing capacities. A finite element model was created and validated against experimental results on restrained high-strength steel columns. Parametric studies were conducted using the validated numerical models, and the influencing factors included the maximum temperature experienced in fire, load ratio, axial stiffness ratio, slenderness ratio, and steel grade. The load ratio is defined as the ratio of the applied load on a structural member to its ultimate bearing capacity, and the axial stiffness ratio is defined as the ratio of the axial stiffness of the axial restraint to that of the member. A simplified calculation method was finally proposed to determine the residual compressive strength of restrained high-strength steel columns, accounting for these influencing factors.

10.3.1 Test setup and specimens Three axially restrained high-strength steel columns, denoted as HC-1, HC-2, and HC-3, were designed and tested. Specimen HC-1 was loaded until failure at ambient temperatures, while the other two specimens (HC-2 and HC-3) were heated in a furnace and were loaded until failure after cooling down to ambient temperatures to determine their residual load-bearing capacities. The two specimens (HC-2 and HC3) have the same dimension, as listed in Table 10.4, but different load ratios of 0.27 and 0.35 respectively, as presented in Table 10.5. These load ratios are relatively smaller than those for columns in practical design at ambient temperature. This is because that they were basically determined based on a loading combination of 1.2D 1 0.5L for the case of fire (ASCE 2010), where D is the dead load and L is the live load. They were also adjusted by conducting numerical simulations to avoid the failure of the column specimens during heating, and thus their residual strength could be measured after fire. A load ratio range of 0.25
0.4 was found from the numerical results, and a load ratio of 0.27 and 0.35 was finally achieved in the test for the specimens HC-2 and HC-3, respectively, to cover this range.

Table 10.4 Cross-sectional properties of the specimens. Specimen no. HC-1 HC-2 HC-3 HB

Width of flange (mm)

Height of section (mm)

Thickness of web or flange (mm)

Length of specimen (mm)

Area of cross section (mm2)

Second moment of area (mm4)

Radius of gyration (mm)

80.5 79.6 79.2 140.1

101.0 100.5 100.4 219.8

8.21 8.21 8.21 8.21

1431.1 1430.6 1429.8 1999.8

2016.2 1997.2 1989.2 3970.2

7.2E 1 05 6.9E 1 05 6.8E 1 05 3.2E 1 07

18.9 18.6 18.5 89.1

Note: HB 5 restraining beam.

Table 10.5 Key experimental results of the specimens. Specimen no.

HC-2 HC-3

Load (kN) (load ratio)

196 (0.27) 253 (0.35)

Maximum temperature ( C)

309 279

Buckling temperature ( C)

268 252

Residual axial displacement (mm)

2.1 1.8

Residual deflection (mm)

30.2 25.3

Residual capacity (kN)

Residual axial stiffness (kN/mm)

Value

Reduction factor

Value

Reduction factor

385 398

0.54 0.56

101.3 104.7

0.32 0.33

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Figure 10.10 Layout of fire tests: (A) schematic of test layout; (B) on-site test layout.

The test setup is shown in Fig. 10.10. All specimens had a net length (Le) of 1.4 m. The load-bearing capacity and axial stiffness of HC-1 were measured as 723 kN and 314.4 kN/mm, respectively. The axial restraint of the columns was provided by a restraining beam with a length of 2 m. No rotational restraint was generated by the restraining beam since a pinned connection between the column and the beam was used. The beam was connected to the reaction frame by friction-type

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high-strength bolts. This is to ensure no relative slide between the beam ends and the reaction frame. The axial restraint stiffness provided by the beam was measured as 116.5 kN/mm, corresponding to a ratio of the restraint stiffness to the column stiffness of 0.37. Pinned supports at the two ends of the columns were simulated by using cambered bearing supports, as shown in Fig. 10.10. The columns were designed to rotate around the weak axis. The column specimens and restraining beam had an I-shaped section of H100 3 80 3 8 3 8 (mm) and H220 3 140 3 8 3 8 (mm), respectively, as shown in Fig. 10.11. The section properties of the columns and restraining beam (HB) are listed in Table 10.1. The dimensions of the web and flange of the columns were designed to satisfy the requirements in a Chinese code [3] to ensure that global buckling of the columns occurred before local buckling. The initial imperfections of the three specimens (HC-1, HC-2, and HC-3) was measured as 0.25, 1.25, 0.8 mm, respectively, which were smaller than Le/1000.

Figure 10.11 Dimension of specimens: (A) restraining beam; (B) column specimen (all units in mm).

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Table 10.6 Material properties of the specimens. Specimen no. HC-1 HC-2 HC-3 Average

Yield strength (MPa)

Tensile strength (MPa)

Elongation ratio (%)

733.2 734.4 732.6 733.4

764.6 765.7 763.8 764.7

18.7 17.9 19.2 18.6

The steel grade Q550 was used for the column specimens and restraining beam. Axial tensile tests were conducted according to a Chinese code [11] to determine the material properties, as listed in Table 10.6. The three specimens showed similar material properties, with an average yield strength of 733 MPa, average tensile strength of 765 MPa, and average elongation ratio of 18.6%.

10.3.2 Instrumentation The following parameters were measured in the tests: axial force in the column, temperature of the column, restraining beam and reaction frame, axial displacement at the top of the column, lateral displacement at mid-span of the column, displacement at the ends of the restraining beam.

10.3.2.1 Strain gauges A dynamic-static resistance strain measurement system DH3817 was used in the test to measure the strain in the column specimen. The system has a strain measuring range and strain resolution of 0.1 and 0.1με, respectively. It has a maximum sampling rate of 1 kHz, and a total of 128 measurement points can be controlled per computer. Three groups of electrical resistance strain gauges made of wire-type conductors were placed at the quarter points along the length of the column, as shown in Fig. 10.12A. In each group, six strain gauges were arranged, of which the gauges (S1
S4) were to measure the strain of flanges, and two gauges (S5, S6) were to measure the strain of the web. The measured strains were used to calculate and monitor the initial axial force in the column before heating and load eccentricity of the column. This is to control the loading quality, and to ensure that the column was under axial compression with negligible eccentricities.

10.3.2.2 Thermocouples The temperatures along the length of the columns were measured by four thermocouples (T1
T4), as shown in Fig. 10.12B. The temperature of the restraining beam and reaction frame was measured by the thermocouple T5 and T6, respectively. One thermocouple (T7) was used to measure the gas temperature in the furnace.

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Figure 10.12 Layout of measurements: (A) strain gauges; (B) thermocouples and displacement meters.

10.3.2.3 Displacement meters The linear variable displacement transducers (LVDTs) were used to measure the displacements. The layout of LVDTs is shown in Fig. 10.12B. The mid-height lateral displacement of the column and beam was measured by D5, D6 and D3, D4, respectively. It was assumed that the axial displacement at the top of the column was equal to the mid-span deflection of the restraining beam (i.e., D3, D4). The relative slide at the two ends of the restraining beam was measured by D1, D2.

10.3.3 Test procedure and results 10.3.3.1 Test procedure The test protocol is as follows: Phase I: loading at ambient temperature. The specimen was loaded until the target load (Table 10.5) was reached. The target axial load of HC-2 and HC-3 was 196 and 253 kN, respectively, corresponding to a load ratio of 0.27 and 0.35, respectively. Phase II: heating process. By keeping the target load constant the specimen was heated in an electric furnace until the top of the column returned to its initial position before heating, that is, the column reached its critical temperature. Phase III: cooling process. After reaching the critical temperature the heating was stopped and the furnace was opened to cool down the specimens to room temperature via natural air circulation. During the entire cooling phase, the load was still kept constant, and the data acquisition remained operative. Phase IV: loading to failure at ambient temperature. After cooling down to room temperatures, the restraining beam was removed and the specimens were loaded until failure to determine their postfire load-bearing capacity. A load-control

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Figure 10.13 Temperature
time curves of the specimens during fire: (A) HC-2; (B) HC-3.

method was used before the load reaches 80% of the estimated load capacity. After that, a displacement-control load was imposed.

10.3.3.2 Test results of fire behavior (heating and cooling) The measured temperature
time curves of the column (T1-T4), restraining beam (T5), reaction frame (T6), and gas in the furnace (T7) are shown in Fig. 10.13. For specimen HC-2 the temperature data of T2 was missed in the test. Note that the gas temperature T7 (25 C) of specimen HC-2 as shown in Fig. 10.13A was slightly lower than the specimen temperature T1 (20 C). This small temperature difference may be because during the long preparation period relatively hot gas went up and accumulated in the top region of the closed furnace, leading to a higher temperature T1. The restraining beam and reaction frame were at a low temperature of 107 C and 20 C, respectively. The temperature distribution along the length of the columns was nonuniform. This was partly due to the malfunction of heating wires at the bottom of the furnace. It was also partly due to the convection of hot air from the bottom of the furnace to the top. A linear temperature distribution between the measurement points was assumed. The average maximum temperature of the column HC-2 and HC-3 was then calculated as 309 C and 279 C. The measured displacement
time curves of the columns are shown in Fig. 10.14. The key measured results are listed in Table 10.5. The initial axial displacement of specimens HC-2 and HC-3 was 20.71 and 20.89 mm, respectively. The initial mid-height lateral displacement was 1.29 and 0.82 mm, respectively. During the heating phase, the axial displacements of the columns varied positively due to the thermal expansion of the columns. The two columns followed a similar axial displacement trend due to the same dimension of the columns and the restraining beam (Fig. 10.14A). The columns HC-2 and HC-3 buckled when their temperature reached 268 C and 252 C, respectively. After buckling the axial displacement

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Figure 10.14 Displacement
temperature curves of the specimens during fire: (A) axial displacement at top of the columns; (B) mid-height lateral displacement of the columns.

and mid-height lateral displacement varied significantly. The earlier buckling of HC-3 than HC-2 was because of its higher load ratio. The displacements of the columns experienced a relatively smooth variation period after buckling. This is due to the plastic deformation of the columns. The load-bearing capacity of the columns reduced, leading to a decrease of the axial displacements. After cooling down to the room temperature, the residual axial displacement of the columns HC-2 and HC-3 was measured as 2.1 and 1.8 mm, respectively. The residual mid-height lateral displacement was 30.2 and 25.3 mm, respectively. This can be observed in the deformed shape of the specimens after the cooling phase. The relative slides at the ends of the restraining beam were also measured, which were smaller than 0.5 mm. This indicates a good connection between the restraining beam and reaction frame.

10.3.3.3 Test results of postfire behavior (residual capacity) The displacement
force curves for the residual capacity tests are shown in Fig. 10.15. The effect of residual deformation after cooling down the specimens was accounted for. Compared to the load-bearing capacity of 723 kN at ambient temperatures for HC-1, the postfire residual capacity of HC-2 and HC-3 was 385 and 398 kN, as listed in Table 10.5. This corresponded to a capacity reduction factor of about 0.55 for these two columns. The axial stiffness of the columns was reduced to 101.3 and 104.7 kN/mm, respectively, resulting in a stiffness reduction factor of about 0.32. The experience of heating significantly reduced the ultimate bearing capacity and axial stiffness of high-strength steel columns. This reduction is mainly due to the buckling behavior during fire and the postfire properties of steel material itself [12]. The failure modes of the three specimens are shown in Fig. 10.16. Global buckling around the weak axis of the columns occurred with obvious mid-height lateral displacements.

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Figure 10.15 Displacement
force curves of the specimens for the residual capacity tests: (A) axial displacement at top of the columns; (B) mid-height lateral displacement of the columns.

Figure 10.16 Failure mode of the specimens: (A) HC-1; (B) HC-2; (C) HC-3.

10.3.4 Numerical simulation 10.3.4.1 Development of numerical models A numerical model of a column was created in finite element software ABAQUS. The S4R element was used to simulate the column. A mesh of 40 elements was used along the length of the column. The web and flange were divided into 20 elements, respectively. A Poisson’s ratio of 0.3 was used. The temperature-dependent values of Young’s modulus and yield strength as well as the thermal expansion coefficient of steel at elevated temperatures were directly taken from the reference by Li et al. [13]. The postfire residual values of Young’s modulus and yield strength were used as proposed by Lyu [14].

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10.3.4.2 Validation of numerical models Specimens HC-2 and HC-3 were first simulated to validate the numerical models. The Young’s modulus and yield strength of Q550 steel at ambient temperature were taken as 214 GPa and 733.4 MPa. The axial restraint provided by the restraining beam was simulated by “SPRING” with a spring stiffness of 116.5 kN/mm. Pinned boundary conditions were assigned at the ends of the specimen. A measured initial imperfection of 1.25 and 0.8 mm was applied in the numerical model of specimens HC-2 and HC-3, respectively. The measured temperatures in Fig. 10.13 were used in the numerical model, where the column was divided into four regions equally and the measured temperatures T1
T4 were assigned to these four regions correspondingly. Since the temperature data of T2 for specimen HC-2 was missed in the test, the temperature T1 was used for it as the temperature distribution was relatively uniform in the top half region of specimen HC-3 as shown in Fig. 10.13B. A comparison of the measured and predicted axial displacements and mid-span deflections during the heating and cooling phase for specimens HC-2 and HC-3 is shown in Fig. 10.17. A comparison of the measured and predicted residual displacement
force curves for these two specimens is shown in Fig. 10.18. Table 10.7 lists a comparison of the measured and predicted key temperatures, displacements, and capacities. The predicted buckling temperatures of specimens HC-2 and HC-3 were 257 C and 246 C, respectively, compared to the measured values of 268 C and 252 C. The predicted and measured maximum axial displacements of specimen HC-2 were 3.16 and 2.9 mm, respectively. For specimen HC-3, these values were 2.83 and 3.12 mm, respectively. The residual load-bearing capacity of the columns was also determined by loading the column to failure after fire. The numerical results of residual capacity of HC-2 and HC-3 were 359 and 372 kN, respectively, compared to the measurements of 385 and 398 kN. A reasonable agreement was achieved since all the errors were less than 10%.

10.3.5 Parametric studies The validated numerical model was used to conduct parametric studies on a highstrength steel column (Q690) with a length of 2 m and a cross section of H130 3 120 3 6 3 10 (mm). The slenderness ratio around the weak axis is 66. An initial imperfection of L/1000 (2 mm) was defined for the column model. It was assumed that the column had a uniform temperature distribution across its section and along its length. It was also assumed that the temperature of the column increased linearly to the target maximum temperature, followed by a linear decrease to ambient temperature. The effect of five influencing factors on the behavior of restrained high-strength columns was investigated, including (1) maximum temperature experienced during a fire (T), represented by the relative temperature factor η defined below (in a range of 0
1.0); (2) load ratio (ρ), in a range of 0.1
0.7; (3) axial stiffness ratio (β), in a range of 0.1
0.5; (4) slenderness ratio (λ), in a range of 42
82; (5) steel grade (Q550, Q690, Q890). A passion ratio of 0.3 was used. The temperature-dependent material properties of Q550, Q690, Q890 steels were

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Figure 10.17 Comparison of measured and predicted displacements during the heating and cooling phase: (A) axial displacement of HC-2; (B) mid-height lateral displacement of HC-2; (C) axial displacement of HC-3; (D) mid-height lateral displacement of HC-3.

taken from the reference [15], while the postfire properties of these steels were taken from Lyu [14]. Some key definitions are provided herein: G

G

G

Buckling temperature (Tbu): the temperature at which a restrained steel column buckles in the heating phase (i.e., occurrence of maximum axial force), as shown in Fig. 10.19. Critical temperature (Tcr): the temperature at which the axial force in the heated column again reaches its starting value (P0) at ambient temperature, rather than the maximum axial force, as shown in Fig. 10.19. This is because the axial restraint prevents the sudden failure of the heated column in the declining phase following the point of maximum axial forces. Reduction factor for residual loading bearing capacity (α): α 5 Ncr0 /Ncr where Ncr0 is the residual compression capacity after exposure to fire; Ncr is the initial compression capacity before fire.

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Figure 10.18 Comparison of measured and predicted displacements after cooling phase: (A) axial displacement of HC-2; (B) mid-height lateral displacement of HC-2; (C) axial displacement of HC-3; (D) mid-height lateral displacement of HC-3. G

G

G

Relative temperature factor η: η 5 (T 2 Tbu)/(Tcr 2 Tbu). Load ratio ρ: ρ 5 P0/Ncr where P0 is the initial compressive force in a column. Axial stiffness ratio β: β 5 k0/kc where k0 is the axial stiffness of the axial restraint; kc is the axial stiffness of steel columns at ambient temperatures, which is determined as kc 5 EA/L where E is the modulus of elasticity; A is the area of cross section of columns; L is the length of columns.

10.3.5.1 Effect of maximum temperatures In this section the effect of maximum temperatures on the reduction coefficient of ultimate strength of restrained high-strength steel columns (α) was investigated. The maximum temperature was represented by the relative temperature factor (η).

Table 10.7 Comparison of measured and predicted key structural behavior of specimens. Specimen no.

HC-2 HC-3

Buckling temperature ( C)

Maximum axial displacement (mm)

Maximum mid-span deflection (mm)

Maximum residual deflection (mm)

Postfire residual capacity (kN)

Test

Num.

Error (%)

Test

Num.

Error (%)

Test

Num.

Error (%)

Test

Num.

Error (%)

Test

Num.

Error (%)

268 252

257 246

4.1 2.4

2.90 3.12

3.16 2.83

9.0 9.3

49.4 45.7

51.8 49.3

4.8 7.9

30.9 29.3

29.3 29.8

5.1 1.9

385 398

359 372

6.8 6.5

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Figure 10.19 Typical axial force
temperature curve of restrained steel columns at elevated temperatures.

The variation of η for different load ratios and axial stiffness ratios is shown in Fig. 10.20. It was found that there was no reduction in residual compressive strength of column (α 5 1) when η , 0 (i.e., T , Tbu). This is because the column experiences elastic deformation before buckling (temperature is smaller than the buckling temperature), and thus this deformation can be recovered after cooling. In addition, since the buckling temperature of the restrained columns is always less than 500 C, the effect of the maximum temperature smaller than this buckling temperature is little. Therefore, when the maximum temperature of columns is less than the buckling temperature, the restrained column has the same residual strength as its initial value. For η 5 0 (i.e., T 5 Tbu), the strength reduction coefficient decreased rapidly. The reduction coefficient at this point is defined as the maximum strength reduction coefficient (αmax). The significant reduction in strength is because the steel column buckled as its temperature reached the buckling temperature where plastic deformation developed which cannot be completely recovered. The obvious residual deformation will lead to a significant reduction in residual strength of columns. For 0 , η , 1 (i.e., Tbu , T , Tcr), the strength reduction coefficient reduced linearly with increasing η. This is because that over the buckling temperature, the higher the maximum temperature reached, the more significant plastic deformation was generated by the thermal expansion, and the more significant residual deformation remained after cooling, and the lower residual strength. The strength reduction coefficient decreased to its minimum value when the maximum temperature reached the critical temperature (T 5 Tcr, η 5 1), which was defined as the minimum reduction coefficient (αmin). At this temperature, the restrained column was in an ultimate limit state of failure, and thus the critical temperature was defined as the limit maximum temperature with respect to the residual strength. As shown in Fig. 10.20, the strength reduction coefficient was in a nearly linear relation with the relative temperature factor under different load ratios and axial stiffness ratios. This means the relation between α and η can be determined from

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Behavior and Design of High-Strength Constructional Steel

Figure 10.20 Variation of α against relative temperature factor β for different axial stiffness ratios: (A) 0.1; (B) 0.3; (C) 0.5.

αmax and αmin. Therefore the following subsections will focus on the effect of the influencing factors on αmax and αmin.

10.3.5.2 Effect of load ratios Fig. 10.21 shows the effect of load ratios (ρ) on the αmax and αmin for different axial stiffness ratios (β). The load ratio had significant effects on αmin, but negligible effects on αmax. The αmin increased linearly with increasing β. This is because that a larger initial load leads to an earlier occurrence of failure, and thus lower critical temperature and smaller bending deformation, which lead to smaller residual deformation and less reduction of material properties, and thus higher residual strength. The αmin varied in a range of 0.26
0.67, corresponding to a load ratio of 0.08 and 0.66, respectively.

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Figure 10.21 Variation of αmax and αmin against load ratios for different axial stiffness ratios: (A) αmax
ρ curve; (B) αmin
ρ curve.

Figure 10.22 Variation of αmax and αmin against axial stiffness ratios for different load ratios: (A) αmax
ρ curve; (B) αmin
ρ curve.

10.3.5.3 Effect of axial stiffness ratios The effect of axial stiffness ratio (β) on αmax and αmin of restrained columns under different load ratios is shown in Fig. 10.22. It was found that the axial stiffness ratio had significant effects on αmax, but negligible effects on αmin. The αmax increased with increasing axial stiffness ratios. This is because a larger axial stiffness can result in a smaller residual bending deformation and lower buckling temperature, and thus a higher residual strength. The larger the axial stiffness ratio, the slower the increment of αmax. The minimum and maximum αmax reached 0.34 and 0.75 for an axial stiffness ratio of 0.1 and 0.5, respectively.

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Behavior and Design of High-Strength Constructional Steel

10.3.5.4 Effect of slenderness ratios The above analyses showed that αmax and αmin significantly depended on the axial stiffness ratio and load ratio, respectively. This section presents the effect of slenderness ratios (λ) on the αmax
β and αmin
ρ curves, as shown in Fig. 10.23. The αmax
β and αmin
ρ had a similar trend for different slenderness ratios (Fig. 10.23A and B). As shown in Fig. 10.24A, αmax decreased as slenderness ratio increased for λ , 60, while αmax increased with increasing slenderness ratio for λ . 60. Fig. 10.24B shows that slenderness ratio had little effect on αmin, since the average standard deviation was 0.012 and average relative standard deviation was 3.4%.

Figure 10.23 Effect of slenderness ratio on (A) αmax
β curve; (B) αmin
ρ curve.

Figure 10.24 Variation of αmax and αmin against axial stiffness ratios for different axial stiffness ratios: (A) αmax
λ curve; (B) αmin
λ curve.

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10.3.5.5 Effect of steel grades Three steel grades of Q550, Q690, Q890 were considered, and the effect of these steel grades on αmax
β, αmax
λ, αmin
β is investigated. The results are shown in Fig. 10.25. Fig. 10.25A shows that the αmax
β curve followed a similar trend for different steel grades. For the same axial stiffness ratio the αmax followed the rules: Q890 . Q550 . Q690. Fig. 10.25B shows the variation of αmax against λ for a given axial stiffness ratio of 0.2. The three αmax
λ curves were similar. From Fig. 10.25A and B, it can be seen that the steel grade had little effect on αmax. Fig. 10.25C shows the effect of steel grades on αmin
ρ. The three curves showed a similar trend. There was obvious difference in αmin of Q550 and Q690 for ρ , 0.3, and they became similar for ρ . 0.3. For Q890, the difference of αmin to Q550 and Q690 was large for ρ , 0.4, but became small for ρ . 0.4. The local difference in αmin for different steel grades is due to the difference in their residual material properties after fire. Fig. 10.26 shows the effect of residual material properties on αmin
Tcr curves. For Q550 steel as shown in Fig. 10.26A the residual material

Figure 10.25 Effect of steel grades on (A) αmax
β curve; (B) αmax
λ curve; (C) αmin
ρ curve.

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Behavior and Design of High-Strength Constructional Steel

Figure 10.26 Effect of residual material properties after fire on αmin
Tcr curves for (A) Q550 steel; (B) Q690 steel; (C) Q890 steel.

properties had no effect on αmin for Tcr , 600 C. This is because if the maximum temperature is smaller than 600 C, the heating process had no effect on the yield strength of materials. After 600 C, the elevated temperatures had significant effect on the residual strength due to material degradation. A similar phenomenon was seen for Q690 and Q890 steel. The branch point of temperature for these two steel grades was 500 C and 600 C, respectively. The reduction magnitude of these three grades is different. It was found that if the difference between initial and residual material properties was not considered, steel grades had little effects on αmin.

10.3.6 Simplified formulation The above parametric studies showed that the residual strength of restrained highstrength steel columns after fire depended on the maximum temperature, load ratio,

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443

axial stiffness ratio, slenderness ratio, and residual material properties. As the residual material properties had different effects for different steel grades, the effect of the former four factors and the fifth factor was considered separately in this study. A formula to calculate the reduction coefficient (α) of the residual strength of steel columns is shown in Eq. (10.4). The function γ took into account of the effect of maximum temperatures, load ratios, axial stiffness ratios, and slenderness ratios, while μ(T) is the reduction coefficient of the residual yield strength of steel materials. α 5 γ ðη; ρ; β; λÞUμðT Þ

(10.4)

As presented in Section 10.3.4, the residual strength was only dependent on residual material properties for η ,0, while α changed linearly with η for 0 , η , 1. The value of α for a given load ratio and axial stiffness ratio can be calculated through linear interpolation from αmax and αmin. Thus Eq. (10.4) can be further simplified to Eq. (10.5) as:
μ η,0 α5 (10.5) αmax ð1 2 ηÞ 1 αmin η 0 # η # 1 It was found in Section 10.3.4 that the residual yield strength of Q550, Q690 and Q890 steels degraded when the maximum temperature was over 600 C, 500 C, 600 C, respectively. For the presence of axial restraints the buckling temperature of restrained columns is always lower than 500 C, and thus the effect of degraded residual material properties has not to be considered for determining the maximum reduction coefficient αmax. From Eq. (10.4), we obtained that αmax 5 γ max. This means the dependence of influencing factors on αmax also applies to γ max, that is, γ max is mainly dependent on axial stiffness ratio β and slenderness ratio λ but is independent of load ratio ρ. In addition, the steel grade has little effects on αmax. It is therefore conservative to determine γ max by curve fitting of γ max
β and γ max
λ of Q690 columns as expressed in Eq. (10.6). γ max 5 Aðλ258Þ2 1 B


40 # λ # 90

A 5 2 0:005β 2 1 0:0032β 2 1:8 3 1025 B 5 0:9β 1 0:25

(10.6) 0:1 # β # 0:5

(10.7)

From Eq. (10.4), the αmax can be determined as αmax 5 γ maxUμðTbu Þ

(10.8)

The determination of αmin is presented as follows. The numerical results showed that the load ratio had a significant effect on αmin. As shown in Fig. 10.26, αmin has a linear relation with Tcr. This means that one αmin correspond to a unique Tcr, and thus a unique μ(Tcr). Form Eq. (10.4), it can be seen that the rules for αmin also

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Behavior and Design of High-Strength Constructional Steel

applies to γ min, that is, γ min only depends on the load ratio. To determine γ min, the effect of residual material properties was also excluded. This is because that critical temperature increased with decreasing load ratio, and the effect of residual material properties is significant only for Tcr . 600 C. The curve fitting of the αmin
ρ curve for Q690 columns in Fig. 10.25C was used to determine the γ min
ρ relation, as expressed in: γ min 5 0:85ρ2 2 0:02ρ 1 0:31

0#ρ#1

(10.9)

Thus γ min can be determined as: αmin 5 γ minUμðTcr Þ

(10.10)

A flowchart to determine the residual strength of axially restrained high-strength steel columns is shown in Fig. 10.27. A general determination process is presented as follows: (1) for a given column, calculate its buckling temperature, critical temperature, load ratio, axial stiffness ratio, and slenderness ratio. (2) Based on a prespecified maximum temperature, calculate the relative temperature factor η. If η , 0, the reduction coefficient of residual strength can be calculated as the reduction coefficient of residual yield strength of steel after fire. Otherwise, for η $ 0, the maximum and minimum value of α should be determined following the next steps. (3) Based on residual mechanical properties of steel materials, calculate the reduction coefficient of residual material properties μ(Tbu) and μ(Tcr). (4) Based on the axial stiffness ratio and slenderness ratio, calculate αmax by Eq. (10.6). Based on the load ratio, calculate γ min by Eq. (10.9). (5) From γ max and μ(Tbu), calculate αmax by Eq. (10.8). From γ min and μ(Tcr), calculate αmin by Eq. (10.10). (6) Calculate α by Eq. (10.5) based on η, αmax, and αmin.

10.4

Creep buckling experiments of high-strength steel columns at elevated temperatures

The creep buckling is a phenomenon in which the buckling load for a column depends not only on slenderness and temperature but also on the duration of the temperature and applied load ratio. When a steel building is subjected to a fire, creep strain can significantly influence the behavior of the column, resulting in an increase in the column deformation and a decrease in the strength. However, the effect of creep in steel structures has not been fully considered in the current fireresistance design codes. Study on creep bucking of steel columns in fire is a critical issue for the fire-safety design of steel structures since steel columns may experience buckling under a very small applied load ratio if the column is subjected to elevated temperature for a relatively long period of time. To address this issue, this section presents the comprehensive experimental investigations on the creep

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445

Figure 10.27 Flowchart for determining residual capacity of restrained high-strength steel columns after fire.

buckling behavior of Q460 and Q690 high-strength steel columns subjected to a long-time ( . 2 hours) constant temperature and compressive load.

10.4.1 Specimen preparation Twelve welded H-shape steel columns with a length of approximately 2700 mm were fabricated using 14-mm-thick steel plates, of which six columns were made of Q460 steel and the other six were made of Q690 steel. At each end of the specimen, a 15-mm-thick steel end plate was welded to the specimen and the plate was connected to the test set-up. As shown in Tables 10.8 and 10.9, two different temperatures and applied load ratios (ratio of the applied load to the load-bearing capacity of the steel columns at a given temperature) were chosen for the tests. The loadbearing capacity of the steel columns at elevated temperatures was obtained from a calculation procedure specified in GB 51249-2017 [1]. The selected applied load level has to be relatively low to avoid the premature buckling prior to the occurrence of creep at the elevated temperatures. The mechanical properties of highstrength steel at elevated temperatures were obtained from the tests conducted by Wang and coworkers [8,9]. The yield strength, tension strength, and elastic modulus

Table 10.8 Specimen details of Q460 high-strength steel columns. Specimen no. S-1 S-2 S-3 S-4 S-5 S-6

Temperature ( C)

Cross-sectional dimensions

Length (mm)

Slenderness ratio λ

Load (kN)

Load ratio R

Initial crookedness (mm)

800 800 800 800 600 600

H200 3 150 3 14 3 14 H200 3 150 3 14 3 14 H200 3 200 3 14 3 14 H200 3 200 3 14 3 14 H200 3 150 3 14 3 14 H200 3 200 3 14 3 14

2730 2730 2730 2730 2730 2730

86 86 62 62 86 62

45 35 62 45 113 192

0.12 0.09 0.12 0.09 0.12 0.12

2.2 2.83 1.74 1.38 2.18 1.39

Table 10.9 Specimen details of Q690 high-strength steel columns. Specimen no. N800-0.3 N800-0.6 W800-0.3 W800-0.6 N700-0.6 W700-0.6

Temperature ( C)

Load (kN)

Load ratio R

Cross-sectional dimensions

Initial crookedness (mm)

800 800 800 800 700 700

30 50 60 100 92 171

0.3 0.6 0.3 0.6 0.3 0.6

H200 3 150 3 14 3 14 H200 3 150 3 14 3 14 H200 3 200 3 14 3 14 H200 3 200 3 14 3 14 H200 3 150 3 14 3 14 H200 3 200 3 14 3 14

0.47 2.61 2.55 2.23 3.61 3.00

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of Q460 steel plate are 587, 628, and 199,440 MPa, respectively; and those of Q690 steel plate are 821, 840, and 187,000 MPa, respectively. The effect of initial crookedness of the specimens on the load capacity of steel columns is accounted for in the investigations. The geometric imperfection of the specimens was defined as the maximum deviation of the column longitudinal axis from the straight-line connecting centroids of both ends of the column, and the values at a quarter, mid-height, and three-quarter points were measured, respectively. The measurement procedure was repeated twice through rotating the specimen by 180 degrees along the column axis in the buckling plane, and the measured results were averaged to reduce possible errors in measurements [16]. The maximum measured initial crookedness of the specimens, which is located at the midheight of the specimens, are shown in Tables 10.8 and 10.9.

10.4.2 Test setup A test setup was designed to investigate the creep buckling behavior of the highstrength steel columns under constant load at elevated temperatures, as shown in Fig. 10.28. The setup consists of a reaction frame (constructed by two columns and two reaction beams) with an overall dimension of a 5.85-m height and a 2-m width, a transfer girder, an electric furnace, and a hydraulic jack. The specimen was installed between the lower reaction beam and transfer girder with hinge supports. The hydraulic jack with a maximum load capacity of 500 kN was installed underneath the top reaction beam to push the transfer girder downward when the load was applied. The column hinge supports and the ends of transfer girder were lubricated to minimize friction between contact surfaces. Both the ends of the specimen and the hinge supports were protected by fireproof blanket. The electric furnace was lifted using a height-adjustable steel table to ensure the entire specimen being heated. The maximum temperature produced by the furnace is approximately

Figure 10.28 Test setup: (A) components of test set-up, (B) photo of test setup.

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Behavior and Design of High-Strength Constructional Steel

1200 C and the temperature elevation can be set up according to the predefined temperature
time curves.

10.4.3 Instrumentation In the test the furnace temperature, specimen temperature, actual load applied on the specimen, axial displacement, and lateral deflection at the mid-height of the specimen were recorded. The location and arrangement of the instrumentation are shown in Fig. 10.29. As for the tests on Q460 steel columns, three thermocouples, labeled as FT1, FT2, and FT3, were placed at different height of the furnace to measure the furnace temperature during heating, and nine thermocouples, labeled as CT1
CT9, were mounted on the specimen to monitor the temperature variations at the locations of C-1, C-2, and C-3. As for the tests on Q690 steel columns, five thermocouples, labeled as FT1
FT5, were placed at different height of the furnace to measure the temperature in the furnace and three thermocouples, labeled as CT1
CT3, were mounted on the specimen to monitor the temperature variations at the locations of C1, C-2, and C-3. The axial displacement of the specimen was recorded by using four linear variable differential transducers (LVDTs) installed vertically around four sides of the specimen shown as D1
D4. Another LVDT, labeled as D5, was installed horizontally to measure lateral deflection at mid-height of the specimen.

10.4.4 Test procedures The test was conducted based on the following procedure: G

G

G

The specimen was first installed into the reaction frame by connecting two end plates of the specimen to the hinge supports. The furnace was moved to encase the specimen with the instrumentation such as LVDTs and thermocouples being installed before the furnace were closed. The specimen was gradually loaded up to 10% of its ultimate load capacity and the load was maintained for two minutes before the load was released. This process is important to ensure the specimen and instrumentation were installed and worked appropriately.

Figure 10.29 Measurement arrangement: (A) Q460 steel columns, (B) Q 690 steel columns.

Behavior of high-strength steel columns under and after fire

G

G

G

G

449

At a given temperature the temperature increase rate of 30 C/min was set up before the furnace was turned on to heat the specimen. When the temperature of the specimen reached the target temperature, the hydraulic jack was turned on with a loading rate of 4
25 kN/min until the apply load reached the specified level. The loading rate was adjusted according to the magnitude of the specified load to ensure the loading phase was completed within 10 minutes. Once the applied load reached the specified level, both of the applied load and specimen’s temperature were maintained as constant and the axial and lateral deflection were recorded until the creep buckling occurred. If the lateral displacement at the mid-height of the specimen reached L/20, the test was stopped and the data were saved.

10.4.5 Experimental results 10.4.5.1 Furnace temperature The measured furnace temperature
time curves for the creep buckling tests on Q460 and Q690 high-strength steel column specimens are plotted in Figs. 10.30 and 10.31. Due to mechanical problem of the electronic furnace, the temperature of specimen W800-0.6 (Q690 steel column) shows an unusual heating phase. The middle furnace temperature is the highest among measured results, followed by top and bottom furnace temperatures. The reason is that hot air would flow upward to transfer heat, and meanwhile, the top and bottom furnace cannot be completely sealed, resulting in an amount of heat loss. The malfunction of some thermal couples leads to abnormal measured results, like FT5 in Fig. 10.31C and FT1 in Fig. 10.31D. Generally, the furnace temperatures could reach the target level within 30 minutes and then maintained constantly with discrepancies less than 20 C, which could provide ideal environment for creep buckling tests. As for the creep buckling tests on Q460 steel columns, the temperature of the furnace increased linearly at the beginning of the heating phase and then the heating rate decreased gradually until the temperature approaching the target temperature. The maximum temperature difference along the height of the furnace is about 150 C during the heating phase.

10.4.5.2 Column temperature and compression load The temperature and the compressive load of Q460 and Q690 steel column specimens in the creep buckling tests are presented in Figs. 10.32 and 10.33, respectively. The temperature evolution trend of the specimen is similar to that of furnace. The temperature distributions of the cross section along the column length are almost uniform and the corresponding temperature differences are not significant for the majority of specimens. However, it was observed the temperature distribution of cross section of specimen S-2 (Q460 steel columns) was not quite uniform. A temperature fluctuation of 70 C was observed during the loading phase due to the insufficient sealing of fire protection at the top of the furnace. Once the column temperature was kept as constant, the load was applied almost linearly until it reached the specified level within about 25 minutes. The scale of

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Behavior and Design of High-Strength Constructional Steel

Figure 10.30 Furnace Temperature in creep buckling tests for Q460 steel column specimens: (A) S-1, (B) S-2, (C) S-3, (D) S-4, (E) S-5, (F) S-6.

the load is shown by the left vertical axis of each figure. The interior vertical lines in Fig. 10.32 define the three phases of the loading process, namely, loading, load maintaining, and creeping buckling (descending segment in load
time curve), in which the vertical line on the right indicates when creep buckling occurred. As demonstrated in Figs. 10.31 and 10.32, all specimens experienced creep buckling with both compressive load and temperature being kept as constant for a certain period. Once the creep buckling occurred, the load dropped quickly due to the lost load-bearing capacity of the specimen.

10.4.5.3 Lateral and axial displacement Fig. 10.34 presents the measured axial and lateral displacements of the six specimens made of Q460 steels. In general, the creep deformation was significant at

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Figure 10.31 Furnace temperature in creep buckling tests for Q690 steel column specimens: (A) N800-0.3, (B) N800-0.6, (C) W800-0.3, (D) W800-0.6, (E) N700-0.6, (F) W700-0.6.

temperature 800 C even the applied load ratio was as low as 10%. In each figure the curves can be divided into three stages: heating, loading and creeping. In the heating stage, the axial displacement increased due to thermal expansion of the specimens. Once the temperature reached target level and kept constant, the axial displacement of the specimen became almost constant because of the completion of the thermal expansion. The lateral displacement for most specimens in this stage

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Behavior and Design of High-Strength Constructional Steel

Figure 10.32 Temperature and compression load of Q460 steel column specimens: (A) S-1, (B) S-2, (C) S-3, (D) S-4, (E) S-5, (F) S-6.

was negligible due to the uniform temperature distribution in the column and little thermal bending. In the loading stage, the axial displacement decreased gradually due to compression of the column and the lateral displacement increased gradually due to the second-order effect. The displacements in this stage were not obvious because the load ratio was very low. In the creeping stage, the lateral displacement increased gradually in the beginning and became significantly later. In this stage the deformation of the specimen was mainly caused by the creep strain developed in the steel column. Fig. 10.35 presents the measured lateral displacements of the six specimens made of Q690 steels. The lateral deflection curves of each specimen followed the similar variation trends to that of Q460 steel columns, wherein the boundary (red line) indicates beginning of creep buckling phase, that is, the axial load reaches a constant level. The change of lateral deflection is not obvious before the boundary (heating phase and load increasing phase), then lateral deflection increases

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Figure 10.33 Temperature and compression load of Q690 steel column specimens: (A) N800-0.3, (B) N800-0.6, (C) W800-0.3, (D) W800-0.6, (E) N700-0.6, (F) N700-0.6.

gradually and accelerates sharply when increasing to L/20, which can be seen as a failure criterion of creep buckling test. The evolution of axial displacements of Q690 steel column specimens in four directions (D1
D4 as plotted in Fig. 10.29B) and average values are presented in Fig. 10.36. In the test, slight rotation would occur to transfer girder, which leads to a larger discrepancy between D1 and D3 than that between D3 and D4. As to N700-0.6, some reversed changes may result from the loosening of bolts attached on the plate, which was used to fix the hinge support. Due to the axial displacement in D3 beyond the range of LVDT, the mean value of D2 and D4 was taken as average in Fig. 10.36E. Therefore average of axial displacements is regarded as analysis

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Behavior and Design of High-Strength Constructional Steel

subject in later content, more consistent with realistic axial displacement of steel column. It can be seen that, in the early stage, the evolution of axial displacement undergoes the expansion phase due to heating temperature and slight decrease caused by the increasing applied load. When reaching to creep buckling phase, axial displacements decrease with a slow and steady rate and then speed up until failure. The general variation trends of axial displacement
time curves of Q690 steel column specimens are quite close to that of Q460 steel column specimens.

Figure 10.34 Displacement
time curves of Q460 steel column specimens: (A) S-1, (B) S-2, (C) S-3, (D) S-4, (E) S-5, (F) S-6.

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Figure 10.35 Lateral deflection versus time curve of Q690 steel column specimens: (A) N800-0.3, (B) N800-0.6, (C) W800-0.3, (D) W800-0.6, (E) N700-0.6, (F) W700-0.6.

10.4.5.4 Creep bucking time In these investigations the creep buckling time is defined as the period between the time when the applied load reaches the specified level and the time when the specimen column failed by creep buckling. In order to compare the evolutions of both axial and lateral displacements among the specimens, the time
displacement relationships of the Q460 steel column specimens are presented in Fig. 10.37. Since the creep buckling time of individual specimen varies with a large range between 21.5 and 1643 minutes, the natural logarithm of time is adopted to present the scale of X coordinate axis. The creep buckling time of each column was also shown in

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Behavior and Design of High-Strength Constructional Steel

Figure 10.36 Axial displacement versus time curves of Q690 steel column specimens: (A) N800-0.3, (B) N800-0.3, (C) W800-0.3, (D) W800-0.6, (E) N700-0.6, (F) W700-0.6.

Fig. 10.37. First, it can be observed that the creep buckling time increases as the decrease of the load ratio. For example, the cross section, length, and temperature of specimen S-3 and S-4 were similar but the load ratio was different, respectively 0.12 and 0.09. The corresponding creep buckling time of specimen S-3 and S-4 was 56.5 and 115 minutes, respectively. Second, the creep buckling time increases as the decrease of the slenderness ratio of the specimen. For example, the cross section, load ratio, and temperature of specimen S-1 and S-3 were similar but the

Behavior of high-strength steel columns under and after fire

457

slenderness ratio was different, respectively 86 and 62. The creep buckling time of S-1 and S-3 was 21.5 and 56.5 minutes, respectively. The comparison on the creep buckling time between specimen S-5 and S-6 at temperature 600 C resulted in a similar conclusion. Third, the buckling time decreases as the elevation of temperature. This trend can be illustrated by a comparison between specimen S-3 and S-6 as all the parameters such as the cross section, length, and load ratio are similar for both specimens except the target temperature is 800 C and 600 C, respectively. The creep buckling time of S-3 and S-6 is 56.5 and 1643 minutes, respectively. As shown in Fig. 10.38, similar curves were constructed to express the relationship between axial displacement and creep buckling time for Q690 steel column specimens. Those curves demonstrate quite close variation trends to that of Q460 steel column specimens as shown in Fig. 10.37. As for Q690 steel column specimens, a decrease in the load ratio (e.g., from W800-0.6 to W800-0.3), or slenderness ratio (e.g., from N700-0.6 to W700-0.6), or temperature (e.g., from W800-0.6 to W700-0.6) resulted in an increase of creep buckling time. These variation trends were similarly found for Q460 steel column specimens as shown in Fig. 10.37.

Figure 10.37 Comparison of creep buckling time of Q460 steel column specimens: (A) lateral displacement
creep time curve, (B) axial displacement
creep time curve.

Figure 10.38 Comparison of creep buckling failure time of Q690 steel column specimens. (A) axial displacement
creep time curve, (B) lateral displacement
creep time curve.

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Behavior and Design of High-Strength Constructional Steel

Figure 10.39 Failure pattern of the Q460 steel column specimens: (A) S-1, (B) S-2, (C) S-3, (D) S-4, (E) S-5, (F) S-6.

10.4.5.5 Failure patterns and visual observation The failure patterns of the specimens made of Q460 and Q690 steels are illustrated in Figs. 10.39 and 10.40, respectively. All steel columns buckled globally and eventually failed about the weak axis and no local buckling was observed. The maximum bending appeared in the mid-height of the column where the initial imperfection is the largest. The failure mode has no relation with load ratio, temperature. The lateral displacement of all specimens associated with creep bucking exceeded 135 mm which was 1/20 of the column length.

10.5

Creep buckling prediction of high-strength steel columns at elevated temperatures

According to the studies in the previous sections, the load-bearing capacity of highstrength steel columns may be overestimated without considering creep of steels at elevated temperatures. Therefore it is necessary to determine the effect of creep on the fire resistance of high-strength steel columns quantitatively. This section presents both numerical and theoretical approaches to calibrating creep buckling capacity of Q690 high-strength steel columns. Parametric studies were performed based on a theoretical approach to investigate the effect of temperature level, slenderness ratio, and material strength on the creep buckling behavior of high-strength

Behavior of high-strength steel columns under and after fire

459

Figure 10.40 Failure pattern of the Q690 steel column specimens: (A) N690-700-0.6, (B) W690-700-0.6, (C) N690-800-0.3, (D) N690-800-0.6, (E) W690-800-0.3, (F) W690-800-0.6.

steel columns. Finally, by using the proposed theoretical analysis approach, modification factors for the creep buckling load were proposed based on EC3 (2005).

10.5.1 Numerical prediction of creep buckling test on Q690 highstrength steel column 10.5.1.1 Numerical model of Q690 high-strength steel column Creep buckling tests on Q690 high-strength steel columns in section 10.4 were used to validate the numerical prediction of its creep buckling behavior. A numerical model was created in [17] to simulate the tested column, as shown in Fig. 10.41. A three-dimensional four-node doubly curved shell element (S4R) was used to model the column, which is capable of simulating the buckling failure mode with reasonable accuracy and saves more computational time compared to solid elements. The selection of mesh size was based on a mesh sensitivity study, and it is approximately 20 mm in this example. As for the boundary condition, axial and transverse movements of the bottom of the column were restrained, while axial displacement of the top end was unrestrained. The external load was applied to the top end of the column.

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Behavior and Design of High-Strength Constructional Steel

Figure 10.41 On-site column specimen and numerical model of Q690 steel columns: (A) on-site column specimen in furnace, (B) numerical model.

10.5.1.2 Creep model of Q690 steels To simulate the effect of creep, a theoretical model of creep strain was input in the numerical model, as shown in Eqs. (10.11)
(10.13). The theoretical creep model is denoted as a three-stage creep model and calibrated from the creep tensile test results reported by Li et al. [18] as follows: εcp 5 c1 αc2 ðT 2Ta Þc3 t1=2 ð0 # t # t1 Þ εcp 5

ðt  t1



c6 c4 α exp 2 T 2 Tb c5

(10.11)

 dt 1 ε1 ðt1 , t # t2 Þ

  c9 c10 εcp 5 c7 αc8 exp 2 2 1 ε2 ðt2 , tÞ T t 2 t2 1 c11

(10.12)

(10.13)

where εcp is the creep strain; α is the stress ratio (applied stress divided by the yield stress at temperature T). Note that when the temperature is 800 C, α is modified to consider the change in steel microstructure. The modification formula of α is explained in the following section; T is the exposure temperature for steel in  C; t is the time in minutes; t1 is the demarcation time point between the primary and secondary stages of creep as shown in Fig. 5.11; t2 is the demarcation time point between the secondary and tertiary stages of creep; ε1 and ε2 are the creep strains at the end of the primary and secondary creep stages, respectively;

Behavior of high-strength steel columns under and after fire

461

Table 10.10 Three-stage creep model parameters of high-strength steel calibrated by Li et al. [18]. Creep parameters 

21

20.5

c1 ( C min c2 c3 c4 (min21) c5 c6 ( C) c7 c8 c9 ( C) c10 (min) c11 (min) Ta ( C) Tb ( C)

)

Q550 steel

Q690 steel

Q890 steel

1.31E-03 3.05 0.328 0.00246 4.19 2 2.53 3.4 2 5.47 2 552 850 105 399.97 400.54

1.10E-06 2.64 1.52 0.03 4.34 832.41 1.96E 1 10 4.81 3545.6 10,064 466.23 362.32 295.51

2.23E-05 2.57 1.03 2.29 3.97 4290 5150 3.55 1996 1078 117.85 391.08 43.23

Table 10.11 Elastic modulus and yield strength of Q690 steel at high temperatures (MPa). Temperature ( C) 400 550 700 800

Elastic modulus

Yield strength (f1  0,T)

192,920 155,720 58,110 33,280

669.2 354.3 67.7 47.2

Ta andTb ; andc1
c11 are the parameters in the creep model, which should be calibrated from test results. The calibration results are shown in Table 10.10.

10.5.1.3 Material properties of Q690 steel columns The temperature-dependent material properties of Q690 steel as listed in Table 10.11 were used in the model. Based on the properties of elastic modulus and yield strength, true stress
strain curves were calibrated from the results of hightemperature tensile tests [19] and were input to the model, as shown in Fig. 10.42.

10.5.1.4 General analysis steps Before performing numerical creep buckling analysis in ABAQUS, several other types of analysis are required (i.e., linear and nonlinear buckle analysis). First, the buckling mode should be determined using the “linear perturbation buckle analysis” approach, in which the nonlinearity of material and geometry is not modeled. The mode shape characterized by weak axis buckling is obtained, and imperfection in terms of a fraction of the column length is introduced into the subsequent nonlinear buckling analysis in ABAQUS, in which both material and geometry nonlinearity

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Behavior and Design of High-Strength Constructional Steel

Figure 10.42 True stress
strain curves of Q690 steel at 700 and 800 C.

are modeled. Second, nonlinear buckling analysis is conducted to calibrate the time-independent buckling load of columns where no creep effect is considered. In the creep buckling analysis, an external constant load is applied to the column with a loading level lower than the time-independent buckling load. In addition to the material constitutive model, the creep behavior of materials in Eqs. (10.1)
(10.3) must be input into the material properties. Moreover, to consider the time effect on creep buckling behavior, a specific analysis approach named “Visco” in ABAQUS is utilized, in which the actual time effect can be modeled in the analysis steps. According to the creep buckling tests, the time-independent buckling load of the column is approximately 285 and 183 kN at temperatures of 700 C and 800 C, respectively. At 700 C in the creep buckling test, one load level of 171 kN (approximately 60% of the time-independent buckling load) was applied to the columns. At 800 C, two load levels of 50 and 100 kN (representing 30% and 60% of the timeindependent buckling load) were applied to the columns, respectively. These values of load levels and temperature levels were simulated in the finite element model.

10.5.1.5 Validation against tests results A comparison of displacement
time curves from experiments and numerical analyses (with and without creep effect) is shown in Fig. 10.43A
D. If the creep effect was not considered in the numerical model, the lateral deflection and axial deformation of columns under a load level lower than the time-independent buckling load was very small and no buckling occurred. If the creep effect was included, the column buckled at 225 minutes for a given temperature level of 800 C and load level of 50 kN. However, for a higher load level of 100 kN, the column buckled after 90 minutes, indicating that the larger the load level the sooner the buckling occurs. Fig. 10.44 shows a comparison of the creep buckling mode from experiments and numerical analysis, and Table 10.12 summarizes the measured and predicted buckling time at each load level. It is clear that the predicted creep buckling modes and

Behavior of high-strength steel columns under and after fire

463

buckling time from the numerical analysis were in a reasonable agreement with those obtained from the experiments since the error for buckling time was within 10%. According to the above experimental and numerical results, it can be concluded that the creep buckling loads of steel columns are time-dependent. To demonstrate the reduction of the buckling load due to creep more effectively, the creep buckling load factor (ϕcp), defined as the ratio of creep buckling load to time-independent buckling load at a given temperature, was plotted against the time of exposure to temperatures, as shown in Fig. 10.43E. To build up creep buckling load factor
time curves from the numerical analysis and experiments, it is required to

Figure 10.43 Comparison of deflection
time curves and creep buckling load factor
time curves from numerical analysis and experiments for Q690 steel columns: (A) 800 C—30% time-independent buckling load, (B) 800 C—60% time-independent buckling load, (C) 700 C—60% time-independent buckling load, (D) Creep effect on axial displacement, (E) Creep buckling load factor
time curves at 800 C.

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Behavior and Design of High-Strength Constructional Steel

perform a series of creep buckling tests and finite element analyses at each temperature and load level. In the test and numerical analyses, to avoid buckling failure at an initial time, the load level applied to the column was lower than the timeindependent buckling load at a given temperature. Subsequently, the buckling time as shown in Fig. 10.43A
D and Table 10.12 was determined when the variation rate of lateral deflection and axial deformation of columns had a very abrupt increase compared with that at the previous time step. The creep buckling load factor was obtained by calibrating the ratio of the applied load level and the timeindependent buckling load obtained from short-time buckling tests. The creep buckling load factor
time curves in Fig. 10.43E show that the creep of steels imposes a significant influence on the buckling load of steel columns subjected to long-time exposure of high temperature. By considering the effect of creep, the buckling load

Figure 10.44 Comparison of creep buckling mode from numerical analysis and experiments for Q690 steel columns: (A) 800 C—30% time-independent buckling load, (B) 800 C—60% time-independent buckling load, (C) 700 C—60% time-independent buckling load.

Table 10.12 Comparison of buckling time from numerical analysis and experiments for Q690 steel columns. Temperature ( C)—load level (kN) Buckling time (min) Errors of buckling time

Numerical results—no creep Numerical results—creep Experimental results (predicted results—measured results)/measured results (%)

700
171

800
50

800
100

N 1030 1000 3

N 220 200 10

N 80 75 6.7

Behavior of high-strength steel columns under and after fire

465

of columns may be reduced to approximately 60% of its initial load capacity after approximately 50 minutes.

10.5.2 Numerical prediction of creep buckling test on ASTM A992 steel column 10.5.2.1 ASTM A992 steel column model in ABAQUS and experiment The numerical model and prediction approach were further validated against tests on normal-strength steel columns with a yield strength lower than 460 MPa. Morovat [20] performed experimental studies on creep buckling behavior of W4 3 13 steel columns with a length of 1295 mm, as shown in Fig. 10.45A. The column has a cross section of 103 mm 3 106 mm 3 8.76 mm 3 7.11 mm (width 3 depth 3 flange thickness 3 web thickness). The column was pin-ended and was made of American structural steels ASTM A992 with a yield strength of approximately 345 MPa. The time-independent buckling load of the column was first determined at a target temperature of 600 C and 700 C, respectively. This was achieved by heating the column to the target temperature followed by loading it until buckling occurred. Subsequently, the creep buckling tests were performed where the column was heated to the same target temperature and was then imposed with a constant load lower than the corresponding time-independent buckling load. The lateral deflection
time curves of the columns were recorded. A numerical model was created to simulate the column specimen, as shown in Fig. 10.45B. The

Figure 10.45 On-site column specimen and numerical model of ASTM A992 steel columns: (A) ASTM A992 steel column in tests [20], (B) ASTM A992 steel columns in simulation.

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Behavior and Design of High-Strength Constructional Steel

same element type and boundary condition as those in Section 10.5.1 were used in this case.

10.5.2.2 Creep model of ASTM A992 steels The creep strain model in the form of a power-law function as shown in Eq. (10.14) was used in this study for ASTM A992 steels. The creep parameters (A, m, and n) in the equation were calibrated from creep strain
time curves of A992 steels at the different stress ratios and temperature levels (see Fig. 10.46A) reported by Morovat et al. [21]. The results are shown in Table 10.13. εcp 5

A σn t11m 11m

(10.14)

where A is the power-law multiplier parameter; n is the stress order parameter; σ is the stress applied to material; t is the time of exposure to load and high temperature; and m is the time order parameter.

10.5.2.3 Material properties of ASTM A992 steel The material properties of ASTM A992 steels, including the elastic modulus and true stress
strain at 600 C and 700 C, were calibrated from the tensile test results of ASTM A992 steels at the elevated temperatures reported by Lee et al. [22]. Note that the true stress
strain curves shown in Fig. 10.46B were not calibrated directly

Figure 10.46 Material properties of ASTM A992 steel columns input in ABAQUS: (A) Creep strain
time curves of ASTM A992, (B) True stress
strain curves of ASTM A992.

Table 10.13 Creep model parameters for ASTM A992 input to ABAQUS. Temperature ( C) 600 700

A

n

m

3.76E 2 7 7.47E 2 9

1.46 2.70

2 0.87 2 0.41

Behavior of high-strength steel columns under and after fire

467

from tensile test results. The engineering stress
strain curves obtained from tensile tests were converted into true stress
strain curves before the onset of necking using Eqs. (10.15) and (10.16). However, due to the complicated stress state, the true stress
strain curves after the onset of necking cannot be calculated from these equations and were determined from the approach reported by Cai et al. [23]. σT 5 σe ð1 1 εe Þ

(10.15)

εT 5 lnð1 1 εe Þ

(10.16)

where εe is the engineering strain; σe is the engineering stress; εT is the engineering strain; and σT is the true stress.

10.5.2.4 General analysis steps A similar numerical analysis approach to that in Section 10.5.1 was used herein. According to the experimental results, the time-independent buckling loads of the column specimen are approximately 271 and 138 kN at temperatures of 600 C and 700 C, respectively. In experimental and numerical creep buckling studies, to scrutinize the relationship between buckling time and buckling load at high temperatures, four load levels of 258, 249, 245, and 240 kN were applied to the W4 3 13 columns at 600 C, while four load levels of 129, 116, 102, and 94 kN were applied to the columns at 700 C. The same temperature and load levels were also simulated in the numerical model.

10.5.2.5 Validation against test results Fig. 10.47A
D and Table 10.14 demonstrate the comparison of maximum lateral deflection
time curves from experiments and numerical analyses. The lateral deflection
time curves and buckling time predicted from the numerical analyses were relatively close to those obtained from experiments, because the difference between the predicted and measured buckling time is within 11.8%. It is also observed that, without the creep effect, the lateral deflection and axial deformation of columns under the load level lower than the time-independent buckling load were very small and buckling was not observed in the simulated columns. With the creep effect, the column buckled at a load level lower than the time-independent buckling load. At 600 C and under a load of 258 kN, the column buckled after 10 minutes. However, when a load of 245 kN was applied to the column, it buckled after 55 minutes. Fig. 10.47E and F depicts a comparison of the creep buckling mode from experiments and numerical analyses. Due to the lack of creep buckling mode from the test at 700 C, only a comparison of buckling modes at 600 C is demonstrated. The creep buckling load factor
time curves shown in Fig. 10.47G and H were plotted using a similar approach to that demonstrated in Section 10.5.1. It is shown that creep has a significant influence on the buckling load of steel

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Behavior and Design of High-Strength Constructional Steel

Figure 10.47 Comparison of creep buckling behavior of ASTM A992 steel columns between tests and simulations: (A) Comparison of lateral deflection
time curves—600 C (ASTM A992), (B) Comparison of lateral deflection
time curves—700 C (ASTM A992), (C) Comparison of axial deflection
time curves—600 C (ASTM A992), (D) Comparison of axial deflection
time curves—600 C (ASTM A992), (E) Creep buckling mode of test— 600 C (ASTM A992 steel column), (F) Creep buckling mode of simulation—600 C (ASTM A992 steel column), (G) Creep buckling load factor
time curves at 600 C (ASTM A992 steel column), (H) Creep buckling load factor
time curves at 700 C (ASTM A992 steel column).

Behavior of high-strength steel columns under and after fire

469

Table 10.14 Comparison of buckling time obtained from computational analysis and experiments for ASTM A992 steel columns. Temperature ( C)

600

Load level (kN) Buckling time (min)

Errors of buckling time

Numerical results—no creep Numerical results— creep Experimental results (predicted results— measured results)/ measured results (%)

700

258

249

245

240

129

116

102

94

N

N

N

N

N

N

N

N

9

11

52

95

11

25

60

130

10

10

50

85

10

27

55

120

10

10

4

11.8

10

7.4

9.1

8.3

columns and the creep buckling load is time-dependent. The comparison again validates the accuracy and reasonability of numerical predictions.

10.5.3 Theoretical study on creep buckling behavior of steel columns As a more straightforward method compared with numerical and experimental studies, a theoretical approach was conducted in this section, which is capable of simultaneously constructing the creep buckling load factor
time curves at different temperature levels. On one hand, the creep buckling load factor of steel columns with an initial imperfection and creep effect was derived based on a combination of tangent modulus theory reported by Morovat [20] and Perry’s formula. On the other hand, the secant modulus theory was also utilized to calibrate the creep buckling load factor of steel columns and was compared with that calibrated from the tangent modulus theory.

10.5.3.1 Theoretical formulation As shown in Fig. 10.48, with an initial imperfection, the shape of the steel columns can be expressed as the sinusoidal equation, as follows: y0 5 δ0 sin

πx l

(10.17)

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Behavior and Design of High-Strength Constructional Steel

where y0 is the initial bending along the length of steel columns; δ0 is the initial deflection at the middle height of steel columns; l is the effective length of steel columns. Then, the bending deformation shape due to axial compression can be derived as follows:     δ0 πx y5 (10.18) sin l ð1 2 N=NET Þ where N is the axial load applied to column; NET is the Euler buckling load at high temperature and is further calibrated from the following equations: NET 5 σET A

(10.19)

π 2 ET λ2

(10.20)

σET 5

where σET is the critical stress due to Euler buckling at a given temperature; ET is the elastic modulus at high temperature; A is the gross cross-sectional area of steel column; and λ is the slenderness ratio of the steel column. It is assumed that the maximum deflection (δmax ) occurs at the middle height of the column, which is therefore expressed as: δmax 5

δ0 ð1 2 N=NET Þ

Figure 10.48 critical stress calculation model for axial compression steel members.

(10.21)

Behavior of high-strength steel columns under and after fire

471

Due to the bending of the columns, the maximum stress applied to the cross section of columns is generated not only by the external load (N) but also by the bending moment (M). It is the multiplication of external load (N) and maximum deflection (δmax ), which is expressed as σmax 5

Nδmax N 1 A W

(10.22)

where W is the section modulus of column. Then, substituting Eq. (10.21) into Eq. (10.22) yields Eq. (10.23) as:   N δ0 A=W 11 σmax 5 A 1 2 N=ðAσET Þ

(10.23)

It is assumed that: e0 5 δ0 A=W

(10.24)

Then, Eq. (10.23) can be converted into Eq. (10.25) as σmax 5

  N e0 11 A 1 2 N=ðAσET Þ

(10.25)

It is then stipulated that the column approaches its ultimate limit state when the maximum stress in the column reaches its yield strength. This assumption is expressed as: ! fy;T 5 σcr;T

e0



11 1 2 σcr;T =σET

(10.26)

where σcr;T is the critical stress of columns at the ultimate limit state, without considering the effect of creep; fy;T is the yield strength of columns at high temperatures. To determine the critical stress (σcr;T ), Eq. (10.26) is transformed into Eq. (10.27) as:   q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 1 ð1 1 e0 ÞσET σET 1 fy;T 2 ð11e0 ÞσET σET 1fy;T 2 ½4fy;T σET σET  σcr;T 5 2 (10.27) The equation is also defined as Perry’s formula for calculating the critical stress of axially compressed columns under high temperatures without considering any creep effect.

472

Behavior and Design of High-Strength Constructional Steel

10.5.3.2 Tangent modulus approach By combining the tangent modulus theory with Perry’s formula, the hightemperature critical stress of axially compressed columns can be obtained as follows. The total strain generated by creep deformation and stress applied to a column is expressed as: ε 5 εcp ðσ; tÞ 1 εσ ðσ; tÞ

(10.28)

where ε is the total strain; εcp ðσ; tÞ is the creep strain caused by creep deformation; and εσ ðσ; tÞ is the strain caused by stress. Taking the derivation of stress (σ) of both sides of Eq. (10.28) yields: @εcp @ε @εσ 5 1 @σ @σ @σ

(10.29)

According to the tangent modulus theory, tangent modulus considering the creep effect is expressed as EcpT 5

@σ @ε

(10.30)

where EcpT is the tangent modulus with consideration of creep effect. Substituting Eq. (10.30) into Eq. (10.29) yields EcpT 5

E T

1 1 ET @εcp =@σ

(10.31)

When the maximum stress in a column approaches its critical stress, EcpT 5

ET

 1 1 ET @εcp =@σ σ5σcr;cpT

(10.32)

where σcr;cpT is the critical stress considering creep effect. According to the tangent modulus theory, we have: σcpET EcpT 5 σET ET

(10.33)

Substituting Eq. (10.32) to Eq. (10.33) yields σcpET 5

σET

 1 1 ET @εcp =@σ σ5σcr;cpT

(10.34)

Behavior of high-strength steel columns under and after fire

473

where σcpET is the Euler buckling stress with creep effect. Considering the creep behavior of materials, Perry’s formula expressed by Eq. (10.35) can be then modified as follows:  ffi q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 1 ð1 1 e0 ÞσcpET 1 fy;T 2 ð11e0 ÞσcpET 1fy;T 2 4fy;T σcpET 2

σcr;cpT 5

(10.35) where σcpET is calibrated from Eq. (10.23).

10.5.3.3 Secant modulus approach If the secant modulus is utilized to calibrate the critical stress of columns with creep effect, both sides of Eq. (10.28) will be divided by stress (σ) as εcp ε εσ 5 1 σ σ σ

(10.36)

The secant modulus (EcpS ) considering creep effect is expressed as EcpS 5

σ ε

(10.37)

Substituting Eq. (10.37) into (10.36) yields EcpS 5

ET

1 1 ET εcp =σ

(10.38)

When the column approaches its ultimate limit state, the second modulus with the creep effect is expressed as EcpS 5

E T

1 1 ET εcp =σcr;cpT

(10.39)

According to the secant modulus theory, σcpET EcpS 5 σET ET

(10.40)

By substituting Eq. (10.39) into (10.40), Euler buckling stress with creep effect is calibrated as follows: σcpET 5

σ ET

1 1 ET εcp =σcr;cpT

(10.41)

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Behavior and Design of High-Strength Constructional Steel

Based on the secant modulus theory, substituting Eq. (10.41) into Eq. (10.35) yields the critical stress in the column with creep effect as: σcr;cpT 5

 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi

2 1 ð1 1 e0 ÞσcpET 1 fy;T 2 ð11e0 ÞσcpET 1fy;T 2 4fy;T σcpET 2 (10.42)

where σcpET is calibrated from Eq. (10.41). A theoretical study on the steel column models similar to those in the numerical and experimental study was performed herein. The tangent modulus and secant modulus approach were utilized, respectively, together with Perry’s formula to calibrate the critical strength at buckling considering the creep effect. The creep strain of a column (εcp ) was determined from the three-stage creep model shown in Eqs. (10.11)
(10.13). By solving Eqs. (10.35) and (10.42) in Matlab, the critical stress of the column was obtained. To keep results consistent with those obtained from numerical study and tests, instead of critical stress, the creep buckling load factor (ϕcp ) was calibrated and plotted against the time of exposure to temperatures as ϕcp 5

σcr;cpT σcr;T

(10.43)

where σcr;cpT is the critical buckling strength of a column with creep effect calibrated from Eqs. (10.34), (10.35) or (10.41), (10.42) along with the three-stage creep model shown in Eqs. (10.11)
(10.13), σcr;T is the critical buckling strength of column with creep effect calibrated from Eq. (10.27).

10.5.3.4 Validation of theoretical results against experimental results Fig. 10.49 presents the comparison of the creep buckling load factor
time curves from the experiments in Sections 10.5.1 and 10.5.2 and the proposed theoretical approach. Overall, there were limited differences (within 15%) between the theoretical analyses results and test results, indicating the reasonability of the theoretical approach. It was also found that the buckling load was time-dependent if the creep effect was considered, and the creep buckling load factor decreased with exposure time to high temperatures. In addition, the creep buckling load factor predicted from the theoretical analyses was smaller than those from experiments. This is probably due to the difference in determination of the failure criterion from each approach. In the theoretical study, the buckling time is determined as the time when the maximum compressive stress at the farthest location from the neutral axis of a column’s cross section just approaches the yield strength of steel at a given temperature, and the remainder of the cross-sectional zone may still be inelastic. However, in the experimental study, buckling is achieved when the rate of change of maximum lateral deflection in a column increased suddenly, and at that point a relatively

Behavior of high-strength steel columns under and after fire

475

Figure 10.49 Comparison of creep buckling behavior from experimental and theoretical studies: (A) ASTM A992 steel column—600 C, (B) ASTM A992 steel column—700 C, (C) Q690 steel column—800 C.

large plastic zone may have already been generated in its cross section. Therefore the creep buckling load factor determined in the experimental study is relatively larger than that determined in the theoretical study. Moreover, at a given time, the creep buckling load factor predicted in the theoretical analysis based on the tangent modulus theory is smaller than that based on the secant modulus theory, and the results based on the secant modulus theory are in better agreement with the test results.

10.5.3.5 Validation of theoretical results against numerical simulation Due to the lack of sufficient creep buckling test results for high-strength steel columns, the accuracy of the proposed theoretical approach for high-strength steel columns was further validated against validated numerical simulations of creep buckling analyses on Q550, Q690, and Q890 high-strength steel columns. For the three high-strength steel columns, it was assumed that they all had a length of 3860 mm, a cross-sectional dimension of 200 mm 3 200 mm 3 14 mm 3 14 mm, and an initial imperfection of approximately 1/1000 of the column length. The material properties of Q550, Q690, and Q890 steels (including yield strength and elastic modulus) are shown in Tables 10.11, 10.15 and 10.16.

476

Behavior and Design of High-Strength Constructional Steel

Table 10.15 Elastic modulus of high-strength steels (MPa). T ( C) 400 550 700 800

Q550

Q890

218,680 135,030 48,660 34,160

200,850 105,400 50,270 30,630

Table 10.16 Yield strength of high-strength steels in terms of f1  0,T (MPa). T ( C)

Q550

Q890

400 550 700 800

521.0 261.6 55.1 44.9

765.0 399.6 70.6 48.0

The creep behavior of the three high-strength steels employed in both the numerical and theoretical analyses is expressed in Eqs. (10.11)
(10.13), denoted as a threestage creep model [18]. The procedure for determining the creep buckling load factor
time curves from the numerical analysis is similar to that in Sections 10.5.1 and 10.5.2. In the theoretical analysis, the creep buckling load factor
time curves were calibrated from Perry’s formula, secant modulus theory, and tangent modulus theory expressed by Eqs. (10.43), (10.35), (10.27), respectively. Fig. 10.50 shows the comparison of creep buckling load factor
time curves from the numerical analysis and calibrated from the theoretical study. Overall, there were maximum differences (within 12.5%) between the theoretical analysis results and numerical results, indicating the reasonability of the theoretical analysis approach for calibrating the creep buckling load of high-strength steel columns. It was found that, at a given time, the creep buckling load factor predicted from the theoretical analysis based on the tangent modulus theory was smaller than that based on the secant modulus theory. The results based on the secant modulus theory were relatively close to the numerical results. In addition, for all three high-strength steel columns, the creep buckling load factor decreased with the increasing time of exposure to high temperatures.

10.5.4 Parametric study for creep buckling behavior of highstrength steel columns The creep buckling load of high-strength steel columns may be affected by several other key factors, such as temperature, the strength of steel, and the slenderness ratio. Since the theoretical approach based on the secant modulus theory predicted better results compared to those based on the tangent modulus theory, it is utilized

Behavior of high-strength steel columns under and after fire

477

Figure 10.50 Comparison of creep buckling behavior from numerical and theoretical studies: (A) Q550 steel column, (B) Q690 steel column, (C) Q890 steel column.

herein to investigate the effect of these key factors on the creep buckling behavior of high-strength steel columns. The high-strength steel columns used for the parametric study have dimensions of 200 mm 3 200 mm 3 14 mm 3 14 mm, slenderness ratios in the range of 30 to 160, and an initial imperfection of approximately 1/1000 of the column length. For each slenderness ratio, four temperature levels (400 C, 550 C, 700 C, and 800 C) and three types of high-strength steels (Q550, Q690, and Q890) were analyzed, respectively. The material properties of these high-strength steels are listed in Tables 10.11, 10.15, and 10.16. In the parametric study the creep buckling load factor
time curves can be determined directly by solving the proposed three-stage creep model and Perry’s formula described in Section 10.5.3.

10.5.4.1 Effect of slenderness ratio Fig. 10.51 presents a summary of the creep buckling load factor
time curves of steel columns for different slenderness ratios within the range of 30
160 at given temperature levels. It shows that the difference in creep buckling load factor between the different slenderness ratios at 400 C was relatively small compared with that at other temperatures. At a given time, there was no unified variation

Figure 10.51 Slenderness ratio on creep buckling behavior of high-strength steel column: (A) Q550—400 C, (B) Q550—550 C, (C) Q550—700 C, (D) Q550—800 C, (E) Q690— 400 C, (F) Q690—550 C, (G) Q690—700 C, (H) Q690—800 C, (I) Q890—400 C, (J) Q890—550 C, (K) Q890—700 C, (L) Q890—800 C.

Behavior of high-strength steel columns under and after fire

479

trend of the creep buckling load factor against the slenderness ratio. For instance, the creep buckling load factor decreased with the slenderness ratio up to 50 or 80, and then it increased with the slenderness ratio. When the slenderness ratio of the columns increased up to a certain level, the creep buckling load factor increased correspondingly, even approaching 1.0. This indicates that the effect of creep on the buckling behavior of columns became slight. This could be because elastic buckling controls the failure mode of steel columns with large slenderness ratios. Therefore plastic deformation, including creep deformation, does not develop in steel columns before the occurrence of elastic buckling.

10.5.4.2 Effect of temperature levels Fig. 10.52 depicts the representative comparison of the creep buckling load factor
time curves for different temperature levels. It was found that the curves for Q550 steel columns were relatively close when the temperature level was higher than 400 C. This indicates that temperature levels higher than 400 C have a limited effect on the creep buckling load factor of Q550 steel columns. However, for both the Q690 and Q890 steel columns, the creep buckling load decreased with increasing temperature.

10.5.4.3 Effect of material strength Fig. 10.53 summarizes the creep load buckling factor
time curves with different steel strengths. It shows that, at relatively low-temperature levels (#550 C), the buckling load factor due to creep decreased correspondingly with the decrease in material strength of the column at a given time point. However, at higher temperature levels ( . 550 C), the creep buckling load factor decreased with increasing material strength, indicating a higher reduction of buckling loads for higher strength steel columns at high temperatures.

10.5.5 Creep buckling load factor for current codes of the practices From Section 10.5.4 the effects of the slenderness ratio, temperature level, and material strength on the creep buckling behavior of high-strength steel columns are interrelated. Consequently, to predict the creep buckling behavior of highstrength columns, all these factors have to be considered comprehensively. In this section, to utilize the above analyses results in fire-safety design, the modified creep buckling load factor based on the current design code Eurocode 3 [24] is proposed. Eurocode 3
Section 1.2 [24] provides an approach to calibrate the timeindependent buckling load for steel columns with yield strengths up to 460 MPa.

480

Behavior and Design of High-Strength Constructional Steel

The same approach was employed herein to assess the buckling load of highstrength steel columns over 500 MPa by considering the effect of creep. Therefore the reduction factor due to buckling obtained from Eurocode 3 has to be modified to account for the creep effect. The procedure of calibration of the modified buckling load factor with creep effect based on Eurocode 3 [24] is demonstrated as next. The critical buckling strength (σcr;T ) without creep effect is calibrated as follows: σcr;T 5 χðTÞσy;T

(10.44)

Figure 10.52 Temperature level effect on creep buckling behavior of high-strength steel column: (A) Q550—λ50, (B) Q550—λ120, (C) Q690—λ50, (D) Q690—λ120, (E) Q890—λ 50, (F) Q890—λ120.

Behavior of high-strength steel columns under and after fire

481

Figure 10.53 Steel strength effect on creep buckling behavior of high-strength steel column: (A) 400 C—λ50, (B) 400 C—λ120, (C) 550 C—λ50, (D) 550 C—λ120, (E) 700 C—λ50, (F) 700 C—λ120, (G) 800 C—λ50, (H) 800 C—λ120.

482

Behavior and Design of High-Strength Constructional Steel

with, χðTÞ 5

1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ψðT Þ 1 ψðT Þ2 2 λ T

(10.45)

and   2 ψðT Þ 5 0:5 1 1 αλ T 1 λ T

(10.46)

and sffiffiffiffiffiffiffiffi 235 α 5 0:65 fy

(10.47)

The nondimensional slenderness ratio (λ T ) at high temperature T is calibrated as follows: sffiffiffiffiffiffiffiffi ky;T λT 5 λ kE;T

(10.48)

with λ5 and

λ λ1

sffiffiffiffi E λ1 5 π fy

(10.49)

(10.50)

where ky;T and kE;T are the reduction factor of yield strength and elastic modulus of steel columns at high temperatures as shown in Table 10.17; and for high-strength steel columns over 500 MPa, they can be obtained from creep tensile tests reported by Li et al. [18]; λ is the slenderness ratio; E is the elastic modulus of steel columns and fy is the yield strength of steel columns at room temperature. According to the previous studies, the creep buckling load of steel columns is time-dependent and is affected by several key factors including temperature, slenderness ratio, and material strength. Therefore the creep buckling strength considering these key factors based on Eurocode 3 [24] was proposed as follows: σcr;cpT 5 φðT; λ; tÞχðT Þσy;T

(10.51)

Behavior of high-strength steel columns under and after fire

483

with φðT; λ; tÞ 5

σcr;cpT σcr;T

(10.52)

where σy;T is the yield strength of steel columns at high temperatures as shown in Tables 10.11, 10.15, and 10.16; σcr;cpT is the critical strength of steel columns calibrated from Eqs. (10.43) and (10.35) together with the three-stage creep model shown in Eqs. (10.11)
(10.13); σcr;T is calibrated from Eq. (10.44) obtained from Eurocode 3 [24]; φðT; λ; tÞ is the Eurocode 3-based creep buckling load factor as shown in Eq. (10.52) and is a function of the slenderness ratio and the time of exposure to high temperatures. The modified buckling load factor (φðT; λ; tÞ) with creep effect is plotted against the buckling time (of the order of minutes). The representative results of the modified buckling load factor
buckling time for Q550 steel columns is shown in Fig. 10.54. Based on the results shown in Fig. 10.54 along with the parametric study results, it was found that beyond slenderness ratio levels of 50 or 80, the creep buckling factor decreased with increasing slenderness ratio, and the creep buckling load factor decreased with increasing temperature. Therefore the modified function of creep buckling load factor for EC3 (2005) can be divided into different stages based on the slenderness ratio and temperature level. It should point out that when the calibrated creep buckling load factor for EC3 is larger than 1.0, it is assumed to be equal to 1.0. As shown in Fig. 10.54, taking the Q550 steel column as an example, by performing nonlinear data regression analysis in Matlab, a theoretical formula was first derived at each slenderness ratio and temperature level. Then, for different slenderness ratios and temperatures, the modified creep buckling load factor (φ (T, λ, t)) can be calibrated as follows: As for 20 C # T # 400 C, φ400 ðλ; tÞ 5 1

(10.53)

Table 10.17 Reduction factor of yield strength and elastic modulus for Q550, Q690, and Q890 steel columns. Temperature ( C)

Reduction factor Q550

400 550 700 800

Q690

Q890

ky;T

kE;T

ky;T

kE;T

ky;T

kE;T

0.8 0.4 0.08 0.07

1 0.64 0.23 0.16

0.8 0.45 0.08 0.06

0.916 0.74 0.276 0.158

0.8 0.4 0.08 0.05

0.968 0.508 0.242 0.148

484

Behavior and Design of High-Strength Constructional Steel

Figure 10.54 EC3-based creep buckling load factor
time curves for Q550 steel column: (A) 400 C, (B) 550 C, (C) 700 C, and (D) 800 C.

φ1 ðT; λ; tÞ 5 φ400 ðλ; tÞ 5 1

(10.54)

As for 400 C # T # 550 C, for different slenderness ratios, the modified buckling factor is expressed as:  φ55021 ðλ; tÞ 5 0:0045ð50 2 λÞ 2 0:116lnðtÞ 1 1:0938ðλ # 50 (10.55)  φ55022 ðλ; tÞ 5 0:003ðλ 2 50Þ 2 0:116lnðtÞ 1 1:0938ðλ $ 50 (10.56) Therefore for different temperatures and slenderness ratios, the modified buckling factor can be further calibrated as: φ221 ðT; λ; tÞ 5

550 2 T φ400 ðλ; tÞ 2 φ55021 ðλ; tÞ 1 φ55021 ðλ; tÞðλ # 50Þ 550 2 400 (10.57)

φ222 ðT; λ; tÞ 5

550 2 T φ400 ðλ; tÞ 2 φ55022 ðλ; tÞ 1 φ55022 ðλ; tÞðλ $ 50Þ 550 2 400 (10.58)

Behavior of high-strength steel columns under and after fire

485

As for 550 C # T # 700 C, the same approach was used herein to derive the modified buckling factor as a function of slenderness ratio, temperature, and time as follows: φ70021 ðλ; tÞ 5 0:0017ð80 2 λÞ 2 0:081lnðtÞ 1 0:862ðλ # 80 φ70022 ðλ; tÞ 5 0:0017ð80 2 λÞ 2 0:081lnðtÞ 1 0:862ðλ $ 80

 (10.59)  (10.60)

φ321 ðT; λ; tÞ 5

700 2 T φ ðλ; tÞ 2 φ70021 ðλ; tÞ 1 φ70021 ðλ; tÞðλ # 50Þ 700 2 550 55021 (10.61)

φ322 ðT; λ; tÞ 5

550 2 T φ ðλ; tÞ 2 φ70021 ðλ; tÞ 1 φ70021 ðλ; tÞð50 # λ # 80Þ 550 2 400 50022 (10.62)

φ323 ðT; λ; tÞ 5

700 2 T φ ðλ; tÞ 2 φ70022 ðλ; tÞ 1 φ70022 ðλ; tÞð80 # λÞ 700 2 550 55022 (10.63)

As for 700 C # T # 800 C, the modified buckling factor can be derived as follows:  φ80021 ðλ; tÞ 5 0:0031ð80 2 λÞ 2 0:081lnðtÞ 1 0:8663ðλ # 80 (10.64) φ80022 ðλ; tÞ 5 0:0023ðλ 2 80Þ 2 0:081lnðtÞ 1 0:8663ðλ $ 80

 (10.65)

φ421 ðT; λ; tÞ 5

800 2 T φ70021 ðλ; tÞ 2 φ80021 ðλ; tÞ 1 φ80021 ðλ; tÞðλ # 80Þ 800 2 700 (10.66)

φ422 ðT; λ; tÞ 5

700 2 T φ ðλ; tÞ 2 φ80022 ðλ; tÞ 1 φ80022 ðλ; tÞð80 # λÞ 700 2 550 70022 (10.67)

where φ400(λ, t) is the modified creep buckling load factor for EC3 at 400 C; φ5501(λ, t) and φ550-2(λ, t) are the modified creep buckling load factors for EC3 (2005) at 550 C at the first stage (λ # 50) and at the second stage (λ $ 50), respectively; φ700-1(λ, t) and φ700-2(λ, t) are the modified creep buckling load factors for EC3 (2005) at 700 C at the first stage (λ # 50) and at the second stage (λ $ 50), respectively; φ800-1(λ, t) and φ800-2(λ, t) are the modified creep buckling load factors for EC3 (2005) at 700 C at the first stage (λ # 80) and at the second stage (λ $ 80); φ1(T, λ, t) is the modified creep buckling load factor for EC3 (2005) at a

486

Behavior and Design of High-Strength Constructional Steel

temperature level lower than 400 C; φ2-1(T, λ, t) and φ2-2(T, λ, t) are the modified creep buckling load factors for EC3 (2005) within a 400 C
550 C temperature range at the first stage (λ # 50) and at the second stage (λ $ 50), respectively; φ31(T, λ, t), φ3-2(T, λ, t) and φ3-3(T, λ, t) are the modified creep buckling load factors for EC3 (2005) within a 550 C
700 C temperature range at the first stage (λ # 50), at the second stage (50 # λ # 80), and at the third stage (λ $ 80), respectively; φ4-1(T, λ, t) and φ4-2(T, λ, t) are the modified creep buckling load factors for EC3 (2005) within a 700 C
800 C temperature range at the first stage (λ # 80) and at the second stage (λ $ 80), respectively; T is the temperature in the unit of  C; t is time of exposure to high temperature in the unit of minutes; and λ is the slenderness ratio. Therefore, according to the above calibration procedure, the simplified function to determine Eurocode 3-based creep buckling load factor (φ (T, λ, t)) for different column types is demonstrated as follows: For Q550 steel columns: 20 C # T # 400 C; φðT; λ; tÞ 5 1

(10.68)

400 C # T # 550 C;

φðT; λ; tÞ 5

8 0:002125T 1 ð0:012 2 0:00003T Þλ 1 ð0:3093 2 0:0007733T Þ >  > > > < lnðtÞ 1 0:15ðλ # 50 2 0:0003747T 2 ð0:008 2 0:00002T Þλ 1ð0:3093 2 0:0007733T Þ > > > > : lnðtÞ 1 1:15ðλ $ 50 (10.69)

550 C # T # 700 C; 8 2 0:0021T 2 ð0:015 2 0:000019T Þλ 2ð0:2443 2 0:00023T Þ > > > > > lnðtÞ 1 2:5ðλ # 50 > > > > > < 0:00036T 1 ð0:025 2 0:000031T Þλ 2 ð0:2443  2 0:00023T Þ φðT; λ; tÞ 5 lnðtÞ 1 0:75ð50 # λ # 80 >  > > > > 2 0:0013T 1 0:0089 2 0:000011TÞλ 2 ð0:2443 2 0:00023T Þ > > >  > > : lnðtÞ 1 1:65ðλ $ 80 (10.70) 700 C # T # 800 C;

Behavior of high-strength steel columns under and after fire

φðT; λ; tÞ 5

487

8 0:001163T 1 ð0:0081 2 0:000014T Þλ 2ð0:088 2 0:00001T Þ > > > > < lnðtÞ 1 0:18ðλ # 80 2 0:000677T 2 ð0:0049 2 0:000009T Þλ2 ð0:088 2 0:00001T Þ > > > > : lnðtÞ 1 1:22ðλ $ 80 (10.71)

where φ(T, λ, t) is the modified creep buckling load factor for EC3; T is the temperature in the unit of  C; t is time of exposure to high temperature in the unit of minutes; and λ is the slenderness ratio. For Q690 steel columns: 20 C # T # 400 C; φðT; λ; tÞ 5 1

(10.72)

400 C # T # 550 C;

φðT; λ; tÞ 5

8 2 0:000455T 1 ð0:0075 2 0:0000187T Þλ1 ð0:224 2 0:00056T Þ > > > > < lnðtÞ 1 1:18ðλ # 50 2 0:002288T 2 ð0:0072 2 0:000018T Þλ1 ð0:224 2 0:00056T Þ > > > > : lnðtÞ 1 1:92ðλ $ 50 (10.73)

550 C # T # 700 C; 8 2 0:0011T 2 ð0:0087 2 0:000011T Þλ 2 ð0:161 2 0:00014T Þ > > > > > lnðtÞ 1 1:52ðλ # 50 > > > > > < 0:000764T 1 ð0:017 2 0:000026T Þλ 2 ð0:161  2 0:00014T Þ φðT; λ; tÞ 5 lnðtÞ 1 0:24ð50 # λ # 80 >  > > > > 2 0:00046T 1 0:0086 2 0:000011TÞλ 2 ð0:161 2 0:00014T Þ > > >  > > : lnðtÞ 1 0:91ðλ $ 80 (10.74) 700 C # T # 800 C; 8 0:001075T 2 ð0:0033 2 0:000003T Þλ 1ð0:021 2 0:00012T Þ > > > > < lnðtÞ 1 0:019ðλ # 80 φðT; λ; tÞ 5 > > 0:000595T 2 ð0:0052 2 0:000009T Þλ 1ð0:021 2 0:00012T Þ > > : lnðtÞ 1 0:17ðλ $ 80 (10.75)

488

Behavior and Design of High-Strength Constructional Steel

For Q890 steel columns: 20 C # T # 400 C; ( φðT; λ; tÞ 5

0:0026ðλ 2 120Þ 1 1:0ðλ # 120 1:0ðλ $ 120Þ

 (10.76)

400 C # T # 550 C; 8 2 0:00038T 1 ð0:0103 2 0:000019T Þλ 1 >  ð0:149 2 0:00041T Þ > > > > ln ð t Þ 1 0:94ðλ # 50 > > > > < 2 0:0021T 2 ð0:0035 1 0:000015T Þλ 1 ð0:149 2 0:00041T Þ  φðT; λ; tÞ 5 lnðtÞ 1 1:63ð50 # λ # 120 > > > > > 2 0:004198T 2 ð0:013 2 0:000033T Þλ 1 ð0:149 2 0:00041T Þ >  > > > : lnðtÞ 1 2:78ðλ $ 120 (10.77) 550 C # T # 700 C; 8 0:00061T 1 ð0:0111 2 0:0000207T Þλ 2ð0:101 2 0:000047T Þ > > > > < lnðtÞ 1 0:39ðλ # 50 φðT; λ; tÞ 5 0:00078T 1 ð0:0181 2 0:000024T Þλ 2 ð0:101 2 0:000047T Þ > > > > : lnðtÞ 1 0:037ðλ $ 50 (10.78) 700 C # T # 800 C; 8 0:00131T 1 ð0:0057 2 0:000013T Þλ 2  ð0:082 2 0:00002T Þ > > > > < lnðtÞ 1 0:099ðλ # 50 φðT; λ; tÞ 5 0:00071T 1 ð0:002 2 0:000001T Þλ 2 ð0:082 2 0:00002T Þ > > > > : lnðtÞ 1 0:086ðλ $ 50 (10.79) It should be pointed out that the proposed creep buckling factor based on Eurocode 3 [24] is applicable for the Q550, Q690, and Q890 high-strength steel columns under the time of exposure to high temperatures up to 800 C within 200 minutes. Beyond these ranges, the creep buckling load factor has to be reevaluated based on the theoretical approach proposed in Section 10.5.3. In addition, the cross section of the steel column evaluated in this part is assumed to be class 1
3 for which local buckling may not affect the plastic development within the cross section of columns.

Behavior of high-strength steel columns under and after fire

10.6

489

Summary

Through discussion hereinabove, the following remarks can be summarized: 1. The restrained high-strength columns will be buckled during heating when its maximum temperature exceeded a certain temperature limit. The buckling may generate obvious residual plastic bending deformation in the column after cooling down to the ambient temperature. 2. The postfire residual capacity and axial stiffness of high-strength columns may be much lower than their initial values. This reduction is mainly due to the presence of residual bending deflection in the column after heating. 3. The maximum temperature of a restrained high-strength steel column experienced in a fire is the key influencing factor for determining its residual capacity. The residual capacity is independent of the maximum temperatures when it is lower than the buckling temperature of the column, but can be significantly reduced when the maximum temperature is over buckling temperature. Generally, the residual capacity of the column decreases proportionally with the heated maximum temperatures. 4. The increase of load ratio reduces the minimum reduction factor of the residual capacity of the restrained high-strength steel columns but has a little effect on the maximum reduction factor. However, the increase of the axial stiffness ratio increases on the maximum reduction factor of residual capacity of the restrained high-strength steel columns but has a limited effect on the minimum reduction factor. 5. Creep buckling load is time-dependent for both high-strength and normal-strength steel columns. In general, creep buckling load capacity is decreased with increase in time of exposure to high temperature. Creep effects on buckling load become remarkable at the temperature level higher than 400 C. 6. The effect of temperature on creep buckling behavior of columns varies with the grades of steel used. As for Q550 steel column, at temperature higher than 400 C, the temperature level has limited effect on the reduction of buckling load due to creep. As for both Q690 and Q890 steel column, however, with increase in temperature level, the reduction degree of buckling load due to creep is increased. 7. Steel strength effects on the creep buckling load factor of column vary with temperatures. At high-temperature level (˃550 C), with the increase in steel strength, there is more reduction of buckling load observed in higher strength steel column. However, at a relatively low-temperature level (#550 C), the opposite is observed. 8. The current code of practices may overestimate the buckling load capacity of steel columns with slenderness ratio up to 200 under long-time fire conditions.

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Behavior and Design of High-Strength Constructional Steel

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6 [in Chinese]. [14] Lyu HB. Research on mechanical properties of domestic 500MPa up high strength structural steel after fire [Ph.D. thesis]. Shanghai: Tongji University; 2017 [in Chinese]. [15] Huang L. Research on mechanical properties of domestic high strength structural steel at elevated temperatures [Ph.D. thesis]. Shanghai: Tongji University; 2017 [in Chinese]. [16] Ban HY. Research on the overall buckling behaviour and design method of high strength steel columns under axial compression [Ph.D. thesis]. Tsinghua University; 2012. [17] ABAQUS Version 6.14
4 user’s manual. Pawtucket, RI: Hibbitt, Karlsson, & Sorensen, Inc, Simulia, ABAQUS/CAE, 2015. [18] Li GQ, Wang XX, Zhang C, Cai WY. Creep behavior and model of high-strength steels over 500 MPa at elevated temperatures. J Constr Steel Res 2020;168:105989. [19] Huang L, Li GQ, Wang XX, Zhang C, Choe L, Engelhardt M. High temperature mechanical properties of high strength structural steels Q550, Q690 and Q890. Fire Technol 2018;54:1609
28. [20] Morovat M. Creep buckling behavior of steel columns subjected to fire [Ph.D. thesis]. Austin, TX: Department of Civil, Architectural, and Environmental Engineering, University of Texas at Austin; 2014. [21] Morovat M, Lee J, Engelhardt M, Taleff E, Helwig T, Segrest V. Creep properties of ASTM A992 steel at elevated temperatures. Adv Mater Res 2012;446
449:786
92. [22] Lee J, Morovat MA, Hu G, Engelhardt MD, Taleff EM. Experimental investigation of mechanical properties of ASTM A992 steel at elevated temperatures. Eng J Am Inst Steel Constr 2013;50(4):249
72. [23] Cai WY, Morovat M, Engelhardt M. True stress-strain curves for ASTM A992 steel for fracture simulation at elevated temperatures. J Constr Steel Res 2017;139:272
9. [24] BS EN1993
1
2. Eurocode 3: design of steel structures
Part 1.2: General rules
structural fire design. Brussels: European Committee for Standardization; 2005.

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11

Yan-Bo Wang, Guo-Qiang Li, Yi-Fan Lyu and Kun Chen Tongji University, Shanghai, P.R. China

11.1

Introduction

Connection is an important topic in the design of steel structures. With proper design of connections the external load will be effectively transferred through different structural members. Loss of a connection is dangerous for normal function of the whole structure, which may cause catastrophic results (e.g., progressive collapse of the whole building) in some extreme cases. Thus a careful consideration is needed for connections when it comes to the design of high-strength steel (HSS) structures [1]. Compared to the welded connection, bolted connection has two unique advantages [2
4]. First, with the application of bolted connection, the effect of heat input on the mechanical properties of HSSs can be avoided. No extra attention and considerations are needed for “heat-affected zone,” which is a complicated problem in the publications of welded connections. Second, fabrication and erection of bolted connection is more convenient with less usage of energy and resources. Based on the abovementioned aspects, this chapter covers the bearing resistance and slip resistance of HSS connections using grade 12.9 and grade 10.9 bolts.

11.2

Bearing-type bolted connections for high-strength steels

Bearing-type connection is a widely used connection at the construction site of steel structures. For this type of connection the load is primarily transferred by bearing between the bolt shank and hole wall of steel plate. If failure of the bolt is prevented, resistance of bearing-type connection is mainly determined by the steel plate. For connections fabricated with HSS plates, higher resistance of the whole connection is expected compared to those fabricated with conventional steels. This may be the most promising potential for application of HSSs into bearing-type bolted connections. Despite higher resistance of connections in HSS, several issues still need to be considered. The drilled or punched holes weaken the continuity of steel plates. In the vicinity of bolt hole, stress level of the material is usually higher than those far from the bolt hole, which is called “stress concentration.” Under the bearing action between bolt shank and Behavior and Design of High-Strength Constructional Steel. DOI: https://doi.org/10.1016/B978-0-08-102931-2.00011-5 © 2021 Elsevier Ltd. All rights reserved.

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Behavior and Design of High-Strength Constructional Steel

Figure 11.1 Connection types: (A) type a: single-bolt connection, (B) type b: two-bolt connection in parallel, and (C) type c: multibolt or multibolt connection in tandem.

hole wall, development of bearing resistance is accompanied with significant plastic deformation of materials near bolt hole. For bearing-type connections fabricated from normal strength steels, the favorable ductility of material is usually sufficient to mitigate the negative effect of stress concentration. Premature failure of material near bolt hole can be avoided. However, HSS is less ductile than normal strength steel, along with significantly reduced deformation capacities and elongation at fracture. This may lead to premature failure of material near bolt hole. In addition, load distribution in multibolt connection is essentially connected with deformation of material near bolt hole. Compatibility of deformation among the group of bolts should be satisfied when structural load is transferred though multibolt connection. As a result, whether the load distribution pattern of multibolt connection changes due to reduced ductility of HSS is an inevitable issue for practical design. To this end, a comprehensive research is necessary for bearing-type connections in HSSs. In the following sections a detailed study on the behavior of the bolted connections with different number of bolts, as illustrated in Fig. 11.1, is presented, respectively.

11.2.1 Behavior of single-bolt connection 11.2.1.1 Experimental design 1. Basic material properties

Three different grades of quenched and tempered HSS plates, Q550D, Q690D, and Q890D with nominal thickness of 10 mm were used in the test, which are produced by Wuyang Iron & Steel Co., Ltd. According to the technical delivery

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Table 11.1 Material properties. Steel grade Q550D Q690D Q890D

fy (MPa)

fu (MPa)

fy/fu

E (GPa)

εu

Δ (%)

677 825 1022

757 859 1064

0.894 0.960 0.960

205 203 203

0.0642 0.0511 0.0590

18.5 13.5 14.5

Figure 11.2 Stress
strain relationship.

conditions steel in Chinese code high-strength structural steel plates in the quenched and tempered condition GB/T 16270-2009 [5], these three grades of HSS are equivalent to S550Q, S690Q, and S890Q in EN 10025-6 [6]. Tensile coupon tests were carried out for each steel plate to determine the stress
strain characteristics steel plate in accordance with GB/T 228-2002 [7]. The measured material properties are summarized in Table 11.1, where fy is yield strength, fu is tensile strength, E is elasticity modulus, εu is the strain at tensile strength, and Δ is the elongation ratio at fracture. Fig. 11.2 shows the typical stress
strain relationship of HSSs. 2. Design of geometric parameters.

Single-bolt connections in double shear were investigated in the tests. Eight specimens for each HSS and a total of 24 specimens of HSS with various end distance and edge distance were prepared. Wire-electrode cutting was adopted to reduce the effect of heat on material properties. The specimens are named in terms of SD-e1/ d0-e2/d0-steel grade, where SD represents single bolt in double shear and e1, e2, and

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Behavior and Design of High-Strength Constructional Steel

Figure 11.3 Definition of geometric parameters.

d0 are defined in Fig. 11.3. The end distance e1 is varied from 1.0d0 to 2.5d0, including the end distance e1 lower than the 1.5d0 limit specified in GB 500172003 [8] and the 1.2d0 limit according to Eurocode 3 [9]. The edge distance e2 of 0.8d0, 1.0d0, 1.5d0, and 3.0d0 are evaluated in the test. The measured dimensions are summarized in Table 11.2, where t is the plate thickness and Anet is the area of net section. 3. Test setup

The tests were carried out on a hydraulic servo-controlled machine with loading capacity of 1000 kN. Grade 12.9 M24 bolts were used to connect the test specimens to the support with enough shear resistance to avoid bolt shear failure, as shown in Fig. 11.4. No pretension of bolt was applied so that load was transferred primarily by bearing not by friction. A 10 kN load was applied and unloaded before actual loading to make bolt shank bear on hole wall. Then, the specimens are loaded at a prescribed displacement rate of 1.5 mm/min until failure of specimens. Two linearly variable displacement transducers (LVDTs) were positioned along both edges of specimen to measure the bolt hole elongation and plate deformations in force direction, as shown in Fig. 11.4. The applied load was recorded by built-in sensor of the 1000 kN machine. A strain gage was attached at the end surface of each specimen to measure the tensile strain perpendicular to load direction, as shown in Fig. 11.4.

11.2.1.2 Test results 1. Failure mode The comparison of failure modes for specimens with fixed edge distance of 3.0d0 and different end distance from 1.0d0 to 2.5d0 is shown in Fig. 11.5. Fig. 11.6 compares the failure modes for specimens with fixed end distance of 1.5d0 and varying edge distance from 0.8d0 to 1.5d0. The typical ultimate states of plate bearing resistance are schematically shown in Fig. 11.7. However, the specimens usually failed in mixed modes. The effect of the three parameters, end distance, edge distance, and mechanical properties of steel plate, on the failure mode is observed as follows: a. End distance: For specimens with large edge distance of 3.0d0 and relatively small end distance up to 1.5d0, the tearout failure mode is shown in bending deformation

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Table 11.2 Measured geometries. Specimen

SD-10-30-550 SD-12-30-550 SD-15-30-550 SD-20-30-550 SD-25-30-550 SD-15-15-550 SD-15-11-550 SD-15-08-550 SD-10-30-690 SD-12-30-690 SD-15-30-690 SD-20-30-690 SD-25-30-690 SD-15-15-690 SD-15-11-690 SD-15-08-690 SD-10-30-890 SD-12-30-890 SD-15-30-890 SD-20-30-890 SD-25-30-890 SD-15-15-890 SD-15-11-890 SD-15-08-890

Measured dimensions e1/d0

e2/d0

0.99 1.19 1.49 1.99 2.49 1.49 1.49 1.49 1.00 1.20 1.50 2.01 2.51 1.50 1.49 1.50 1.00 1.19 1.50 2.00 2.49 1.49 1.50 1.50

2.99 2.99 2.99 3.00 2.99 1.50 1.09 0.80 3.00 3.00 3.00 3.00 3.00 1.50 1.10 0.80 3.00 3.00 3.00 3.00 3.00 1.49 1.10 0.80

Figure 11.4 Test setup.

d0 (mm) 26.0 26.0 26.0 26.0 26.1 26.0 26.0 25.9 26.0 26.0 26.0 26.0 26.0 26.0 26.0 26.0 26.0 26.0 26.0 26.0 26.0 26.0 26.0 26.0

t (mm)

Anet (mm2)

9.8 9.8 9.9 9.9 9.9 9.9 9.9 10.0 10.1 10.1 10.0 10.2 10.2 10.2 10.2 10.2 10.3 10.3 10.3 10.4 10.3 10.4 10.3 10.3

1268.9 1268.9 1281.9 1287.0 1286.8 514.8 303.7 156.0 1313.0 1313.0 1300.0 1326.0 1326.0 530.4 318.2 159.1 1339.0 1339.0 1339.0 1352.0 1339.0 535.4 321.4 160.7

Figure 11.5 Failure modes of specimens with e2 5 3.0d0: (A) SD-10-30-550, (B) SD-12-30-550, (C) SD-15-30-550, (D) SD-20-30-550, (E) SD25-30-550, (F) SD-10-30-690, (G) SD-12-30-690, (H) SD-15-30-690, (I) SD-20-30-690, (J) SD-25-30-690, (K) SD-10-30-890, (L) SD-12-30-890, (M) SD-15-30-890, (N) SD-20-30-890, and (O) SD-25-30-890.

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Figure 11.6 Failure modes of specimens with e1 5 1.5d0: (A) SD-15-15-550, (B) SD-15-11550, (C) SD-15-08-550, (D) SD-15-15-690, (E) SD-15-11-690, (F) SD-15-08-690, (G) SD15-15-890, (H) SD-15-11-890, and (I) SD-15-08-890.

Figure 11.7 Typical ultimate states of bearing resistance.

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Behavior and Design of High-Strength Constructional Steel

manner. With the increase in end distance, the tearout failure of specimens shows more evident bolt hole elongation due to bearing deformation. The specimens of series SD-25-30 show an excessive hole elongation that may not meet the serviceability limit. b. Edge distance: Splitting failure is observed when the edge distance is reduced but still adequate to avoid net section failure. With the same edge distance and end distance of 1.5d0, the specimen series SD-15-15 shows the splitting failure mode, instead of the tearout failure with evident hole elongation of the series SD-15-30. If the edge distance is reduced unreasonably, net cross-sectional failure will happen, such as the series SD15-11 and SD-15-08. Generally, net cross-sectional failure should be avoided because it cannot give full use of connection resistance. c. The specimens with identical dimensions but different steel grades show similar failure modes, which indicates the failure mode is more sensitive to geometric parameters rather than steel grades. 2. Effect of end distance e1

In order to evaluate the effect of end distance on bearing resistance and deformation capacity, the fixed edge distance e2 5 3.0d0 was adopted to prevent yielding and necking of net cross section. The main parameter herein is the end distance e1, which is varying from 1.0d0 to 2.5d0. The measured ultimate strength Fu and corresponding deformation Du are summarized in Table 11.3. Fig. 11.8 shows the load
displacement curves of the specimens with varying end distance e1 and fixed edge distance e2 of 3.0d0. It can be seen that both ultimate strength and deformation capacity increase with end distance e1 regardless of what grade of steel is tested. Fig. 11.9A shows that, with wide enough edge distance, the ultimate bearing strength is proportional to the end distance. In the form of normalized ultimate strength by fudt, the specimens with various tensile strength are in the same straight line. Different from Fu, the ultimate displacement Du demonstrates an exponential function relationship with the end distance e1, as shown in Fig. 11.10. Since excessive hole elongations may affect the serviceability of building and the performance of structure under design loads, a limit value should be specified when the deformation around the bolt hole is a design consideration. The limit values of 6.35 mm and d0/6 were suggested Moˇze and Beg [10,11], respectively. For M24 bolts the value of d0/6 is 4.33 mm in this test. Accordingly, the bearing resistance based on the hole deformation limits of 6.35 mm and d0/6 are illustrated in Fig. 11.8. At the hole deformation of 6.35 mm, the specimens with end distance up to 2.0d0 achieves no less than 96% of the maximum strength. For the hole deformation limit of d0/6, the threshold value of end distance increases to 1.5d0, where about 98% of the maximum strength can be expected. 3. Deformation manners

A strain gage was attached at the extreme outer edge of the specimen end to measure the tensile strain perpendicular to load direction. Fig. 11.11 shows the axial load versus perpendicular strain curves up to failure of the strain gage. The extreme

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Table 11.3 Test results of specimens with varying end distance e1. Specimen SD-10-30-550 SD-12-30-550 SD-15-30-550 SD-20-30-550 SD-25-30-550 SD-10-30-690 SD-12-30-690 SD-15-30-690 SD-20-30-690 SD-25-30-690 SD-10-30-890 SD-12-30-890 SD-15-30-890 SD-20-30-890 SD-25-30-890 

Fu (kN) 183 226 292 382 456 210 249 332 437 542 262 328 416 550 662

Fu/fudt 1.01 1.24 1.61 2.10 2.52 1.02 1.21 1.61 2.12 2.63 1.02 1.28 1.63 2.15 2.59

Du (mm) 4.61 5.00 6.96 10.92 16.46 4.15 4.52 5.37 10.03 16.59 4.28 4.81 5.52 9.42 14.40

F6.35/Fu 

1.00 1.00 1.00 0.96 0.89 1.00 1.00 1.00 0.97 0.90 1.00 1.00 1.00 0.98 0.90

Fd0/6/Fu 1.00 1.00 0.98 0.90 0.81 1.00 1.00 0.99 0.92 0.82 1.00 1.00 0.99 0.92 0.83

Failure mode Bending TO Bending TO Bending TO Shear TO Shear TO Bending TO Bending TO Bending TO Shear TO Shear TO Bending TO Bending TO Bending TO Shear TO Shear TO

The corresponding displacement at ultimate strength is lower than 6.35 mm.

Figure 11.8 Load
displacement curves of specimens with e2 5 3.0d0: (A) Q550D specimens, (B) Q690D specimens, and (C) Q890D specimens.

502

Behavior and Design of High-Strength Constructional Steel

Figure 11.9 Relationship between ultimate strength and end distance e1: (A) ultimate strength and (B) ultimate strength fudt.

Figure 11.10 Relationship between ultimate deformation Du and end distance e1.

outer edge of the specimen end was found yielded before ultimate strength for all specimens. Similar as the load
deformation curves, the strain response curves have an initial elastic region and a hardening region. If the initial tangent modulus of load
displacement curve and load
strain curve are denoted as Ed and Es, the proportional limit point of the curves is defined as the point with the secant modulus of 0.9Ed or 0.9Es, and Fd,p and Fs,p represent the applied load at the proportional limit points of load
displacement curves and load
strain curves, respectively. Since the net cross-sectional yielding of the specimens with e2 5 3.0d0 is prevented and the elastic deformation is insignificant, the measured displacement is mainly contributed by the plastic deformation in the vicinity of bolt hole subjected to bolt bearing, including shear deformation and bending deformation. In contrast, the measured strain of extreme outer edge can be recognized as the result of bending

Figure 11.11 Load versus bending caused strain curves of specimens with e2 5 3.0d0: (A) Q550D, (B) Q690D, and (C) Q890D.

Figure 11.12 Relationship between proportional loads and end distance: (A) Q550D specimens, (B) Q690D specimens, and (C) Q890D specimens.

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Behavior and Design of High-Strength Constructional Steel

Figure 11.13 Schematic diagram of deformation manners for bolt hole bearing.

deformation. Thus Fig. 11.12 compares Fd,p with Fs,p to reveal the dominating manner of plastic deformation of the specimens with different end distance. For the specimens with small end distance up to 1.5d0, Fd,p is higher than Fs,p, indicating a bending deformation manner. Because the end distance is relative small, the materials in front of bolt hole act as an intermediate beam under bolt bearing, as schematically shown in Fig. 11.13. With the increase in end distance, Fd,p becomes less than Fs,p when the end distance e2 is higher than about 2.0d0. In other words, the steel in front of bolt hole acts as a deep beam when the end distance is large enough. In this case, shear deformation is the dominating manner. 4. Effect of edge distance e2

The effect of edge distance e2 was investigated through the tests of the specimen series SD-15, with fixed end distance of 1.5d0 and varying edge distance from 0.8d0 to 3.0d0. The test results are summarized in Fig. 11.14 and Table 11.4. Three typical failure modes were observed. Net cross-sectional failure was found in the specimen series SD15-08 and SD-15-11, where the edge distance was not wide enough compared to such end distance. Tearout failure occurred in the specimen series SD-15-30 when the yielding of net cross section was avoided by the wide enough edge distance. Splitting failure happened when the end distance was adequate to avoid net section failure but not wide enough to prevent the rotation at net cross section. As observed in the specimen series SD-15-15, although the net cross-sectional yield strength is higher than the ultimate strength of plate bearing with Fu/fyAnet of 0.74
0.77, the partial yielding of net cross

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Figure 11.14 Failure modes with decreasing of edge distance e2: (A) tearout failure, (B) splitting failure, and (C) net cross-sectional failure.

Table 11.4 Ultimate strength (e2 varies). Specimen SD-15-30-550 SD-15-15-550 SD-15-11-550 SD-15-08-550 SD-15-30-690 SD-15-15-690 SD-15-11-690 SD-15-08-690 SD-15-30-890 SD-15-15-890 SD-15-11-890 SD-15-08-890

Fu (kN)

Fu/fudt

Du (mm)

292 271 242 125 332 317 284 149 416 392 347 183

1.61 1.49 1.33 0.69 1.61 1.54 1.38 0.72 1.63 1.54 1.36 0.72

6.96 6.53 3.61 1.81 5.37 5.47 2.96 1.66 5.52 5.11 2.87 1.45

fyAnet (kN) 880 352 211 106 1073 429 257 129 1328 531 319 159

Fu/fyAnet 0.33 0.77 1.15 1.18 0.31 0.74 1.11 1.16 0.31 0.74 1.09 1.15

section due to stress concentration allowed the plastic rotation of net cross section without evident necking. This makes possible of the splitting fracture from the extreme edge of specimen end. The splitting failure can be recognized as a transition between tearout failure and net section failure. The load versus deformation curves of the specimen series SD-15 with fixed end distance of 1.5d0 and different edge distances are shown in Fig. 11.15. The specimen series SD-15-15 with a splitting failure mode shows a 5%
7% lower strength and comparable deformation capacity around bolt hole than the specimen series SD-15-30 with a tearout failure mode. However, the specimens failed at net cross section show a significant reduction in both strength and deformation capacity. 5. Effect of steel grades

The effect of steel grade was investigated based on the comparison of the experimental results among three steel grades. The ultimate strength of specimens with different steel grades is normalized by actually measured f u dt and

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Behavior and Design of High-Strength Constructional Steel

Figure 11.15 Response curves (e2 varies): (A) Q550D specimens, (B) Q690D specimens, and (C) Q890D specimens.

shown in Fig. 11.16. Compared with end distance and edge distance, the effect of tensile strength on the normalized ultimate strength of the bearingtype bolted connections is negligible, which is no more than 5.5%. For the influence of material ductility on the ultimate displacements D u , Fig. 11.17 compares the test results of the specimens fabricated from three grades of steel. For tearout failure the difference of D u between different steel grades is up to 23%. Similarly, the difference of D u between Q550D and Q890D specimens is 20%
23% for splitting failure and net cross-sectional failure. It is observed that the crack in splitting failure can be classified into shear fracture segment and tensile fracture segment, as shown in Fig. 11.18. The straight crack propagated from the wall of bolt hole and toward the free edge of plate is caused by shear failure, while the cup-and-cone fracture formed from the extreme edge fiber of the plate is caused by tensile failure due to bending deformation. The tensile crack length of the Q550 specimen (0.86 mm) is less than that of the Q690 and Q890 specimens (6.36 and 6.11 mm). The reason is revealed by the test result of tensile coupons in Section 11.2.1. The elongation ratio decreases with the increase in steel strength. Based on those observations, steel grade has an evident influence on the deformation capacity of the specimens.

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Figure 11.16 Effect of steel grade on Fu/fudt.

Figure 11.17 Effect of steel grade on Du.

11.2.1.3 Discussions 1. Splitting failure

Splitting failure was observed as a transitional failure mode between tearout failure and net cross-sectional failure, when the end distance is adequate to avoid net section failure but not wide enough to prevent the rotation at net cross section. It was found that

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Behavior and Design of High-Strength Constructional Steel

Tensile fracture 0.86 mm

Shear fracture

(A) SD-15-15-550

Tensile fracture 6.11mm

Tensile fracture 6.36 mm

Shear fracture

(B) SD-15-15-690

Shear fracture

(C) SD-15-15-890

Figure 11.18 Tensile fracture depth of the splitting failure specimens: (A) SD-15-15-550, (B) SD-15-15-690, and (C) SD-15-15-890.

Figure 11.19 Ultimate bearing strengths between tearout failure and splitting failure.

the ultimate bearing strength of specimens with splitting failure mode is lower than that with tearout failure mode, which is attributed to the tensile failure at the extreme end tip. Fig. 11.19 compares the ultimate bearing strengths between tearout failure and splitting failure. In addition to the 3 specimens of splitting failure observed in this chapter, the 13 test data of splitting failure with various end distance available in references are included in Fig. 11.19 to regress a trend line. The comparison between the two trend lines shows that the ultimate bearing strength of splitting failure is lower than that of tearout failure by 8%
19% with end distance varying from 4.0d0 to 1.0d0. In both Eurocode 3 and AISC 360-10, the splitting failure caused reduction of the ultimate bearing strength is not considered. The formula based on the test data of tearout failure may overestimate the bearing strength of connections in splitting failure. Thus a reduction coefficient for splitting failure is recommended, which can be interpolated by end distance between 4.0d0 and 1.0d0.

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Figure 11.20 Dimensions of shearing section and net cross section.

To this end, it is necessary to find out the boundaries among the transition of failure modes. The typical phenomenon of the transition from tearout failure to splitting failure can be recognized as the rotation of net cross section, which finally results in the tensile splitting from the extreme end tip of bearing plate. Due to the stress concentration around the bolt hole, the rotation occurs before fully yielding of the net cross section. The boundary between tearout failure and splitting failure can be expressed in the following form: Fs 5

Fn;y K

(11.1)

where Fs is the shear resistance, Fn,y is the yield resistance of net cross section, and K is the stress concentration factor. According to schematic failure mode based on experimental observation (Fig. 11.20), Fs and Fn,y can be expressed as fu Fs 5 2ðe1 2 03d0 Þt pffiffiffi 3 Fn;

y

5 2ðe2 2 0:5d0 Þtfy

(11.2) (11.3)

The stress concentration factor K is calibrated by the test data summarized in Fig. 11.21. The rotation of net cross section is observed subjected to 71% of net cross-sectional yield resistance. Consequently, K 5 1.41 is adopted here. Substituting Eqs. (11.2) and (11.3) into Eq. (11.1) and rewriting in form of e1/e2, we have pffiffiffi pffiffiffi   3 fy 3 fy d0 e1 5 1 0:3 2 e2 K fu 2K fu e2

(11.4)

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Behavior and Design of High-Strength Constructional Steel

Figure 11.21 Boundaries of splitting failure mode in e1/e2.

In consideration of the increasing yield to tensile strength ratio with steel strength, the boundary between tearout failure and splitting failure in terms of e1/e2 can be obtained as follows: For normal strength steel   fy e1  (11.5a) 5 0:80 5 0:82je2 5 1:2d0 0:86je2 5 1:5d0 fu e2 For HSS   fy 5 0:96 fu

e1  5 0:94je2 5 1:2d0 0:99je2 5 1:5d0 e2

(11.5b)

On the other hand, the boundary between net cross-sectional failure and splitting failure can be expressed as Fs 5 Fn;u

(11.6)

where Fn,u is the tensile resistance of net cross section. An estimation of this boundary by Moˇze and Beg [10] assumed the length of the shear plane as e1 2 0.5d0 for Fs. In Eq. (11.13) the measured length of the shear plane, e1 2 0.3d0, is adopted, as shown in Fig. 11.20 and Eq. (11.2). Fn;u 5 2ðe2 2 0:5d0 Þtfu

(11.7)

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Substituting Eqs. (11.2) and (11.7) into Eq. (11.6) and rewriting in form of e1/e2, we have pffiffiffi  3 d0 e1 pffiffiffi 5 3 1 0:3 2 e2 2 e2

(11.8)

Consequently, the boundary between net cross-sectional failure and splitting failure in terms of e1/e2 can be obtained as follows: e1  5 1:26je2 5 1:2d0 1:35je2 5 1:5d0 e2

(11.9)

Fig. 11.21 compares the derived lower boundary of Eq. (11.5) and upper boundary of Eq. (11.16) for splitting failure with experimental results of bolted connections with HSSs. The experimentally observed failure modes agree well with the derived transition boundaries. 2. Bearing strength considering excessive hole elongation

With the increase in end distance, the specimens show more evident bolt hole elongation due to bearing deformation before finally tearout failure, which may not meet the requirements of serviceability. Thus the excessive hole elongation is recognized as a virtual failure mode, when the deformation around bolt hole is a design consideration under service load. American design code AISC 360-10 choses 6.35 mm as the limit of hole vocalization at service load, based on the founding by Frank and Yura that the developed bearing force is generally increased higher than 2.4dtFu. Thus Eq. (11.10) is specified in AISC 360-10 to limit the excessive deformation of connections. Fb; Rd 5 1:2lc tfu # 2:4fu dt

(11.10)

Although excessive hole elongation is not separately listed out in Eurocode 3, it is implicitly included in the limited maximum bearing strength of 2.5dtFu. However, there is no limit value of excessive bolt hole elongation given by Eurocode 3. Moˇze and Beg [10] proposed a ultimate strength based method to consider excessive bolt hole elongation based on the threshold value of d0/6: For connections with e1/d0 smaller than 2.5, 80% of ultimate strength is adopted; For connections with e1/d0 no less than 2.5, 2.0fudt is adopted as an upper limit. The experimental bearing strengths at d0/6 (4.33 mm) and 6.35 mm of the tested high-strength specimens are summarized in Table 11.5 and compared with design formulae specified in AISC 360-10 and Eurocode 3 and the ultimate strength-based method proposed by Moˇze and Beg. Similar percentage of ultimate bearing strength could be achieved at the specified value of hole elongation for the specimens with different steel grades. It is found that the hole elongation at ultimate bearing strength is less than d/6 and 6.35 mm for the specimens with end distance of no more than 1.2d0 and 1.5d0, respectively. Although AISC 360-10 aims to control the hole elongation within 6.35 mm, the bearing strength of the specimens with end distance no more than

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Behavior and Design of High-Strength Constructional Steel

Table 11.5 Bearing strength at limited hole elongation. Specimen

SD-10-30-550 SD-10-30-690 SD-10-30-890 SD-12-30-550 SD-12-30-690 SD-12-30-890 SD-15-30-550 SD-15-30-690 SD-15-30-890 SD-20-30-550 SD-20-30-690 SD-20-30-890 SD-25-30-550 SD-25-30-690 SD-25-30-890

Experimental strength

Predicted strength

Fu (kN)

Fd/6/ Fu

F6.35/ Fu

AISC 360/ F6.35

183 210 262 226 249 328 292 332 416 382 437 550 459 542 662

1.00 1.00 1.00 1.00 1.00 1.00 0.98 0.99 0.99 0.90 0.92 0.92 0.81 0.82 0.83

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.95 0.96 0.97 0.88 0.89 0.89

0.64 0.64 0.63 0.73 0.76 0.71 0.81 0.81 0.80 0.98 0.96 0.94 1.08 1.02 1.04

Eurocode 3/Fd/6 0.83 0.82 0.81 0.81 0.83 0.78 0.80 0.79 0.78 0.88 0.85 0.84 1.02 0.96 0.96

Moˇze & Beg/Fd/6 0.80 0.80 0.80 0.80 0.80 0.80 0.82 0.81 0.81 0.89 0.87 0.87 0.98 0.93 0.93

1.5d0 is underestimated, where excessive hole elongation is not expected to be observed. For the specimens with end distance of 2.0d0
2.5d0, the prediction of AISC 360-10 is close to the measured strength at 6.35 mm. The predicted curve of Moˇze and Beg method based on the elongation of d0/6 is very close to that of Eurocode 3 with end distance no more than 2.5d0, as shown Fig. 11.22. However, both predictions are conservative for the specimens up to 22% when the end distance is decreasing from 2.5d0 to 1.2d0. Due to different threshold values are proposed to limit excessive hole elongation and the ultimate hole elongation Du is a function of end distance, further investigation on the relationship between hole elongation and bearing strength is needed. 3. Effect of end distance e1

To investigate the effect of end distance e1 on the ultimate bearing resistance, the mixed effect of edge distance e2 should be exclude. Insufficient edge distance e2 will cause the reduction of the ultimate bearing resistance. Thus parameter e1/ e2 5 0.2 is chosen as the case for sufficient edge distance e2. Connection with e1/ e2 5 0.2 is rarely applied in the practical design but may serve as a typical case for the excluded effect of e2. The ultimate bearing resistance for each e1 under the condition of e1/e2 5 0.2 is shown in Fig. 11.23. Normalization by fudt where fu is material tensile stress, d is bolt diameter, and t is plate thickness is applied. It is found that normalized ultimate bearing resistance is almost identical for different grades of HSS when the end distance e1 is the same. An approximate linear relationship can be found between the normalized ultimate bearing resistance and e1/d0. With the method of linear fitting based on the least square method,

Bolted connections

513

Figure 11.22 Comparison between experimental results and predictions.

Figure 11.23 Ultimate strength related to e1/d0

Fu;TO 5 1:04

e1 fu dt d0

  e1 Fu;TO 5 1:04 1 0:18 fu dt d0

(11.11)

(11.12)

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Behavior and Design of High-Strength Constructional Steel

Eq. (11.11) is established, with R2 of 0.97, as shown in Fig. 11.23. In Ref. [3], Eq. (11.12) is proposed to predict the ultimate bearing resistance. Main difference between Eqs. (11.11) and (11.12) is the constant term. Eq. (11.12) is established based on the experimental results from Ref. [12]. Due to limited techniques to exclude the friction in the experiments, Eq. (11.12) incorporates the effect of friction. However, for Eq. (11.11), no friction is considered in the numerical model. As a result, Eq. (11.11) provides the ultimate bearing resistance contributed only by the plate bearing.

11.2.2 Behavior of two-bolt connection in parallel 11.2.2.1 Experimental design The two-bolt connection is named TH connection, which is abbreviation for “twobolt arranged horizontally” and shown in Fig. 11.24. Five geometric parameters are designated, including end distance e1, edge distance e2, bolt spacing p2, bolt hole diameter d0, and plate thickness t. All the TH specimens are cited in terms of THe1/d0-e2/d0-p2/d0-steel grade. Two different end distances e1 as 1.2d0 (minimum requirement in Eurocode 3) and 1.5d0 are adopted. Six combinations of edge distance e2 and bolt spacing p2 are prepared for each e1, as shown in Table 11.6. The measured dimensions for the specimens are summarized in Table 11.6. The tests are carried out on a servo-controlled hydraulic machine with the loading capacity of 2000 kN. Experimental setup is shown in Fig. 11.25. Grade 11.9 highstrength bolt M24 (bolt diameter d 5 24 mm) of is used in the experiments to provide adequate shear resistance and avoid bolt shear failure. No pretension of bolt is applied and a 1.6 mm gap is reserved, as shown in Fig. 11.25. These two precautions ensure that the load is primarily transferred by bearing instead of friction. A 10-kN preloading is applied and unloaded to get the bolt shank to bear on the hole wall. Then, the specimens are loaded at the prescribed rate of 1.5 mm/min until failure. Two LVDTs are positioned along left edge and right edge of the specimen to measure the bolt hole elongation and plate deformations in the loading direction.

Figure 11.24 Geometric parameters in TH connection.

Bolted connections

515

Table 11.6 Geometrical dimensions for TH specimens. Specimen

e1/d0

e2/d0

p2/d0

d0 (mm)

t (mm)

TH-12-12-24-550 TH-12-12-27-550 TH-12-12-30-550 TH-12-15-24-550 TH-12-15-27-550 TH-12-15-30-550 TH-15-15-30-550 TH-15-15-35-550 TH-15-15-40-550 TH-15-20-30-550 TH-15-20-35-550 TH-15-20-40-550

1.20 1.20 1.20 1.19 1.19 1.20 1.49 1.50 1.49 1.49 1.50 1.49

1.20 1.19 1.20 1.49 1.49 1.50 1.50 1.49 1.49 1.99 1.99 1.99

2.39 2.70 3.00 2.39 2.68 3.00 3.00 3.49 3.98 2.98 3.48 3.98

25.9 25.9 25.9 26.0 26.1 25.9 26.0 26.0 26.1 26.0 26.0 26.0

10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0

TH-12-12-24-690 TH-12-12-27-690 TH-12-12-30-690 TH-12-15-24-690 TH-12-15-27-690 TH-12-15-30-690 TH-15-15-30-690 TH-15-15-35-690 TH-15-15-40-690 TH-15-20-30-690 TH-15-20-35-690 TH-15-20-40-690

1.20 1.20 1.20 1.20 1.20 1.20 1.49 1.50 1.50 1.51 1.50 1.50

1.20 1.20 1.20 1.50 1.50 1.50 1.50 1.50 1.50 2.00 2.00 2.00

2.40 2.70 3.00 2.40 2.70 3.00 3.00 3.50 3.99 3.00 3.50 4.00

26.0 26.0 26.0 26.0 26.0 26.0 26.0 26.0 26.0 26.0 26.0 26.0

10.2 10.2 10.2 10.2 10.2 10.2 10.2 10.2 10.2 10.2 10.2 10.2

TH-12-12-24-890 TH-12-12-27-890 TH-12-12-30-890 TH-12-15-24-890 TH-12-15-27-890 TH-12-15-30-890 TH-15-15-30-890 TH-15-15-35-890 TH-15-15-40-890 TH-15-20-30-890 TH-15-20-35-890 TH-15-20-40-890

1.20 1.20 1.20 1.20 1.20 1.19 1.49 1.50 1.50 1.50 1.50 1.50

1.20 1.20 1.20 1.49 1.51 1.50 1.50 1.50 1.50 2.00 2.00 1.99

2.40 2.69 3.00 2.40 2.70 3.00 3.00 3.50 4.00 2.99 3.50 4.00

26.0 26.0 26.0 26.0 26.0 26.0 26.0 26.0 26.0 26.0 26.0 26.0

10.3 10.3 10.3 10.3 10.3 10.3 10.3 10.3 10.3 10.3 10.3 10.3

11.2.2.2 Test results 1. Failure mode

The failure modes of TH specimens for each HSS are shown in Fig. 11.26. According to steel grades and observed failure modes, the specimens in Fig. 11.27 are

516

Behavior and Design of High-Strength Constructional Steel

Figure 11.25 Experimental setup for TH connection.

divided into six blocks. Each row of a block presents specimens with the same end distance e1 and edge distance e2 and varying spacing p2. The observed failure modes are tearout failure and splitting failure, which are schematically shown in Fig. 11.27. Both failure modes show the shear fracture lines positioned in front of the bolt. The main difference between them is in the deformation in net cross section. For splitting failure, obvious rotation of net cross section can be observed while the net cross section shows negligible rotation in tearout failure. The effects of the geometric parameters (e1, e2, and p2) and steel grade on the failure mode are observed as follows: a. End distance e1: The tearout failure and splitting failure are both observed in the specimens with varying end distance e1. The specimen with end distance e1 5 1.5d0 (e.g., TH15-20-40-690) has an increased length of shear fracture by approximate 33% compared to the specimen with e1 5 1.2d0 (e.g., TH-12-15-30-690). b. Edge distance e2 (e1/e2): For specimens with end distance of 1.2d0 and 1.5d0, two values of e1/e2 are used. For e1/e2 5 1.0 (e.g., TH-12-12-24-550: e1 5 1.2d0 and e2 5 1.2d0), significant rotation of net cross section is observed. For e1/e2 5 0.75 (e.g., TH-15-20-30-690: e1 5 1.5d0 and e2 5 2.0d0), no significant rotation of net cross section is found. c. Bolt spacing p2: There are six values of p2, increasing from 2.4d0 to 4.0d0. The 2.4d0 is the minimum spacing distance specified in Eurocode 3. Similar profile of failure mode is observed no matter what value of p2 is used. The negligible distortion of the grid drawn between the two bolts indicates that the minimum value of p2 specified by Eurocode 3 is sufficient to avoid the net cross-sectional failure between the two bolt holes. d. Steel grade: The specimens with identical dimensions but of different steel grades show a similar failure mode. This phenomenon indicates that, if matching or overmatching bolt is used in the connection, the effect of steel grade on the failure mode is negligible.

For comparison the failure modes of single-bolt connection are reproduced in Fig. 11.28. The single-bolt connections are fabricated from the same HSS plates as

Bolted connections

517

Figure 11.26 Failure mode of TH connection.

Figure 11.27 Typical failure modes.

the specimens in this chapter. Tearout failure and splitting failure were also observed in the test of single-bolt connections. For the tearout failure, negligible difference is observed between the two-bolt connection and the single-bolt connection when the end distance e1 is same (e.g., TH-15-15-30 and Fig. 11.28D). For the

518

Behavior and Design of High-Strength Constructional Steel

Figure 11.28 Failure mode of single-bolt connection: (A) e1 5 1.5d0, e2 5 3.0d0, Q550D; (B) e1 5 1.5d0, e2 5 3.0d0, Q690D; (C) e1 5 1.5d0, e2 5 3.0d0, Q890D; (D) e1 5 1.5d0, e2 5 1.5d0, Q550D; (E) e1 5 1.5d0, e2 5 1.5d0, Q690D; and (F) e1 5 1.5d0, e2 5 1.5d0, Q890D.

splitting failure the rotation of net cross section occurs at the two edge sides of both single-bolt and two-bolt connections. Due to the geometric symmetry of two-bolt connection, there is no rotational deformation at the spacing section. 2. Ultimate resistance Fu and related deformation Du

The measured ultimate resistances of the specimens are summarized in Table 11.7. The end distance e1 is one of the geometric factors influencing the ultimate resistance. With the increasing of e1, more materials are involved for bearing against the bolt. Compared to the specimen with e1 5 1.2d0, ultimate resistance of the specimen with e1 5 1.5d0 increases by approximate 27%. Meanwhile, the end distance to edge distance ratio e1/e2 determines the failure modes. The specimens with e1/e2 5 1.0 show a splitting failure while those with e1/e2 5 0.75 show a tearout failure. Table 11.7 indicates that the ultimate resistance of the specimens in splitting failure is generally lower than those in tearout failure by up to 6%. For the bolt spacing p2, its influence can be ignored when it is enough to avoid net crosssectional failure. A 3% difference of the ultimate resistance is observed for the specimens with varying p2. The ultimate resistance is increased by about 30% when the steel grade is increased from Q550D to Q890D. In terms of the normalized ultimate resistance by fudt, (where fu is the tensile stress, d is the bolt diameter, and t is the plate thickness), similar normalized ultimate resistances with the relative difference lower than 4% are observed for the specimen fabricated from different grades of steels. Comparison with the single-bolt connection (FuSingle) fabricated from the same steel grade is shown in Table 11.7. For the two-bolt specimen in tearout failure, ultimate resistance of the single-bolt connection with the same end distance e1 and e2 5 3.0d0 (prevent the rotation of net cross section) is used. For the two-bolt specimen in splitting failure, ultimate resistance of the single-bolt connection with the

Bolted connections

519

Table 11.7 Ultimate resistance of TH connections. Specimen

Failure mode

Fu (kN)

Fu/ fudt

Du (mm)

Fu/ (2 3 FuSingle)

Du/ (DuSingle)

TH-12-12-24-550 TH-12-12-27-550 TH-12-12-30-550 TH-12-15-24-550 TH-12-15-27-550 TH-12-15-30-550 TH-15-15-30-550 TH-15-15-35-550 TH-15-15-40-550 TH-15-20-30-550 TH-15-20-35-550 TH-15-20-40-550

SP SP SP SP SP SP SP SP SP TO TO TO

443 439 438 452 442 441 561 562 552 575 576 570

2.44 2.41 2.41 2.49 2.43 2.43 3.09 3.09 3.04 3.16 3.17 3.14

5.25 5.26 5.37 5.31 5.29 5.11 6.66 6.64 6.38 6.63 6.44 6.34

0.98 0.97 0.97 1.00 0.98 0.97 0.96 0.96 0.95 0.98 0.99 0.98

1.04 1.04 1.07 1.05 1.05 1.01 0.96 0.96 0.92 0.95 0.93 0.91

TH-12-12-24-690 TH-12-12-27-690 TH-12-12-30-690 TH-12-15-24-690 TH-12-15-27-690 TH-12-15-30-690 TH-15-15-30-690 TH-15-15-35-690 TH-15-15-40-690 TH-15-20-30-690 TH-15-20-35-690 TH-15-20-40-690

SP SP SP TO TO TO SP SP SP TO TO TO

511 516 516 525 523 516 639 646 663 678 654 654

2.43 2.45 2.45 2.49 2.49 2.45 3.04 3.07 3.15 3.22 3.11 3.11

4.55 4.57 4.64 4.60 4.46 4.61 5.50 5.52 5.63 5.41 5.38 5.55

1.03 1.04 1.04 1.05 1.05 1.04 0.96 0.97 1.00 1.02 0.99 0.98

1.01 1.01 1.03 1.02 0.99 1.02 1.02 1.03 1.05 1.01 1.00 1.03

TH-12-12-24-890 TH-12-12-27-890 TH-12-12-30-890 TH-12-15-24-890 TH-12-15-27-890 TH-12-15-30-890 TH-15-15-30-890 TH-15-15-35-890 TH-15-15-40-890 TH-15-20-30-890 TH-15-20-35-890 TH-15-20-40-890

SP SP SP TO TO TO SP SP SP TO TO TO

621 620 615 648 643 637 807 794 800 835 834 828

2.36 2.36 2.34 2.47 2.44 2.42 3.07 3.02 3.04 3.17 3.17 3.15

4.79 4.61 4.44 4.75 4.61 4.77 5.45 5.54 5.34 5.62 5.79 5.38

0.95 0.95 0.94 0.99 0.98 0.97 0.97 0.95 0.96 1.00 1.00 1.00

0.99 0.95 0.91 0.98 0.95 0.98 0.98 1.00 0.96 1.01 1.04 0.97

same end distance e1 and e1/e2 5 1.0 (rotation of net cross section occurs) is adopted. The relative difference is found within 6 5%. The deformation at ultimate resistance Du is the average of the measured connection deformation at the ultimate resistance by the LVDTs along both edges.

520

Behavior and Design of High-Strength Constructional Steel

The effect of end distance e1 on the Du is found to be significant. Up to 24% increase of Du is observed in Table 11.7 when the end distance e1 increases from 1.2d0 to 1.5d0. The effect of e1/e2 and bolt spacing p2 is found to be less significant with the relative difference within 7%. For the effect of steel grade, up to 16% decrease in Du is observed when steel grade changes from Q550D to Q890D. This phenomenon may be caused by different deformation capacity of the material observed between Q550D and Q890D. Comparison of Du with the single-bolt connection shows that Du of the two-bolt connection is approximate 91%
107% of the result of single-bolt connection with the same e1, as shown in Table 11.7. 3. Load
displacement curves

The load
displacement curves of all the specimens are shown in Fig. 11.29. For each figure, there are two clusters of curves. One is for the specimens with e1 5 1.2d0 and the other is for e1 5 1.5d0. The typical load
displacement curve consists of a linear stage, a strengthening stage, and a smoothly descending stage. The specimens both in tearout failure and splitting failure show a favorable ductile failure. Even for Q890D HSS, the connections show a sufficient ductility for the application in the two-bolt connection arranged perpendicularly to the load direction. The load
displacement curves of the two-bolt connection are compared with those of the single-bolt connection in Fig. 11.30. For the two-bolt connection a half of the applied load is used to represent the resistance contributed by each bolt. A good agreement is observed from the load
displacement curves between the two types of connections. The comparison in Fig. 11.30 indicates that behavior of an individual bolt in the two-bolt connection may be represented by a single-bolt connection with the same end distance e1 and reasonable edge distance e2 and spacing distance p2.

11.2.2.3 Optimization of e2 to p2 ratio Ductile behavior is favorable for the connection during the transferring structural load. The previous findings show that different combinations of e2 and p2 result in different failure modes. Tearout failure and splitting failure show a significant bolt hole elongation development and a ductile behavior of the connection, while the mixed failure shows a localized deformation in the necking net cross section, which may cause a brittle failure of the connection without obvious deformation. Thus a parametric study is conducted to optimize the e2 to p2 ratio for the positioning design of bolt holes. Considering the negligible effect of steel grade on failure mode, HSS Q550D is chosen for the parametric study. Geometric parameters are summarized in Table 11.8. Three different end distance e1 are set as 1.2d0, 2.0d0, and 2.5d0. The edge distance e2 and the bolt spacing p2 are designed according to “2e2 1 p2 5 8.0d0,” which aims to simulate the fixed width of plate in practical design. The edge distance e2 varies from 1.2d0 to 2.8d0 at an interval of 0.2d0, while the parameter p2 varies from 5.6d0 to 2.4d0 at an approximate interval of 0.4d0. The

Bolted connections

Figure 11.29 Load
displacement curves: (A) Q550D specimens, (B) Q690D specimens, and (C) Q890D specimens.

Figure 11.30 Comparison between TH connection and single-bolt connection.

521

522

Behavior and Design of High-Strength Constructional Steel

Table 11.8 Parameters for the parametric study. Steel Q550D

e1

e2

p2

d

d0

t

1.2d0, 2.0d0, 2.5d0

1.2d0
2.8d0

e2 1 p2 5 8d0

24 mm

26 mm

10 mm

Figure 11.31 Typical failure modes for different e1, e2, and p2.

values of 1.2d0 and 2.4d0 are the minimum distance requirement for e2 and p2 in Eurocode 3. Bolt shank diameter d and hole diameter d0 are set as 24 and 26 mm. Plate thickness t adopts 10 mm. The convention of the specimens still follows “THe1/d0-e2/d0-p2/d0-steel grade.” The failure modes of the parametric study are shown in Fig. 11.31. Three typical combinations of e2 and p2 are illustrated, which are “2e2 extremely smaller than p2 (2e2{p2),” “2e2 close to p2 (2e2  p2)” and “2e2 extremely larger than p2 (2e2cp2).” When e1 is small (e.g., e1 5 1.2d0), the mixed failure does not occur even for the two extreme cases as “2e2{p2” and “2e2cp2.” However, for larger e1 (e.g., e1 5 2.0d0 or 2.5d0), the mixed failure occurs in the two extreme cases (“2e2{p2” and “2e2cp2”). This phenomenon indicates that the minimum e2 and p2 required in current Eurocode 3 cannot fully exclude the mixed failure for different end distance e1. As shown in Fig. 11.32, whether the mixed failure occurs is determined not only by e2 and p2 but also by e1. The relationship between the ultimate resistance and the ratio 2e2/p2 is shown in Fig. 11.32. The parameter 2e2/p2 is used to quantify the relative magnitude between e2 and p2. For different end distance the boundaries of mixed failure are also illustrated in Fig. 11.32.

Bolted connections

523

1200 Boundary

Boundary

Ultimate load (kN)

1000 800 Optimal 600 400 200

e1=1.2d0:

TO or SP

e1=2.0d0:

Mixed failure;

TO or SP

e1=2.5d0:

Mixed failure;

TO or SP

0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Parameter 2e2/p2 Figure 11.32 Effect of 2e2/p2 on the ultimate load.

Based on the observation of Fig. 11.32, the ultimate resistance of the connection depends on the combination of e2 and p2. With the constant width of “2e2 1 p2 5 8.0d0,” the connection in the case of “2e2  p2” shows the largest ultimate resistance. For the other two extreme cases as “2e2{p2,” and “2e2cp2,” an up to 21% reduction in the ultimate resistance is found compared to the case of “2e2  p2.” Thus “2e2  p2” is the most economic rule for positioning design of bolt holes in the direction perpendicular to load, which achieves the maximum ultimate resistance under the same plate width.

11.2.3 Behavior of multibolt connection in tandem The conventions for two-bolt and three-bolt connections in tandem are denoted as TV-e1/d0-e2/d0-p1/d0-steel grade and TP-e1/d0-e2/d0-p1/d0-steel grade, respectively. Related definitions of geometries are shown in Fig. 11.33. For TV connection the end distance e2 is fixed as 4.5d0 to avoid failure on the net cross section. Four values of e1/p1, including 0.66, 1.0, 1.25, and 1.5, are designed. Compared to TV connection, one more bolt is designed in the TP specimens. The bolt end distance e2 is also fixed as 4.5d0. Three values of e1/p1, including 0.66, 1.25, and 1.50, are designed. The plate thickness is 10 mm while d0 5 26 mm. The failure mode found in the TV connection (Q550D results) is shown in Fig. 11.34. The effect of steel grades is negligible on the profile of failure mode. The shear fracture along the load direction is observed in front of the two bolts. Plate deformation is localized around bolt hole, which is similar to SD connection and TH connection. Similar profile of failure can be observed in the TP connection,

524

Behavior and Design of High-Strength Constructional Steel

Figure 11.33 Definition of geometries for multibolt connection in tandem.

Figure 11.34 Failure mode of TV connection: (A) TV-20-45-20-550, (B) TV-25-45-20-550, (C) TV-30-45-20-550, and (D) TV-20-45-30-550.

which is illustrated in Fig. 11.35. The obvious bolt elongation in the failure mode of TP connection indicates a ductile behavior of a group of fasteners. The load
displacement curves of TV and TP connections (Q550D) are shown in Fig. 11.36. The shape of the load
displacement in TV and TP is similar to that of the SD and TH connections. Although different (e1 1 p1) and (e1 1 2p1) are set for TV and TP connections, the deformation capability of different specimens shows no significant difference, as shown in Fig. 11.36. For convenient description the bolt near plate edge in TV or TP connection is defined as end bolt and other bolts are defined as inner bolts. The Du of SD-20-30 series is compared with TV and TP connections, as shown in Fig. 11.36. The results of SD-20-30 series are chosen since the value of e1 and p1 is 2.0d0 for all the TV and TP connection. For TV-2045-30 and TP-20-45-30 series with e1 5 2.0d0, the load
displacement curve levels off at the Du of SD-20-30. The subsequent increase in the resistance of the whole connection is negligible. The underlying reason for this phenomenon is that the end bolt has reached the ultimate bearing resistance. Behavior of the end bolt enters into the decreasing stage. As a result, the overall behavior of the whole connection shows a plateau branch. It is also found that, at the Du, the behavior of SD-20-30 for TV and TP connections with p1 5 2.0d0 is similar to the single-bolt connection

Bolted connections

525

Figure 11.35 Failure mode of TP connection: TP-20-45-30-550, TP-25-45-20-550, and TP30-45-20-550.

Figure 11.36 Load
displacement of TV and TP connections: (A) TV specimens and (B) TP specimens.

with e1 5 2.0d0. Such similarity indicates that the inner bolt with p1 5 2.0d0 has a similar deformation capability to the end bolt with e1 5 2.0d0. The ultimate bearing resistances of TV and TP connections are summarized in Table 11.9. The effect of steel grade on the normalized ultimate bearing resistance is negligible in the TV and TP connections. It can be found from Table 11.9 that the specimens with larger (e1 1 p1) or (e1 1 2p1) show higher resistance. For TV-20-45-20-550 and TV30-45-20-550, an increase of (e1 1 p1) by 1.0d0 results in the increase in the ultimate bearing resistance of 142 kN. This phenomenon indicates that the increasing of e1 and p1 is an effective strategy to achieve improved connection resistance.

11.2.4 Comparison with current design codes 11.2.4.1 Brief introduction of current codes 1. Eurocode 3 The design bearing resistance of individual fastener in Eurocode 3 is calculated by Eq. (11.13). Two parts as αbk1fudt and 2.5fudt in Eq. (11.13) are set for different conditions. The first part αbk1fudt is aimed to exclude the tearout failure while the second part 2.5fudt is aimed to limit bolt hole elongation. The parameter αb is set to consider effect of geometry parallel to load direction (e.g., end distance e1) while the parameter k1 is set to consider the effect of geometry perpendicular to load direction (e.g., edge distance e2).

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Behavior and Design of High-Strength Constructional Steel

Table 11.9 Ultimate bearing resistance of TV and TP connections. Q550D (kN)

Norma

Q690D (kN)

Norm

Q890D (kN)

Norm

TV-20-45-20 TV-25-45-20 TV-30-45-20 TV-20-45-30

671 746 813 785

3.69 4.11 4.48 4.32

766 850 932 895

3.64 4.04 4.43 4.26

961 1062 1151 1115

3.65 4.04 4.37 4.24

TP-20-45-30 TP-25-45-20 TP-30-45-20

1178 1031 1124

6.48 5.67 6.19

1364 1174 1279

6.49 5.58 6.08

1648 1502 1623

6.27 5.71 6.17

Specimen

a

Norm: the normalized ultimate bearing resistance.

Fb;EC3 5 αb k1 fu dt # 2:5fu dt

(11.13)

where αb and k1 are calculated according to end distance e1 and edge distance e2 as follows: For edge bolts

  e1 fub ; ;1 ; αb 5 min 3d0 fu

  e2 k1 5 min 2:8 2 1:7; 2:5 d0

 For inner bolts αb 5 min

 p1 1 fub 2 ; ;1 ; 4 fu 3d0

(11.14a,b)

  p2 k1 5 min 1:4 2 1:7; 2:5 d0 (11.14c,d)

where fub is the ultimate tensile strength of bolt. 2. AISC 360-16

The bearing resistance formulae in current AISC 360-16 are shown in Eqs. (11.15a)
(11.15f) and (11.16). Clear distance denoted as lc is used in AISC 360-16. Unlike Eurocode 3, AISC 360-16 sets two cases according to whether deformation around bolt hole is a design consideration at service load. a. When the deformation around bolt hole is not a design consideration at service load, the bearing strength at bolt hole is calculated as   Fb;AISC 5 min Ftearout ; Fbearing

(11.15a)

Tearout:

Ftearout 5 1:5lc tfu

(11.15b)

Bearing:

Fbearing 5 3:0fu dt

(11.15c)

b. When the deformation around bolt hole is a design consideration at service load

Fb;AISC 5 minð1:2lc tfu ; 2:4fu dtÞ

(11.15d)

Bolted connections

Tearout:

527

Ftearout 5 1:2lc tfu

Bearing:Fbearing 5 2:4fu dt

(11.15e) (11.15f)

where lc is the clear distance in the direction of the force, between the edge of the hole and the edge of the adjacent hole or edge of the material. 3. Chinese code GB50017-2017

The bearing resistance formula in current Chinese code GB50017-2017 is shown in Eq. (11.16). A parameter named bearing strength fcb is used in Eq. (11.16). This parameter is related only to the tensile stress of steel. No effect of geometry (e.g., end distance e1) is considered in the fcb , which show a different design methodology of Chinese code compared to current Eurocode 3 and AISC 360-16. Fb;CH 5 d

X

tU fcb

(11.16)

where the parameter fcb 5 1:26fu .

11.2.4.2 Comparisons The abovementioned three codes are used to predict the bearing resistance of single-bolt connections. The measured geometric dimensions and material properties are used in the calculation while all partial factors were set as 1.0. The comparison between the test results and the prediction by existing codes is shown in Fig. 11.37. For AISC 360-16 Eq. (11.15a) is used to evaluate the ultimate bearing resistance of different connections. The predictions for the SD connection, TH connection, TV connection, and TP connection are shown in Fig. 11.37A
D, respectively. Evaluations are summarized in terms of connection types. 1. Single-bolt connection (SD) For Eurocode 3 the predicted resistance lies on the safe side. The relative difference between the test result and prediction is from 216% to 223% according to end distance e1. Similar conservative predictions can be found in the research by Moˇze and Beg [10]. For AISC 360-16, it is found that the safety margin of the prediction varies according to the value of end distance e1. If e1 , 1.5d0, the predicted resistance by AISC 360-16 is lower than the test result. These data are distributed on the safe side. For specimen with e1 5 1.5d0, the prediction by AISC 360-16 agrees very well with the test result. The relative difference is within 6 5% for different grades of steel. If e1 . 1.5d0, the AISC 36016 provides unsafe prediction. The largest difference is observed close to 120%. For Chinese code GB50017-2007, it is found that the predicted resistance has excellent agreement with the test result when e1 is around 1.2d0. For e1 , 1.2d0, prediction by GB50017-2007 is unsafe. For e1 . 1.2d0, prediction by GB0017-2007 is far more conservative than Eurocode 3. The largest difference is up to 250%. According to the

528

Behavior and Design of High-Strength Constructional Steel

Figure 11.37 Distribution of prediction for different connection types: (A) single-bolt connection (SD), (B) two-bolt connection perpendicular to load direction (TH), (C) two-bolt connection parallel to load direction (TV), and (D) three-bolt connection parallel to load direction (TP).

GB50017-2007, the minimum requirement of e1 is 2.0d0. Thus the prediction by GB00172007 is conservative for practical design. 2. Two-bolt connection in parallel (TH) For Eurocode 3, two typical distributions of data are found. One is the distribution with the relative difference close to 245% and the other is around 220%. The distribution with 245% difference is found in the series TH-12-12-27 with three grades of steel. Due to e2 5 1.2d0, the parameter k1 for TH-12-12-27 in Eq. (11.15b) equals 1.66. Compared to e2 5 1.5d0 (e.g., TH-12-15-27), an extra reduction of 34% is introduced for the series TH-12-12-27. The parameter k1 in Eurocode 3 considers the effect of edge distance e2 on the resistance. However, such reduction is unnecessary because the predicted resistance is already lower than the test result by about 220%. Thus it is recommended that the parameter k1 can be set constant as 2.5 for different values of edge distance e2. For AISC 360-16 the predicted resistance is lower than or close to the test result. For GB50017-2017, predicted resistance is close to the test result for TH connection with e1 5 1.2d0. For TH connection with e1 5 1.5d0, the difference is about 220%. For TH connection with e1 5 1.2d0 and e1 5 1.5d0, the difference between the prediction and the test result is close to that of SD connection. 3. Multibolt in tandem (TV and TP)

For Eurocode 3 the predicted resistance lies on the safe side. The difference between the prediction and the test result is about 225%, which shows a stable feature compared to SD and TH connection.

Bolted connections

529

For AISC 360-16 the predicted resistance lies on the unsafe side. The largest difference is over 125%. The unsafe prediction by AISC 360-16 is caused by the unsafe prediction in SD connection when e1 . 1.5d0. For TV and TP connections the parameters e1 and p1 both exceed the 1.5d0. As a result, an unsafe prediction is provided by the AISC 360-16. For GB50017-2007 the prediction is far more conservative than Eurocode 3. The conservative prediction by GB50017-2007 can also be explained by the prediction of SD connections. For e1 $ 2.0d0, prediction by GB50017-2007 is lower than the test result with relative difference close to 250%. For TV and TP connections with e1 and p1 higher than 2.0d0, very conservative predictions by GB50017-2007 can be expected. 4. Summary of the evaluation

Based on the previous observations, the applicability of current codes to HSSs can be summed as follows. Eurocode 3 and GB50017-2007 are applicable for the design of bearing-type bolted connections in HSSs. But potential resistance exists for both two codes, which can be further improved to make the design more economic and efficient. For AISC 360-16 the prediction is unsafe. An extra safety factor should be considered if the AISC 360-16 is used for the design of bearing-type bolted connections in HSSs.

11.3

Slip critical
type bolted connections for high-strength steels

11.3.1 Introduction HSS with a nominal yield stress not less than 460 N/mm2 has been increasingly used in spatial structures, high-rise buildings and bridges. Slip critical connection is a typical bolted connection that transmits forces by friction between the contacted members rather than bolt shear or bolt bearing. The importance of slip performance of bolted joint for structural design has been well demonstrated [13]. The resistance of slip critical joints is determined by the slip factor of faying surfaces and pretension force applied the installed high-strength bolt. The slip resistance is calculated by the following equation [9]: FS 5 μnf

nb X

Fp;i

(11.17)

i51

where μ is the slip factor, nf is the number of slip planes, nb is the number of highstrength bolts, and Fp is the pretension force of the high-strength bolt. Two methods are usually used to apply pretension force: strain monitor method, torque
tension indicator. The pretension force, in general, is approximately equal to 60%
70% of the ultimate tensile strength of the bolt. If the pretension forces of all bolts are identical, Eq. (11.17) can be simplified as FS 5 μnf nb Fp

(11.18)

530

Behavior and Design of High-Strength Constructional Steel

Accordingly, the slip factor μ plays the most significant role on the resistance of slip critical joints if the numbers of the slip planes and high-strength bolts are determined. It is generally accepted that the slip factor depends on the treatment effect of faying surfaces. For different surface treatments the surface classification and corresponding slip factors are specified in design standards. The American Institute of Steel Construction Specifications, ANSI-AISC 360-2016 [14] adopts two-level surfaces classification: Class A surfaces (μ 5 0:30) include unpainted clean mill scale steel surfaces, surfaces with Class A coatings on blast-cleaned steel, hotdipped galvanized surfaces and roughened surfaces; Class B surfaces (μ 5 0:50) include unpainted blast-cleaned steel surfaces and surfaces with Class B coatings on blast-cleaned steel. Three classes of faying surfaces are proposed by Research Council on Structural Connections (RCSC) for structural joints using ASTM A325 and A490 bolts [15]: Class A surface with μ 5 0:33 (blast-cleaned steel surfaces with Class A coatings of unpainted clean mill scale faying surfaces); Class B surface with μ 5 0:50 (blasted-cleaned steel surfaces with Class B coatings or unpainted blast-cleaned steel surfaces); Class C surface with μ 5 0:35 (the galvanized faying surface prepared by roughening). In the Canadian standard for the design of highway bridges CAN/CSA-S6-06 [16], hot-dip galvanized with wirebrushed surfaces with a slip factor of 0.4 (Class C) is specified, in addition to the same definition of Class A (μ 5 0:33) and Class B (μ 5 0:50) in RCSC specification. In the Parts 1
8 of Eurocode 3, four classes of friction surfaces are defined: Class A with μ 5 0:50 (shot or grit blasted with loose rust removed and not pitted); Class B with μ 5 0:40 (shot or grit blasted and spray-metalized or -coated); Class C with μ 5 0:30 (wire-brushed or flame-cleaned and loss rust removed); Class D with μ 5 0:20 (no surface treatment). In Chinese standard for design of steel structures [17], the influence of steel strength grade on the slip factor is considered; however, the steel strength grade is ranging from Q235 to Q460. For the surface with shotblasting treatment, the influence of strength grade is ignored and the corresponding slip factor is 0.40. In the present study the slip factor of shot-blasted surface is investigated and corresponding slip factors in different standards are summarized in Table 11.10. In current design codes the slip factor depends on faying surface treatments. It is based on the assumption that the slip factor is only determined by the roughness faying surfaces. However, the influence of steel strength on slip factor is not considered in current design standards. The design codes mentioned previously are based on the fundamental research on mild steel, and the applicability of these codes for HSS has not been investigated well. Heistermann et al. carried out a series of experiments to investigate the slip factor for different surface preparations and steel grades in range between S275 and S690 [18]. And it was demonstrated that the steel grade does not have a significant influence on slip factor. Cruz et al. tested the slip factor for S690 steel and concluded that the steel grade has small influence on the slip factor [19]. However, this is contrary to the background literatures of Eurocode, where the influence of steel strength on slip factor was investigated by Kulak et al. [20]. Test results showed that, with the same faying surface treatment, the slip factor of HSS is reduced compared to that of mild steel. However, only

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531

Table 11.10 Code specifications of slip factors. Faying surface treatment

Shot-blasted

GB 50017-2017

EC3

AISC 360

RCSC

Q460

Mild steel

Mild steel

Mild steel

0.40

0.50

0.50

0.50

RCSC, Research Council on Structural Connections.

ASTM A514 constructional alloy steel (equivalent to S690 HSS according EN 10025 [21]) was tested and the test database was very limited. On the other hand, no matter the influence of the steel strength on the slip factor is weak or strong, the mechanism of this phenomenon has not been studied clearly in slip critical connection. However, without corrosion resistant treatment, the slip resistance of the joint is possible to decrease due to the corrosion of faying surfaces. In order to avoid the risk of rusting of shot-blasted surfaces, which threatens the structure safety, shotblasted surfaces are generally coated with specific corrosion resistance treatments. Inorganic zinc-rich coating method, characterized by high corrosion resistant, has been widely adopted in recent years. However, the slip factor of faying surfaces with coating is generally lower than that of uncoated surfaces. In addition, creep deformation of coating generally occurs, which induces loss of pretension force in bolts. Therefore, in the slip test procedure, stability of slip factor should be considered. The influence of coating process on slip factor is studied by some researchers, which reports that compared with brush, better bonding effectiveness can be obtained when spray method is used. The influence of coating type on slip factor is significant. Because of good performance on anticorrosive and stability, there are three coating types widely used: hot-dip galvanizing, aluminum coating, and zinc-rich coating. The slip factor of hot-dip galvanizing surface after pickling cleaning ranges from 0.15 to 0.25. However, the deviation of slip factor is high. When shot-blasting method rather than pickling cleaning method is used, significantly higher of slip factor of hot-dip galvanizing surface can be obtained. The zinc-rich coating is generally classified into inorganic type and organic type. Compared with shot-blasting surface, zincrich coating surface has lower slip factor but better anticorrosive performance. Kulak summarized the experimental results on slip factor of zinc-rich coating surfaces and reported that the slip factor of inorganic type is higher than that of organic one. The applicability of inorganic zinc-rich coating method has been investigated in many previous researches. The influence of coating thickness on slip factor has been investigated. The coating thickness has been set as one significant parameter in the calibration procedure of slip factor. In RCSC the influence of coating thickness is considered for longterm tests but negligible for short-term tests. In general, the creep phenomenon becomes more significant with increasing of coating thickness, which induces

532

Behavior and Design of High-Strength Constructional Steel

decreasing of pretension force in bolts and slip loads. Cruz et al. conducted a series test to compare the slip factor of coating surfaces with 70 µm and given the similar conclusions. However, contrary conclusions are drawn by some researchers. According to experimental investigation on slip factor of inorganic zinc-rich coating with various thicknesses, Kulak et al. report that the slip factor of surface with 50
80 µm thickness coating is higher than that of surface with 20
30 µm thickness coating. Frank Philip Bowden and David Tabor investigated the physical mechanisms of friction at a microscopic level in the 1950s and proposed friction model as illustrated in Fig. 11.38 [22]. This model emphasizes that static friction consists of adhesion force and furrow force, which are the products of adhesion effect between contact area and interlock action between furrows, respectively. It is generally accepted that with the increasing of the hardness of the material, the roughness of the faying surface is lower after shot-blasting treatment. Consequently, the contribution of the furrow force decreases with the increase of hardness. Furthermore, Bowden demonstrated that contact area between faying surfaces caused by asperities is only a small part of the apparent area, as shown in Fig. 11.39, and when the pressure is fixed, the contact area increases with the increasing of the deformation capacity of material, which induces the increasing of the contribution of the adhesion force. As well known the hardness increases with the steel strength while the plastic deformation capacity decrease with the increasing of steel strength. According to the friction mechanism theory, the influences of the hardness and plastic deformation capacity of HSS on the slip factor are necessary to be investigated. In order to investigate the influence of the steel strength grade and surface treatment on the slip factor, 15 specimens were fabricated from HSS (Q550, Q690, and Q890) to test the slip factors of shot-blasted surface with cleanliness level of SA

Figure 11.38 Friction model of adhesion force and furrow force.

Bolted connections

533

Figure 11.39 Asperities between the contacted surface pair.

2.5. Moreover, the effect of rusting of shot-blasted surface was investigated, since the rusting of steel plates is inevitable during the steel construction period and service period for uncoated surfaces. Therefore, in order to investigate the influence of the rusting treatment on the slip factor, 15 specimens are fabricated from HSS (Q550, Q690, and Q890) to test the slip factors of rusted surface after blasting shotblasted. For comparison, five specimens fabricated from mild steel (Q345) with shot-blasting surface treatment are tested as well. The slip factors of Q550, Q690, and Q890 steels with shot-blasted surface are measured and the influences of surface roughness and plastic deformation capacity of steel on the slip factor are evaluated based on the friction mechanism theory [23]. Moreover, the slip resistance of mild steel (Q345) and HSS (Q550, Q690, and Q890) surfaces with zinc-rich coating is investigated for varying values of coating thickness and loading duration time [24]. A number of uncoated shot-blasted surfaces are also tested for comparison. Creep tests are carried out to evaluate long-term effects on the inorganic zinc-rich coating surface. According to experimental results, the recommended values of slip factors for zinc-rich coating surfaces are given.

11.3.2 Experimental programs 11.3.2.1 Test specimens Standard test specimens fabricated from HSS (Q550, Q690, and Q890) are designed according to Annex G of EN 1090-2 [25], as shown in Fig. 11.40. Each specimen is composed of two core plates in 100 mm wide and 10 mm thick, and two cover plates in 100 mm wide and 10 mm thick, assembled by four M20 bolts. The bolt grade is selected to offer matching strength with steel plates (grade 10.9 for Q550 steel plates and grade 12.9 for Q690 and Q890 steel plates). The test specimen fabricated from mild steel (Q345) is composed of two core plates in 100 mm wide and 12 mm thick, and two cover plates in 100 mm wide and 12 mm thick, and assembled by four M20 bolts (10.9 grade).

534

Behavior and Design of High-Strength Constructional Steel

For specimens with coating the zinc-rich coating with two different thicknesses (60 and 80 µm) is applied on the surface through an electric arc. Before the steel plates are assembled, the coating thickness is measured using a positector magnetic gage and a total of five different spots are selected for each plate to determine the mean value of coating thickness. According to Chinese standard GB/T 8923.1-2011 [26], the shot-blasted surface is manufactured by shot-blasting machine with G70 steel grid to a quality SA 2.5. After shot blasting the steel plate has a gray metallic surface with some residual rust spots. In addition, to consider the effect of rusting of the slip resistance, a part of the shot-blasted steel plates was exposed to laboratory condition for 20 days to get rusted faying surfaces. The average room temperature of the laboratory is about 20 C. The relative humidity is 60%
70%. The untreated surface, shot-blasted surface, and rusted surface after shot blasting are shown in Fig. 11.41. According to steel grades and surface treatments, 35 test specimens are divided into 7 series as summarized in Tables 11.11 and 11.12. According to Metallic Materials—Tensile Testing—Part 1: Method of Test at Room Temperature (GB/T 228.1-2010) [27], mechanical properties of four grades of steel (Q345, Q550, Q690, and Q890) are investigated using uniaxial tensile coupon tests. The dimension of the tensile coupon is shown in Fig. 11.42. In the loading process, in order to eliminate the effect of strain rate, the tensile specimen is pulled at a rate of 0.5 kN/s. The load is recorded by the MTS test machine and the elongation of the specimen is measured by strain gauges. For each strength grade steel, three coupons were tested to investigate the material properties.

11.3.2.2 Material properties Fig. 11.43 shows the engineering stress
strain curves of four grades of steels obtained from uniaxial tensile tests. Different from Q550 and Q345 steels, there is no clearly defined yield plateau in stress
strain curves of Q690 and Q890 steels. Table 11.13 shows mechanical properties of four grades of steel. The yield strength is defined as 0.2% offset yield strength for Q690 and Q890 steels. In Table 11.13, εu is the strain corresponding to the ultimate strength and εy is the yield strain. The plastic deformation ability can be compared in terms of εu to εy ratio. It is generally accepted that the increasing of steel strength leads to the decrease in plastic deformation capacity. However, due to the different rolling technologies and chemical compositions adopted by the two steel and iron companies, the plastic deformation ability of Q890 steel is even better than Q690 steel in this test.

11.3.2.3 Preloading of bolt According to EN 1993-1-8 [9], the pretension force of the high-strength bolt FP;C is calculated as follows: FP;C 5 0:7fu Ae

(11.19)

Figure 11.40 Dimensions of standard test specimens (unit: mm).

536

Behavior and Design of High-Strength Constructional Steel

Figure 11.41 Faying surface treatments.

Table 11.11 Classification of specimens. Specimen

Steel grade

Surface treatment

Bolt grade

Thickness

Repeat

B-345 B-550 RB-550

Q345 Q550 Q550

10.9 10.9 10.9

12 10 10

5 5 5

B-690 RB-690

Q690 Q690

12.9 12.9

10 10

5 5

B-890 RB-890

Q890 Q890

Shot blasting Shot blasting Rusting after shot blasting Shot blasting Rusting after shot blasting Shot blasting Rusting after shot blasting

12.9 12.9

10 10

5 5

Table 11.12 Classification of specimens with inorganic zinc-rich coating. Specimen

Bolt grade

Steel grade

Coating thickness (µm)

Repeat

Q345-60 Q550-60 Q690-60 Q890-60 Q345-80 Q550-80 Q690-80

10.9 10.9 12.9 12.9 10.9 10.9 12.9

Q345 Q550 Q690 Q890 Q345 Q550 Q890

60

5 5 5 5 5 5 5

80

Figure 11.42 Tensile test coupon geometry (unit: mm).

Figure 11.43 Stress
strain curves of steel.

where fu is the ultimate tensile strength of the bolt (fu 5 1040 MPa for grade 10.9 bolts and fu 5 1220 MPa for grade 12.9 bolts); Ae is effective area of the bolt (Ae 5 245 mm2 for bolts with the diameter of 20 mm). According to Eq. (11.19), the target pretension forces of grade 10.9 bolts and grade 12.9 bolts are 155 and

538

Behavior and Design of High-Strength Constructional Steel

Table 11.13 Mechanical properties. Steel grades Q345 Q550 Q690 Q890

Yield strength (MPa) 371.6 677.0 767.7 1021.7

Ultimate strength (MPa) 531.3 757.3 865.7 1063.7

Elongation (%) 28.8 18.5 13.8 14.5

εu/εy 22.7 21.4 10.7 12.7

180 kN, respectively. In general, the reduction of the pretension force is low for the uncoated faying surface. The preload is applied by using a torque wrench. Two strain gauges are symmetrically attached on the notched sides near bolt head to measure the axial strain, as shown in Fig. 11.44. Consequently, the axial strain of bolt shank is monitored to control the pretension force. The applied pretension force Fp can be obtained by the following equation: Fp 5 E 3 εave 3 An

(11.20)

where E 5 206 GPa is elastic modulus of the bolt steel; εave is the average axial strain measured by the two strain gauges; and An is the net area of the notched section.

11.3.2.4 Test setup Slip tests are carried out on a 1000 kN universal testing machine at Tongji University, as shown in Fig. 11.45. Two LVDTs are attached to each end to measure the relative slip between core plates and cover plates, as shown in Fig. 11.46. To prevent the influence by bolt bearing, the pretension of the bolt is applied following the three steps shown in Fig. 11.47. (1) Insert bolt and fasten nut by hand with little visible gap between cover plates and core plate; (2) apply a 5 kN compression force to make sure that the bolt shanks are contacted with hole walls, in other words, the gap between the bolt and hole is reserved in one side to prevent bearing force under tension force; and (3) preload all bolts to the target value according to EN 1993-1-8 and then the compression force is unloaded. In the formal loading process the applied tension force is in the rate of 0.5 kN/s until slips are observed at both ends and the load transfer is changed from friction to bolt bearing. The duration of each test ranges from 10 to 15 minutes.

11.3.3 Experimental results In order to assist the identification of slip, two white lines are marked on each end of the specimen, as shown in Fig. 11.48. Based on the observation, in most cases, two ends slip one by one, except for one specimen of series B-550 with two ends

Bolted connections

Figure 11.44 Arrangement of strain gauges.

Figure 11.45 Slip test setup.

539

540

Behavior and Design of High-Strength Constructional Steel

Figure 11.46 LVDT displacement transducers’ setup. LVDT, Linearly variable displacement transducer.

slipping simultaneously, and show the typical load
slip displacement curves obtained from tests. Based on observation, the test procedure can be characterized by four stages: (1) development of slip resistance up to the first peak load; (2) observation of slip at one end with sharp drop in testing load; (3) regain resistance up to the second peak load with one end (the slipped end) in bolt bearing and the other end in friction; and (4) the observation of slip at the second end companied with sharp drop in testing load. For Q345 and Q550 specimens the slip displacement increases steadily and the load
slip displacement curve is smooth after the first peak load is reached. On the contrary, for Q690 and Q890 specimens, slips are accompanied with huge noises. Fig. 11.49 shows that there is a sudden drop at this moment and then shows a serrated load
slip displacement curve after the occurrence of the first slip. This phenomenon indicates the effect of the steel ductility on the friction behavior of slip critical bolted connections (Figs. 11.50 and 11.51).

11.3.3.1 Long-term effect For the specimens with inorganic zinc-rich coating, the fifth specimen of each group is loaded with a specific load of 90% of the mean value of slip loads obtained from first four tests for 3 hours. The typical slip displacement
time curve of Q550 is plotted in Fig. 11.52. Based on experimental results shown in Table 11.14, it can be found that the for the fifth test of each group, the difference between the measured slip displacement at 5 minutes and at 3 hours with target load exceeds 2 µm. Consequently, the other three extended creep tests shall be tested.

Bolted connections

541

Figure 11.47 Installation and loading diagram of bolt.

Figure 11.48 Two slips of contacted plate pair: (A) before loading, (B) the first slip, and (C) the second slip.

542

Behavior and Design of High-Strength Constructional Steel

Figure 11.49 Typical load
slip displacement curves for Q345 and Q550 steels.

Figure 11.50 Typical load
slip displacement curves for Q690 and Q890 steels.

The duration time of creep tests is 40 days in this study. The typical slip displacement
log time of Q690 is plotted in Fig. 11.53. The linear fitting is conducted using these data with 0.967 fitting precision. The linear extrapolation of the slip displacement
log time curve is shown in Fig. 11.54. It can be observed that the 0.3 mm slip displacement is not reached in the extrapolation curve up to 50 years. Therefore the slip factor shall be determined from previous four short-term tests. According to Annex G of EN 1090, the slip loads are defined as the load corresponding to the slip of 0.15 mm. However, according to the friction theory, the static friction force is defined as the maximum possible friction force between two surfaces before sliding begins. In order to investigate the influence of the steel strength grade on the slip factor, it is more meaningful to use the peak resistance to investigate the slip critical behavior. Therefore the slip load is defined as the load at the peak point in the following sections. In most cases, two ends of the specimens do not slip simultaneously. Thus two different values of slip load can be obtained from one specimen, as shown in Fig. 11.55. A total of 10 slip loads can be obtained for each group of tests.

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543

Figure 11.51 Typical load
slip displacement curves for specimens with inorganic zinc-rich coating.

The slip factor μi for the specimen i, the mean value of slip factor μm, and the standard deviation of slip factor Sμ are calculated according to Annex G of EN 1090-2 as the following equations: μi 5

FSi n f n b Fp

(11.21)

P μm 5

μi n

(11.22)

544

Behavior and Design of High-Strength Constructional Steel

Figure 11.52 Slip displacement
time curve (Q550).

Table 11.14 Slip displacement of long-term test with a duration of 3 h. Specimen

End

Q550-60

Top Bottom Top Bottom Top Bottom Top Bottom Top Bottom

Q550-80 Q690-60 Q690-80 Q890-60

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 P μi 2μm Sμ 5 n21

Slip displacement (µm) 21.95 18.7 17.1 10.9 18.85 17.85 10.4 9.45 10.7 25.75

Mean value (µm) 20.33 14 18.35 9.93 18.225

(11.23)

where FSi is the slip load defined as the load at the peak point for the specimen i; Fp is the pretension force of the bolt; nf 5 2 for the number of slip planes; nb 5 2 for the number of bolts; and n 5 10 for the number of experimental data.

Bolted connections

545

Figure 11.53 Slip displacement
log time curve (Q690).

Figure 11.54 Linear extrapolation of the slip displacement
log time curve.

The mean value of slip load and the standard deviation of the slip load are calculated by the following equations: P FSi (11.24) FSm 5 n

546

Behavior and Design of High-Strength Constructional Steel

Figure 11.55 Definition of peak load in tests.

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P ðFSi 2FSm Þ2 SFs 5 n21

(11.25)

The characteristic value of the slip factor with a guaranteed value of 97.5% is given by the following equation: μcar 5 μm 2 2:05Sμ

(11.26)

Slip loads obtained from the test results, average μm and standard deviation Sμ of slip loads are summarized in Table 11.15. The characteristic values of the slip factor are also given in Table 11.15. The relation between the mean value of slip factor and the steel grade is illustrated in Fig. 11.56. It can be found that the influence of steel grade on the slip factor is not negligible, especially for HSS, which is neglected in most design standards. For shot-blasted surfaces the characteristic values of slip factors of Q550, Q690, and Q890 steel specimens are lower than Q345 by 2.2%, 16.4%, and 11.4%, respectively. It should be mentioned that the characteristic values of slip factors of Q690 and Q890 are lower than 0.5 given by most of design standards. To evaluate the effect of rusting on the slip factor of shot-blasted surface, the test results of series RB-550, RB-690, and RB-890 are compared with the shot-blasted specimens in Fig. 11.56. It can be observed that for Q550 and Q690 steels, the rusted surface after blasting develops the higher roughness than that of unrusted surface, which results in the higher slip factor. For Q890 steel, however, the slip factor of rusted shot-blasted surface is lower

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547

Table 11.15 Slip factors. Series

B-345 B-550 B-690 B-890 RB-550 RB-690 RB-890

Steel grade

Q345 Q550 Q690 Q890 Q550 Q690 Q890

Slip load

Slip factor

FS,max (kN)

FS,min (kN)

FSm (kN)

μm

Sμ (%)

μcar

395.6 380.8 399.4 419.9 396.4 454.2 416.4

351.3 325.0 344.1 359.0 353.8 335.4 351.5

369.9 364.7 360.8 394.6 377.2 377.0 386.0

0.594 0.591 0.490 0.532 0.610 0.509 0.520

2.5 2.9 1.7 2.5 2.1 4.7 2.9

0.543 0.531 0.454 0.481 0.567 0.412 0.461

Note: FS,max is maximum value of slip loads in one series of test; FS,min is minimum value of slip loads in one series of test.

Figure 11.56 Influence of steel grade on slip factors (mean value).

than that of unrusted shot-blasted surfaces by 4%. Moreover, it is worth noting that, for the specimens with rusted surface, the standard deviations of the test results are higher (up to 4.7%) than the results of the specimens with shot-blasted surface (up to 2.9%). This is mainly attributed to the uncertainty of the environment effect, which results in the random distribution of rust on the faying surface after exposure. Therefore, in design procedure, it is not recommended to consider the positive influence of rusting on the slip factor in design.

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Behavior and Design of High-Strength Constructional Steel

11.3.4 Discussion 11.3.4.1 Vickers hardness-steel grade According to Metallic Materials-Vickers hardness test-part 1(GB/T 4340.1-2009) [28], the Vickers hardness tests are carried out to measure the Vickers Pyramid Number (HV) of steel specimens used in this research. The Vickers hardness tester with a diamond indenter is used, as shown in Fig. 11.57. The loading force F is 500 g with a duration time of 10 seconds. After loading the length of the test inden-

Figure 11.57 Vickers hardness test machine.

Figure 11.58 Typical test indentation.

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Figure 11.59 Relationships of tensile strength and Vickers hardness.

tation d is measured by microscope, as shown in Fig. 11.58. The HV is calculated by the formula: HV 5

 F kgf=mm2 A

(11.27)

where A is indentation area, which can be obtained by A5

d2   2sin 136 =2 

(11.28)

The Vickers hardness of specimen is defined as the mean value of 10 times testing results. Fig. 11.59 shows that the Vickers hardness of steels has a positive correlation with the steel tensile strength. The Vickers hardness of Q550 steel is slightly higher than Q345 mild steel by 4%. For Q690 and Q890 steels the Vickers hardness are higher than Q345 steel by 32% and 68%, respectively.

11.3.4.2 Vickers hardness
roughness The roughness of the shot-blasted surface is defined to qualify the irregularities of the surface. The surface roughness is determined by the deviations in the direction of the normal vector of a real surface from its ideal form. According to international organization standard ISO 8503-4 [29], the surface roughness is calculated by the arithmetical mean value of deviation of the roughness profile Ra , as shown in Fig. 11.60. Ra 5

Ðl   ydx 0

l

(11.29)

550

Behavior and Design of High-Strength Constructional Steel

Figure 11.60 Computation method of surface roughness.

Figure 11.61 Diagrammatic sketch of roughness measurement.

where l is sampling length and y is the vertical distance from the mean line to the integration point. A profilometer with a diamond stylus is used to measure surface roughness of steel plates in this study. For a selected sampling area a total of 10 testing paths, including four paths in transverse direction, four paths in longitudinal direction, and two paths in diagonal direction are measured, as shown in Fig. 11.61. The mean value is adopted to represent the roughness of the faying surface. The tensile strength, Vickers hardness and surface roughness of the specimens fabricated from different grades of steels are summarized Fig. 11.62. It is found that the surface roughness has a negative liner correlation with the Vickers hardness. Under the same shot-blasting process, the higher steel grade (hardness), the

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Figure 11.62 Relationships of surface roughness and steel hardness.

Figure 11.63 Relationships of slip factors, surface roughness and steel grade.

lower surface. According to the friction mechanism theory, the friction consists of furrow force and adhesion force. The furrow force is produced by the interlock action between furrows, which depends on the roughness of the treated surface of steel plate. Since it is difficult to separate the furrow force part from the static friction, Fig. 11.63 shows the relationships between slip factor and surface roughness. For Q345, Q550, and Q690 steels the slip factor decreases with the surface roughness. The trend of slip factor agrees with that of the furrow force part. For Q890 steel plate the slip factor is higher than Q690 steel specimens by 8.6%, although the surface roughness of Q890 steel plate is lower than that of Q690 steel plate. Therefore, as the other part of friction, adhesion force should be investigated.

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Behavior and Design of High-Strength Constructional Steel

Figure 11.64 Relationships of slip factor and plasticity index εu/εy.

11.3.4.3 Effect of plasticity deformation capacity on slip factor In addition to the furrow force part, the adhesive force is the other constituent part of friction based on the friction mechanism theory. The adhesion force is the product of the adhesion effect between contact area. During the loading process, the contacted area increases due to the plastic deformation of faying surfaces under transverse force. In this case the increase in plastic deformation capacity of steel results in the increasing of contacted area and the higher adhesive force, which means a higher slip factor. On the contrary, if the steel with low plasticity deformation capacity is used, under transverse loading, the initial contacted peaks may be destroyed rather than forming new contacted area, which induces the decreasing of the slip factor. In the present study the deformation capacity of steel plates is measured by plasticity index, which is defined as the ratio of εu to εy. The relationships between steel grade and slip factors and plasticity index are shown in Fig. 11.64. It is worth noting that the Q690 and Q890 steel plates are produced by two different companies. The different rolling technologies and chemical compositions adopted by the two steel and iron companies result in the higher plasticity index for Q890 steel than that of Q690 steel. Thus the plasticity index curve shows a drop with the steel grade up to Q690, followed by a recovery for Q890 steel. Compared with the slip factor curve, the similar trend can be observed. However, the trend is different from the surface roughness curve shown in Fig. 11.64. This means the higher adhesive force caused by the better deformation capacity of Q890 steel can complement the reduced furrow force due to the lower surface roughness and provide a comparable slip factor with an 8.6% increase.

11.3.5 Design recommendation Based on the observation of test results, the slip factor is dependent on the steel strength grade for shot-blasted surfaces and inorganic zinc-rich coating surfaces. However, in

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Table 11.16 Slip factors according to EN 1090. Series

B-550 RB-550 B-690 RB-690 B-890 RB-890

Steel grade

Q550 Q550 Q690 Q690 Q890 Q890

Slip load

Slip factor

FS,max (kN)

FS,min (kN)

FSm (kN)

μm

Sμ (%)

μcar

332.2 358.9 356.9 362.9 365.5 381.9

285.7 267.9 328.7 314.9 304.2 313.3

305.2 306.1 338.4 341.1 336.4 360.2

0.502 0.494 0.457 0.461 0.455 0.487

2.1 3.9 1.2 1.9 2.7 2.7

0.459 0.415 0.431 0.422 0.400 0.432

Table 11.17 Recommended slip factor for high-strength steels. Faying surface treatment

Steel grade

Shot-blasted

Q550

Q690

Q890

0.40

0.40

0.40

Table 11.18 Slip factors for specimens with inorganic zinc-rich coating. Specimen

Slip load FS,max (kN)

Q345-60 Q550-60 Q550-80 Q690-60 Q690-80 Q890-60

229.5 212.6 233.9 253.3 278.3 261.2

Slip factor

FS,min (kN)

FSm (kN)

μm

Sμ (%)

μcar

190.1 189.2 215.7 215.2 251.0 237.9

206.1 202.9 225.0 238.9 264.2 250.2

0.332 0.327 0.363 0.323 0.357 0.338

2.407 1.580 1.156 1.579 1.293 0.980

0.283 0.295 0.339 0.291 0.331 0.318

most of current design codes, the effect of the steel strength grade on the slip factor is not considered. According to Annex G of EN 1090, the slip loads are defined as the values of load corresponding to 0.15 mm of slip displacement rather than the ultimate slip resistance. Consequently, the slip loads at slip displacement of 0.15 mm and corresponding the slip factor are calculated and summarized in Tables 11.16 and 11.17. Based on the characteristic values of the slip factor, the slip factors for HSSs are recommended, as shown in Tables 11.18 and 11.19. For the specimens with rusted surface after blasting, there may be an increment in the slip factor owing to the increased surface roughness by

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Behavior and Design of High-Strength Constructional Steel

Table 11.19 Recommended slip factor for specimens with inorganic zinc-rich coating. Coating thickness (mm)

60 80

High-strength steel grade Q550

Q690

Q890

0.25 0.25

0.25 0.25

0.25 0.25

rusting. However, the positive influence of rusting on the slip factor is not recommended to be considered in the connection design due to the nonuniform rusting and its sensitivity to storage conditions. Therefore the recommended slip factor of the specimen with rusted surface after blasting takes the same value as the shot-blasted surface.

11.3.6 Summary Based on the investigations on the slip critical
type bolted connections for HSSs Q550, Q690, and Q890, the following conclusions can be summarized: 1. The slip factor is not only influenced by surface roughness (furrow force) but also can be affected by the plastic deformation capacity of the contacted materials (adhesion force). With the increasing of plastic deformation capacity of steel, the slip factor tends to be higher. 2. According to the experimental results, the recommended slip factor for the tested HSSs Q550, Q690, and Q890 with shot-blasted surface is 0.40. 3. For the rusted surface after shot blasting, there may be an increment in the critical slip resistance owing to the increased surface roughness by rusting. However, the positive influence by rusting is not recommended to be considered in the design of slip critical connection due to the dispersed test results. Therefore the recommended slip factor of the specimen with rusted surface after blasting is also 0.40. 4. With the same surface treatment of shot blasting, the surface roughness tends to decrease with the increase of steel hardness (steel strength), which results in decreasing of the furrow force between the faying surfaces. 5. The slip factor of steel surface with thick inorganic zinc-rich coating (thickness 5 80 mm) is higher than that of surface with thin coating (thickness 5 60 mm) by 10%. 6. Extended creep tests are carried out to study the long-term effect on slip resistance and stability of HSS surface with inorganic zinc-rich coating. Experimental results report that the 0.3mm slip displacement will not be reached in the extrapolation curve up to 50 years. As a result, the related slip factor shall be determined from short-term tests. 7. The recommended value of slip factor of HSS with inorganic zinc-rich coating is 0.25, which is lower than that in current design codes.

11.4

Experimental study on slip factor of hybrid connections

11.4.1 Introduction To achieve both safety and economy, HSS structures in seismic zones are generally recommended to be designed as dissipative structures, which allow for the

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formation of plastic hinges in the expected locations. Since the ductility decreases with the increase in yield strength, the mild steel members with good ductility are recommended to be used combined with HSS members to satisfy the demands of structural plastic behavior during earthquake actions, where HSS members are used as the load-bearing structural elements (columns), and mild steel members are designed as energy-dissipating components (beams and braces). Consequently, hybrid connections between mild steel and HSS are inevitable in HSS structures. The investigation on the slip resistance of hybrid slip critical connection between HSS and mild steel is limited. The effect of discrepancies of hardness and plastic deformation capacity between mild steel and HSS on the slip factor should be examined rather than inconclusively taking the recommended values in existing codes for mild steel. In order to evaluate the slip resistance of hybrid connections between HSS and mild steel with shot-blasted surfaces, 30 specimens fabricated from HSS (Q550, Q690, and Q890) and mild steel (Q235 and Q345) are tested [30]. According to the experimental results, the influences of steel grade on the slip factor are evaluated in terms of furrow force and adhesion force according to friction mechanism. The slip factors of shot-blasted hybrid connections between HSS and mild steel are recommended.

11.4.2 Specimen of slip critical test The specimens for slip critical test are designed according to Annex G of EN 10902, as shown in Fig. 11.40. Each test specimen is assembled by two core plates and two cover plates with four M20 bolts. The two core plates are fabricated from HSS with the width of 100 mm and thickness of 10 mm. The two cover plates are fabricated from mild steel with the same width of 100 mm. The thicknesses of Q235 and Q345 steel plates are 12 and 10 mm, respectively. The grade 10.9 bolt and grade 12.9 bolt are selected to offer matching strength with Q550 and Q690/Q890 steel core plates. The surfaces of both core and cover plates a12.0re treated by shot-blasting machine to achieve the cleanliness level of SA 2.4. After shot-blasting, rust, corrosion products, oxides, and other foreign matter have been thoroughly removed, except for very light shadows or streaks caused by rust stain. The shot-blasted steel plate has a near white metal surface, as shown in Fig. 11.65. With three grades of core steel plate (Q550, Q690, and Q890) and two grades of cover steel plate (Q235 and Q345), six groups of specimens are prepared, as summarized in Table 11.20.

11.4.3 Experimental results 11.4.3.1 Load
slip curve In order to monitor slips, two straight lines are marked on top and bottom parts of the specimen, as shown in Fig. 11.66. The typical load
slip displacement curves for specimens obtained from tests are shown in Figs. 11.67 and 11.68.

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Behavior and Design of High-Strength Constructional Steel

Figure 11.65 Shot-blasted surface.

Table 11.20 Classification of specimens. Specimen HC550-235 HC-550-345 HC-690-235 HC-690-345 HC-890-235 HC-890-345

Grade of bolt 10.9 10.9 12.9 12.9 12.9 12.9

Grade of core plate

Grade of cover plate

Q550 Q550 Q690 Q690 Q890 Q890

Q235 Q354 Q235 Q345 Q235 Q345

Repeats 5 5 5 5 5 5

11.4.3.2 Slip factor All characteristic of slip loads and slip factors mentioned previously are listed in. For comparison, results of 15 specimens fabricated from the same grade of HSS in reference are reproduced in Table 11.21.

11.4.3.3 Analysis of test result The mean values of slip factor with different strength grades of steels are illustrated in Fig. 11.69. It can be found that for hybrid connection between HSS and mild steel, the influence of steel strength grade on the slip factor should be considered, which is verified in previous publication but ignored in most design codes. The slip factors of hybrid

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Figure 11.66 Two slips of contacted plate pair: (A) before loading, (B) the first slip, and (C) the second slip.

Figure 11.67 Typical load
slip displacement curves for hybrid connection specimen.

Figure 11.68 Typical load
slip displacement curves for HSS connection specimen. HSS, High-strength steel.

connections are higher than that of pure HSS connections up to 15%. For hybrid connections between Q235 steel and HSS, the mean value of slip factors decreases with the increasing of HSS grade. The mean value of slip factors of Q690
Q235 and

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Behavior and Design of High-Strength Constructional Steel

Table 11.21 Slip factors. Specimen

Slip load 

HC-550-235 HC-550-345 HC-690-235 HC-690-345 HC-890-235 HC-890-345 SC-550 SC-690 SC-890 

FS,max (kN) 400.8 392.9 436.8 432.0 443.8 430.0 380.8 399.4 419.9

Slip factor 

FS,min (kN)

FSm (kN)

μm

Sμ (%)

μcar

346.4 347.8 395.0 387.5 358.1 368.2 325.0 344.1 359.0

375.8 368.0 418.9 408.6 396.8 398.6 364.7 360.8 394.6

0.608 0.598 0.564 0.550 0.544 0.541 0.591 0.490 0.532

2.8 2.4 2.1 2.1 4.6 2.6 2.9 1.7 2.5

0.551 0.549 0.520 0.508 0.449 0.488 0.531 0.454 0.481

FS;max and FS;min are maximum and minimum values of slip loads in one group of slip tests, respectively.

Figure 11.69 Slip factors for hybrid connections and HSS connections. HSS, High-strength steel.

Q890
Q235 specimens are lower than Q550
Q235 by 7.8% and 11.8%, respectively. Similar trend is observed in Q345
HSS hybrid connections. The mean value of slip factors of Q690
Q345 and Q890
Q345 specimens are lower than Q550
Q345 by 8.7% and 10.5%, respectively. For hybrid connections with the same HSS but different mild steels, the mean value of slip factor of the Q235 steel combination tends to be slightly higher (up to 2.5%) than the Q345 steel combination.

11.4.4 Discussion 11.4.4.1 Friction mechanism In the 1950s, Frank Philip Bowden and David Tabor studied the friction theory microscopically and proposed a friction model, as shown in Fig. 11.39. The static

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Figure 11.70 Diagram of effect of furrow force between contact surfaces: (A) smooth surface and (B) rough contact surface.

Figure 11.71 Relationships of tensile strength and steel hardness.

friction consists of furrow force and adhesion force, which are the products of interlock action between furrows of faying surface and adhesion effect between contact surfaces, respectively. Accordingly, the influence of steel grade on the slip factor of hybrid connection between HSS and mild steel is discussed from following two aspects: furrow force and adhesive force.

11.4.4.2 Effect of furrow force The furrow force is the product of interlock action between furrows. It is generally accepted that furrow force depends on the roughness of the contacted surface pair, as shown in Fig. 11.70. Under the same surface treatment of shot blasting, the roughness of the treated surface may be affected by the harnesses of different grade of steels. This may result in the variation of the contribution of furrow force.

11.4.4.3 Hardness of steel plates The mean value of Vickers hardness results is taken as the Vickers Pyramid Number of each strength grades of steel plates, as shown in Fig. 11.71. It can be

560

Behavior and Design of High-Strength Constructional Steel

Figure 11.72 Relationships of slip factors, surface roughness, and steel grade.

observed that there is a positive correlation between the Vickers hardness of steels and the steel tensile strength. The Vickers Pyramid Numbers of Q345, Q550, Q690, and Q890 steels are higher than that of Q235 steel by 18%, 23%, 56%, and 98%, respectively.

11.4.4.4 Roughness of steel plates The Vickers hardness and surface roughness of different strength grades of steels are plotted in Fig. 11.59. It can be observed that there is a negative correlation between Vickers hardness and surface roughness, which means that with the same surface treatment, the lower surface roughness is obtained when steels with the higher hardness (strength) are used. It is generally accepted that the decreasing of surface roughness induces the decreasing of contribution of furrow force to friction. Accordingly, as shown in Fig. 11.72, for hybrid connections, the slip factor tends to be lower with the increasing of the steel strength.

11.4.4.5 Effect of adhesive force Based on friction model proposed by Bowden, the adhesion force is the product of the adhesion effect between the contact area. Under normal pressure and transverse shear force, with the plastic deformation of the faying surfaces, the contacted area may increase, as shown in Fig. 11.73. In the present experimental study, the increase of the contacted area between the surface pair is characterized by relative critical slip displacement when slip failure occurs. The increase in critical plastic slip deformation between the contacted steel plates before slip failure results in the increasing of contact area as well as adhesion force. Consequently, a higher slip factor may be achieved. The relationship between slip factors and relative critical slip displacements for different hybrid connections

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Figure 11.73 Asperities between the contacted surface pair.

Figure 11.74 Relationships of slip factors, surface roughness, and steel grade.

is shown in Fig. 11.74. For hybrid connections between Q235 steel and HSS, the relative critical slip displacement and the slip factor decreases with the increasing of HSS strength grades. The relative critical slip displacement of the specimen HC550-235 is higher than those of the specimens HC-690-235 and HC-890-235 by 26.1% and 36.8%. Similarly, the slip factor of the specimen HC-550-235 is higher than those of the specimens HC-690-235 and HC-890-235 by 7.8% and 11.8%. The hybrid connections between Q345 steel and HSS show the similar trend. This is because the strength discrepancy between the mild steel and HSS leads to the appearance of plastic deformation earlier in the mild steel than in the HSS. The higher grade of HSS, the more likely localization of plastic deformation in mild steel rather than HSS, which results in a lower contacted area as well as a lower slip factor.

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Behavior and Design of High-Strength Constructional Steel

Table 11.22 Slip factors according to EN 1090. Specimen

Slip load

Slip factor

FS,max (kN)

FS,min (kN)

FSm (kN)

μm

336.1 333.6 408.9 386.5 390.0 388.7

299.3 285.2 354.0 316.3 326.8 335.3

318.0 302.9 384.1 360.7 354.7 364.7

0.513 0.488 0.519 0.487 0.479 0.493

HC-550-235 HC-550-345 HC-690-235 HC-690-345 HC-890-235 HC-890-345

Sμ (%) 2.2 2.5 2.7 2.9 2.6 2.2

μcar 0.467 0.437 0.463 0.428 0.426 0.447

Table 11.23 Recommended slip factor for hybrid connection. Mild steel grade

Q235 Q345

High-strength steel grade Q550

Q690

Q890

0.45 0.40

0.45 0.40

0.40 0.40

11.4.5 Design recommendation The test results show that the slip factor of hybrid connections between mild steel plate and HSS plate is related to the strength/hardness steel and strength difference. However, the influence of these factors is not considered in existing design codes. In Annex G of EN 1090 the slip loads are defined as the load at the slip of 0.15 mm instead of the peak load. Therefore the slip loads at the slip of 0.15 mm are summarized in Table 11.22. The mean values and characteristic values of the slip factors are calculated and listed in Table 11.22. According to the characteristic values of the slip factor, the slip factors for hybrid connection between HSS and the mild steel with shot-blasted surfaces are recommended in Table 11.23.

11.4.6 Summary A total of 30 hybrid connection specimens are tested to investigate the influences of hardness and strength difference of steel plates on the slip factor. Five strength grades of steel are investigated, including mild steel (Q235 and Q345) and HSSs (Q550, Q690, and Q890). The conclusions can be drawn as follows: 1. The slip factors of hybrid connections between HSS and mild steel are lower than those of connection fabricated by the same grade of mild steel. Therefore it is not conservative to adopt the slip factors for mild steel in existing design code to design the hybrid connections.

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2. With the same shot-blasting surface treatment, lower surface roughness is obtained when higher hardness (strength) steel was used, which impairs the contribution of furrow force to friction and reduces the slip factor for hybrid connection between HSS and mild steel. 3. For hybrid connections between HSS and mild steel, the higher strength difference, the lower relative slip deformation between the contacted surface pair before slip failure. Attributed to the large discrepancy of strength between mild steel and HSS, the plastic defamation may be localized in the mild steel and impair the potential increase in contacting are of the surface pair. This may influence the slip factor by reducing the contribution of adhesion force. 4. The slip factor of 0.45 is recommended for hybrid connections between Q550/Q690 HSS and Q235 mild steel. The slip factor of 0.40 is recommended for hybrid connections between Q550/Q690 HSS and Q345 mild steel. For hybrid connections between Q890 HSS and mild steel Q235/Q345, slip factor of 0.40 is recommended.

References [1] Li GQ, Wang YB, Chen SW. The art of application of high-strength steel structures for buildings in seismic zones. Adv Steel Constr 2015;11(4):492
506. [2] Wang YB, Lyu YF, Li GQ, Liew JR. Bearing-strength of high strength steel plates in two-bolt connections. J Constr Steel Res 2019;155:205
18. [3] Wang YB, Lyu YF, Li GQ, Liew JR. Behavior of single bolt bearing on high strength steel plate. J Constr Steel Res 2017;137:19
30. [4] Lyu YF, Wang YB, Li GQ, Jiang J. Numerical analysis on the ultimate bearing resistance of single-bolt connection with high strength steels. J Constr Steel Res 2019;153:118
29. [5] National Standardization Technical Committees. (in Chinese) High strength structural steel plates in the quenched and tempered condition. Beijing: China Standard Press; 2009. [6] European Committee for Standardization. Hot rolled products of structural steels—Part 6: Technical delivery conditions for fiat products of high yield strength structural steels in the quenched and tempered condition. Brussels; 2004. [7] National Standardization Technical Committees. (in Chinese) Metallic materials: tensile testing at ambient temperature. Beijing: China Standard Press; 2002. [8] GB50017-2013 (in Chinese) Code for design of steel structures. Beijing: China Architecture & Building Press; 2018. [9] European Committee for Standardization. EN 1993-1-8 Eurocode 3: design of steel structures—Part 1-8: Design of joints. Brussels; 2005. [10] Moˇze P, Beg D. High strength steel tension splices with one or two bolts. J Constr Steel Res 2010;66(8
9):1000
10. [11] Moˇze P, Beg D. A complete study of bearing stress in single bolt connections. J Constr Steel Res 2014;95:126
40. [12] Snijder HH, Ungermann D, Stark JWB, Sedlacek G, Bijlaard FSK, HemmertHalswick A. Evaluation of test results on bolted connections in order to obtain strength functions and suitable model factors Part A: Results. 1988. [13] Zafari B, Qureshi J, Mottram JT, Rusev R. Static and fatigue performance of resin injected bolts for a slip and fatigue resistant connection in FRP bridge engineering. Structures 2016;7:71
84.

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[14] AISC Committee on Specifications. Specification for structural steel buildings. Chicago; 2010. [15] Research Council on Structural Connections. Specification for structural joints using high-strength bolts. 2009. [16] Canadian Standards Association. Canadian highway bridge design code. Mississauga; 2006. [17] GB50017-2017. National code for design of steel structures. China Planning Press; 2017. [18] Heistermann C, Veljkovic M, Simo˜es R, Rebelo C, Da Silva LS. Design of slip resistant lap joints with long open slotted holes. J Constr Steel Res 2013;82:223
33. [19] Cruz A, Simo˜es R, Alves R. Slip factor in slip resistant joints with high strength steel. J Constr Steel Res 2012;70:280
8. [20] Kulak GL, Fisher JW, Struik JH. Guide to design criteria for bolted and riveted joints. 2nd ed. 2001. [21] European Committee for Standardization. Hot rolled products of structural steels—Part 1 to 6. Brussels; 2004. [22] Yamamoto T. History of science on friction. J Jpn Soc Mech Eng 2005;108:256
60. [23] Wang YB, Wang YZ, Chen K, Li GQ. Slip factors of high strength steels with shot blasted surface. J Constr Steel Res 2019;157:10
18. [24] Wang YB, Wang YZ, Chen K, Jin HJ. Slip factor of high strength steel with inorganic zinc-rich coating. Thin-Walled Struct 2020;148:106595. [25] European Committee for Standardization. Execution of steel structures and aluminum structures—Part 2: Technical requirements for the execution of steel structures. Brussels; 2008. European Committee for [26] GB/T 8923.1-2011. Preparation of steel substrates before application of paints and related products-visual assessment of surface cleanliness. Beijing: China Standard Press; 2012. [27] GB/T 228.1-2010. Metallic materials—tensile testing—Part 1: Method of test at room temperature. Beijing: Standards Press of China; 2011. [28] Metallic materials—vickers hardness test—Part 1: Test method. Beijing: China Standard Press; 2009. [29] International Organization for Standardization. Preparation of steel substrates before application of paints and related products—surface roughness characteristics of blastcleaned steel substrates—Part 4 Method for the calibration of ISO surface profile comparators and for the determination of surface profile—stylus instrument procedure. Switzerland; 2012. [30] Wang YB, Wang YZ, Chen K, Li GQ. Slip factor between shot blasted mild steel and high strength steel surfaces. J Constr Steel Res 2020;168:105969.

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12

Fei-Fei Sun1, Ming-Ming Ran1,2, Guo-Qiang Li1 and Yan-Bo Wang1 1 Tongji University, Shanghai, P.R. China, 2Sichuan University, Chengdu, P.R. China

12.1

Introduction

Chemical and metallurgical changes involved in the welding process are critical considerations in the welded connections for steel structures. In recent years more and more researchers [1
9] have noted that the heat-affected zone (HAZ) of highstrength steels (HSSs) trends to result in a softened region, impairing the strength and the ductility of the welded joints. In essence, the emergence of softened zone is dependent on the applied strengthening mechanisms of the steel plate and their behavior and susceptibility to thermal heat treatment. Two common strengthening mechanisms for high-strength low-alloy steels are thermomechanical controlled processing (TMCP) and quenched and tempered (QT) processing. Softening may be classified as being the result of peak temperature: tempering softening and transformation softening [1]. QT steels usually consist of a metastable transformation microstructure of martensite or bainite, which results in both tempering softening and transformation softening occurring in HAZ, while as softening in the HAZ of TMCP steels is mainly restricted to transformation softening, resulting width of softened zone in TMCP steels is smaller in comparison to QT [1,10]. As for external cause, with the increase of heat input for a given HSS [4,11
16], the softened zone width increases almost linearly while the strength in softened zone drops until approaching a saturation value. It is well established that [1,17
24] if the relative width of softened zone Xsz (Fig. 12.1) decreases, the strength will increase and may reach the strength of the base metal (BM). Furthermore, very small width of the softened zone does not impair the global strength, which can be explained by the occurrence of the socalled contact strengthening due to the constraints by adjacent unyielded materials [1,35]. Experimental and numerical studies showed that this constraint effect is only effective if Xsz do not exceed certain value. de Meester [24], Maurer et al. [5] and Hochhauser et al. [1] separately provided this value are 1.0, 0.25, and 0.33, respectively. Meanwhile, all of them clearly pointed out that the limitation value of Xsz is affected by the strength of its adjacent material. Notwithstanding this research on softening mechanisms and effects of softened zone, the experimental studies in the existing publications are somewhat sparse. First, only one or two steels have been tested in one article [5,26,27] and the strengthening mechanisms have not been studied comprehensively; even though the Behavior and Design of High-Strength Constructional Steel. DOI: https://doi.org/10.1016/B978-0-08-102931-2.00012-7 © 2021 Elsevier Ltd. All rights reserved.

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Behavior and Design of High-Strength Constructional Steel

Figure 12.1 Welded joint with a softened zone and explanation for denotations [13].

results from different articles could be compared, the welding procedures and loading conditions may differ. Second, undermatched filler metals are first allowed to be used in welding HSSs in Eurocode 3 Part 1
12 (EC3) [28], the test data for validating the feasibility of the specification are not sufficient. Third, although hardness distribution has always popular to characterize softening, little attention has been focused on hardness distribution classification and quantitative analysis. Perhaps most important, most discussions in present publications are focused on overall behavior (load and deformation), while the strain distribution over the specimen during the loading process is merely analyzed due to technique issue. Regarding that Rodrigues et al. [29] presented strain and stress distributions for butt joints with different softening ratio and softened zone width based on simulation, its feasibility is deserved to be validated because the finite element model has not been verified by tests. Even though the softening phenomenon can be observed directly from experiments, the experimental analysis of the true stress and true strain distribution in a welded joint is very difficult to achieve. The nonuniform strain and stress distribution in HAZ and the welding affected material properties make all possible experimental analysis very complicated. Numerical simulation is an effective alternative way to do a one-by-one evaluation of the parameters that may affect the tensile strength of a welded joint. Numerical simulation has already become an essential tool to further clarify and explain experimental observations as well as provide detailed analysis of the mechanical behavior of composite materials in different patterns. Based on the numerical analysis, Rodrigues et al. [29] evaluated the influence of the strength and width of softened HAZ (SHAZ) on the plastic behavior of butt joint with the same material model between BM and welded metal (WM). It was observed that the overall strength of the joint was determined by the tensile strength of SHAZ and the joint strength showed a linearly relationship with the width of SHAZ. More detailed numerical analysis was performed by Maurer et al. [5]. They studied the influence of the strength and width of SHAZ and WM and the angle of bevel. The key influence factors were ranked empirically as follows: relative width

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of softened zone . softening ratio . mismatch ratio . welding groove type and the angle of bevel (negligible). Present research was focused on the qualitative relation between the relative width of softened zone and the load-carrying capacity. It is well established that [2,16,18
21,24], [1,17
24,27,28], if the relative width of softened zone Xsz decreases, the strength will increase and may reach the strength of the BM. This can be explained by the occurrence of the so-called contact strengthening due to the constraints by adjacent unyielded materials [1,25]. Experimental and numerical studies showed that this constraint effect is able to achieve matching strength connection of butt joint with SHAZ, if Xsz do not exceed a threshold value. The threshold values of 1.0, 0.25, and 0.33 were proposed by Meester [24], Maurer et al. [5], and Hochhauser et al. [1], respectively. The conflicts on the proposed threshold value of the relative width of softened zone by different researchers indicate that it is not the unique parameter and there may be other parameters affecting the constraint effect. Further study is necessary to achieve a quantitative relation between the influence factors and the constraint effect. Based on the recently work by the authors [30], three hardness distribution patterns of butt joints were defined for HSS. Considering using undermatched, evenmatched, and overmatched weld metals, there are nine different combinations of hardness distribution pattern of butt joints. The effect of mechanical parameters and geometry parameters on the load-bearing capacity of butt joints with different hardness distribution patterns should be investigated. Although recent studies have examined factors underlying the presence of the softened zone [4,17,21,31
33], the width and strength of softened zone [2,4,21,22,34], and the influence of the softened zone on the strength of the whole butt joint [5,18,35,36], no quantitative formula for calculating the strength of the butt welded joint with softened zone is available. In both SHAZ after welding and undermatched welds (the strength of filler metal is less than BM), a soft interlayer is generated within the butt joint. When such a butt joint is loaded in tension normal to the weld direction, as soon as yielding occurs, stress constraint is developed at the interface between the soft material and the adjacent materials that remain unyielded [7,29]. This restricts strain in the transverse direction and the soft material develops a hydrostatic stress component. Thus relatively larger tensile stresses are required to further increase plastic strain in the soft material. Amraei et al. [7] studied the effect of constraint on ultra-HSS material through experimental, analytical, and numerical investigations, showing that the HAZ constraint created a hydrostatic stress component and thus increased the yield stress of HAZ. Satoh and Toyoda [19,37] proposed formula for calculating the tensile strength of undermatched butt joint under plane strain and axisymmetric conditions based on the method of Davidenkov and Spiridonova [38], in which the BM is assumed rigid. Current design methods for steel structures, such as Eurocode 3 [28], ANSI/ AISC 360-10 [39], and GB 50017-2017 [40] are primarily based on the research on conventional mild steels. Although the upper limit of yield strength is 700 MPa (S700 steel) in EC3 or 690 MPa (A514 steel) in AISC code, the design strength of

568

Behavior and Design of High-Strength Constructional Steel

butt joint is based on the strength of filler metal. These approaches are questionable because neither strength reduction due to softened zone nor strength increase owing to constraint is considered in these methods, due to the unavailability of experimental or analytical research specifically addressing this issue. Motivated by the limitation of current research, the study has been conducted at Tongji University to provide insights on the mechanical properties of butt joints based on the analysis of strain data obtained by using digital image correlation (DIC) technique. Systematic experiments, 134 butt joints consisting of parameters such as HSS grades, two strengthening mechanisms of HSS, and mismatch ratio were conducted. Mechanical properties and chemical components of HSSs and filler metals as well as the geometry and manufacturing process of the specimens are shown. The hardness distribution curve for each assembly, the load-carrying capacity and deformation capacity were displayed intensively. Then the applicability of Eurocode 3 was discussed based on the test results. Subsequently, explanation for strength loss and strength increase and the indication for design theory were discussed. Moreover, based on the proposed three hardness distribution patterns of butt welded joints of HSS, a systematic numerical analysis considering geometric and mechanical parameters on mismatched butt joints of HSS is performed with a total of 938 simulated specimens. An extensive analysis of the effect of eight parameters on the mechanical behavior of mismatched butt joints is preformed, including width of weld metal, width of softened zone, width of hardened zone, the plate width-tothickness ratio, the bevel angle, the strength of weld metal, the strength of softened zone, and the strength of hardened zone. Through significance analysis, the contribution of each parameter is discussed. Finally, a strength model, which considers the effect of undermatched and the SHAZ, is proposed based on the combination of theoretical and finite element analysis on 798 specimens. The comparison between test data, predicted results by Eurocode 3 (EC3) and those by the calculation formula, indicates that the predictions from the proposed strength model provide reasonable agreement with test data across a range of parameters (mismatch ratio, the width of SHAZ, relative width of the plate), whereas EC3 overestimates the capacity of butt joints in situations with a wide SHAZ. In addition, a design formula for butt joints considering the constraint effect on undermatched and softened zone is proposed.

12.2

Experimental investigation

12.2.1 Material information In order to complement test data and extend the scope of the strength of HSS, nine steel plates with four grades and two strengthening mechanisms were investigated. Meanwhile, in order to study the mechanical properties of butt joints with mismatched filler metal and verify the applicability of EC3, four kinds of filler metals, composing mismatch ratio between 0.70 and 1.33, were used. In addition, the relative width of the plate was also considered as a major factor due to the observation

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from Satoh and Toyoda [19,28]. Tables 12.1 and 12.2 summarize the measured chemical composition and mechanical properties of the BMs and the filler metals used in this study. To obtain the mechanical properties of each material, ancillary tests were conducted for the BMs [41] and for the filler metals [42]. The mechanical properties of all steels fulfill the requirements of GB/T 16270 [43] and the electrodes shown in Table 12.3 are according to GB/T 8110-2008 [44]. In addition to yield strength, ultimate strength, and the average hardness of BM ðHb Þ and filler metal ðHw Þ, the parameters n and K of the constitutive model proposed by Ref. [8] are calibrated, as shown in Tables 12.3 and 12.4.

12.2.2 Gas metal arc welding All 108 butt joint specimens were fabricated by gas metal arc welding process with a combination of 80% argon (Ar) and 20% carbon dioxide (CO2). Four and six welding passes were contained in 10 and 20 mm thickness butt joints, respectively, as indicated in Table 12.5. Two key parameters are mismatch ratio Swm and the thickness-to-width ratio u ðXw Þ. Five replicates cut from one assembly shown in Fig. 12.2 were tested for most parameter sets. The plates were fixed by chuck jaws, and run-in and runoff plates were added. After welding the weld reinforcement was removed in order to make the welded zone have the same thickness as BM. All specimens were tested in monotonic tension. A constant feed rate of displacement loading 1 mm/min was used to simulate quasistatic loading conditions. Test setup and deformation measurements are shown in Fig. 12.3.

12.2.3 Digital image correlation measurement and calibration DIC was used to measure the full-field strain over the side surface of the specimen. It is a noncontact optical 3D deformation measuring system. Two charge-coupled device cameras measurement could identify three-dimensional surface contours and the strain distribution of the surface by taking into account the displacement in the depth direction. DIC measurement can recognize the surface characteristics of an object in digital camera images and allocates coordinates to the image pixels. Then, DIC measurement records further images during loading of the object and compares the digital images and calculates the deformation of the object characteristics based on the first image (i.e., undeformed state of the object). The detailed working mechanism is presented in the user manual [45]. Even though DIC technique measurement has been used in mechanics and industry fields [46
48] in recent years, the accuracy for measuring the deformation of our specimens required verification. A benchmark specimen was tested and the deformation and strain were recorded by traditional measurement and DIC measurement as shown in Fig. 12.3, and then the comparison of the deformation development recorded by those two methods is displayed in Fig. 12.4. The agreement of these curves validates the accuracy of DIC measurement and the advantage of DIC measurement is also evident due to its large strain range compared to strain gauge.

Table 12.1 Measured chemical composition of base metals (mass %). No.

Grade

Thickness (mm)

Strengthening mechanisms

C

Si

Mn

P

S

Al

Nb

Ti

Cr

Mo

V

Ni 1 Cu

CEVa

A1 A2 B1 B2 C1 C2 D1 D2 SC1b

Q460 Q460 Q550 Q550 Q690 Q690 Q890 Q890 Q690

10 20 10 20 10 20 10 20 10

QT QT TMCP TMCP TMCP TMCP TMCP TMCP QT

0.150 0.104 0.052 0.087 0.092 0.103 0.114 0.160 0.175

0.272 0.267 0.239 0.242 0.244 0.248 0.284 0.330 0.187

1.384 1.311 1.232 1.215 1.241 1.236 1.015 1.250 1.450

0.025 0.022 0.026 0.029 0.030 0.025 0.019 0.011 0.012

0.008 0.006 0.006 0.006 0.003 0.005 0.008 0.002 0.005

0.010 0.030 0.034 0.021 0.026 0.018 0.032 0.041 0.017

0.012 0.022 0.002 0.008 0.018 0.020 0.015 0.021 0.020

0.007 0.017 0.020 0.017 0.017 0.012 0.017 0.017 0.018

0.024 0.020 0.303 0.226 0.206 0.194 0.231 0.270 0.015

0.003 0.003 0.003 0.080 0.099 0.096 0.509 0.560 0.003

0.064 0.004 0.006 0.005 0.007 0.006 0.048 0.048 0.005

0.041 0.014 0.015 0.014 0.018 0.016 0.472 0.130 0.022

0.402 0.329 0.320 0.352 0.363 0.370 0.472 0.553 0.422

QT, Quenched and tempered; TMCP, thermomechanical controlled processing. a CEV 5 C 1 Mn/6 1 (Cr 1 Mo 1 V)/5 1 (Ni 1 Cu)/15 [39]. b SC1 refers to the base metal introduced in Ref. [16].

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Table 12.2 Chemical composition of filler metals (mass %) [8]. No. Filler 1 Filler 2 Filler 3 Filler 4

Electrode

C

Mn

Si

P

S

Ni

Mo

Cr

ER50-6 ER59-G ER76-G ER96-G

0.07 0.07 0.01 0.08

1.47 1.71 1.61 1.76

0.86 0.71 0.58 0.78

0.021 0.009 0.01 0.01

0.009 0.004 0.001 0.001



1.38 2.31


0.28 0.3 0.6

0.02
0.28 0.35

Table 12.3 Measured average mechanical properties of base metals. No.

fyb (MPa)

fub (MPa)

Hb (Hv0.1)

Ab (%)

n

K (MPa)

A1 A2 B1 B2

483.9 503.9 700.4 648.3

609.5 640.3 770.6 719.7

194 199 245 245

24.9 25.7 17.2 17.7

0.171 0.162 0.087 0.062

984 1001 1043 900

No.

fyb (MPa)

fub (MPa)

Hb (Hv0.1)

Ab (%)

n

K (MPa)

C1 C2 D1 D2 SC1

703.0 807.2 1074.1 1012.8 807

767.2 831.8 1107.8 1050.9 856

251 286 345 345 270

14.6 20.3 13.8 19.1 11.3

0.076 0.073 0.073 0.076 0.035

997 1101 1445 1382 1001

Table 12.4 Measured mechanical properties of filler metals [8]. No. Filler 1 Filler 2 No. Filler 3 Filler 4

fyw (MPa)

fuw (MPa)

Hw (Hv0.1)

Aw (%)

n

K (MPa)

547 641

627 727

192 249

23.3 23.2

0.133 0.133

931 1095

fyw (MPa)

fuw (MPa)

Hw (Hv0.1)

Aw (%)

n

K (MPa)

688 886

771 956

265 311

21.0 22.1

0.091 0.107

1033 1377

12.2.4 Measured load-carrying capacity and deformation capacity of butt joints Representative load versus displacement curves (for specimen D1) are shown in Fig. 12.5. The peak load Fmax (denoted as Fjoint and Fhom in Table 12.6 for the loadcarrying capacity of butt welded joint and corresponding BM, respectively) for each specimen is obtained from such a load
displacement curve. Deformation

572

Behavior and Design of High-Strength Constructional Steel

Table 12.5 Welding parameters. Base steel

Welding pass no.

Current I (A)

Voltage U (V)

Welding speed S (mm/min)

t8/5 (s)

Arc energy (kJ/mm)

Average heat input [16] (kJ/mm)

A1, B1, C1, D1

1 2 3 4 1 2 3 4 1 2 3 4 5 6

240 240 240 250 240 240 270 250 240 240 250 250 240 250

27 27 27 27 27 27 28 27 27 27 27 27 27 27

350 320 320 320 350 320 320 320 350 320 320 320 320 320

2 2 2 2 2 2 2 2 2 2 2 2 2 2

1.11 1.22 1.22 1.27 1.11 1.22 1.42 1.27 1.11 1.22 1.27 1.27 1.22 1.27

1.20

SC1L

A2, B2, C2, D2

1.25

1.22

Figure 12.2 Geometry of weld preparation for multipass butt joints; dimensions in mm.

capacity Umax (denoted as Ujoint and Uhom in Table 12.6 for the deformation capacity of butt welded joint and corresponding BM, respectively) is defined as the deformation when the load capacity drops to 85% maximum load. T1, T2, and T3 in Fig. 12.5 refer to three typical loading time instants along the whole loading process. Time T1 adverts to the loading time instant at the end of elastic loading; Time T2 refers to the loading time instant at the peak load ðFmax Þ; and Time T3 denotes the loading time instant corresponding to the deformation capacity ðUmax Þ.

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Figure 12.3 Test setup and deformation measurements.

Figure 12.4 Comparison between the traditional deformation measurements and DIC measurement: (A) deformation development and (B) strain development. DIC, Digital image correlation.

Figure 12.5 Typical load
displacement curves of butt joint and corresponding base metal (D1).

574

Behavior and Design of High-Strength Constructional Steel

Table 12.6 Test data from butt joint tension tests. Test number

Xw a

b Swm u

Fjoint =Fhom c

Ujoint =Uhom d

Failure locatione

A11-2 A11-3 A11-4 A11-5 A13-1 A13-2 A13-3 A13-4 A13-5 B11-1 B11-2 B11-3 B11-4 B11-5 B12-1 B12-4 B12-5 B14-1 B14-2 B14-3 B14-4 B14-5 C11-1 C11-2 C11-3 C11-4 C11-5 C13-1 C13-2 C13-3 C13-5 C14-1 C14-2 C14-3 C14-4 C14-5 D13-1 D13-2 D13-3 D13-4 D13-5 D14-1 D14-2 D14-3

0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4

1.029 1.029 1.029 1.029 1.265 1.265 1.265 1.265 1.265 0.814 0.814 0.814 0.814 0.814 0.943 0.943 0.943 1.241 1.241 1.241 1.241 1.241 0.817 0.817 0.817 0.817 0.817 1.005 1.005 1.005 1.005 1.246 1.246 1.246 1.246 1.246 0.696 0.696 0.696 0.696 0.696 0.863 0.863 0.863

1.026 1.030 1.042 1.032 1.042 1.058 1.045 1.033 1.042 0.824 0.800 0.849 0.840 0.833 0.929 0.948 0.975 1.015 1.008 1.036 1.015 1.018 0.875 0.796 0.828 0.838 0.865 0.987 0.977 0.991 0.995 1.013 1.055 1.067 1.030 1.025 0.774 0.776 0.792 0.767 0.784 0.863 0.895 0.881

0.691 0.553 0.613 0.661 0.793 0.814 0.771 0.800 0.770 0.216 0.267 0.369 0.464 0.277 0.641 0.341 0.523 0.707 0.829 0.782 0.903 0.869 0.393 0.376 0.283 0.391 0.383 0.616 0.560 0.882 0.596 0.618 0.798 0.700 0.687 0.920 0.388 0.290 0.354 0.433 0.401 0.334 0.373 0.668

WM WM WM WM BM BM BM BM BM WM WM WM WM WM HAZ HAZ HAZ BM BM BM BM BM WM WM WM WM WM HAZ HAZ WM HAZ HAZ BM HAZ HAZ BM WM WM WM WM WM WM WM WM (Continued)

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Table 12.6 (Continued) Test number

Xw a

b Swm u

Fjoint =Fhom c

Ujoint =Uhom d

Failure locatione

D14-4 D14-5 SC11L-1 SC11L-2 SC11L-3 SC11L-4 SC12L-1 SC12L-2 SC12L-3 SC12L-4

0.4 0.4 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

0.863 0.863 0.733 0.733 0.733 0.733 0.850 0.850 0.850 0.850

0.877 0.880 0.743 0.757 0.737 0.735 0.835 0.825 0.814 0.824

0.656 0.575 0.443 0.666 0.600
0.408 0.407
0.470

WM WM WM WM WM WM HAZ HAZ HAZ HAZ

Test number

Xw

Swm u

Fjoint =Fhom

Ujoint =Uhom

Failure location

A21-1 A21-3 A21-4 A23-1 A23-2 A23-3 A23-4 A23-5 B21-1 B21-2 B21-3 B21-4 B21-5 B22-4 B22-5 B24-1 B24-2 B24-3 B24-4 B24-5 C21-1 C21-2 C21-3 C21-4 C22-1 C22-2 C22-3 C22-4 C23-1 C23-2

0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8

0.979 0.979 0.979 1.204 1.204 1.204 1.204 1.204 0.871 0.871 0.871 0.871 0.871 1.010 1.010 1.328 1.328 1.328 1.328 1.328 0.754 0.754 0.754 0.754 0.874 0.874 0.874 0.874 0.927 0.927

0.986 0.988 0.990 0.992 0.993 0.992 0.994 0.994 0.919 0.909 0.917 0.907 0.914 1.022 1.006 1.005 1.041 1.062 1.024 1.015 0.842 0.860 0.871 0.859 0.917 0.903 0.916 0.904 0.929 0.922

0.881 0.851 0.828 0.876 0.872 0.794 0.887 0.859 0.219 0.219 0.243 0.213 0.245 0.583 0.612 1.013 0.913 0.967 0.938 0.962 0.469 0.436 0.392 0.415 0.595 0.565 0.625 0.356 0.687 0.633

BM BM BM BM BM BM BM BM WM WM WM WM WM WM WM BM BM BM BM BM WM WM WM WM WM WM WM WM WM WM (Continued)

576

Behavior and Design of High-Strength Constructional Steel

Table 12.6 (Continued) Test number

Xw

Swm u

Fjoint =Fhom

Ujoint =Uhom

Failure location

C23-3 C23-4 C24-1 C24-2 C24-3 C24-4 D23-1 D23-2 D23-3 D23-4 D23-5 D24-1 D24-2 D24-3 D24-4 D24-5 SC13L-1 SC13L-2 SC13L-3 SC13L-4 SC14L-1 SC14L-2 SC14L-3 SC14L-4

0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

0.927 0.927 1.149 1.149 1.149 1.149 0.734 0.734 0.734 0.734 0.734 0.910 0.910 0.910 0.910 0.910 0.901 0.901 0.901 0.901 1.117 1.117 1.117 1.117

0.933 0.922 1.009 1.007 1.009 1.016 0.904 0.897 0.885 0.903 0.893 1.004 0.982 0.993 0.995 0.994 0.832 0.850 0.842 0.834 0.842 0.826 0.842 0.835

0.612 0.674 0.721 0.734 0.651 0.671 0.375 0.326 0.302 0.389 0.359 0.731 0.302 0.366 0.364 0.616 0.391 0.354 0.383 0.462 0.342 0.379 0.263 0.390

WM WM HAZ HAZ HAZ HAZ WM WM WM WM WM WM WM WM WM WM HAZ HAZ HAZ HAZ HAZ HAZ HAZ HAZ

a

Xw : thickness-to-width ratio. Su : mismatch ratio. c Fjoint and Fhom : maximum load of butt welded joint and corresponding base metal, respectively. d Ujoint and Uhom : deformation when the load capacity drops to 85% maximum load of butt welded joint and corresponding base metal, respectively. e BM: Base metal; HAZ, heat-affected zone; WM, weld metal. b wm

All the loading capacities and deformation capacities of 108 butt welded joints are summarized in Table 12.6. The naming rule for the specimen Xyz(L)-n is as follows: X denotes the grade of HSS, while A, B, C, and D denote Q460, Q550, Q690, and Q890, respectively, and SC denotes another Q690 grade HSS introduced in Ref. [16]; y refers to the thickness of the plate, whereas 1 and 2 represent 10 and 20 mm, respectively; z denotes the grade of filler metal, with 1, 2, 3, and 4 referring to Filler 1, Filler 2, Filler 3, and Filler 4, respectively; n denotes the replicate number; and with or without L means the width of the plate is 100 or 25 mm, respectively.

12.2.5 Linear correlation between strength and hardness To facilitate examination of spatial variation in material strength in the vicinity of the weld, hardness tests were conducted along the welded section according to the Vickers

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measurement method [45]. The instrument for the microhardness test was shown in Fig. 12.6. The indentations for the hardness determination were positioned approximately 2 mm away from the edge, in successive positions at 0.25 mm intervals, to measure spatial variations of hardness with high resolution, as displayed in Fig. 12.7. The Vickers hardness of plate SC1 (Q690 grade steel with yield strength of 807 MPa) after heating to 400 C
900 C and then cooling in air was measured to validate the linear correlation between hardness and strength of postfire HSS. Corresponding yield strength and tensile strength of each postfire situation were tested and presented in Ref. [46]. Regression analysis was used to determine the correlation of the yield strength and tensile strength to the Vickers hardness values. The fitting curves and correlation coefficient, R2 , are shown in Fig. 12.8. Both yield

Figure 12.6 Vickers indentation test machine: (A) HVS-1000ZCCD microhardness tester and (B) indentation.

Figure 12.7 Location for hardness test on specimens with (A) 10 and (B) 20 mm thickness.

Figure 12.8 Plots of (A) yield strength and (B) tensile strength of various situations as a function of hardness.

578

Behavior and Design of High-Strength Constructional Steel

strength and tensile strength of the HSS exhibit a linear correlation with the hardness over a range of tensile strength from 600 to 1100 MPa with heated up from 400 to 900. Therefore the strength distribution over the etched surface can be represented by hardness distribution due to the continuity and testability of hardness.

12.2.6 Measured hardness distribution curves of butt joints A total of 27 measured hardness distribution curves (6 curves from other researchers concluded) for each assembly are exhibited in Fig. 12.9. Uniform hardness distribution in both welded zone and BM zone was observed, while hardness distribution in HAZ has a severe variability. For QT steel plates with lower yield strength (i.e., A1 and A2), hardness increased in HAZ due to heat input while welding. In other words, the strength in HAZ increased after welding. For TMCP steel plates with higher yield strength (i.e., B1, B2, C1, C2, D1, and D2), the hardness in HAZ ascended first then descended from welded zone to BM zone. For QT steel plate SC1, severe hardness loss in HAZ was observed. The zones in HAZ with hardness increase and decrease are denoted as hardened HAZ (HHAZ) and SHAZ, respectively. Moreover, take steel plate C2 for example (Fig. 12.9F), the hardness in HAZ, as expected, is uncorrelated with the filler material. sz In addition, the definition and value range of hardened ratio Shz H , softened ratio SH , relative HHAZ width Xhz , and relative SHAZ width Xsz are displayed and summarized in Table 12.7. Except steel plate SC1, the total width of HAZ is approximately 4 mm, with either only HHAZ (for A1 and A2) or both HHAZ and SHAZ (B1
D2), due to similar heat input shown in Table 12.5. The value of Swm H mostly depends on the mismatch ratio, for undermatched, matched, and overmatched butt joints, the value of Swm H will be lower, equal to and higher than 1.0, respectively. Generally, hardness distribution in HAZ is not uniform and it could be a triangle-shape (i.e., A21) or parabolic-shape (i.e., SC12), or rectangular-shape (D23), as shown in Fig. 12.9. In order to simplify calculation model and propose a conservative predicted value, the hardness distribution in HAZ was supsz posed to be uniform and the values of Shz H and SH are the highest measured hardness in sz HHAZ and the lowest measured hardness in SHAZ, respectively. The values of Shz H , SH , wm and SH are the average value of the measured value corresponding to each zone.

12.3

Summarization of experimental results

12.3.1 Three hardness distribution patterns Due to the linear relation between hardness and strength, based on 27 measured hardness distribution curves presented previously, 3 simplified hardness distribution patterns are identified herein, as shown in Fig. 12.10. The specific distribution pattern for each curve is summarized in Table 12.7. G

G

G

Pattern I: There is only HHAZ and it occurs in the lower strength steel, such as A1 and A2. Pattern II: For steels B1, C1, C2, D1, and D2, there are both HHAZ and SHAZ. Pattern III: For the steel SC1, there is only SHAZ.

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Figure 12.9 Hardness distribution curves of (A) A1, (B) A2, (C) B1, (D) B2, (E) C1, (F) C2, (G) D1, (H) D2, and (I) SC1 [16] over the etched surface, 2 mm from the top edge. Each figure contains hardness distribution and etched surface for corresponding butt joint.

Table 12.7 Summary of hardness information. Hb a (Hv0.1)

b Swm H

c Ssz H

d Shz H

Xwm e

Xsz f

Xhz g

XHAZ h

WHAZ i (mm)

Patternj

Failure location

A11 A13 A21 A23 B11 B12 B14 B21k B22k B24k C11 C13

193 195 196 201 245 245 246 238 251 246 251 251

0.97 1.30 1.11 1.27 0.81 0.93 1.33 0.86 0.98 1.44 0.78 0.95





0.75 0.79 0.80


0.82 0.81

1.18 1.08 1.59 1.30 1.16 1.07 1.07


1.15 1.08

0.75 0.83 0.90 0.98 0.93 0.98 0.90 0.98 1.00 1.00 1.05 1.15





0.15 0.23 0.18


0.15 0.20

0.40 0.40 0.15 0.15 0.18 0.18 0.18


0.18 0.18

0.40 0.40 0.15 0.15 0.33 0.40 0.36


0.33 0.38

4.0 4.0 3.0 3.0 3.3 4.0 3.6


3.3 3.8

I I I I II II II I I I II II

C14

251

1.22

0.78

1.08

1.03

0.28

0.18

0.45

4.5

II

C21 C22 C23 C24 D13 D14 D23 D24

287 288 286 281 335 354 346 344

0.74 0.82 0.86 1.06 0.71 0.87 0.72 0.92

0.84 0.76 0.79 0.76 0.87 0.79 0.87 0.82

1.08 1.00 1.03 1.05 1.19 1.16 1.23 1.25

1.18 1.15 1.23 1.33 1.18 1.25 1.13 1.03

0.08 0.08 0.10 0.10 0.18 0.30 0.06 0.08

0.08 0.10 0.14 0.14 0.13 0.15 0.14 0.16

0.15 0.18 0.24 0.24 0.30 0.45 0.20 0.24

3.0 3.5 4.8 4.8 3.0 4.5 4.0 4.8

II II II II II II II II

WM BM BM BM WM SHAZ BM WM WM BM WM WM/ SHAZ BM/ SHAZ WM WM WM SHAZ WM WM WM WM

Assembly no.

SC11l SC12l SC13l SC14Al SC14B SC14C

277 273 280 266 264 262

0.83 0.88 0.97 1.12 1.07 1.14

0.68 0.68 0.69 0.68 0.62 0.63









1.50 1.55 1.40 1.25 1.13 1.00

0.85 0.70 0.53 0.75 0.68 0.55









0.85 0.70 0.53 0.75 0.68 0.55

8.5 7.0 5.3 7.5 6.8 5.5

III III III III III III

BM, Base metal; HHAZ, hardened heat-affected   zone; SHAZ, softened heat-affected zone. n P a Hb : hardness of base metal Hb 5 Hbi =n . i51

 Hwi =Hb =n).  c sz SH : relative hardness of softened zone (Ssz H 5 min Hszi =Hb ).   d hz SH : relative hardness of hardened zone (Shz H 5 max Hhzi =Hb ). e Xwm : relative width of welded zone. f Xsz : relative width of softened zone ðXsz 5 maxðXsz1 ; Xsz2 ÞÞ. g Xhz : Relative width of hardened zone. h XHAZ : relative width of heat-affected zone ðXHAZ 5 Xsz 1 Xhz Þ. i WHAZ : width of heat-affected zone ðWHAZ 5 XHAZU t0 Þ. j Hardness distribution patterns are shown in Fig. 12.10. k For steel plate B2, it seems no obvious SHAZ or HHAZ in HAZ, due to mere effect on the strength, the hardness distribution pattern is considered as Pattern I. l SC1L and SC1 are from the same assembly. SH : relative hardness of welded zone (Swm H 5

b wm

n  P

i51

WM SHAZ SHAZ SHAZ SHAZ SHAZ

582

Behavior and Design of High-Strength Constructional Steel

Figure 12.10 Hardness distribution patterns of butt joints (A-1) Pattern I: with only hardened zone (specimen A21); (B-1) Pattern II: with both softened and hardened zones (specimen D23); (C-1) Pattern III: with only softened zone (specimen SC12). (A-2), (B-2), and (C-2) only show the schematic diagram, and actually the butt joints can be undermatched, evenmatched, or overmatched.

12.3.2 Strain distribution for each hardness distribution pattern Based on the previous analysis, the three hardness distribution patterns imply three strength distribution patterns, which lead to nonuniform strain distribution with a tension load. Representative strain distribution for each hardness distribution pattern is illustrated in Fig. 12.11. For each representative specimen the strain measured by DIC and hardness distribution curve at the same location are shown in top figure; strain cloud at representative loading time instant (shown in Fig. 12.5) is exhibited

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583

Figure 12.11 Strain development for three patterns of hardness distribution: (A) Pattern I (A21-3), (B) Pattern II (C13-3), and (C) Pattern III (SC14B-3). Thereinto, (A-1), (B-1), and (C-1) indicate the relation between hardness distribution and strain distribution; (A-2), (B-2), and (C-2) display the strain distribution cloud over the sample at some typical loading time (T1, T2, and T3); and (A-3), (B-3), and (C-3) indicate the strain development along the loading time at typical points.

in middle figure; the bottom figure indicates the strain development of several typical points in WM, SHAZ, HHAZ, and BM (shown in middle figure). For Pattern I as shown in Fig. 12.11A, there is only HHAZ, which means the strength in HHAZ is the highest, whereas the nominal strain in HHAZ is the lowest. The strain distribution at time T2 indicates clearly the X-shaped HHAZ with lower strain, with the consequence that strain localization and failure occur in the BM. At this moment the strain in right-side BM (PBM3) increases sharply, while the strains in welded zone (PWM) and HHAZ (PHHAZ) stabilize. For other specimens with Pattern I hardness distribution, strain localization and failure will occur in BM or welded zone, dependent on the mismatch ratio. For the most common hardness distribution pattern of HSS-Pattern II as shown in Fig. 12.11B, there is almost no discrepancy on strain distribution at SHAZ or HHAZ. Strain distribution rests on mismatch ratio and the distance to the strain localization zone. The strain localization occurs in welded zone due to undermatching ratio. Meanwhile, as being closer to failure location, the strain in HHAZ (PHHAZ) is higher than that in SHAZ (PSHAZ). For other specimens with Pattern II hardness distribution, the failure location will lie on BM or welded zone in most cases, dependent on the mismatch ratio. However, according to our test results, several specimens (C13-n, C14-n, and C24-n) failed in SHAZ. No strength loss occurred for these specimens. For Pattern III as shown in Fig. 12.11C, there is only SHAZ. In contrast to Pattern I, the strength in SHAZ is lowest, whereas the nominal strain in SHAZ is

584

Behavior and Design of High-Strength Constructional Steel

highest. Low strength promotes strain localization in SHAZ starting from initial yielding (time T1). Bilateral SHAZs had larger strain compared to welded zone and BM until plastic deformation localized in one of them. The strain distribution cloud at time T3 indicates clearly the failure location in SHAZ. Generally, for this hardness distribution pattern due to the wide softened zone the failure location is in SHAZ. Furthermore, the strength loss of this kind of welded joint is common. However, for specimens C11L and SC11, though the strength of welded zone is slightly higher than SHAZ, failure occurred in welded zone for these specimens, due to the combination of low strength and large region of the welded zone.

12.3.3 Strength loss for specimens with different softened heat-affected zone width As outlined previously, the strength of butt joints may decrease due to the existence of SHAZ zone. However, although there is SHAZ zone in specimens with Pattern II or Pattern III hardness distribution, only specimens with Pattern III hardness distribution had a strength loss. Fig. 12.12 indicates the strain development of two specimens with thin SHAZ ðXsz 5 0:18Þ and wide SHAZ ðXsz 5 0:68Þ, respectively. Due to low strength of SHAZ, the strain in SHAZ is highest during whole loading process for both of them. However, for specimen with thin SHAZ, as soon as yielding occurs in SHAZ, constraint is developed at the interface between the SHAZ and the adjacent materials that remain unyielded. This restricts strain in the transverse direction and the SHAZ develops a hydrostatic stress component. Thus relatively larger tensile stresses are required to further increase plastic strain in the SHAZ, which results in stress redistribution in SHAZ and welded zone and final strain localization occurs in welded zone. However, for specimen with wide SHAZ, both stress and strain localization occur in SHAZ owing to the large weak zone. From test results, no strength loss occurs for specimen with width-to-thickness ratio of SHAZ smaller than 0.30 (D14); the strength decreases almost 10% for specimen with width-tothickness ratio of SHAZ is 0.55 (SC14C).

12.3.4 Strength increase due to the constraint Constraint at the interface of the soft interlayer and the adjacent material can be used to explain the positive effect on strength of plate width. When a butt joint loaded normal to the weld axis, the soft interlayer (WM or SHAZ) will yield first. As strain localizes in the soft interlayer, it will begin to deform in both the width and thickness directions, in addition to tension in the longitudinal direction due to the applied load. Since the adjacent material is unyielded, it will constrain the deformation of the soft interlayer. For the relatively thin plates (25 mm thickness and less), the constraint through the thickness does not develop significantly. In the absence of constraint through the thickness, the maximum constraint is that associated with a plate of infinite width in the direction transverse to the axial loading

Welded connections

585

Figure 12.12 Development of strain distribution of butt joints with (A) Pattern II: Xsz 5 0:18 (D13-3) and (B) Pattern III: Xsz 5 0:68 (SC14B).

and therefore a state of plane strain can be obtained. In this case, the strength for  pffiffiffin11 the soft interlayer beginning to yield will be 2= 3 times of yield strength of its homogenous material. Take hardening exponent of Filler 1 (n 5 0.13) if failure for example, the weld metal begins to yield at 1.18 times of its yield strength. In other words, the weld metal will not yield until the axial stress approaches 118% of the uniaxial yield strength due to constraint in a state of plane strain.

12.3.5 Ductility loss due to mismatched connections Fig. 12.13 plots the ratio Ujoint =Uhom between the deformation capacity in butt welded joints ðUjoint Þ and that attained in a sample of homogeneous base plate

586

Behavior and Design of High-Strength Constructional Steel

Figure 12.13 Ductility defined as ðUjoint =Uhom Þ as a function of mismatch ratio ðSwm u Þ for (A) eight HSS plates studied in this chapter and (B) plate from Ref. [16].

ðUhom Þ against the mismatch ratio to provide an understanding of joint ductility. Greater variability of ductility can be observed for specimens coming from the same assembly compared to strength, due to the defect sensitivity of ductility from the material or welding process. The results grouped in Fig. 12.13A allow us to conclude that the mismatch, no matter undermatched or overmatched, in the material properties induces a decrease of ductility, which is more enhanced for low values of mismatch ratio ðSwm u Þ. However, for butt joints with wide SHAZ (Pattern III), it is possible to observe an increase in ductility with the reduction in mismatch ratio as shown in Fig. 12.13B. To better understand the reasons underlying the observed ductility of the specimens, and its relation to mismatch ratio and the failure location, the strain

Welded connections

587

Figure 12.14 Strain distribution of (A) C1, (B) B2, and (C) SC1_Ref. [16] at the load reduce to 85% of the peak load (time T3 shown in Fig. 12.8), for explaining the deformation capacity for different failure locations at (A) WM, (B) BM, and (C) SHAZ.

distribution along the specimen, at time T3, is plotted in Fig. 12.14, grouped by failure location at (A) welded metal (WM), (B) BM, and (C) SHAZ. The deformation capacity ðUmax Þ can be considered approximately as area under the curve (e.g., C11-3 shown in Fig. 12.14A). From Fig. 12.14A, it can be seen clearly that for the situation that failure at WM, in the cases of undermatched ðSwm u 5 0:82Þ and evenmatched ðSwm u 5 1:0Þ, the deformation is strongly localized inside the WM, and the

588

Behavior and Design of High-Strength Constructional Steel

strain values in BM are very close to zero, especially for undermatch. This clearly demonstrates that for the situation of failure in WM, the ductility of the overall sample strongly depends on the hardening behavior of the filler metal. Based on test results (shown in Fig. 12.13A), due to approximate value of hardening component (n) of four kinds of filler metals, filler metal with higher strength will increase the strength of butt joint, which will provide more plastic deformation over the specimen. In other words, for the same parent metal, increase of mismatch ratio can increase the deformation capacity for undermatched or evenmatched specimen. On the other hand, for overmatched joints, two possible failure locations are SHAZ wm (Fig. 12.14A, Swm u 5 1:25) and BM (Fig. 12.14B, Su 5 1:33). The deformation localization inside the SHAZ decreases the ductility, whereas ductility loss cannot occur for the situation of failure in BM. For the wide SHAZ widths (Fig. 12.14C, hardness distribution Pattern III), the strain is strongly localized in side SHAZ, resulting in a loss of ductility compared to homogeneous base plate. Meanwhile, decrease of mismatch ratio brings a spread of deformation in welded metal, explaining the results in Fig. 12.13B where an increase of ductility is detected for the cases with undermatched tensile strength.

12.4

Applicability of Eurocode 3 Part 1
12

In Fig. 12.15 the test results of the maximum load attained by the various patterns of butt welded joints ðFjoint Þ, normalized with the maximum load attained in a tensile test of homogeneous base plate sample ðFhom Þ, are shown. The results are grouped according to the same base plate and are plotted relative to the mismatch ratio ðSwm u Þ. According to Eurocode 3 Part 1
12 [28], for HSSs the filler metal may have lower strength than the base material and the ultimate strength of butt joint equals to the weaker strength between BM and filler metal. From our test data, as shown in Fig. 12.15A, most of test data (92 specimens) are located closely above the EC3 estimated line. In other words, this graph shows that for eight plates studied in this chapter, the maximum load of butt joints increases almost linearly with increasing values of mismatch ratio. When the value of mismatch ratio reaches 1.0, the load capacity of the joint is equal to that of the homogeneous base plate. However, as shown in Fig. 12.15B, the strength of specimens with steel plate SC1 is significantly lower than the estimated value from EC3 due to the existence of SHAZ, which exposes the weakness and nonconservative of EC3. Meanwhile, as shown in Fig. 12.15B, increase of the plate width from 25 to 100 mm (Xw drops from 0.4 to 0.1) can bring in a 4%, 10%, and 5% increase of load-carrying capacity for specimens with mismatch ratio 0.73, 0.85, and 0.9, respectively. However, the observation was not considered in EC3, which may result in a conservative design. In conclusion, strength reduction was observed in several specimens due to SHAZ zone, resulting in overestimation on the load-carrying capacity of butt joints when using design theory in EC3; meanwhile, the positive effect of the width of the steel plate was also not considered in EC3.

Welded connections

589

Figure 12.15 Normalized load values ðFjoint =Fhom Þ as a function of mismatch ratio ðSwm u Þ. (A) Specimens with Pattern I and II hardness distribution pattern; (B) Specimens with Pattern III hardness distribution pattern.

12.5

Strength model of butt welds

12.5.1 Theoretical model The following four assumptions were adopted by Satoh and Toyoda to develop a theoretical strength model for butt welded joints [19,37]. 1. The welded joint consists of only two kinds of metals, or BMs and a soft interlayer, each of which is homogeneous and isotropic. 2. The BM is assumed rigid throughout loading; the contacting surfaces to the soft interlayer always remain perpendicular to the loading direction.

590

Behavior and Design of High-Strength Constructional Steel

3. At any stage of loading the volume of soft interlayer is constant. 4. The equivalent stress and equivalent strain are related through a power law.

Following Davidenkov’s analysis [18], Eqs. (12.1) and (12.4) were proposed by Satoh and Toyoda [19,37] for the plane strain and axisymmetric conditions (Fig. 12.16), respectively. σPu

 n11 2 K ðlnð11εÞÞn 5 pffiffiffi ð1 1 Y t Þ ð1 1 εÞ 3

(12.1)

where σPu is tensile stress for plane strain condition; ε is nominal strain; Yt is a variable that equals a=3R; and a and R are geometrical parameters shown in Fig. 12.16 where a denotes half of the instantaneous plate thickness at the narrowest section and R is the radius of curvature of the neck. Enforcing compatibility along with the assumption of constant volume, the following relationships may be derived between a and R as shown in the following equations: h2 5 ða0 2 aÞð2R 2 a0 1 aÞ 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  2 1@ h h Rh 1 2 1 R2 sin21 A 5 a0 h0 ða 1 RÞh 2 2 R R

(12.2)

(12.3)

where a0 , h0 , and h are geometrical parameters shown in Fig. 12.17B. a0 refers to half of the initial plate thickness; h0 and h indicate half of the initial thickness of the interlayer and half of the instantaneous plate thickness, respectively.where σCu is tensile stress for axisymmetric condition; ε, n, K, and R have the same meaning as in Eq. (12.1); r denotes the instantaneous radius at the smallest section. Again,

Figure 12.16 Sketch of the stress trajectories in the neck of a tensile specimen: (A) plane strain condition and (B) axisymmetric condition.

 n r ε 2 σCu 5 K lnð11εÞ 11 12 4R 2

(12.4)

Welded connections

591

Figure 12.17 Schematic of butt joints under (A) rectangular condition ðW0 . t0 Þ, (B) plane strain condition ðW0 ct0 Þ, and (C) axisymmetric condition ðW0 5 t0 Þ.

enforcing compatibility and the constant volume condition result in the following relationship between r and R as shown in the following equations: h2 5 ðr0 2 r Þð2R 2 r0 1 r Þ 

(12.5)

( rffiffiffiffiffiffiffiffiffiffiffiffiffiffi )  h3 h2 2 21 h 2R 1 2rR 1 r h 2 2 ðR 1 r Þ 3 Rh 1 2 2 1 R sin 5 r02 h0 R 3 R 2

2

(12.6) where h0 and h are geometrical parameters shown in Fig. 12.17C. In this section, based on the functional forms for butt joint with rigid constraint proposed by Satoh and Toyoda [16,17] and by means of simulation results, a strength model for calculating the tensile strength of butt joint with nonrigid constraint is presented.

12.5.2 Finite element model 12.5.2.1 Material model In order to validate the strength model by comparing the prediction results with experimental results (data from Ref. [50]), material model used in finite element model is calibrated of the real BMs and filler metals used in Ref. [50]. The calibration method of material model shown in Sun et al. [36] is used herein. The true stress versus strain relationship is expressed by the following equation: σ 5 Kεnp 0 1 0 σ 2 σ yA Uεp 1 σy σ5@ ε0p

εp . ε0p 0 , εp # ε0p

(12.7)

592

Behavior and Design of High-Strength Constructional Steel

Table 12.8 Constitutive model parameters for base metal and filler metals. No. A1a A2a B1 B2 C1 C2 D1 D2 Filler 1a Filler 2 Filler 3 Filler 4

Material

σ0 (MPa)

ε0p

εup

n

K (MPa)

Q460 Q460 Q550 Q550 Q690 Q690 Q890 Q890 ER50 ER59 ER76 ER96

505 511 717 612 722 824 1096 1025 564 667 693 930

0.0214 0.0199 0.0154 0.0032 0.0144 0.0190 0.0154 0.0190 0.0253 0.0240 0.0120 0.0250

0.178 0.162 0.087 0.056 0.075 0.074 0.074 0.077 0.133 0.134 0.091 0.106

0.171 0.162 0.087 0.062 0.076 0.073 0.073 0.076 0.133 0.133 0.091 0.107

984 1001 1043 900 997 1101 1445 1382 931 1095 1033 1377

a

The true stress versus strain curve of material with yield plateau. Source: Detailed information of these materials are displayed in Ref. [55].

where σ is the true stress and εp is the plastic strain, σy is the yield stress, K is a constant and n is the strain hardening exponent, E is elastic modulus. ε0p and σ0 is the plastic strain and true stress at the end of yield plateau, respectively. If there is no obvious plateau, then it is assumed that the point corresponding to ðε0p ; σ0 Þ coincides with the intersection of the test curve and the straight line σ 5 Eεp . The average elastic modulus of all materials studied in this chapter is 213 3 103 MPa. Table 12.8 summarizes the material model parameters and true stress
plastic strain curves for each material.

12.5.2.2 Numerical experiments Fig. 12.17A shows schematic of two-material model under rectangular condition considering mismatch ratio ðSwm u Þ, relative width of welded zone ðXwm Þ and thickness-towidth ratio ðXw Þ. According to the research of Satoh and Toyoda [19,37], the upper limit strength of rectangular section condition with rigid constraint shown in Fig. 12.17A may be simplified as plane strain condition, as shown in Fig. 12.17B when the plate width W0 is large enough compared with plate thickness t0 , while the lower limit strength condition in Fig. 12.17A can be simplified as axisymmetric condition shown in Fig. 12.17C with the plate width W0 equals to plate thickness t0 . For rectangular condition the finite element discretization makes use of the eight-node linear brick element with reduced integration (C3D8R) and only oneeighth of the sample was simulated, considering geometrical and material symmetry. For plane strain condition a four-node bilinear plane strain quadrilateral with reduced integration (CPE4R) mesh was used and only one-fourth of the sample was simulated. For axisymmetric condition a four-node bilinear axisymmetric quadrilateral with reduced integration (CAX4R) mesh was used. The boundary conditions used in the simulations are also schematized in Table 12.9. Similar finite element model with same boundary condition and

Table 12.9 Summary of all simulation cases in this chapter. Case

Value of each parameter

Simulation model

Case

Value of each parameter

Case 1 (79a)

r0 : 3, 5, 10 mm Xtz : 0.05, 0.1, 0.15, 0.2, 0.3, 0.5, 1, 1.5, 2, 3, 4, 5, 6, 8, 10 Target zone: F1b, F2, F4, A1, D1

Case 2 (54)

a0 : 5 mm Xtz : 0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.5, 0.6, 0.8, 1, 1.25, 2, 2.5, 4, 5, 8, 10 Target zone: F1, F2, F4

Case 3 (36)

a0 : 5 mm Xtz : 0.05, 0.1, 0.2, 0.5, 1, 1.25, 2, 2.5, 4, 5, 8, 10 Target zone: F1, F2, F4

Case 4 (70)

a0 : 5 mm Xtz : 0.05, 0.1, 0.2, 0.5, 1, 1.25, 2, 2.5, 4, 5; 1=Xw : 1, 1.05, 1.1, 1.25, 1.5, 1.6, 1.8, 2, 3, 4, 5, 6, 8, 10; Target zone: F1

Case 5 (54)

r0 : 5 mm Xwm : 0.05, 0.1, 0.15, 0.2, 0.3, 0.5, 0.6, 0.8, 1.0, 1.25, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0; Base metal: D1; Weld metal: F1, F2, F4

Case 6 (51)

a0 : 5 mm Xwm : 0.05, 0.1, 0.15, 0.2, 0.3, 0.5, 0.6, 0.8, 1.0, 1.25, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5; Base metal: D1; Weld metal: F1, F2, F4

Simulation model

(Continued)

Table 12.9 (Continued) Case

Value of each parameter

Simulation model

Case

Value of each parameter r0 : 5 mm Xwm : 0.6, 1.0, 2.0; Xsz : 0.05, 0.1, 0.15, 0.2, 0.3, 0.5, 0.6, 0.8, 1.0, 1.25, 1.5, 2; Base Weld Soft metal metal zone Case8-1 D1 F2 F1 Case8-2 B1 F4 F1

Case 7 (70)

a0 : 5 mm Xwm : 0.05, 0.1, 0.2, 0.5, 1.0, 1.25, 2.0; 1=Xw : 1, 1.05, 1.1, 1.25, 1.5, 1.6, 1.8, 2, 3, 4, 5, 6, 8, 10; Base metal: D1; Weld metal: F1

Case 8 (72)

Case 9 (72)

a0 : 5 mm Xwm : 0.6, 1.0, 2.0; Xsz : 0.05, 0.1, 0.15, 0.2, 0.3, 0.5, 0.6, 0.8, 1.0, 1.25, 1.5, 2;

Case 10 (240)

Case 9-1 Case 9-2 a

Base metal D1

Weld metal F2

Soft zone F1

B1

F4

F1

The number in the bracket means the total number of the simulation specimen in corresponding case. F1, F2, F3, and F4 are abbreviations of Filler 1, Filler 2, Filler 3, and Filler 4, respectively.

b

a0 : 5 mm Xwm : 0.5; Xsz : 0.1, 0.2, 0.5, 0.75, 1.0; 1=Xw : 1, 2, 4, 10; α: 60, 67.5, 75, 90 Base Weld Soft metal metal zone Case D1 F2 F1 10-1 Case B1 F4 F1 10-2 Case D1 F4 F1 10-3

Simulation model

Welded connections

595

element type has been used in references [5,29,36]. Especially in Ref. [36], the validation of the finite element model was verified by comparing the load
deformation curve predicted by simulation results with test results. As listed in Table 12.9, 10 numerical cases are simulated, including 798 physical experiments in total. Cases 1
3 are considered for validating the theoretical formula for plane strain condition and axisymmetric condition under rigid constraint proposed by Satoh and Toyoda [19,37] as presented in Section 12.5.1. Xtz denotes as the relative width of target zone. Case 4 is used to get the strength model considering the effect of thickness-to-width ratio of plate Xw under rigid constraint. Cases 5
7 are used to obtain the strength model for butt joints considering only two materials (BM and weld metal) under deformable constraint, in which the rigid body shown in Cases 1, 2, and 4 is replaced by nonrigid BM. Case 8 and Case 9 are applied to propose a strength model for butt joints with three materials (BM, weld metal and SHAZ) with deformable constraint under axisymmetric and plane strain condition, respectively. Finally, Case 10 is used to validate the strength model proposed based on the simulation results of those Cases 1
9, considering softened zone relative width ðXsz Þ, welded zone relative width ðXwm Þ, thickness-to-width ratio ðXw Þ, weld bevel ðαÞ, and the relative strength between BM and filler metal ðSwm u Þ. The constitutive model of Filler 1 is used as material model of softened zone in Cases 8
10 due to two reasons: (1) the true constitutive model and the strength data of softened zone are unavailable and (2) the strength of Filler 1 is the lowest among the filler metals, so it can be used for simulating the softened phenomenon.

12.5.3 Formula modification for butt joints with rigid constraint under axisymmetric condition In Satoh and Toyoda’s formula, several assumptions and simplifications in curvature of stress trajectory have been made, while square bar is assumed as axisymmetric condition. Therefore formula (12.4) is verified first. There are four possible equivalent radii relevant with square section (shown in pffiffiffi Fig. 12.18): inscribed radius R1 5 a0 , circumscribed radius R 5 2 a , equivalent 2 0 pffiffiffi pffiffiffi area radius R3 5 2a0 = π, and gyration radius R4 5 2a0 = 6. From the simulation results of Fig. 12.19, relative strength Fjoint =Ftz (ratio of the strength of butt joint to the strength of target zone) of butt joints under cylindrical condition has no relationship with the diameter of round bar. The ratio between the length of target zone ðh0 Þ and the radius of round bar ðr0 Þ, Xtz , is the only factor that affects the relative strength. Besides, when Xtz is more than 2.0, the relative strength is less than 1%, indicating that the effect of constraint is too small and can be neglected. Taking Filler 1 for example, the simulation results (Fig. 12.20) of Case 1 and Case 3 demonstrate that R1 matches best with axisymmetric condition and the equivalent radius r0 equals to the square length a0 . Hence, the parameter r0 and r in Eqs. (12.5)
(12.7) can be replaced by a0 and a, respectively. In order to simplify the calculation formula of rectangular section specimens, the formula of plane strain condition and axisymmetric condition should be unified.

596

Behavior and Design of High-Strength Constructional Steel

Figure 12.18 Section equivalent.

Figure 12.19 Simulation results from Case 1.

pffiffiffi Considering that the yield strength in plane strain condition is approximately 2= 3 times of that in axisymmetric condition, the calculation formula for the ultimate strength in axisymmetric condition is assumed as the following equation:  n11 2 σCu 5 σPu = pffiffiffi 3

(12.8)

Fig. 12.21 shows the comparison between simulation results of Cases 1
3 and theoretical predicted curves. It indicates that Eqs. (12.1) and (12.8) in calculating the ultimate strength of soft interlayer with rigid constraint have good accuracy compared to simulation results.

Welded connections

597

Figure 12.20 Comparison between axisymmetric condition (Case 1) and square section condition (Case 3).

Figure 12.21 Comparison between present theoretical formula and simulation results (Cases 1
4, Filler 1, r0 5 a0 5 5).

12.5.4 Unified formula for butt joints with rigid constraint under all conditions Fig. 12.21 demonstrates that the axisymmetric condition (the curve ①) and the plane strain condition (the curve ②) produce lower and upper limit strengths of rectangular section condition, respectively. When the width W0 has a certain finite value larger than thickness t0 , the tensile strength of the weld joint will lie between the curves ① and ②. For example, when the width W0 changes from t0 to an

598

Behavior and Design of High-Strength Constructional Steel

infinite value under a constant value Xtz , for example, 0.2 and 0.5 (shown in Fig. 12.21), the relative strength will rise from B to A for Xtz 5 0:2 or D to C for Xtz 5 0:5. Curves A0 B0 and C0 D0 show the tensile strength being an approximately linear function of the ratio Xw . Therefore, by taking the thickness-to-width ratio Xw into the theoretical formula as a parameter and after combining Eqs. (12.1) and (12.8) together, a new formula for rectangular condition can be obtained as the following equation: ( σRu

5

2 pffiffiffi 3

) n11 K ðlnð11εÞÞn U ð 1 2 Xw Þ 1 X w ð1 1 Yt Þ ð1 1 ε Þ

(12.9)

where Xw means the thickness-to-width ratio (t0 =W0 or a0 =b0 ). Noting that εup  n [13], the following term can be simplified as: n u K ðlnð11εÞÞn K εp f tz  5 u ð1 1 ε Þ 11n 11n

(12.10)

where futz is the tensile strength of target zone. Noting the similarity of Eqs. (12.1), (12.8), and (12.9), a unified calculation formula for the previous three different conditions with rigid constraints can be obtained as the following equation: σu 5 αwU

futz ð1 1 Yt Þ ð 1 1 nÞ

(12.11)

where αw is a geometric parameter, 8  pffiffiffi n11 > U ð 1 2 X w Þ 1 Xw > < 2= 3 αw 5 1 > > :  pffiffiffin11 2= 3

0 , Xw , 1

Rectangular condition

Xw 5 1

Axisymmetric condition

Xw 5 0

Plane strain condition (12.12)

12.5.5 Unified formula for butt joint with nonrigid constraint under all conditions The formula mentioned previously (Eq. 12.11) is applicable for butt joints with rigid constraint. However, for butt joint in undermatched cases or in softened zone cases, the adjacent material of soft interlayer is deformable and yieldable, resulting in an upper bound for the strength of the butt joint, especially when the relative width of target zone ðXtz Þ reduces to zero. In other words, the tensile strength of butt joint corresponding to Xtz 5 0 is not infinite, different from the prediction by Eq. (12.11). Therefore the following piecewise function is

Welded connections

599

constructed to represent the upper bounded strength of butt joint with nonrigid constraint, σ1u .

σ1u 5

80 > >

> :

1 Xt A 0 Xt 1  Uσu 1 Uσ X 50:5 12 0:5 0:5 u t

α1wUfuref

Xt , 0:5 Xt $ 0:5

where Xt is the relative width of soft interlayer between nonrigid constraints in coma0 0 parison with its counterpart Xtz between rigid constraints; σ0u 5 α1wUσa0 u , σu and σu are the ultimate strengths of butt joint under axisymmetric condition and rectangular condition, respectively, when Xt 5 0; furef is an appropriate strength value as a reference for integrating constraint effect, as listed in Table 12.10 for different cases. The term α1w is a modified geometric parameter to represent nonrigid constraint, such that: α1w 5

ðαw 1 1Þ=2 1

Xt $ 0:2 Xt , 0:2

(12.14)

It is important to note that the constraining material itself has finite strength, with the implication that the strength of the butt joint as determined by Eq. (12.13) must be further bounded by the strength of the butt joint under rectangular condition when Xt 5 0, to provide a unified formula as shown in the following equation:   σu 5 min σ0u ; σ1u

(12.15)

The next subsections discuss that Eq. (12.13) is applicable for both undermatched conditions without HAZ and conditions with SHAZ. However, it is necessary to note that the input values of Xt and σa0 u in each condition should be assigned as per Table 12.10. More specifically, the soft interlayer in the undermatched case without HAZ is the welded zone and the limiting strength is controlled by the BM. As there are three materials in the SHAZ condition, the softened zone is assumed (for convenience) to be the soft interlayer, that is, taking Xt 5 Xsz , while σa0 u corresponds to the lesser of the BM and the filler metal strengths, that is,   fuweaker 5 min fub ; fuw .

12.5.6 Undermatched cases without softened heat-affected zone Only BM and filler metal are considered, herein, in undermatched butt joints without HAZ, in which the strength of the filler metal is lower than that of the BM, as simulated in Cases 5
7 (Table 12.9). Fig. 12.22 shows the relationship between the relative strength and the relative width obtained from simulation and theoretical equations. The x-marks, &-marks, and 3 -marks represent the relative strength obtained from undermatched butt joints under plane strain condition, axisymmetric condition, and rectangular condition, respectively. The curves ① and ② represent

Table 12.10 Assignment of the parameters in Eq. (12.13) under different cases. Cases

Schematic diagrams

Xt

σa0 u

Undermatched

Xwm

fub

Softened zone

Xsz

fuweaker





furef

 fuw =ð1 1 nw Þ U ð1 1 Yt Þ

 fusz =ð1 1 nsz Þ U ð1 1 Yt Þ

    sz sz b sz wm fuw 5 fubU Swm H ; fu 5 fuU SH ; nw 5 nbU Hb =Hw ; nsz 5 nbU Hb =Hsz where Hb , Hw , and Hsz denote the hardness of base metal, weld metal, and softened HAZ, respectively; SH and SH denote the relative hardness of weld metal and softened HAZ to base metal.

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601

Figure 12.22 Comparison between results predicted by Eq. (12.13) and simulation results (Cases 5
7, Filler 1, r0 5 a0 5 5): (A) the effect of Xwm and (B) the effect of Xw .

the relations calculated by Eq. (12.11) for cases with rigid constraints, agreeing well with the simulation results (x-marks, &-marks) when the relative width is large ðXwm $ 0:5Þ. As shown in Fig. 12.22, when Xwm is smaller than 0.5, the simulation results appear to be linearly correlated with Xwm , somewhat different than the estimates as per Eq. (12.11). The upper bound of the linear relation at Xwm 5 0 may be interpreted as the relative strength of BM, αw fub =fuw . More specifically, the bounds pffiffiffin11  upper for plane strain (Case 5) and axisymmetric (Case 6) conditions are 2= 3 fub =fuw b w (Point F) and fu =fu (Point E), respectively, as marked in Fig. 12.22. The effect of thickness-to-width Xw is noted as a nonlinear influence on the relative strength according to results from Case 7 ( 3 -marks), as shown in the enlarged view of Xwm 5 0:2 and Xwm 5 0:5 in Fig. 12.22A. Fig. 12.22B further shows a varying dependence of αw on Xw with nonrigid constraint, where the modified geometric parameter α1w in Eq. (12.14) provides a conservative prediction. Therefore linear interpolation is applied for the first branch of the piecewise function Eq. (12.13), providing satisfactory prediction as curves ③ and ④ in Fig. 12.22.

602

Behavior and Design of High-Strength Constructional Steel

Figure 12.23 Comparison between results predicted by Eq. (12.13) and simulation results of butt joints when the strength of base metal is (A) greater and (B) lower than that of filler metal (Case 8 and Case 9).

12.5.7 Softened heat-affected zone cases The strength relationship between the BM and the filler metal is considered for two situations, that is, undermatched and overmatched, taking evenmatched as their special case. Similar to the observation in last subsection, Eq. (12.11) provides satisfactory prediction for Xsz . 0:5, as depicted in Fig. 12.23 for both undermatched and overmatched cases under both plane strain and axisymmetric conditions. The simulation results are (again) approximately linearly related with Xsz ð p ,ffiffi0:5 Þ. To conduct ffin11 linear interpolation in this range, four horizontal lines of 2= 3 fustronger =fusz ,  pffiffiffin11 weaker sz stronger sz weaker sz fu =fu , 2= 3 fu =fu , and fu =fu are plotted in Fig. 12.23 to enable the selection of an appropriate upper bound value at Xsz 5 0. Here, fustronger and fuweaker represent the larger and the smaller strength between the BM and the filler one. It is apparent that fuweaker can be conservatively and satisfactorily chosen as the upper bound value for σa0 u in Eq. (12.13).

12.5.8 Interpretation of Eq. (12.15) In order to unify constraint effects with various factors in a relatively simple formula of Eq. (12.13), two simplifications have been introduced, that is, the transition value of Xt being chosen as 0.5 and the strength of softened zone in SHAZ case being assumed as the lowest one. Overestimation due to these simplifications may be avoided if the following considerations are Eq. (12.15), as interpreted next.

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603

Figure 12.24 Possible strength distribution from Eq. (12.13): (A) σ0u . σ1u jXt 50:5 , (B) σ0u # σ1u jXt 50:5 , and (C) fuw , fusz

As illustrated in Fig. 12.24A, in most situations σ0u can be observed being larger than the transition value of σ1u at Xt 5 0:5. The contrary situations shown in Fig. 12.24B are due to that the actual transition point of Xt is larger than 0.5. As can been seen in Fig. 12.24A and B, the thicker solid line of Eq. (12.15) can avoid the overestimation for the latter. Fig. 12.24A and B is valid for both the undermatched cases without HAZ and the SHAZ cases. As plotted in Fig. 12.24C, when the strength of the filler metal is smaller than that of the softened zone, this is contrary to the assumption of the lowest strength for the SHAZ case. In this situation, furef in Eq. (12.13) is fusz and fuw is smaller than the values predicted by Eq. (12.13). In fact, the strength of the butt joint is no larger than fuw , which is conservatively provided by Eq. (12.15).

12.5.9 Verification of strength model Table 12.11 summaries the test results and corresponding estimates as per Eqs. (12.15) and (12.16). The specimens are summarized into five groups and the features of each group are listed in the table. For Groups 1
4 the predicted values by Eq. (12.15), EC3 and Eq. (12.16) are almost the same, most of which are close but a little bit lower than the test results, which indicate high precision. For Group 5 the strength decrease due to wide softened zone is neglected in EC3, resulting in

Table 12.11 Comparison of theoretical predicted results and test results. Group no.

Applied case

Group 1

Undermatched cases

Group 2

SHAZ cases with Xt # 0:2 fuw # fusz

Group 3

SHAZ cases with Xt # 0:2fuw . fusz

Assemble no.a A11 B21 B22 Mean CV B11 C11 C21 D13 D23 Mean CV B12 B14 C13 C22 C23 D24 Mean CV

Xt

furef b (MPa)

σa0 u (MPa)

γ

λ

Average test resultsc (MPa)

0.75 0.98 1.00

574 578 661

610 627 720

0.97 0.86 0.98

1.037 0.917 1.053

623 657 723

0.14 0.15 0.15 0.16 0.11

1527 1517 1637 2078 2829

627 627 627 771 771

0.95 1.08 1.14 1.23 1.21

1.154 1.218 1.250 1.295 1.284

639 645 714 863 942

0.19 0.18 0.18 0.15 0.18 0.14

1192 1325 1276 1558 1351 2261

727 771 767 727 771 956

0.84 0.83 0.86 0.97 0.92 0.92

1.102 1.095 1.109 1.167 1.141 1.138

733 785 757 757 771 1044

Predicted value Eq. (12.15)

EC3

Eq. (12.16)

0.971 0.895 0.931 0.932 0.033 0.981 0.972 0.878 0.894 0.818 0.909 0.067 0.992 0.982 1.013 0.960 1.000 0.915 0.977 0.033

0.978 0.954 0.995 0.976 0.017 0.981 0.972 0.878 0.894 0.818 0.909 0.067 0.992 0.982 1.013 0.960 1.000 0.915 0.977 0.033

0.978 0.875 0.995 0.950 0.056 0.981 0.972 0.878 0.894 0.818 0.909 0.067 0.992 0.982 1.013 0.960 1.000 0.915 0.977 0.033

Group 4

SHAZ cases with 0:2 , Xt # 0:5fuw . fusz

Group 5

SHAZ cases with Xt . 0:5fuw . fusz

C14 C24 D14 C14Cd Mean CV C11d C12d C13d C14Ad C14Bd Mean CV

0.26 0.39 0.23 0.44

897 771 1495 610

767 832 956 856

0.79 0.78 0.95 0.65

0.973 0.963 1.168 0.799

796 840 974 788

0.76 0.65 0.53 0.69 0.64

583 596 644 601 558

627 727 771 856 856

0.82 0.78 0.74 0.70 0.64

0.882 0.832 0.795 0.744 0.685

611 642 685 716 764

1.003 1.007 1.001 0.898 0.977 0.047 1.001 0.973 0.986 0.880 0.767 0.921 0.048

0.964 0.990 0.982 1.086 1.005 0.047 1.026 1.132 1.126 1.196 1.120 1.120 0.048

   pffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi      Xt 5 1= 3 ð1 2 εa Þ 2εa =Yt 2 3ð1 2 εa Þ 3 2εa 1 1=3 2 ð12εa Þ2 Yt =2εa where εa 5 1=ð1 1 εÞ. a The overmatched cases (A12, A21, A22, and B12) are not listed in this table. b The variable Yt for calculating furef in Eq. (12.12) is obtained from the following equation. c For each assemble, five replicates were tested and the data in this column are the average strength of these five specimens. d The data are from the references [36,50]. Source: Test data from Sun F, Ran M, Li G, et al. Experimental and numerical study of high-strength steel butt weld with softened HAZ. Proc Inst Civ Eng Struct Build 2017:1-15 and Ref. [50].

0.937 0.953 0.982 0.868 0.935 0.067 0.905 0.942 0.895 0.890 0.767 0.880 0.072

606

Behavior and Design of High-Strength Constructional Steel

Figure 12.25 The comparison between theoretical predicted results and simulation results: (A) the effect of Xw (Case 10-1, α 5 0); (B) the effect of Xw (Case 10-3, α 5 0); (C) the effect of Xt (Xw 5 0:1; α 5 0); (D) The effect of Xt (Xw 5 0:25; α 5 0); (E) the effect of welded bevel α (Xw 5 0:25; Xt 5 0:5); and (F) The effect of welded bevel α (Xw 5 0:25; Xt 5 0:75).

an overestimation of the ultimate strength of butt joints up to 120% (C13d), which is unconservative and undesirable. Similar observations arise from the comparison between the theoretical predicted results and the simulation results (Fig. 12.25). Larger values of Xw result in an increase in accuracy of the estimates as per Eq. (12.15). The increase of Xt results in the overestimation of strength up to 140% by EC3, which is unacceptable. Meanwhile, it is also noted that Xt does not significantly affect the prediction of Eq. (12.15) (Fig. 12.25C

Welded connections

607

and D). The parameter α has a modest on the prediction accuracy of EC3 and Eq. (12.15), as shown in Fig. 12.25E and F. Overall, the prediction accuracy of Eq. (12.15) is much higher than that of EC3 for butt joints with SHAZ.

12.6

Design proposal

12.6.1 Design formula According to the investigation presented hereinabove, EC3 overestimated the ultimate strength of HSS butt joint when the softened zone is wide enough ðXsz . 0:5Þ. Even though the calculation formula (12.15) is more accurate than EC3, it is somewhat cumbersome to apply in a design setting. A simplified piecewise formula is proposed as shown in Eqs. (12.16) and (12.17), along with recommended values for various parameters to be used within this formula (in Table 12.12): fubutt 5 λσa0 u

(12.16)

where λ is the strength reduction factor obtained by Eq. (12.17) and λ # 1. 8 0:5Uγ 1 0:68 Xt , 0:2 > > < 1:23Uγ 0:2 # Xt , 0:5 λ5 0:5 # Xt , 1:0 > 1:07Uγ > : γ Xt $ 1:0

(12.17)

where γ is the relative strength of the softest zone in butt joint. HSS is a new constructional material, so welding procedure qualification should be done before welding. Since softened phenomenon is common when welding HSS, the width of softened zone Xsz and hardness of softened zone Hsz should be added as two new performance parameters for welding procedure qualification, just as tensile strength and impact toughness. Then strength reduction factor λ can be calculated by Eq. (12.17) and based on the design strength of BM and filler metal, the strength of butt joints can be obtained from Eq. (12.16). The relative accurate strength can be calculated by this way. However, if the present information about HSS cannot be collected or the welding procedure qualification cannot be done, a conservative value of λ can be considered as γ and the Vickers hardness of softened Table 12.12 Assignment of the parameters in Eqs. (12.16) and (12.17) under different cases. Cases Undermatched cases Softened HAZ cases HAZ, Heat-affected zone.

Xt

σa0 u

Xwm Xsz

fub weaker fu

γ 

Swm u  Hsz =minðHb ; Hw Þ or 190=minðHb ; Hw Þ

608

Behavior and Design of High-Strength Constructional Steel

zone can be taken as 190 [55], which is the hardness of the weakest metallographic phase (i.e., ferrite). Besides, if necessary, a designer can designate a strength of butt joint, which is smaller than fuweaker , and through Eqs. (12.16) and (12.17) propose the lower limitation of the width and the hardness of the softened zone. Then through selecting different filler materials and welding process (i.e., heat input energy) make the softened zone meet the requirements.

12.6.2 Design strength A resistance factor, φ, needs to be identified to determine the design strength of HSS butt joint as butt fdu 5 φfubutt

(12.18)

for guaranteeing the suitable reliability of the joint. The reliability of the strength obtained with EC3, Eqs. (12.15) and (12.16), can be evaluated through a reliability analysis, with the objective of characterizing the resistance factor required to obtain the target safety index (or failure probability). Traditionally [56], the target safety index, β, is selected as 4.0
4.5 for welded connections that show limited ductility and undesirable modes of failure. For a given resistance factor the safety index for each predictive model is different because the procedure to obtain the joint capacity varies between models. The resistance factor, φ, that provides the required safety index β is commonly obtained from a relationship originally proposed by Ravindra and Galambos [57], φ 5 CρR expð 2βαR VR Þ

(12.19)

where C is a correction factor when the resistance factor is not 3.0; αR is a separation variable taken as 0.55, as originally proposed by Ravindra and Galambos [53]; and ρR and VR are the bias coefficient and coefficient of variation for the resistance, respectively. An equation for C, derived using a procedure proposed by Fisher et al. [52] for welded and bolted connections, is adopted to calculate the adjustment factor for a live load to dead load ratio of 3.0. C 5 0:0078β 2 2 0:156β 1 1:400

(12.20)

The bias coefficient for the resistance, ρR , is determined as: ρR 5 ρG ρM ρP

(12.21)

The associated coefficient of variation is given by: VR2 5 VG2 1 VM2 1 VP2

(12.22)

Welded connections

609

The geometric factor ρG is defined as the ratio between the effective weld size based on pretest measurements of the weld profiles and the nominal weld sizes. Based on previous study [36], for butt welds, the reinforcement in welded zone, which is thicker than plate, is usually removed before test and uniform thick along the specimen is presumed. Published data regarding measured size of butt welds are relatively sparse. Consequently, the mean value of ρG and its coefficient of variation VG in fillet weld according to Kwan et al. [54] was used here (ρG 5 1:07 and VG 5 0:15). The term ρM is the material parameter accounting for the variation of material strength, which addresses the variation in the filler metal tensile strength or yielding strength of the plate. Here, ρM is selected as the mean value of the measured to nominal static yield strength of the plate (ρM 5 fyb =Fyb , ρM 5 1:164, VM 5 0:013, where fyb is the measured yield strength, and Fyb is the nominal yield strength). The value is obtained from the material experiments on a series of HSS tests (360 specimens). The professional factor, ρP , is the average ratio of the observed test capacity to the predicted capacity, and VP is the corresponding coefficient of variation. The predicted capacity is calculated using each of the predictive models with the measured values of the relevant material and geometric properties. Those three strength predictive models described earlier are analyzed to determine the safety index offered by each one. The results of the analysis and the resistance factors for different values of the safety index are presented in Table 12.13. The bias coefficient for the geometric parameter and the material parameter and the professional factor for all the models are also listed in the table. According to EC3, the design resistance of butt joint for steels with grades greater than S460 up to S700 is estimated as: Nt;Rd 5

0:9Anet fuweaker γ M12

(12.23)

Table 12.13 Summary of safety indices for butt joint. Professional factor Xsz ,0.2 0.2
0.5 .0.5 Eq. (12.15) ,0.2 0.2
0.5 .0.5 Eq. (12.16) ,0.2 0.2
0.5 .0.5

EC3

Resistance coefficient

ρP

VP

ρR

VR

1.061 0.997 0.944 1.061 1.073 1.068 1.061 1.039 1.176

0.066 0.045 0.077 0.066 0.082 0.089 0.066 0.030 0.091

1.301 1.241 1.115 1.298 1.285 1.236 1.302 1.409 1.432

0.165 0.161 0.162 0.167 0.166 0.193 0.165 0.168 0.171

Resistance factor

Safety index

φðβ 5 4:0Þ φðβ 5 4:5Þ β ðφ 5 0:72Þ 0.82 0.78 0.70 0.81 0.80 0.73 0.82 0.88 0.89

0.74 0.71 0.64 0.73 0.73 0.66 0.74 0.8 0.8

4.69 4.45 4.03 4.69 4.65 4.58 4.69 4.72 5.07

The following value for each parameter is used in all models: ρG 5 1:07, VG 5 0:154, ρM 5 1:164, VM 5 0:013.

610

Behavior and Design of High-Strength Constructional Steel

where Nt;Rd is design resistance; Anet means the area of net section; γ M12 is partial factor for HSS, and the value γ M12 5 1:25 is recommended. From Eq. (12.23), the resistance factor is: φ5

0:9 5 0:72 γ M12

(12.24)

As listed in Table 12.13, for a resistance factor φ 5 0:72, the safety indices provided by the calculation formula (12.15) and the design formula (12.16) are higher than the target value of 4.5. EC3 has a poor safety index in wide softened zone, which is only 3.87.

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8.

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Behavior and Design of High-Strength Constructional Steel

[34] Maurer W, Ernst W, Rauch R, et al. Electron Beam Welding of A TMCP Steel With 700 MPa Yield Strength[J]. Welding in the World 2012;56(9):85
94. [35] Panda SK, Sreenivasan N, Kuntz ML, et al. Numerical Simulations and Experimental Results of Tensile Test Behavior of Laser Butt Welded DP980 Steels[J]. Journal of Engineering Materials and Technology 2008;130(4):41003. [36] Sun F, Ran M, Li G, et al. Experimental and numerical study of high-strength steel butt weld with softened HAZ[J]. Proceedings of the Institution of Civil Engineers Structures and Buildings 2017;1
15. [37] Satoh K, Toyoda M. Size effect on static tensile properties of welded joints including soft interlayer[J]. Journal of the Japan Welding Society 1970;37(11):58
70. [38] Davidenkov N.N., Spiridonova N.I. Mechanical methods of testing-analysis of the state of stress in the neck of a tension test specimen[C]. AMER SOC TESTING MATERIALS 100 BARR HARBOR DR, W CONSHOHOCKEN, PA 19428-2959, 1946. [39] ANSI/AISC 360-10 Specification for Structural Steel Buildings[S]. 2010. [40] Code for design of steel structures[S]. First Edition ed. Beijing, 2003. [41] Standard test methods for tension testing of metallic materials[S]. West Conshohocken, PA, 2008. [42] Society A.W. Specification for Carbon Steel Electrodes for Flux Cored Arc Welding [S]. Standard A A N, 2005. [43] High strength structural steel plates in the quenched and tempered condition[S]. First Edition ed. China, 2009. [44] Welding electrodes and rods for gas shielding arc welding of carbon and low alloy steel[S]. First Edition ed. China, 2008. [45] ARAMIS User Manual - Software[M]. Germany: GOM mbH Mittelweg 7-8 D-38106 Braunschweig, 2011. [46] Kamaya M, Kawakubo M. A procedure for determining the true stress
strain curve over a large range of strains using digital image correlation and finite element analysis [J]. Mechanics of Materials 2011;43(5):243
53. [47] Ghaednia H, Cermik O, Marghitu DB, et al. Collision measurements using digital image correlation techniques[J]. International Journal of Mechanical Sciences 2017;131-132:836
46. [48] Johnson DK, Botha TR, Els PS. Real-time side-slip angle measurements using digital image correlation[J]. Journal of Terramechanics. 2018. [49] Li G, Lv H, Zhang C. Experimental research on post-fire mechanical properties of Q690 steel[J]. Journal of Building Structures 2017;05:109
16. [50] Ran M, Sun F, Li G, et al. Experimental study on behavior of mismatched butt welded joints of high strength steel[J]. Journal of Constructional Steel Research 2018. [51] Rahman M, Maurer W, Ernst W, et al. Calculation of hardness distribution in the HAZ of micro-alloyed steel[J]. Welding in the World 2014;58(6):763
70. [52] Fisher JW, Ravindra MK, Kulak GL, et al. Load and resistance factor design criteria for connectors[J]. Journal of the Structural Division 1978;104(9):1427
41. [53] Ravindra MK, Galambos TV. Load and resistance factor design for steel[J]. Journal of the Structural Division 1978;104(9):1337
53. [54] Kwan YK, Gomez IR, Grondin GY, et al. Strength of welded joints under combined shear and out-of-plane bending[J]. Canadian Journal of Civil Engineering 2010;37(2):250
61. [55] Sun F, Ran M, Li G, et al. Strength model for mismatched butt welded joints of high strength steel[J]. Journal of Constructional Steel Research 2018;150:514
27.

Application of high-strength steels in seismic zones and case studies

13

Guo-Qiang Li and Yan-Bo Wang Tongji University, Shanghai, P.R. China

13.1

Introduction

With the recent development of high-strength steel and advances in welding techniques, high-strength steel (HSS) members can be produced at a reasonable cost and quality. However,the use of HSS was limited by current codes for seismic resistance usage, such as National Code for Seismic Design of Buildings GB 50011-2010 [1] which requires a higher ductility of steels due to the expectation of inelastic behavior of structural elements and connections under rare earthquakes. However, with the increase of material strength, the yield to tensile strength ratio (Y/T ratio) and elongation ratio of HSS which affect the structural ductility could hardly meet the requirements of GB 50011-2010. Therefore it is important to determine whether HSS could be used in seismic structures and how to use HSS in seismic structures [2].

13.2

Limits related to application of high-strength steel in seismic structures

13.2.1 Effect of material properties on the ductility of structural members It is expected that structures will deform inelastically under rare earthquake action. Thus structural members must process adequate ductility. Previous researches [3 5] indicate Y/T and elongation ratios play fundamental rules in influencing the ductility of structural members. It is obviously the elongation ratio of material has a direct influence on the deformability of structural members. Meanwhile, Y/T ratio has an indirect influence on the ductility of structural members, as shown in Fig. 13.1. Fig. 13.1(A) shows a perforated or reduced tensile member, where fy is the yield strength, fy , fp # fu , fu is the tensile strength, Ny 5 Afy , Np 5 An fp , A is the area of cross-section, and An is the net area of reduced cross-section. If the Y/T ratio is high enough to result in Ny . Np, the inelastic deformation will localized in the reduced zone. Although the material itself has a good deformability, the Behavior and Design of High-Strength Constructional Steel. DOI: https://doi.org/10.1016/B978-0-08-102931-2.00013-9 © 2021 Elsevier Ltd. All rights reserved.

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Figure 13.1 The effect of yield ratio on the ductility of members.

ductility of the member will be significantly impaired by the localized deformation. Fig. 13.1(B) shows a flexure member, where My is yield bending moment, Mpi is plastic bending moment. If the Y/T ratio is close to 1, the development of plastic zone in beam ends will be limited because the values of Mp1 and Mp2 are very close. Consequently the rotation capacity of the beam, which represents the ductility of flexure members, will be jeopardized.

13.2.2 Limits of current design codes The limits of the Y/T and elongation ratios have been specified in GB 50017-2003 [6] and GB 50011-2010 [1] to ensure the ductility and deformability of structural members for nonseismic and seismic usage, respectively, as shown in Table 13.1. Obviously compared to nonseismic usage specified in GB 50017-2003, seismic usage specified in GB 50011-2010 has a more restricted limits on the Y/T and elongation ratios. The tensile coupon test results of various grades of steel [7] demonstrated that the increase in strength will result in increasing Y/T ratio and decreasing elongation, which mean the HSS could hardly meet the material requirements of seismic design. The limits of material properties specified in Standard on Steel Plates for Building Structure GB/T 19879-2005 and Standard on High Strength Low Alloy Structural Steels GB/T 1591 2008 are shown in Tables 13.2 and 13.3, respectively. It can be seen that steel with nominal yield strength greater than 420 MPa could not meet the requirements of GB 50011-2010 on properties of steel for application of structures to resist seismic actions.

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Table 13.1 Comparison of requirements of material properties. GB 50011-2010

GB 50017-2003

1. Yield to tensile strength ratio # 0.85 2. Ratio of elongation after failure $ 20% 3. Well defined yielding plateau

1. Yield to tensile strength ratio # 0.83 2. Ratio of elongation after failure $ 15% 3. εu $ 20εy

Table 13.2 Requirements of material properties in GB/T 19879-2005. Steel grade

Yield strength (MPa)

Tensile strength (MPa)

Elongation (%)

Y/T raito

Q235GJ Q345GJ Q390GJ Q420GJ Q460GJ

$ 235 $ 345 $ 390 $ 420 $ 460

400B510 490B610 490B650 520B680 550B720

$ 23 $ 22 $ 20 $ 19 $ 17

# 0.80 # 0.83 # 0.85 # 0.85 # 0.85

Table 13.3 Requirements of material properties in GB/T 1591 2008. Steel grade Q345 Q390 Q420 Q460 Q500 Q550 Q620 Q690

13.3

Yield strength (MPa)

Tensile strength (MPa)

Elongation (%)

Y/T raito

$ 345 $ 390 $ 420 $ 460 $ 500 $ 550 $ 620 $ 690

470B630 490B650 520B680 550B720 610B770 670B830 710B880 770B940

$ 20 $ 20 $ 19 $ 17 $ 17 $ 16 $ 15 $ 14

# 0.73 # 0.80 # 0.81 # 0.84 # 0.82 # 0.82 # 0.87 # 0.90

Proposed methods for application of high-strength steel in seismic structures

The current seismic design philosophy is based on the assumption that structural members and connections can undergo a certain plastic deformation without a loss in structural bearing capacity under expected rare earthquakes. This assumption is simply ensured by the specified material property requirements in current seismic design code. However, as HSS could not meet such material property requirements, HSS members are hard to be used in seismic structures. However, the application of HSS in seismic structures could be reconsidered in the level of seismic design philosophy to give up the property requirements of steel in current seismic design

618

Behavior and Design of High-Strength Constructional Steel

Figure 13.2 The effect of ductility on the requirement of design seismic actions.

code. The methodologies for application of HSS in seismic structures are proposed hereinafter.

13.3.1 Determination of design earthquake action In view of structural performance under earthquakes, design earthquake action could be reduced in accordance with the ductility of structures, as shown in Fig. 13.2. Structures with good ductility could be designed under a reduced seismic action, so the inelastic behavior of structures is expected under actual earthquakes and the seismic energy could be dissipated by plastic deformation of members and connections. Instead, brittle structures should be designed under higher seismic action than ductile structures to reduce the requirement of structural ductility.

13.3.2 Selection of structural systems The elastic design of steel structures under expected rare earthquakes are usually uneconomical and unreasonable. Thus seismic resistant buildings are mostly designed as energy dissipative structures which allow for the formation of plastic hinges in the expected locations, such as “strong column, weak beam” philosophy. It is important to note that the ductility of structures, in addition to the ductility of materials and members, also depends on the selection of structural systems. Since the ductility of HSS members is poor to satisfy the demands of structural plastic behavior during earthquakes, an appropriate structural system to use HSS members is to isolate the plastic deformations in the specified ductile members to prevent the yield of HSS members, as shown in Fig. 13.3 Semi-rigid frames could provide sufficient inelastic deformation with large rotation capacity of beam-to-column connections and also large lateral stiffness and load-bearing capacity with energy dissipative members, such as buckling-restrained braces (BRBs) and buckling-restrained steel plate shear

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619

Figure 13.3 Structure system providing elastic design of high-strength steel member.

Table 13.4 Classification of ductility for steel members in term of material properties. Material properties

Ductile member

Semi-ductile member

Brittle member

Elongation (%) Y/T

$ 20 # 0.85

$ 14 # 0.9

, 14 , 0.9

wall, under seismic action. The yield of HSS beams and columns can be prevented by the limited capacity of beam-to-column connections.

13.3.3 Adjustment of reliability index Moreover, under probability-based design criteria, the targeted reliability index of members with insufficient ductility could be strengthened to enhance the safe of those members. Structural members are classified as ductile members and brittle members in Unified Standard for Reliability Design of Building Structures GB 50068-2001 [8]. Although HSS members could not meet the material property requirements of ductile members, it is uneconomical to be used as brittle members. Therefore a new class member named semi-ductile member is suggested, as shown in Tables 13.4 and 13.5. If HSS could not meet the material property requirements of ductile member while can meet the material property requirements of semiductile member, the target reliability index of semi-ductile member could be adopted in the design of HSS members.

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Behavior and Design of High-Strength Constructional Steel

Table 13.5 Target reliability index of various ductility classifications of steel members. Member classification

Ductile Semi-ductile Brittle

Important category I

II

III

3.7 4.2 4.7

3.2 3.7 4.2

2.7 3.2 3.7

Figure 13.4 Perspective view of the building.

13.4

Information of the case study

This case study is based on the structure for a building located in Shanghai. It was designed for a complex function combining a cinema and spaces for commercial and office uses. The centrally chevron-braced steel frame structure is used. The building is 50.8 m in height with 14 stories above the ground and 1 stories below the ground, as shown in Fig 13.4 The plane size of the building at bottom is 36 3 39.2 m2 and at the standard floor is 22.2 3 24.9 m2. The three-dimensional model and elevation of the structure for the building are shown in Fig 13.5(A) and (B), respectively. The live loads were determined in accordance to Chinese Code for Loads of Building Structures GB50009-2012 [9], as shown in Table 13.6. The design seismic intensity is 7, the seismic peak ground acceleration of which is 0.10 g with occurrence of once in 475 years. The site category is IV and the site characteristic period is 0.90 seconds. The seismic peak ground accelerations for

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621

Figure 13.5 Structural model and elevation of the building.

Table 13.6 Live loads on the building. Number

Location

Live load (kN/m2)

1 2 3 4 5 6 7 8 9 10 11

Nonaccessible roof Accessible roof Office Restroom Cinema Restaurants Gymnasium Book storage room Elevator machine room Stair Lobby

0.5 2.0 2.0 2.0 3.0 2.5 4.0 5.0 7.0 3.5 2.5

frequent and rare earthquakes are scaled down to 0.035 and up to 0.20 g, respectively. The basic wind pressure is 0.55 KN/m2 with occurrence of once in 50 years. Various steels, including Q690, Q460, and Q345 grades, were selected for the structure for comparison. The sizes of the main structural members are shown from Tables 13.7 13.9. It should be noted that, to dissipate energy under rate earthquake and protect high-strength steel columns and beams, BRBs are used in 1 8 floors of the Q690 structural solution, as shown in Table 13.9.

Table 13.7 Sizes of structural members of Q345 steel solution. Floor

Middle column

Edge column

Girder (X-direction)

Girder (Y-direction)

Beam

Brace

1 5 6 9 10 14

&750 3 32 &550 3 26 &450 3 20

&650 3 22 &550 3 22 &450 3 18

H550 3 180 3 14 3 22 H550 3 220 3 12 3 16 H550 3 220 3 12 3 16

H550 3 200 3 12 3 16 H550 3 240 3 10 3 16 H550 3 240 3 10 3 16

H450 3 200 3 10 3 14 H500 3 200 3 10 3 16

PipeΦ377 3 14

Table 13.8 Sizes of structural members of Q460 steel solution. Floor

Middle column

Edge column

Girder (X-direction)

Girder (Y-direction)

Beam

Brace

1 5 6 9 10 14

&650 3 30 &450 3 28 &450 3 18

&550 3 22 &450 3 18 &400 3 14

H550 3 200 3 10 3 14 H500 3 180 3 10 3 14 H500 3 180 3 10 3 14

H550 3 240 3 10 3 16 H500 3 200 3 10 3 16 H550 3 200 3 10 3 16

H400 3 150 3 8 3 14 H400 3 200 3 10 3 14

PipeΦ377 3 14

Table 13.9 Sizes of structural members of Q690 steel solution. Floor

Middle column (Q690 & Q345)

Edge column (Q690 & Q345)

Girder in X-direction (Q460)

Girder in Y-direction (Q460)

Beam (Q460)

Brace

1 5

&500 3 3 28 (Q690) &400 3 18 (Q690)

&500 3 22 (Q690) &400 3 18 (Q690)

H550 3 200 3 10 3 14

H550 3 240 3 10 3 16

H400 3 150 3 8 3 14 H400 3 200 3 10 3 14

BRB (1 8 floors) PipeΦ377 3 14 (9 14 floors)

&400 3 3 14 (Q460)

&400 3 14 (Q460)

H500 3 180 3 10 3 14

H550 3 200 3 10 3 16

H500 3 3 180 3 10 3 14

H500 3 200 3 10 3 16

6 9

10 14

BRB, Buckling restrained brace.

Application of high-strength steels in seismic zones and case studies

13.5

623

Comparison of structural performance between normal strength steel solution and high-strength steel solution

The overall performance of the structures with different steel solutions can be obtained with using finite element method analysis. The followings are the main results.

13.5.1 Period and period ratio The first six periods of the three types of steel structures are shown in Table 13.10. The fundamental torsional-to-translational period ratios, Tt/T1, of the three types of steel structures are 0.66, 0.66, and 0.67, respectively, which is within the upper limit value of 0.9 [10]. It can be seen that the period of the structure increases slightly with the increase of steel strength due the size reduction of structural members. However, it is not unfavorable considering that the seismic action on the structure will be reduced with increase of structural period.

13.5.2 Performance of structures under frequent earthquakes and wind The inter-story drift angles of the structures with various steels under frequent seismic actions and basic wind loads are illustrated in Fig. 13.6(A) (D), respectively. It can be found that the inter-story drift angle of the structures with various steels increases slightly with the increase of steel strength, but within the limit of 1/250 specified in the code. Because the sizes of the beams and columns used in the Q690 steel structure are smaller than that of the Q460 steel and Q345 steel structures, and the stiffness of the BRBs adopted in the Q690 steel structure in 1 10 stories is less

Table 13.10 Periods of three types of steel structures (s). Modes

Q345 steel structure

Q460 steel structure

Q690 steel structure

1 (Translational motion) 2 (Translational motion) 3 (Torsional motion) 4 (Translational motion) 5 (Translational motion) 6 (Torsional motion)

2.358

2.681

2.956

2.256

2.607

2.931

1.547 0.725

1.758 0.8

1.97 0.903

0.7

0.783

0.891

0.509

0.56

0.639

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Behavior and Design of High-Strength Constructional Steel

Figure 13.6 Comparison of structural inter-story drift ratio.

Table 13.11 The displacements of the structures. Structures

Q345 steel

Q460 steel

Q690 steel

Directions

X

Y

X

Y

X

Y

Maximum top displacement (mm) Torsional displacement ratio Stiffness to weight ratio

91.0

92.9

110.5

110.3

124.8

120.9

1.09 3.06

1.02 2.84

1.08 2.25

1.03 2.16

1.02 1.75

1.03 1.72

than that of ordinary braces used in the Q460 steel and Q345 steel structures, the inter-story drift angles of the Q690 steel structure are greater than that of other two types of steel structures. The displacements of the steel structure in Q345, Q460, and Q690 solutions under the frequent earthquake are shown in Table 13.11. The load-bearing capacities of the structural members are checked with the forces induced by the combined actions under frequent earthquake. The forcecapacity ratio of the structural members is in the range of 0.80 0.85 for the side columns and around 0.90 for the middle columns under frequent earthquakes.

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625

13.5.3 Performance of structures under rare earthquakes According to the requirements of Chinese Standard for Seismic Design, Coaling345 seismic wave is selected for severe seismic evaluation. The main ground motion of coaling-345 seismic wave is shown in Fig 13.7 The maximum base shear force under frequent and rare earthquakes obtained by time history analysis, compared with that obtained by response spectrum method (complete quadratic combination, CQC), for the structures with various steels are shown in Table 13.12. The base shear force of Q690 steel structure is 84% of that of Q345 steel structure under frequent earthquake. This is mainly attributed to the reduced self-weight. Under rare earthquake, the additional favorable reduction in base shear force is achieved by the excellent energy absorption capacity of BRBS in the Q690 steel structures. The time histories of the base shear force and top story lateral displacement of the structures with various steels under the rare earthquake are shown in Figs. 13.8 and 13.9, respectively. It can be found that the maximum top displacement of the Q690 steel structure with BRBs decreases by 17% and 7% compared with the Q345 steel structure and the Q460 steel structure, respectively.

Figure 13.7 The ground motion of Coaling-345 seismic wave.

Table 13.12 Base shear forces of the structures with various steels obtained by different methods (kN). Structures with various steels

Q345 steel structure Q460 steel structure Q690 steel structure

Base shear force CQC

Frequent earthquake

Rare earthquake

4774 4289 3986

4804 4668 3230

25279 21174 13153

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Behavior and Design of High-Strength Constructional Steel

Figure 13.8 Base shear force history of three structures.

Figure 13.9 Top displacement of three structures.

The inter-story drift angle of the structures under the rare earthquake is shown in Fig 13.10. Due to the use of BRBs in the 1 10 stories of Q690 steel structure, large amount of seismic energy input to the structure is consumed by the plastic deformation of BRBs under the rare earthquake, which greatly reduces the seismic response of the structure. So, the inter-story drift angle of the Q690 steel structure under the rare earthquake is significantly decreased compared with that of the structures using other two grades of steels. Compared with the Q345 steel structure, the average inter-story drift angle of the Q690 steel structure with BRBs under the rare earthquake is reduced by 11.7% with smaller sizes of column sections.

13.6

Economic evaluation of high-strength steel structures

Economic evaluation of a building structure can be made in terms of steel price, steel consumption, occupied area of structural members, overall quality of the structure, storage, and transportation costs, etc. The aforementioned aspects are discussed separately as follows.

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627

Figure 13.10 Variation of inter-story drift angle of the structures under rare earthquake.

Figure 13.11 Steel price analysis.

13.6.1 Evaluation of the prices of structures using different steels Through field investigation and telephone contact, the price of steel products with different strength grades (I-beam and steel plate) in Shanghai was investigated. The results are shown in Fig. 13.11. From the results, it can be seen that with the increase of steel strength, the price of steel per unit weight (normalized price) increases, but the price of steel per unit strength decreases continuously, which provides necessary conditions for the economic feasibility of using high-strength steel structure. If the strength of steel can be fully utilized, it will be helpful to generate significant economic benefits.

13.6.2 Evaluation of steel consumption The steel consumption and the cost of the structures with various steels is listed in Table 13.13. The steel consumption of the structure with Q460 high-strength steel solution is about 80% of the structure with Q345 steel solution. Considering the

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Behavior and Design of High-Strength Constructional Steel

Table 13.13 Steel consumption and cost of the structures with various steels. Structure types

Steel for column (102t)

Steel for beam (102t)

Steel for brace (102t)

Steel for unit area (kg/m2)

Material cost (yuan/ m2)

Q345 steel Q460 steel Q690 steel

3.93 2.88 2.23

4.42 3.70 3.70

0.77 0.77 0.14

108.28 87.49 72.08

517 477 545 (with BRB)

price difference of different grades of steel, the material cost of the Q460 steel structure is 7.7% lower than that of the Q345 steel structure. For the Q690 steel structure, the material cost for steel beams and columns is 418 yuan/m2 and the cost of BRBs is approximately 18,000 yuan/pc. The total material cost of the Q690 steel structural scheme is approximately 545 yuan/m2, which is 5.4% higher than that of the Q345 structural scheme.

13.6.3 Occupied area of structural members Through statistical analysis of the area occupied by the column grids of three different steel structures, the area of Q460 and Q690 steel structural columns occupied is decreased, compared with Q345 steel structural scheme, from 82 to 60, and 46 m2, by 25% and 44%, respectively. The cumulative height of beams for the structures is decreased from 7.15 m for Q345 steel structure to 6.75 m for Q460 and Q690 steel structures, by 5%. Therefore the high-strength steel structure scheme can effectively increase the available area of the building, which is conducive to the investors to obtain greater economic benefits.

13.6.4 Foundation construction cost In general, the total investment of a building project, including structural system cost, enclosure system cost, mechanical system cost, and so on, in which the structural construction cost is approximately 30% of the overall construction cost. The cost of structure construction can be further divided into superstructure construction cost and foundation construction cost. Normally the cost of superstructure construction is only 50% 70% of the cost of foundation construction. When Q460 highstrength steel is used for the structure, the consumed quantity of the structural materials is approximately 80% of that of the Q345 steel structure, which greatly reduces the self-weight of the structure and further reduce the cost of foundation construction. In addition, the decrease of foundation construction cost and difficulty is also beneficial to the decrease of the overall construction duration.

Application of high-strength steels in seismic zones and case studies

13.7

629

Summary

1. Although the ductility of high-strength steel is not so good as the that of normal structural steels and not so favorable to be used in seismic zones for engineering structures, it is still applicable for structures to resist earthquakes with limiting the plastic deformation demand or increase the seismic action in the structural design to reduce the probability of the excessive plastic deformation in the high-strength steel elements. 2. Although the unit price of high-strength steel is high compared with that of normal structural steels, the material price per unit strength decreases with the increase of the strength. Using high-strength steel in steel structure can greatly reduce the amount of steel consumed, and it still has certain economic advantages. If we can make full use of its advantages of high-strength steels, and ensure the reasonable design, the construction cost of high-strength steel structures can be reduced significantly. 3. The use of high-strength steel as structural material is conducive to reducing the weight of the structure, increasing the structural natural periods, and further greatly reducing the seismic response of the structure in general. In the region requiring the higher seismic resistance level, the effect is even more obvious, which is also conducive to improving the economic significance of using high-strength steel structures.

References [1] GB 50011-2010. Code for seismic design of buildings. Beijing: China Architecture & Building Press; 2010. [2] Li GQ, Wang YB, Chen SW. The art of application of high-strength steel structures for buildings in seismic zones. Advanced Steel Construction 2015;11(4):492 506. [3] Kato B. Deformation capacity of steel structures. J Construct Steel Res 1990;17:33 94. [4] Kato B. Role of strain-hardening of steel in structural performance. ISI J Int. 1990;30:1003 9. [5] Earls CJ. Influence of material effects on structural ductility of compact I-shaped beams. J Struct Eng 2000;126:1268 78. [6] GB 50017-2003. Code for design of steel structures. Beijing: China Architecture & Building Press; 2003. [7] Fukumoto Y. New constructional steels and structural stability. Eng Struct. 1996;18:786 91. [8] GB 50068-2001. Unified standard for reliability design of building structures. Beijing: China Architecture & Building Press; 2001. [9] Load Codes for The Design of Building Structures: GB 50009-2012. Beijing: China Architecture and Building Press, 2012. [10] Technical Specification for Steel Structure of Tall Buildings: JGJ 99-2015 Beijing: China Architecture and Building Press, 2015.

Index

Note: Page numbers followed by “f” and “t” refer to figures and tables, respectively. A Adhesive force, 552 effect of, 560 561 ANSI/AISC 360 16, 321 322, 350 351, 526 Antisymmetric welding sequence, 214 215 Applied pretension force, 538 Architectural and structural advantages, 5 Argon (Ar), 569 ASTM A992 steel column model in ABAQUS, 465 466 Axial compressive force in specimen, 421 422 Axial displacement, 419 421, 419f, 450 454 Axial force ratio, 396 397 Axial restraint ratio, 414 Axial stiffness ratio, 435 effect of, 439 B Backstresses, superposition of, 78 Base metal (BM), 565 Bauschinger effect, 94 Beam residualstresses effect on bearing capacity of, 292 294 weldedI-section, 271 272 Bearing resistance, 496 500, 499f Bearing strength considering excessive hole elongation, 511 Bearing-type bolted connections for HSSs, 493 529 behavior of multibolt connection in tandem, 523 525 behavior of two-bolt connection in parallel, 514 523 optimization of e2 to p2 ratio, 520 523

behavior of single-bolt connection, 494 514 experimental design, 494 496 geometric parameters, 496f material properties, 495t measured geometries, 497t test results, 496 506, 501t test setup, 497f comparison with current design codes, 525 529 Behavior of HSS columns, 224 248 experiment program, 224 231, 225f test setup and test procedures, 228 231, 229f test specimen data and fabrication procedure, 225 228 overall buckling behavior of Q460 columns, 232 240 of Q690 columns, 240 248 Bias coefficient for resistance, 608 Bird’s Nest. See Chinese National Stadium BM. See Base metal (BM) Bolt spacing, 516 Bolted connections, 493 bearing-type bolted connections for HSS, 493 529 slip critical type bolted connections for HSS, 529 554 Bounding surface (BS), 93 94, 96 101 Box section columns assessment of test result comparison of test results with design codes, 333 334 overall buckling behavior, 329 333 of HSS design recommendation, 353 354 material properties, 324

632

Box section columns (Continued) out-of-straightness and loading eccentricity, 327 328 specimen design and fabrication, 325 327 test setup and loading procedure, 328 329 numerical investigation, 343 346 effect of eccentricity ratio, 344 345 effect of slenderness, 343 344 effect of width to thickness ratio, 345 346 for welded Q460 steel sections, 220, 221t for welded Q690 steel sections, 223, 223f, 223t BRBs. See Buckling-restrained braces (BRBs) Bridgman modified function, 41 42 Buckling behavior creep buckling behavior of steel columns, 469 476 of Q460 columns, 232 240 of Q690 columns, 240 248 Buckling temperature (Tbu), 434 Buckling-restrained braces (BRBs), 618 619 Burger’s model, 173 175, 175f, 176f Butt joints, 567 deformation capacity, 571 576 measured hardness distribution curves, 578 Butt welds, strength model of, 589 607 finite element model, 591 595 formula modification for butt joints with rigid constraint, 595 596 interpretation, 602 603 softened heat-affected zone cases, 602 theoretical model, 589 591 undermatched cases without softened heat-affected zone, 599 601 unified formula for butt joints withnonrigid constraint, 598 599 with rigid constraint, 597 598 verification of strength model, 603 607 C Calibration initial values and value ranges of cyclic model parameters, 131t

Index

method, 129 130 of model parameters, 130, 133t, 134t result verification, 130 132 Carbon dioxide (CO2), 569 Chaboche model, 63 64, 74 78, 76f, 77f, 79t, 80f, 96 Checking point method, 57 China application in, 7 10 applications outside of, 10 13 China Central Television Headquarters, 7 8, 8f Chinese code GB50017 2017, 527 Chinese constructional steel market, 3 Chinese National Stadium, 8 9, 9f Chinese specifications Code, 237 CIA. See Cooled in air (CIA) CIW. See Cooled in water (CIW) Coking coal, 3 4 Column temperature, 449 450 Combined time hardening model in ANSYS, 178 179, 181f, 182f, 183f Composite time hardening model, 178 Compression load, 449 450 Compressive residual stresses, 221 222 Constant amplitude, 67 loading history, 65 66 Constitutive model, 93 capability, 120f IHSES, 117 KHES, 118 119 cyclic behaviors of steels, 119 120 evolution laws of parameters, 132 143 framework, 105 106 modeling methodology based upon experimental observations, 107t three-stage damage evolution law, 108f Contact strengthening, 565, 567 Cooled in air (CIA), 186 Cooled in water (CIW), 186 Correction methods, 30 32 Cost-efficiency, 3, 4f CPS. See Cumulative plastic strain (CPS) CPS elastic modulus curve (CPS E modulus curve), 129 130 CPS stress curves (CPS S curves), 121, 122f, 129 130 Creep behavior of HSSs

Index

creep rate curves and demarcation point for three stages of HSSs, 170 172 creep strain of Q550, Q690, and Q890, 172 173 creep-time curves at stress levels, 165 170 at elevated temperatures creep phenomenon and curves, 161 162 creep test procedure, 163 165 setup and specimens in creep test, 163 numerical creep models, 173 183 Creep bucking time, 455 457 Creep buckling experiments of high-strength steel columns, 444 458 experimental results, 449 458 instrumentation, 448 specimen preparation, 445 447 test procedures, 448 449 test setup, 447 448 Creep buckling load factor (jcp), 463 465, 474 for current codes of practices, 479 488 Creep buckling prediction of high-strength steel columns, 458 488 creep buckling load factor for current codes of practices, 479 488 numerical prediction on ASTM A992 steel column, 465 469 ASTM A992 steel column model in ABAQUS and experiment, 465 466 creep model of ASTM A992 steels, 466 general analysis steps, 467 material properties of ASTM A992 steel, 466 467 validation against test results, 467 469 numerical prediction on Q690 highstrength steel column, 459 465 creep model of Q690 steels, 460 461 general analysis steps, 461 462 material properties of Q690 steel columns, 461 numerical model of Q690 high-strength steel column, 459 validation against tests results, 462 465 parametric study, 476 479 theoretical study on creep buckling behavior of steel columns, 469 476

633

secant modulus approach, 473 474 tangent modulus approach, 472 473 theoretical formulation, 469 471 validation of theoretical results against experimental results, 474 475 validation of theoretical results against numerical simulation, 475 476 Creep effect, 462 463, 473 Creep phenomenon and curves, 161 162, 162f Creep rate curves and demarcation point, 170 172, 171f Q550 steel, 172t Q690 steel, 172t Q890 steel, 172t Creep strain of Q550, Q690, and Q890, 172 173, 174f Creep test procedure, 163 165 Creep-time curves at stress levels, 165 170, 166f, 167f, 168f creep strain curves of, 170f Q550 steel, 168t Q690 steel, 168t Q890 steel, 170t Critical buckling strength, 480 483 Critical stress, 471 Critical temperature (Tcr), 434, 437 Cumulative damage mechanics model, 81 82 Cumulative plastic strain (CPS), 96 Cyclic backbone, 63 64 Cyclic behavior in constitutive model of steels, 119 120 experimental program cyclic test specimens, 64 65 loading protocols, 65 66 test materials, 64 of HSSs, 64 74 Cyclic coupon test, 129 130 Cyclic hardening behavior, 106 108 Cyclic hardening/softening behavior of YS and BS, 96 101 CS CP curves of cyclic coupon tests, 100f, 101f cyclic stress ratios against cumulative plastic strain, 102f decomposition of individual cyclic loop, 99f

634

Cyclic hardening/softening (Continued) peak stress, flow stress, and back stress, 98f predicted EMES using proposed evaluation method, 98f constitutive model, 121 128 calibration procedure for parameters of kinematic components, 123f relationships between critical strain and yield-to-tensile ratio, 128f Cyclic loading coupon tests, 95 96 Cyclic model parameters, 140 143, 145f simplified evaluation approach for, 143 145, 147f Cyclic parameter calibration calibration method, 129 130 calibration of model parameters, 130 verification of calibration result, 130 132 Cyclic peak stress, 67 Cyclic plasticity constitutive models, 93 94 two-surface constitutive model, 121 Cyclic skeleton curves, 70 72 Cyclic softening capacity, 67 69 phenomenon, 114 115 Cyclic stress (CS), 96 99 evolution law, 63 64 indexes, 117 ratio, 99 101 Cyclic stress strain data, 128 Cyclic test specimens, 64 65, 64f, 65f, 66f Cylinder specimen compression test, 38 39 FE model of cylinder specimen, 39f load-elongation curves for Q690, 39f load-elongation curves of, 38f D Damage model, 81 82 Damage-based cyclic constitutive model, 95 Data retrieval and statistics of material properties, 49 50 proportion of origin, 53f statistic parameters of yield strength, 53t Data source, 95 96 Decomposed cyclic softening process, 116 117 Deformation capacity of butt joints, 571 576

Index

manners, 500, 504f at ultimate resistance, 518 519 Demarcation point for three stages of HSSs, 170 172, 172t, 173t Design earthquake action, 618 adjustment of reliability index, 619 selection of structural systems, 618 619 Design formula, 334, 511 512, 568, 607 608 DIC. See Digital image correlation (DIC) Diffusional-flow mechanism, 162 Digital image correlation (DIC), 568 measurement and calibration, 569 570 Dislocation climb mechanism, 162 Dislocation glide mechanism, 162 Displacement meters, 429 Displacement transducers, 310 Dong Shen model, 63 64, 81 86, 82f, 84t calibration results of hardening parameters for, 85t of stiffness parameters for, 85t Ductility, 63 loss due to mismatched connections, 585 588 Dynamic-static resistance strain measurement system, 428 E Earthquake resistance, 93 EC3. See Eurocode 3 (EC3) Eccentricity ratio effect on eccentrically loaded columns behavior, 344 345 ECCS. See European Convention for Constructional Steelwork (ECCS) Edge distance, 500, 516 effect of, 504 Effective length method, 276 277 8-node solid element (C3D8R), 47 Elastic module parameters, 86 Elastic modulus, 119, 157, 157t, 158t, 159 degradation, 104, 105f parameters, 130 Elastic range, 82 83 Elastic stiffness degradation, 116 117 Elastic-buckling behavior, 233 234 Elastic-ideally plastic model, 339, 339f Elastic-linear hardening model, 339 Elasto-plastic behavior, 17 Elevated temperatures

Index

creep behavior of HSSs at, 161 183 creep buckling experiments of high-strength steel columns, 444 458 prediction of high-strength steel columns, 458 488 engineering stress strain curves of HSS at, 155 fracture modes of HSS at, 154 155, 155f HSSs at, 153 161 mechanical properties of HSSs at, 153 161 ultimate strain of HSS at, 157, 157t, 158t EN 1993 1-1, 318 320, 347 352 ratios of FE to design buckling resistances about major axis, 349t about minor axis, 349t End distance, 496 500, 516 effect of, 500, 512 Energy dissipation behavior, 63 64, 72 74 capacity, 72, 74f, 75f Engineering stress strain curve, 30 of HSS at elevated temperatures, 155 relationship of high-strength bolts after fire, 197, 198f Environmental friendly steel, 3 4 Equivalent constitutive model, 95 Equivalent viscous damping, 72 Euler buckling stress, 473 474 Eurocode 3 (EC3), 238 239, 266 267, 267f, 525 526, 568 applicability, 588 Eurocode 3-based creep buckling load factor, 486 European Convention for Constructional Steelwork (ECCS), 207 Evolution laws of constitutive model parameters, 132 143 of cyclic model parameters, 140 143 of monotonic parameters, 132 140 Exponential stress strain relationship, 102 103 Exponential-form function, 70 Extensometer, 36 F Fabrication procedure, 225 228

635

Failure mode, 422 423, 422f, 496 500, 498f, 515 with decreasing of edge distance, 505f of high-strength bolts after fire, 196, 196f, 197f Failure patterns and visual observation, 458 FE model. See Finite element model (FE model) Field & Field model, 173 178, 179f, 180f, 181f Fillet weld, 609 Finite element analysis (FE analysis), 32 34, 357 358 Finite element model (FE model), 32 34, 271, 285 289, 591 595 boundary conditions and mesh, 288 of cylinder, 38 initial geometric imperfections and residual stresses, 287 material model, 288, 591 592 numerical experiments, 592 595 verification, 288 289, 384 Fire, mechanical properties of HSSs after, 186 192 behavior of HSS after fire, 183 186 mechanical properties of HSS bolts after fire engineering stress strain relationship of high-strength bolts, 197 failure modes of high-strength bolts, 196 reduction factors of high-strength bolts, 198 200 specimens of high-strength bolts, 192 194 tests procedure, 194 196, 196f Fire resistance, 423 424 Flame-cut component plates, 225 228 welded H-sections, 221 222 Flat grooved plate tensile test, 47 49 dimension of flat grooved plate, 49f FE model of flat grooved plate, 51f load-elongation curves, 50f Flexural torsional buckling, 271 failure, 422 423 of Q460 HSS-welded I-section beams, 273 Foundation construction cost, 628

636

Fracture modes of HSS at elevated temperatures, 154 155, 155f Friction mechanism, 558 559 model, 532 Furnace temperature, 449 Furrow force, effect of, 559 G Gas metal arc welding (GMAW), 211 213, 306 307, 569 GB 50017 2003 code, 264 266, 265f, 266f, 268f, 322 324, 351 352 Geometric and boundary conditions, 382 383 factor, 609 imperfection, 228 sensitivity, 251, 252f Giuffre Menegotto Pinto model (GMP model), 63 64, 74 78, 80f, 81t Global warming, 4 GMAW. See Gas metal arc welding (GMAW) GMP model. See Giuffre Menegotto Pinto model (GMP model) Gravity loading method, 276 277 Gravity-load simulator, 276 277 H H section columns of HSS design recommendation, 346 352 ANSI/AISC 360 16, 350 351 EN 1993 1-1, 347 352 GB 50017 2003, 351 352 initial out-of-straightness, 310 311 material properties, 308 numerical investigation, 334 343 residual stresses, 335 339 tensile to yield strength ratios, 339 343, 340t specimen fabrication, 306 308 tensile effect to yield strength ratios on buckling resistances, 342f test procedures, 311 test program, 306 test results applicability of design rules, 318 324 failure modes and failure loads, 311 313

Index

initial loading eccentricity, 316 318 load deformation relationships, 313 load strain relationships, 313 316 test setup, 308 310 Hardened heat-affected zone (HHAZ), 578, 583 Hardness of steel plates, 559 560 Heat-affected zone (HAZ), 493, 565 HHAZ. See Hardened heat-affected zone (HHAZ) High-performance steel. See High-strength constructional steel High-strength bolts after fire engineering stress strain relationship, 197 failure modes, 196 reduction factors, 198 200 specimens of, 192 194, 195t High-strength constructional steel, 3 advantages and limits architectural and structural advantages, 5 cost-efficiency, 3 environmental friendly, 3 4 limits, 5 applications, 6 13 in China, 7 10 outside of China, 10 13 High-strength steel (HSS), 17, 93, 153, 207, 305, 357 358, 413, 493, 565 axial force ratio, 396 397 behavior of restrained high-strength steel columns, 413 424 columns under combined compression and bending box sections, 324 334 design recommendation, 346 354 H sections, 306 324 numerical investigation, 334 346 comparison with current design codes, 299 302 creep behavior at elevated temperatures, 161 183 creep buckling experiments of high-strength steel columns, 444 458 prediction of high-strength steel columns, 458 488

Index

cyclic loading amplitudes, 368t cyclic lateral loading, 369f protocol, 368f experimental phenomenon, 368 374 experimental setup, 360 362 hysteretic model, 397 410 hysteretic response, 374 381 loading amplitude, 373t loading protocols, 364 368 measurement arrangement, 362 364 mechanical properties, 18 at elevated temperatures, 153 161 after fire, 183 200 members under bending moment experimental setup and instrumentation, 276 277 failure mode and experimental procedures, 277 279 finite element modeling, 285 289 initial geometric imperfections, 274 275, 276t investigation of effective length, 283 285 I-section of flexural torsional buckling, 272f I-section of specimen, 274f load deflection curves, 280 load strain curves, 281 283 material properties, 272 specimens, 273 274 members under compression behavior of HSS columns, 224 248 material properties, 208, 209t parametric analysis and design recommendation, 249 257 residual stresses in welded HSS box sections and H-sections, 209 224 numerical simulation, 381 384 geometric and boundary conditions, 382 383 initial imperfection, 383 material model and mesh, 381 verification of finite element model, 384 parameter design, 384 387 parametric study and analysis, 289 297 initial geometric imperfections and residual stresses, 289 291

637

effect of residual stress, 292 294 effect of width-to-thickness ratio and height-to-thickness ratio, 295 297 plate component details, 364t postfire residual capacity of high-strength steel columns, 424 444 prediction of current codes, 297 299 Q460C, 10 Q690 steel beam column specimens, 388t solution economic evaluation of, 626 628 normal strength steel solution vs., 623 626 specimens, 358 360, 361t statistical analysis of Q690 HSS, 49 59 tensile specimens of Q550, Q690, and Q890 HSS, 17f 3D constitutive model, 18 49 web height thickness ratio, 394 396 width-to-thickness ratio of flange, 387 394 Hole-drilling method, 211 213 Hollow structural section (HSS), 271, 615 Hooke’s law, 215 219 Horizontal self-reaction loading system, 415 417 H-sections, 209 224, 212f, 224f, 224t measured dimensions of, 218t ratios of residual stresses, 222t specimens, 215 219 HSS. See High-strength steel (HSS); Hollow structural section (HSS) Hysteretic behavior cyclic skeleton curves, 70 72 energy dissipation behavior, 72 74 of HSSs under cyclic loading cyclic behavior of HSSs, 64 74 hysteretic model and verification, 74 90 stress strain hysteretic curve, 67 70 Hysteretic model, 74 86 Chaboche model, 74 78 with damage behavior, 397 405 Dong Shen model, 81 86 GMP model, 78 81 for HSS beam column, 409f simplified hysteretic model, 405 410 Hysteretic response, 374 381

638

I IHSES. See Isotropic hardening/softening evolution surface (IHSES) Incompressible assumption, 30 32 Increase stable-decrease process, 103 Independent differential equations, 93 94 Industrialization, 6 Inelastic behavior, 5 Inelastic cyclic behavior, 63 Influence index of residual stress, 292 294 Initial imperfection, 383 Initial loading eccentricity, 316 318 Isotropic and kinematic hardening model, 95 Isotropic constitutive models, 94 Isotropic hardening parameter, 105 106 Isotropic hardening/softening, 106 Isotropic hardening/softening evolution surface (IHSES), 117, 118f, 119f Isotropic softening components, 124 K KHES. See Kinematic hardening evolution surface (KHES) Kinematic constitutive model, 94 Kinematic hardening, 99 model, 114 116 rule, 102 104 evolution laws of kinematic parameters, 104f Kinematic hardening evolution surface (KHES), 102 103, 118 119, 120f L Landmark Tower in Yokohama, 10 12, 11f Lateral deflection, 419 421, 420f Lateral displacement, 450 454 Latitude Tower in Sydney, 10 12, 13f LBD. See Local-buckling-driven deterioration (LBD) LFD. See Low-cycle fatigue driven deterioration (LFD) Linear variable differential transformers. See also Linear varying displacement transducers (LVDTs) Linear varying displacement transducers (LVDTs), 228, 230f, 277, 281f, 328 329, 362 364, 417, 429, 448, 496 Load effect, statistical analysis of, 54 55

Index

targeted reliability index under ultimate limit states, 56t updated statistic parameters of loads, 55t Load ratio, 414, 435 effect, 438 Load-carrying capacity, 571 576 Load deflection curves, 280 Load deformation relationships, 313 Load displacement curves, 500 504, 520, 524 525 Loading process, 449 450 protocols, 65 66 Load shortening curves, 228 Load slip curve, 555 Load strain curves, 281 283, 500 504 relationships, 313 316 Local-buckling-driven deterioration (LBD), 397 398 Lode angle, 27, 40 43 Lode Code for Design of Building Structures (GB50009 2001), 55 Logarithmic stress rate based constitutive model, 94 Low-cycle fatigue driven deterioration (LFD), 397 398 Low-yield-point steel (LYP steel), 93 Lubricant, 27 29 LVDTs. See Linear varying displacement transducers (LVDTs) LYP steel. See Low-yield-point steel (LYP steel) M Manson Coffin equation, 109, 121, 143, 401 Material model and mesh, 381 Material strength effect, 479 Material-dependent parameter, 110 Maximum temperatures, effect of, 435 438 Measured residual stress, 213 219. See also Residual stress(es) experimental results of box section, 219f of H-section, 220f measured dimensions of box section, 218t of H-section, 218t

Index

simplified residual stress pattern, 222f welding parameters, 218t Mechanical properties behaviors of HSSs at elevated temperature elastic modulus, yield strengths, ultimate strength, 157 engineering stress strain curves, 155 fracture modes of HSS at elevated temperatures, 154 155 tests and specimens, 154 yield strength, 155 157 of HSSs after fire, 183 200 of HSSs at elevated temperatures, 18 comparative study, 159 161 methodology, 153 154 temperature-dependent elastic modulus and yield strength, 157 159 Metal plasticity, 17 Microcrack incubation, 106 108 Miner’s rule, 401 Mismatch ratio, 569 Modified buckling load factor, 483 486 Monotonic coupon test, 129 130 Monotonic parameters, evolution laws of, 132 140, 142f MTS 880, 64 65 MTS machine, 29 30 Multibolt in tandem, 528 Multiple random variables, 51 53 Multisurface model, 93 94 N Nonbuckling steel members, 3 Nondamage elastic modulus, 104 Nondimensional slenderness ratio, 482 Normal strength steel solution, 623 626 Normal-strength steel, 93 Norton model, 173 176, 177f, 178f Norton Bailey law, 175 Notched cylinder compression test, 45 47 dimension of notched cylinder specimen, 45f, 45t FE model of notched cylinder specimen, 45f notched cylinder load-elongation curves, 48f stresstriaxiality on center of minimum cross section, 47f Notched round bar tensile test, 34 37, 37f

639

dimensions of notched round bar, 34f, 34t evolution of stress triaxiality, 36f FE model of notched round bar, 34f NTV Tower in Tokyo, 10 12, 11f Numerical creep models, 173 183 Burger’s model, 175 combined time hardening model in ANSYS, 178 179 Field & field model, 176 178 Norton’s model, 175 176 three-stage creep model, 180 183 Numerical models, 432 validation, 433 Numerical simulation, 566 HSS, 381 384 geometric and boundary conditions, 382 383 initial imperfection, 383 material model and mesh, 381 verification of finite element model, 384 P Parameter design, 384 387 Parametric analysis and design recommendation design of welded Q460 steel columns, 256 257 parametric analysis of Q460 columns, 249 255 of Q690 columns, 258 263 dimensions of box columns, 258t dimensions of H-columns, 259t elastic-ideally plastic stress strain relation, 259f residual stresses effects, 260 response of steel grades to initial deflections, 262 263 sectional symbols of columns, 259f width-to-thickness effects of sections without residual stress, 260 261 Partial coefficients expected reliability index, 58t of resistance and design strength, 56 59, 58t Peak load, 571 572 Period and period ratio, 623 Perry’s formula, 473 Plastic

640

Plastic (Continued) modulus, 94 range, 82 83 strain, 3 amplitude, 143 Plasticity deformation capacity effect on slip factor, 552 index, 552 model of HSS, 43 44, 43t cylinder specimen load-elongation curves, 44f pure shear specimen load-elongation curves, 44f model validation for HSS, 45 49 flat grooved plate tensile test, 47 49 notched cylinder compression test, 45 47 Postfire behavior, 153 mechanical properties of HSS, 153 Postfire residual capacity of HSS columns, 424 444 instrumentation, 428 429 numerical simulation, 432 433 parametric studies, 433 442 simplified formulation, 442 444 test procedure and results, 429 431 test setup and specimens, 424 428 Postyield surface, 105 106 Power law model, 176 Preloading of bolt, 534 538 Professional factor, 609 610 Pudong Financial Plaza, 9f, 10, 10f Pure shear test, 39 40 dimension of pure shear specimen, 40f FE model of pure shear specimen, 41f load-elongation curves of, 40f specimen load-elongation curves for Q690, 41f Pure softening process, 99 101 Q Q460 columns HSS columns, 417 overall buckling behavior of, 232 240, 232t comparison and column curves, 238f, 239f

Index

load deflection curves, 232f, 235f, 241f load strain curves, 234f, 236f local buckling of specimens, 233f measured ultimate strength of specimens, 236t test results with design codes, 237 240 parametric analysis, 249 255, 251t geometric imperfection sensitivity, 251 effect of residual stresses, 251 255 Q550D, 64 Q690 columns, overall buckling behavior, 240 248 exposed to flame, 244f initial geometric imperfections, 243t load deflection curves, 242f load strain curves, 245f, 246f material properties, 243t support rotation, 242f test results and ANSI/AISC 360 10, 250f andEurocode, 249f and GB 50017 2003, 248f ultimate strength of columns, 241t Q690 steel, 167 169, 305 behavior and design of Q690 steel columns design of welded Q690 steel columns, 263 267 parametric analysis of Q690 columns, 258 263 Q890D, 64 Quenched and tempered processing (QT processing), 565 Quenching process, 6 R Ramberg Osgood model, 63 64, 70, 72t, 73f Random variables, 51 53 Rate-independent plasticity, 105 RCP. See Reserved capacity parameter (RCP) RCSC. See Research Council on Structural Connections (RCSC) Real-time load deflection, 228 Reduction factor, 159, 160f, 161f, 186 187, 188t, 254. See also Slip factor of high-strength bolts after fire, 198 200

Index

ofpostfire elastic modulus of HSSs, 191f ultimate elongation of HSSs, 194f ultimate strength of HSSs, 190f yield strength of HSSs, 190f ofpostfire mechanical properties of Q550 steel, 192t of Q690 steel, 193t of Q890 steel, 193t values and reduction factors, 189t for residual loading bearing capacity, 434 of strength and elasticity modulus, 199t, 200t, 201f Relative temperature factor, 435 Reliability index, 58 adjustment of, 619 Research Council on Structural Connections (RCSC), 530 Reserved capacity parameter (RCP), 108 109 Residual stress(es), 287, 287f. See also Measured residual stress effect, 251 255, 252f, 260, 260f, 261f effect on bearing capacity of beams, 292 294 model model for welded Q690 steel sections, 223 224 for welded Q460 steel sections, 219 222 in welded HSS box sections and Hsections, 209 224 assessment in welded Q460 steel sections, 211 214 assessment in welded Q690 steel sections, 214 219 residual stress model for welded Q460 steel sections, 219 222 residual stress model for welded Q690 steel sections, 223 224 sectioning method, 209 211 Resistance factor, 246 247, 608 Resistance partial coefficients, 56 57 Restrained HSS column behavior under fire, 413 424 column specimen and restraining beam, 415f comparison with restrained mild steel columns, 423 424

641

experimental loading system layout and specimen set-up, 416f information of specimens, 415t material properties of steels, 414t specimen preparation, 413 414 test procedure, 417 test results, 417 423 test set-up and measurements, 415 417 Rheological model. See Burger’s model Rotational restraint ratio, 414 Roughness of steel plates, 560 S SAW. See Submerged arc welding (SAW) Sawing machine, 210 Secant modulus approach, 473 474 Sectioning method, 209 211, 210f, 222 with a wire cutting machine, 211f initial measurement, 210f of measuring zone, 211f sectioned strips, 212f Seismic behavior, 63 Seismic structures, HSS in adjustment of reliability index, 619 case study, 620 622 design earthquake action determination, 618 economic evaluation of high-strength steel structures, 626 628 effect of material properties, 615 616 limits of current design codes, 616 normal strength steel solution vs. HSS solution, 623 626 selection of structural systems, 618 619 Seismic zone, 5 Self-equilibrium conditions, 215 219 Semi-ductile member, 619 SHAZ. See Softened HAZ (SHAZ) Shrinkage deformations, 225 228 Single-bolt connection, 527 528 Slenderness effect on eccentrically loaded columns behavior, 343 344 ratio effect, 440, 477 479 Slip critical test, specimen of, 555 Slip critical type bolted connections for high-strength steels, 529 554 design recommendation, 552 554 experimental programs, 533 538

642

Slip critical type bolted connections for high-strength steels (Continued) material properties, 534 preloading of bolt, 534 538 test setup, 538 test specimens, 533 534, 535f experimental results, 538 547 long-term effect, 540 547 effect of plasticity deformation capacity on slip factor, 552 Vickers hardness-steel grade, 548 549 Vickers hardness roughness, 549 551 Slip factor, 529 533, 543 544, 547t, 553t, 556. See also Reduction factor characteristic value, 546 code specifications, 531t deviation, 531 experimental study on slip factor of hybrid connections, 554 563 design recommendation, 562 experimental results, 555 558 specimen of slip critical test, 555 plasticity deformation capacity effect on, 552 for specimens with inorganic zinc-rich coating, 553t Slip loads, 542, 546, 552 554 Slip resistance, 529 530 Smooth round bar tensile test, 29 34 FE model of flat specimen and smooth round bar, 32f load-elongation curves between experiments and FE analyses, 33f load-elongation curves of flat specimen, 31f mechanical properties of HSS, 32t Softened HAZ (SHAZ), 566 567, 578, 583 584 cases, 602 strength loss for specimens with different SHAZ width, 584 undermatched cases without, 599 601 Softening, 565 Sony Centre in Berlin, 10 12, 12f Specimens, 27 29, 211 215 actual dimensions, 213t chemical composition and hardness value of steels, 28t experimental specimens, 28t

Index

of high-strength bolts, 192 194 photographs of representative specimens, 29f welding parameters, 213t Splitting failure, 507, 509 SSRC. See Structural Stability Research Council (SSRC) Stable load deflection relationship, 232 Star City in Sydney, 10 12, 12f Static friction, 532 Statistical analysis of load effect, 54 55 of Q690 HSS, 49 59 of structural resistance, 51 54 Steady-state creep, 175 Steady-state test, 154, 154f, 163f Steel consumption, 627 628 Steel grades, 108 109, 516 effect of, 441 442, 505 response to initial deflections, 262 263 Steel members under cyclic deformation, 93 Steel production and consumption, 6 annual crude steel production, 6f global use of steel, 7f percentage of steel grade, 8f US steel shipments by market classification, 7f Steel-making process, 3 4 Stepwise cyclic loading history, 67 Strain, 99, 153 154 amplitude, 108 109 distribution for hardness distribution pattern, 582 584 gauges, 277, 417, 428 Strain-based partition coefficient, 110 113 Strain-hardening behavior, 157 159 coefficient, 86, 88f modulus, 78 81 parameters, 86 Strain stress relation, 76 77 Strength reduction factor, 607 Stress concentration, 493 494 factor, 509 510 deterioration, 114 115 state, 18 25, 27 triaxiality, 40 43, 42f

Index

initial yield strength data point on deviatoric plane, 43f Stress strain characteristics, 208 curves, 67, 197 of steels, 18 hysteretic curve, 63 64, 67 70, 68f evolution law of cyclic peak stress, 70f mechanical characteristics from cyclic loading tests, 71t relationship of normalized cyclic peak stress and normalized cycles, 69f relationships, 82 83 Structural reliabilities using Q690 steel, 51 55 determination of structural reliability, 51 statistical analysis of load effect, 54 55 of structural resistance, 51 54 Structural resistance, statistical analysis of, 51 54 statistic parameters of resistance uncertainties, 54t of yield strength, 53t Structural Stability Research Council (SSRC), 350 351 Structural steels, prediction results on, 146 147 predicted EMES using proposed evaluation method, 150f predicted IHSES, 148f predicted KHES, 149f Structural systems, selection of, 618 619 Submerged arc welding (SAW), 306 307 T Tangent modulus approach, 472 473 Target reliability, 56 Tearout failure, 509 Temperature effect, 105 evolution, 417 419 level effect, 479 temperature-dependent elastic modulus and yield strength of HSSs, 157 159 Tempering process, 6 Tempering softening, 565 Tensile steel members, 3 Tension coupons, 208

643

Tension compression asymmetry, 86, 117, 124 Tertiary creep, 162 Test specimen data and fabrication procedure, 225 228, 225f, 226t, 227t Thermocouples, 417, 428 Thermomechanical controlled processing (TMCP), 565 Thickness-to-width ratio, 569 Three hardness distribution patterns, 578 581 Three-dimensional constitutive model of HSSs applicability of von Mises yield criterion for HSS, 27 40 cylinder specimen compression test, 38 39 notched round bar tensile test, 34 37 pure shear test, 39 40 smooth round bar tensile test, 29 34 specimens, 27 29 effect of stress triaxiality and lode angle, 40 43 fundamental definition, 18 27 proposed plasticity model of HSS, 43 44 validation of proposed plasticity model for HSS, 45 49 Three-stage creep model, 180 183, 184f creep parameters for Q890 steel, 186t curves for Q890 steel, 185f parameters for Q550 steel, 185t steel Q690, 185t Three-step softening hardening/softening model, 106 114, 111f CS CP curves of cyclic coupon tests, 100f, 101f cyclic stress evolution law, 114f ratios against cumulative plastic strain, 102f distribution of cyclic stress envelop with respect, 115f Three dimensional FE model (3D FE model), 285 Time effect, 105 Time-dependent deformation, 161 162 TMCP. See Thermomechanical controlled processing (TMCP)

644

Torsion compression experimental analysis, 93 94 Transformation softening, 565 Transient creep, 175 Transient-state test, 153 154 Transition range, 82 83 True stress vs. strain relationship, 591 592 True stress strain curve, 30 32 Two-bolt connection in parallel, 528 Two-step hardening/softening model, 106 114 Two-surface model, 93 94 U Ultimate resistance, 518 519 Ultimate strain of HSS at elevated temperature, 157, 157t, 158t Ultimate strength, 157, 157t, 158t Uniaxial cyclic coupon tests, 96 99 Uniform constitutive model, 95 Uniform material model for constructional steel capability of constitutive model, 117 120 cyclic hardening/softening constitutive model, 121 128 cyclic parameter calibration, 129 132 Dafalias Popov two-surface constitutive model, 94f evolution laws of constitutive model parameters, 132 143 evolution surfaces of back stress for various steels, 103f experimental observations cyclic hardening/softening behavior, 96 101 data source, 95 96 degradation of elastic modulus, 104 kinematic hardening rule, 102 104 mechanical properties of target structural steels, 97t prediction results on different structural steels, 146 147 simplified evaluation approach for cyclic model parameter, 143 145 theoretical modeling, 105 117 elastic stiffness degradation, 116 117 framework of constitutive modeling, 105 106 kinematic hardening model, 114 116

Index

tension compression asymmetry, 117 two-step hardening and three-step softening hardening/softening model, 106 114 Unloading reloading process, 119 strain stress path, 78 81 Urbanization, 6 V Verification, 86 90, 89f of calibration result, 130 132 Vickers hardness-steel grade, 548 549 Vickers hardness roughness, 549 551 vonMises criteria, 105 vonMises equivalent stress, 27 vonMises yield criterion applicability for HSS cylinder specimen compression test, 38 39 notched round bar tensile test, 34 37 pure shear test, 39 40 smooth round bar tensile test, 29 34 specimens, 27 29 chemical composition and hardness value of steels, 28t experimental specimens, 28t W Web height thickness ratio, 394 396 Welded box-section columns, 256 257 numerical results vs. Eurocode-3, 256f numerical results vs. GB 50017 2003, 257f Welded connections, 565 568 applicability of Eurocode 3, 588 design proposal, 607 610 design formula, 607 608 design strength, 608 610 ductility loss due to mismatched connections, 585 588 experimental investigation, 568 578 digital image correlation measurement and calibration, 569 570 gas metal arc welding, 569 linear correlation between strength and hardness, 576 578 material information, 568 569

Index

measured hardness distribution curves of butt joints, 578 measured load-carrying capacity and deformation capacity, 571 576 strain distribution for hardness distribution pattern, 582 584 strength increasing due to constraint, 584 585 loss for specimens with different SHAZ width, 584 model of butt welds, 589 607 three hardness distribution patterns, 578 581 Welded H-section columns, 257, 258f Welded HSS box sections and H-sections, 209 224 Welded I-section beams, 271 272 Welded metal (WM), 566 567, 586 588 Welded Q460 steel columns design welded box-section columns, 256 257 welded H-section columns, 257 Welded Q460 steel sections, 219 222 box sections, 220 H-sections, 221 222 measured residual stress, 213 214 specimens, 211 213 Welded Q690 steel columns design, 263 267 comparison with Eurocode 3, 266 267 comparison with GB 50017 2003 code, 264 266

645

Welded Q690 steel sections measured residual stresses, 215 219 residual stress model for, 223 224 box sections, 223 H-sections, 223 224 specimens, 214 215 Welding parameters, 214 215, 218t Width-to-thickness effects of sections without residual stress, 260 261, 262f Width-to-thickness ratio of flange, 387 394 WM. See Welded metal (WM) World Coal Association, 3 4 World steel production, 3 4 Y Yield strength, 155 157, 156f, 157t, 158t of HSSs, 157 159 Yield stress parameters, 86 Yield to tensile strength ratio (Y/T ratio), 615 Yield-to-tensile ratio (YTR), 99 101, 140 Yielding surface (YS), 96 101 Young’s modulus, 213 Z Zienkiewicz and Cormeau’s unified theory, 178 Zinc-rich coating, 531