Single Skin and Double Skin Concrete Filled Tubular Structures: Analysis and Design 0323855962, 9780323855969

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SINGLE SKIN AND DOUBLE SKIN CONCRETE FILLED TUBULAR STRUCTURES

Woodhead Publishing Series in Civil and Structural Engineering

SINGLE SKIN AND DOUBLE SKIN CONCRETE FILLED TUBULAR STRUCTURES ANALYSIS AND DESIGN MOHAMED ELCHALAKANI

University of Western Australia, Perth, Australia

POURIA AYOUGH

University of Malaya, Kuala Lumpur, Malaysia

BO YANG

Chongqing University, Chongqing, China

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 © 2022 Elsevier Inc. 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. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-323-85596-9 For information on all Woodhead Publishing publications visit our website at https://www.elsevier.com/books-and-journals

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Contents Acknowledgments

vii

1. Introduction 1.1 Introduction 1.2 Advantages of CFST and CFDST members 1.3 Erection of CFST and CFDST columns 1.4 Applications of composite members 1.5 International design guidelines 1.6 Material References

2 2 5 6 15 15 24

2. Experimental tests 2.1 Introduction 2.2 Cross-sectional shapes 2.3 Experiments 2.4 Effective parameters 2.5 Failure modes 2.6 Stiffeners in composite columns 2.7 Size effects 2.8 Hollow ratio 2.9 Effects of the confinement factor 2.10 Effects of the material properties 2.11 Load eccentricity 2.12 The role of the inner steel tube 2.13 Material imperfections 2.14 Effects of preload and long-term sustained load References

30 33 33 39 41 72 90 100 102 105 113 119 125 153 163

3. Analytical methods 3.1 3.2 3.3 3.4 3.5 3.6 3.7

Introduction Stressestrain response of materials Creep model for concrete Creep analysis of CFST columns A constitutive model for computing the lateral strain of confined concrete Axial and lateral stressestrain response of CFST column An analytical axial stressestrain model for circular CFST columns

v

168 168 279 284 298 305 308

vi

Contents

3.8 Path dependent stressestrain model for CFST columns 3.9 Elastic-plastic model for the stressestrain response of CFST columns 3.10 Strength enhancement induced during cold forming References

314 321 332 335

4. Numerical methods 4.1 Introduction 4.2 Numerical modeling of confined concrete 4.3 Investigating the behavior of composite members through numerical analysis References

343 343 384 533

5. Design rules and standards 5.1 Limitations of design regulations on the strength of materials and section slenderness 5.2 Compressive design strength of composite members based on design guidelines 5.3 Moment design strength of composite members based on design guidelines 5.4 CFST members under combined axial loading and bending 5.5 Strength of composite members based on research works 5.6 Local buckling of steel plates 5.7 Further discussion on local buckling of steel plates in rectangular CFST columns under axial compression 5.8 Compressive strength of CFST stub columns with stiffeners 5.9 Compressive strength of CFST stub columns with local corrosions 5.10 Strain compatibility between the steel tube and the concrete core References

542 546 577 609 635 678 706 732 776 785 786

6. Future research 6.1 Introduction 6.2 Material properties 6.3 Geometric properties 6.4 Nonuniform confinement 6.5 Fire performance of composite members 6.6 Stiffened composite members 6.7 Environmentally sustainable material References

Index

793 795 798 799 801 801 805 808

813

Acknowledgments The authors would like to acknowledge the people who helped to prepare and review the manuscript: Laila, Aya, Yaseen and Farouk Elchalakani for reviewing Chapters 1 to 6; Prof Mostafa Fahmi Hassanein for providing materials in Chapters 1 to 6; Dr Minhao Dong for proving experimental results in Chapter 2; A/Prof Sabrina Fawzia for reviewing Chapters 5 and 6; and Dr Shovona Khursu for reviewing Chapters 4 and 5. The authors are grateful for the advice on composite structures received from eminent researchers in civil engineering from different parts of the world, including Prof Sherif El-Tawil from Michigan University; Prof Alaa Morsy from Arab Academy for Science, Technology, and Maritime Transport; Prof Metwali Abu Hamad from Cairo University; Prof Sherif Safar and Prof Ezz-Eldin Sayed Ahmed from the American University in Cairo; Prof Xiao-Ling Zhao from UNSW/Monash University; Prof Gangadhara Prusty and Prof Serkan Saydam from UNSW; Prof Nie Shidong from Chongqing University; Dr Shagea Alqawzai, Prof Kang Chen, Prof Le Shen, and Dr Miao Ding from Chongqing University; A/Prof Nor Hafizah Ramli Sulong and Dr Sabrina Fawzia from QUT; A/Prof Zainah Binti Ibrahim from the University of Malaya; Prof Allan Manalo from USQ; Prof Hua Yang, Prof Lanhui Guo, and Prof Wei Zhou from Harbin Institute of Technology; A/Prof Muhamad Hadi from Wollongong University; Dr Mohamed Ali from the University of Adelaide; Prof Emad Gad, Prof Riadh Al-Mahaidi, and Prof Jay Sanjayan from Swinburne University; Prof Mostafa Hassanein from Tanta University; Prof Tong-Bo Shao from Sichuan University; Prof Dilum Fernando and Dr Chris Becket from the University of Edinburgh; Prof Hong Hao, Dr Thong Pham, and Dr Wensu Chen from Curtin University; Prof Jingsi Hu from Hunan University; Prof Brian Uy and Dr Michael Bambach from the University of Sydney; Dr Afaq Ahmed from Taxila University; Prof Sherif Yehia from American University in Sharjah; and finally Prof Ali Karrech and Dr Minhao Dong from the University of Western Australia; and Prof Tianyu Xie from South China University of Technology. Finally, we wish to thank our families for their support and understanding during the many years that we have been undertaking research on composite structures at the University of Western Australia, the University of Malaya, and Chongqing University during the preparation of this book.

vii

viii

Acknowledgments

Praise be to Allah, the Lord of the worlds, the Beneficent, the Merciful, Master of the Day of Requital, Thee do we serve and Thee do we beseech for help. Guide us on the right path, the path of those upon whom Thou hast bestowed favours. Not those upon whom wrath is brought down, nor those who go astray. The Holy Quran, Surah Al-Fatiha.

C H A P T E R

1 Introduction O U T L I N E 1.1 Introduction

2

1.2 Advantages of CFST and CFDST members

2

1.3 Erection of CFST and CFDST columns

5

1.4 Applications of composite members

6

1.5 International design guidelines

15

1.6 Material 1.6.1 Steel 1.6.1.1 AS5100 part 6 1.6.1.2 BS5400 part 5 1.6.1.3 DBJ13-51 1.6.1.4 Eurocode 4 1.6.1.5 AISC 360-16 1.6.2 Concrete 1.6.2.1 AS5100 1.6.2.2 BS5400 1.6.2.3 DBJ13-51 1.6.2.4 Eurocode 4 1.6.2.5 AISC 360-16 1.6.3 About this book

15 16 16 16 17 17 17 17 17 21 21 21 22 22

References

24

Single Skin and Double Skin Concrete Filled Tubular Structures https://doi.org/10.1016/B978-0-323-85596-9.00001-9

1

© 2022 Elsevier Inc. All rights reserved.

2

1. Introduction

1.1 Introduction The term ‘composite structures’ refer to structures in which different materials such as timber, steel, concrete, and masonry are used simultaneously for construction. The most common type of composite construction is the simultaneous use of steel and concrete to form steelconcrete composite structures. Steel-concrete composite structural members are known as economic members and have been used to construct various structure types. It is a very well-known fact that steel members are susceptible to buckling, while their tensile strength is remarkable. Conversely, plain concrete members can withstand a large magnitude of compressive force; however, their tensile strength is deficient. Therefore, the simultaneous use of steel and concrete allows the structural designers to take advantage of steel and concrete and neutralize each material’s drawback by using the other material. By taking this viewpoint, most structural members such as slabs, columns, beams, and trusses can be constructed using steel-concrete composite members. The main focus of the current book is to understand the structural behavior of concrete-filled steel tubular (CFST) members and concrete-filled double skin steel tubular (CFDST) members as one of the most recent forms of composite members. Other structural members are beyond the scope of this book.

1.2 Advantages of CFST and CFDST members Prior to the development of structural design codes and regulations, a deep understanding of the behavior of structural members and knowing the performance and their failure mechanism under the considered loading condition are required. Using the available codes to design structural members without understanding their material and structural behavior turns mathematical formulas into sets of vague concepts. Therefore, the results of the experimental tests performed on CFST and CFDST members are presented in this book to show the real structural behavior of members. Certainly, experimental tests provide valuable information regarding the behavior of structural members. However, for thoroughly assessing the influence of different parameters such as members’ geometric and material properties, loading conditions, initial geometric imperfections, and residual stress, it is required to fabricate a large number of specimens in the advanced structural engineering laboratories, which can be time-consuming and expensive. Recent improvements in the capacities of computers in performing complex mathematical calculations, as well as the development of finite element (FE) software like ABAQUS and ANSYS, allow researchers and structural

1.2 Advantages of CFST and CFDST members

3

engineers to profoundly investigate the structural performance of members and study their physical behavior. As a result, the behavior of CFST and CFDST members are studied through the results captured from the nonlinear FE analysis in this book. After a basic understanding of CFST and CFDST members’ behavior, the available international design regulations for designing CFST and CFDST columns, beams, and beamcolumns are introduced. Additionally, the most recent design models developed based on the experimental and numerical analysis of composite members are included in this book. Design examples of composite members are utilized widely throughout the book. CFST members consist of a hollow steel tube filled with the concrete core, with or without steel reinforcement bars. Compared with hollow steel sections or the reinforced concrete (RC), the structural performance of CFST members, such as their ductility, compressive, bending, torsional strengths, fire resistance, and energy absorption capacities, are remarkably better. Besides, steel tubes can also act as concrete formwork during the construction process to reduce the time and cost of construction. These advantages of CFST members over conventional steel tubes have recently attracted the attention of civil engineers and have led to their wide use in recent structures. The ideology behind the use of CFST members is that the concrete core can avoid or delay the local buckling in the steel tubes. Besides, the brittle behavior of the concrete material can be highly enhanced by the confinement effect provided by the steel tube. Fig. 1.1 shows the comparison between different steel tubes such as square hollow section (SHS), circular hollow section (CHS), square CFST, and circular CFST, based on their axial strength under compression. The compressive strength of the concrete used in these steel tubes was around 40 MPa. Besides, the schematic view of the axial load-displacement

FIGURE 1.1 columns.

Ultimate axial strengths of SHS, CHS, square CFST, and circular CFST

4

1. Introduction

FIGURE 1.2

Comparison between the axial compressive behavior of steel hollow sections, CFST, RC, and plain concrete. Developments and advanced applications of concrete-filled steel tubular (CFST) structures: members. J Construct Steel Res 2014;100:211e228. https://doi.org/ 10.1016/j.jcsr.2014.04.016.

responses of CFST, RC, plain concrete, and hollow steel section is presented in Fig. 1.2. It can be recognized from the figures that filling the steel tube with concrete could highly enhance the compressive capacity of the column. In addition, confining the concrete with the steel tube improved the residual strength and ductility of the column. Despite the advantages of CFST members, they also have some disadvantages, which are as follows: 1. The outer steel usually takes a large part of the external axial load as compared to the concrete core because of its higher stiffness under composite action. 2. The concrete core close to the neutral axis has an insignificant contribution to the flexural strength. 3. The contribution of the concrete core in enhancing the torsional strength is insignificant. 4. The initial elastic dilation of concrete under axial compression is relatively small, and thus the confining pressure provided by the steel tube to concrete is relatively low during the elastic stage. 5. The heavy self-weight of the concrete can limit the performance of CFST due to the strength-to-weight ratio. From the above discussion, it is quite clear that the central part of the concrete core of the CFST column can effectively be replaced by another smaller hollow steel tube with similar axial, flexural, and torsional

1.3 Erection of CFST and CFDST columns

5

strength. This form of column construction is known as the CFDST columns. CFDST members possess several advantages over that of the CFST column, which could be summarized as follows: 1. CFDST have higher flexural and torsional strength as compared to the CFST members. Additionally, their strength-to-weight ratio is improved significantly by replacing the central concrete with a steel tube of a much smaller cross-sectional area. Moreover, the inner tube expands laterally under compression loading, and, hence the confining pressure provided to the concrete also increases. Consequently, the initial confining pressure builds up more rapidly in CFDST as compared to CFST members so that their elastic strength and stiffness are enhanced. 2. CFDST members contain less concrete, which creates a more sustainable environment by reducing the embodied energy levels of the member. 3. The cavity inside the inner tube provides a dry atmosphere for facilities or utilities such as power cables, telecommunication lines, and drainage pipes. Therefore, CFDST members are chiefly useful for maritime structures.

1.3 Erection of CFST and CFDST columns During the construction, the inner tube of the CFDST column is erected first. This is then followed by the erection of the outer tube. After the erection of both tubes, the concrete is poured in between them. Fig. 1.3

FIGURE 1.3 Erection of CFDST columns. (a) during construction and (b) connection details. Developments and advanced applications of concrete-filled steel tubular (CFST) structures: members. J Construct Steel Res 2014;100:211e228. https://doi.org/10.1016/j.jcsr.2014.04.016.

6

1. Introduction

FIGURE 1.4 Different shapes of composite members.

from [7] provides the erection process of CFDST columns used in transmission towers. The plates and bolts are used to fix the position of the tubes, while the flanges on both ends of each tube are used to connect it with its extension. A similar procedure is used to construct CFST columns, except that the inner steel tube is not installed. Fabrication of hollow steel tubes can be done by welding steel sheets or using the hot-rolled or cold-formed process. Depending on the crosssection of the steel tubes, different shapes of composite members can be constructed, as shown in Fig. 1.4.

1.4 Applications of composite members Appropriate structural performance of CFST and CFDST members has convinced structural engineers to use them in different types of structures like industrial structures, structural frames, electricity transmitting poles, and spatial constructions. In China, for example, CFST members have been used for more than 50 years. In 1996, structural designers of subway stations in Beijing used CFST members as the primary columns. The use of CFST members for the construction of power plant buildings backs to the 1970s [7]. Historically, the concept of “double skin” composite construction was devised for use in submerged tube tunnels [12]. A graph presenting the cross-section of the double skin composite construction is shown in Fig. 1.5. This cross-section was used for the first time in the Kobe Minatojima Submerged Tunnel in Japan. In this section, some examples from the use of CFST and CFDST members in practice are presented. In the 1980s, and by accelerating the building high-rise structures, structural engineers designed and built many buildings using CFST members in Beijing and Fujian, China to reduce the columns’ size [9]. CFST members are typically served as the columns of composite frame systems for bearing compressive loads, and they are generally connected to steel or RC beams. To improve the lateral resistance of high-rise buildings, CFST composite frame structures are designed by employing RC core tubes or steel shear walls as the lateral load resisting system. The hybrid structural systems consisting of CFST columns with the core tubes

1.4 Applications of composite members

7

FIGURE 1.5 Example of submerged tube tunnel cross section. https://www. nipponsteel.com/result.html#/?ajaxUrl¼%2F%2Fmf2apr01.marsflag.com%2Fnipponsteel__ all__customelement%2Fx_search.x&ct¼&d¼&doctype¼all&htmlLang&imgsize¼1&page¼ 1&pagemax¼10&q¼Development%20of%20sandwich-structure%20submerged%20tunnel% 20tube%20production%20method&sort¼0.

or shear walls benefit from appropriate stiffness and ductility. The performed cyclic test on the building system depicted in Fig. 1.6 from [8] indicated that the frame’s first-order damping ratios are in the range of 0.03 and 0.035. One of the earliest uses of CFST members in high-rise buildings is at SEG Plaza in Shenzhen, China, shown in Fig. 1.7 from [7]. The building has 71 floors with a height of almost 292m, in which circular hollow sections with a diameter of 1600 mm and the steel wall thickness of 28 mm were fabricated by Q345 steel and filled with C60 concrete. In this project, the steel sections of the columns were transferred to the site. The shipped steel parts were in the lengths of three stories. After the installation of hollow steel sections, steel I-beams were connected to columns using bolts. Later, sets of column-beam were transferred to their exact location. At the same time, the deck floors were fabricated. The use of this strategy allowed engineers to construct two-and-a-half floors each week. The fast construction process of this project proves the efficiency of composite structures. Comparison of results showed that the use of CFST columns enabled structural designers of this project reduce the required steel material and prevent the use of very thick steel plates. If hollow steel sections were used, the required steel material would be twice as much as that used for the fabricated CFST columns. The SEG Plaza is the first super-high-rise building in China in which circular CFST members were used [13].

8

1. Introduction

FIGURE 1.6 Schematic view of a composite structural system for high-rise buildings. Developments and advanced applications of concrete-filled steel tubular (CFST) structures: members. J Construct Steel Res 2004;100:211e228. https://doi.org/10.1016/j.jcsr.2014.04.016.

Despite the better confinement effect provided by circular steel tubes and their aesthetic properties, the ease of fabrication of beam-to-column connections in rectangular composite columns has made them popular in composite frames. Ruifeng International Commercial Building in Hangzhou, China, is shown in Fig. 1.8 reproduced from [7] and [61], in which square CFST members were used as columns. The project consisted of two towers. The west one has 24 floors with a height of 84.3 m, and the east one has 15 floors with a height of 55.5 m. The combination of CFST composite frames and the lateral resistance system of RC shear walls has been used in this hybrid structural system in which the CFST columns have a depth of 600 mm and the steel plate thickness ranging from 28 to 16 mm. The Canton Tower in Guangzhou, China, is a multipurpose observation tower that consists of a space lattice composite frame twisting around an RC core, as shown in Fig. 1.9. The main body of the structure has a 454 m height, and the overall height of the tower is 600 m. Twenty-four inclined CFST members have been employed for the construction of this tower. The maximum diameter of the steel tube is 2000 mm, and the maximum wall thickness of the tube is 50 mm.

1.4 Applications of composite members

9

FIGURE 1.7 SEG plaza in Shenzhen, China. Developments and advanced applications of

concrete-filled steel tubular (CFST) structures: members. J Construct Steel Res 2004;100: 211e228. https://doi.org/10.1016/j.jcsr.2014.04.016.

The massive compressive forces in high-rise structures can lead to a large required column cross-section. The mega composite column can be used to deal with the heavy axial load, as shown in Fig. 1.10 from [7]. The cross section of the column is split into various champers using internal webs. The application of internal web, longitudinal stiffeners, and reinforcing tie bars improves the stability of steel plates. Puring concrete is performed using vent holes and manholes. If the member’s cross-section is large, it may be necessary to use shear connectors to guarantee the appropriate load transfer between steel and concrete. Z15 Tower in Beijing, China, with a height of 528 m, has been constructed using mega CFST columns. The advantages of composite members can be exploited in various kinds of bridges, i.e., arch bridges, suspension bridges, cable-stayed bridges, and truss bridges. They can be used for the construction of piers, towers, arches, as well as the deck system of bridges. Different bridge structures in which composite members are used are shown in Fig. 1.11 from [7].

10

1. Introduction

FIGURE 1.8 Ruifeng International Commercial Building in Hangzhou, China. Developments and advanced applications of concrete-filled steel tubular (CFST) structures: members. J Construct Steel Res 2004;100:211e228. https://doi.org/10.1016/j.jcsr.2014.04.016.

The Wangchang East River Bridge with a main span of 2 m is the first arch bridge in China, where CFST members were used, as shown in Fig. 1.12 reproduced from [7]. The central arch has a dumbbell shape cross-section with a total depth of 2 m. C30 concrete was used for filling the hollow steel sections of both top and bottom chords with a diameter of 800 mm and a wall thickness of 10 mm. A superior benefit of adopting CFST members in arch bridges is that the hollow steel tubes act as the permanent framework for the concrete, which can remarkably decrease the time and cost of construction. Additionally, the inherent stability of the arch tubular structure eliminates the need for a temporary bridge to install the composite arch. The erection of the composite arch bridge can be done using relatively simple construction technology due to the low

1.4 Applications of composite members

11

FIGURE 1.9 Canton tower. Guangzhou Tower. 2009. https://commons.wikimedia.org/wiki/ File:Guangzhou_Tower2009.jpg.

FIGURE 1.10 Mega CFST column cross section. Developments and advanced applications of concrete-filled steel tubular (CFST) structures: members. J Construct Steel Res 2014;100:211e228. https://doi.org/10.1016/j.jcsr.2014.04.016.

12

1. Introduction

FIGURE 1.11 Applications of composite members in bridges. Developments and advanced applications of concrete-filled steel tubular (CFST) structures: members. J Construct Steel Res 2014; 100:211e228. https://doi.org/10.1016/j.jcsr.2014.04.016.

FIGURE 1.12 Wangchang East River Bridge, China. Developments and advanced applications of concrete-filled steel tubular (CFST) structures: members. J Construct Steel Res 2014;100: 211e228. https://doi.org/10.1016/j.jcsr.2014.04.016.

weight of the hollow steel tubes. Cantilever launching techniques and horizontal or vertical “swing” techniques are the most typical techniques for constructing composite arch bridges. Another application of composite members is in subway stations. Columns in subway stations are generally subjected to significant

1.4 Applications of composite members

13

compressive loads, and designing them using hollow steel sections or RC members may lead to large cross-sections. Therefore, structural engineers of the Qianmen subway station in Beijing, Japan, employed circular CFST columns in their design. Similarly, lines 2 and 9 of the subway station in Tianjin, China, have been constructed using circular CFST columns, as shown in Fig. 1.13 reproduced from [7]. Composite members can also be employed in industrial buildings. The use of circular CFST columns in a power plant workshop is shown in Fig. 1.14 from [7]. Compared to hollow steel columns, the use of CFST columns in this project halved the consumption of steel materials [7]. Composite members can be utilized in the construction of electricity transmission pylons. The long-span transmission tower in Zhoushan, China, with a height of 370 m, is the biggest electricity towers globally, as shown in Fig. 1.15 from [7]. For the construction of this tower, a tubular lattice with four CFST columns with a diameter of 2000 mm has been used. Recently, CFDST members have been used in the construction of electrical network infrastructures, as shown in Fig. 1.16 from [7].

FIGURE 1.13 Application of composite members in subway stations. Developments and advanced applications of concrete-filled steel tubular (CFST) structures: members. J Construct Steel Res 2014;100:211e228. https://doi.org/10.1016/j.jcsr.2014.04.016.

14

1. Introduction

FIGURE 1.14 Application of composite members in a power plant workshop. Developments and advanced applications of concrete-filled steel tubular (CFST) structures: members. J Construct Steel Res 2014;100:211e228. https://doi.org/10.1016/j.jcsr.2014.04.016.

FIGURE 1.15 Zhoushan electricity transmission tower. Developments and advanced appli-

cations of concrete-filled steel tubular (CFST) structures: members. J Construct Steel Res 2014;100: 211e228. https://doi.org/10.1016/j.jcsr.2014.04.016.

1.6 Material

15

FIGURE 1.16 Application of CFDST columns in the electricity transmission tower. Developments and advanced applications of concrete-filled steel tubular (CFST) structures: members. J Construct Steel Res 2014;100:211e228. https://doi.org/10.1016/j.jcsr.2014.04.016.

1.5 International design guidelines Different well-known international design regulations like American steel design code AISC 360-16 [1], British bridge code BS 5400 [11], Japanese code AIJ [2], Chinese code DBJ13-51 [3], Australian code AS5100 [10], and European code EC4 [5] support the use of CFST structures. However, no design codes have been developed to cover the design of CFDST members. Therefore, structural engineers typically have to modify the models recommended for CFST members to design CFDST members. For the sake of simplicity, the codes are named “AISC 360-16”, “BS5400,” “DBJ13-51,” “AS5100,” and “EC4” in this book.

1.6 Material As discussed above, a CFST member consists of a hollow steel section with an infilled concrete core. The steel tube’s fabrication can be done

16

1. Introduction

TABLE 1.1 Types of concrete used for composite columns. Concrete

Notation

References

Polymer concrete

PC

[47,48]

Normal strength concrete

NSC

[20,24,30,32,41e44,52,62,63]

High strength concrete

HSC

[15,20,25,29e32,34,39e42,44e46,51e53,62]

Ultra-high-strength concrete

UHSC

[29,30,40e42,49]

Rubberized concrete

RuC

[17e19,22,23,33,59]

Self-compacting concrete

SCC

[14,21,26e29,35,36,50]

Seawater and sea sand concrete

SWSSC

[37]

Grout

GR

[38]

using welding steel plates, hot-rolling, or cold forming steel material. The steel tube can be fabricated using mild carbon steel, high strength steel, stainless steel, high-performance fire-resistant steel, and aluminum. The employed concrete material can be a normal strength concrete (NC), high strength concrete (HSC), ultra-high-strength concrete (UHSC), or selfconsolidating concrete (SCC). Other concrete types, such as polymer concrete (PC) or even rubberized concrete (RuC), can also be applied. Table 1.1 summarizes the most recent experimental and numerical studies conducted on composite columns using different types of concrete material. However, design regulations provide some limitations for using steel and concrete material, which are addressed in this section.

1.6.1 Steel 1.6.1.1 AS5100 part 6 AS5100 only covers structures built by steel material, having a yield strength of less than 450 MPa. Besides, the steel elements should have a thickness of at least 3 mm. Symmetrical steel sections having a yield strength of not more than 350 MPa must be employed. Besides, the slenderness of the steel plate must satisfy the yield slenderness limit. 1.6.1.2 BS5400 part 5 BS5400 allows the use of circular and rectangular hollow steel sections for the fabrication of composite members. The hollow steel sections must be symmetrical, and steel grades of S275 or S355 shall be used in accordance with standards EN10025 [4] or EN10210 [16]. Also, the slenderness of the steel plate must satisfy the yield slenderness limit.

1.6 Material

17

1.6.1.3 DBJ13-51 According to DBG13-51, the steel material used for CFST members shall be in accordance with the steel design code GB50017 [54]. Four steel grades of Q235, Q345, Q390, and Q420 can be employed based on the Chinese code. 1.6.1.4 Eurocode 4 EC4 refers to clauses 3.1 and 3.2 of Eurocode 3 Part 1.1 [6] for the steel material limitation. Based on EC4, the maximum steel yield stress must be less than 460 MPa. 1.6.1.5 AISC 360-16 AS360-16 allows the use of circular and rectangular hollow steel sections for the fabrication of composite members. The steel yield strength for the fabrication of hollow steel sections must be limited to 525 MPa. The minimum allowable yield stress fy and tensile strength fu defined by design codes are listed in Table 1.2 for steel plates [4,17,55,58], Table 1.3 for hot-rolled hollow steel sections [16], and Table 1.4 for cold-rolled hollow steel sections [56,59,61]. It can be recognized from tables that fy ranging from 200 to 460 MPa and . fu ranging from 300 to 720 MPa. The tensile strength-to-yield stress fu fy ratio is in the range of 1.11 and 1.96.

1.6.2 Concrete 1.6.2.1 AS5100 According to AS5100, concrete having normal strength and density, based on the requirements of AS5100 Part 5, must be used, and the size of the aggregate should be limited to 20 mm. In addition, if steel reinforcement bars are required, they must meet AS 5100 Part 5. The characteristic compressive cylinder strength at 28 days ranges from 25 to 65 MPa. Besides, the saturated surface-dry density of the concrete should be in the range of 2100e2800/m3. According to AS5100, the modulus of elasticity of the concrete is governed by: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   1:5 ¼ r  0:043 fcm ðMPaÞ EAS5100 (1.1) C where the term r denotes the density of the concrete and fcm is the average amount of concrete compressive strength at the relevant age.

18

TABLE 1.2 Minimum allowable amounts of fy, fu, and fu/fy ratio for steel plates. (a) AS/NZS 3678 fy (MPa)

Grade t (mm)

t8

8 < t  12

12 < t  20

20 < t  32

32 < t  50

50 7 are categorized as long columns. Suppose 3 < L=D  7, buckling mechanism. Based on the length-to-diameter ratio

o

o

columns are considered as medium length. In another classification, columns with 3 < L=D  5 are classified as short columns. This is a o tentative classification for CFST/CFDST columns. However, the slenderness classification mentioned above aligns with the International Institute of Welding proposal [22]. The Structural Stability Research Council (SSRC) guidelines [23] also present similar criteria regarding stub column definition. In the design of stub specimens for performing the axial

40

2. Experimental tests

compression test, the L=D ratio of the column must be controlled based o on the limits determined above to avoid the effects of overall buckling and end conditions. The effects of the column length can also be taken into account through the slenderness ratio ðlÞ defined as: l¼

Le pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i ¼ IDS =ADS

(2.3)

where Le presents the effective buckling length, i is the section radius of gyration, IDS denotes the moment of inertia, and ADS is the area of the cross section of the CFST/CFDST columns. • Nominal steel ratio an : The nominal steel ratio an is applied to determine the proportion of steel to concrete, as follows: Aso (2.4) Ace where Aso and Ace are the cross-sectional area of the steel tube and the nominal cross-section area of the concrete core, respectively. an ¼

• The confinement factor (coefficient) x: The confinement factor can evaluate the composite action between the outer tube and the concrete core. This parameter is defined to assess the influence of passive confining pressure induced by the steel tube on the concrete core’s plasticity as follows: x ¼ an

fsyo 0 fck

(2.5)

where an represents the nominal steel ratio, fsyo is the yield stress of the 0 is the characteristic compressive strength of concrete outer tube, and fck and can be taken as 67% of the concrete compressive strength fc0 . It can be noticed that the role of the inner tube on the confinement effect has been neglected by Eq. (2.5), which may lead to conservative results. Hence, the following equation can also be defined for the confinement factor of CFDST members in which the effect of the inner tube is taken into account: x¼

Aso fsyo þ Asi fsyi 0 fck

(2.6)

in which Asi and fsyi are the cross-sectional area of the inner tube and its yield stress, respectively. The confinement factor parameter considers the influence of the steel tubes’ slenderness ratios and their yield stresses. Therefore, it can be regarded as a proper parameter for estimating steel tubes’ ability to provide confinement effect.

2.5 Failure modes

41

2.5 Failure modes The prominent drawback of steel hollow tubes is their vulnerability to buckling when they are under axial compression. Additionally, concrete is inherently a brittle material. The simultaneous use of steel tubes and concrete can form a composite structural member in which constituent components may reduce or even neutralize each other’s drawbacks. The concrete core prevents the inward buckling of the outer steel tube and improves its stability. Simultaneously, the buckling of the inner steel tubes of CFDST members in the outward direction may be restricted by the presence of the concrete core. The plain concrete is prone to shear failure, whereas steel tubes can cause the confined concrete to fail in a ductile manner or at least postpone its brittle shear or crushing failure. This section will discuss the failure mechanism of hollow steel sections and CFST/CFDST members.

2.5.1 Hollow tube failure modes Residual stresses and initial geometric imperfections of the steel tubes affect their failure mechanism. However, the main factor distinguishing

the buckling mode is the depth-to-thickness ratio D=t of the steel tube. Two primary failure mechanisms are formed in the hollow steel sections. The first one, called the distorted diamond mode, occurs in thin-walled steel cylinders with D=t > 50. In contrast, axisymmetric elephant-foottype buckling is the governing mode of instability in thick-walled CHSs with D=t < 50, as shown in Fig. 2.7a. In the elephant-foot-type buckling mode, the stress concentration happens at one end of the column, leading to an increase in the diameter of the cylinder in this region. It should be noted that the recommended limits for the D/t ratio are approximate. Local buckling can occur anywhere along the column height due to the stochastic distribution of imperfections in steel tubes. However, local buckling of the short hollow steel tube is typically positioned at the end of the tube because of the “end effect”. The “end effect” in hollow steel tubes is due to the enormous magnitude of the axial load and frictional force at the interface of the loading platen and the column. Contrary to short CHSs, the main failure mechanism of long CHSs is bending-collapse [24,25], as displayed in Fig. 2.7b. In this failure shape, the length of the steel cylinder is split into some areas, with joint points of these subdivided areas are plastic hinges. The stress concentration area is developed at the plastic hinge that is the origin spot of the collapse by lateral deflection of the cylinder from this spot. The axial loadedisplacement responses of CHSs with D ranging from 114.3 to 165.1 mm and t ranging from 3.0 to 6.0 mm are shown in Fig. 2.8.

(a)

(b)

FIGURE 2.7 Failure modes of CHSs [24,25]. (a) Short CHS, (b) Long CHS.

FIGURE 2.8 Axial loadedisplacement curves for CHSs [25].

2.5 Failure modes

43

It can be observed from the figure that the peak load is increased by increasing the tube thickness. Similarly, the energy absorbed by the cylinder was enhanced by reducing the D=t ratio. By initiating the buckling at the peak load, the load-carrying capacity decreases remarkably with increasing the axial deformation up to the point that the top and bottom parts of the buckled regions touch each other. The comparison of specimens C1 and C8 indicates that a 65% reduction in the D=t ratio has notable effects on the ductility and energy absorption capacity of the cylinder. Depending on the cylinder’s slenderness, the tube buckling may happen before or after the inelastic region. In thick-walled CHSs with D=t < 50, the ductile elephant-foot-like failure mode usually occurs after reaching the peak point, whereas the brittle unsymmetrical diamond failure mechanism is expected to happen prior to reaching the peak point when D=t > 50. Similar to CHSs, the occurrence of elastic or plastic buckling in SHSs depends on the B/t ratio. The dominant failure mode in thin-walled SHSs with B=t > 100 is the “flip disk” mechanism, whereas the roof-type mechanism mainly occurs in thick-walled SHSs with B=t < 100. Fig. 2.9 shows the roof-type mechanism formed in an SHS with B=t < 25. The effect of the wall thickness of steel hollow sections on their failure mode is summarized in Fig. 2.10.

2.5.2 Composite member failure modes 2.5.2.1 Circular stub columns The failure mechanism of circular CFST/CFDST stub columns with the outer circular steel tube can be divided into two types. The first one is the

FIGURE 2.9 The roof-type mechanism in SHSs [25].

44

2. Experimental tests 250

200

D/t

150

100

50

0 0.1

Elephant 0.2foot like buckling

Distorted 0.3 diamond

Roof 0.4type

Flip 0.5 disk

0.6

FIGURE 2.10

Typical failure modes of hollow steel sections based on their width-tothickness ratio [26].

spiral-shaped local buckling of the tube and the diagonal slip shear failure of the sandwiched concrete. The second failure mechanism is the local outward buckling of the tube (elephant-foot-like buckling) along with the crushing of the sandwiched concrete adjacent to the steel tube buckled area. For brevity, the first and second modes are named “SLB þ SC” and “LOC þ CC”, respectively. The failure mode “SLB þ SC” occurs in circular CFST/CFDST stub columns with a thin-walled outer tube (large Do=t ratio). Based on the experimental works of Ekmekyapar et al. [27], o Uenaka et al. [28,29], and Han and Li [13], the criterion of Do=t > 45 or o

nominal steel ratio of an < 3% can be defined as an approximate limit for the occurrence of the “SLB þ SC” failure mode. This failure mode is accompanied by large diagonal shear deformation of the concrete core, which is a common failure mode in plain concrete. Therefore, this failure mode can be known as a brittle mechanism that is undesirable. The reason is that a thin-walled outer tube under axial compression is inherently susceptible to local buckling. Besides, the volume expansion of the concrete core in the lateral direction induces hoop tension on the outer tube. Hence, the tube cannot prevent the lateral deformation of the concrete core, leading to the brittle behavior of the whole member. Fig. 2.11 presents the failure mode “SLB þ SC” in a specimen with Do=t ¼ 105:2. o The failure mode “LOC þ CC” mainly occurs in circular CFST/CFDST columns with a thick-walled outer tube (small Do=t ratio) where the o concrete core is crushed adjacent to the bulged area of the outer steel tube. Due to the large thickness of the outer tube, “LOC þ CC” can be identified as a ductile failure mechanism. The reason is that this failure mode mainly occurs after reaching the ultimate strength since a thick-walled outer tube can highly confine the brittle concrete. Fig. 2.12 depicts the failure mode

45

2.5 Failure modes

Shear failure of the concrete core

FIGURE 2.11

Failure mode ‘SLB þ SC’ [29].

Local outward buckling of the steel tube

Crushing of the concrete

FIGURE 2.12 Failure mode ‘LOC þ CC’ [30].

“LOC þ CC” in a CFDST stub column with an austenitic stainless steel outer tube with Do=t ¼ 48:8 and an inner carbon steel tube with di=t ¼ o

i

22:9. It can be observed from the figure that the failure mechanism of the outer tube is similar to the elephant-foot-like buckling, and the concrete crushing takes place adjacent to the buckled area of the tube. The same failure mode is possible in circular CFST columns with a thick-walled steel tube. Another factor that can affect the failure mechanism of CFDST columns is the di=D ratio. As discussed above, shear failure of the concrete core o occurs in stub columns with a thin outer steel tube. However, in a CFDST

46

2. Experimental tests

column with a thin outer steel tube and a large di=D ratio (large inner o steel tube), it is expected that the thin outer tube buckles while no shear failure occurs in the concrete core. This type of failure mechanism can be found in the experimental work [29], where CFDST stub columns with Do=t ¼ 80 and di=D < 0:33 failed in the mode “SLB þ SC”. By contrast, o

o

elephant-foot-like buckling without any shear failure of the concrete was reported as the failure mechanism of the CFDST stub column with the same Do=t ratio, but di=D > 0:33. No shear failure of the sandwiched o o concrete in the column with thin outer steel and a large inner tube can be due to the significant rigidity of the inner steel tube, leading to sustaining the in-filled concrete. This phenomenon can be explained by investigating the multiaxial stress condition of the outer tubes. Based on the von Mises yield criterion, the yield stress of a steel plate with perfectly plastic behavior under plane stress is governed by: ss ¼ s2l  sl sq þ s2q  s2y

(2.7)

where sl and sq are the axial and hoop stresses, respectively and sy indicates the yield stress of the steel tube. During the elastic behavior, the stresses sl and sq increase linearly and ss < 0. Whereas in the case of plastic behavior, ss ¼ 0. Eq. (2.8) can be used to calculate the stress increment with respect to the strain increments.

dsl dsq

! ¼

2 8 1 > E > s > 4 > > > 1  w2s w > > s
> > > 1 < E > > s > 4 > > 2 :> :1  ws ws

ws

30 5@

dεl

1 A

if ss < 0

dεq

1 ws 1

3

2

51 6 4 S

S2l Sl Sq

390 1 Sl Sq > = dε 7 @ lA 5 > ; dεq S2q

if ss ¼ 0 (2.8)

Es Es ðsl þ ws sq Þ,Sq ¼ ðsq þ ws sl Þ, andS ¼ sl Sl þsq Sq , 1  w2s 1  w2s Es is Young’s modulus of steel tube, and ws is the steel Poisson’s ratio. sq and sl present the stress deviators of circumferential and longitudinal directions, respectively. The biaxial stress histories of the outer circular steel tubes of CFDSTstub columns with similar material properties but different di=D ratios o are given in Fig. 2.13. The vertical and horizontal axes in Fig. 2.13 are normalized by the yield stress of the outer tube, fy ¼ 200 MPa, and the tensile region is considered negative. From Fig. 2.13, two trends can be expressed. Firstly, thick-walled outer tubes are entirely under axial compression stress, and there is no trace of circumferential stress before they yield. By reaching the yield strength, the in whichSl ¼

47

σl / σy

2.5 Failure modes

1 di/Do = 0 di/Do = 0.17 di/Do = 0.33 0.5

di/Do = 0.5

σθ / σy

0 -1.25

-0.75

-0.25

0.25

σl / σy

(a)

1

di/Do = 0 di/Do = 0.17

0.5

di/Do = 0.33 di/Do = 0.5

σθ / σy

0 -1.25

-0.75

-0.25

0.25

(b) FIGURE 2.13 Biaxial stress condition of outer circular steel tube [29]. (a) to ¼ 1:0 mm (thin), (b) to ¼ 1:6 mm (thick).

circumferential stress moves into the tensile zone due to the volumetric dilatation of the sandwiched concrete. Specimens with thin-walled outer steel tubes and small di=D ratios swell in the initial stage of loading. o However, no circumferential tensile stress can be observed in columns with large values of di=D ratios, namely di=D ¼ 0:33 and 0:5. Besides, in o

o

48

2. Experimental tests

a column with a large di=D ratio, the circumferential stress is likely to o proceed into the compression region (see specimens with di=D ¼ 0:33). o The reason is that the load-bearing contribution of constituent components in the CFDST column is directly proportional to their cross-sectional area. As the cross-sectional area of the concrete core decreases (increasing the diameter of the inner tube di ), the load carried by the outer tube increases, making it prone to buckling at the initial loading stage. Therefore, the confinement effect is decreased by increasing the di=D o ratio. Besides, decreasing the concrete contribution ratio in load-bearing capacity due to reducing its area and increasing the rigidity of the inner tube by increasing di reduces the volumetric dilation of the concrete. Concerning the elliptical CFST/CFDST columns, their failure mode is similar to the circular cross-section, where the outer tube presents the bulging around the crushed area of the concrete. This is also true for round-ended CFST/CFDST columns. Comparing the buckling modes of inclined CFST/CFDST columns with their straight counterparts indicates that the impacts of the inclined angle and oblique planes on the failure mechanism of CFST/CFDST columns are negligible. 2.5.2.2 Rectangular/square stub columns Unlike circular CFST/CFDST stub columns with different failure mechanisms, stub columns with a rectangular cross section usually show only one failure mode. The typical failure mechanism of CFST/CFDST stub columns with the outer rectangular steel tube is the outward folding of the steel plates and crushing of the concrete, as shown in Fig. 2.14a. A similar failure mechanism can be expected in CFST/CFDST columns with an outer square steel tube, as indicated in Fig. 2.14b. It can be observed from Fig. 2.14 that the failure mechanism of CFST/ CFDST stub columns with rectangular and square outer tubes are the same. The only difference between the buckling mechanisms of rectangular and square sections is that the buckling of all sides of the square steel tube occurs almost simultaneously. By contrast, the shorter sides of the rectangular steel tube mainly buckle after the longer sides. This can be justified by comparing the buckling strengths of different sides of rectangular steel tubes. The elastic local buckling stress of steel plates subjected to uniform compression sol is given by: sol ¼

Kp2 Es  
 b 2 12 1  w2s t

(2.9)

where ws represents the Poisson’s ratio of steel, b is the depth of steel plate (see Fig. 2.15), t indicates the thickness of steel plate, Es is the steel elastic modulus, and K is the buckling coefficient and depends on the boundary

2.5 Failure modes

49

(a)

(b)

FIGURE 2.14

Typical failure mechanism in rectangular and square composite columns. (a) Rectangular CFST and CFDST stub columns [31], (b) Square CFDST stub column [32].

FIGURE 2.15 Boundary conditions of the flat steel plate in rectangular tube.

50

2. Experimental tests

conditions and loading configuration. The value of K for rectangular steel plates in contact with the concrete can be taken as 10.3, whereas it is taken as 4 for bare plates [30]. It can be observed from Eq. (2.9) that the buckling stress is reduced by increasing the depth of the steel plate b. The corners of the rectangular steel tubes can be considered as supports of the plates. By increasing the width of the steel plate b, the weakly constrained areas placed in the flat regions of the plate expand and increase the susceptibility of steel plates to local buckling. For instance, the buckling stress of a steel plate with b ¼ 50 mm is nine times greater than a steel plate with b ¼ 150mm when other properties are the same. Therefore, the longer sides of the rectangular steel tube buckle earlier than the shorter ones. 2.5.2.3 Corrugated stub columns Compared with flat steel tubes, corrugated steel tubes are lighter and more economical. Moreover, their buckling strength, load-carrying capacity, and energy absorption capacity are higher than flat plates. Also, they can preserve their rigidity in one direction while they are bent in the other direction. Fabrication costs of structural members with corrugated steel panels are usually less than those with the stiffened plates [4]. The longitudinal load may not fully be transferred by corrugated steel tubes due to their low axial rigidity, which can be identified as an advantage to confine the concrete core. A robust interlocking effect between the corrugated steel tube and the concrete core occurs because of the corrugation of the steel plate. This issue prevents the steel tube from being detached from the concrete core. These desirable advantages of corrugated steel plates have convinced structural engineers to apply them in their designs. Fig. 2.16 shows different shapes of corrugated steel plates made from flat plates, namely sinusoidal, trapezoidal, triangular, or rectangular. Among them, trapezoidal and sinusoidal are the most common profiles.

FIGURE 2.16 Variation of corrugated steel plates: (a) trapezoidal, (b) sinusoidal, (c) triangular, and (d) rectangular.

51

2.5 Failure modes

Buckled region Buckled region

(a)

(b)

FIGURE 2.17 Comparison of failure mechanism in CFDST stub columns with outer square and corrugated steel tubes [5]. (a) Outer square steel tube, (b) Outer corrugated steel tube.

In general, no noticeable difference can be found between the buckling shapes of rectangular/square CFST/CFDST stub columns made with flat and corrugated steel plates. However, the intensity of plastic deformation resulting from the local buckling in flat rectangular steel skins is more than corrugated steel tubes. This issue can be found in Fig. 2.17, where the failure behavior of a square CFDST stub specimen is compared with its counterpart fabricated with a corrugated outer steel tube. Concerning the circular composite column with a corrugated outer tube, it can be expected that the failure mechanism of normal and corrugated columns is almost similar. For instance, the “SLB þ SC” failure mode is the dominant failure mechanism of corrugated CFST/CFDST stub columns with thinwalled outer steel tubes or a small nominal steel ratio. In other words, buckling of the outer tube and failure of the concrete core in corrugated CFST/CFDST stub columns occur at a larger axial shortening than the counterparts made with a normal outer tube. This is due to the stiffening action of the corrugated profiles. The failure mechanism of a circular stub column with a corrugated steel tube having a nominal steel ratio an < 3% is shown in Fig. 2.18. It can be observed from the figure that the crimping edge of the steel tube is torn (Fig. 2.18a), and the concrete core experiences stagger displacement (Fig. 2.18b). The overall failure mode of the column is the shear failure mechanism, as depicted in Fig. 2.18c.

52

2. Experimental tests

Dislocation

Rupture

(a)

(b)

Shear failure

(c)

FIGURE 2.18 Typical failure mode of the corrugated circular CFST columns [33]. (a) Failure of the corrugated steel tube, (b) Deformation of the concrete core, (c) Shear failure mode of the concrete core.

2.5.2.4 Polygonal stub columns The failure mode of CFST/CFDST stub columns with polygon outer steel tubes depends on the number of sides. The greater the number of sides, the more the buckling mechanism is inclined toward the behavior of the circular column. For instance, the failure mechanism of the dodecagonal CFST/CFDST stub column is the local outward deformation

53

2.5 Failure modes

Outward local buckling

Outward local buckling Outward local buckling

Weld cracks Crushed concrete

(a)

(b)

(c)

FIGURE 2.19 The failure mechanism of polygon CFDST stub columns. (a) Hexagonal [34], (b) Octagonal [35], (c) Dodecagonal [36].

of the outer tube with the crushing of the concrete core, as shown in Fig. 2.19. By contrast, the typical failure mode in the hexagonal CFST/ CFDST stub column is the outward folding of the outer steel tube as well as the crushing of the concrete that is similar to the rectangular column, as presented in Fig. 2.19. Since the polygon steel tubes are mainly fabricated by welding steel plates, the weld fracture due to the poor welding quality is possible, especially around the buckled region of the steel tube due to the large out-of-plane deformation of the tube developed by the volumetric dilatation of the concrete core. 2.5.2.5 Inner steel tube of CFDST stub columns Up to this point, the typical failure modes of the outer steel tube and the in-filled concrete of CFST/CFDST stub columns with different cross sections were discussed. According to the content provided above, it can be concluded that regardless of the inner steel tube cross section, the failure behavior of the outer tube and the sandwiched concrete in CFDST stub columns is similar to their conventional CFST counterparts. This issue shows that the failure mechanism of CFDST stub columns depends on the outer tube’s geometric properties, and the dimension of the inner tube does not significantly influence the overall failure mode of columns. In general, the failure behavior of the inner steel tube in CFDST columns depends on the di=t ratio and the hollow ratio. The presence of the i sandwiched concrete changes the failure mechanism of inner steel tubes compared with hollow tubes. In the case of a small hollow ratio, the rigidity of the sandwiched concrete prevents the outward folding of the inner tube. Hence, the inner steel tube is only prone to experiencing inward buckling; this issue depends on the di=t ratio. In the case of a large i

54

2. Experimental tests

di=t ratio, the rigidity of the sandwiched concrete prevents the outward i

folding of the inner tube, and inward folds are formed in the inner tube. By contrast, the inner steel tube with a small di=t ratio can remain intact. It i

is noteworthy that depending on the value of the hollow ratio, the inner steel tube may also present outward folding. This issue can occur when the sandwiched concrete thickness is insufficient to provide strong support for the inner steel tube. As an approximate criterion, both outward and inward folding of the inner circular and square steel tubes may occur if the hollow ratio is higher than 0.6 and 0.8, respectively. 2.5.2.6 Stub columns under partial compression load Axial compression load is usually applied to the entire cross-sectional area of the column in structures. However, exceptional cases exist in which the column is under axially partial compressive loading conditions. For example, transferring the axial load from the deck to the pier of the girder bridge is done through a bearing plate that subjects the column to partial compression, as shown in Fig. 2.20. CFST/CFDST columns under axially partial compression are depicted in Fig. 2.21. In the experiment, the partial load is induced on the in-filled concrete using a circular or square steel ring as the bearing plate. As shown in Fig. 2.21, the central lines of the ring and the in-filled concrete are coincident. The most direct factor affecting the performance of partially loaded CFST/CFDST columns is the partial compression area ratio ðbÞ, which is defined as: b¼

Ac Ap

(2.10)

FIGURE 2.20 A schematic view of the partially loaded composite column.

55

2.5 Failure modes

Ring bearing plate

Inner tube

Concrete core Outer tube

(a)

(b)

FIGURE 2.21 CFDST columns under partial compression. (a) Circular cross section, (b) Square cross section.

where Ap is the cross-sectional area of partially compressive load, and Ac represents the cross-sectional area of in-filled concrete, as shown in Fig. 2.21. The presence and the thickness of the endplate between the top end of the column and the bottom surface of the ring bearing plate influence the behavior of partially loaded CFST/CFDST columns. It is expected that the outer steel tubes undergo substantial outward buckling, and no severe cracks happen in the sandwiched concrete when b is small. By contrast, the sandwiched concrete may split, and the range of plastic deformation in the outer steel tube is reduced by increasing b. The reason is that the microcracking development in the sandwiched concrete of the column with a smaller b is more extensive than the counterpart with a bigger b. As a result, the confinement effect provided by the outer steel tube enhances with decreasing b. It is noteworthy that

56

2. Experimental tests

Ring bearing plate Top end-plate

Crushing of concrete

Inner tube

Outer tube

(a)

(b)

FIGURE 2.22 Typical failure modes of partially loaded CFDST columns. (a) Circular cross section, (b) Square cross section.

reducing b increases the load-carrying proportion of the tubes due to the bond between the concrete core and steel tubes. In other words, the longitudinal stresses in the steel tubes are increased by reducing b. Therefore, a larger ring bearing plate improves the composite action within the concrete core and steel tubes. Fig. 2.22 shows the schematic view of the failure mechanism in circular and square CFDST stub columns under partial compression. It can be seen that the plastic deformation of the steel tubes in partially loaded columns is located close to the top end of the members. Besides, the plastic deformation of the circular specimen is more evident than the square specimen. However, the range of plastic deformation in square tubes is greater than that of circular ones. This is because the confinement effect provided by the circular steel tube is more significant than the square tube. It is expected that the outer steel tubes undergo substantial outward buckling when b is small. By contrast, the range of plastic deformation in the outer steel tube is reduced by increasing b. The reason is that the microcracking development in the concrete core of the column with smaller b is more extensive than the counterpart with bigger b. As a result, the confinement effect provided by the outer steel tube enhances with reducing b. It is noteworthy that reducing b increases the load-carrying proportion of the tubes due to the bond among the concrete core and steel tubes. Therefore, the composite action within the concrete core and

2.5 Failure modes

57

steel tubes is improved by using the larger ring bearing plate. Fig. 2.22 shows the schematic view of the failure mechanism in circular and square CFDST stub columns under partial compression. It can be seen that the plastic deformation of the steel tubes is located close to the top part of the CFDST columns. This is similar to the position of the steel tube of CFST columns under partial compression. Comparison of the buckling modes of the inclined CFDST columns with their straight counterparts indicates that the impacts of the inclined angle and oblique planes on the failure mechanism of CFDST columns are negligible. Imposing the partial compression on the top endplate forms a Ushaped indentation on the plate because of the crushing of the sandwiched concrete under the ring bearing plate. As a result, four corners of the endplate are deformed upward, whereas the plate’s middle part is deformed downwards. Applying a thick top endplate increases the plastic deformation of the steel tubes. By contrast, the local failure of a thin top endplate can even prevent the buckling of the steel tubes. Therefore, increasing the rigidity of the top endplate by increasing its thickness prevents denting of the endplate, and hence, it improves the composite action of the column by efficiently transferring the load to the column. If the endplate is not used and the ring bearing plate is placed directly on the top surface of the sandwiched concrete, the imposed axial load in the partially loaded column is transferred from the sandwiched concrete to the steel tubes due to the bond action between them. Therefore, steel tubes do not directly undertake the axial loads. Besides, the sandwiched concrete on both sides of the ring bearing plate presents upward deformation instead of lateral deformation since the rigidity of steel tubes in the lateral direction is greater than that of the sandwiched concrete. Consequently, it can be expected that disusing the endplate leads to smaller plastic deformations in the steel tubes than in the case where the endplate is employed. Similar to the fully loaded steel-concrete composite columns, the failure mechanisms of partially loaded CFST and CFDST columns are the same. The amount of b also affects the failure mechanism of the in-filled concrete. Fig. 2.23 illustrates the schematic view of the concrete’s failure modes in partially loaded CFST stub columns with square and circular cross sections and different amounts of b. It can be observed in Fig. 2.23a that in the square concrete core, the cracks expand from the corners of the ring bearing plate to the corners of the concrete core. By contrast, the spread of the cracks in the circular concrete core is uniform, as depicted in Fig. 2.23b. The cracks’ extensions pattern shows that the confinement effects in the square steel tube chiefly arise at the corner points, and the induced confining pressure on the middle part of the square concrete core is negligible. By contrast, the circular steel tube can effectively and uniformly confine the whole perimeter of the concrete core. Concerning

58

2. Experimental tests

(a)

(b)

FIGURE 2.23 Failure modes of the concrete core in partially loaded CFST columns. (a) Square cross section, (b) Circular cross section.

the value of b, the thickness of cracks is reduced, and they become dense by increasing the cross-sectional area of the ring bearing plates (reducing b). In comparison, reducing the cross-sectional area of the ring bearing plates (increasing b) increases the cracks’ thickness and makes them sparse, similar to the plain concrete. This issue again reinforces the fact that the confinement effect provided by the steel tube is enhanced by reducing b. 2.5.2.7 Tapered stub columns Tapered structural members refer to components in which the cross section is varied along the longitudinal axis. Utilizing tapered columns in some particular structures like station platforms, exhibition halls, bridges, and electricity transmission towers may be a more economical option than straight ones due to their lighter weight. Besides, architectural design and esthetic aspects may lead to the use of tapered columns. Consequently, their behavior and design methods have attracted increasing interest from researchers and structural engineers. The failure

59

2.5 Failure modes

Crush of concrete Crush of concrete

Inward folding

Outward folding

Outward folding

Inward folding

(a)

(b)

FIGURE 2.24 Typical failure modes of straight and tapered CFDST columns. (a) Straight, (b) Tapered.

mechanism of tapered CFST/CFDST stub columns is similar to their straight counterparts. The only difference is the location of the buckling. Like CFDST stub columns under partial compression, the buckling occurs adjacent to the top section of the column in the form of outward and inward buckling for the outer and inner tubes, respectively, with crushing the concrete at these areas. Fig. 2.24 illustrates the comparison of the failure modes in straight and tapered CFDST stub columns. The dominant failure mode in tapered CFST/CFDST column with circular, round-end rectangular, and elliptical cross sections is the elephant-foot-like buckling of the outer steel tube and crushing of the concrete adjacent to the buckled region of the steel tube. For the square tapered CFST/CFDST column, the outward local buckling of the outer tube, along with the crushing of the concrete, is likely to occur. The failure mechanism of the outer tube and the sandwiched concrete of the tapered CFDST columns is precisely similar to the conventional CFST tapered columns. Compared with tapered composite columns, the failure mechanism of tapered steel hollow sections without the in-filled concrete is outward and inward buckling at the middle of the columns. 2.5.2.8 Medium-length and slender columns The length of the column can profoundly affect its behavior. From the above discussion, it can be concluded that the dominant failure mode of CFST/CFDST stub columns is local buckling, yielding of the steel tube, and the crushing of the in-filled concrete and the intensity of the failure mode depends on the shape and the amount of the diameter-to-thickness ratio D=t of the steel tube. By contrast, slender composite columns are

60

2. Experimental tests

FIGURE 2.25

Failure mode of slender CFST/CFDST columns [37,38].

more prone to overall buckling, and they mainly fail by forming a halfsine wave. Fig. 2.25 shows the failure modes of circular and square CFST/CFDST slender columns. It can be seen from the figure that regardless of the cross-section, flexural buckling with remarkable lateral deflections occurred in columns. The occurrence of the global failure mode reduces the load-bearing capacities of columns compared to the case where only local buckling occurs. This is because a lateral displacement happens at the midheight of slender columns that causes a midheight secondary moment. The secondary bending moment is increased by increasing the length of the column. Consequently, the reduction influence of global buckling on the capacity must be considered in the design of slender columns. The failure mode “L” shows that the maximum material use is achieved by concrete crushing, fracture of the weld, and steel yielding. Therefore, the strength degradation because of the global buckling effect becomes less significant by reducing the slenderness ratio. The in-filled concrete of medium-length CFST/CFDST columns is typically under compressive stresses. However, the intensity of the compressive stress is reduced gradually by moving from the compression area to the opposite area. Conversely, the longitudinal stress status of in-filled concrete in long columns may change from compression to tension. As displayed in Fig. 2.25, the slender column is under bending. As a result, part of the column is under compression, and the rest is under tension. Due to the low resistance of the square steel tube against local buckling, the square CFST/CFDST slender columns are prone to a

61

2.5 Failure modes

combination of local buckling developed on the compression face and global buckling. Depending on the compactness of the square steel tube, the local buckling can occur prior to or after reaching the peak load. Generally speaking, depending on the length of the square and circular CFST/CFDST columns, three main failure modes can be expected, as follows: (1) Local buckling (L): typically happens in stub columns. (2) Global buckling (G): typically happens in slender columns (see the slender circular column shown in Fig. 2.25). (3) Interaction between local and global buckling (L þ G): typically happens in medium-length columns (see the medium-length circular column shown in Fig. 2.25). Additionally, weld fracture (F) is likely to occur due to poor welding quality and intensive out-of-plane deformation of the flat steel plates. Fig. 2.26 compares the occurrence of different buckling modes in square CFST columns according to their slenderness ratio L=r in which L is the length of the columns ranging from 1512 to 3512 mm, and r presents the radius of gyration. The radius of gyration r in composite symmetrical sections is given by: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E s Is þ E c Ic (2.11) r¼ E s A s þ Es A s where Es and Ec are the moduli of elasticity for steel and concrete, respectively, and As and Ac are the cross-sectional area of the steel tube and the in-filled concrete. Is is the second-moment area for the steel tube, and Ic indicates the second-moment area for the uncracked concrete core. It can be seen from Fig. 2.26 that the failure mode “L þ G” is the predominant failure mechanism of slender intermediate columns with 140 120

L/r

100 80 60 40 20 0 0

1L

2 L+F

G3

4 L+G

5 G+L+F

6

FIGURE 2.26 Failure modes of square CFST/CFDST columns according to their slenderness [37].

62

2. Experimental tests

P

Plastic stage

Elastic-plastic stage Elastic stage λ

FIGURE 2.27 Strength-slenderness ratio relationship of composite columns.

42 < L=r < 68, while flexural buckling (failure mode “G”) mainly occurs in columns with L=r > 68. Columns with low slenderness ratios L=r < 42 fail due to local buckling of the steel tubes. Fig. 2.27 illustrates the typical axial strength P-slenderness ratio l curve for columns under axial compression. Based on the slenderness ratio l of the column, the overall buckling mode of CFST/CFDST slender columns may occur elastically or in-elastically. For distinguishing the occurrence of elastic and plastic buckling, the longitudinal strain εl in the steel tube can be compared with its yield strain εy . Plastic buckling occurs if εl > εy ; otherwise, the column experiences overall elastic buckling. Fig. 2.28 3

λ=44

2.5

Hassanein and Kharoob [34]

εl / εy

2

Hassanein et al. [35]

1.5 1 0.5 0 0

50

100

150

200

λ

FIGURE 2.28 Effect of the slenderness ratio on buckling mechanism of long CFST/ CFDST columns. Ayough P, Sulong NHR, Ibrahim Z. Analysis and review of concrete-filled double skin steel tubes under compression. Thin-Walled Struct 2020;148:106495.

63

2.5 Failure modes

presents the ratio of

εl as the function of the column slenderness ratio l εy

εl ratio is more than unity in columns with the slenderness εy ε ratio l  44, whereas, in columns with l > 44, the l ratio becomes less εy than unity. Hence, elastic buckling occurs in long columns with l > 44, while the associated buckling in intermediate-length columns occurs plastically. Eq. (2.12) can be used for predicting the occurrence of elastic, elastic-plastic, or plastic buckling of CFDST columns. εl ¼ 27:32lð0:878Þ (2.12) εy [26,39,40]. The

2.5.2.9 Columns under eccentric load The ideal mode of column design is based on its placement under the concentric axial load. However, the eccentricity of the applied load on CFST/CFDST columns is likely to occur in composite structures. In this case, members are subjected to axial compression and bending moment and must be designed as beam-columns. When the column is subjected to concentric loading conditions, the in-filled concrete experiences compression stresses only. By contrast, applying eccentric load on a composite column develops both compression and tension zones in the in-filled concrete. The deformed shape of the column under eccentric compression is shown in Fig. 2.29. It can be observed from the figure that

Tension region

Compression region

Separation

Buckle

Horizontal cracks

Crushed

0.2-0.4 mm wide

FIGURE 2.29

The failure mechanism of composite columns under eccentric compression [41].

64

2. Experimental tests

due to the bending moment developed by the eccentric compression load, the column experiences unsymmetrical deformation with the half-sine wave shape. The failure mechanism of the column in the compression zone resembles the column under concentric compression, where the steel tube experiences local buckling at its midheight, and the cavity occurs between the steel tube and the in-filled concrete. Therefore, the buckled area of the steel tube is unable to confine the in-filled concrete, and the concrete is crushed in the compression side. Obviously, no local buckling happens in the tension side of the column. Compared with the compression region, narrow horizontal flexural cracks are developed in the concrete tension region. This proves that the circular steel tube can highly confine the concrete in this area. By contrast, wide flexural cracks can be expected in the tensile region of the concrete core confined by the rectangular steel tube, as illustrated in Fig. 2.30. As mentioned above, the behavior of composite columns under eccentric compression is similar to beam-columns. Hence, considering their bending stiffness is crucial in their design. Placing an internal tube inside the CFST beam-column and filling it with the concrete can significantly improve the section’s bending stiffness and moment capacity. Figs. 2.31 and 2.32 show the failure mechanism of circular and square eccentrically loaded concrete-filled double steel tubular columns, respectively. It can be observed from the figures that the behavior of eccentrically loaded CFDST columns in which the inner steel tubes are filled with concrete are similar to the eccentrically loaded CFST/CFDST columns. Local buckling occurs in the compression zone near the midheight of the column, and the concrete crushes at the buckled area of the steel tube. Figs. 2.31 and 2.32b show that the in-filled concrete can protect the inner steel tubes from local buckling; thus, only overall buckling is likely to happen in the inner tubes. Besides, the concrete core is directly confined by the inner steel tube and indirectly by the outer steel tube. Consequently, no crush occurs in the concrete core.

FIGURE 2.30 compression [42].

Severe tensile crack in the in-filled concrete in columns under eccentric

65

2.5 Failure modes

Local buckling

Concrete crushing

Local buckling (a)

(b) Separation

(c)

FIGURE 2.31 The failure mechanism of eccentrically loaded circular concrete-filled double steel tubular columns [43]. (a) Outer steel tube, (b) Inner steel tube and the concrete core, (c) Separation of the concrete from the inner steel tube.

Furthermore, the compression region of the sandwiched concrete in circular columns experiences minor damage. Hence, it can be concluded that the presence of the in-filled concrete improves the confinement effect on the sandwiched concrete by preventing the inner steel tube from local buckling. Comparing the concrete cores of the square and circular eccentrically loaded CFDST columns with in-filled concrete (Figs. 2.31 and 2.32b) indicates the damage intensity in the sandwiched concrete of circular cross-section is less than the square cross-section. The reason lies in the fact that the confinement effect provided by the outer circular tube is better than the square tube. It can be observed from Figs. 2.31 and 2.32c that the tension region of the concrete core may be separated from the inner steel tube. Therefore, the top part of the inner steel tube adjacent to the separated concrete area may buckle outward due to the pressure which is induced by the in-filled concrete. This is more likely in a rectangular steel tube than in a circular one.

66

2. Experimental tests

Local buckling

Local buckling

(a)

concrete crushing

(b) Separation

(c)

FIGURE 2.32 Failure mechanism of eccentrically loaded rectangular concrete-filled double steel tubular columns [44]. (a) Outer steel tube, (b) Inner steel tube and the concrete core, (c) Separation of the concrete from the inner steel tube.

2.5.2.10 Columns with outer stainless steel tube Replacing the conventional carbon steel tube with a stainless steel tube can lead to a more ductile behavior of CFST/CFDST members. In general, the failure mechanism of CFST or CFDST columns with stainless steel tubes is similar to their counterparts with carbon steel tubes. However, the amplitude of the outward buckling in outer stainless steel tubes is more significant than mild carbon steel tubes due to the higher ductility of stainless steel than carbon steel. The schematic failure mechanism of CFDST columns with an outer carbon steel tube, as well as a stainless steel tube, are given in Fig. 2.33. 2.5.2.11 Columns with rubberized in-filled concrete Replacement of natural aggregates of concrete by rubber particles from waste tires can lead to an environmentally friendly and sustainable material with better ductility, dynamic energy absorption capacity, and postfailure response than normal concrete (NC). The overall failure mechanism of outer and inner steel tubes of rubberized CFST/CFDST (RuCFST/CFDST) columns is similar to composite columns filled with NC, as shown in Fig. 2.34. However, the rubberized concrete (RuC)

67

2.5 Failure modes

(a)

(b)

FIGURE 2.33 Schematic failure modes of square CFDST columns. (a) Outer mild carbon steel tube, (b) Outer stainless steel tube.

Local buckling

(a)

(b)

(c)

FIGURE 2.34

Failure mechanism in RuCFDST columns [45]. (a) Circular cross section, (b) Square cross section, (c) Inner steel tube.

68

2. Experimental tests

FIGURE 2.35 Concrete and outer steel tube bonding [45].

behavior is different from NC. Compared with NC, Poisson’s ratio of RuC is closer to Poisson’s ratio of steel material. As a result, the transverse elongations of RuC and steel tubes nearly match. This issue causes the RuC core to fill the voids around the buckled area of the steel tube faster than that of NC (see Fig. 2.35). Filling of the buckled area resumes up to the point that the buckled section’s stiffness increases enough to resist further loading. This issue can significantly enhance the column’s ductility and lead to a smoother postpeak region than CFST/CFDST columns since NC requires more time for filling the steel tube buckled area. A drawback of RuC is its fragile nature of postfailure. As shown in Fig. 2.35, the bond between NC and the inner face of the outer tube is adequate. The failure mechanism of eccentrically loaded RuCFST/RuCFDST columns is similar to the counterparts with NC. Applying eccentric loading causes the RuCFST/RuCFDST column to experience overall buckling. In addition, local buckling in the compression region of the column may develop, especially in columns with an outer square steel

69

2.5 Failure modes

tube, and the tensile region of the steel tube is prone to fracture. The buckled region of circular and square RuCFST columns under eccentric compression is illustrated in Fig. 2.36. The intensity of the damaged area in the compression region of NC is negligible. By contrast, RuC undergoes severe crushing on the compression surface, and the intensity of the crushed area becomes severe by increasing the rubber replacement ratio. Concerning the tension region, the tensile cracks distribute uniformly throughout NC. In contrast, the tensile cracking in RuC is more distributed in the buckled region. Fracture planes in the tension region of RuC are expected to be broader and more numerous. Due to the low tensile and compression strengths of RuC with high content of rubber particles, the concrete core may crush into powders. Markedly, the steel tube of the RuCFST column with high rubber particles undergoes more severe local

Tension face

NSC

RuC15%

RuC30%

(a)

Compression face

NSC

RuC15%

RuC30%

Tension face

NSC

RuC15%

RuC30%

(b)

FIGURE 2.36 The failure mechanism of the RuCFST under eccentric compression [46,47]. (a) Circular cross section, (b) Square cross section.

70

2. Experimental tests

buckling. A good bond between the concrete core and the inner surface of the steel tube can be expected for RuCFST column, especially when a high rubber content is used in the concrete, as shown in Fig. 2.36. 2.5.2.12 Composite beams The typical momenterotation curves of hollow tubes and composite beams having the same material and geometric properties under the fourpoint bending test are depicted in Fig. 2.37. The stresses in steel fibers of both the hollow tube and the composite beam are increased until the elastic stage is finished. During the elastic stage, the response of the hollow tube beam coincides with the composite beams, as shown in Fig. 2.37. However, the elastic area of CFST/CFDST beams normally develops and extends beyond those of the hollow tubes. It is noteworthy that the initial stiffness of the beam is increased by increasing the tube wall thickness. By reaching the elastic-plastic behavior, the response of the composite beam exceeds the hollow tube. Bending deformation in the beam starts at the outset of the nonlinear behavior. By deceleration of the load, several buckles appear on the compression side, and they grow by increasing the load. By reaching the peak load, multiple buckles appear. However, usually, one buckle at or around the midspan will immediately expand into a dominant buckle. After the peak, the moment of the composite beam plateaus. This issue does not occur in hollow tubes. The in-filled concrete may deform and fill the outward buckled area, leading to efficiently delaying the failure. Increasing the cross-sectional area increases the plastic section modulus at midspan, enhancing the moment capacity of composite beams. Consequently, CFST/CFDST beams can

40

Moment (kNm)

35 30 CFST beam

25 20 15 10

CHS beam

5 0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Rotation (rad)

FIGURE 2.37

Typical momenterotation response of the hollow and composite beams [46].

71

2.5 Failure modes

sustain the same amount of the moment until failure, as shown in Fig. 2.37. The cross section of the hollow steel tubes is reduced and turns into an elliptical shape due to inward buckling. This issue reduces the moment capacity of hollow tubes, which can be observed in a remarkable reduction in the moment of the hollow tubes during the postpeak region, as displayed in Fig. 2.37. Buckling of both the composite beam and the hollow tube usually occurs at the same locations (at midspan). However, the presence of the concrete core makes the compression side of the steel tube buckle outwardly. Fracture of the steel tube is likely to occur in the tension side at large deformation. Fig. 2.38 illustrates the cut section of the composite beam at midspan. Also, the effects of applying RuC on the failure mechanism of the composite beam are shown in Fig. 2.38. It can be observed that the Compression side

Tension side

NC

RuC15

RuC30

(a)

Top flange

Top flange

Tension side

Tension side Bottom flange

Bottom flange

Tensile rupture

NC

Top flange Compression side

Compression side

RuC15

Tensile rupture covered by RuC

Bottom flange

RuC30

(b)

FIGURE 2.38 The failure mechanism of circular composite beams filled with NC and RuC [46,47]. (a) Circular cross section, (b) Square cross section.

72

2. Experimental tests

compression face of NC is crashed, and the tensile failure occurs on the tension face where the concrete is split up. The tension side of the steel tube may experience tensile fractures. Compared with NC, the crushing of the concrete on the compression region and the shear failure on the tension region are more evident in composite beams with RuC, and the intensity of failures becomes severe by increasing the rubber replacement ratio. Besides, RuC may bond to the buckled region of the steel tube, especially in RuC with high rubber content. The concrete core has the same curvature as the confining tube regardless of the composite beam’s cross-sectional shape. Compared with composite beams filled with NC, composite beams filled with RuC have smaller strength but higher ductility. The failure modes of CFDST beams are similar to that of CFST beams. The outer tube experiences the initial local buckling on the compression side, followed by a notable buckle at the midspan of the beam. Besides, the tensile fracture may be developed on the tension face of the outer tube. Concerning the inner steel tube, typically, inward buckling occurs on the compression side of the tube. The primary role of the in-filled concrete in composite members under bending is to prevent the inward buckling of the steel tube. Hence, the contribution of the in-filled concrete becomes more significant as the steel tube becomes more sensitive to buckling. In other words, the concrete core’s contribution increases with increasing the slenderness of the steel tube. Compared with the square section, the concrete core’s contribution to circular composite beams is more significant. As a general trend, increasing the rubberized replacement ratio reduces the load-carrying capacity of the composite members. However, this issue is less prominent in composite beams than composite columns. This is due to the low contribution of the concrete core in bending.

2.6 Stiffeners in composite columns Steel stiffeners can be applied to ensure an adequate bond between the steel tubes and the concrete core, as well as to improve the local stability and buckling strength of steel tubes in composite members. Using appropriate steel stiffeners can improve the structural performance of composite members by amplifying the concrete confinement effects. Hence, the required construction material can be reduced by decreasing the steel tube’s wall thickness and/or reducing the concrete strength. Despite the excellent structural performance of circular CFST/CFDST members, rectangular CFST/CFDST members are still a widely used cross-sectional shape in practice. This is because the fabrication of beamto-column connections in polygonal composite members is more

73

2.6 Stiffeners in composite columns

convenient, as mentioned previously. Additionally, the significant moment of inertia of rectangular composite members makes them a useful option for bearing high lateral loads. Despite the advantages of rectangular composite members, adequate measures should be applied to improve their bond action and prevent premature buckling of steel tubes, mainly when a thin-walled steel tube is used. Consequently, the use of stiffening methods is more common in rectangular composite members. However, they are also some methods for stiffening circular composite members. Fig. 2.39 illustrates different types of stiffeners. The most conventional type of stiffeners is longitudinal plate stiffeners, called ordinary longitudinal ribs, fabricated by welding longitudinal steel plates on the inner or outer surface of steel tubes. Depending on the column’s width, one to several stiffeners can be welded on each side of the column. Fig. 2.40 shows the buckling mechanism of square CFST stub columns with single or double longitudinal ribs. No visible buckling occurs on the steel tube at the locations of the longitudinal ribs. Besides, the intensity of local buckling in the steel tube can be reduced by increasing the number of stiffeners. Like unstiffened rectangular CFST/CFDST columns, local buckling first occurs at the longer side of the stiffened rectangular columns. Concerning the buckling mode of stiffeners, crushing the

Longitudinal rib

(a) Longitudinal ribs

Inclined plates

(b) Inclined plates

Curling ribs

(c) Curling ribs

FIGURE 2.39 Different types of internal stiffeners. (a) Longitudinal ribs (b) Inclined plates (c) Curling ribs.

74

2. Experimental tests

Undamaged stiffener

(a)

(b)

(c)

Twisted stiffener

(d)

FIGURE 2.40 Effect of longitudinal ribs on buckling mode of square CFST stub columns. (a) No stiffener, (b) Single stiffener, (c) Double stiffeners, (d) Concrete core and stiffeners. Tao, Z., Han, L.-H., & Wang, D.-Y. (2008). Strength and ductility of stiffened thin-walled hollow steel structural stub columns filled with concrete. Thin-Walled Structures, 46(10), 1113e1128. doi: https://doi.org/10.1016/j.tws.2008.01.007.

concrete core can squeeze the stiffeners and twist them, as shown in Fig. 2.40. To postpone or prevent twisting of the longitudinal ribs, binding bars can be welded on stiffeners. Reducing the spacing between binding bars increases the efficiency of this stiffening method. Theoretically, using ordinary longitudinal ribs and increasing their number and height can enhance the buckling strength of steel tubes, leading to improved confinement effect provided by the steel tube, the load-bearing capacity, and the ductility of the column. Therefore, using stiffeners can be considered as a method for reducing the required wall-thickness of steel tubes in steel-concrete composite columns. However, welding the ribs to the steel tube can induce residual stresses on the tube, which can counteract the beneficial influence of stiffeners. External stiffeners are less efficient than inner stiffeners since the concrete core can highly restrict the inner longitudinal ribs and increase their buckling strength. Outer stiffeners are a practical option for stiffening cold-formed steel tubes produced in the factory. Saw-shaped longitudinal ribs can be used to reduce the utilized steel materials and interlock the stiffeners with the concrete core. Besides, several circular holes can be cut from ordinary longitudinal ribs, leading to a new type of longitudinal ribs, known as Perfobond Leister ribs (PBR). Using saw-shaped stiffeners and PBR in composite columns can also improve the local stability and buckling strength of steel tubes. However, their efficiency in improving the performance of thinwalled composite members is less than ordinary longitudinal ribs. Applying stiffeners can significantly postpone the occurrence of buckling in rectangular CFST/CFDST columns. However, the efficiency of

2.6 Stiffeners in composite columns

75

longitudinal ribs in improving the mechanical performance of rectangular composite columns highly depends on the rigidity of the stiffeners. Increasing the stiffeners’ rigidity delays the buckling of stiffeners, postponing the buckling of the steel tube. The required stiffener rigidity for ordinary longitudinal rib in thin-walled steel hollow sections under axial 
compression Is; h depends on the width, thickness, and yield stress of the steel plate, and it is expressed as [48]: 8 > >



> u=t > 100 >

 50 for 0:5 u  u < 1:5 u=t > = = > t a t > a > < u=t a (2.13) Is; h ¼

> 257 u=t > > >  285 for u=t  1:5 u=t > > a > u > > : =t a where t is the thickness of the steel tube, u represents the width of the subpanel plate and can be taken approximately as u ¼ b=2, in which b

pffiffiffiffi presents the width of the steel tube. u=t in Eq. (2.13) is equal to 580 fy , a where fy is the yield stress of the steel tube in MPa. Eq. (2.13) can be simplified as follows [48]:  2 0:045 u t fy Is; h ¼

280

t4

(2.14)

All the parameters in Eq. (2.14) are as same as the ones introduced in Eq. (2.13). Eq. (2.14) can be used for calculating the required stiffness of ordinary longitudinal stiffeners in rectangular or square hollow steel tubes. In CFST or CFDST columns, the outer tube imposes confining stress on the concrete core, and the dilation of the concrete core develops counter stress and promotes local buckling of the outer tube. However, the local buckling of the outer tube due to lateral pressure of the concrete core can be counteracted by using stiffeners, especially in columns with a thin-walled outer tube (large D/t ratio). This is because the stiffeners are embedded in the concrete core. Consequently, the required rigidity of ordinary longitudinal rib stiffeners welded on the inner surface of the steel tube in CFST/CFDST columns is smaller than the hollow section. Therefore, Eq. (2.14) is modified for composite columns as [48]: Is; c ¼ 3:1 

104 ðu=tÞ3:5 fy 4 t 280

(2.15)

76

2. Experimental tests

Another type of stiffener is inclined plates, as shown in Fig. 2.39. Similar to the longitudinal ribs, inclined stiffener ribs can delay the local buckling of the steel tube. Besides, increasing the number of inclined stiffeners enhances the mechanical performance of composite columns. The buckled regions of steel tubes with inclined stiffener ribs with adequate rigidity are located between the stiffened and unstiffened areas of the steel plate. Fabrication of rectangular steel tubes with longitudinal stiffeners requires heavy welding. In addition to the high amount of the required welding material, this issue can induce residual stress in the steel plate. Moreover, crack development is possible in the welded corners in rectangular hollow steel sections fabricated by welding steel sheets. During the Mexico Valley earthquake, cracks in welded corners and buckling of steel box columns resulted in the entire collapse of two buildings at the Pino Suarez station. A possible solution to overcome problems related to welding corners of rectangular steel plates is CFST/CFDST columns with round corners. Like the welded rectangular steel tubes, stiffeners can be welded on the inner or outer surface of round-end steel tubes. Welded built-up square steel tubes formed by cold-formed L-shaped steel plates were introduced in Korea. For connecting the L-shaped plates, the center of the tube is welded. Therefore, the effect of residual stress developed by steel tube edge bending and the heat of welding is minimized. This fabrication method leads to a new kind of stiffener called curling ribs. The ribs can have a straight or oblique shape, as shown in Fig. 2.39. The overall failure mechanism of composite members with curling ribs and oblique curling ribs is similar to their counterparts stiffened with longitudinal ribs, displayed in Fig. 2.40. Adopting the curling ribs stiffening methods can efficiently enhance the ultimate axial strength of CFST/CFDST columns by improving the buckling strength of the tube and participating in ultimate load-bearing capacity. Compared to the conventional composite columns with longitudinal plate stiffness, this kind of stiffening method can diminish the required amount of welding material. Using longitudinal rib stiffeners can improve the ductility of the column, but not to a satisfactory extent. Accordingly, another stiffening scheme called binding bars (tie rods) has been developed, as shown in Fig. 2.41. Binding bars are regularly applied along the longitudinal axis of rectangular CFST/CFDST columns. Binding bars can efficiently restrict the lateral deformation of the steel tube and the in-filled concrete, leading to a higher ultimate axial strength of the column at a larger axial

77

2.6 Stiffeners in composite columns

(a) Binding bars

(b) Inclined tension bars

(e) Rhombus bars

(f) U-shaped tie bars

(c) Orthogonal tension bars

(g) Battlement tie bars

(d) Spiral tension bars

(h) Triangular indentedshaped tie bars

FIGURE 2.41 Different types of tension bar stirrup stiffeners in square CFST columns.

deformation than the unstiffened counterpart. Reducing the longitudinal distance between binding bars can lead to the faster and more adequate development of hoop strain in the steel tube. Hence, the confinement effect provided by the rectangular steel tube can be enhanced by reducing the spacing between binding bars, leading to the higher ultimate axial

78

2. Experimental tests

Clamp Tie bar

(a)

(b)

FIGURE 2.42 Schematic view of the buckling modes of steel plates with and without binding bars. (a) Without tie bars, (b) With tie bars.

strength. The influences of confinement effects provided by binding bars on the behavior of rectangular CFST stub columns can be discussed in two aspects. The first one is the effect of binding bars on the buckling of the steel tube. The second one is the effect of binding bars on restraining the lateral deformation of the in-filled concrete. Fig. 2.42 illustrates the local buckling of steel plates in rectangular CFST columns with and without binding bars in which a is the wavelength of the local buckling and b is the width of the steel plate. The buckling coefficient k [49] in Eq. (2.16) is governed by:   2 6 2 þ 64 þ 4 (2.16) K¼ 3 42 where 4 represents the wavelength parameter and is equal to 4 ¼ a=b. It can be observed in Fig. 2.42a that in the case of the steel plate without tie bars, the wavelength of local buckling a is almost equal to b (a ¼ b). Hence, the buckling coefficient of the steel plate with no tie bar is Kno tie bars ¼ 10:67. Local buckling in steel plate with tie bars takes place in the space between the tie bars, as shown in Fig. 2.42b. The rows of level tie bars can be considered clamped. Therefore, the wavelength is equal to the longitudinal space of binding bar arrangement bs (a ¼ bs ). Assuming the width of the square column is D, bs ¼ 0:5D, and b ¼ D leads to the buckling coefficient Ktie bars ¼ 19:67. Substituting Kno tie bars and Ktie bars sol; tie bars into Eq. (2.16) results in ¼ 1:84, indicating that applying tie sol; no tie bars bars can positively improve the buckling strength of square steel tubes, and therefore, postpone or even prevent local buckling. Fig. 2.43 shows the internal forces in the square CFST column with binding bars in which the as is the transverse space of binding bars, Fs is the stress of the binding bars, ss is the steel stress, and fl is the lateral confining stress of the concrete core. Using the equilibrium of forces and assuming that the separated part has a depth equal to the longitudinal spacing of binding bars bs , it can be concluded that:   D fl bs ðD  2tÞ  ss bs $2t   1 Fs ¼ 0 (2.17) as

2.6 Stiffeners in composite columns

79

FIGURE 2.43 Internal forces in a square CFST stub column with tie bars.

Fs ¼ Es εs As

(2.18)

where Es is Young’s modulus of the binding bar, εs is the strain of the binding bar, and As is the area of the binding bar. Replacing Eq. (2.18) into Eq. (2.17) leads to the following equations: fl ¼

ss þ

D  as A s $ Es ε s 2r as bs D=ð2tÞ  1

(2.19)

Typically, the binding bars reach their yield strength when the whole member achieves the ultimate axial strength. Therefore, Eq. (2.19) can be revised as follows: D  as A s $ fy 2r as bs fl ¼ þ D=ð2tÞ  1 D=ð2tÞ  1 ss

(2.20)

The second part of Eq. (2.20) represents the contribution of binding bars to lateral confining pressure. It can be observed from Eq. (2.20) that the lateral confining stress on the concrete core is increased by reducing as and bs . Another type of stiffening system is tension bars. In this method, different shapes of steel bars are welded to the inner face of the outer steel tube, as shown in Fig. 2.41. In the inclined tension bars scheme, a set of four angled steel bars is welded to the corners of the square tube. Depending on the arrangement of inclined tension bars, they can reduce or even prevent the load degradation during the postpeak region of the axial forceedisplacement response of the column due to their effect on enhancing the buckling strength of the steel tube. The use of inclined tension bars reduces the intensity of local buckling in the steel tube, and no local buckling occurs at the stiffened areas. Increasing the diameter or

80

2. Experimental tests

reducing the longitudinal spacing can highly improve the confinement effect provided by the square steel tube. Similar conclusions can be drawn for the effects of using other types of tension bars, including orthogonal tension bars, rhombus tension bars, and z-shape tension bars. The mechanical performance of binding bars, inclined tension bars, orthogonal tension bars, spiral tension bars, rhombus tension bars, and Z-shaped tension bars are based on the steel-steel bond action in which stiffeners can fully connect two sides of the steel tube. The failure mechanism of rectangular steel tubes in CFST stub columns with and without orthogonal tension bars is illustrated in Fig. 2.44. Comparing the intensity of the local buckling in specimens indicates that stirrups are able to restrain the lateral deformation of the steel tube. It is noteworthy that similar to applying the stirrups in the reinforced concrete members, in CFST/CFDST

Stiffened

Unstiffened

Unstiffened

(a)

Unstiffened

(b)

Stiffened

(c)

Stiffened

Unstiffened

Stiffened

(d)

FIGURE 2.44 Comparison of local buckling in rectangular CFST stub columns with and without tension bars [50]. (a) B=D ¼ 1, (b) B=D ¼ 1.5, (c) B=D ¼ 2, (d) B=D ¼ 3.

2.6 Stiffeners in composite columns

81

columns, stirrups can provide direct lateral restraint to the concrete core. Lateral expansion of the concrete core imposes tensile stress on the stirrups. Appropriate dimensions and spacing of stirrups can ensure their full utilization, restrict lateral deformation of the steel tube, and mitigate local buckling. The reason for the excellent performance of CFST columns with tension bars lies in the fact that steel bars can profoundly connect two planes of the steel tubes and provide the steel-steel bond mechanism. Therefore, contrary to the steel-concrete bond in CFST columns with internal plate stiffeners, the tension bars do not depend on the mechanical interlock with concrete. Other types of tension bars stiffeners, namely U-shaped tie bars, battlement-shaped tie bars, triangular indented-shaped tie bars, and inclined battlement-shaped tie bars, are based on the steel-concrete bond mechanism. The advantages of stiffeners with steel-concrete bond mechanisms can only be achieved if they are sufficiently fixed in the concrete. Consequently, the embedment depth plays a crucial role in their design. Despite the positive effects of using tension bars on the load-bearing capacity and ductility of rectangular composite columns, their installation can be challenging. For instance, tension bars with the steel-steel bong mechanism or inclined battlement-shaped tie bars cannot be placed prior to welding steel tube sheets. For comparing the influence of different types of stiffeners, the buckling index ðBIÞ can be defined as the ratio of the axial load corresponding to the first tube
buckling ðPb Þ to the load corresponding to the yielding of  the steel tube Py . The BI of a thin-walled steel tube usually is smaller than unity, indicating that the buckling of the tube occurs before reaching steel yielding strength. Adding stiffeners can increase BI sine they can postpone buckling of the tube (increase Pb ). Generally speaking, spiral ring bars have the highest BI and the most improvement on the compressive strength of composite columns than any other type of tension bars stiffeners. The BI of thin-walled composite columns with spiral ring bars can even be larger than unity, meaning that without increasing the thickness of the square tube, stresses in the extreme compression fiber of the tube can achieve the yield strength. This is because parts of the spiral stiffeners welded to the steel tube can restrain the out-of-plane deformation of the tube. Besides, they can function as stirrups and confine the concrete core. Adding stiffeners to the steel tube also alters the failure mechanism of the concrete core. A thin-walled square steel tube is susceptible to local buckling. By the occurrence of the local buckling in the steel tube, the concrete core loses the confinement effect around the steel buckled area and may experience local brittle crushing as well as diagonal cracks, mostly with a shallow depth that means the concrete is not able to exploit its full compressive strength. By contrast, the concrete core in stiffened composite members is prone to minor cracks and a crush mode with no

82

2. Experimental tests

diagonal cracks, which shows the concrete can fully achieve its compressive strength. The crushed area is usually located adjacent to the stiffeners. Fig. 2.45 compares the failure modes of the concrete cores in unstiffened and stiffened CFST stub columns. Stiffening the square steel tubes using internal stiffeners still has some drawbacks. Firstly, access to the inner area of the steel tube for welding the stiffeners may be challenging, especially when the size of the tube is small. Moreover, drilling the steel plates to provide the holes for the arrangement of the binding or tension bars can generate stress concentration in the tube. Also, welding the stiffeners to the steel tube’s inner surface can induce residual stress on the steel plate. Additionally, internal stiffeners can prevent the flow of fresh concrete throughout the member, leading to the development of voids around the stiffeners and the weak performance of the column. A possible solution is to use high-performance

Diagonal cracks

Concrete crush

(a)

Trivial cracks Concrete crush

(b)

FIGURE 2.45 Failure modes in unstiffened and stiffened square CFST columns [51]. (a) Unstiffened CFST stub column, (b) Stiffened CFST stub column.

83

2.6 Stiffeners in composite columns

concrete. However, this can impose high costs on the project. If the facade design of the column allows, external stiffeners can be adopted. Although they can positively affect the column’s overall performance by delaying or even preventing the local buckling of the steel tube, they require greater flexural rigidity than the inner stiffeners. This is because embedding the stiffeners inside the concrete core can delay or even prevent the stiffeners’ local buckling. The confinement effect provided by the circular steel tube is sufficient and uniform. Also, the local stability of the circular steel tube is greater than the square one. Therefore, stiffening methods may not be necessary for the design of circular CFST or CFDST columns. However, in case of necessity, stiffening methods such as binding bars, orthogonal tensile bars, and bidirectional tensile bars can be used in circular composite columns, as shown in Fig. 2.46. A beneficial effect of using stiffeners in circular columns is that they can improve the steel-concrete composite interaction, especially during the linear behavior where there is no trace of interaction between the steel tube and the concrete core. Due to the strong confinement effect provided by the circular tubes and their appropriate buckling strength than the square tube, the use of internal stiffeners may only slightly improve the ultimate axial strength or ductility of the entirely loaded column. In contrast, stiffening circular columns using internal stiffening schemes can remarkably improve the performance of partially loaded CFST/CFDST columns. As discussed above, in the partially loaded columns where the axial load is applied only to the in-filled concrete or the steel tube, the contribution of the unloaded component to the load-bearing capacity is due to the bond action between the tube and concrete. Internal stiffeners can efficiently enhance the bond between the tube and the in-filled concrete and guarantee satisfactory load transferring between the constituent components of columns. Consequently, the use of internal stiffeners in circular composite columns is an efficient method for improving the structural performance of partially loaded columns.

(a)

FIGURE 2.46

(b)

(c)

(d)

Different types of tension bar stiffeners in circular CFST columns. (a) Binding bars, (b) Orthogonal tension bars, (c) Orthogonal tension bars, (d) Bidirectional tensile bars.

84

2. Experimental tests

During the initial loading stage, the axial load increases linearly by increasing the axial strain. By entering the elastoplastic stage, internal microcracks will start to develop inside the in-filled concrete, leading to a remarkable increase in the Poisson’s ratio of the concrete. From then onwards, the lateral expansion of the in-filled concrete pushes the steel tube. Hence, the confining pressure provided by the steel tube is activated. This issue makes circular CFST columns fail in a ductile manner. The sufficient confinement mechanism in circular CFSTcolumns with binding bars can be detected by observing numerous oblique lines developed at the positions of tie caps due to the stress concentration at these areas, as displayed in Fig. 2.47. The development of oblique lines shows the yielding of the steel tube under large deformation. Fig. 2.48 shows the failure mode of circular CFST stub column with binding bars. The typical failure mode of circular CFST stub columns with tie bars is the fracture of the bars at large axial load level. Besides, longitudinal cracks may be formed and extend on the midheight of the steel tube, leading to fracture of the steel tube at large axial displacement. This failure mode of the steel tube is highlighted by increasing the strength of the concrete due to the brittle behavior of HSC. By failing the concrete core, the stresses on the concrete are decreased abruptly, leading to transferring a considerable amount of axial load to the steel tube, which increases the axial and lateral strains. The large induced axial strains significantly increase the splitting tensile stress at the crack areas, which eventually causes the tube to rupture.

Oblique lines

FIGURE 2.47 Developed oblique lines in circular CFST columns with binding bars [52].

85

2.6 Stiffeners in composite columns

Fracture of the tie bars due to large axial force

Steel tube rupture due to the stress Concentration around the tie caps Local buckling

FIGURE 2.48 Failure mode of circular CFST columns with binding bars [52].

Other internal stiffeners, namely orthogonal tensile bars and bidirectional tensile bars, do not change the failure mechanism of circular CFST/ CFDST columns. However, orthogonal stiffeners can provide a better confinement effect, leading to higher enhancement of the ultimate axial strength and the ductility of circular CFST/CFDST columns than bidirectional tensile bars. The composite action, compressive capacity, and the ductility of a circular composite column stiffened with tensile bars are increased by increasing the volume-stirrup ratio ðrsv Þ. The volume-stirrup ratio of circular CFST stub columns is governed by: ffi 8   sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2 2 > > d D D s >8 > pffiffiffi p  > > 2 2 6 > 8 2d2s > > for orthogonal stirrups ¼  2 > > 3sD > D > > ps > > > 2 > > > > >  2 > > d > > < 4 s Dp 4d2s 2 rsv ¼ for bidirectional stirrups ¼   2 > sD > D > > ps > > > 2 > > > > >  2 > > ds > > > Dp2 > > pd2 2 > > ¼ s for loop stirrups >  2 > > sD D > > : ps 2 (2.21)

86

2. Experimental tests

FIGURE 2.49 Different types of external stiffeners in circular CFST columns.

Like square composite columns, welding and installing internal stiffeners in circular composite columns with small diameters is challenging. Hence, different types of external stiffness, such as ring bar stiffeners, jacket strip stiffeners, and spiral bar stiffeners, can be employed for stiffening circular composite columns, as shown in Fig. 2.49. The use of external stiffeners in circular CFST/CFDST columns improves the ultimate capacity, stiffness, and ductility of the column. Improving the mechanical performance of circular columns is due to the additional confinement effect provided by the external stiffening measures. Therefore, reducing the spacing between the external stiffeners enhances the performance of the column. Moreover, the external stiffeners can reduce the different dilatations of the steel tube and the concrete core in the initial elastic phase. Fig. 2.50 shows the failure mechanism of circular CFST stub columns with ring bars. The failure mechanism of circular CFST/CFDST stub columns with ring bars is the local buckling of the tube between ring bars because the ring bars can efficiently restrict lateral deformation of the in-filled concrete and reduce the effective depth of tube between the spacing of rings. Depending on the

87

2.6 Stiffeners in composite columns

Extra confinement provided by steel rings Local buckling between steel rings

(a)

(b)

(c)

FIGURE 2.50 Failure modes of circular CFST stub columns with steel rings [53]. (a) Normal strength concrete, (b) High-strength concrete, (c) Ultra-high-strength concrete.

thickness of the tube, the local buckling can occur in the form of elephantfoot-like buckling or shear failure. In the case of filling the tube with HSC, the tube and steel rings may not be able to restrict the sudden increase in lateral expansion of the concrete due to its brittle behavior. Consequently, the failure shape of columns with HSC may not be as uniform as that of columns fabricated with NSC. The use of UHSC can even cause fractures of the tube and rings. Since ring bars and jacket strip stiffeners can restrict lateral dilation of steel tube and in-filled concrete; therefore, the intensity of the concrete failure in the stiffened column is less than the unstiffened counterpart. Fig. 2.51 compares the concrete shear failure in circular CFST columns with and without jacket strip stiffeners. The D=t ratio of both columns is 116.25, and all the material and geometric properties are the same. As shown in the figure, the angle of rupture in the unstiffened column

88

2. Experimental tests

Shear failure of the concrete core

Shear failure of the concrete core

Angle of rupture = 73°

Angle of rupture = 47°

(a)

(b)

FIGURE 2.51 The effect of external stiffeners on the shear failure of circular CFST stub column [54]. (a) Unstiffened, (b) Stiffened.

is 73 , whereas it is reduced to less than 50 in the stiffened column, showing the effectiveness of using external stiffeners. In general, possible failure modes of in-filled concrete are splitting failure, shear failure, and crushing failure, and the occurrence of each one depends on the compressive strength of the concrete and the intensity of confinement effect provided by the steel tube. The failure mode alters from the compression crush to shear failure and splitting failure, respectively, by increasing the concrete strength or reducing the confinement effect. Consequently, it is advisable to use external stiffeners in circular CFST/CFDSTcolumns with HSC and UHSC to enhance the ductility and prevent sudden failure mode. As discussed above, stiffeners can efficiently improve the confinement effect provided by the steel tube, especially in rectangular CFST/CFDST columns. The confinement distribution in the concrete core of circular and square CFST columns is shown in Fig. 2.52. Arching curves describe the boundary between effective and ineffective confinement regions. It can be observed from Fig. 2.52a that the confinement effect provided by the square steel tube is ununiform and concentrates around the corners and in the central area of the concrete. By contrast, the center of the steel plate cannot provide a significant confinement effect because of the plate’s insufficient rigidity to bear the stress induced by the concrete core’s expansion. Applying steel stiffeners in composite columns can be considered a practical method to improve the pattern of confinement stress distribution and supply extra confinement, thereby enhancing the column’s mechanical performance. The effect of the stiffeners on the concrete confinement stress distribution in square CFST columns is presented in Fig. 2.52bed. Fig. 2.52b shows the confinement effect provided by the square steel tube stiffened using longitudinal ribs, U-shaped tie bars, battlement-shaped tie bars, or triangular

89

2.6 Stiffeners in composite columns

Weld

Ineffective confinement

(a)

Effective confinement

(b)

(c)

(d)

Weld

(e)

FIGURE 2.52

(f)

Confinement distribution of unstiffened and stiffened CFST columns.

indented-shaped tie bars. It can be observed from the figure that the middle part of the steel plate with longitudinal stiffeners is appropriately restrained. In other words, adding a stiffener converts the weak confined area into two smaller sections, in which the stiffener weld is the boundary point. Consequently, the previous weak confined area is turned into a robust confined area adjacent to the welds. Fig. 2.52c relates to the confinement effect corresponding to the use of inclined tension bars. The confinement effect provided by the square steel tube with spiral tension bars is displayed in Fig. 2.52d, in which additional confined area benefits from the circular spiral tension bars, generating uniform circular confinement effect on the in-filled concrete. It can be seen that the in general, the use of stiffeners relatively improves confining stress. Contrary to the square cross section, unstiffened circular CFST/CFDST columns can provide a stronger confinement effect which is distributed uniformly over the infilled concrete, as shown in Fig. 2.52e. The schematic view of the failure mechanism along the longitudinal direction of composite columns with and without stiffeners is shown in Fig. 2.53. The concrete core crushes adjacent to the areas where the bulge buckling occurs in the steel tube. Using steel stiffeners restricts the outward steel bulges, enhancing the concrete confinement and the overall performance of composite columns. The analytical modeling of stiffened composite columns is discussed in Chapter 3.

90

2. Experimental tests

(a)

(b)

FIGURE 2.53 Local buckling mode of CFST columns with and without stiffeners. (a) Unstiffened column, (b) Stiffened column.

2.7 Size effects CFSTs and CFDSTs are widely utilized in high-rise buildings, large-span bridges, and underground structures because of their outstanding performance, such as load-bearing capacities, excellent plasticity, outstanding seismic performance, and ease of construction. Over the past 2 decades, CFST members have been widely used as the arch ribs of around 300 arch bridges with a maximum span of 530 m. Based on a survey [21] performed in China on more than 230 CFST arch bridges, steel tubes with an outer diameter greater than 600 mm have been used to construct more than 221 of them, among which more than 53% of ribs’ diameters are greater than 900 mm. As depicted in Fig. 2.54, the largest steel tubes used to construct the surveyed bridges have a diameter of 1800 mm. Similarly, CFST members with diameters of greater than 600 mm are usually used in high-rise buildings. For instance, the constructed CFST members used in the Jin Tower’s outer frame, Tianjin, China, have diameters in the range of 1200 and 1600 mm. The width of the CFST columns in the first level of the Union Square building complex in Seattle, WA, USA, is more than 3000 mm [55].

91

2.7 Size effects

2100

Diameter (mm)

1800 96.6%

1500

53.2%

1200 900 600 300 0 0

100

200

300 Span (m)

400

500

600

FIGURE 2.54 Variety of the columns’ size in arch bridges.

In the Guangzhou CTF Finance Centre, Guangzhou, China, CFST columns with multicavity with a section size of 3.5  5.6 m have been used [56]. The size effect is the fundamental characteristic of brittle or quasi-brittle materials, like rock, concrete, ceramics, and glass. By contrast, materials like steel which has a ductile behavior, do not display this property. In general, the compressive strength of the plain concrete is reduced by increasing the size of concrete. Eqs. (2.22a and b) gives the compressive
 strength of general cylinders ð fcy ðdÞÞ and cubes fcu ðdÞ according to the
 compressive strength of standard concrete cylinders fc0 . 0:49fc0 ffi þ 0:81fc0 fcy ðdÞ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffi d 1þ 2:6 1:17fc0 ffi þ 0:62fc0 fcu ðdÞ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffi d 1þ 2:6

(2.22.a)

(2.22.b)

where d represents the diameter or width of the specimen. The in-filled concrete is an essential element of composite columns. Therefore, the size effect of the concrete can influence the performance of CFST or CFDST columns. The importance of this issue becomes more apparent in the large columns. All international design regulations for CFST members, such as the European code EC4 [57], Japanese provision AIJ [58], or American guideline AISC 360-16 [59], have been developed based on the experimental test results achieved from small-dimension specimens. Therefore, their accuracy in predicting the performance of CFST members with a large cross-section is questionable. Contrary to unconfined concrete columns, the size effect in composite columns is more complicated since the confinement effects provided by steel tubes

92

2. Experimental tests

Axial load

D4

D3 D2 D1

Axial displacement

FIGURE 2.55

Typical axial loadedisplacement curves of CFST/CFDST columns.

can hinder crack propagation. Fig. 2.55 shows the typical axial loade displacement curves of CFST/CFDST columns with different widths D but the same material properties, length, cross section, and nominal steel ratio an , in which D4 > D3 > D2 > D1 . Increasing the width of the steel tube enlarges the concrete area. Therefore, the column’s ultimate axial strength is enhanced by increasing the width of the steel tube. Notably, the postpeak region of the curve becomes steeper as the size of the column increases. This issue can be explained by evaluating the size effect on the confinement provided by the steel tube. During the initial loading stage, when the axial stress is proportional to the strain, the steel tube and the infilled concrete bear the load independently, and there is no trace of interaction between them. This is because Poisson’s ratio of steel is higher than that of concrete at this stage. The proportional coefficient corresponding to the initial loading stage is termed the composite elastic modulus and is defined by Esc ¼ N=½ðA þ A Þε , in which N represents s c v the axial load and εv is the longitudinal strain of column. By increasing the axial compression, materials begin to present nonlinear behavior. Hence, Poisson’s ratio of concrete exceeds that of the steel tube until a specific load magnitude. At this point, the lateral deformation of the concrete induces hoop tensile stress on the steel tube. This issue imposes triaxial compressive confinement on the in-filled concrete. The biaxial stress state of the steel

 tube obeys the von Mises yield criterion. Therefore, the vertical stress fsl of the steel tube is reduced while the hoop tensile stress fsq is increased. Considering columns with the same material properties and D=t ratios (same thickness), when they reach the peak point of the curve, it can be stated that by increasing the column size, fsq is reduced, and fsl is increased. The intensity of the size effect on fsq and fsl is increased by increasing the D=t ratio. By knowing the ultimate axial strength of the column ðNu Þ, vertical stress of the steel tube fsl , and cross-sectional areas

93

2.7 Size effects

of the in-filled concrete
 ðAc Þ and the steel tube ðAs Þ, the stress of the confined concrete fcc at the ultimate axial strength can be calculated by: Nu  fsl As (2.23) Ac The stress of the confined concrete fcc corresponding to the peak point is diminished by increasing the diameter of the column.
 It should be noted that the reduction effect of the confining stress fr becomes significant with the increase in the D=t ratio. The stress of the confined concrete fcc comprises two parts: the compressive strength of the concrete and the confinement strength provided by the steel tube. To assess the impact of size and the D=t ratio on these two parts, they should be separated from the stress of the confined concrete. A precise model [60] has been developed for calculating the prism fcc ¼

compressive strength of the concrete ð fcp Þ by considering the size effect based on the modified size effect law (MSEL) [60] as follows: 2 fcp ¼ fc0

3

6 7 6 7 0:4 6 7  60:8 þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7  4 HD 5 1þ 50

(2.24)

where H and D are the height and depth of the column, respectively. Eq. (2.24) is based on standard prismatic concrete samples with a dimension of 150  150  300 mm. Therefore, the shape conversion coefficient of concrete compressive strength equal to 1.073 must be multiplied by Eq. (2.24) to convert it to the strength of a cylinder

compressive with a dimension of 150  300 mm fcp .

Therefore, the confinement stress provided by the steel tube at the ultimate axial strength can be obtained as: fr ¼ fcc  fcp

(2.25)

The size effect is more influential on the confinement stress fr than the concrete strength fcp . Besides, the intensity of the size effect on the confinement stress is increased by increasing the D=t ratio. This is because the volume expansion of in-filled concrete reduces with increasing the column size. The confinement effect on the in-filled concrete depends on the steel tube’s deformation with that of the core concrete. Therefore, reducing the hoop expansion of in-filled concrete decreases the confinement effect provided by the steel tube. As a result, the concrete becomes brittle with increasing D, and hence, the descending branch of the axial loadedisplacement curve becomes steeper (Fig. 2.55).

94

2. Experimental tests

For considering the size effect on reducing the concrete compressive strength in composite columns, a strength reduction factor ðgc < 1:0Þ is usually multiplied by the cylindrical compressive strength of concrete. gc for unconfined concrete can be defined as the ratio of the concrete core’s compressive strength with an arbitrary depth of d to the compressive strength of the benchmark concrete with a depth of d ¼ 150 mm. The strength reduction factor, recommended by Sakino et al. [61], is governed by   d 0:112 (2.26) gc ¼ 150 Compared with quasi-brittle failure modes in unconfined concrete, the confinement effect provided by the steel tube can convert the brittleness of the concrete into plasticity, depending on the D=t ratio of the steel tube. The brittleness of concrete principally causes the size effect. Increasing the nominal steel ratio enhances the confinement effect provided by the steel tube. Therefore, the ductility of in-filled concrete improves as the nominal steel ratio increases. However, the shear failure is still likely to occur in the concrete confined by a thin steel tube. Hence, it can be stated that the nominal steel ratio has a critical influence on the intensity of in-filled concrete damage since the confinement effect enhances with increasing the nominal steel ratio, leading the damage status of in-filled concrete to be transferred from macrocracks to microcracks, as shown in Fig. 2.56.

(a)

(b)

(c)

FIGURE 2.56 Effect of the steel ratio on the concrete damage status [55]. (a) Unconfined concrete, (b) Confined concrete an ¼ 4:1%, (c) Confined concrete an ¼ 10:3%.

2.7 Size effects

95

Therefore, the effect of the nominal steel ratio on the size effect of CFST/ CFDST columns should be taken into account. To gain a better insight into the effect of the nominal steel ratio on the failure mechanism of confined concrete, the failure modes of circular and square CFST stub columns, captured by nonlinear finite element analysis, are shown in Fig. 2.57. All columns have the same concrete diameter (or depth) of 200 mm, steel yield strength of 360 MPa, and concrete compressive strength of 52.7 MPa. Varying wall thickness of steel tubes from 0 mm (no steel tube) to 12 mm leads to the confinement coefficients of 0 (unconfined concrete), 0.38, 0.77, 1.12, and 1.53. As displayed in Fig. 2.57, all columns experience shear failure, indicating that the steel tube cannot change the brittle behavior of the concrete entirely into plasticity. The angle of rupture for the plain concrete is greater than 45 , showing the typical brittle shear failure. By increasing the confinement coefficient, the development of cracks in the concrete is reduced, exhibiting the transition of the failure behavior from brittle manner to ductile manner since the increase in the lateral restriction curbs the development of internal cracks in the concrete core.

Damage 0.99

0

FIGURE 2.57

The effect of confinement coefficient on the failure pattern of circular and square CFST stub columns [62].

96

2. Experimental tests

Damage 0.99

0

FIGURE 2.58 The effect of column size on the failure pattern of circular and square CFST stub columns [62].

Fig. 2.58 compares the failure patterns of circular and square CFST stub columns with different diameters of 200 mm, 400 mm, 800 mm, and 1000 mm, but the same slenderness ratio l ¼ 3:0, confinement coefficient of x ¼ 0:77, steel yield strength of 360 MPa, and concrete compressive strength of 52.7 MPa. It is evident that the angle of rupture is not affected by the size of the column, showing that CFST/CFDST columns with various structural sizes (but the same confinement coefficient) display the same failure patterns. gc for the in-filled concrete confined by the steel tube is governed by Ref. [55]: an   d 0:112ð1600 Þ (2.27) gc ¼ 150 Fig. 2.59 compares the strength reduction coefficient recommended by Sakino et al. [61] with the one developed by Wang et al. [55]. The graph 1.1

Confined concrete

γu

1

0.9

0.8 Unconfined concrete

0.7 0

500

1000 d (mm)

1500

2000

FIGURE 2.59 Comparison between the recommended strength reduction factor models.

2.7 Size effects

97

indicates that by increasing the diameter of unconfined concrete from 150 mm to 900, the compressive strength declines by 18.2%, whereas with the same increase in the diameter of confined concrete, the compressive strength decreases by 13.9%, 11.3%, and 9.1%, respectively, for an ¼ 4%, 7%, and 10% and the difference between the unconfined concrete and concrete core of CFST/CFDST columns keeps enlarging with the increase of nominal steel ratio an . During the elastic stage, the longitudinal stress of the stub column is proportional to the strain. Hence, the composite elastic modulus can be defined as follows: Esc ¼ N=½ðA þ A Þε  s c v

(2.28)

in which N is the axial load and εv is the longitudinal strain in the column. In general, the composite elastic modulus is not affected by the diameter of the column. In other words, the elastic modulus relates to the behavior of the concrete during the elastic step when the propagation of microcracks is slow. Consequently, the elastic modulus can be known as a constant value, and it is independent of the size effect. This is because the size effect is associated with the concrete core fracture, while no crack happens in the concrete during the initial loading stage. It is noteworthy that increasing the nominal steel ratio an enhances composite elastic modulus. The compressive strength of the composite column can be expressed by defining the peak axial stress ðsu Þ, as follows: su ¼ Nu=ðA

so

þ Asi þ Ac Þ

(2.29)

Two parameters are first introduced to discuss the size effect on the peak stress of CFST/CFDST stub columns. The first parameter is g < 1:0, which is the ratio of the ultimate axial stress of the CFST/CFDST column with an arbitrary diameter d to the ultimate axial stress of the benchmark CFST/CFDST column with the diameter of in-filled concrete d ¼ 150 mm. The second parameter is gc < 1:0, called the strength reduction coefficient of unconfined concrete, and is defined as the ratio of the ultimate axial stress of the unconfined concrete column with an arbitrary diameter d to the ultimate axial stress of the benchmark unconfined concrete column with the diameter D1 ¼ 150 mm. It can be expected that by increasing the diameter of the column, both g and gc are reduced. Suppose there are four CFST columns with different diameters of the concrete core (D4 > D3 > D2 > D1 ¼ 150 mm), while the rest of the material and geometric properties are the same. It can be expected that g4 < g3 < g2 < g1 . In other words, the peak axial stress of columns with the same material properties

98

2. Experimental tests

and nominal steel ratio is reduced by increasing the diameter of the column due to the size effect. However, the intensity of the size effect on the peak axial stress can be reduced by increasing the nominal steel ratio. Compared with unconfined concrete columns, composite columns present a smaller size effect (g < gc ). The reason lies in the fact that the confinement effect provided by the tube can restrain the development of macrocrack, leading to the reduction of the size effect on the in-filled concrete. The difference in size effect between confined and unconfined concrete becomes meaningful with increasing the nominal steel ratio. Consequently, available models for considering the size effect on the strength of unconfined concrete should not be applied directly for predicting the size effect of composite columns. Additionally, the influence of the nominal steel ratio on the size effect of the in-filled concrete should be counted. Similar to the peak axial stress, the peak axial strain is reduced by increasing the diameter of the column. The size effect of the peak axial strain is due to two reasons. The first reason is the effect of the diameter on the concrete failure mode. The failure of the concrete is generated by a local fracture. The height ratio between the fracture region and the column is reduced by enlarging the diameter. The second reason is that the efficiency of the steel tube in providing the confinement effect diminishes with increasing the diameter. In general, the axial compressive strength of concrete is enhanced by increasing the confinement coefficient. However, there is a particular confinement coefficient beyond which the compressive strength would no longer increase with increasing the confinement coefficient. This is because when the restraint effect provided by the tube achieves a specific level, the confinement coefficient no longer affects the strength enhancement. Compared with the square column, the strength enhancement due to increasing the confinement effect is more prominent in the circular column. Regarding the relationship between the size effect and the confinement coefficient for a column with a specific concrete and column size, the size effect on the compressive strength is reduced by increasing the confinement coefficient. However, there is a critical confinement coefficient ðxcr Þ beyond which the compressive strength would no longer be dependent on the structural dimension of the column, meaning that the size effect is entirely curbed. This is similar to suppose that the concrete that is confined by a very thick steel tube is no longer a brittle material (the concrete is no longer a quasi-static material). Therefore, the size effect is no longer the property of the highly confined concrete. Consequently, the size effect on the compressive strength of CFST/CFDST columns with x > xcr can be ignored. Regarding the material strengths, the size effect is reduced by increasing the steel yield strength. Conversely, increasing the concrete compressive strength increases the size effect. This can be explained mathematically by observing the effects of material strengths on the confinement coefficient.

2.7 Size effects

99

It can be observed from Eq. (2.5) that the confinement coefficient is increased by increasing the steel yield strength or reducing the concrete compressive strength. Meanwhile, the size effect is reduced by increasing the confinement coefficient. Therefore, in addition to the diameter of the column and the nominal steel ratio, the strength of materials also influences the size effect. Eq. (2.27) is a function of the diameter and nominal steel ratio. However, it was discussed that in addition to the diameter of the column and the nominal steel ratio, the strength of materials also influences the size effect. For taking into account the influences of all variables on the size effect, the confinement coefficient can be used in the size effect model [63], as follows:   su D KðxÞ ¼ (2.30) s150 150 where su and s150 are the peak stresses of the columns with diameters D and 150 mm, respectively. KðxÞ in Eq. (2.30) is a function of confinement coefficient, and it takes into account the effect of confinement on the size effect of axial compression strength. KðxÞ is given by Eq. (2.31). It should be noted that Eq. (2.30) has been developed based on the circular CFST stub columns with D  2000 mm and is not applicable for rectangular/ square composite columns. From Eq. (2. 31), it can be noticed that for unconfined concrete (x ¼ 0), the value of KðxÞ is 0.12, which is slightly higher than the exponent of Eq. (2.27) for unconfined concrete. The value of KðxÞ is gradually increased and approached zero by increasing the confinement coefficient. For the confinement coefficient 1.5, KðxÞ has a value of 0.009. Consequently, the strength reduction is smaller than 3% as the diameter of the column enlarges from 150 to 2000 mm. Therefore, the size effect of CFST/CFDST columns with a confinement factor greater than 1.5 can be ignored. ( 0:092x0:46  0:12 if x  1:5 (2.31) KðxÞ ¼ 0 if x > 1:5 Similar to the concrete core, the size effect can also affect the steel tube. The relations of the nominal stress versus the strains of the steel tubes of circular and square CFST columns with the width ranging from 200 to 800 mm and the confinement coefficients x ¼ 0:38 and 1.53 are illustrated in Fig. 2.60 [62] where εh and εv are the horizontal and longitudinal strains at the middle of the columns, respectively. All columns have the same slenderness ratio l ¼ 3:0, steel yield strength of 360 MPa, and concrete compressive strength of 52.7 MPa. Based on Fig. 2.60, εh and εv gradually increase during the elastic phase with increasing axial load. During the elastic phase, Poisson’s ratio of the steel tube is greater than the concrete core. Therefore, the steel tube’s lateral expansion is more than the concrete

100

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80

140

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60

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Steel tube strain ε

FIGURE 2.60 Vertical stress versus strain relationship of circular and square steel tubes. Jin L, et al. Size effect of axial-loaded concrete-filled steel tubular columns with different confinement coefficients. Eng Struct 2019;198:109503.

core, and no interaction occurs. This issue results in the faster growth of the longitudinal strain than the horizontal strain at the elastic stage. As the load increases, the strain growth trend becomes nonlinear. The steel tube is yielded by reaching the peak stress (reaching the axial compressive strength). In columns with x ¼ 0:38, the smaller columns have larger strains in both directions when the peak stress is reached, particularly for the square section. Compared with larger columns, smaller columns have a faster increase in εh and εv after the peak load. This shows that steel tubes in smaller columns can fully constrain the concrete core. Increasing the confinement coefficient reduces the size effect on the steel tubes’ longitudinal and horizontal strains, especially in circular columns, due to their ability to better confine the concrete core.

2.8 Hollow ratio The hollow ratio c is an influential parameter that can profoundly affect the behavior of CFDST columns. This parameter can be used to assess the effects of the size of the sandwiched concrete on the behavior of CFDST members. The axial loads carried by constituent components of CFDST stub columns as a function of the axial displacement are presented in Fig. 2.61. The majority of the axial load in CFDST stub members is carried by the sandwiched concrete and the outer steel tube, respectively. In comparison, the contribution of the inner steel tube in load-bearing capacity is the least. Therefore, changing the cross-sectional area of the sandwiched concrete (altering c) can impose significant effects on the behavior of the column. It is noteworthy that the load distribution between the in-filled concrete and steel tubes, shown in Fig. 2.61 is true only for columns subjected to axial compression, and it can be altered for other loading conditions. For instance, the load-bearing proportions for the steel tubes and the sandwiched concrete of CFDST columns under pure torsion

101

2.8 Hollow ratio

Axial force

CFDST column

Concrete core Outer tube Inner tube

Axial displacement

FIGURE 2.61 The load bearing contribution of constituent components of CFDST stub columns under concentric axial compression.

are totally different from the results displayed in Fig. 2.61. For instance, when a CFDST column is under torsional load, outer and inner tubes carry the majority of the load, while the sandwiched concrete bears the least. The practical range of the hollow ratio is between 0 and 0.8. Suppose the hollow ratio c is equal to 0. In this case, the column is a conventional CFST. The upper limit of 0.8 is considered to guarantee that the inner steel tube can achieve its full yield capacity. Therefore, it can be expected that the inner steel tube can efficiently restrict the inner indent of the sandwiched concrete; hence, the behavior of the sandwich concrete confined by outer and inner steel tubes is similar to the concrete core that is in a fully in-filled tube without the inner void. Depending on the length of the column, the hollow ratio can affect the load-bearing capacity. In stub CFDSTcolumns, increasing the hollow ratio from 0 to 0.3 has a negligible effect on the ultimate axial strength. Consequently, the design of CFDST columns with small hollow ratios of less than 0.3 is not recommended since they can increase the weight of the column without any significant effect on the compressive strength. By contrast, a hollow ratio beyond 0.3 can reduce the compressive strength of the column. A similar trend can be expected for the effect of the hollow ratio on the elastic energy absorption capacity of CFDST stub columns. Compared with short CFDST columns, the hollow ratio has an insignificant effect on the ultimate axial strength of intermediate-length CFDST columns. The relationship between the hollow ratio and the axial load-bearing capacity of very long CFDSTcolumns is opposite to the stub and intermediatelength columns. Contrary to stub and intermediate length CFDST columns, the ultimate axial strength of very long CFDST columns may increase with enlarging the hollow ratio. The contradictory effects of the hollow ratio on the ultimate axial strengths of the stub, intermediate, and very long CFDST columns are due to the difference between their mechanical performance.

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As shown in Fig. 2.61, sandwich concrete is the main element of the column in axial load-bearing. Therefore, any increase in the hollow ratio beyond 0.3 can remarkably reduce the cross-sectional area of the sandwiched concrete, leading to the reduction of the load-bearing capacity of stub CFDST columns. The role of the in-filled concrete on the compressive capacities of CFST/CFDST columns is declined by increasing the length of the column. This issue is discussed in detail in Section 2.11.1 and Chapter 4. Besides, the failure mode of very long CFST/CFDST columns is overall elastic buckling. Therefore, the flexural rigidity of very long CFST/CFDST columns dominates their compressive strengths. By increasing the hollow ratio, the inner steel tube is placed farther from the center of the column, leading to an increase in flexural rigidity. Consequently, the contribution of the inner tube to the moment of inertia of the column increases as the hollow ratio enlarges. In intermediate-length columns, the decline in the strength caused by reducing the cross-sectional area of the sandwiched concrete is neutralized by enhancing the strength caused by the extra flexural rigidity obtained from the inner tube, leading to an insignificant influence of the hollow ratio on the ultimate axial strength of intermediate-length columns. Increasing the hollow ratio can enhance the ductility and the strength s

of CFDST columns. The cross-sectional of the sandto-weight ratio w wiched concrete, which is inherently a brittle material, is reduced with increasing the hollow ratio. Therefore, the ductility of the column can be increased by increasing the hollow ratio. Also, the total weight of the member is reduced by reducing the volume of the sandwiched concrete. Typically, reducing the column’s weight by increasing the hollow ratio outweighs the reduction of the strength. Hence, it can be expected that the s ratio is improved by using a large hollow ratio. This issue can become of w importance in the design of structures located in high-risk seismic zones. Another effect of the hollow ratio is on the confining pressure. Increasing the hollow ratio reduces the induced hoop stress on the outer steel tube, thus reducing the confining pressure on the sandwiched concrete. Therefore, the confinement effect is reduced by increasing the hollow ratio (see Fig. 2.13). However, altering the hollow ratio has an insignificant impact on the developed hoop stress on the inner tube. In general, the effects of the hollow ratio are more prominent on the behavior of circular CFDST members than rectangular ones. This is because of the uniform and efficient confinement effect provided by the circular steel tube.

2.9 Effects of the confinement factor The confinement factor x is a comprehensive parameter that can evaluate the composite action of CFST/CFDST members. This parameter takes into account the material and geometric properties of the member. Therefore,

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2.9 Effects of the confinement factor

Axial load

C B D

A

Axial displacement

FIGURE 2.62

General load-displacement curve of CFST/CFDST columns.

it can be known as a suitable factor for assessing the steel tube’s ability to develop passive confinement and hence, the increase in the plasticity of the concrete core. The typical axial loadedisplacement curves of CFST/CFDST columns with three different confinement factors are illustrated in Fig. 2.62. Based on the elastic and plastic behavior of constituent components, the load-displacement history can be divided into four steps: Stage 1: Elastic phase (Point OeA): During this stage, the in-filled concrete and steel tubes have elastic behavior, and no noticeable change occurs in the shape of the column. Therefore, the steel tubes and the infilled concrete bear the external load independently. In this stage, Poisson’s ratio of the steel tube is ws ¼ 0:3, which is higher than the in-filled concrete wc ¼ 0:2. Therefore, the outer steel tube experiences a larger lateral expansion than the in-filled concrete and can even be detached from the concrete. Hence, no interaction occurs between the outer tube and the in-filled concrete at this stage. By contrast, the inner steel tube may impose slight pressure on the in-filled concrete due to restricting the lateral dilation of the concrete. However, the imposed pressure by the inner tube does not last long. This is because of the fast propagation of cracks in the concrete that transfer its behavior to the plastic zone. Stage 2: Elastic-plastic phase (Point AeB): The initial damages occur at this stage. With increasing the external load, the concrete cracks and shows remarkable expansion in the lateral direction. Therefore, the Poisson’s ratio of in-filled concrete exceeds the Poisson’s ratio of the outer tube. This is the moment when the interaction between the outer steel tube and the in-filled concrete develops. The confinement enhances with increasing the longitudinal deformation. However, no interaction is expected between the inner steel tube and the in-filled concrete at this point. Yielding or buckling of the outer tube occurs at Point B. Stage 3: Plastic phase (Point BeC): The steel tubes exceed their yield limits. Besides, the concrete core experiences much higher longitudinal

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stress than its cylindrical strength fc0 . The distribution of longitudinal stresses in the circular column is different from columns with polygonal cross sections. In the circular cross section, the stress distribution is layered and increases by moving from the inner steel tube to the outer steel tube. This is because the confinement effect in CFDST columns is mostly provided by the outer tube, and the role of the inner tube in confining the concrete core is insignificant. In polygonal cross sections, the maximum longitudinal stress occurs at the outer perimeter of the concrete. However, the distribution of the stress is not uniform, and the stress concentration occurs at corners. More details regarding the longitudinal stress distribution over the cross section of the concrete are provided in Chapter 4. Stage 4: Softening or hardening phase (Point CeD): The column reaches the ultimate axial strength at Point C. Because of the development of plasticity in the steel tube, the load increment is mainly tolerated by the infilled concrete. The behavior of the column during the postpeak region depends on the value of the confinement factor x. It can be expected that the plasticity of the concrete keeps increasing if the confinement factor is large enough (x  x0). By contrast, if the amount of confinement factor is small ðx < x0 Þ, the value of the longitudinal stress in the concrete reduces compared with the previous stage (Point BeC). The physical reason for this behavior is the ability of the outer steel tube to develop strain hardening behavior. A thin-walled outer steel tube is prone to buckle at this stage or even before reaching the ultimate axial strength. In this case, the in-filled concrete will lose the confinement effect, and the column shows a descending postpeak region. Contrarily, the strain-hardening phenomenon of the steel tube can help the tube to continue restraining the concrete. The increment in load-bearing capacity of the concrete due to the provided confinement effect can compensate or even exceed the decrease in the axial internal force of the tubes, leading to the strengthening step of the loaddisplacement response. The residual strength of the column is diminished by reducing the confinement factor. According to the effect of the confinement coefficient on the postpeak behavior, it can be stated that the confinement coefficient is the main parameter for ductility. The amount of x0 depends on the cross-sectional shape of the outer steel tube. x0 in CFST/ CFDST columns with the circular outer steel tube equals 1, whereas it is increased to 4.5 for the columns with the square outer steel tube. By comparing the postpeak region of the curves, it can be deduced that increasing the confinement factor x can lead to the following results: (1) (2) (3) (4)

Increasing the confinement effect provided by the outer steel tube Increasing the load-bearing capacity of the column Improving the residual strength and ductility of the column Increasing energy absorption capacity and plastic deformation of column (increasing the area below the load-displacement curve)

2.10 Effects of the material properties

105

2.10 Effects of the material properties CFST/CFDST members are fabricated using steel tubes and concrete. Hence, their material properties can influence the overall performance of composite members, but with varying degrees. As shown in Fig. 2.61, the contribution of CFST/CFDST columns constituent components under axial compression in the load-bearing capacity in descending order is: in-filled concrete > outer steel tube > inner steel tube. Hence, the effects of the concrete compressive strength are remarkably more than other column components, whereas the material properties of the inner steel tube are the least. The effects of material properties on the behavior of CFST/CFDST columns under axial compression are discussed in this section.

2.10.1 Concrete The influence of the concrete compressive strength on the performance of CFST/CFDST columns under axial compression can be discussed in various aspects. The first issue is the impact of the concrete compressive strength on the ultimate axial strength of the column. In general, the column strength is highly enhanced by increasing the concrete compressive strength. This is due to the significant contribution of in-filled concrete to the load-bearing capacity of the column and the appropriate behavior of the concrete material under compression. Besides, increasing the concrete strength remarkably increases its contribution and reduces the role of the steel tubes in the load-bearing capacity of the column. Consider two circular and square columns with outer steel tubes having the same dimension of Do  to and the steel yield stress of fyo . Increasing the compressive strength of the in-filled concrete in both columns with the same ratio enhances their ultimate axial strengths. However, the rate of strength increment in the square column is more than the circular one. This is because the cross-sectional area of the concrete in the square column is more than the circular one with the same width and thickness. However, circular CFST/CFDST columns are more cost-effective than columns with the square section to achieve the required design resistance. Suppose that a CFST column is required to bear the axial compressive load of 2300 kN. The applied cubic compressive strength at 28 days is 49.5 MPa, and the utilized steel grade is S235. The design of the column with a circular cross section to carry the imposed external load leads to a column with Do  to ¼ 220  4 mm, whereas the case of using a square cross section with the same dimension ðBo  to ¼ 220  4 mmÞ leads to a column with a compressive strength of more than 2500 kN that is more than the required design resistance. Despite a slight difference in compressive capacities, the required quantities of the

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2. Experimental tests

steel and the concrete materials in the 22%  circular CFST column are almost  As; circular Ac; circular less than their square counterpart ¼ ¼ 0:78 . As; square Ac; square Another aspect is the efficiency of improving the column strength by increasing the concrete compressive strength. It seems that increasing the concrete compressive strength can be known as an efficient method of improving the column strength. With the development of the construction industry, the use of HSC and UHSC has become popular among civil engineers, and they can be considered as a desirable material for filling the hollow steel sections. However, the efficiency analysis proves that increasing the concrete compressive strength cannot always be considered as an efficient method for improving the column’s ultimate axial strength. For instance, in a CFDST stub column with an outer square steel tube with Bo  to ¼ 300  10 mm and fyo ¼ 350 MPa, and an inner circular steel tube with di  ti ¼ 150  7 mm and fyi ¼ 350 MP, increasing the compressive
 strength of the concrete from fc0 ¼ 30 MPa to fc0 ¼ 100 Dfc0 ¼ 233:33% leads to only a 64% improvement in the ultimate axial strength of the column. The achieved rate of improvement in the ultimate axial strength of the column compared with a more than 233% increase of the concrete compressive strength is negligible. Therefore, the cost-safety balanced criterion must be taken into account in designing CFST/CFDST columns with HSC and UHSC. For any compressive strength of the concrete, the ultimate axial strength of the column is enhanced by increasing the steel tube wall thickness. Compared with NSC, using HSC reduces the intensity of increment in the compressive strength of the column caused by increasing the tube wall thickness (reducing the D=t). In other words, the use of HSC in CFST/CFDST columns becomes more cost-effective by enlarging the D=t ratio of the steel tube (i.e., reducing the cross-sectional area of the outer tube). The third aspect is the impact of the concrete strength on the confinement effect provided by the steel tubes. Generally, the confinement effect is reduced by increasing the compressive strength of the in-filled concrete. This is because the stiffness of the concrete is enhanced as its compressive strength increases, leading to its smaller lateral expansion. Consequently, increasing the concrete compressive strength postpones the generation of interaction between the steel tubes and the core concrete. This issue can also be described mathematically by the confinement factor x (Eq. 2.5). The confinement factor reduces upon increasing the compressive strength of the in-filled concrete. The confinement effect provided by steel tubes enhances the structural performance of CFST/CFDST columns by improving the ultimate axial strength, axial stiffness, ductility, residual strength, and energy absorption capacity of the in-filled concrete. The question that may arise is whether the confinement effect is more important for NSC or HSC.

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2.10 Effects of the material properties

f'cc

90

Amount of increases

55.9

60 45.9

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50 40

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30

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Ratio of increase

0

FIGURE 2.63

Comparison of the compressive strengths of concrete in a CFDST column having different unconfined strengths [26].

Fig. 2.63 presents the relationship between the compressive strengths of unconfined concrete with their counterparts confined in a CFDST column. The outer circular steel tube has a diameter of 2000 mm and a thickness of 10 mm, yield strength of 196.2 MPa, and modulus of elasticity of 206 GPa. The diameter and thickness of the inner circular steel tube are 1400 and 5 mm, respectively, with the yield strength of 294.3 MPa and Young’s modulus of 206 GPa. As shown in Fig. 2.63, the confinement effect of steel tubes can remarkably enhance the concrete compressive strength. However, the rate of increase under the same conditions is nearly constant, even though their unconfined concrete compressive strengths are different. For instance, the compressive strength of the unconfined concrete with 0 ¼ 45.89 MPa. fc0 ¼ 30 MPa is increased by 15.89 MPa and reaches fcc 0 Meanwhile, the unconfined concrete with fc ¼ 60 MPa experiences the 0 ¼ 75.89 MPa. same increase in its strength (15.89 MPa) and reaches fcc The compressive strength of lower-strength concrete is increased by 53%, whereas the rate of increase in the compressive strength of higher-strength concrete is reduced to 26.5%. This issue signifies that the confinement effect for improving the compressive strength of concrete is more important for the low-strength unconfined concrete. The fourth aspect is the influence of the concrete compressive strength on the mechanical properties of CFST/CFDST columns. Initial axial stiffness, elastic energy absorption capacity, and ductility are useful quantitative parameters for evaluating the performance of structural members. Developing microcracks in the concrete reduces its secant modulus. The secant modulus corresponding to 40% of the unconfined concrete compressive strength (0:4fc0 ) can be considered as the modulus of elasticity of the concrete. The increase of the stress in the CFST/CFDSTcolumn due to imposing the external force develops cracks in the concrete. This issue reduces the secant compressive stiffness of the column. Therefore, similar to the unconfined concrete, the secant stiffness corresponding to 40% of the column

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ultimate axial strength ð0:4Nu Þ can be defined as the compressive stiffness of CFST/CFDST columns ðEc Þ. The fourth aspect is the influence of the concrete compressive strength on the mechanical properties of CFST/CFDST columns. Initial axial stiffness, elastic energy absorption capacity, and ductility are useful quantitative parameters that can be used for evaluating the performance of structural members. Developing microcracks in the concrete reduces its secant modulus. The secant stiffness
corresponding to 40% of the unconfined  concrete compressive strength 0:4fc0 can be considered as the modulus of elasticity of the concrete. The stress increase in CFST/CFDST columns due to imposing the external force develops cracks in the concrete. This issue reduces the secant compressive stiffness of the column. Therefore, similar to unconfined concrete, the secant stiffness corresponding to 40% of the column ultimate axial strength ð0:4Nu Þ can be defined as the compressive stiffness of CFST/CFDST columns ðEc Þ. The area under the axial loadedisplacement curve can be calculated to determine the absorbed energy, as shown in Fig. 2.64. The area below the axial loadedisplacement curve of the column, from the outset of loading up to the point when the member reaches 75% of its axial compressive strength in the prepeak region ðP0:75u Þ can be defined as the elastic absorbed energy ðEe Þ. Ee is actually the absorbed energy at the displacement from which the nonlinear behavior of the column starts and can be defined as follows: DZ 0:75u 1 FdD ¼ ðP0:75u  D0:75u Þ (2.32) Ee ¼ 2 0

Force

where D0:75u is the axial shortening of the column corresponding to P0:75u .

Displacement

FIGURE 2.64 Determination of the absorbed energy for CFST/CFDST columns.

2.10 Effects of the material properties

109

Eu in Fig. 2.64 is the energy corresponding to the ultimate axial strength governed by: ZDu Eu ¼

FdD

(2.33)

D0:75u

where Du is the axial shortening corresponding to Pu . The ductility of CFST/CFDST columns can be defined in two ways. The first method is the energy-based ductility factor ðmÞ and can be obtained as follows: Eu (2.34) Ee Another method is the strain ductility index ðDIÞ that is expressed by: m¼

εu (2.35) εe where εu represents the axial strain of the column when the load drops to 90% of its ultimate axial strength in the postpeak region. For columns with x  x0 , εu is considered as the ultimate axial strain. Ex represents the yield ε0:75 , where ε0:75 shows the axial strain strain of the column and is taken as 0:75 when the load reaches 75% of its ultimate axial strength in the prepeak stage. In general, the compressive stiffness of the column increases upon increasing the concrete core compressive strength. Similarly, a column with a higher fc0 can absorb more energy than the counterpart with a smaller fc0. One of the most noticeable effects of the concrete core strength is on the ductility of the column. Increasing the concrete strength can significantly reduce the ductility and the residual strength of the column. The effect of the concrete strength on the ductility of the column can be found in the postpeak region of the axial loadedisplacement curve (Point CeD in Fig. 2.62): as the concrete strength increases, so does the slope of the descending region. This issue is an undesirable phenomenon in designing structures. The reason is that once the ultimate axial strength of the column is achieved, applying any further external load on the column will lead to a sudden cracking and shattering of the concrete that is an immediate brittle failure with limited warning. The negative effect of the concrete strength on the ductility of the column is due to its inverse effect on the confinement effect. As discussed above, the confinement effect is reduced by improving the concrete strength. Hence, the in-filled concrete loses restriction and may show brittle behavior. It can be expected that the effect of the concrete strength on the ductility of circular columns would be more than columns with polygonal cross sections. Partial replacement of natural aggregates with rubber particles changes the performance of the concrete in two ways. RuC is defined by a DI ¼

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remarkable decline in its compressive, tensile, and flexural strengths. Different factors are involved in the low resistance of RuC. The first reason is the poor adherence between the surface of rubber particles and the cement, leading to the weaker strengths of RuC. A feasible solution to enhance the workability of the RuC is performing the NaOH pretreatment of rubber particles to improve the adhesion of rubber to cement paste. Another reason for the weaker strengths of the RuC is the difference between the Poisson’s ratio and Young’s modulus of rubber particles relative to concrete. The Poisson’s ratio of rubber particles is twice that of concrete material, and their Young’s modulus is almost 33% lower than the concrete. Therefore, large relative deformations between the concrete and rubber particles are likely to occur, leading to premature concrete cracking. Besides, the low modulus of elasticity of rubber particles leads to the development of significant internal tensile stresses perpendicular to the direction of the axial compression load. As a result, depending on the replacement ratio and size of rubber aggregates, the reduction in RuC strength can even reach 80%. Despite the drawbacks mentioned above, RuC is characterized by considerable improvements in its ductility, fracture toughness, impact resistance, energy absorption capacity, and seismic resistance compared to the normal Ordinary Portland Cement concrete (NC). Approximately every 5% increase in rubber particles enhances the ductility of the concrete by 15%. In addition, the ratio of flexural to compressive strengths of RuC is higher than NC. This means that the use of rubber particles can improve anticracking performance. Besides, the viscous damping and kinetic energy can be improved by adding rubber particles to concrete. Hence, replacing the normal concrete core as the main component in the load-bearing capacity of CFST/CFDST columns under axial compression with RuC can profoundly change the columns’ overall behavior. In general, replacing the natural aggregates with rubber particles reduces the axial stiffness and compressive strength of the column, and the intensity of this reduction depends on the percentage of replacement. However, confining RuC with steel tubes can make it possible to use RuC as a sustainable and environmentally friendly material in construction projects. Additionally, employing RuC in CFST/CFDST columns can effectively delay the axial shortening corresponding to the peak load and improve the ductility of columns. The effects of RuC on the mechanical performance of CFST/CFDST columns depend on the geometric properties of members. As a general trend, the following conclusions can be drawn: (1) The reduction influence of the rubber replacement ratio ðrvr Þ on the ultimate axial strength of columns increases with increasing the column width. This is because the in-filled concrete bears most of the external load due to its significant contribution to the loadbearing capacity compared with other constituent components of the columns. Therefore, increasing the in-filled concrete crosssectional area by increasing the depth of the outer steel tube increases the impacts of the concrete properties on the column’s

2.10 Effects of the material properties

111

behavior. Similar to the compressive strength of the columns, the effect of RuC on the ductility is more significant in specimens with a larger Do=t ratio. This is due to the fact that the confinement effect o is reduced by increasing the Do=t ratio. Therefore, increasing rvr for o

neutralizing the brittle behavior of concrete is more efficient in columns with a large Do=t ratio. o (2) The effect of the rubber replacement ratio on the compressive capacity is inversely related to the size of the hollow ratio. This is owing to the fact that the smaller the hollow ratio, the larger the cross-sectional area of the in-filled concrete, which generally contributes to the majority of the load-bearing capacity. Therefore, the effects of the rubber particles on the compressive strength of the column decline with increasing c. Concerning the impact of rvr on the ductility of columns, adding rubber particles to the in-filled concrete for improving ductility is more efficient when the hollow ratio is small due to the larger cross-sectional area of the concrete. (3) The impact of Ro on reducing compressive capacities of columns with to ¼ ti is more than columns with to > ti . In other words, the effect of rubber particles on the ultimate axial strength of the RuCFST/RuCFDST column reduces with increasing the wall thickness of the outer steel tube. Regarding the effect of rvr on the ductility of columns, increasing rvr is more remarkable on the ductility of the column with a thinner outer steel tube than the counterpart with a thicker outer steel tube. This issue indicates that irrespective of the axial strength, instead of increasing the thickness of the steel tube to improve the column’s ductility, which can increase the cost and weight of construction, using RuC as the in-filled concrete can be considered an efficient method. The next point is the influence of RuC on the confinement effect. Regardless of the shape of the column, the confining pressure is reduced by adding rubber particles to the in-filled concrete. As mentioned above, RuC has more ductile behavior than NC. Consequently, the development of lateral expansion in NC occurs quicker than RuC. Filling the steel tube with NC reduces the local second-order effects in the tube due to restricting the inward buckling of the tube, leading to sufficient confinement to concrete. By contrast, filling the tube with RuC increases the susceptibility of the tube to local buckling. As a result, the confinement effect provided by the steel tube is reduced. Another reason is the difference in the dilation angle (j) between NC and RuC. The dilation angle j controls the nonassociated flow rule and affects the plastic volumetric strain of material generated during plastic shearing. The dilation angle of     NC is between 20 and 30 , whereas it is reduced to j ¼ 15 and 10 by adopting rvr ¼ 5% and rvr ¼ 15%, respectively. This is because the high Poisson’s ratio and hyperelasticity of rubber particles make them deform

112

2. Experimental tests

easily without failure. As a result, using them in the concrete reduces crack propagation. Usually, the Poisson’s ratio of rubber particles is taken as 0.5 due to their incompressible behavior that can also contribute to the smaller amount of the RuC dilation angle than that of NC. During the initial loading stage, there is no trace of interaction between the in-filled concrete and the steel tube. Whereas by turning into the elastoplastic phase, the concrete begins expanding. The dilation angle j influences the concrete’s expansion in the lateral direction and, therefore, the confinement effect. A smaller dilation angle leads to a smaller passive confining pressure provided by the steel tube to the in-filled concrete. Analysis of composite action in RuCFST/CFDST stub columns is further discussed in Chapter 4. Compared with circular CFST/CFDST columns, RuC has smaller impacts on the overall performance of square CFST/CFDST columns. Besides, the efficiency of steel tubes in providing the confinement effect reduces with increasing rvr . This is because rubber particles enhance the concrete ductility and reduce the dilation angle.

2.10.2 Steel tube Like the in-filled concrete, the mechanical properties of the steel tubes affect the behavior of CFST/CFDST columns, however, to a smaller degree. This is because the cross-sectional area of steel tubes is less than the in-filled concrete, leading to smaller contributions of steel tubes in load-bearing capacity than the concrete. Also, the properties of the inner steel tube in CFDST members have the least impact on the performance of members. Regardless of the column’s cross-sectional shape, the effect of steel yield stress on the initial axial stiffness of the column is negligible. By contrast, increasing the outer steel tube’s yield stress can remarkably enhance the column’s ultimate axial strength. Besides, the steel tube with high yield stress can provide a better confinement effect and improve the column’s ductility. The reason is that the local stability and buckling strength of the tube improves with increasing its yield stress, leading to the better restriction of lateral expansion of in-filled concrete. These issues can also be justified by assessing the impact of the tube yield stress on the confinement factor x. As the steel yield stress increases, so does the confinement factor. As a result, the confining pressure induced on the concrete by the steel tube is enhanced. The use of high-strength steel (HSS) tube can delay Point A in Fig. 2.62. As a result, the longitudinal stresses in the in-filled concrete of a CFST/CFDST column fabricated with an
HSS tube can exceed the  compressive strength of unconfined concrete slc > fc0 when materials still have elastic behavior (at Point A). This issue is expected to occur in columns with normal-strength steel (NSS) tubes when the column enters the plastic phase. Even during the plastic phase, the amount of slc in the infilled concrete confined by the HSS tube is more than its counterpart fabricated with the NSS tube ðslc ÞHSS > ðslc ÞNSS > fc0 .

2.11 Load eccentricity

113

Another effect of increasing the steel tube yield stress is on the loadcarrying proportions of constituent components. Increasing the yield stress of the outer tube raises the role of the tube in the load-bearing capacity of the column. Besides, the axial load carried by the sandwiched concrete of the column fabricated with the high-strength outer tube is more significant than the counterpart made with the normal strength steel tube. This is because the concrete strength is improved as the high-strength steel tube can generate stronger confinement. The strength of the outer tube has a negligible influence on the contribution of the inner tube in the loadbearing capacity of CFDST columns under axial compression. Contrary to the outer steel tube, the strength of the inner steel tube in CFDST columns has no significant effect on the behavior of the member. The reason is the small contribution of the inner tube in the column’s loadbearing capacity compared with other components. Additionally, the inner tube provides minor confinement. It can be concluded from the above discussion that using HSS as the outer steel tube is more economical. By contrast, the least possible yield stress should be considered for the inner tube due to its minor load-bearing contribution compared with the outer tube. Besides, it is recommended to use HSS as the outer tube of CFST/ CFDST columns fabricated with HSC or UHSC since it can improve the ductility and residual strength of the column.

2.11 Load eccentricity CFST/CFDST columns in composite buildings are likely to be subjected to eccentric loads. Hence, understanding their behavior under eccentric loading is essential for practical design purposes. Fig. 2.65 illustrates the schematic view of the column under eccentric compression load. It can be seen that when the column is under eccentric load, the second-order effect is developed in the column. Therefore, part of the column is under tension, and the rest of it experiences compression. In this case, the column must be designed for the combination of axial load and bending moment. Fig. 2.66 shows the four-point interaction curves for section capacities of circular CFST members under combined loading conditions. The member is under pure compression at Point A, and the amount of moment at this point is MA ¼ 0. The behavior of the member at Point A is similar to the column under concentric axial compression. By contrast, the member is under pure bending at Point B, and the amount of axial compression is PB ¼ 0. The behavior of the member at Point B is similar to the beam. The amount of moment at Point C is equal to Point B, and the load equals the load capacity of the confined concrete infill. The column reaches the maximum moment resistance at Point D, whereas the amount of load at this point is half of that at Point C.

114

2. Experimental tests

P

Lateral displacement ( )

FIGURE 2.65

Compression face

Tension face

Deformed shape

Eccentricity ( )

A schematic view of the deformed shape of CFST/CFDST members under

eccentric load.

N

A

C

D

B

M

FIGURE 2.66 The typical interaction diagram for section capacities of the CFST members.

115

2.11 Load eccentricity

Various parameters influence the structural behavior of CFST/CFDST beam-columns, namely, the slenderness ratio, compressive strength of the concrete, steel tube yield stress, nominal steel ratio, concrete confinement, and load eccentricity ratio. The effects of each parameter will be discussed in the following sections.

2.11.1 Effects of column slenderness ratio The fundamental behavior of CFST/CFDST beam-columns is highly affected by the column slenderness ratio (L=r). As a general trend, increasing the slenderness ratio of beam-columns enhances the effect of the second-order moment. Therefore, increasing the slenderness ratio diminishes the flexural and axial stiffness, as well as the load-bearing capacities of beam-columns under eccentric load. By contrast, as the slenderness ratio increases, so does the lateral displacement ðDÞ and displacement-ductility of the column. Fig. 2.67 shows the schematic view of beam-column interaction curves with different slenderness ratios

ððL=rÞ

4

> ðL=rÞ > ðL=rÞ > ðL=rÞ 3

2

1

Þ under the constant load eccentricity of

e. In Fig. 2.67, the ultimate axial strength ðNn Þ is normalized to the ultimate axial load ðNo Þ of the axially loaded beam-column section, while the ultimate moment ðMn Þ is normalized to the ultimate pure bending moment ðM0 Þ of the beam-column. Besides, it is supposed that the material properties and other geometric properties of beam-columns are the

Nu (L/r)4 (L/r)3 (L/r)2 (L/r)1

Mu

FIGURE 2.67

Effect of the beam-column slenderness ratio on the interaction curve.

116

2. Experimental tests

same. It can be observed from Fig. 2.67 that reducing the slenderness ratio enlarges the interaction curve of the beam column. For beam-columns subjected to the equal axial load level, increasing the L=r ratio decreases ultimate axial strengths and ultimate bending strengths. However, the slenderness ratio does not affect the ultimate pure bending strength. Concerning the load distribution in beam-columns, increasing the member slenderness ratio reduces the load carried by the in-filled concrete, whereas the contribution of the steel tube increases with increasing the slenderness ratio. As discussed in Section 1.5.2.8, CFST/CFDST beam-columns generally fail at their midheight. Therefore, the load-bearing capacity at the midheight of beam-columns plays an important role in the behavior of the member, especially when the load eccentricity is high. The cross-sectional strength, which is affected by the concrete confinement, influences the strength at the midheight of slender beam-column. The slenderness ratio highly affects the confinement effect, and therefore, the strength at the midheight of a slender beam-column, especially circular ones. As a general trend, the confinement effect is reduced by increasing the column slenderness ratio up to the point that the circular steel tube in a very slender column with L=r > 60w70 cannot provide any confinement effect on the infilled concrete. Therefore, the confinement effect does not affect the ultimate axial strength of a very slender CFST/CFDST column. In other words, the confinement effect in very slender columns can be ignored. This issue is further explained in Chapter 4.

2.11.2 Effects of eccentricity ratio The eccentricity ratio of the applied axial compression load ðe =DÞ has notable impacts on the behavior of beam-columns. In general, increasing the load eccentricity reduces the flexural stiffness and the ultimate axial strength of the beam-column. This is because increasing the load eccentricity ratio increases the bending moment at the ends of the column, and therefore, reduces the load-carrying capacity as a result of the strength interaction. Additionally, deflections at the midheight of the beam-column, when it is achieved its maximum axial load, increases as the eccentricity ratio increases. Similarly, the displacement-ductility is enhanced by increasing the eccentricity ratio. By increasing the eccentricity ratio in the beam-column, the impact of axial load decreases, and the beam-column shifts to a more flexural member. Another impact of the load eccentricity ratio is its effect on the distribution of the load on constituent components of beam columns. Increasing the load eccentricity ratio reduces the load-bearing contribution of the steel tubes, whereas the load carried by the in-filled concrete increases with increasing the eccentricity ratio.

117

2.11 Load eccentricity

Similar to the slenderness ratio, increasing the load eccentricity ratio of the beam-column e=D decreases the efficiency of the confinement effect provided by the steel tube. The confinement effect in the circular beame > 2:0 can be column subjected to the large load eccentricity ratio D ignored. The influence of the eccentricity ratio on the confinement effect is further discussed in Chapter 4.

2.11.3 Effects of the concrete compressive strength The concrete compressive strength affects the fundamental behavior of beam-columns. Under the same circumstances, the flexural stiffness of the beam-column improves with increasing the concrete strength. Similarly, the ultimate axial strength enhances as the concrete strength increases. The compressive strength of the concrete has the most pronounced effect on the ultimate axial load than the bending capacity. However, its effect reduces with increasing the slenderness of the member. Consequently, the use of HSC in very slender CFST/CFDST columns is not efficient. By contrast, the ductility of beam-columns declines with increasing the concrete compressive strength due to the brittle behavior of the HSC. The incremental influence of the concrete strength on the ultimate axial strength of beamcolumns depends on the slenderness ratio of the member L=r . The effect of the concrete strength is the most when the slenderness ratio is L=r < 10, whereas its influence reduces with increasing L=r. Concerning the axial loademoment interaction diagrams of CFST/ CFDST beam-columns, increasing the concrete compressive strength reduces the distance between Points C and D in Fig. 2.66 and hence, shifts the position of balanced point inwards and downwards. Fig. 2.68 shows Nu

(f'c)1 (f'c)2 (f'c)3 (f'c)4

Mu

FIGURE 2.68

Effect of the concrete strength on the interaction curves of beam-columns.

118

2. Experimental tests

the schematic view of beam-column different

 interaction

curves
 with  concrete compressive strengths fc0 1 < fc0 2 < fc0 3 < fc0 4 while other parameters are constant. It can be seen that the concrete compressive strength has significant impacts on the interaction curves of beam-columns. The interaction curve is enlarged by increasing the concrete strength. Besides, pure ultimate axial load, ultimate bending moment, and ultimate pure bending moment are enhanced by increasing the concrete strength. However, due to the better performance of the concrete in compression than the tension, the concrete compressive strength is more influential on the pure ultimate axial load than the pure bending moment.

2.11.4 Effects of the steel tube yield stress The behavior of beam-columns is affected by the yield stress of steel tubes. In general, increasing the yield stress enhances the ultimate axial strength of the column. However, the effect of steel strength on the bending stiffness is negligible. Similar to the CFDST columns under concentric compression, the inner tube yield stress barely affects beam-columns overall performance. In contrast, the outer tube yield stress significantly affects the performance of beam-columns. It should be noted that the impacts of the steel tube yield stress are reduced by increasing the slenderness ratio of the member. A schematic view of the axial loademoment interaction diagrams of CFST/CFDST beam-columns with different outer steel tube yield stresses




 are presented in Fig. 2.69. It is supposed fy < fy < fy < fy 1

2

Nu

3

4

(fy)1 (fy)2 (fy)3 (fy)4

Mu

FIGURE 2.69 Effect of the steel tube yield stress on the interaction curves of beam-columns.

2.12 The role of the inner steel tube

119

that all the beam-columns have the same concrete compressive strength and geometric properties. As shown in the figure, increasing the steel tube’s yield stress enhances both the pure ultimate axial and bending resistances of beam-columns. Contrary to the concrete compressive strength, which has a more noticeable influence on the pure ultimate axial strength, the effect of steel yield stress is more prominent on the bending resistance.

2.12 The role of the inner steel tube 2.12.1 The effects of the inner steel tube The load-bearing contribution of the inner steel tube in a CFDST column is the least compared with the outer steel tube and the sandwiched concrete. Consequently, the geometric and material properties of the inner steel tube do not significantly impact the overall performance of CFDST members. However, the inner tube is a critical member that motivates the extension of the use of CFDST members. Replacing the center of the core concrete in conventional CFST members with a hollow steel tube reduces the weight of the member. Besides, the central part of the concrete core adjacent to the neutral axis has weak flexural and torsional resistance. Hence, replacing this part with an inner steel tube can enhance the member’s bending stiffness and flexural rigidity. Another advantage of using the inner steel tube is the superior performance of CFDST members compared with CFST members in a fire scenario. The outer steel tube and the sandwiched concrete decrease the temperature increment of the inner steel tube. Therefore, the member can sustain the load for a longer time before it collapses. Additionally, the inner tube can work as a ventilation channel during a fire accident. The thickness of the inner steel tube plays a significant role in the performance of CFDST members. If the inner steel tube buckles while the outer steel tube is still in the elastic phase, the inner side of the sandwiched concrete loses the confinement effect. By contrast, preventing the failure of the inner steel tube can guarantee that the behavior of the CFDST member is almost similar to a CFST member. If no inner steel tube is used or the premature buckling of the inner steel tube is not prevented, the outer tube cannot impose proper hoop stress on the sandwiched concrete and confine it properly. As a result, the member presents a brittle failure. Therefore, the optimum performance of CFDST members is achieved when the premature failure of the inner tube is prevented. This is because the lateral expansion of the inner tube restricts the inward deformation of the sandwiched concrete and increases the confinement effect. More discussions regarding the effects of the inner tube thickness are provided in Chapter 4. Regardless of premature buckling of the inner steel tube, when a thickwalled outer tube is utilized, the thickness of the inner tube does not have

120

2. Experimental tests

a significant effect on improving the ultimate strength of the member. By contrast, when a thin-walled outer tube is applied, increasing the thickness of the inner tube can highly improve the bearing capacity of the member compared with the former case. Considering different configurations of the tubes’ thicknesses in CFDST columns, two comparisons are possible. The first case is to use a thick-walled outer tube and alter the thickness of the inner tube. With the same cross section, length, and material properties, using a thick-walled outer tube can increase the ultimate axial strength of

¼ a . The second case is to use a thin-walled the column by a1 NNthickthick 1 thickthin outer tube and alter the thickness of the inner tube. With the same cross section, length, and material properties, using a thick-walled inner tube can

¼ a increase the ultimate axial strength of the column by a2 NNthinthick 2 . The thinthin rate of improvement in the column strength in the latter case is more than in the former case ða2 > a1 Þ. This indicates that when a thin-walled outer tube is utilized, the influence of the inner tube becomes more prominent.

2.12.2 Filling the inner steel tube with the concrete For increasing the bearing capacity of CFDST columns, dual-skin concrete-filled steel tube (DCFST) columns can be used in which instead of increasing the thickness of the steel tubes or the width of the column, the inner steel tube can be filled by the concrete, as shown in Fig. 2.70. Compared with CFST and CFDST members, DCFST members have various advantages, as listed below: (1) The inner steel tube with the HSC core can design to resist applied load during the fire, while the outer steel tube and the low-strength sandwiched concrete are sacrificial.

FIGURE 2.70

Typical cross section of circular DCFST members.

2.12 The role of the inner steel tube

121

(2) For achieving a required design strength, DCFST columns occupy less space and are more economical because satisfying a required strength in practice by increasing the thickness or width of the steel tube can lead to unpractical dimensions that are not economical. Besides, sourcing thick-walled or very wide tubes can be highly challenging. (3) The central part of the concrete core in CFST members has a low flexural and torsional load-bearing contribution. Therefore, this part of the concrete can be replaced by weaker concrete to reduce cost. (4) Adding the inner steel tube to the column improves the confining pressure on the concrete core, enhancing the overall performance of the member, such as ductility, load-bearing capacity, and postpeak resistance. The compressive strength of the sandwiched concrete and the concrete core can be different. For instance, UHSC can be used for the sandwiched concrete, whereas the inner steel tube can be filled with NC and vice versa. However, the influence of the compressive strength of concrete components on the behavior of the column is not the same. In addition, the intensity of the effects of increasing the thickness of the inner or outer tube on the behavior of DCFST columns differs from that of CFDST columns. The concrete-steel contribution ratio ðCSCRÞ can be used to compare the behavior of CFDST and DCFST columns with different configurations of steel thicknesses and concrete compressive strength with the classical CFST column. CSCR is a mechanical parameter by which the effect of adding the inner steel tube and filling it with the concrete to improve the performance of the CFDST column with respect to the CFST counterpart can be assessed. CSCR is defined as the ratio of the ultimate axial strength of the DCFST (or CFDST) column to the ultimate axial strength of the corresponding CFST column. Considering the same cross-sectional area of steel tubes in both CFST and DCFST columns are used, a CSCR greater than unity means that the new redistribution of steel tubes along with filling the inner steel with the concrete core improves the ultimate axial strength of the conventional CFST column. In addition to CSCR, the inner concrete contribution ratio ðICCRÞ can be used to assess the effect of filling the inner steel tube of the CFDST column with the concrete on the ultimate axial strength of the column. ICCR is calculated as the relation between the ultimate axial strength obtained for the DCFST and the ultimate axial strength measured for the corresponding CFDST column. Therefore, ICCR greater than unity shows the efficiency of filling the inner steel tube with the concrete. CSCR and ICCR can be determined as: CSCR ¼

Nu; DCFST Nu; CFST

(2.36a)

122

2. Experimental tests

ICCR ¼

Nu; DCFST Nu; CFDST

(2.36b)

where Nu; DCFST , Nu; CFST , and Nu; CFDST are the ultimate axial strengths of the DCFST and the corresponding CFST and CFDST columns, respectively. To assess the effects of steel tube configurations and fill the inner steel tube with the concrete, the behavior of slender DCFST columns with the length of 3315 mm under concentric axial compression is discussed in this section [64]. Fig. 2.71 [64] shows CSCR and ICCR values for different configurations of steel tubes and concrete compressive strengths, where the geometric and material properties of specimens are given in Tables 2.1. In Fig. 2.71, the dark gray concrete means HSC, whereas the light gray concrete indicates NSC. It can be observed from Fig. 2.71a that in the thin-thick configuration, the values of CSCR are less than unity when NSC is used as the sandwiched (outer or ring) concrete. Therefore, the configuration of thin outer tubeeNSCethick outer steel tube does not improve the behavior of the classical CFST column. However, filling the inner steel tube with concrete increases CSCR, especially when UHSC is used for the concrete core (inner concrete). As shown in Fig. 2.71a, the thick-thin configuration can highly improve CSCR values, indicating that the thickness of the outer tube is more important than the inner tube. Concerning the use of UHSC as the sandwiched concrete, both configurations of steel tubes show almost the same CSCR values. This indicates that ultra-high-strength sandwiched concrete dominates the mechanical behavior of these columns. Compared with the case when HSC is used for the concrete core, values of CSCR are much higher when UHSC is employed for the ring concrete. Another point is that the value of CSCR in CFDST columns with UHSC as the sandwiched concrete is close to unity. This shows that the current columns have almost the same ultimate axial strength as the benchmark

FIGURE 2.71

The effects of steel tubes and the concrete compressive strength configurations on CSCR and ICCR values [64]. (a) CSCR, (b) ICCR.

TABLE 2.1 Geometric and material properties of examined steel-concrete composite columns for investigating CSCR and ICCR [64]. Label

Do 3 to (mm)

fyo (MPa)

fc;0 e (MPa)

di 3 ti (mm)

fyi (MPa)

fc;0 i (MPa)

1

1e1

200 3

300

36

114.3 8

377

0

1e2

200 3

332

45

114.3 8

403

42

1e3

200 3

272

43

114.3 8

414

134

1e4

200 6

407

35

114.3 3

343

0

1e5

200 6

377

44

114.3 3

329

40

1e6

200 6

386

43

114.3 3

343

123

2e1

200 3

300

138

114.3 8

377

0

2e2

200 3

332

139

114.3 8

403

43

2e3

200 3

272

142

114.3 8

414

140

2e4

200 6

407

137

114.3 3

343

0

2e5

200 6

377

137

114.3 3

329

45

2e6

200 6

386

146

114.3 3

343

140

2

2.12 The role of the inner steel tube

Series

CFST specimens 1

1

193.7 8

444

40

e

e

e

2

2

193.7 8

444

142

e

e

e

123

124

2. Experimental tests

CFST column but with less self-weight, therefore, a better s=w ratio. As shown in Fig. 2.71b, series one specimens have higher ICCR values than series two specimens, meaning that the use of concrete core (inner concrete) is more efficient for CFDST columns having NSC as the sandwiched concrete. Concerning the steel tubes configuration, employing the thickthin configuration leads to higher ICCR values than the thin-thick configuration. By contrast, the concrete core contribution is less significant for thick-thin columns fabricated with ultra-high-strength sandwiched concrete. This again reinforces the fact that the outer part of the section governs the behavior of the DCFST column, particularly when a thick-walled outer steel tube is used. As a general trend, adding the inner steel tube to the conventional CFST column enhances the ultimate axial strength of the column. This is due to two reasons. The first reason is that the total cross-sectional area of the steel tube is increased by adding the inner tube. The second reason is the additional confinement effect provided by the inner steel tube. Besides, due to the extra confinement effect, the ductility and the residual strength of the DCFST column are better than the CFST counterpart. However, the difference between the initial axial stiffness of CFST and DCFST columns is negligible, indicating that, similar to the conventional CFST column, the DCFST column has excellent resistance to elastic deformation. Compared with CFDST columns, the ultimate axial strength and initial axial stiffness of DCFST columns are remarkably higher due to the presence of the concrete core. Concerning the ductility, the DCFST column presents the best result compared with CFST and CFDST columns. The geometric material properties of the inner tube affect the loadbearing capacities of DCFST columns. For instance, the ultimate axial strength of the column enhances as the width of the inner tube increases. This is because increasing the width of the inner tube reduces the crosssectional area of the sandwich concrete but increases the cross-sectional area of the inner tube and the concrete core. Additionally, more area of the concrete core is confined by using a larger inner steel tube. Contrary to the ultimate axial strength, the width of the inner steel tube has negligible influences on the axial stiffness and the ductility of DCFST columns. The same effects can be drawn regarding the influence of the inner steel tube thickness on the compressive capacity of DCFST columns. However, the effect of the inner tubes width on the compressive capacity of CFST columns is more significant than that of its thickness. Besides, the ductility of DCFST columns is enhanced by increasing the thickness of the inner tube. The effect of the dimensions and material properties of a circular inner tube on the behavior of DCFST columns is greater than that of a square inner tube. Concerning the material strengths, increasing the compressive strength of the sandwiched concrete and concrete core can significantly improve the column strength but declines the ductility. However, the

2.13 Material imperfections

125

ultimate axial strength and the ductility of DCFST columns are more dominated by the properties of the sandwiched concrete. A similar trend is true for the yield stresses of the steel tubes. The load-bearing capacity and the ductility of DCFST columns improve as the steel tubes’ yield strengths increases. However, the influences of the outer tube yield stress on the behavior of the column are more than the yield stress of the inner tube. The difference between the effects of outer and inner steel tubes yield strengths, and the sandwiched concrete and the concrete core compressive strengths on the load-bearing capacities become apparent in DCFST columns subjected to eccentric compression, especially when the slenderness ratio of the column is large. It can be concluded from the above discussions that DCFST columns possess better structural performance, such as higher load-bearing capacity and ductility than the conventional CFST columns. High-strength steel tubes can be used to fabricate the inner steel tube to further enhance the capacity of the column. In addition to the positive effects of the inner steel tube on the behavior of DCFST columns, using them instead of steel reinforcing bars can accelerate the construction.

2.13 Material imperfections 2.13.1 Geometrical imperfections Initial geometric imperfections mean the deviation of a member from ‘ideal’ geometry. Geometric imperfections are an inevitable characteristic of steel members that may come from manufacturing, fabrication, welding, and transportation, affecting the stability and load-bearing capacity of hollow steel sections. Different imperfections can occur in hollow steel sections, such as twisting, bowing, warping, and local deviations. Local deviations are described by indent and regular undulations in the plate. Depending on the length of the member, two types of imperfections can generally occur: (1) local imperfections in short members, (2) global imperfections in long members. The initial geometric imperfections can remarkably reduce the buckling strength and stiffness of steel plates. In other words, local imperfections cause the local buckling in hollow steel tubes to begin at a load level smaller than the theoretical buckling strength, especially in thin-walled steel sections. The influences of initial geometric imperfections reduce with increasing the wall thickness of steel tubes. The distribution of the initial geometric imperfections over the cross section of the steel plate is random. However, various parameters can affect the distribution of imperfections, i.e., rolling, fabrication, construction, material strength, and geometric properties. Concerning CFST/ CFDST stub columns, the initial imperfections generally reduce the confinement effect provided by the steel tube and the load-bearing

126

2. Experimental tests

capacity of the stub column. However, the effect of initial imperfections on the behavior of CFST/CFDST stub columns is less significant than that of hollow steel sections. The reason lies in the fact that the presence of the in-filled concrete postpones the tube local buckling. Additionally, most of the compressive capacity of CFST/CFDST stub columns is governed by its in-filled concrete. Unlike stub columns, the initial geometric imperfection is one of the factors that can remarkably influence the behavior of slender columns. The overall geometric imperfections can cause load eccentricity, which can become of importance in slender columns. One of the key test parameters is loading eccentricity (e), which strongly influences the behavior of composite beam-columns. Accurate measurement of the load eccentricity is an important step in the experimental test so that the loadbearing capacity of the member achieved from the test can be compared directly with the design strengths determined from the measured loading eccentricity since the actual initial loading eccentricity may be different from the intended amount of the eccentricity. Fig. 2.72 shows a slender composite column under eccentric compression with the overall geometric imperfections of d at its midlength. It can be observed from the figure that the overall geometric imperfections d increases the eccentricity of the applied load. The imperfection of members will cause the P  d moment,

FIGURE 2.72

Slender column subjected to eccentric axial compression.

127

2.13 Material imperfections

leading to the instability problem. Therefore, ignoring the effect of overall geometric imperfections on the behavior of slender CFST/CFDST columns can lead to inaccurate predictions of the behavior of members. A solution to estimate the actual initial loading eccentricity about the buckling axis is to induce a relatively small elastic load on the column subjected to a specific nominal eccentricity. The applied elastic load typically has a magnitude of 10%e15% of the estimated failure load ðNu Þ. After applying the elastic load, the maximum and minimum strains from two strain gauges attached to the extreme fibers of the cross section at the midheight are recorded. Besides, the overall deflection achieved from the LVDT placed at the midheight of the column is needed. The recorded longitudinal strains at two extreme fibers of the steel tube consist of two ðεmax þ εmin Þ and parts: (1) strains emerged by the compression load εc ¼ 2 ðεmax  εmin Þ (2) strains emerged by the bending moment εm ¼ , where 2 εmax and εmin denote respectively the maximum and minimum longitudinal strains recorded from two strain gauges at the extreme fibers. εm is developed because of the initial loading eccentricity, initial midheight global geometric imperfection, and midheight lateral deflection). Based on statics and EulereBernoulli beam theory, the actual bending moment at the midheight of the column is equal to ðEIÞeff $k, where k ¼ εm=0:5D refers to the curvature at midheight, in which D is the relevant outer dimension of the cross section. It is also known that the bending moment at the midheight of the column is equal to Pðd þ eÞ þ D. Consequently, the actual initial load eccentricity is governed by: ðEIÞeff ðεmax  εmin Þ Dd (2.37) ND where N is the applied load, and other parameters have been introduced above. Assume that the goal is to apply the load eccentricity of L 1000. If the measured ðd þ eÞ be larger than the desired eccentricity L 1000, the position of the column must be readjusted, following by repeating the preloading procedures. This trial and error process must be repeated until reaching ðd þ eÞ ¼ L 1000. Upon fulfillment of the alignment of the column, the final test can be conducted. The failure moment ðMu Þ, inelastic moment ðMi Þ, and second-order elastic moment ðMe Þ can be calculated by knowing the overall geometric imperfection at midlength of the column ðdÞ, initial load eccentricity ðeÞ, the overall horizontal deflection at midlength of the column at peak load ðDu Þ, the ultimate axial strength ðNu Þ, and the elastic buckling load ðNe Þ of the column. e¼

=

=

=

Mu ¼ Nu ðe þ dÞ

(2.38)

128

2. Experimental tests

Mi ¼ Nu ðe þ d þ Du Þ Me ¼

Mu

1 þ Nu=N

(2.39) (2.40)

e

2.13.2 Corrosion of steel tubes CFST/CFDST members are among the most useful structural members for constructing coastal infrastructure, offshore structures, bridges, and tower structures. Thus, their steel tubes are exposed to extreme environmental conditions, making them vulnerable to environmental corrosions during the service life span. In addition to the marine corrosive chloride environment, acid rain attacks can corrode steel tubes. Acid rain is among one of the most problematic environmental concerns worldwide due to industrial and urban development. The wall thickness of the steel tube is reduced by corrosion. Consequently, corrosion diminishes the buckling strength of hollow steel sections. Although filling hollow sections with concrete reduces the corrosion effects on the overall performance of the members, corrosion of steel tubes cannot be ignored in the design and seismic rehabilitation of CFST/CFDST members. This is because the corrosion of the steel tube will decrease the structural size, and therefore, reduce the resistance of the tube. Besides, the composite action between the steel tubes and the in-filled concrete will also be affected. Both issues deteriorate the structural performance compared with the intact composite member. Therefore, researchers have aimed to assess the performance of steel-concrete composite members subjected to different types of loadings and chloride corrosion conditions. International design codes for the design of steel structures provide guidance on protecting marine steel structures from corrosion to assure that structures can perform well during 50 years’ service life. Table 2.2 summarizes the guidance recommended by design codes worldwide. In general, most design codes recommend using protective coatings such as paints, sacrificial coatings, galvanizing, or electrochemical protection like impressed current cathodic protection. However, the recommended methods are not always efficient, especially for long-term horizons and when the operational or maintenance circumstances are less than ideal. Consequently, some structural engineers consider sacrificial corrosion for designing steel structures exposed to marine environments [73]. In this method, the designer predicts the future possible corrosion extension and allows reasonable steel corrosion to be developed at a constant rate. Metal corrosion can be classified into (1) uniform corrosion and (2) local corrosion. The effects of each of them on the behavior of CFST/ CFDST members are discussed in the following sections.

2.13 Material imperfections

129

TABLE 2.2 Summary of methods recommended by design codes for corrosion protection [65]. Country

Code

Methods

Australia

AS 4100-1998 [66]

Protection methods; Painting of steelwork; Protection of members’ corrosion protection.

Britain

BS EN 1090e92:2008 [67]

Thermal spraying; Hot-dip galvanized coatings; Paints; Varnishes; Zinc and aluminum

Canada

CAN/CSA-S16-01 [68]

Proper alloying elements; Protective coatings; Other effective means.

China

GB 50017-2003 [69]

Use of weathering steel; Paint systems.

Germany

DIN 18800-1 [70]

Allowance for corrosion; Coatings; Cathodic corrosion protection; Use of suitable nonrusting materials; Covering with suitable materials.

Hong Kong, China

Code of practice for the structural use of steel [71]

Paint systems; Galvanizing; Use of a corrosion resistant alloy; Use of a sacrificial corrosion allowance (As a guide, an allowance of 0.25e0.5 mm per year may be considered); Use of cathodic protection.

USA

ANSI/AISC 360-05 [72]

Use of a specific amount of tolerance; Protection systems (for instance, coatings, cathodic protection); Maintenance programs.

2.13.2.1 Uniform corrosion Typically, long-term sustained load conditions are used to investigate the behavior of CFST/CFDST members in chloride environments. The performance of steel-concrete composite members under long-term sustained loading conditions is discussed individually in Section 2.14. Specimens are tested in two stages. Phase one of the experiments simulates the combined events of long-term sustained loading conditions and chloride corrosion. During this stage, sustained loads ðNl Þ are initially imposed on members. The imposed sustained load Nl is the ratio of the

130

2. Experimental tests

ultimate axial strength of the column subjected to short-term loading conditions ðNu Þ. This ratio is named the long-term sustained load ratio, N and it is governed by m ¼ l . The same long-term sustained load ratio Nu M can be introduced for beams as g ¼ l in which Ml is the applied Mu moment to the beam during the corrosion test, and Mu represents the flexural strength of the beam under short-term loading conditions. In general, the effect of the long-term sustained load is more prominent on the axial strength of composite members than their flexural strength. The reason is the difference in stress distribution in the in-filled concrete of the beam compared to the column. When the CFST/CFDST stub column is under concentric axial compression, the entire cross section of the core concrete is under compression only. By contrast, part of the in-filled concrete of the CFST/CFDST beam is under compression, while the rest is under tension. Therefore, the concrete of the beam has a smaller compressive area than the column. Test setups for applying long-term sustained loads on CFST/CFDST columns and beams are displayed in Figs. 2.73 and 2.74, respectively. The magnitude of the sustained load is measured by load transducers. Besides, strain and deformation of specimens are recorded using strain gauges and LVDTs, respectively. Specimens subjected to constant sustained loads are immersed inside the sodium chloride (NaCl) water tank to perform accelerated corrosion tests. Typically, between 3% and 4% NaCl by the water weight is utilized as an electrolytic solution. For making a path for the corrosion flow, the anode of the power source is connected to the steel tube, while the cathode is connected to the NaCl water tank. The ends of specimens must be coated

LVDT

High-strength bolt Load transducer

N NaCl solution Strain gauges

NaCl water tank Loading rod

N

Power source

FIGURE 2.73 Test setup for applying long-term sustained load conditions on CFST/CFDST stub columns immersed in corrosive solution [65].

2.13 Material imperfections

LVDT

N

131

Spread beam

NaCl solution

Strain gauges

NaCl water tank

Power source

FIGURE 2.74 Test setup for applying long-term sustained load conditions on CFST/CFDST beams immersed in corrosive solution [65].

with epoxy to protect them from chloride ions. This is to ensure that only the external surface of the steel tube is corroded. Besides, the vertical distance between the water tank and each steel tube surface must be the same so that the corrosion flow can develop consistent results over the outer surface of the specimen. The amount of NaCl solution and the water temperature must remain constant throughout the test. If the PH of the solution is between 4 and 10, it does not play a significant role in the rate of immersion corrosion [74]. However, it is recommended that the average PH of the utilized NaCl solution be close to the PH of seawater (between 8 and 8.3) to simulate the corrosion process of offshore structures with better accuracy. Based on ASTM G 31e72 [75], the corrosion solution should be replaced every 20 days during the test to prevent the unwanted effect of the corrosion products. At the same time, the specimens must be washed to be cleaned of any corrosion products and rust. The current test setup only concerns the corrosion loss of the outer surface of CFST/CFDST members. Generally, the effect of chloride ions on the inner surface of steel tubes and in-filled concrete is moderate. The reason is that the spread of chloride ions is limited according to the current examination methods (there are no NaCl solution exchanges between the inner and outer surfaces of the steel tube). On the other hand, the infilled concrete of CFST/CFDST members in which no steel reinforcing bars are used is typically less prone to chloride. NaCl solution has been widely employed in the examination of reinforcement corrosion, as well as other issues concerning structural corrosion. However, using NaCl solution for the simulation of the corrosion conditions

132

2. Experimental tests

of the seawater is insufficient. Also, the results presented by the accelerated corrosion testing method are less accurate than field corrosion observations. This is because the natural environment of seawater cannot be accurately simulated using accelerated corrosion tests since the corrosion rate in the real seawater environment is quite different. The most accurate test method is the in-field test condition. However, in-field tests have low corrosion rates, which makes them time consuming. Besides, in-field tests are generally expensive. Also, the corrosion rate is not an important factor because the main concern is to simulate the uniform thickness loss made by corrosion. The accelerated corrosion testing method using NaCl solution is faster and less expensive than the field corrosion observation. Besides, it can provide prompt solutions for construction projects. The test setups introduced above can effectively lead to the uniform corrosion of steel tubes in CFST/CFDST members and prepare them to be examined under uniform corrosion distribution without simulating all the in-field conditions. NaCl solution has been widely employed in the examination of reinforcement corrosion, as well as other issues concerning structural corrosion. However, using NaCl solution for the simulation of corrosion conditions of seawater is insufficient. Also, the results presented by the accelerated corrosion testing method are less accurate than field corrosion observations. This is because the natural environment of seawater cannot be accurately simulated using accelerated corrosion tests since the corrosion rate in the real seawater environment is quite different. The most accurate test method is the in-field test condition. However, in-field tests have low corrosion rates, which makes them time consuming. Besides, in-field tests are generally expensive. Also, the corrosion rate is not an important factor because the main concern is to simulate the uniform thickness loss made by corrosion. The accelerated corrosion testing method using NaCl solution is faster and less expensive than the field corrosion observation. Besides, it can provide prompt solutions for construction projects. The test setups introduced above can effectively lead to the uniform corrosion of steel tubes in CFST/CFDST members and prepare them to be examined under uniform corrosion distribution without simulating all the in-field conditions but with acceptable accuracy. The loss of wall thickness from the outer surface of the steel tube due to corrosion is called the designed corrosion depth. Two methods can be used to verify the desirable designed corrosion depth in the experimental test. The first method is to measure the weight of the tube, and the second method is to control the diameter of the tube. Usually, the strain and deformation magnitudes because of the long-term sustained load effects in CFST/CFDST members tend to stabilize after around 100 days. This issue is discussed in detail in Section 2.14. Therefore, Stage one of the test should last for at least 100 days. To ensure the full implementation of preloading of members, they can remain under Stage one conditions for

2.13 Material imperfections

133

120 days. During the first phase, the applied load and the corrosion flow must remain constant, and the values of strains and deformations must be recorded. The second step can be started after stabilizing strains and deformations values (after 100e120 days). The corrosion test setup must be removed to start the second step, while the sustained axial load must still be kept constant. Outer surfaces of specimens must be cleaned before attaching new strain gauges to them. Next, specimens must be loaded until failure to define their load-bearing capacity. Applying long-term sustained load to CFST/CFDST columns develops long-term sustained deformation in columns ðDlc Þ over time. However, the magnitude of Dlc over the testing time for specimens subjected under corrosion conditions grows faster than the counterpart under normal conditions. Besides, the corrosion conditions increase the magnitude of Dlc . The same results are reasonable for the effects of long-term sustained load and chloride corrosion on the midspan deflection ðumlc Þ of CFST/ CFDST beams. Corrosion conditions have a moderate influence on the failure mechanism of CFST/CFDST members. In other words, it can be expected that the failure modes of corroded CFST/CFDST members are similar to their intact counterparts. For instance, local outward buckling and the crushing of the concrete core adjacent to the steel tube buckled area may occur in corroded square CFST stub column under sustained load. Corrosion conditions can remarkably affect the load-displacement histories of CFST/CFDST members. In general, initial axial stiffness, flexural stiffness, load-bearing capacities, ductility, residual strength, and energy absorption capacity of members are diminished by corrosion of the steel tube. The effects of corrosion on the behavior of CFST/CFDST members can be discussed from the confinement coefficient x aspect. Under chloride corrosions, the wall thickness, and hence, the crosssectional area of the steel tube is reduced, leading to the reduction of the confinement coefficient. Consequently, the confinement effect provided by the steel tube is reduced by the corrosion. The reduction of the composite action and the cross-sectional area of the steel tube are the reasons for diminishing the structural performance of CFST/CFDST members subjected to corrosion. Under the same corrosion conditions, the magnitude of load applied to the column can affect the impacts of the corrosion environment on CFST/CFDST members. Suppose that two CFST stub columns with the same geometric and material properties are subjected to the long-term sustained load ratios of m1 and m2 ðm1 > m2 Þ, respectively. Both columns are under the same corrosion conditions. Therefore, they both experiences the same steel wall thickness loss ðDts1 ¼ Dts2 Þ. Due to the difference between the applied long-term sustained load ðNl1 > Nl2 Þ, the first column experiences a more significant loss of ultimate axial strength than the second column.

134

2. Experimental tests

Loss of strength due to corrosion of hollow steel sections is more remarkable than their counterparts filled with concrete. This is because part of the load, which was initially carried by the intact steel tube, will be transferred to the in-filled concrete by the corrosion of the tube. Therefore, filling the hollow steel sections can reduce the effects of corrosion on the load-bearing capacities of CFST/CFDST members. Corrosion deteriorates the structural performance of both circular and square CFST/CFDST columns but with different intensities. The influence of corrosion on the ultimate axial strength of corroded CFST/CFDST columns depends on the geometric properties of the section. Suppose two circular and square CFST stub columns with the same steel ratio, cross-sectional area, and material properties. The ultimate axial strength of the circular CFST column will be higher than that of square one. This is because the confinement effect provided by the circular tube is higher than square one. It is reasonable to conclude that the steel-concrete interaction force reduction in corroded circular CFST column will be more significant than square one under similar corrosion conditions. Consequently, reducing the load-bearing capacity in the circular CFST column will be more significant than square one. That is to say that circular CFST/CFDST members are more sensitive to corrosion environments than square ones. Similar to the geometric properties, the material strengths can also affect the influence of corrosion on the ultimate axial strength of corroded CFST/CFDST columns. The effect of corrosion on the compressive capacity of the CFST/CFDST column can be represented by defining the residual axial strength ratio RSc as follows: Nuc (2.41) Nu where Nuc is the compressive capacity of the corroded CFST/CFDST column and Nu is the ultimate axial strength of the noncorroded counterpart. The influence of corrosion on RSc increases as the steel yield strength of the steel tube and steel ratio increases. By contrast, increasing the concrete compressive strength reduces the influence of corrosion on RSc . The reason lies in the fact that the contribution of the steel tube to the load-bearing capacity increases with increasing the steel ratio and steel yield strength or reducing the concrete compressive strength. Therefore, the ultimate strength of the column will be more sensitive to the steel wall thickness loss. Same as the residual axial strength ratio RSc , the residual flexural ratio RSm can be defined for assessing the influence of corrosion on the flexural capacity of the CFST/CFDST beam: RSc ¼

Muc (2.42) Mu where Muc represents the flexural strength of corroded CFST/CFDST beam load and corrosion, and Mu is the flexural strength of the noncorroded RSm ¼

2.13 Material imperfections

135

counterpart. Contrary to the CFST/CFDST columns, material strengths have a negligible effect on RSm . Although corrosion of steel tubes diminishes load-bearing capacities of CFST/CFDST members, bending moment capacity is more sensitive to corrosion than compressive capacity. The reason is the difference in the distribution of load between constituent components of CFST/CFDST members. As discussed before, most of the load is carried by the in-filled concrete, while the contribution of the steel tubes to the load-bearing capacity is remarkably less than the concrete. Therefore, steel tubes are not the dominant component in bearing the axial load, and their corrosion leads to a relatively small loss of axial strength of the column. The behavior of CFST/CFDST beams is quite different from CFST/CFDST columns. The contribution of the concrete core in bearing the sectional moment is significantly less than that of the steel tubes due to its low tensile strength. Therefore, contrary to the composite columns in which the concrete core is the dominant component in bearing the axial load, the steel tubes are the governing elements of the flexural strength of composite beams. Approximately almost 66%e75% of the sectional moment is carried by steel tubes. Hence, the corrosion of steel tubes has a more substantial effect on the bending capacity than the compressive strength of CFST/ CFDST members. 2.13.2.2 Local corrosion Contrary to the uniform corrosion, the damaged area in steel tubes suffered from local corrosion is concentrated in a specific part of the steel surface. Although uniform corrosion spreads over a larger area, local corrosion can induce more detrimental effects on the performance of CFST/ CFDST members because it generates asymmetric cross-sectional properties, leading to the eccentric loading condition while jeopardizing the confinement on the in-filled concrete. Additionally, steel plates are more prone to local corrosion than uniform corrosion [76]. More than 80% of the reported corrosion accidents belong to local corrosion [77]. Different types of local corrosion can be named, such as pitting corrosion and through thickness pitting corrosion which can be caused by the degradation of the paint coating surface or grooving corrosion that usually occurs at steel welds or in areas where the anticorrosion coating failed. Fig. 2.75 shows the local corrosion of steel plates. In the experimental tests, artificial notches throughout the steel tube wall thickness are used to represent the material imperfections from local grooving corrosion. These notches can have various lengths, depths, widths, and orientation angles. The artificial notches can be developed using the Computerized Numerical Control (CNC) machine. Spark technique can also be used for creating notches with high accuracy with the least imperfections. Creating a rectangular notch can cause sharp discontinuities

136

2. Experimental tests

(a)

(b)

(c)

FIGURE 2.75 Local corrosion of steel members [77e79]. (a) Pitting corrosion, (b) Through thickness pitting corrosion, (c) Grooving corrosion.

in the steel tube, making stress concentration throughout the notch corners. This issue affects the overall behavior of the steel tube, and hence, the CFST/CFDST member. Therefore, it is recommended to design the ends of the notch in an arch. For preventing concrete leakage while pouring it, notches should be filled with polyvinyl chloride (PVC) boards. The behavior of CFST/CFDST columns with the notch in steel tubes depends on the orientation, length, depth, and width of the notches. Fig. 2.76 shows the failure modes of CFST stub columns with different notch orientations ranging from q ¼ 0 to 90 . It can be observed from Fig. 2.76a and b that steel tubes are ruptured at the notch location with orientations of q ¼ 60 and 90 . Besides, two symmetrical cracks are developed in the steel tube at the ends of notches. By contrast, no cracks are generated in steel tubes of columns with notch orientations of q ¼ 0 and 30 . Also, the width of notches is reduced by increasing the axial compression, and they are closed at the end of the loading, as indicated in Fig. 2.76c and d. Regardless of the orientation angle of the notch, local buckling can be found in the steel tubes at the ends of notches. Fig. 2.77 shows the concrete failure modes of circular columns with the notch on the tube. The concrete crushes near the notches with the

137

2.13 Material imperfections

Crack

Crack

(a)

(b)

Notch Notch

(c)

(d)

FIGURE 2.76 Failure modes of circular CFST stub columns with different notch orientations [80]. (a) q ¼ 60 , (b) q ¼ 90 , (c) q ¼ 0 , (d) q ¼ 30 .

orientation angles of 0 and 30 . In columns with orientation angles of 60 and 90 , the concrete is crushed up around the notches with an inclined path. The shear failure mode, shown in Fig. 2.77c and d, represents that increasing the notch degree reduces the confinement effect provided by the steel tube.

138

2. Experimental tests

Damaged concrete

(a)

Damaged concrete

Damaged concrete

(b)

(c)

FIGURE 2.77 Failure modes of concrete core in CFST stub columns with different notch orientations [80]. (a) q ¼ 0 , (b) q ¼ 60 , (c) q ¼ 90 .

Increasing the depth, length, or width of the notch worsens the failure of steel tubes and the in-filled concrete in CFST/CFDST columns. For instance, the failure modes of circular CFST stub columns with a vertical notch having the same length and width but different depths are shown in Fig. 2.78. Both columns have the same geometric and material properties. As shown in Fig. 2.78, both steel tubes are ruptured and display local buckling at the notch location. Besides, overall buckling can be seen in both columns due to the pin-pin boundary conditions. Because of the rupture and local buckling of the steel tube, the concrete core adjacent to the notch loses the confinement effect provided by the steel tube, leading to the crushing of the concrete. Compared with the column with a shallower notch, the length and the width of the ruptured area in the steel tube with the deeper notch are longer, leading to a more severe crushing of the concrete in this specimen. The failure mechanisms of square CFST stub columns with horizontal and vertical notches are shown in Fig. 2.79. Like circular CFST/CFDST columns, local buckling occurs at the ends of the horizontal notch in square CFST/CFDST columns, while the notch is closed under axial compression, as shown in Fig. 2.79a and b. The outward buckling happens in the middle of the vertical notch, as shown in Fig. 2.79c and d. Fig. 2.80 shows the failure mechanism of the in-filled concrete in square CFST/CFDST stub columns with horizontal and vertical notches. It can be observed that the concrete core is crushed near the notch area. The failure mode of the square CFST stub column with an oblique notch ðq ¼ 45 Þ is displayed in Fig. 2.81. The specimen has pin-pin boundary conditions, and it is subjected to concentric axial compression. It can be observed from the figure that local buckling occurs at the ends of the notch.

139

2.13 Material imperfections

(1) The 1st side

(2) The 2st side

(3) The 3rd side

(4) The 4th side

(3) The 3rd side

(4) The 4th side

(a)

(1) The 1st side

(2) The 2st side (b)

FIGURE 2.78 The effects of the notch depth on the failure mode of circular CFST stub columns [77]. (a) Depth of the notch ¼ 4 mm, (b) Depth of the notch ¼ 3 mm.

Notch

Notch

Notch

(a)

(b)

(c)

(d)

FIGURE 2.79 Failure modes of square CFST stub columns with horizontal and vertical notches [81]. (a) Horizontal notch on the side wall, (b) Horizontal notch on the corner, (c) Vertical notch on the side wall, (d) Vertical notch on the corner.

140

2. Experimental tests

Damaged concrete

Damaged concrete Notch N

Notch

Notch

FIGURE 2.80

Damaged concrete

Notch

Damaged concrete

Failure modes of square CFST stub columns with horizontal and vertical

notches [81].

(1) The 1st side

FIGURE 2.81

(2) The 2st side

(3) The 3rd side

(4) The 4th side

The failure mode of the square CFSTstub column with an oblique notch [82].

Besides, outward local buckling happens on the other surfaces of the column. Due to the orientation of the notch, the concrete core fails in shear. Fig. 2.82 compares the failure modes of the square CFST stub column with a vertical notch with the intact counterpart under the concentric axial compression with pin-pin end conditions. As shown in Fig. 2.82a,

141

2.13 Material imperfections

(1) The 1st side

(2) The 2st side

(3) The 3rd side

(4) The 4th side

(3) The 3rd side

(4) The 4th side

(a)

(1) The 1st side

(2) The 2st side (b)

FIGURE 2.82

Comparison of the failure modes in intact and imperfect square CFST stub columns [82]. (a) Intact square CFST stub column, (b) Square CFST stub column with vertical notch.

local buckling occurs in the bottom of the intact steel tube. Similarly, the local buckling happens in the steel tube with the vertical notch, as shown in Fig. 2.82b. In addition, the steel tube bulges at the location of the vertical notch. Comparison of the intact specimen with the one having a vertical notch shows that the existence of the notch alters the failure mode of the square CFST column. Steel plates of the square steel plate have a simply supported boundary condition. In contrast, at the area of the steel plate with the notch, the previous boundary condition is changed to freesupported at one end. The development of the local buckling around the notch is because part of the steel tube near the notch is unable to provide circumferential stress, and the steel tube deformation cannot be restricted.

142

2. Experimental tests

Notch characteristics, including the length, depth, width, and orientation, can affect the axial loadedisplacement histories of CFST/CFDST stub columns. In general, the ultimate axial strength of the column is reduced by increasing the length of the notch while other geometric and material properties of the column are constant. However, the intensity of the notch length on the behavior of columns depends on its orientation. Compared with the columns having sloping notches, the effect of the notch length on the compressive capacity of the column is less notable in the columns with horizontal notches. The reason is that a longer sloping notch in the column generates a more extended tube segment that produces the inadequate confinement effect for the in-filled concrete, leading to a lesser peak resistance in the column. By contrast, the length of the tube segment generating inadequate confinement on the in-filled concrete does not change by the length of the horizontal notch. Consequently, the ultimate axial strength of the column having the horizontal notch is less sensitive to the length of the notch. Regarding the postpeak regions of columns, the ductility of the intact column and the counterpart with the horizontal notch is almost the same. The effect of the slanted notch on the ductility of the column depends on the length, depth, and orientation of the notch and the wall thickness of the steel tube. If the depth and the orientation of the notch are not large enough or the steel tube is stocky, the existence of the slanted notch cannot diminish the ductility. Otherwise, the slanted notch can lead to shear failure in the column, which is known as a brittle response. Regarding the vertical notch, both the ultimate axial strength and the ductility of columns are diminished by increasing the notch length. This is because a longer vertical notch corresponds to a more extended part in the tube that generates an inadequate confinement effect on the in-filled concrete, jeopardizing both the compressive capacity and ductility of columns. Similar to the length of the vertical notch, its depth has significant effects on the behavior of columns. Both the peak resistance and ductility reduce as the depth of the vertical notch increases. It is obvious that the thickness of the steel tube is reduced by increasing the depth of the notch, leading to providing a less efficient confinement action of the in-filled concrete. Contrary to the length and depth of the vertical notch, its width has negligible influence on both the peak resistance and ductility of columns. Suppose there are three circular CFST stub columns with the same geometric and material properties. The length, width, and depth of notches on the steel tubes are the same, while their orientations are vertical ðq ¼ 90 Þ, slanted ðq ¼ 45 Þ, and horizontal ðq ¼ 0 Þ. The peak resistance and the ductility are enhanced by changing the angle of the notch from vertical to horizontal, while other parameters of the notch remaining the same. The reason can be find by comparing the effect of the notch angle on

2.13 Material imperfections

143

the failure mechanism of columns, presented in Figs. 2.76 and 2.77. The buckling of the steel tube and the crushing of the concrete core are intensified by increasing the notch angle.

2.13.3 Concrete core imperfections 2.13.3.1 Compaction of the concrete core Another possible material imperfection in CFST/CFDST members is concrete imperfections. In general, three types of concrete imperfections are likely to occur in CFST/CFDST members. The first imperfection of the concrete is due to insufficient compaction of the in-filled concrete. One of the challenges in the construction of CFST/CFDST members is to assure proper concrete compaction. Like reinforced concrete members, compaction of the concrete in steel-composite members affects the mechanical properties of the concrete itself. Besides, the sufficient bond strength and composite action between the in-filled concrete and steel tubes highly depend on the concrete compaction. This is because appropriate compaction enhances microlocking and macrolocking in the in-filled concrete, leading to the better performance of composite members. Compaction of the in-filled concrete can be done using two methods: the first method is compacting the concrete core by hand, and the second method is using mechanical vibration such as poker vibration. The experimental test results clarified that the poker vibrator method is more efficient than the other method. The method used for compaction of the in-filled concrete can alter the overall behavior of the composite member. Concerning the failure mode, local buckling of the steel tube in the member compacted by hand happens earlier and more severely than the counterpart compacted using the poker vibrator. Changing the buckling mode due to the compaction method have a notable impact on the load-bearing capacity of the column. Critical buckling loads are enlarged by excellent concrete compaction. The improved structural performance of the member in which the concrete is compacted using the poker vibrator is due to the full composite action between the constituent components of the member. By contrast, the full achievement of composite action is not expected for the member with the concrete infill compacted by hand. The compaction method of the in-filled concrete may only have little influence on the performance of the composite member during the initial loading stage (elastic stage). This is because no interaction between the in-filled concrete and the steel tubes is expected to happen at the elastic stage. By contrast, it can affect the load-bearing capacity, ductility, and residual strength of the member. The strength lost index ðSLIÞ can be defined as follows to assess the effect of uncompaction of the in-filled concrete on member strength loss.   Nu;P  Nu;H (2.43) SLI ¼ Nu;P

144

2. Experimental tests

where Nu; P and Nu; H are the column ultimate axial strengths compacted by poker vibration and by hand, respectively. The compaction effects may not be substantial if the dimension of the cross section is small, while it becomes evident upon increasing the crosssectional area of in-filled concrete. The reason is that the contribution of the in-filled concrete to the load-bearing capacity is increased by increasing its cross-sectional area. Moreover, the amount of SLI increases with increasing the column’s slenderness ratio and the load eccentricity ratio. In other words, the in-filled concrete becomes more sensitive to compaction by increasing the length and load eccentricity of the member. 2.13.3.2 Circumferential and spherical-cap gaps Columns

In real CFST/CFDST structures, two kinds of gaps may exist between the steel tubes and the in-filled concrete, i.e., circumferential gap and sphericalcap gap, as presented in Fig. 2.83. Typically, the concrete shrinkage causes the circumferential gap in the circumferential direction of vertical members such as columns and piers. Another possible reason for generating the circumferential gap is the temperature difference between the steel tube and the in-filled concrete. However, in real CFST/CFDST structures, the circumferential gap developed by the shrinkage of in-filled concrete is very small. Based on the experimental tests performed by Han [83], after 3 years, the shrinkage strain of the in-filled concrete in a circular CFST member was only 180 mε, meaning that the concrete shrinkage only generated a circumferential gap ratio of around 0.018%. The spherical-cap gap can be generated in horizontal members like beams, arch bridges, and truss structures during the construction process. The steel hollow sections of an arch or truss are typically fabricated and closed firstly, following by pumping the concrete inside them. Therefore, the existence of residual air in the in-filled concrete is possible. The residual air and the settlement of concrete can cause the spherical-cap gap at the top part of the concrete section.

FIGURE 2.83 Schematic view of concrete gaps in CFST members. (a) Spherical-cap gap, (b) Circumferential gap.

2.13 Material imperfections

145

The effects of the concrete gap on the performance of CFST/CFDST members can be assessed by the gap ratio parameter cgap governed by: 8 2tc > > for CFST with circumferential gap < D cgap ¼ > > : ts for CFST with spherical  cap gap D

(2.44)

where tc and ts are the thicknesses of the circumferential gap and spherical-cap gap, respectively, and D indicates the outer diameter of the steel tube. As discussed in Section 1.5.2, the typical failure mode of CFST/ CFDST stub columns without gaps is outward buckling, mainly at the midheight of the steel tube. In CFST/CFDST stub columns with the circumferential gap, the elephant-foot-like buckling at the top and bottom, along with the outward buckling at the midheight of the column, is the typical failure mechanism as displayed in Fig. 2.84. Compared with CFST/CFDST stub columns without gap or with the spherical-cap gap, the intensity of local failure in columns with the circumferential gap is more pronounced. The typical axial loadedisplacement history of CFST/ CFDST stub columns with a circumferential gap is illustrated in Fig. 2.85. Five characteristic points are also marked on the curve. The response of CFST/CFDST stub columns with the circumferential gap can be generally divided into five stages as follows: Stage 1: Ascending phase (from Point O to Point A): During this stage, the axial load keeps increasing until the ultimate axial strength of the column is obtained at Point A. Due to the existence of the gap and the difference between the Poisson’s ratios of the steel tube and the in-filled concrete, no interaction occurs between them in this stage. Stage 2: The crushing of the concrete (from Point A to Point B): After reaching the ultimate strength, the concrete is crushed, and hence, the load

Elephant foot like buckling

Crushed concrete

FIGURE 2.84 The failure mechanism of CFST columns with the circumferential gap [84].

146

2. Experimental tests

A

E C

Axial load

B

D

O

Displacement

FIGURE 2.85

Typical axial loadedisplacement response of CFST/CFDST columns with the circumferential gap.

is suddenly dropped to Point B. The load at Point B is approximately 20%e 30% of the ultimate axial strength. The cracking of the in-filled concrete is because of the lack of confinement effect caused by the circumferential gap. Due to the existence of the gap between the steel tube and the in-filled concrete, the proper composite action cannot be developed, leading to the brittle behavior of the in-filled concrete. Stage 3: Development of the confining pressure (from Point B to Point C): At this stage, the volume of the crushed in-filled concrete increases in lateral directions, and the interaction between the concrete and the steel tube happens. Therefore, the steel tube provides the confinement effect for the concrete. The provided confinement effect restricts the in-filled concrete and prevents any further crushing. Hence, the ductility and the strength of the concrete increase, leading to a gradual increase in the column’s axial loadedisplacement response. The elephant-foot-like-buckling of the steel tube occurs at this stage, as presented in Fig. 2.84. Stage 4: The second local crushing of the in-filled concrete (from Point C to Point D): By reaching Point C, the concrete usually experiences the second local crushing, which leads to another load drop in the axial loadedisplacement response of the column. It can be seen in Fig. 2.84 that the in-filled concrete is crushed at two points. One happens at Point A, and the other one occurs at Point C. Stage 5: The second contact between concrete and steel tube (from Point D to Point E): At this stage, the crushed concrete contacts the steel tube for the second time, and therefore, the axial load tends to develop again. The elephant-foot-like buckling at the ends of the column becomes more significant. Additionally, a slight outward local buckling may occur in the middle of the steel tube, while no inward buckling can be observed since the in-filled concrete can efficiently support the steel tube.

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147

It is noteworthy that the loads at Points C and E may be smaller or higher than at Point A, depending on the amount of cgap . However, the ultimate axial strength of CFST/CFDST columns with the circumferential gap in practice is considered the load at Point A, even if it is smaller than the loads at points C and E. The reason is that the deformations corresponding to points C and E are large and not permissible in real composite structures. Compared with a column without a gap, the ultimate axial strength and the ductility of the column with the circumferential gap are remarkably smaller, especially in the column with a large amount of cgap . This is because when the gap is large, the steel tube cannot efficiently confine the in-filled concrete since the tube may be prone to early local buckling due to the lack of contact between the tube and the in-filled concrete. In addition, the in-filled concrete may require further lateral expansion to contact the steel pipe. This issue postpones the generation of interaction between the tube and the concrete. Compared with the circumferential gap, the spherical-cap gap’s effects on the overall performance of CFST/CFDST members are more moderate. When the spherical-cap gap is large enough, a significant void is created between the concrete and the steel tube. Therefore, the steel tube around the gap area suffers from a lack of internal support, and both inward and outward buckling is likely to develop at the location of the gap. Besides, the in-filled concrete may be crushed in this area. By contrast, the failure mechanism of CFST/CFDST stub columns with a small spherical-cap gap is similar to their counterparts without a gap. In general, the overall shape of the column’s load-displacement response with the spherical-cap gap is similar to the intact counterpart. Although the existence of the sphericalcap gap reduces the load-bearing capacity and the ductility of CFST/ CFDST members, its effects are less than the circumferential gap. This is because most perimeter of the in-filled concrete in the column with the spherical-cap gap is in contact with the steel tube, and therefore, the confinement effect is more reliable than the column with the circumferential gap. Both the load-bearing capacity and the ductility of the column tend to reduce by increasing the gap ratio cgap . The reason is that a large gap reduces the confinement effect provided by the steel tube and the supporting effect of in-filled concrete on the steel tube. In general, the effect of the circumferential gap on reducing the ultimate axial strength of the column is more substantial than that of the spherical-cap gap. For comparing the ultimate axial strength of CFST/ CFDST columns with and without gaps, the strength index of the column ðSICÞ is defined as follows: SIC ¼

Nu; with gap Nu; without gap

(2.45)

where Nu; with gap and Nu; without gap are the load-bearing capacities of the columns with and without gaps, respectively. The amount of SIC for both

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types of gaps is sensitive to the cgap . Increasing the gap ratio cgap leads to the reduction of the column strength, and therefore, the SIC factor. Considering the same material and geometric properties for three columns, one without the gap, one with the spherical-cap gap, and one with the circumferential gap with the same cgap , it can be stated that ðSICÞcircumferential gap < ðSICÞsphericalcap gap . The reason is that in the column with the circumferential gap, no interaction happens between the tube and the concrete before reaching Point A. By contrast, the confinement effect occurs earlier in the column with the spherical-cap gap, and hence, the effect of the gap on the load-bearing capacity of the column is less than the counterpart with the circumferential gap. The behavior of CFST/CFDST stub columns with concrete imperfections using the nonlinear finite element analysis is furthered discussed in Chapter 4. Beams

The failure mechanism of CFST/CFDST beams with the gap is a halfsine wave deformation. However, the beam with the circumferential gap may show different deformation at the initial loading stages. This issue will be discussed later. In addition, outward buckling on the compression side of the beam is expected to happen. The outward local buckling of the tube becomes significant with increasing the gap ratio cgap . Concerning the failure modes of the in-filled concrete in CFST/CFDST beams with gaps, the concrete is crushed around the midsection of the compressive zone, while several cracks are developed along the longitudinal direction of the concrete in the tension zone. Compared with the beam with the spherical-cap gap, the tensile cracks in the concrete of the beam with the circumferential gap are less but more severe. For instance, the maximum width of the cracks developed in the tensile area of the concrete core in the circular CFST stub column with the circumferential gap is 17 mm, while the largest crack formed in the counterpart specimen with the sphericalcap gap is slightly more than 6 mm, as shown in Fig. 2.86. This is because the concrete core with the circumferential gap suffers from the lack of confinement during the initial loading stage, leading to considerable crack development. As discussed before, the final deformation of composite beams with and without the gap is a half-sine wave. Suppose a horizontal CFST beam with a circumferential gap is subjected to the bending moment using a four-point bending rig. At the initial stage of loading, the steel tube’s tension side presents an upward deformation, whereas the direction of the external load is downward. The existence of the circumferential gap allows the steel tube to perform without any support from the concrete core. This issue leads to the slight ovalization and flattening of the tube’s tension zone. Later, with the increasing bending moment, the concrete

149

2.13 Material imperfections Crushed concrete

Local buckling

Compression size

Tension side (a)

Crushed concrete

Local buckling

Compression size

Tension side (b)

FIGURE 2.86 Failure modes of concrete cores and steel tubes of circular CFSTstub columns with circumferential and spherical cap gaps [84]. (a) Circumferential gap cgap ¼ 2:2%, (b) Spherical-cap gap cgap ¼ 6:6%.

core contacts the tube, and therefore, the tube’s ovalization is supported, and the deformation of the tube’s tension zone starts to switch to the downward direction. Compared with the beam with the circumferential gap, the overall deformation of the beam with the spherical cap gap is almost similar to the counterpart beam without any gap during the whole loading process. By contrast, the deformation shape of the composite beam with the circumferential gap shows several features under different loading steps. In the initial loading stage, when the bending level is

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2. Experimental tests

Moment

relatively small, the ovalization of the steel tube deflects the beam deformation from the half-sine wave. However, the deformation of the beam coincides with the half-sine wave when the bending level is high. To further discuss the effects of gaps on the behavior of CFST/CFDST beams, the relationships between bending moment versus midspan deflection at the bottom of the steel tube of two circular CFST beams are presented in Fig. 2.87. The response of the corresponding circular hollow section without the concrete core is also presented. It should be noted that both members have the same geometric and material properties, and the midspan deflection in a downward direction is taken as positive and vice versa. It can be observed from the figure that the bending moment of the hollow steel section dropped after the peak point. By contrast,

Negative

0

Positive

Mid-span deformation

Moment

(a)

Mid-span deformation (b)

FIGURE 2.87

Moment-mid-span deformation responses of composite beam with gaps [84]. (a) Circumferential gap, (b) Spherical-cap gap.

2.13 Material imperfections

151

no descending response can be observed for the composite beams. This issue shows the efficiency of the in-filled concrete in improving the performance of the beam. During the initial loading stage, ovalization of steel tube results in upwards deformation of both hollow steel section beam and CFST beam with the circumferential gap, although the direction of applied load is downward. The concrete core can restrain any further ovalization by increasing the loading due to the interaction between the in-filled concrete and the steel tube in the CFST beam with the circumferential gap. Consequently, the development of the midspan deflection of the beam reverses to the downward direction. Similar to CFST/CFDST columns with gaps, the strength index can be defined for the beam ðSIBÞ to evaluate the effects of circumferential and spherical cap gaps on their ultimate bending moments, as follows: SIB ¼

Mu; with gap Mu; without gap

(2.46)

Moment

where Nu; with gap and Nu; without gap are the bending moment capacities of the columns with and without gap, respectively. The initial section flexural stiffness ðki Þ and the serviceability-level section flexural stiffness ðks Þ of CFST/CFDST beams can be defined from the moment M-curvature 4 curve, as shown in Fig. 2.88. ki and ks are defined as the secant stiffness corresponding to 0:2Mu and 0:6Mu , respectively. As depicted in Fig. 2.88, the moment M-curvature 4 curve of CFST/CFDST beams consists of the elastic response followed by the inelastic behavior with a gradual decrease in stiffness until the ultimate moment is achieved. The experimental test results show that after the steel tensile strain ðεl Þ achieves 10,000 mε, the moment of beams tends to stabilize. Therefore, for practical issues, the ultimate moment of CFST/ CFDST beams is taken as the moment corresponding to the maximum fiber strain ofεlt ¼ 10; 000 mε.

Mu

Ki Ks

0.6Mu 0.2Mu

Curvature

FIGURE 2.88 The typical moment M-curvature 4 response of CFST/CFDST beams.

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2. Experimental tests

The initial section flexural stiffness index ðFSIi Þ and serviceability-level flexural section stiffness index ðFSIs Þ are defined to assess the impacts of gaps on the flexural stiffness of CFST/CFDST beams. FSIi ¼

Ki; with gap Ki; without gap

(2.47a)

FSIs ¼

Ks; with gap Ks; without gap

(2.47b)

where Ki; with gap and Ks; with gap are the initial section flexural stiffness and serviceability-level section flexural stiffness of the beams with gaps, respectively, and Ki; without gap and Ks; without gap are the initial section flexural stiffness and serviceability-level section flexural stiffness of the beams without gaps, respectively. Section modulus and the second moment of area of the steel tube are reduced by ovalization, leading to a gradual deterioration of bending resistance and bending stiffness. In an intact CFST/CFDST beam, the infilled concrete contributes to the moment capacity of the member. Besides, it can delay local buckling and brace the steel tube to resist ovalization. Consequently, the plasticity along the longitudinal direction of the steel tube will be fully developed, leading to a significant enhancement of ultimate strength and flexural stiffness. The circumferential gap postpones and reduces the support that the steel tube receives from the concrete core. This issue leads to ovalization and early local buckling in the tube, preventing the tube from reaching its full plastic behavior. Additionally, the circumferential gap diminishes the confinement effect provided by the steel tube in the compression region of the concrete core. The confinement effect notably enhances the strength and plasticity of in-filled concrete and plays a crucial role in the moment capacity of CFST/CFDST beams after yielding the compression area of the steel tube. As a result, the circumferential gap reduces SIB, and the rate of reduction increases with increasing the gap ratio cgap . A similar trend can be expected for the effect of the spherical-cap gap on SIB. Compared with the circumferential gap, the intensity of the spherical-cap gap on reducing the ultimate moment of CFST/CFDST beams is less significant. When there is no circumferential gap in the beam, the concrete core can efficiently brace the tube to hinder its ovalization. Besides, the bond strength and the friction among the interface of tube and concrete can inhibit the development of cracks in the concrete along the depth direction of the beam. Therefore, CFST/CFDST beams can have a large initial section flexural stiffness ki and serviceability-level section flexural stiffness ks . The circumferential gap results in ovalization in the tube and reduces the second-moment area of the tube. Simultaneously, the bond at

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153

the interface of the tube and concrete is prevented by the circumferential gap, leading to severe cracking of concrete. As a result, the flexural stiffness of the beam with the circumferential gap is smaller than the counterpart without the gap. Compared with the circumferential gap, the intensity of the spherical-cap gap on reducing the flexural stiffness of CFST/CFDST beams is less critical.

2.14 Effects of preload and long-term sustained load 2.14.1 Effects of preload During the construction process of multistory composite structures, the frames are first assembled by placing hollow steel columns, beams, metal floor decking, braces, and other elements. After completing several stories, the hollow steel tubes are filled by pumping the concrete inside them. Consequently, before the concrete core hardens and can carry the external loads, the steel tubes are mainly subjected to preloading resulting from self-weight, the gravity of the wet concrete core, and construction load. This issue induces initial stresses and deformations in tubes that can influence the load-bearing capacity of CFST/CFDST columns after construction. Therefore, the hollow steel columns must be designed for the gravity and construction loads as bare steel sections. In general, CFST/ CFDST columns of a tall building must withstand the loads imposed during three stages of loading throughout their life span, including constriction loading, service loading, and ultimate loading. For the design of composite columns, it must be assured that the steel tubes can withstand different levels of construction loads before pumping the concrete. Steel hollow sections must have enough strength and rigidity during this loading stage. Consequently, the design of hollow steel sections is based on local and overall buckling. By pumping the wet concrete inside the hollow sections, they will be subjected to hydrostatic and axial loads. Upon hardening the in-filled concrete and starting the composite action in columns under service loads, creep and shrinkage strains will expand in the concrete. This issue can affect the behavior of the steel tubes. Increasing the strain in the tube may cause yielding or local buckling. The most wellunderstood step of composite columns constructions is the ultimate loading stage, and most studies have been performed regarding the behavior of composite members at this stage. The schematic view of the concrete-filled steel tubular columns throughout construction is depicted in Fig. 2.89 [85]. The pumping operation of the concrete applies hydrostatic lateral pressure on the hollow steel sections. In this stage, the steel tubes will be under both axial compressions induced by the construction operations and hydrostatic loads generated by the wet concrete pumping. The hydrostatic lateral

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2. Experimental tests

FIGURE 2.89 CFST columns construction process in tall buildings.

pressure may develop excessive lateral deflections. Consequently, slenderness limits must be determined to assure the deflections are maintained at an acceptable limit. The serviceability of the structure may be affected by the lateral deflections since they can induce unsightly deformations and lead to an increased volume of concrete being pumped. However, the impact of the hydrostatic pressures on strains and stresses is not significant enough to membrane stretch the steel plates since the deformations are uncoupled. If the thickness of the steel plates is greater than the deviations, they will have a linear behavior. Generally, the thickness of the steel plates in composite columns with a large cross section is more than the acceptable deflection limits. Therefore, considering the geometric linear behavior is proper.
 For subjecting CFST/CFDST columns under the construction load Np in the laboratory, the preloading is applied on the hollow steel sections using pre-stressing bars in which the load magnitude is controlled employing a load cell, as depicted in Fig. 2.90. Once the desired construction load Np is reached, the load remains constant during the test by adjusting the bars. This circumstance must remain constant until the amount of the axial strain obtained from the vertical strain gauges attached to the outer surface of the steel tube tends to stabilize. Once the longitudinal strains of the steel tube are stabilized, the tube must be filled by the concrete in different layers. Each layer must be compacted using the poker vibrator. Next, the column must be placed vertically until the test day to air-dry. If longitudinal shrinkage occurred during the curing, high-strength epoxy should be used to fill the circumferential gap. In the

155

2.14 Effects of preload and long-term sustained load

FIGURE 2.90 Test setup of imposing the preload on CFST/CFDST columns.

end, the specimen must be subjected to the compression testing machine until its failure. Fig. 2.91 illustrates the comparison between the typical axial loade displacement responses of a CFST/CFDST column without preload with the counterpart with preload on steel tubes. Four characteristic points are also marked on the curve. The response of CFST/CFDST columns with preload on steel tubes can generally be divided into four stages as follows: Stage 1: Preload stage (from Point O to Point A): At this stage, steel tubes are subjected to preload. Therefore, initial longitudinal stress and deflection develop in the tubes.

Ultimate axial load without preload C Axial load

D B

Ultimate axial load with preload

A

O Axial displcament

FIGURE 2.91 The typical axial loadedisplacement curve of CFST/CFDST columns with preload on steel tubes.

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2. Experimental tests

Stage 2: Elastic stage (from Point A to Point B): During this stage, the axial load is applied to the whole composite column. However, due to the elastic behavior of steel tubes and the in-filled concrete, they carry the load independently. No interaction happens between the tubes and the concrete core. Comparison of the slope of the curves demonstrates that preloading of steel tubes leads to an increase in the column deformation and declines the stiffness of the column. Stage 3: Elastic-plastic stage (from Point B to Point C): By increasing the axial compression, the concrete cracks and steel tubes reach their yield stresses at Point B. Therefore, the lateral expansion of the concrete becomes more than the outer steel tube, leading to imposing the confining pressure on the in-filled concrete. Compared with the column without preload, the ultimate axial strength of the column with preload (Nup ) is smaller. Also, the axial shortening corresponding to the peak load is delayed by preloading steel tubes. Concerning the confinement effect, preloading steel tubes generally reduces the confinement effect provided. Compared with the square cross section, the influence of preloading the steel tubes is more prominent on the behavior of CFST/CFDST columns with the circular cross section. Preloading reduces the confinement effect because the Mises stress in the steel tube of the preloaded CFST/CFDST column develops more rapidly than the counterpart without preload. Therefore, the preloaded steel tube yields quicker, reducing the confinement effect. Due to the reducing effect of steel tubes preloading on the confinement effect, the axial strength in the column with preload is less than the counterpart without the preload. Stage 4: Hardening or descending stage (from Point C to Point D): At this stage, depending on the confinement factor, the column can present an ascending or descending postpeak response. This is similar to the behavior of the CFST/CFDST columns without preload, as discussed previously. Different key parameters may affect the performance of CFST/CFDST columns with preload on steel tubes, i.e., initial stress ratio, slenderness ratio, the compressive strength of the concrete, the hollow ratio of CFDST members, load eccentricity ratio, and nominal steel ratio. For ease of analysis, the strength index ð4s Þ is defined to evaluate the impact of the preloading of steel tubes on the CFST/CFDST columns. The strength index 4s is defined by: Nup (2.48) Nu where Nup and Nu are the ultimate axial strengths of members with and without preload. In short square CFST/CFDST columns under axial loads, the confinement effect provided by the square steel tubes is weak. Therefore, it can be assumed that the load-bearing capacity is equal to the sum of the strengths of the steel tubes and the in-filled concrete. The initial stresses 4s ¼

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157

induced by the preloading of steel tubes do not alter the steel yield strength or the concrete compressive strength. Besides, the influence of the preloading on the confinement effect in square CFST/CFDST columns is negligible. Consequently, the influence of preloading of steel tubes in square CFST/CFDST stub columns is insignificant, especially when the preload ratio is small. Contrary to square CFST/CFDST columns, the infilled concrete of circular CFST/CFDST columns is strongly confined by the steel tube. Therefore, the load-bearing capacities of short circular CFST/CFDST columns depend on both the strengths of materials and the confining pressure. Similar to the short square CFST/CFDST columns, the initial stresses cannot affect the strengths of materials in short circular CFST/CFDST columns. However, initial stresses may reduce the confinement effect provided by the circular steel tube. Therefore, the initial stresses can reduce the ultimate axial strength of circular CFST/CFDST stub columns. In general, the effects of key parameters on the strength index 4s of short circular and square CFST/CFDST columns are almost the same and are moderate. The strength index can be considered independent of the material strengths and hollow ratio, even for the large preload ratio. However, for the sake of completeness of discussion, the relations between the key parameters and the strength index of both short and slender columns are discussed in the following. Although the preloading can decline the load-bearing capacities of short CFST/CFDST columns, its effect is not significant and is less than 5%. Consequently, the preloading effects in steel-concrete composite columns can be ignored if the slenderness ratio is Lr < 20w22 or the preload ratio is less than 0.2. Contrary to the short CFST/CFDST columns, the initial stresses have a more pronounced impact on intermediate and slender columns’ load-bearing capacity. This is because the initial global imperfection in slender columns subjected to axial loading can cause load eccentricity. In addition, preloading can develop lateral deflection in the steel tube and accelerate the plastic response. Therefore, preloading slender composite columns intensifies their lateral deformation, reducing the confinement effect and the ultimate axial strength. The effect of the preload ratio on the ultimate axial strength of the column increases with increasing slenderness ratio. The strength index 4s is reduced by increasing the steel tube yield stress, meaning that loss of strength because of the preload effects is increased by increasing the steel yield strength. Usually, the value of the elastic modulus of steel tubes is constant. Therefore, steel tubes having different steel yield strengths possess the same flexural rigidity. Considering a constant amount of NP, the absolute value of the initial stresses increases with increasing the steel tube yield stress. Therefore, the deformation of the in-filled concrete generated by the initial stresses becomes larger when working together with the tube, reducing the ultimate axial strength. Increasing the concrete strength reduces the strength index 4s and hence, increases the effect of initial stresses. The contribution of concrete

158

2. Experimental tests

to the load-bearing capacity increases with increasing its compressive strength. Additionally, increasing the concrete strength reduces the confinement effect. Therefore, under the same amount of Np , the strength index 4s is reduced by increasing the concrete strength. Compared with the steel strength, the influence of the concrete strength is moderate. The nominal steel ratio and load eccentricity ratio have less effect on the strength index 4s of short and slender CFST/CFDST columns under preload than the previous key parameters. However, increasing both of them leads to an increase in the strength index 4s . This reveals that the influence of initial stresses becomes less significant by increasing the nominal steel ratio and the load eccentricity ratio. Regarding the effect of the void size on 4s in CFDST stub columns, 4s is nearly independent of the hollow ratio. As discussed before, the inner steel tube can normally provide substantial support to the sandwiched concrete before reaching the compressive strength. Therefore, the behavior of CFDST columns is similar to the conventional CFST columns to some extent. As a result, the void size does not affect 4s . Out of all key parameters, the slenderness ratio has the most significant influence on the behavior of CFST/CFDST columns under preload. The strength index 4s is reduced by increasing the slenderness ratio. Therefore, increasing the slenderness ratio intensifies the impact of initial stresses. Considering the same amount of Np , the lateral deflection of the steel tube due to the initial stresses becomes more significant by increasing the column’s slenderness ratio. As a result, the load-bearing capacity of the column under preload reduces with increasing the slenderness ratio.

2.14.2 Effects of long-term sustained load CFST/CFDST columns in buildings or bridges must be under service loads for the long term, making them suffer from the impacts of creep and shrinkage of the in-filled concrete. The long-term sustained load does not influence the failure mechanism of CFST\CFDST members. However, it can affect the mechanical properties of composite members, such as axial stiffness and ultimate axial strength. It is noteworthy that compared with the column under short-term loading, the long-term sustained load effects can reduce the ultimate axial strength of the column by 20%. The test setup for applying the long-term service load is similar to the experimental test setup for CFST/CFDST columns with preload on steel tubes. The test consists of two stages. The long-term service load test must be performed 28 days after the fabrication of specimens. It must be assured that the temperature and humidity of the test environment are constant during the experiment to minimize the effect of the temperature gradient. The long-term sustained load ðNL Þ is then applied to the column using prestressing bars. The magnitude of the load NL is adjusted by the

2.14 Effects of preload and long-term sustained load

159

load cell and must be kept constant. The axial deformation of the column is measured using the strain gauges attached to the center of the outer surface of the steel tube. The axial strain increases at early loading ages and tends to stabilize after about 100 days [86]. The trend of strain stabilization is similar between CFST and CFDST columns. Proportional to the desired duration, the column is subjected to the constant long-term service load. Next, the column must be removed from the long-term loading devices and subjected to the compression testing machine. The long service load develops creep and shrinkage strain in the column. Because of the creep effects over time, concrete stress is decreased while steel stress is increased. Therefore, when CFST/CFDST columns are subjected to long-term service loads, the force from the concrete is transferred to the steel tube over time due to the longitudinal creep deformation of the in-filled concrete. As a result, the amount of force in the steel tube ðNs Þ increases gradually, while the load in the concrete core ðNc Þ decreases. This issue is illustrated in Fig. 2.92. The long-term strain ðεL Þ because of the creep and shrinkage of axially loaded column increases over time, and its amount depends on the age time at loading, time under sustained loading, long-term sustained load

L ratio m ¼ N Nu , nominal steel ratio an , the concrete strength, the steel tube yield stress, and the depth-to-thickness ratio Dt. The long-term strain εL is governed by:

Axial load

(2.49) εL ¼ εcr þ εsh where εcr is the creep component of strain and εsh is the shrinkage component of strain. Based on the experimental test results, the long-term strain εL is reduced by increasing the age time of loading. Similarly, the stain of εL declines as the nominal steel ratio an increases. This is because increasing

Time

FIGURE 2.92

Axial load versus time relation for steel tube and the in-filled concrete.

160

2. Experimental tests

the steel ratio reduces the in-filled concrete cross-sectional area and its contribution to the composite column. By contrast, increasing the time that the column is sustained under axial load, enhancing the concrete compressive strength, improving the steel yield strength, or increasing the long-term load level m increases the strain of εL . Under the same load level, increasing the steel yield strength increases the applied load. This issue leads to an increase in the initial concrete and steel stresses, resulting in a larger value of the long-term strain. Enhancing the concrete strength increases the concrete stress due to two reasons. The first reason is that the elastic modulus increases, and therefore for a considered constant load, the stress of the concrete would be slightly higher. The second reason is that the applied load is increased by increasing the concrete strength because it is a constant fraction of the failure force. Consequently, the initial stress of the concrete is again higher, leading to a greater amount of long-term strain. Fig. 2.93 illustrates the typical axial load N-deformation D curves of CFST/CFDST columns under short-term and long-term conditions. It can be observed from the figure that under long-term loading conditions, the column deformation is increased, and the stiffness of the column is reduced since the creep of the concrete core over time increases the deflection of the column. The ultimate axial strength of the column under long-term sustained load is less than the counterpart under short-term loading conditions, especially when pin-pin boundary conditions are used or when the slenderness of the column is large. This is because longterm sustained loading generates deformation in the column, leading to

FIGURE 2.93

Typical axial loadedeformation curves of CFST/CFDST columns with and without long-term effects.

2.14 Effects of preload and long-term sustained load

161

an extra moment to the CFST/CFDST column subjected to the axial compression. Two characteristic points of A and B are defined on the curve. Point A indicates the moment when the sustained load is imposed, and Point B shows the moment when the sustained load is finished. During Point A and Point B, the amount of load is constant, while deformation of the column continues to increase because of shrinkage and creep of the in-filled concrete. The length of A-B indicates an increase in deformation due to long-term effects, and it depends on the loading ratio and the sustaining time. Generally, increasing the load ratio and sustaining time increases the length of A-B, meaning a more increment of deformation. During the long-term sustained loading stage, part of the external load carried by the in-filled concrete is transferred to the outer tube in the CFST column or both inner and outer steel tubes with a similar proportion in the CFDST column. Consequently, the concrete stress declines due to the deformation compatibility among the tubes and the infilled concrete. Compared with the CFST column, the decrease of the force carried by the sandwiched concrete in the CFDST column is more significant. Besides, the load increment in the steel tube of the CFST column is greater than that of the CFDST column. This is because the inner tube of the CFDST column carries part of the load transferred from the core concrete. After Point B and by terminating the long-term sustained loading, the axial compressive load is imposed, and the deformation keeps increasing with increasing the axial load. Usually, the outer steel tube reaches its ultimate strength before the in-filled concrete. After that, the load on the in-filled concrete still keeps increasing because of the confinement effect provided by the tubes. Between CFST and CFDST columns, the influences of long-term sustained loading on the compressive capacity and the corresponding deformation are more significant for the conventional CFST column. This contributes to the fact that the inner steel tube of the CFDST column participates in the load-bearing capacity. The existence of the inner steel tube can reduce the compressive stress level in the sandwiched concrete, and hence, can decline the effects of the long-term sustained loading on the performance of the CFDST column. This can be considered as an advantage of using the inner steel tube in steel-concrete composite members and again reinforces the fact that the inner tube plays an important role in column behavior. The general mechanism of the confinement effect in CFST/CFDST columns subjected to long-term sustained loading is similar to their counterparts under short-term loading. During the initial loading stage, when both steel tubes and the in-filled concrete have elastic behavior, the outer steel tube tends to separate from the in-filled concrete, and no interaction occurs between them because of the larger Poisson’s ratio of the steel than the concrete. By increasing the deformation, the in-filled

162

2. Experimental tests

concrete starts to crack, which leads to the development of its plasticity. Therefore, the lateral expansion of the concrete begins to exceed that of the steel tube. At this moment, the outer steel tube imposes confining pressure on the in-filled concrete and restricts its lateral deformation. As discussed above, the long-term sustained loading condition reduces the stress on the in-filled concrete. Therefore, the crack development in core concrete is delayed, postponing the interaction between the steel tube and the concrete. Usually, the interaction between the steel tube and the in-filled concrete in the column under short-term loading conditions occurs during the ascending region of the axial loadedisplacement response before reaching the ultimate axial strength, whereas, in the counterpart under long-term sustained loading, the interaction does not happen before the compressive strength of the column is achieved. Compared with the CFST/CFDST column under short-term loading, the amount of confining stress in the counterpart subjected to long-term sustained loading is smaller. Contrary to the outer steel tube, the long-term sustained loading effect on the interaction stress between the inner steel tube and the concrete is moderate. Different key parameters can affect the behavior of CFST/CFDST columns subjected to long-term sustained loading, i.e., slenderness ratio, the compressive strength of the concrete, the hollow ratio of CFDST members, load eccentricity  and nominal steel ratio. For ease of analysis, the
ratio, strength index 4s;L is defined to evaluate the impact of long-term sustained loading on the ultimate axial strength of CFST/CFDST columns. The strength index 4s;L is governed by: 4s; L ¼

NuL Nu

(2.50)

where NuL denotes the ultimate axial strength of members under longterm sustained loading conditions and Nu is the ultimate axial strength of members under short-term loading conditions. Concerning the slenderness ratio and nominal steel ratio, the amount of 4s;L in columns with a slenderness ratio smaller than 60 reduces with increasing the slenderness ratio. By contrast, in columns with a slenderness ratio greater than 60, the sustained-loading conditions have no significant impact on 4s;L . For a short column with a slenderness ratio of less than 10, the long-term sustained load has almost no effect on the ultimate axial strength. Increasing the nominal steel ratio increases the amount of 4s;L in the columns with a slenderness ratio greater than 40, whereas it has no influence on 4s;L in the columns with a slenderness ratio less than 40. Increasing the yield stress of the outer steel tube in CFST/CFDST columns with a slenderness ratio greater than 60 increases 4s;L . By contrast, the outer tube yield stress has a negligible influence on the 4s;L when the

References

163

slenderness ratio is less than 60. Contrary to the outer steel tube, the yield stress of the inner steel tube of CFDST columns has a negligible effect on 4s;L . 4s;L reduces with increasing the compressive strength of in-filled concrete in columns with a slenderness ratio greater than 40, while the concrete strength almost has no impact on 4s;L of columns with a slenderness ratio smaller than 40. The effect of both width-to-thickness ratios of the outer and inner steel tubes on the 4s;L of columns with any slenderness ratio is small. By contrast, 4s;L increases as the hollow ratio c of CFDST columns increases. This is due to the fact that the cross-sectional area of the sandwiched concrete reduces upon increasing the hollow ratio. Therefore, the effect of long-term sustained loading is declined.

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[40] Hassanein MF, Kharoob OF. Analysis of circular concrete-filled double skin tubular slender columns with external stainless steel tubes. Thin-Walled Struct 2014;79:23e37. [41] Yang L, et al. Experimental behavior of concrete-filled corrugated steel tubular short columns under eccentric compression and non-uniform confinement. Eng Struct 2020;220:111009. [42] Yuan F, Huang H, Chen M. Effect of stiffeners on the eccentric compression behaviour of square concrete-filled steel tubular columns. Thin-Walled Struct 2019;135:196e209. [43] Ahmed M, et al. Behavior of eccentrically loaded double circular steel tubular short columns filled with concrete. Eng Struct 2019;201:109790. [44] Ahmed M, et al. Experimental and numerical investigations of eccentrically loaded rectangular concrete-filled double steel tubular columns. J Constr Steel Res 2020;167: 105949. [45] Elchalakani M, et al. Experimental tests and design of rubberised concrete-filled double skin circular tubular short columns. Structures 2018;15:196e210. [46] Dong M, et al. Circular steel tubes filled with rubberised concrete under combined loading. J Constr Steel Res 2019;162:105613. [47] Dong M, et al. Behaviour and design of rubberised concrete filled steel tubes under combined loading conditions. Thin-Walled Struct 2019;139:24e38. [48] Tao Z, Han L-H, Wang Z-B. Experimental behaviour of stiffened concrete-filled thinwalled hollow steel structural (HSS) stub columns. J Constr Steel Res 2005;61(7): 962e83. [49] Cai J, He Z-Q. Axial load behavior of square CFT stub column with binding bars. J Constr Steel Res 2006;62(5):472e83. [50] Ding F-X, et al. Mechanical behavior of stirrup-confined rectangular CFT stub columns under axial compression. Thin-Walled Struct 2018;124:136e50. [51] Yang Y, Wang Y, Fu F. Effect of reinforcement stiffeners on square concrete-filled steel tubular columns subjected to axial compressive load. J. Thin-Walled Struct. 2014;82: 132e44. [52] Ho JCM, Lai MH. Behaviour of uni-axially loaded CFST columns confined by tie bars. J Constr Steel Res 2013;83:37e50. [53] Lai MH, Ho JCM. Confinement effect of ring-confined concrete-filled-steel-tube columns under uni-axial load. Eng Struct 2014;67:123e41. [54] Lai MH, Ho JCM. Axial strengthening of thin-walled concrete-filled-steel-tube columns by circular steel jackets. Thin-Walled Struct 2015;97:11e21. [55] Wang Y, et al. Size effect of circular concrete-filled steel tubular short columns subjected to axial compression. Thin-Walled Struct 2017;120:397e407. [56] Hong Z, et al. Engineering application of 8 columns mega frame-core wall system in Guangzhou East Tower 2012;42(10):1e6. [57] CEN. EN 1992-1-1. Eurocode 2: design of concrete structures e part 1e1: general rules and rules for buildings. Brussels, Belgium: European Committee for Standardization; 2010. [58] AIJ. Recommendations for design and construction of concrete filled steel tubular structures. Tokyo (Japan): Architectural Institute of Japan; 1997 [in Japanese]. [59] ANSI/AISC 360-16. Specification for structural steel buildings. Chicago: AISC; 2016. [60] Kim JK, Yi ST, Park CK, Eo SH. Size effect on compressive strength of plain and spirally reinforced concrete cylinders. ACI Struct J. n.d.;96(1). [61] Sakino K, et al. Behavior of centrally loaded concrete-filled steel-tube short columns. J Struct Eng 2004;130(2):180e8. [62] Jin L, et al. Size effect of axial-loaded concrete-filled steel tubular columns with different confinement coefficients. Eng Struct 2019;198:109503. [63] Chen P, Wang Y, Zhang S. Size effect prediction on axial compression strength of circular CFST columns. J Constr Steel Res 2020;172:106221.

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[64] Romero ML, et al. Influence of ultra-high strength concrete on circular concrete-filled dual steel columns. Structures 2017;9:13e20. [65] Han L-H, Hou C, Wang Q-L. Square concrete filled steel tubular (CFST) members under loading and chloride corrosion: experiments. J Constr Steel Res 2012;71:11e25. [66] AS 4100-1998. Australian standard: steel structures. Sydeny: Standard Australia; 1998. [67] BS EN 1090e2:2008. Execution of steel structures and aluminium structures-part 2: technical requirements for the execution of steel structures. Brussels: European Committee for Standardization; 2008. [68] CAN/CSA-S16-01. National standard of Canada: limit state design of steel structures. Ontario: Canadian Standards Association; 2005. [69] GB 50017e2003. Code for design of steel structures. Ministry of Construction of the People’s Republic of China; 2003 [in Chinese]. [70] DIN 18800-1. Structural steelwork: design and construction. Berlin: Deutsches Institut fu¨r Normung e.V; 1990. [71] Code of practice for the structural use of steel. Hong Kong: Building Department of the Government of the Hong Kong Special Administrative Region; 2005. [72] AISC 360-05. Specification for structural steel buildings. Chicago: American Institute of Steel Construction (AISC), An American National Standard; 2005. [73] Melchers RE. Recent progress in the modeling of corrosion of structural steel immersed in seawaters. J Infrastruct Syst 2006;12(3):154e62. [74] Melchers RE. Modelling immersion corrosion of structural steels in natural fresh and brackish waters. Corrosion Sci 2006;48(12):4174e201. [75] ASTM G 31e72. Standard practice for laboratory immersion corrosion testing of metals. West Conshohocken: ASTM International; 2004. [76] Zhang J, Shi XH, Guedes Soares C. Experimental analysis of residual ultimate strength of stiffened panels with pitting corrosion under compression. Eng Struct 2017;152: 70e86. [77] Huang H, et al. Tests of circular concrete-filled steel tubular stub columns with artificial notches representing local corrosions. Eng Struct 2021;242:112598. [78] Ahn J-H, et al. Shear buckling experiments of web panel with pitting and throughthickness corrosion damage. J Constr Steel Res 2015;115:290e302. [79] Ahn J-H, et al. Residual shear strength of steel plate girder due to web local corrosion. J Constr Steel Res 2013;89:198e212. [80] Chang X, et al. Behaviors of axially loaded circular concrete-filled steel tube (CFT) stub columns with notch in steel tubes. Thin-Walled Struct 2013;73:273e80. [81] Ding F-X, Fu L, Yu Z-W. Behaviors of axially loaded square concrete-filled steel tube (CFST) stub columns with notch in steel tube. Thin-Walled Struct 2017;115:196e204. [82] Guo L, et al. Axial behavior of square CFST with local corrosion simulated by artificial notch. J Constr Steel Res 2020;174:106314. [83] Han LH. Concrete filled steel tubular structures. 2nd ed. China Science Press; 2007 [Beijing China Science Press]. [84] Liao F-Y, Han L-H, He S-H. Behavior of CFST short column and beam with initial concrete imperfection: experiments. J Constr Steel Res 2011;67(12):1922e35. [85] Uy B, Das S. Wet concrete loading of thin-walled steel box columns during the construction of a tall building. J Constr Steel Res 1997;42(2):95e119. [86] Han LH, Tao Z, Liu W. Effects of sustained load on concrete-filled hollow structural steel columns. J Struct Eng 2004;130(9):1392e404.

C H A P T E R

3 Analytical methods O U T L I N E 3.1 Introduction

168

3.2 Stressestrain response of materials 3.2.1 Carbon steel 3.2.2 Stainless steel 3.2.2.1 RambergeOsgood model 3.2.2.2 Mirambell and Real’s model 3.2.2.3 Gardner and Nethercot’s model 3.2.2.4 Quach’s model 3.2.2.5 The generalized multistage model 3.2.2.6 Inversion of the stressestrain relationship 3.2.3 Confined concrete 3.2.3.1 Constitutive stressestrain model for confined concrete

168 168 179 180 183 185 185 187 189 190

3.3 Creep model for concrete

279

3.4 Creep analysis of CFST columns

284

3.5 A constitutive model for computing the lateral strain of confined concrete

298

3.6 Axial and lateral stressestrain response of CFST column

305

3.7 An analytical axial stressestrain model for circular CFST columns

308

3.8 Path dependent stressestrain model for CFST columns

314

3.9 Elastic-plastic model for the stressestrain response of CFST columns

321

191

3.10 Strength enhancement induced during cold forming

332

References

335

Single Skin and Double Skin Concrete Filled Tubular Structures https://doi.org/10.1016/B978-0-323-85596-9.00005-6

167

© 2022 Elsevier Inc. All rights reserved.

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3. Analytical methods

3.1 Introduction Recent advances in computers’ capabilities in performing numerical computing, in addition to the increase in the use of finite element (FE) software, have expanded the use of analytical and numerical analysis of structural members. A fundamental step in performing analytical methods is defining models that can accurately predict the behavior of materials. Accordingly, researchers have always attempted to develop numerical expressions to evaluate materials’ behavior precisely and develop reliable analysis-based models for concrete-filled steel tubular/ concrete filled double skin steel tube (CFST/CFDST) members. Firstly, in this chapter, models for predicting the stressestrain responses of structural steel and stainless steel will be discussed. Next, models for defining the stressestrain response of concrete confined by various shapes of steel tubes will be covered. Then, a model for evaluating the effects of creep on the behavior of the confined concrete will be explained. After that, different models for estimating the stressestrain response of composite columns under axial compression will be presented.

3.2 Stressestrain response of materials 3.2.1 Carbon steel The engineering tensile stressestrain curve for structural carbon steels is shown in Fig. 3.1. It can be observed from the figure that the depicted tensile stressestrain curve consists of three phases: During the first step, Upper yield point

Ultimate strength

Stress

Plastic flow (Yield plateau)

Necking

Fracture

Strain hardening Lower yield point

Elastic

Strain FIGURE 3.1 Simplistic characterization of a typical engineering stressestrain curve of low carbon steel.

3.2 Stressestrain response of materials

169

steel material shows a linear behavior up to a specific proportionality limit. This linear behavior is due to the locking of dislocations in structural carbon steels. The stress then drops to a lower yield value. The amount of stress is constant during the plastic flow, and the strain increases along a yield plateau. At this stage, Lu¨ders bands of plastic deformation develop in the steel. The yield plateau continues up to the point that the entire steel material reaches the Lu¨ders strain ðεL Þ. Later, the stress gradually increases and reaches the ultimate tensile stress ðsu Þ due to the strain hardening. Finally, the local necking makes the steel material unstable and leads to fracture. The length of the yield plateau does not depend only on the material’s intrinsic property, and the fabrication process and the strain history of the steel can affect it. More precisely, several factors influence the length of the yield plateau, i.e., chemical composition, strain aging, grain size, heat treatment, loading rate of the test, stiffness of the test rig, and specimen alignment. The existence of a yield plateau, as well as the difference in the slope of each stage, makes it impossible to define a continuous characterization for the steel material. Hence, stressestrain curves of carbon steels are usually defined by simplified piecewise functions (Fig. 3.2) in computational and analytical investigations. The most straightforward and widely used function is the idealized piecewise-linear model, as follows:

(3.1)
where sy is the steel yield stress, and εy ¼ sy E is the yield strain corresponding to sy . The length of the yield plateau is defined by multiplying n by the yield strain. Therefore, the linear strain hardening region starts at

FIGURE 3.2

The idealized piecewise-linear model for structural carbon steel.

170

3. Analytical methods

the stain of ð1 þnÞ εy . The acceptable strains in classical structural engineering are generally not very large. Therefore, strain hardening can be simplified by a linear behavior. In this case, the strain hardening modulus of Eh ¼ hEnom can be used in which Eh is the initial strain hardening tangent modulus and derived from calculated stressestrain curves, and h is the relation of Eh to the elastic value. This simplified model does not attempt to cover the true stressestrain curves at large strains. The ultimate strain ðεu Þ corresponding to the ultimate tensile stress ðsu Þ is not an independent parameter; however, it can be defined as:   su  sy (3.2) εu ¼ ð1 þ nÞ εy þ Eh Real carbon steels display a progressive loss in stiffness, whereas in the defined piecewise-linear model, a constant strain hardening stiffness is assumed. Therefore, this model does not lead to entirely accurate results and can give an unconservative prediction of the true ultimate tensile strain. However, its impacts on structural resistance calculations are negligible. Depending on whether the yield plateau and strain hardening behavior of carbon steels are taken into account or not, different variants of stressestrain curves can be developed. The first model is the classical elastic-perfectly plastic curve, as shown in Fig. 3.3a. According to Hooke’s law, the steel has linear behavior in this model up to its yield point. Also, the strain hardening behavior is neglected. Therefore, Eh ¼ 0, and the yield plateau is infinity ðn /NÞ. Hence, the steel is assumed to fail by reaching the yield strength and cannot bear any additional loading. This 2fu

arctan(Es)

arctan(Es)

εy

0 5 10 Strain ε (a) Elastic-perfectly plastic model fu Stress

arctan(Esh)

0

εy

5 Strain ε10 (b) Bilinear model

0

15

εp εy εsh

εu 1Strain ε 2 (d) Idealized multilinear model

1200 fu 1 0.8 0.6

Stress

800

arctan(Esh)

Stress

fy 400

1.5fy f 1p

Small B/t Moderate B/t Large B/t

fy 800 0.9fy 600

Stress

f 400 y

arctan(Esh)

Stress

Stress

Esh=0

400

arctan(Es) εy εsh εu εu 0 5 ε 10 Strain 0 (e) Bilinear plus nonlinear hardening model

2Strain ε 4 (f) Sakino model

0

εp

εsh Strain 2 ε (g) Liang model

εu

4

FIGURE 3.3 Existing stressestrain model for hot-rolled carbon steel material. (a) Elasticperfectly plastic model (b) Bilinear model (d) Idealized multilinear model. (e) Bilinear plus nonlinear hardening model (f) Sakino model (g) Liang model.

171

3.2 Stressestrain response of materials

model is the simplest because it only needs two inputs, i.e., the steel yield strength sy and the nominal elastic modulus Enom . The classical elasticperfectly plastic model can be considered an appropriate stressestrain relationship for scenarios when the steel tube’s strain-hardening properties are not expected to occur. For instance, steel tubes with a sizable depth-to-thickness ratio, particularly in rectangular tubes where local buckling dominates tubes’ resistance. This model forms the concept of the design methods recommended by EN 1993-91-1 [1]. The second model is the elastic-linear hardening model (Fig. 3.3b), which is the simplest model that considers the strain hardening of steel. The yield plateau, however, is ignored by this model. The elastic, linear hardening model, which is recommended in Annex C of EN 1993-91-5 [2], is proper for advanced computational and design methods like the strainbased continuous strength method (CSM) because it takes into account the beneficial impact of strain hardening in designing structural steel elements, i.e., structural carbon steel, stainless steel, and aluminum. Like the classical elastic-perfectly plastic model, this model only requires two inputs, i.e., the steel yield strength sy and the nominal elastic modulus Enom . Nevertheless, this model may not be considered as a suitable model for the analytical study of CFST/CFDST members built with hot-rolled carbon steel tubes due to the existence of a yield plateau in their behavior. The next level of complexity of steel material models is the multilinear model, as illustrated in Fig. 3.3c, in which both the yield plateau and strain-hardening characteristics of steel are considered. However, this model cannot lead to accurate strain-hardening behavior due to the gradual loss of stiffness in the real behavior of the material. This is because the multilinear model considers a constant strain hardening modulus Es , similar to the previous model. Another model is the multilinear stressestrain curve consisting of five stages: elastic behavior, plastic behavior, yield plateau, strain-hardening behavior, and final plastic deformation, as depicted in Fig. 3.3d. The multilinear stressestrain curve is defined as follows:

(3.3)

in which A1 ¼

0:2s

y 2 2 , A2 ¼ 0:2εp , A3 ¼ 0:8sy þ A1 εp . In this model, the ðεy εp Þ steel has linear behavior from the beginning to the proportional stress
limit fp ¼ 0:8sy with the corresponding strain of εp ¼ 0:8sy E . εy is

nom

taken as 1:5εp , εsh is the strain at the beginning of the strain hardening and

172

3. Analytical methods

is equal to 10εp , and εu is the strain corresponding to the ultimate tensile strength and is taken as 100εp while su is taken as 1.6sy . As discussed above, during the strain hardening, the stiffness of steel material reduces gradually in a nonlinear trend. For increasing the accuracy of results, the bilinear plus nonlinear hardening model can be used in which the linear strain hardening portion in the idealized multilinear model can be replaced with a rounded behavior, as shown in Fig. 3.3e. This model is suitable for incorporation into advanced numerical simulations in which modeling the progressive loss of stiffness is required and can provide accurate inputs for predicting the behavior of hot-rolled steel. The bilinear plus nonlinear hardening model is defined by:

(3.4)

where p is the strain hardening exponent and it is governed by:   εu εsh p ¼ Eh su  sy

(3.5)

in which Eh is taken as 0:02Enom . εsh and εu can be expressed as follows: (3.6)

(3.7) Also, the ultimate tensile strength of steels su can be determined from sy by using Eq. (3.8). (3.8) Various parameters affect the strain corresponding to the strain hardening εsh and the strain hardening modulus Eh including the crosssectional shape, thermal effects, chemical composition of the steel, and residual stresses induced during the fabrication process. The calibration of the testing machine used for measuring the stressestrain curve can also affect these two parameters. Over the past decades, several investigations have been conducted to determine the values of εsh and Esh . For instance, the values of

3.2 Stressestrain response of materials

173

Esh ¼ 2%Enom and εsh ¼ 0:025 have been proposed according to the interpretation of stressestrain curves of steel material with yield stress in the range of 235 and 460 MPa obtained from experimental tests [3]. An elastic hardening model with Eh ¼ 1%Enom has been permitted by EN 1993-91-5 [2] for limit state design in computational techniques. Strain hardening modulus of Esh ¼ 2%Enom and εsh ¼ 10εy have been recommended by the ECCS publication [4]. The existing divergence between the proposed amounts of εsh and Esh is because each reference has used almost inadequate stressestrain curves obtained from the experimental test data of a relatively limited set of steel grades. Assuming a constant amount for Esh according to the primary postyield tangent slope considered at εsh is practically appropriate for the very initial steps of the strain hardening region. Very recently, two standardized stressestrain models, including the quad-linear stressestrain model and the bilinear plus nonlinear hardening model, have been developed [5]. The quad-linear stressestrain model consists of four steps and is able to cover the full range of tensile strains. The quad-linear model is suitable for simplified design or analytical methods in which the strain hardening must be taken into account. Besides, it can provide precise inputs for numerical simulations. This model is governed by:

(3.9)

where a1εu is the strain at the intersection point of the third stage of the stressestrain curve, and fa1εu represents the corresponding stress. a1 is the material coefficient that is applied to prevent overpredictions of material strength. Material coefficient a2 is also utilized for predicting the slope of strain hardening modulus Esh : su  sy Esh ¼ (3.10) a2 εu  εsh For defining Esh , two specific points on the stressestrain curve are required: the yield stressand the strain corresponding to the outset of the strain hardening εsh ; sy , and the ultimate tensile stress and the strain a2 εu ða2 εu ; su Þ. This approach has been employed in developing the CSM for defining models for aluminum and stainless steel. Material coefficients for different steel materials are listed in Table 3.1. Amounts of three basic parameters, i.e., E, sy , and su , are recommended by design codes, such as EN 1993-91-1 [1]. Therefore, defining material models using these parameters is desirable. Besides, other extra

174

3. Analytical methods

TABLE 3.1 Material coefficients for the CSM material model. a1

a2

a3

a4

Very high strength structural steel

0.40

0.45

0.60

0

Cold-formed structural steel

0.40

0.45

0.60

0

Austenitic and duplex stainless steel

0.1

0.16

1.00

0

Ferritic stainless steel

0.40

0.45

0.60

0

Aluminum

0.50

0.50

0.13

0.06

material parameters, i.e., εsh , εu , a1 , and a2 , can be formulated in terms of three basic material parameters. The amount of εu depends on the ratio of
the steel yield stress to its ultimate tensile stress sy s . εu can be predicted u

by using Eq. (3.11):

  sy þ a4 εu ¼ a 3 1  su

(3.11)

The magnitude of εu in the hot-rolled and cold-formed steels generally follow a similar trend, shown in Fig. 3.4. The slope of the curve, presented in Fig. 3.4, is the same between the ferritic stainless steel and hot-carbon steel due to the similar basic microstructure. The only difference between hot-rolled and cold-formed steels is that once the amount of
sy s for hot-rolled steel (usually for high-strength material) is more than u

0.6 0.5

εu

0.4 0.3 0.2 0.1 0 0

0.2

0.4

0.6

0.8

1

σy / σu

FIGURE 3.4 The amount of εu for hot-rolled and cold-formed carbon steels.

175

3.2 Stressestrain response of materials

0.9, εu can be taken as a constant amount of εu z 0:06. Therefore, Eq. (3.12) can be defined for calculating the value of εu for hot-rolled carbon steels. 8
> 0:06 sy s  0:09 > < u   (3.12) εu ¼ s
y > > 0  sy s  0:09 : 0:6 1  s u u
Similar to εu , the amount of εsh depends on the sy s ratio. Eq. (3.13) u can be used for predicting the εsh : sy εsh ¼ 0:1  0:055 but 0:015  εsh  0:03 (3.13) su
Fig. 3.5 illustrates εsh as the function of the sy s ratio. u The strain-hardening region of the curve from the experimental test data can be accurately expressed by a seventh order polynomial, as presented by Eq. (3.14). sðεÞ ¼ sy þ

7 X

ai ðε  εsh Þi

εsh < ε  εu

(3.14)

i¼1

where ai is a set of trial coefficients. Using the least-squares procedure on the experimental test data, the material coefficients a1 and a2 can be driven by the following equations. a1 ¼

εsh þ 0:25ðεu  εsh Þ εu

(3.15)

εsh þ 0:4ðεu  εsh Þ εu

(3.16)

a2 ¼

0.04

εsh

0.03

0.02

0.01

0 0.4

0.6

0.8

σy / σu

FIGURE 3.5 The amount of εsh for hot-rolled carbon steels.

1

176

3. Analytical methods

Replacing Eq. (3.16) with Eq. (3.10), the formula for Esh can be expressed as follows: su  sy (3.17) Esh ¼ 0:4ðεu  εshÞ By defining the four coefficients K1 ¼ 0:4; K2 ¼ 2; K3 ¼ 400; and K4 ¼ 5, the accuracy of the bilinear plus nonlinear hardening model described in Eq. (3.4) can be improved, as follows:

(3.18) Fig. 3.6 compares the experimental stressestrain curve of steel material of a square hollow section with the steel grade S355, modulus of elasticity of E ¼ 215; 200 MPa, yield strength of sy ¼ 456 MPa, and ultimate strength of sy ¼ 545 MPa [6] with the predicted ones obtained from different models [5]. It can be found that considering a constant value of Esh ¼ 2%Enom , recommended by ECCS, leads to a substantial deviation from the experimental results. The amounts of εu , εsh , and Esh , calculated by Eqs. (3.12), (3.13) and (3.17), are based on nominal material properties and nominal element thickness smaller than 40 mm of a series of standard hot-rolled structural steel grades according to EN 1993-91-1 [1] listed in Table 3.2. 600

Stress (MPa)

500 400 300

Experiment Quad-linear model

200

Nonlinear model ECCS model

100 0 0

0.02

0.04

0.06 0.08 Strain

0.1

0.12

0.14

FIGURE 3.6 Comparison of experimental stressestrain curves with different material models. Yun X, Gardner L. Stress-strain curves for hot-rolled steels. J Constr Steel Res 2017; 133:36e46.

177

3.2 Stressestrain response of materials

TABLE 3.2 Values of key parameters for a series of standard structural steel grades. Steel grade

E (MPa)

sy (MPa)

su (MPa)

εy %

εu %

εsh %

Esh (MPa)

a1

S235

210,000

235

360

0.11

20.83

1.50

1616

0.33

S275

210,000

275

430

0.13

21.63

1.50

1925

0.35

S355

210,000

355

490

0.17

16.53

1.75

2283

0.38

S450

210,000

440

550

0.21

12.00

2.50

2895

0.41

As discussed in Chapter 2, the resistance of square steel tubes is governed by local buckling. However, stocky square steel tubes are likely to present strain hardening and reach stresses larger than their yield stress. The Sakinos model [7] can be applied for square steel tubes, as shown in. According to this model, three scenarios can be expected for the stresse qffiffiffiffi s strain curve of square steel tubes based on their slenderness ratio Bt Ey . The first scenario occurs for the square tube with a small Bt ratio in which the steel tube reaches stresses greater than its yield stress because of the strain hardening. In the second scenario, the steel tube with a moderate Bt ratio achieves its yield stress. The last scenario happens for the steel tube with a thin steel tube (large Bt ratio). In this case, the steel tube cannot obtain its yield stress due to its susceptibility to local buckling. Table 3.3 summarizes the required input parameters for defining the stressestrain curve of square steel tubes based on the Sakino model [7].

TABLE 3.3 Required input parameters for defining the stressestrain curve of square steel tubes based on the Sakino model.

  sy Max   εy Max

εyE S1 ¼

Small Bt ratio qffiffiffiffi sy B E £ 1:54 t sy S1 0

1

C B B6:06  r1ffiffiffiffiffi 2 0:801  r1 ffiffiffiffiffi  þ1:10Cεy A @ B sy B sy t E t E

εy   2 1 s 0:698 þ 0:128 Bt Ey

Moderate Bt ratio qffiffiffiffi s 1:54 < Bt Ey < 2:03

Large Bt ratio qffiffiffiffi sy B E ‡ 2:03 t

sy

sy S2

εy

sy E

εy

sy E

S2 ¼

1   2 s 4 0:698 þ 0:128 Bt Ey  6:97

178

3. Analytical methods

Compared with normal-strength steel, high-strength and cold-formed steels present a moderate strain hardening and are defined by a rounded stressestrain curve. This behavior of high-strength steel tubes and coldformed steel tubes can be taken into account using Liang’s model, as shown in Fig. 3.3g. The round region of the stressestrain curve is governed by:  1 ε  0:9εy 45 for εy < ε  εsh (3.19) sðεÞ ¼ sy εsh  0:9εy For ease of calculation, the curved part of high-strength steel tubes can be replaced with a straight line, as shown in Fig. 3.3g. In this model, the amount of εsh for mild carbon steel tubes is similar to the recommendation of ECCS publication, and it should be taken as 0.005 for cold-formed or high-strength steel tubes. Besides, the ultimate axial strain of mild carbon steel tubes and high-strength and cold-formed steel tubes are taken as 20% and 10%, respectively. This is to distinguish the ductility of mild carbon steel tubes with high-strength and cold-formed steel tubes. The reduction coefficient of 0.9 in Eq. (3.19) is applied to take into account the effect of hoop stress on the yield strength of the steel tube. In CFST/ CFDST columns subjected to axial compression, the steel tubes experience the biaxial stress state. Therefore, the imposed hoop compression or tension due to confining the concrete core reduces the tube yield strength. The stresses corresponding to strains in the range of εsh and εu are defined by:    εu  ε n  (3.20) su  sy for εsh < ε  εu sðεÞ ¼ su  εu  εsh   εu  εsh (3.21) n ¼ Esh su  sy In practical design calculations and analytical formulations of CFST/ CFDST members, where strains of general structural interest (typically smaller than 5%) are considered, the stressestrain relationship of steel has an insignificant impact on the ultimate strength. However, it can have a slight influence on the postpeak region of the curve. This is because the steel material hardly shows strain hardening at small strains. By contrast, strain hardening of steel and the election of an appropriate model for predicting steel material behavior becomes important in analyzing structural members’ behavior that undergoes severe deformation. The influences of the steel model in the outputs of nonlinear FE analysis on CFST/CFDST columns are discussed in Chapter 4.

179

3.2 Stressestrain response of materials

3.2.2 Stainless steel The stressestrain behavior of stainless steel is different from that of carbon steel. Stainless steel usually shows excellent ductility. After fracturing of austenitic stainless steel, the steel can present a remarkable elongation under tension. Fig. 3.7 compares the stressestrain behavior among typical carbon steel grades, stainless steel, high-strength steel, and aluminum up to 2% strain. It can be found from Fig. 3.7 that at small strains, the behavior of carbon steel material can be described by an idealized bilinear model. By contrast, applying this model to stainless steel, high-strength steel, and aluminum leads to inaccurate results. Generally, carbon steel exhibits a linear elastic response up to the yield strength, and there is usually a yield plateau before encountering the strain hardening. By contrast, stainless steel demonstrates a more rounded behavior, and its yield stress is not well defined. For materials such as stainless steel, it is required to define an equivalent yield point for design due to the lack of an obvious yield point. A common method is to select the stress at 0.2% plastic strain. Different grades of stainless steel are available based on their chemical compositions and heat treatment. The use of three kinds of stainless steel for structural applications has been recommended by the European design regulations EN 10088 [8,9], and American code ASTM A959 [10], namely, ferritic, austenitic, and duplex (austenitic-ferritic). Compared with austenitic and duplex stainless steels, Ferritic stainless steels are generally cheaper but have the weakest corrosion resistance due to the limited nickel content in their composition. Besides, ferritic stainless steels’ weldability is less among the three categories due to the heataffected zones’ embrittlement. Nevertheless, they generally demonstrate better engineering properties compared with austenitic alloys. The most 700 High-strength steel

600

Stainless steel

Stress

500 400 300

Carbon steel

200

Aluminum

100 0 0

0.005

0.01 Strain

0.015

0.02

FIGURE 3.7 Typical stressestrain curves of steel materials.

180

3. Analytical methods

popular stainless steels among Civil Engineers are austenitic stainless steel. This is because they generally have appropriate corrosion resistance and desirable forming and fabrication properties. Duplex stainless steels are composed of a combination of ferrite and austenite stainless steel microstructures. Compared with austenitic stainless steels, duplex stainless steels usually have higher strengths, the same weldability, and similar or higher corrosion resistance. However, their formability is lower than austenitic stainless steel. Different grades of structural stainless steels recommended in different design regulations, i.e., EN 10088-1 [8], ASTM A959 [10], and GB/T 20878 [11], are summarized in Table 3.4. Besides, the main chemical compositions of each grade have been presented in this table,   based on EN 10088-1. Table 3.5 summarizes the nominal yield stress sy and ultimate strength ðsu Þ for each stainless steel grade, according to EN 10088-2. The comparison of the stressestrain responses of carbon steel and various kinds of stainless steel is depicted in Fig. 3.8. Choosing the most suitable grade of stainless steel depends on the environmental condition of the structure, the method of fabrication, the finishing of the surface, and the structure maintenance. 3.2.2.1 RambergeOsgood model For predicting the buckling resistance of tubes fabricated by stainless steel, iron, and brass, a polynomial definition can be used   to represent the behavior of the material after the proportional limit sp as follows: 8 s for s  sp > > < E0   (3.22) ε¼ s  sp n > s > : þ εy for s > sp sy  sp E0 where the parameter n is used to define the nonlinearity of the curve. Considering n/N leads to the perfect elastic-plastic behavior. Based on Eq. (3.22), a model was developed for aluminum alloys by Ramberg and Osgood [12].   s s (3.23) ε¼ þ K E0 E0 where K is governed by: K ¼ εy

 n E0 sy

(3.24)

Conventionally, the amount of sy is taken as 0.2% proof stress s0:2 , and it is widely recommended by different design regulations such as

TABLE 3.4 Grades and chemical compositions of stainless steels.

Austenitic

Duplex

Ferritic

Grade

Main chemical composition according to EN 10088

EN 10088e1:2005

ASTM A989-2009

GB/T 20878-2007

C

Cr

Ni

Mo

1.4301

304

06Cr19Ni10

0.07

17.5e19.5

8.0e10.5

e

1.4404

316

06Cr17Ni12Mo2

0.07

16.5e18.5

10.0e13.0

2.0e2.5

1.4541

321

06Cr18Ni10Ti

0.08

17.0e19.0

9.0e12.0

e

1.4162

S32101

e

0.04

21.0e22.0

1.35e1.9

0.1e0.8

1.4462

S32205/S31803

e

0.03

21.0e23.2

4.5e6.5

2.5e3.5

1.4003

S40977

0.22Cr12

0.03

10.5e12.5

0.3e1.0

e

1.4521

444

019Cr19Mo2NbTi

0.025

17.0e20.0

e

1.8e2.5

3.2 Stressestrain response of materials

Type

181

182 TABLE 3.5

3. Analytical methods

Nominal yield strength and ultimate strength of stainless steels based on EN 10088. Cold rolled strip: t £ 8 mm su (MPa)

sy (MPa)

su (MPa)

210

520

210

520

530

220

530

220

520

220

520

200

520

200

500

1.4162

530a

700a

480b

680b

450

650

1.4462

500

700

460

700

460

640

c

450c 420d

Grade

sy (MPa)

Austenitic

1.4301

230

540

1.4404

240

1.4541

280

450

280

450

250

1.4521

300

420

280

400

280d

b: t  10 mm

c: t  25 mm

Sress

a: t  6:4 mm

1.4003

d: t  12 mm

Stress

Ferritic

sy (MPa)

Hot rolled plate: t £ 75 mm

su (MPa)

Type

Duplex

Hot rolled strip: t £ 13:5 mm

Carbon steel Austenitic stainless steel Duplex stainless steel Ferritic stainless steel Strain (a) Initial stage of loading

Carbon steel Austenitic stainless steel Duplex stainless steel Ferritic stainless steel Strain (b) Full range of loading

FIGURE 3.8 Schematic view of stressestrain curves of carbon steel and stainless steel alloys. (a) Initial stage of loading (b) Full range of loading. Han, L.-H., Xu, C.-Y., & Tao, Z. (2019). Performance of concrete filled stainless steel tubular (CFSST) columns and joints: Summary of recent research. Journal of Constructional Steel Research, 152, 117e131. doi: https://doi.org/10. 1016/j.jcsr.2018.02.038.

AS/NZS 4373:2001 [13], Eurocode 3, Part 1e4 [14], and SEI/ASCE [15]. Therefore, Eq. (3.23) can be written as follows:  n s s (3.25) ε ¼ þ 0:002 E0 E0

3.2 Stressestrain response of materials

183

The 0.01% proof stress s0:01 is usually applied for calculating the nonlinear constant n: n¼

lnð20Þ   s0:2 ln s0:01

(3.26)

3.2.2.2 Mirambell and Real’s model The modified RambergeOsgood model (Eq. 3.25) is relatively simple and can accurately predict the stressestrain behavior of material before reaching the 0.2% proof stress s0:2 . However, using this model for defining material behavior for stresses greater than s0:2 can overestimate the results, which are higher than the actual values, as shown in Fig. 3.9. For increasing the accuracy of the RambergeOsgood model, various improvements are recommended. Mirambell and Real [16] developed a smooth two-stage material model by introducing a second Ramberge Osgood curve, as depicted in Fig. 3.10. As illustrated in Fig. 3.10, the first stage of the curve is defined using the modified RambergeOsgood model, and the second stage which is generated from the outset of the 0.2% proof stress s0:2 is governed by:    s  s0:2 s  s0:2 s  s0:2 m ε¼ þ εu  ε0:2  (3.27) E0:2 E0:2 su  s0:2 where su and εu is the ultimate strength and the corresponding strain, respectively, ε0:2 represent the strain at s0:2 , E0:2 is the tangent modulus of elasticity at the second stage, and m represents the degree of nonlinearity of the second stage.

Stress

Ramberg–Osgood model

Real behavior

200

0 .

Strain

FIGURE 3.9 Comparison between the real stressestrain behavior of stainless steel and the RambergeOsgood model.

184

3. Analytical methods

The 1st stage

The 2nd stage

0.002

FIGURE 3.10 Two-stage model of Mirambell and Real.

E0:2 in Mirambell and Real’s model must be equal to the tangent modulus of elasticity at the last point of the first step. Hence, E0:2 can be obtained by: E0:2 ¼

E0



E0 1 þ 0:002n s0:2



(3.28)

Assuming that the ultimate plastic strain and the total ultimate strain are approximately equal and depend on the ss0:2u ratio, the second stage of Mirambell and Real’s model can be revised for austenitic, ferritic, and duplex stainless steels [17]. This model, introduced in Annex C of Eurocode 3, Part 1e4 [14], can be calculated using the following formula.    s  s0:2 s0:2 s  s0:2 m þ 1 þ ε0:2 (3.29) ε¼ E0:2 su su  s0:2 where the second-stage nonlinear parameter m is governed by: s0:2 m ¼ 1 þ 3:5 su The ultimate strength su of stainless steels is governed by:   8 s0:2 > for austenitic alloys > > 0:2 þ 185 E0:2 > s0:2 <   ¼ s0:2 su > 0:2 þ 185 > > > E0:2 : for all stainless steel alloys 1  0:0375ðn  5Þ

(3.30)

(3.31)

3.2 Stressestrain response of materials

185

To improve the accuracy of the second stage and make sure that the predicted stressestrain curve passes through proof stresses of s1:0 and s2:0 , the value of m can be calculated using the following equations.         s1:0  s0:2 s1:0  s0:2 su  s0:2 ln 0:008 þ   ln εu  ε0:2  E0 E0:2 E0:2 m¼ lnðs1:0  s0:2 Þ  lnðsu  s0:2 Þ (3.32)         s2:0  s0:2 s2:0  s0:2 su  s0:2 ln 0:018 þ   ln εu  ε0:2  E0 E0:2 E0:2 m¼ lnðs2:0  s0:2 Þ  lnðsu  s0:2 Þ (3.33) 3.2.2.3 Gardner and Nethercot’s model Generally, the ultimate strain corresponding to su occurs at large strains that are usually greater than the general structural interest strains. Hence, a significant deviation between the predicted material behavior and the experimental results occurs when a lower strain is adopted. Additionally, due to the absence of the necking phenomenon, and hence, the lack of ultimate stress su , the Mirambell and Real model is not useful for predicting the compressive stressestrain behavior. These issues can be considered the main drawbacks of the Mirambell and Real model for defining the second stage. Therefore, it is proposed to end the second portion of RambergeOsgood’s curve at the 1.0% proof stress s1:0 [18]. Accordingly, Eq. (3.27) can be rewritten as follows:    s  s0:2 s1:0  s0:2 s  s0:2 n2:01:0 þ 0:008  ε0:2  þ ε0:2 (3.34) ε¼ E0:2 E0:2 s1:0  s0:2 where n2:01:0 is the nonlinear parameter expressing the stressestrain curve that crosses through s0:2 and s1:0 . The amount of n2:01:0 can be defined from measured stressestrain curves or predicted using the following equations.    8 E0:2 s1:0 > > 12:225 þ 1:037 for tension coupons > < E0 s0:2 (3.35) n2:01:0    > E0:2 s1:0 > > : 6:399 þ 1:145 for compression coupons E0 s0:2 3.2.2.4 Quach’s model Gardner and Nethercot’s model (Eq. 3.34) can predict the stressestrain behavior of stainless material both in compression and tension up to

186

3. Analytical methods

strains of general structural interest. However, this model is not useful for predicting the behavior of stainless steel at high strains. Understanding the material behavior at high strain levels is usually not required for the primary applications of civil engineering. This is because, in members such as columns, in which overall buckling generally governs the design strength, the amounts of stresses are typically less than the 1.0% proof stress. By contrast, in some cases, such as the design of connections under tension, it is required to define the complete stressestrain behavior up to the ultimate strength. The three-stage model of Quach et al. [19] can be used to assess the behavior of members fabricated with stainless steel subjected to severe deformation. In this model, the first and second steps are defined by using Eqs. (3.25) and (3.34) respectively, and the third stage is based on the true stress and strain values, as follows:

(3.36) where the upper and lower signs of the third stage correspond to tension and compression, respectively. Constants a and b in Eq. (3.36) can be achieved from values at the boundary points ðst2:0 ; εt2:0 Þ and ðstu ; εtu Þ in which the superscript “t” indicates true. The true stress-nominal strain equation at ε > ε2:0 is governed by: st ¼ a þ bε

(3.37)

Eqs. (3.38) and (3.39) can be used for both compression and tension to convert the nominal stress and strain to the true stress and strain, respectively. st ¼ sð1  εÞ

(3.38)

εt ¼  lnð1  εÞ

(3.39)

Constants a and b can be determined using the boundary values of stresses and strains and replacing Eq. (3.38) into Eq. (3.37). a ¼ s2:0 ð1  ε2:0 Þ  bε2:0

(3.40)

3.2 Stressestrain response of materials

b¼

su ð1  εu Þ  s2:0 ð1  ε2:0 Þ εu  ε2:0

187 (3.41)

in which ε2:0 ¼

s2:0 þ 0:02 E0

(3.42)

3.2.2.5 The generalized multistage model The model of Mirambell and Real based on the modified Ramberge Osgood model can be generalized to achieve a simplified multistage model [20]. Each step is expressed employing the tangent modulus of pl

elasticity Ei, stress si , and plastic strain εi along with the nonlinear parameter ni, as shown in Fig. 3.11. The generalized multistage model is governed by:  ni s  si s pl  s  si þ εi þ εi þ i si < s  siþ1 (3.43) ε¼ Ei Dsi E0 The tangent modulus of the next segment is calculated using the following equation: Ei   Ei 1 þ εi ni Dsi   1 1 pl  εi ¼ Dεi  Dsi Ei E 0 Eiþ1 ¼

FIGURE 3.11 Parameters of the generalized multistage model.

(3.44)

(3.45)

188

3. Analytical methods

A specific number of stresses and corresponding plastic strains or the difference between each point and the next calculated ones are required to define the material’s behavior. pl

pl

pl

Dεi ¼ εiþ1  εi

(3.46)

Dsi ¼ siþ1  si

(3.47)

In defining the successive stresses, they must satisfy the constraint related to each section’s initial tangent modulus. pl

si < siþ1 < si þ Dεi

E0 Ei E 0  Ei

(3.48)

The nonlinear parameter at each step ni can be calculated using the following equation in which an additional point j is used between points i and i þ 1: ! Dεi ln εj  εi ! ni ¼ (3.49) Dsi ln sj  si where Dεi ¼ εiþ1  εi is the total strain difference. The comparison between different material models in predicting the stressestrain curve of 1.4003 ferritic stainless-steel grade (see Table 3.5) is illustrated in Fig. 3.12 for strains lower than 10% and Fig. 3.13 for strains higher than 10%.

400 350 Stress (MPa)

300 250

Multistage

200

Rasmussen

150

Ramberg-Osgood

100

Mirambell-Real

50

Gardner

0 0

0.002

0.004

0.006 0.008 Strain ε

0.01

0.012

0.014

FIGURE 3.12 Comparison of different models at strains smaller than 10%. Hradil P, Talja A, Real E, Mirambell E, Rossi B. Generalized multistage mechanical model for nonlinear metallic materials. Thin-Walled Struct 2013;63:63e9.

189

3.2 Stressestrain response of materials

400

Stress (MPa)

380 360 340

Multistage

320

Rasmussen Ramberg-Osgood

300

Mirambell-Real 280

Gardner

260 0

0.02

0.04

0.06 Strain ε

0.08

0.1

0.12

FIGURE 3.13 Comparison of different models at strains greater than 10%. Hradil P, Talja A, Real E, Mirambell E, Rossi B. Generalized multistage mechanical model for nonlinear metallic materials. Thin-Walled Struct 2013;63:63e9.

3.2.2.6 Inversion of the stressestrain relationship In some cases, it may be preferred to express the stress as a function of strain. The following model defines the explicit form of the stressestrain model [20]: s ¼ si þ

Ei ðε  εi Þ   ðε  εi Þ pi 1 þ ðri  1Þ Dεi

εi < ε  εiþ1

(3.50)

in which Dεi Dsi

(3.51)

Ei 1 þ ðri  1Þni

(3.52)

r i ¼ Ei Eiþ1 ¼

Dεi Dsi

(3.53)

1  si ri  1

(3.54)

si ¼ Eiþ1 pi ¼ ri

All constitutive strains must satisfy the constraint related to each section’s initial tangent modulus: εiþ1  εi þ

Dsi Ei

(3.55)

190

3. Analytical methods

3.2.3 Confined concrete Contrary to the steel tubes, modeling the behavior of infilled concrete in CFST/CFDST members is more complicated due to the confinement effects provided by steel tubes. Fig. 3.14 shows the condition of the internal forces in a circular CFDST column  under the axial compression. Imposing the external axial pressure ðfe on a CFDST column makes the sandwiched concrete expands in the lateral direction. However, steel tubes restrict the lateral expansion of the infilled concrete and impose the  lateral confining pressure ðfl in the radial direction. Besides, the arching  action of the infilled concrete generates the circumferential pressure ðfcir . The confinement effect enhances the structural performance of CFST/ CFDST members by improving the stiffness, strength, ductility, and energy absorption capacity of the infilled concrete. Fig. 3.15 compares the stressestrain curve of the confined concrete with unconfined concrete. Generally, two types of confinement can be generated in composite columns: active and passive. Passive confinement is mainly applied to composite columns by limiting the lateral strain (i.e., the lateral enlargement) of the confined concrete employing different methods such as internal transverse reinforcement or external confining methods like

(a) Cross-section

(b) Internal forces

(c) Stresses on concrete wall

FIGURE 3.14 Free body diagram for CFDST columns. (a) Cross section (b) Internal forces (c) Stresses on concrete wall.

191

3.2 Stressestrain response of materials

Compressive stress

Confined concrete

Unconfined concrete

Compressive strain FIGURE 3.15 Stressestrain relations of confined and unconfined concrete.

using fiber-reinforced polymers (FRPs) or steel tubes. By imposing the compressive load to the composite column, the axial strain of the confined concrete increases, and the lateral strain of the confined concrete reduces because of the effect of Poisson’s ratio. No lateral pressure is applied to the infilled concrete in passive confinement prior to applying the axial compressive load. For achieving sufficient lateral pressure in passive confinement, extensive deformation must be developed in the infilled concrete in the lateral direction. In other words, the confining pressure increases continuously with increasing the expansion of the concrete in the lateral direction. For imposing active confinement, it is required to subject the concrete to lateral prestressing. This lets the concrete be under the active confining stress before developing splitting cracks in the concrete. Compared with passive confinement, active confinement is relatively more effective. As mentioned before, the confining stress in passively confined concrete increases with increasing the axial or lateral deformation, whereas the confining stress for actively confined concrete remains constant during the loading process. Therefore, the amount of axial deformation in actively confined concrete is less than the passively confined concrete at given axial stress due to the more considerable axial stiffness and slower development of tensile splitting cracks in concrete. Alternatively, the axial stress will be smaller in passively confined concrete at given axial strain and confining stress. 3.2.3.1 Constitutive stressestrain model for confined concrete The precise definition of the theoretical stressestrain model for infilled concrete confined by steel tubes has remarkable impacts on the accuracy of outputs achieved from the numerical studies on CFST/CFDST

192

3. Analytical methods

members. Compared to the stressestrain models of steel material that have moderate effects on predicting the behavior of CFST/CFDST members, the constitutive confined concrete model remarkably affects the results. Therefore, researchers have put efforts into developing precise and reliable stressestrain relationships for confined concrete that can take into account the passive confinement effect and the nonlinear behavior of the infilled concrete. For developing such models, it is commonly considered that the confining stress in CFST/CFDST members is constant during the loading history. Therefore, it is assumed that passively and actively confined concrete follow the same axial stressestrain relationship, and the stress path does not change the stressestrain behavior of confined concrete. This is the principal hypothesis for developing the stressestrain models of confined concrete. This section discusses the developed constitutive models for concrete confined by various crosssectional shapes of steel tubes in CFST and CFDST members. Circular and square CFST members The stressestrain model of Mander et al. Mander et al. [21] developed a unified stressestrain model for the concrete confined by circular and rectangular-shaped transverse reinforcements. This model is also applicable for predicting the prepeak and postpeak behaviors of infilled concrete of CFST/CFDST columns under axial compressive load. Mander et al. used an expression recommended by Popovics [22] for predicting the longitudinal stress of infilled concrete subjected to the monotonic axial compression loading with a quasi-static strain rate as follows:  
fcc0 l εc ε0 cc (3.56) sc ¼ l 
l  1 þ εc ε 0 cc

in which the term l refers to the concrete brittleness constant. Based on Ref. [23], l is governed by: l¼

Ec  
Ec  fcc0 ε0

(3.57)

cc

where sc represents the longitudinal stress of the concrete, fcc0 and ε0cc are the compressive strength of confined concrete and the corresponding strain, respectively, εc the longitudinal compressive strain of the concrete, and Ec is Young’s modulus of concrete. For developing the relationship for compressive strength fcc0 and the corresponding strain ε0cc , it is assumed that the concrete core is confined by active hydrostatic fluid pressure. Therefore, fcc0 and ε0cc can be predicted by using the following equations: fcc0 ¼ fc0 þ K1 fr

(3.58)

3.2 Stressestrain response of materials

ε0cc ¼ ε0c

K2 f 1þ 0 l fc

193

! (3.59)

where fc0 and ε0c are the compressive strength of unconfined concrete and the corresponding strain, respectively, fr is the lateral fluid pressure (lateral confining pressure), and coefficients K1 and K2 are functions of the lateral pressure and the concrete mix. The amount of K1 can be taken as 4.1 and K2 ¼ 5K1 . The concrete core’s initial elastic modulus under passive confinement is almost similar to the corresponding unconfined concrete. This is because at the initial loading stage, when both steel tubes and the concrete core have elastic behavior, the concrete core’s lateral expansion is small, and there is no trace of interaction between the concrete core and the steel tubes. Therefore, Ec can be obtained by: qffiffiffiffi ðMPaÞ (3.60) Ec ¼ 5000 f 0c or the equation recommended by ACI [24] as follows: qffiffiffiffi ðMPaÞ Ec ¼ 3320 f 0c þ 6900

(3.61)

Although this model has been developed based on the concrete core confined by steel reinforcement bars, the main stressestrain model can also be applied to predict the concrete core’s behavior confined by steel tubes. The stressestrain model of Attard and Setung A fractional function is applied in this model [25]. This model that is shown in Fig. 3.16 is appropriate for defining the axial stressestrain responses of a broad range

Softening stage

FIGURE 3.16 compression.

Stressestrain model of Attard and Setunge for the concrete under triaxial

194

3. Analytical methods

of normal- and high-strength concretes. However, the main limitation of this method is that it is only valid when the confining pressure is relatively low, and the condition fc0

fr fc0

 0:2 must be met in which fr is the

confinement stress, and is the uniaxial compressive strength of the concrete. This model is defined by: sc A:X þ B:X2 ¼ f0 1 þ C:X þ D:X2 where f0 is the peak stress at ε0 , and X ¼ εε0 cX‡0

(3.62) 0  sf0c  1. Constants

A, B, C, and D are used to define the ascending and descending stages of the curve, and they are different for confined and unconfined concrete. Depending on the ascending or descending stage, the constants should be determined based on the boundary conditions. It is assumed that the behavior of the concrete from the beginning of loading up to the proportional limit fpl ¼ 0:45fc0 is elastic. The boundary conditions for the ascending stage of the curve are as follows: sc ¼ 0;

dsc dsc sc ¼ Eti ; sc ¼ f0 ; ¼ 0; sc ¼ f0 ; ε ¼ ε0 ; sc ¼ fpl ; ε ¼ (3.63) dε dε Ec

where Eti represents the initial tangent modulus at zero stress, and Ec is the secant modulus of concrete calculated at a stress level of fpl z 45%f0 . Ec can be calculated using Eq. (3.64) for crushed aggregates or Eq. (3.65) when weaker aggregates, like vesicular basalt, are used for the mix design of the concrete. qffiffiffiffi (3.64) Ec ¼ 0:043r1:5 f 0c q ffiffiffiffi  r 1:5  Ec ¼ 3320 f 0c þ 6900 (3.65) 2320  
in which r Kg m3 represents the surface dry unit weight and fc0 is the uniaxial compressive strength. Based on the boundary conditions, given in Eq. (3.63), constants A, B, C, and D can be defined as follows: A¼ B¼

Eti f0

ðA  1Þ2 A2 ð1  at Þ !þ !1 fpl fpl fpl 2 at at 1  1 f0 f0 f0

(3.66) (3.67)

C ¼ ðA  2Þ

(3.68)

D ¼ ðB þ 1Þ with either B  0 or jBj < A

(3.69)

3.2 Stressestrain response of materials

195

where at ¼ EEtic . Constants A and B can be simplified by assuming Eti ¼ Ec and fpl ¼ 45%fc0 . E c ε0 f0

(3.70)

ðA  1Þ2 1 0:55

(3.71)

A¼ B¼

The postpeak region of the curve depends on whether there is confinement effect or not. The boundary conditions for the descending part are: sc ¼ f0 ;

dsc ¼ 0; sc ¼ f0 ; ε ¼ ε0 ; sc ¼ fi ; ε ¼ εi dε

(3.72)

when the concrete is The amount of residual stress fres at the final strain   under uniaxial stress condition is taken as zero limε/N fres ¼ 0 . For the confined concrete, an additional point is considered on the postpeak region of the curve at sc ¼ f2i , ε ¼ ε2i ¼ 2εi  ε0 , as shown in Fig. 3.16. Constants A, B, C, and D for the softening stage of the concrete under uniaxial stress conditions are governed by: A¼

fic ðεic  εc Þ εc εic sc  fic

(3.73)

B¼0

(3.74)

C ¼ ðA  2Þ

(3.75)

D¼1

(3.76) fc0

where εc is the strain corresponding to and is governed by: 8 > > fc0 4:24 > > pffiffiffiffi Crushed agregates > > < Ec 4 f 0 c εc ¼ 0 > f 3:78 > > > c pffiffiffiffi Gravel agregates > > E : c 4 fc0

(3.77)

fic and εic are the stress at the inflection point and the corresponding strain, respectively. fic and εic for the concrete under uniaxial stress state can be obtained by:   εic ¼ 2:5  0:3 ln fc0 fc0  20 MPa (3.78) εc   fic ¼ 1:41  0:17 ln fc0 fc0  20 MPa (3.79) 0 fc

196

3. Analytical methods

In this model, it is supposed that the ascending stage of the curve for triaxial and uniaxial conditions are the same. Hence, Eqs. (3.73)e(3.76) can be used to determine constants A, B, C, and D of concrete under confinement. However, the postpeak behavior of concrete under confinement is different from that of under uniaxial state. Constants A, B, C, and D for the softening stage of the confined concrete are defined by:  

ε  εi ε E 4ε E  2i i    i 2i  (3.80) A ¼ 2i ε0 f0  fi f0  f2i 

E 4E2i  B ¼ ðεi  ε2i Þ  i    (3.81) f0  fi f0  f2i C ¼ ðA  2Þ

(3.82)

D ¼ ðB þ 1Þ with B  0 and either A > 0 or jAj < B

(3.83)

f

f

where Ei ¼ εii and E2i ¼ ε2i2i . The compressive strength of the confined concrete f0 is achieved by:  k f0 fr ¼ þ1 (3.84) fc0 ft !  0 0:21 fr k ¼ 1:25 fc 1 þ 0:062 0 (3.85) fc where fr is the confinement stress, and ft represents the tensile strength of concrete and can be taken as 90% of the split cylinder tensile strength fsp . fsp can be predicted from the uniaxial compressive strength, as follows: 8   < 0:32 fc0 0:67 Without silica fume fsp ¼ (3.86)  0:67 : With silica fume 0:62 fc0 The strain ε0co corresponding to the peak point is given by the following expression: !  fr  ε0 0 ¼ 1 þ 17  0:06fc (3.87) εc fc0 in which fr presents the confining stress. The amount of stress fi and the corresponding strain εi at the inflection point can be taken as: εic 2 εi εc þ2 (3.88) ¼ !0:26 ε0 fr þ1 1:12 0 fc

3.2 Stressestrain response of materials

fic 1 fi fc0 ¼ !0:57 f0 fr 5:06 0 þ1 fc

197

(3.89)

Substituting Eqs. (3.78) and (3.79) into Eqs. (3.88) and (3.89) leads to:   0:5  0:3 ln fc0 εi þ2 fc0  20 MPa ¼ (3.90) !0:26 ε0co fr þ1 1:12 0 fc   fi 0:41  0:17 ln fc0 þ1 fc0  20 MPa ¼ (3.91) !0:57 f0 fr þ1 5:06 0 fc The stress f2i is given by the following expression:   f2i 0:45  0:25 ln fc0 þ1 ¼ !0:62 fc0 fr þ1 6:35 0 fc

(3.92)

This model assumes that due to the elastic behavior of materials and small expansion of the concrete core than the steel tube, no interaction occurs between the steel tube and the concrete core at the initial loading stage and before reaching the peak point. Therefore, the impact of the confinement on the compressive strength of the concrete core is negligible and fcc0 ¼ f0 . This assumption, however, can lead to conservative results. The model for defining the softening behavior of concrete neglects the confinement level. Hence, it may lead to predicted results smaller than the experimental values. The model is based on concrete with low confinement. Hence, it may lose its precision when the confinement level is high. The stressestrain model of Binici The model of Binici [26] can be used for defining the stressestrain curve of uniaxial concrete and concrete confined by circular steel tubes, steel spirals, and FRP layers. This model consists of three phases, i.e., the linear elastic step where the concrete is assumed to be isotropic, the ascending hardening step that is a function of the lateral pressure, and the softening step that represents the residual capacity of the concrete remaining due to the internal friction. The stressestrain curve of the concrete defined by this model is given in Fig. 3.17. The hardening step is defined by the model proposed by Mander et al. [21], and the softening step is expressed by using an

198

3. Analytical methods

FIGURE 3.17 Stressestrain model of Binici for the concrete under triaxial compression. Binici B. An analytical model for stressestrain behavior of confined concrete. Eng Struct 2005;27(7): 1040e51.

exponential function incorporating the softening fracture energy. The use of softening fracture energy lets the size effects be included in the model. The model is defined by:

(3.93) f

where ε1e ¼ E1ec and f1e indicate the linear elastic limit of the behavior of concrete, and f0 is the peak stress of the concrete at ε0 , shown in Fig. 3.17. It can be observed from Eq. (3.93) that the descending region is defined using an exponential softening function. The modulus of elasticity of concrete is estimated by using ACI-318-02 [27] as follows: qffiffiffiffi Ec ¼ 4750 f 0c ðMPaÞ (3.94) The expression r is governed by: r¼

Ec Ec  E s

(3.95)

f0  f1e ε  ε1e

(3.96)

in which Es ¼

3.2 Stressestrain response of materials

199

The strain corresponding to the peak stress f0 under confinement effect is defined by Ref. [28]: ! f0 (3.97) ε0 ¼ 5εc 0  0:8 fc where εc is the strain at the peak stress of the concrete under uniaxial compression. Based on Ref. [29], εc is governed by:  (3.98) εc ¼  0:067fc02 þ 29:9fc0 þ 1053 106 The amounts of f1e , f0 , and fres are obtained by employing the loading/ failure surface defined by: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 ! fr fr fr 0 (3.99) f ¼ fc kh c þ m 0  ð1  kh Þ 0 þ 0 fc fc fc in which fr is the lateral stress acting on the concrete core, kh is the hardening parameter and it is taken as 0.1 at the elastic limit and 1 at the peak stress and during the softening step, c is the softening parameter taken as 1 in the ascending step and zero at the residual step, and m is a constant parameter and it is defined by: m¼

fc0  ft fc0 ft

(3.100)

It is noteworthy that one parameter MohreCoulomb friction law and the Rankine criterion with a tension cut-off condition are combined in the loading/failure surface given in Eq. (3.101). Besides, the impact of intermediate principal stress is excluded; consequently, it is a useful method for states where the amounts of confining stresses in lateral directions are the same such as axially loaded columns. Additionally, the elastic limit, ultimate strength, and residual capacity are defined directly in terms of stress components. Therefore, there is no need to determine stress invariants. To develop the stressestrain curve, Binici [26] supposed that the concrete does not fail due to the lateral confining stress before applying the axial stress. This assumption is valid if the lateral pressure is less than the biaxial compressive strength of the concrete before applying the axial compressive stress. Besides, the considered assumption is never violated as long as imposing the passive confinement, which is initiated by applying the axial stress. f

The fr0 ratio in Eq. (3.99) is called the confinement ratio. The confinement c

ratio can be used to predict the brittle or ductile behavior of the concrete. The waterecement ratio is a factor that can affect the confinement ratio and change the behavior of concrete from a brittle to a ductile one [30]. In

200

3. Analytical methods

general, the confinement ratio varies between 0.2 and 0.6. The confinement ratio of 0.4 can be considered as the transition point (TP) beyond which the concrete may have a perfectly plastic behavior without softening trend over the postpeak region. For the development of this model, it is assumed that the area below the softening phase is equivalent to the ratio of the uniaxial postpeak compressive fracture energy

Gfcu

obtained from the uniaxial

compression test to the height of the specimen in the direction of the loading ðlc Þ. This hypothesis indicates that instead of considering a true material law, the described model must be considered as a phenomenological model in which the size of the specimen is incorporated. Accordingly, the strains employed in the derivation discussed above are average amounts over the concrete rather than actual ones. By the initiation of localization, the distribution of strain over the concrete height is no longer uniform. Rather than tracking the nonuniform distribution of strain, average values of strain can be used. Considering the equality described above leads to the following expression: (Z  2 ) 

 N  f0  fres ε  ε0 2 dε þ (3.101) f0  fres exp  Gfcu ¼ lc a 2Ec ε0 Based on Eq. (3.101), parameter a can be defined as:  2 ! 2Gfcu f0  fres 1   a ¼ pffiffiffi lc Ec p f0  fres

(3.102)

In this way, the softening area can approximately be regularized by assuring that specimens with different lengths dissipate the same amount of energy during softening behavior. It is assumed that localization begins after reaching the peak stress under axial loading. Typically, bond cracks in the loading and transverse directions are the main reason for the extent of cracking up to around 85% of the peak stress. After that, these cracks combine into continuous cracks, leading to a remarkable expansion of the concrete volume [31]. For plain concrete subjected to uniaxial compression, strain localization happens at the maximum stress [32]. Hence, localization can be supposed to occur approximately at the peak stress for concrete under uniaxial compression. The same assumption may be considered for the concrete confined by a steel tube under axial compression [33]. However, compared to uniaxial compression, the start of localization can be hindered by the triaxial state of stress. Based on the experimental test results [34], spread microcracking and several macrocracks result in a softening response of the concrete subjected to uniaxial compression. By contrast, no distributed cracking is observed in the

3.2 Stressestrain response of materials

201

concrete confined at a constant rate. In this case, the failure of the confined concrete is because of the spread of a few macrocracks. The friction between macrocracks is the reason for the residual capacity of confined concrete. In general, two types of failure can be expected in concrete, i.e., diffusive and localized. Diffusive failure happens when the confinement effect is low. By contrast, localized failure is likely for highly confined areas [34e36]. Because either diffusive or localized failure modes signify a discontinuity at the material level, Binici [26] did not distinguish between them in his recommended model. In other words, no distinction is made between the energy regularization of low and high confinement states. This assumption used by Binici [26] makes the recommended model not reflect the actual behavior of concrete. However, it makes this model easy to use in practical engineering models. Analytical examinations of failure modes prove that depending on the confinement intensity, failure of the concrete can be diffusive or localized. Diffusive failure occurs for low confinement, whereas localized failure takes place for high confinement regions. Each failure type indicates a discontinuity at the material level. However, no difference is considered between them in this model. In other words, it is assumed that the facture energy of the postpeak region for a confined concrete is similar to the concrete under the uniaxial stress state. Therefore, no distinction is made between the energy regularization of low and high confinement states. This assumption is to consider the size effects in the model. Although this assumption makes implementing this model easy in practical engineering models, it may not necessarily reflect the material’s real behavior. Contrary to the hypothesis used in this model, which considers that the compressive fracture energy per unit area is independent of the confinement level, the compressive fracture energy at the postpeak region is not constant, and it alters with the level of confinement. Beyond the f

confinement ratio fr0 of about 0.1 and 0.2, the compressive fracture energy c

is increased by increasing the confinement. From then onwards, the compressive fracture energy reduces until it becomes zero. The reason is that tensile cracking is nullified by imposing low confinement. Therefore, the peak strength and fracture energy develop beyond the uniaxial level. By increasing the confinement, the failure mode is replaced by barreling dispersed cracking. Barreling failure mode initiates when the amount of confinement is sufficient for dispersing microcracks development over the specimen’s volume. In this case, the localized shear band coincides with the peak strength is no longer formed. Therefore, the compressive fracture energy is no longer exists. The boundary between the brittle and ductile compressive behavior of the concrete can be determined by a TP beyond which the confinement effect is high enough to change the behavior from brittle to ductile.

202

3. Analytical methods

Residual envlope Failure envlope

f0 / f c'

2.5 2

Transition point

Continuous hardening

Brittle

fr / f c'

FIGURE 3.18 Relationship between failure and residual stress and confinement. Samani AK, Attard MM. A stressestrain model for uniaxial and confined concrete under compression. Eng Struct 2012;41:335e49.

Usually, the confining stress equal to around 30% of the uniaxial strength is taken as the TP. As illustrated in Fig. 3.18, the failure and residual strength surfaces match beyond the TP. In Binici’s model, it is assumed that the TP occurs when fr ¼ 40%fc0 . Eq. (3.99) is recommended for the residual stress level; however, it does not consider continuous hardening at the proposed TP, leading to higher results than the experimental residual levels at the low confinement level. These two issues can be considered as the main drawbacks of this model. The stressestrain model of Samani and Attard The model of Samani and Attard [37] can be used for defining the stressestrain curve of uniaxial and confined concrete. In this model, the ascending branch is defined by using the modified equation of Attard and Setung (Eq. 3.62), as follows:

sc A:X þ B:X2 ¼ f0 1 þ ðA  2Þ:X þ ðB þ 1Þ:X2

(3.103)

where X ¼ εε0 and 0  ε  ε0 , 0  f  f0 . Constants A and B are governed by Eqs. (3.66)e(3.71). A power function is developed for defining the softening descending branch. In this model, the postpeak region of the curve passes the inflection point, as defined in Attard and Setunge’s model. The recommended function for the postpeak softening region is: 

f fres fres ¼ þ 1 fo fo fo



fic fc0

 2 ! εε0 εi ε0

ε  ε0

(3.104)

203

3.2 Stressestrain response of materials

The inflexion point stress ratio is governed by: !   fi fres fres fic ¼ þ 1 fo fo fo fc0

(3.105)

where fic indicates the uniaxial inflection point, and it is defined by Eq. (3.79). The above function has limits for the inflection point between the uniaxial case and the fully ductile state subjected to high confinement. In the uniaxial case, the residual stress is taken as zero. Hence, the inflection stress corresponds to the value of the value of uniaxial inflection stress fic . In the fully ductile state where the confinement level is high, the residual stress and the stress at the inflection point are equal to the peak stress. Therefore, the postpeak stage for the uniaxial case is defined as follows: f f ¼ ic0 0 fc fc

 2 ! εεc εic εc

εc  ε

(3.106)

  εic ¼ 2:76  0:35 ln fc0 εc

(3.107)

in which εc can be computed by using Eq. (3.77). The amount of strain corresponding to the peak stress under confinement εo can be defined as: !½0:3124þ0:002fc0   fr  ε0 k1 0 ¼ e ; k1 ¼ 2:9224  0:00367fc (3.108) εc fc0 The inflection point strain ratio εε0i can be obtained using an interpolation function as follows:    εi fres fres εic 2:89 ; k2 ¼ 1:26 þ qffiffiffiffi ¼ kþ 1 (3.109) ε0 f0 fo εc f0 c

in which the term k2 represents the limiting amount for the inflection point ratio as fres /f0 . The expression for the residual stress level ratio

fres f0

is

required for computing Eqs. (3.104), (3.105), and (3.109). The residual stress level ratio

fres f0

is governed by: fres ¼1  f0

fr a 0 fc

1 !k3

(3.110) þ1

204

3. Analytical methods

where a ¼ 795:7  3:291fc0 ! !0:694 fr þ 1:301 k3 ¼ 5:79 0 fc

(3.111) (3.112)

The compressive fracture energy per unit area of the concrete under uniaxial loading can be achieved by the integration of the softening stage of the stressestrain relationship, as follows: 1 pffiffiffi 1 0  2 pðεic  εc Þfc0 f Gfcu 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2 c ¼q     lc Ec ln f 0  ln f c

Gfcu lc

(3.113)

ic

for the confined concrete is governed by:   1 2 1 pffiffiffi f0  fres Gfcu 2 pðεic  ε0 Þ f0  fres ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ2  0   lc Ec ln f  ln f c

(3.114)

ic

Specimens with various lengths will have equal plastic or nonrecoverable inelastic displacements but different computed strains during softening step of the stressestrain curve if the compression fracture is fully localized into a shear band. The compressive fracture energy per unit area will not be the same for the concretes with the same aspect ratio greater than or equal to 2 but with different heights under the uniaxial state. It is commonly assumed that concretes with a similar aspect ratio usually display the same stressestrain response in the compression test. However, this assumption cannot be valid due to fracture localization. According to Fig. 3.17, the model of Samani and Attard can be adjusted to incorporate size effects. The total strain can be split into two components, i.e., elastic strain and inelastic strain. Therefore:    8 wpc þ εd hd f0 f > > þ þ  h > hd ε > < 0 Ec h Ec (3.115) ε¼    f > wpc f0 > > : ε0  þ εd þ þ h  hd Ec Ec h where wpc presents the localized inelastic axial displacement because of the shear band fracture, εd is the extra inelastic strain in the damaged region corresponding with longitudinal tensile cracking, h indicates the height of the specimen, and hd is the height of the damaged area and it can

3.2 Stressestrain response of materials

205

be taken as 2 times the width or diameter of the specimen. The following equation can be used for wpc.   f0  f hr  ε d hr (3.116) wpc ¼ ðε  ε0 Þhr þ Ec Therefore, the total strain εh for the concrete of height h can be expressed as:  

   8 f  f0 hr hr hd hr > > ε þ þ ε  h > hd 1  þ ðε  ε Þ > 0 0 d < h h h h Ec εh ¼ (3.117)  

   > > f  f0 hr hr hr > : ε0 þ ðε  ε0 Þ þ þ εd 1  h  hd 1 Ec h h h For using Eq. (3.117), it is required to determine the extra inelastic strain in the damaged region corresponding with longitudinal tensile cracking εd . Based on Ref. [38], by assuming linear stress versus inelastic strain curve, the ultimate inelastic strain because of the longitudinal cracking for the uniaxial state can be defined as: εdu ¼

2kGft   rð1 þ kÞ fc0

(3.118)

in which Gft represents the tensile fracture energy, the term r is a parameter with the dimension of length proportional to the average distance between consecutive longitudinal cracks, and it can be taken as 1.25 mm for the maximum aggregate size of 16 mm. k is a material constant and is taken as 1 for lightweight aggregate concrete and 3 for normal density concrete. Based on Ref. [39], the tensile fracture energy Gft is governed by: Gft ¼ 0:00097fc0 þ 0:0418 N=mm

(3.119)

It should be noted that the value obtained from Eq. (3.119) is only an estimation of the actual tensile fracture energy. In addition to the concrete compressive strength, the magnitude of Gft depends on various parameters such as the aggregate size, type, and property. Eq. (3.119) is used in the model of Samani and Attard for calculating the tensile fracture energies of confined and unconfined concretes. However, the tensile fracture energy is affected by confinement [40]. Here it is assumed that the damaged zone associated with longitudinal tensile cracking has a nonlinear softening behavior for the inelastic strain.

206

3. Analytical methods

The following power function can be used to define the strain-softening behavior: !0:8 !0:8 2kGft fc0  f fc0  f   ¼ f  fc0 (3.120) εd ¼ εdu fc0 fc0 rð1 þ kÞ fc0 For the concrete under confinement effect, it can be assumed that increasing confinement nullifies the ultimate inelastic strain (Eq. 3.118). Therefore:   2kGft 2kGft fo  fres   ¼ (3.121) εdu ¼ fo rð1 þ kÞfo rð1 þ kÞ fo  fres Therefore, Eq. (3.120) can be rewritten as follows:   2kGft fo  f 0:8   εd ¼ fres  f  fo rð1 þ kÞ fo fo  fres

(3.122)

The stressestrain model of Lim and Ozbakkaloglu This model [41] is practical to predict the behavior of normal weight concrete (NWC) and lightweight concrete (LWC) under triaxial compression. Concrete with a
density rc greater that 2250 Kg m3 can be classified as NWC, whereas
concrete with rc less than 2250 Kg m3 can be known as LWC [29]. In general, the compressive strength of concrete depends on the waterto-cement ratio and the density of the concrete. First, the initial equations for predicting the mechanical properties of unconfined concrete are introduced. Based on Ref. [41], the 28-day compressive strength of unconfined concrete cylinder with a dimension of 152  305 mm can be predicted using the following equation: 1 0  q ffiffiffiffiffiffiffi C rc 1:6 B 21 þ 32 sf=c A (3.123) fc0 ¼ @ 2400 w=c

in which 0:2  w=c  1:3 is the water-to-cement ratio, 0  sf=c  0:2 is the
silica fume-to-cement ratio, and 650  rc  2550 Kg m3 . Eq. (3.123) is

valid for concrete cylinder with fc0 up to 120 MPa. The elastic modulus of concrete Ec is affected by the concrete compressive strength fc0 and the density of the concrete rc . It can be expected that NWC has a larger elastic modulus than LWC. Based on Ref. [41], the elastic modulus of NWC and LWC can be defined as: qffiffiffiffi rc 1:4 (3.124) Ec ¼ 4400 f 0c 2400
in which 50  rc  2550 Kg m3 and fc0  120 MPa.

3.2 Stressestrain response of materials

207

The axial strainεco corresponding to compressive strength of unconfined concrete for NWC and LWC is different. In addition to the effect of rc , diameter of the concrete D and the height-to-diameter ratio (aspect ratio) H D are also affect the axial strain εco . Generally, increasing the density of the concrete rc reduces the axial strain εco , and the decline becomes more notable in higher strength concretes. Likewise, the axial strain εco is reduced by increasing D or the H D ratio of the concrete. Based on Ref. [41], the axial strain εco can be expressed as:  0 0:225kd f ks ka (3.125) εco ¼ c 1000   2400 0:45 (3.126) kd ¼ rc   152 0:1 ks ¼ (3.127) D  0:13 2D ka ¼ (3.128) H in which D ðmmÞ and H ðmmÞ are the diameter and height of the concrete, respectively. kd , ks , and ka , respectively, are coefficients to take into account the influences of the concrete density, size of the concrete, and the aspect ratio of the concrete. Eq. (3.15) is valid for concretes with
50  rc  2550 Kg m3 and fc0  120 MPa, 2  H D  8, 50  D  400 mm, and 100  H  850 mm. Fig. 3.19 illustrates the typical stressestrain curves of NWC and LWC under triaxial compression defined by this model. This model considers

Stress

Inflection point

Strain

FIGURE 3.19 Stressestrain model of Lim and Ozbakkaloglu for NWC and LWC under triaxial compression.

208

3. Analytical methods

the parabolic behavior for the ascending stage of the curve. The concrete stress is declined after the peak stress fcc0 . Similar to the models discussed in previous sections, it is considered that interparticle cohesion in the concrete resumes after the peak stress. Besides, during the descending stage of the curve, the frictional action generates the residual strength fres and develops a stabilized plateau, as shown in Fig. 3.19. The compressive strength of the confined concrete fcc0 and the corresponding strain ε0cc are given by:  0:91 fr fcc0 ¼ fc0 þ 5:2 fc0 fc0 fr ε0cc ¼ εco þ 0:045 0 fc

!ðfc0 Þ0:06 (3.129) !1:15 (3.130)

in which εco is given by Eq. (3.125). Knowing the value of the strain corresponding to the inflection point εci can assist the accurate modeling of the location where the shape of the softening region of the stressestrain curve changes. As shown in Fig. 3.19, the sign of curvature of the postpeak region changes from negative to positive at the inflection point. The amount of the strain εci can be expressed as: ! ! ! fres  0 0:12 fres  0 0:47  rc 0:4 0 0 þ 10εcc 1  0 (3.131) fc fc εci ¼ 2:8εcc 0 fcc fcc 2400 The ascending part of the curve (0  εc  ε0cc ) is defined using the equations recommended by Popovics [22] (Eq. 3.56). For predicting the postpeak behavior of the concrete, the following expression can be used: fcc0  fres #  εc  ε0cc 2 þ1 εci  ε0cc

sc ¼ fcc0  "

for εc > ε0cc

(3.132)

In this model, the residual strength of the concrete fres is predicted by: ! fr0:24 0 (3.133) fres ¼ 1:6fcc 00:32  fcc0  0:15fc0 fc The model of Lim and Ozbakkaloglu [41] can be applied to both unconfined and actively confined concretes. For developing the stress and strain of unconfined concrete, fcc0 and the corresponding axial strain ε0cc in Eqs. (3.56), (3.57), and (3.132) must be taken as fc0 and εco , respectively.

3.2 Stressestrain response of materials

209

Eq. (3.132) is based on the experimental test results of the concrete cylinders with a 152 mm diameter and 305 mm height. Accordingly, modifications are required to be applied to the model to account for the effects of the size and slenderness of specimens on axial strains corresponding to the postpeak region. This is to adjust the axial strains of specimens with arbitrary geometric dimensions that are different from the reference specimens [37]. Inelastic deformation refers to a displacement that happens in the compression damage region ðHd Þ within a part of the specimen height, while the elastic component is a strain that happens over the total height of the specimen ðHÞ. The relative elastic and inelastic components of deformation is different. Therefore, the average axial strain εc alters with changing the height H. Eq. (3.134) can be applied to adjust the concrete’s axial strains with arbitrary geometric dimensions different from the geometric properties of reference specimens (152 mm diameter and 305 mm height). This is to consider the relative contributions of the elastic and inelastic components of the total deformation in the average axial strain.

(3.134) in which εc;h is the axial strain for the concrete of height H, Hr ¼ 305 mm is the height of the reference specimen, Hd represents the height of the compression damage zone and is taken as 2D, and εd is the inelastic strain in the damage zone. Based on Ref. [41], εd is governed by: ! 8 0 0 > 0:02f 0 0:5 fcc  fc > for confined concrete > c > < fcc0  fres (3.135) εd ¼ ! > 0 >   f  f > 0:5 c c > : 0:02 fc0 for unconfined concrete fc0  fres By adjusting the axial strain of the concrete, the axial strain εc in Eq. (3.132) should be substituted with the adjusted axial strain εc;h to generate the full stressestrain relationship of a specimen with given geometric properties. Fig. 3.20 illustrates the stressestrain curve of the reference specimen with D ¼ 152 mm and Hr ¼ 305 mm (Fig. 3.20a), and the adjusted stressestrain curve of the specimen with arbitrary geometric dimensions (Fig. 3.20b). It can be seen from Fig. 3.20a that the axial strain can be divided as follows: The elastic component consists of a linear portion that follows a loading path equal to the concrete’s elastic modulus Ec , and a

210

3. Analytical methods

FIGURE 3.20

Axial stressestrain curve of confined concretes based on the model of Lim and Ozbakkaloglu. (a) Typical stressestrain curves of a reference specimen with height Hr. (b) Adjusted stressestrain curve of a specimen with height H. Lim JC, Ozbakkaloglu T. Stressestrain model for normal- and light-weight concretes under uniaxial and triaxial compression. Construct Build Mater 2014;71:492e509.

parabolic part approaching the axial strain ε0cc corresponding to the compressive strength of the confined concrete fcc0 . An unloading path with linear behavior and a slope equal to the elastic modulus of concrete Ec is supposed for the elastic component of the axial strain by reaching the peak point. The remaining strains occur in the damage zone during   the inelastic displacement ðdc Þ. During the inelastic displacement dc;h , an

211

3.2 Stressestrain response of materials

inelastic strain (εd ) occurs in the damaged area of the concrete (Hd ). Besides, the formation of macrocracks at this stage develops a localized displacement (w). Therefore, an inelastic strain (εd ) and the localized displacement (w) form the inelastic displacement ðdc ¼ εd Hd þwÞ, as shown in Fig. 3.20a. A typical stressestrain curve, shown in Fig. 3.20a, can be formed when the height of the damage zone (Hd ) and the height of the specimen (Hr ) are equal. Axial strain adjustment is required when the concrete height (H) is more than the height of the damage zone (H), as shown in Fig. 3.20b. This is to consider the relative contributions of elastic and inelastic components of the total displacement. It is noteworthy that the model of Lim and Ozbakkaloglu [41] is recommended for specimens with H > 2D. This is because in specimens with H < 2D, the impact of the frictional resistance provided by the loading plates at the ends of the concrete can affect the compressive behavior [42,43]. The stressestrain model of Tao et al. This model [44] that is shown in Fig. 3.21 has been developed based on the models of Samani and Attard [37] and Binici [26] and can be used for predicting the behavior of infilled concrete in circular and rectangular CFST members under axial compression. In this model, it is assumed that the confinement effect does not influence the concrete compressive strength. This is in accordance with the definition of passive confinement. As discussed above, there is no or negligible interaction between the concrete core and the steel tube during the initial loading stage in the passive confinement. Therefore, the ascending branch of the curve (from Point O to Point A) can be defined using the stressestrain relationship of unconfined concrete. By reaching the peak strength fc0 , the interaction occurs between the tube and the concrete. The effect of confining pressure on increasing the peak strain is taken into account by considering a plateau from Point A to Point B. From

A

B

Confined concrete

Softening stage

Unconfined concrete

C

O

FIGURE 3.21 Stressestrain model of Tao et al. for confined concrete.

212

3. Analytical methods

then onwards, a softening branch with enhanced ductility due to the confinement effect is defined, as shown in Fig. 3.21. In this model, the ascending branch of the curve (from Point O to Point A) is defined by the model developed by Samani and Attard (Eq. 3.103). Based on Ref. [45], the strain corresponding to the peak stress of the concrete under uniaxial compression εco is governed by: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   εc0 ¼ 0․00076 þ 0․626f 0c  4․33  107 10 MPa  f 0c  100 MPa (3.136) The strain at Point B ðεcc Þ is defined by Eq. (3.108), developed by Samani and Attard [37] in which fr is the confining pressure at Point B. For developing an equation for calculating the confining stress (fr ), the effects of the shape of the cross section, the width-to-thickness ratio of the steel tube Dt, and the concrete compressive strength fc0 are taken into account. As a general trend, the confinement effect increases with increasing the steel tube yield strength fy , whereas it reduces with increasing the concrete compressive strength fc0 and the width-to-thickness ratio of the tube Dt. 8   > > > 1 þ 0․027f $e0․02Dt > y > > > for circular columns >  4:8 > > < 1 þ 1․6e10 $ fc0 fr ¼ (3.137) ffi   0:02pffiffiffiffiffiffiffiffi B2 þD2 > > > t 0․25․ 1 þ 0:027fy $e > > > > for rectangular columns  4:8 > > > 1 þ 1․6e10 $ fc0 : The descending branch of the curve (from Point B to Point C) is defined by the model developed by Binci [26] with a slight modification to consider the influence of the shape of the cross section, as follows:     0  ε  εcc b ε  εcc (3.138) sc ¼ fres þ fc  fres exp  a where fres is the residual strength as depicted in Fig. 3.21. Terms a and b are defined to distinguish between the behavior of the softening stage of circular and rectangular CFST columns. fres is defined as: (   0․7 1  e1․38x fc0  0․25fc0 for circular columns (3.139) fres ¼ 0․1fc0 for rectangular columns

213

3.2 Stressestrain response of materials

b is taken as 1.2 for the circular cross section and 0.92 for rectangular cross section. The term a is proposed as: 8 0:036 > < 0:04  for circular columns 6․08x3․49 1 þ e (3.140) a¼ > : 0․005 þ 0․0075x for rectangular columns The stressestrain model of Hu et al. The model of Hu et al. [46] has been specifically developed to define the stressestrain behavior of the concrete core of circular and square CFST columns. In this model, it is supposed that the concrete has linear behavior up to 50% of the compressive strength, as shown in Fig. 3.22. The nonlinear behavior of the concrete core when the strain is in the range of εe ¼ 0․5ε0cc and ε0cc can be expressed using the stressestrain relationship developed by Saenz [47] as follows:

fc ¼



εc 1 þ ðR þ RE  2Þ 0 εcc



Ec ε c



εc  ð2R  1Þ 0 εcc

2



εc þR 0 εcc

0 0 3 ; 0․5εcc  εc  εcc

(3.141) where RE ¼ R¼

Ec ε0cc fcc0

RE ðRs  1Þ ðRε  1Þ

2

(3.142) 

1 RE

(3.143)

in which constants Rs and Rε can be taken as 4 [48].

Confined concrete 60

Stress

50 Unconfined concrete

Strain

FIGURE 3.22

Stressestrain model of Hu et al. for confined concrete.

214

3. Analytical methods

The initial modulus of elasticity is obtained based on the recommendation of ACI-318 [24], as follows: qffiffiffiffiffiffi (3.144) Ec ¼ 4700 f 0cc ðMPaÞ In this model, the compressive strength of the confined concrete fcc0 and the corresponding axial strain ε0cc are calculated using Eqs. (3.58) and (3.59), respectively. When the strain of the concrete core exceeds ε0cc , the softening behavior of the concrete is modeled by a linear   descending line. The last point of the curve is taken as K3 f 0cc ; 11ε0cc in which K3 is the material degradation coefficient. For a complete definition of the equivalent uniaxial stressestrain behavior, parameters fl and K3 are required. Parameters fr and K3 can be expressed as follows: • For circular cross section:   8 > 21:7  D=t  47 > < 0:043646  0:000832 D=t . (3.145) fr f ¼   y > > : 0:006241  0:0000357 D 47  D=t  150 =t 8 1 21:7  D=t  40 > > <     2 K3 ¼ > 0:0000339 D > : t  0:010085 D=t þ 1:3491 40  D=t  150 (3.146) • For square cross section: 8   > > < 0:055048  0:001885 D=t 17  D=t  29:2 . fr f ¼ y > > :0 29:2  D=t  150

(3.147)

8  2   > > < 0:000178 D  0:02492 D=t þ 1:2722 17  D=t  70 =t K3 ¼ > > : 0:4 70  D=t  150 (3.148) . Fig. 3.23 shows fr f and K3 as the function of the D=t ratio. It can be y seen from the figure that both the lateral confining pressure and the

215

3.2 Stressestrain response of materials

0.03 Circular cross-section

0.025

Square cross-section

fr /fy

0.02 0.015 0.01 0.005 0 10

30

50

70

90

110

130

150

D/t (a)

versus

1.2 Circular cross-section 1 Square cross-section

K3

0.8 0.6 0.4 0.2 10

30

50

70

90

110

130

150

D/t (b)

FIGURE 3.23 versus D=t.

versus

. . Variation of fr and K3 with respect to D= . (a) fr versus D= . (b) K3 t t fy fy

material degradation coefficient are reduced by increasing the D=t ratio. When the D=t ratio is small, fr and K3 tend to be large because of the appropriate confinement effect provided by the steel tube. The amount of lateral confining stress and the material degradation coefficient in the square cross section is lower than in the circular cross section. It is supposed that the square steel tube with D=t greater than 29.2 cannot provide any confinement effect to the concrete core. This assumption is based on the fact that a thin square steel tube is susceptible to local buckling, and it may buckle before the column achieves its ultimate strength. The smaller material degradation coefficient in the square cross section than the circular one is also due to the better local stability of the circular steel tube and hence, better ductility of the column.

216

3. Analytical methods

A

Stress

Z B

C

0O Strain

FIGURE 3.24 Stressestrain model of Susantha et al. for confined concrete. The stressestrain model of Susantha et al. This model [49] is applicable for defining the uniaxial stressestrain behavior of the concrete core of CFST short columns confined by the circular, square, and octagonal steel tubes. The schematic view of this model is shown in Fig. 3.24. In this model, the nonlinear part O-A is defined by the expression developed by Popovics [22] (Eq. 3.56) in which fcc0 and ε0cc to be calculated by Eqs. (3.58) and (3.59), respectively. Referring to Fig. 3.25a, the radial pressure on the inner face of the circular steel tube is given by the following expression:

fr ¼

(a) Circular section

FIGURE 3.25

2t fsr D  2t

(3.149)

(b) Rectangular section

Free body diagram for steel tubes of CFST columns. (a) Circular section (b) Rectangular section.

217

3.2 Stressestrain response of materials

where fsr denotes the circumference stress in the steel tube. It can be assumed that the maximum lateral pressure corresponds to the steel yield strength. Therefore, Eq. (3.149) can be rewritten as follows: fr ¼

2t fy D  2t

(3.150)

As discussed earlier, no interaction occurs at the initial loading stage due to the difference in the Poisson’s ratios of the steel and the concrete core. The difference between the Poisson’s ratios reduces, and interaction takes place by increasing the axial load. Besides, the presence of the concrete core can change the Poisson’s ratio of the steel tube. The change of the Poisson ratio of steel tube and concrete core with column loading is considered by the coefficient 4 as follows: fr ¼ 4

2t fy D  2t

(3.151)

in which 4 ¼ we  ws

(3.152)

where we is the Poisson’s ratio of the hollow steel section without the infilled concrete, and ws is the Poisson’s ratio of the steel tube filled with the concrete and is taken as 0.5. we is governed by: ! ! !2 0 0 0 f f f (3.153) we ¼ 0:2312 þ 0:3582v0e  0:1524 c þ 4:843v0e c  9:169 c fy fy fy v0e ¼ 0:881  106

 3  2   D 4 D 2 D þ 0:4011  2:58  10 þ 1:953  10 t t t (3.154) f0

The above equations are valid for circular columns with fyc in the range of 0.04 and 0.20. For the box sections, the lateral confining stress is expressed by:  0 1:46  1:46 f fr ¼  6:5R c þ 0:12 fc0 (3.155) fy where R is the width-to-thickness parameter and is governed by: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffisffiffiffiffiffi fy b 12 1  w2s (3.156) R¼ Es t 4p2 It should be noted that Eq. (3.155) has been verified based on columns with fc0 in the range of 10 and 50 MPa. Besides, the width-to-thickness

218

3. Analytical methods

parameter R must be less than 0.85. The limit of 0.85 is according to the relationship recommended by Ge et al. [50] for local buckling resistance of CFST columns. The model of Ge et al. [50] is discussed in detail in Chapter 5. By reaching the peak stress, the curve declines by a linear descending line. The following equation governs part A-B: fc ¼ fcc0  Zðε  εcc Þ

εcc  ε  εcu

(3.157)

in which the term Z is defined for considering the effect of the steel tube cross section on the slope of the softening branch and ductility of the column. The slope Z is given by: • For circular CFST columns

(3.158) in which Rt stands for the radius-to-thickness ratio parameter and can be expressed as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi fy D (3.159) Rt ¼ 3 1  w2s Es 2t • For rectangular CFST columns ! 8 . > 0 > 0 for R fc f  0:0039 > > y < Z¼ ! ! > . > fc0 > 0 > : 23400R  91:26 for R fc f > 0:0039 y fy

(3.160)

The ultimate strain corresponding to Point B ðεcu Þ is defined as follows:

219

3.2 Stressestrain response of materials

• For circular CFST columns εcu ¼ 0:025

(3.161)

• For rectangular CFST columns

(3.162) Regardless of the shape of the cross section, the calculated ultimate strain εcu must satisfy the following imitation: εcu  εcc þ

fcc0 Z

(3.163)

The stressestrain model of Liang et al. Constitutive models of confined concrete for CFST columns with circular and square cross sections developed by Liang et al. [51,52] are discussed in this section. The first model [51] is used to predict the behavior of the concrete core confined by the circular steel tube. Fig. 3.26 illustrates the schematic view of this model. The ascending part of the curve (Part OA) is defined using the equations recommended by Popovics [22] (Eq. 3.56) in which fcc0 and ε0cc to 70 A

Stress

B

C

0O Strain

FIGURE 3.26

Stressestrain model of Liang and Fragomeni for the concrete core of circular CFST columns.

220

3. Analytical methods

be calculated by Eqs. (3.58) and (3.59), respectively. For considering the impacts of the column size, the quality of concrete and the loading rates on the compressive strength of the concrete, the strength reduction factor  gc is multiplied by the strength of the concrete gc fc0 . gc is governed by: gc ¼ 1․85D0․135 ; 0․85  gc  1:0 c

(3.164)

in which Dc is the diameter of the concrete core. More information can be found in Chapter 2 regarding the size effect in CFST/CFDST members. Young’s modulus of concrete is obtained by Eq. (3.63) given by ACI [24]. The strain ε0c corresponding to the compressive strength of unconfined concrete is governed by:

(3.165) Fig. 3.27 illustrates the variation of ε0c with respect to the effective compressive strength of the unconfined concrete gc f 0c . Comparing the numerical results with experimental test results indicated that the lateral confining stress model developed by Susantha et al. [49] leads to acceptable accuracy for CFST members built with normalstrength concrete. However, their model overestimates the lateral confining pressure for CFST columns with high-strength concrete due to range  of concrete strength used for verification of the model  the narrow 10  fc0  50 MPa . Besides, the confining pressure model recommended by Hu et al. [46] overestimates the confining pressure of CFST columns built with a diameter-to-thickness ratio smaller than 47 built with highstrength steel tube. Therefore, the current model employed the model of Susantha et al. [49] and Hu et al. [46] for predicting the lateral confining

FIGURE 3.27

Variation of ε0c in circular CFST columns with respect to the gc f 0 c.

3.2 Stressestrain response of materials

221

pressure of CFST columns with Dt  47 and 47 < Dt  150, respectively, as follows:

(3.166) It is noted that the lateral confining pressure model for CFST columns with Dt  47 used herein is 30% less than the original equation (Eq. 3.151). Eq. (3.166) is appropriate for both normal- and high-strength materials. Based on Ref. [53], the lateral confining pressure in circular CFST columns with aluminum steel tubes is expressed as follows:    2   D D D s0:2  0:002297 þ 0:001269ðs0:2 Þ2  0:5329 fr ¼ 0:00249 t t t  0:7214s0:2 þ 104:24 (3.167) The softening branch of the curve is modeled by the following equations:   8   > < bc fcc0 þ εcu  εc fcc0  bc fcc0 for ε0cc < εc  εcu 0 εcu  εcc (3.168) sc ¼ > : 0 bc fcc for εc > εcu where εcu ¼ 0:02 represents the strain corresponding to Point B, shown in Fig. 3.26, and bc is the strength reduction coefficient to define the impacts of confinement provided by the steel tube on the residual strength and ductility of the column. According to Ref. [46], bc is governed by: 8 D > for  40 > < 1․0 t bc ¼  2   > > : 0․0000339 D  0․010085 D þ 1․3491 for 40 < D  150 t t t (3.169) The second model [52] has been developed for predicting the stresse strain behavior of the concrete core confined by the square steel tube, as shown in Fig. 3.28. Similar to the previous model, the ascending region of the curve (Part OA) is defined using the equations recommended by Popovics [22] (Eq. 3.56). In this model, it is assumed that the square steel tube does not enhance the compressive strength of the concrete core and only improves its ductility. Therefore, the compressive strength of the concrete confined by the square steel tube is equal to the compressive strength of unconfined concrete fcc0 ¼ gc f 0c . The axial strain ε0cc corresponding to the strength fcc0 is governed by Eq. (3.165).

222

3. Analytical methods

Stress

A

B

C

40

O

0.005

D

0.015

Strain

FIGURE 3.28 Stressestrain model of Liang for the concrete core of square CFSTcolumns.

By reaching the peak stress, the model shows a plateau from Point A to Point B and then enters the softening region (Part BC). This is similar to Tao et al.’s model [44] (see Fig. 3.21). The postpeak region of the curve is governed by:

(3.170) The strength reduction coefficient bc for square CFST columns is expressed as: 8 B > > 1․0 for  24 > > > t > > < 1 B B (3.171) bc ¼ 1:5  for 24 <  48 > 48 t t > > > > > B > : 0:5 for > 48 t The stressestrain model of Han et al. This model [54] is practical for defining the stressestrain response of the concrete core confined by both circular and square steel tubes. The schematic view of this model is depicted in Fig. 3.29. The equivalent stressestrain model is defined as follows:   2 8  ε ε ε > > 2  1 > > ε ε ε > 0 0 0 > sc < ε ¼ (3.172) fc0 > ε ε > 0 >  > 1 > h > > ε0 :b ε  1 þ ε 0 ε0 ε0

223

3.2 Stressestrain response of materials

Softening branch:

Ascending branch:

FIGURE 3.29

Stressestrain model of Han et al. for the confined concrete of circular and square CFST columns.

8

ðf c Þ  0:12 > < 0․5 2․36  10 > ðf 0c Þ > : p ffiffiffiffiffiffiffiffiffiffiffi 1․2 1 þ x

(3.173)

(3.175)

for circular columns

0․1

for Square columns (3.176)

The stressestrain model of Ellobody et al. This model [55] uses the same equations developed by Hu et al. [46] to define the stressestrain behavior of the concrete core of circular and square CFST columns. However, slight modifications have been applied to the descending region of the curve in this model. In the model of Hu et al. [46], the stress corresponding to the strain 11ε0cc is taken as K3 f 0cc . However, the proposed model for calculating K3 is practical only for concrete cube strength up to 30 MPa. Therefore the stress K3 f 0cc is replaced by rK3 f 0cc in which the term r is the reduction coefficient and can be taken as 1.0 for concrete cube strengths up to 30 MPa, or as 0.5 for concrete cube strengths higher or equal to 100 MPa. For determining r for concrete cube strength between 30 and 100 MPa, linear interpolation should be used. The stressestrain model of Portoles et al. All the models described above are workable for CFST short members under concentric loading or

224

3. Analytical methods

loading with small eccentricity. In circular CFST short columns under concentric loading, the steel tube can efficiently confine the concrete core and enhance its compressive strength and ductility. However, the concrete’s confinement effect is negligible in slender structural members where the second-order effects are remarkable. On the other hand, the postpeak region of the curve is crucial for describing the softening behavior of the slender CFT columns. Similar to the square columns, in slender circular columns with a high eccentricity ratio, by developing cracks in the concrete, the dilation of the concrete core reaches the steel tube, initiating the interaction between the steel tube and the concrete. The model of Portoles et al. [56] is useful for predicting the stressestrain behavior of the normal-strength and high-strength concrete core in eccentrically loaded circular CFST slender columns. In this model, it is proposed that the behavior of the concrete core during the ascending branch is similar to the unconfined concrete. The prepeak region of the curve is defined using the model recommended by Collins et al. [57], which is practical for both normal-strength concrete and high-strength concrete without confinement. In contrast, the postpeak behavior of the curve is governed by the confinement effect provided by the steel tube. Therefore, the softening region of the curve is determined using the model developed by Tomii and Sakino [58], in which the effect of the Dt ratio on the softening slope of the curve has been incorporated. The stressestrain model is expressed as follows:

(3.177)

where n ¼ 0:8 þ

fc0 17

fc0 n Ec ðn  1Þ sffiffiffiffiffiffiffiffiffiffi f 0c Ec ¼ 3320 6900 εc X¼ εc1 εco ¼

in which εc1 and εc2 are taken as 0.005 and 0.015, respectively.

(3.178) (3.179)

(3.180) (3.181)

3.2 Stressestrain response of materials

225

By reaching the peak stress, the model shows a plateau up to an axial strain of εc1 ¼ 0:005. The postpeak region of the curve depends on the Dt ratio. If Dt  24, the concrete stress remains constant. For Dt > 24, the compressive stress of the concrete is declined by increasing the axial strain to the residual stress fres at a strain εc2 ¼ 0:015. The residual stress fres is governed by:

(3.182)

Round-ended rectangular and elliptical CFST members The stressestrain model of Ahmed and Liang This model has been developed to define the stressestrain curve of the concrete core in roundended concrete-filled steel tubular (RECFST) stub columns [59]. The schematic view of this model is depicted in Fig. 3.30. The ascending branch of the graph is defined by Eq. (3.56). In this model, the elastic modulus of concrete Ec is obtained using an equation recommended by Lim and Ozbakkaloglu [41], as follows: qffiffiffiffiffiffiffiffiffi Ec ¼ 4400 gc f 0c (3.183)

The softening branch of the curve is expressed based on Eq. (3.132) by taking the axial strain corresponding to the inflection point εci as 0.007. The compressive strength of the confined concrete fcc0 and the

Inflection point

FIGURE 3.30 Stressestrain model of Ahmed and Liang for the confined concrete of RECFST columns.

226

3. Analytical methods

corresponding strain ε0cc are represented using Eqs. (3.58) and (3.59), respectively. The strain ε0c corresponding to the compressive strength of unconfined concrete is defined using the equation recommended by De Nicolo et al. [45] and represented in Eq. (3.136). The model for predicting the lateral confining pressure fr in RECFST in B  Ref. [59] is based on experimental test results of specimens with 1:2  D D 0 4:2, 25  t  69:5, 32:4  fc  89:4 MPa, and 0:35  x  0:97. fr is governed by:     8 B > > for 1:2  < 2 < 10:288  0:928x  3:115 B=D  0:048 B=D D fr ¼ > > B : 0:9 for 2   4:2 D (3.184) in which x is the confinement factor and B and D are shown in Fig. 3.31. The developed model for the residual strength fres is the function of the B=D ratio and the confinement factor x. The residual strength is defined as fres ¼ bc fcc0 in which bc is the strength reduction coefficient, and it is governed by:   8 B > > 1:436  0:512 B =D þ 0:015x for 1:2  D  2 < bc ¼ (3.185) > > B : 0:4 for 2 <  4:2 D The stressestrain model of Patel This mode [60] is specifically for predicting the stressestrain response of the concrete core confined by round-ended rectangular steel tubes. In this model, the cross section of the column is split into two areas, i.e., round-end sections and the middle rectangular section, as shown in Fig. 3.32. It is assumed that the

FIGURE 3.31

Cross section of a round-ended rectangular CFST column.

3.2 Stressestrain response of materials

FIGURE 3.32

227

Cross-sectional division of round-ended rectangular CFST columns.

compressive strength and ductility of the round-end concrete core are enhanced by the confinement effect provided by the steel tube. By contrast, the flat parts of the steel tube do not enhance the concrete core compressive strength and only improve its ductility. Therefore, the behavior of the concrete in the round ends is defined using the model developed by Liang and Fragomeni [51] for the confined concrete of circular CFST members. Besides, the stressestrain response of the concrete in the middle rectangular part is defined by using the model recommended by Liang for the confined concrete of square CFST members [52]. The stressestrain model of Ahmed and Liang The proposed model for defining the stressestrain curve of the concrete core confined by the elliptical steel tube [61] is similar to the one proposed by Ahmad and Liang [59] for the round-ended rectangular CFST column (see Fig. 3.30). Fig. 3.33 illustrates the cross section of the elliptical CFST column. For defining the ascending branch of the curve, Eq. (3.56) is employed. The softening branch of the curve is expressed based on Eq. (3.132) by taking the axial strain corresponding to the inflection point εci as 0.007. The compressive strength of the confined concrete fcc0 and the corresponding strain ε0cc are represented using Eqs. (3.58) and (3.59), respectively. The strain ε0c corresponding to the compressive strength of

FIGURE 3.33

Cross section of an elliptical CFST column.

228

3. Analytical methods

unconfined concrete is defined using the equation recommended by De Nicolo et al. [45] and represented in Eq. (3.136). The lateral confining pressure fr is expressed as follows: 8 a > < 1:065x þ 3:449 for 1:5  < 2 b fr ¼ (3.186) > : 1:019x þ 0:304 for a  2 b where a and b are the long radius and short radius of the section, as shown in Fig. 3.33. The residual stress is expressed as fres ¼ bc fcc0 in which the strength reduction coefficient bc is governed by: 8 a > for 1:5  < 2 < 0:5576x þ 0:225 b bc ¼ (3.187) > : 0:0033x3  0:0683x2 þ 0:4575x for a  2 b The stressestrain model of Dai and Lam This model [62] is applicable for predicting the stressestrain response of the concrete core confined by the elliptical steel tube. For developing this model, it has been assumed that the elliptical steel tube can confine the concrete core and increase its compressive strength and ductility. However, the confinement effect provided by the elliptical steel tube differs from that of the circular steel tube. In elliptical CFST columns under axial compression, the highest confining stress occurs at the sharper corners, and it reduces along the perimeter of the concrete to the least amount at the short diameter corner. Considering an elliptical CFST column and a circular CFST column with the same cross-sectional areas of the steel tube and the concrete core, the confining pressure provided by the circular steel tube is in the range of the maximum and minimum confining stress provided by the elliptical steel tube. However, the average confining pressure in the circular CFST column is more than that of the elliptical CFST one. In other words, the confinement effect provided by the elliptical tube is less than that of the circular tube. Fig. 3.34 illustrates the schematic view of the stressestrain curve defined by this model. The ascending part of the curve is defined by the model developed for circular CFST columns by Hu et al. [46]. The compressive strength of the confined concrete fcc0 and the corresponding strain ε0cc are represented using Eqs. (3.58) and (3.59), respectively. However, due to the difference between the confinement effect provided by the elliptical steel tube and the circular steel tube, slight modifications must be implemented in Eq. (3.58). The term K1 in Eq. (3.58) is governed by: a k1 ¼ 6:770  2:645 (3.188) b

Coefficient K2 in Eq. (3.59) should be taken as 20.5.

229

3.2 Stressestrain response of materials

A B

C

0O

FIGURE 3.34 Stressestrain model of Dai and Lam for the confined concrete of elliptical CFST members.

The lateral confining pressure fr is obtained by:    ða þ bÞ fr ¼ fy ðv1  v2 Þ  t

(3.189)

where a and b are depicted in Fig. 3.33. The terms v1 and v2 are governed by: 8 > > < 0:043646 for 21:7  ða þ bÞ=  47 t (3.190) v1 ¼ > > : 0:006241 for 47  ða þ bÞ=t  150

v2 ¼

8 > > < 0:000832

for 21:7  ða þ bÞ=t  47

> > : 0:0000357 for 47  ða þ bÞ=t  150

(3.191)

The second part of the curve is modeled by a quick softening to represent the crushing of the concrete core after reaching the peak stress. This part of the curve begins from fcc0 and terminates at fe . The stress fe is governed by:   fe ¼ v3 fcc0  fu þ fu (3.192) in which the term v3 depends on the geometry of the steel section. v3 can be taken as 0.3 for the cross section with 15  ða þ bÞ=t  30 and a=b ¼ 2. The strain εe corresponding to the stress fe is taken as 10εc where εc is the strain corresponding to the compressive strength of the unconfined concrete fc0 . The last stage of the curve begins at fe and terminates at fu ¼ v4 fcc0 ‡ 30 MPa. The strain corresponding to the stress fu is εu ¼ 30εc . It is expected that the local buckling of the tube will happen at this stage. The amount of the parameter v4 depends on the concrete compressive strength. v4 is taken as 0.7 for the concrete cube strength up to 30 MPa and

230

3. Analytical methods

0.3 for concrete cube strengths higher or equal to 100 MPa. For the determination of v4 for concrete strength between 30 and 100 MPa, linear interpolation is used. The stressestrain model of Patel et al. This model [63] employs the idealized stressestrain curve depicted in Fig. 3.34 for predicting the behavior of the concrete performance in elliptical CFST columns. The ascending branch of the curve is defined by Eq. (3.56). Besides, the compressive strength of the confined concrete fcc0 and the corresponding strain ε0cc are represented by Eqs. (3.58) and (3.59), respectively. However, the term k1 in Eq. (3.58) is governed by: a k1 ¼ 6:7  2:6 (3.193) b

in which a and b are the major and minor radius axis, respectively, as shown in Fig. 3.33. The lateral confining pressure fr is governed by:    ða þ bÞ ða þ bÞ fy 17   29 (3.194) fr ¼ 0:037  0:00755 t t The postpeak region of the curve is modeled by two linear parts in which strains corresponding to part AB are in the range of ε0cc and εe and strains corresponding to part BC are in the range of εe and εu , as shown in Fig. 3.34. Linear segments AB and BC are expressed as:  8 0    > > fcc0 þ fe  fcc0 εc  ε0cc for ε0cc < εc  εe > < εe  εcc sc ¼ (3.195)   >   εc  εe > > : fe þ fcu  fe for εe < εc  εu εcu  εe in which εe and εu are depicted in Fig. 3.34 and are taken as 10εc and 30εc , respectively. The stresses fe and fcu are governed by: fe ¼ ac fcc0

(3.196)

fcu ¼ bc fcc0

(3.197)

in which the terms ac and bc are defined to incorporate the confinement effect on the ductility of the column as follows: ac ¼ 0:889  0:004gc fc0 bc ¼ 0:841429 

0:005714gc fc0

(3.198) (3.199)

Hexagonal and octagonal CFST members The stressestrain model of Hassanein et al. The stressestrain response of the concrete core in a hexagonal CFST column depends on the section’s geometric internal angle q. Depending on the value of the geometric internal angle q, two types of hexagonal CFST columns can be fabricated,

3.2 Stressestrain response of materials

231

135◦

120◦ 90◦

(a) Regular cross-section

(b) Dual-axisymmetric cross-section

FIGURE 3.35 Types of hexagonal CFST columns. (a) Regular cross section (b) Dualaxisymmetric cross section.

as shown in Fig. 3.35. The concrete core behavior in a dual-axisymmetric hexagonal CFST column is similar to the concrete behavior of a rectangular CFST column, in which the confinement effect can only enhance the ductility of the concrete core and does not have any influence on its compressive strength [64]. Therefore, the concrete behavior in hexagonal CFST columns with q ¼ 135 can be defined by using the model developed by Liang [52] for predicting the stressestrain behavior of the concrete core confined by the square steel tube, as shown in Fig. 3.28. For regular hexagonal CFST columns, shown in Fig. 3.35a, the steel tube can enhance both the ductility and the compressive strength of infilled concrete [65]. Hence, the stressestrain behavior of infilled concrete of regular hexagonal CFST columns can be defined using the model represented in Fig. 3.26. The ascending branch of the curve (Part OA) is defined by Eq. (3.56). Besides, the compressive strength of the confined concrete fcc0 and the corresponding strain ε0cc are represented by Eqs. (3.58) and (3.59), respectively. Based on Ref, [66], the lateral confinement pressure of regular hexagonal CFST columns can be expressed as:  8  BþD BþD > fy for 17  < 63 > < 0:0491703  0:0007943 2t 2t (3.200) fr ¼   > > : 0:0065311  0:0000044 B þ D fy for 63  B þ D  103 2t 2t The postpeak region of the curve is governed by Eq. (3.168). The strength reduction coefficient bc is necessary for calculating the postpeak behavior of the curve. bc for the regular hexagonal CFST columns is expressed as: bc ¼ 0:8726  0:006fc0

for 47 

BþD  103 2t

(3.201)

232

3. Analytical methods

The stressestrain model of Patel et al. The overall shape of the stresse strain curve of confined concrete in the octagonal CFST column defined by this model [63] is similar to Fig. 3.34. The ascending branch of the curve (Part OA) is defined by Eq. (3.56). Besides, the compressive strength of the confined concrete fcc0 and the corresponding strain ε0cc are represented by Eqs. (3.58) and (3.59), respectively. The lateral confinement pressure of octagonal CFST columns can be expressed as:  8 D D > 0:02508  0:0003977 fy for 17  < 47 > > > t t > > > 

t t > > > > > > D : 0:0  150 t

The postpeak region of the curve is modeled by Eqs. (3.195)e(3.197). However, coefficients. ac and bc are governed by: ac ¼ 0:9729  0:0091gc fc0

(3.203)

0:0107gc fc0

(3.204)

bc ¼ 0:9987 

The stressestrain model of Susantha et al. The model of Susantha et al. [49] presented in the previous section for determining the behavior of the concrete core of circular and square CFST stub columns is also applicable for defining the stressestrain behavior of the concrete core of octagonal CFST short columns. The overall shape of the stressestrain curve for the concrete core of octagonal CFST columns is similar to the circular and square ones, as shown in Fig. 3.24. In this model, the nonlinear part O-A is defined by the model developed by Popovics [22] (Eq. 3.56) in which fcc0 and ε0cc to be calculated by Eqs. (3.58) and (3.59), respectively. The softening branch of the curve is modeled by Eq. (3.157). The lateral confining stress in octagonal CFST columns is expressed by:  0 1:35  1:02 f fr ¼  35R c þ 0:22 fc0 (3.205) fy

where R is the width-to-thickness parameter and is governed by Eq. (3.156). The slope Z is given by: ! 8 . > 0 > 0 for R fc f  0:018 > > y < (3.206) Z¼ ! ! > 0 > . f > > : 2:85  104 R c  513 for R fc0 f > 0:018 y fy

3.2 Stressestrain response of materials

233

The ultimate strain corresponding to Point B ðεcu Þ is defined as follows: ! 8 . > 0 > 0:035 for R fc f  0:03 > > y < (3.207) εcu ¼ ! ! > . > fc0 > 0 > : 0:566R þ 0:052 for Rt fc f > 0:03 y fy Similar to the circular and square CFST columns, the calculated ultif0

mate strain εcu must be less than εcc þ Zcc . CFDST members

Basically, the inner indent of the sandwiched concrete in CFDST columns can be restricted by the lateral expansion of the inner steel tube if the thickness of the inner tube is high enough or if a small hollow ratio is considered in the member design. In this case, the behavior of the sandwiched concrete in CFDST members can be considered similar to the concrete core of conventional CFST members. Following the mentioned viewpoint, the stressestrain models for predicting the behavior of the concrete core under triaxial compression presented in the last section can also be employed in defining the behavior of the sandwiched concrete of CFDST columns with the small hollow ratio or the thick inner steel tube. However, researchers have always attempted to improve the accuracy of the previously developed model by modifying some parameters and turning them into functional relationships for predicting the stressestrain response of the sandwiched concrete of CFDST members. The recommended models and implemented modifications for predicting the behavior of the sandwiched concrete of CFDST members are discussed in this section. The stressestrain model of Hu and Su This model [67] has been developed explicitly for defining the stressestrain behavior of the sandwiched concrete of circular CFDST columns based on the constitutive model of confined concrete for circular CFST columns developed by Hu et al. [46], shown in Fig. 3.22. The nonlinear behavior of the concrete during the ascending region of the curve is modeled by Eq. (3.141). Besides, the compressive strain of the confined concrete fcc0 and the corresponding strain ε0cc are calculated using Eqs. (3.58) and (3.59), respectively. Similar to the of  model  Hu et al. [46], the last point of the curve is taken as K3 f 0cc ; 11ε0cc . However, the stress corresponding to the last point of the curve K3 f 0cc can be replaced by rK3 f 0cc in which the term r is the reduction coefficient and can be taken as 1.0 for concrete cube strengths up to 30 MPa, or as 0.5 for concrete cube strengths higher or equal to 100 MPa. For determining of r for concrete cube strength between 30 and 100 MPa, linear interpolation is used. This is based on the model of Ellobody et al. [55].

234

3. Analytical methods

The main difference between the model of Hu and Su [67] with the model of Hu et al. [46] is the expressions used for calculating the lateral confining pressure fr and the material degradation coefficient K3 . In the current model, it is supposed that the amounts of the lateral confining pressure fr and the material degradation coefficient K3 depend on the  width-to-thickness ratios of the outer steel tube Do=t and the inner o  steel tube Di=t , and the steel tubes yield stresses. The lateral confining i pressure fr is expressed as:    2   D0 Di D0  0․00897 þ 0․00125 fr1 ¼ 8․525  0․166 t0 ti t0 (3.208)  2     D0 Di Di   0․00550 þ 0․00246 0 t0 ti ti    2   fr2 D0 Di D0 þ 0․00040 þ 0․00001 ¼ 0․01844  0․00055 fyi t0 ti t0 (3.209)  2     D0 Di Di   0․00002 þ 0․00001 0 t0 ti ti    2   fr3 D0 Di D0  0․00013 þ 0․00001 ¼ 0․01791  0․00036 fyo t0 ti t0 (3.210)  2     D0 Di Di   0․00002 þ 0․00001 0 t0 ti ti   The lateral confining pressure should be taken as fr ¼ min fr1 ; fr2 ; fr3 . The material degradation coefficient K3 is governed by:    2   D0 Di D0  0․04731  0․00036 0 < K3 ¼ 1․73916  0․00862 t0 ti t0  2     D0 Di Di   0․00058 þ 0․00134  1:0 t0 ti ti (3.211) The stressestrain model of Liang et al. The overall shape of this model [68] is similar to the one recommended for the circular CFST columns under axial compression, as shown in Fig. 3.26. In this model, the nonlinear part O-A is defined by the model developed by Popovics [22] (Eq. 3.56) in which fcc0 and ε0cc to be calculated by Eqs. (3.58) and (3.59), respectively. The axial strain ε0c corresponding to the compressive strength of unconfined concrete is governed by Eq. (3.165). The postpeak region of the curve can be defined using Eq. (3.168).

235

3.2 Stressestrain response of materials

The model of Hu and Su [67] contains two drawbacks [69]. The first one is that the proposed model for defining the lateral confining pressure fr is based on the experimental test results of circular CFDST columns with 20  Dt00  100 and 15  Dtii  55. Eqs. (3.209) and (3.210) remarkably overestimate the predicted amount of the lateral confining pressure up to 60% and 44%, respectively. Therefore, Eq. (3.208) should be implemented in practice. Another issue is with the expression developed for the material degradation coefficient K3 . As mentioned in Eq. (3.211), the amount of K3 must be in the range of 0 and 1.0 (0 < K3  1:0). However, its value can be negative, zero, or more than one for some dimensions, as depicted in Fig. 3.36. Therefore, for circular CFDST columns with Dt00  40, the material degradation coefficient can be taken as 1.0 and Eq. (3.211) can be considered for circular CFDST columns with Dt00 > 40. Therefore, the descending stage of the curve can be defined using the strength reduction coefficient bc by incorporating the coefficient K3 , as follows: 8 D0 > > for  40 > 1․0 > t0 > > > > < D0 for > 40 (3.212) bc ¼ K3 t0 > > > >  2   > > > > 0․0000339 D0  0․010085 D0 þ 1․349 for K < 0 : 3 t0 t0

2 1.5 1

K3

0.5 0

-0.5 Di/ti=10 Di/ti=20 Di/ti=30 Di/ti=40

-1 -1.5 -2 10

20

30

40

Di/ti=15 Di/ti=25 Di/ti=35

50

60 70 Do/to

80

90 100 110

FIGURE 3.36 Variation of material degradation coefficient (K3 ) with respect to the dimensions of outer and inner steel tubes.

236

3. Analytical methods

The ultimate strain εcu corresponding to Point B (Fig. 3.26) is expressed by Eq. (3.213) [68], in which the amount of εcu is the function of the Dt00 ratio. Fig. 3.37 illustrates the variation of εcu corresponding to the Dt00 ratio. 8 D0 > > 0․03 for  60 > > t0 > > > >   < D0 D0 εcu ¼ 0․023 þ 100  ð0․03  0․023Þ=ð100  60Þ for 60 <  100 > t t0 0 > > > > > D0 > > : 0․02 for > 100 t0 (3.213) The stressestrain model of Ahmed et al. The overall trend of the stressestrain curve of core concrete used by this model [70] is similar to the one shown in Fig. 3.30. The ascending stage of the curve is defined by the model developed by Popovics [22] (Eq. 3.56) in which fcc0 and ε0cc to be calculated by Eqs. (3.58) and (3.59), respectively. The axial strain ε0c corresponding to the compressive strength of unconfined concrete can be defined using the equation recommended by De Nicolo et al. [45] and represented in Eq. (3.136). In general, the confining pressure fr is increased by increasing the thickness or yield strength of the outer tube, whereas increasing the hollow ratio c or the concrete compressive strength fc0 declines the magnitude of fr . Accordingly, Ahmed et al. [70] develop a lateral pressure model which is a function of the Dtoo ratio, fyo , fc0 , x, and c, as follows: !   fyo to  (3.214)  0:096x2  0:067c2 þ 0:411 fr ¼ 23:471  Do gc fc0

0.035

εu

0.03 0.025 0.02 0.015 40

60

80

100

120

Do/to

FIGURE 3.37

Variation of ε0cu

in circular CFDSTcolumns as the function of the Do=t ratio. o

3.2 Stressestrain response of materials

237

in which x¼

Aso fyo þ Asi fyi Ac gc fc0

(3.215)

Di Do  2to

(3.216)

c¼

The descending stage of the curve is defined using Eq. (3.132), developed by Lim and Ozbakkaloglu [41]. The residual strength fres is taken as bc fcc0 . Ahmed et al. [70] determined a new model for the strength reduction factor bc in which the effects of the geometric and material properties, as well as the hollow ratio of the column, have been incorporated: !     fyo to to 2 2  þ 0:302c þ 0:01   0:03x bc ¼ 2:828  Do gc fc0 Do (3.217)  fyi 0:155

Stress

The stressestrain model of Zhao et al. This model [71] has been specifically developed for the square CFDST columns. The typical stressestrain response of the concrete core in the square CFDST columns obtained by this method is presented in Fig. 3.38. The amount of stress at each strain can be computed using the model proposed by Mander et al. [21] as follows:   ε fcc0 0 rm εcc  rm (3.218) sc ¼ ε rm  1 þ 0 εcc

Strain

FIGURE 3.38 members.

Stressestrain model of Zhao et al. for the confined concrete of CFDST

238

3. Analytical methods

in which rm ¼

Ec

f0 Ec  cc ε0cc

(3.219)

The amount of the strain ε0cc corresponding to the compressive strength of the confined concrete is expressed by: !# " fcc0 0 0 εcc ¼ εc 1 þ 5 0  1 (3.220) fc where the strain ε0c is defined by: ε0c ¼ 0:002 þ

fc0  20 ð0:003  0:002Þ 100  20

(3.221)

The compressive strength of confined concrete is governed by: !#   8 " f t yo 0 > > f 0 1 þ 4:9 for circular cross  section > > D0 < c fc0 0 (3.222) fcc ¼ " !#   > > f t > yo > : fc0 1 þ 0 for square cross  section B0 fc0 Dual skin concrete-filled steel tube columns

In terms of the confinement effect provided by the steel tubes, two kinds of concrete can be considered in dual skin concrete-filled steel tube (DCFST) columns, i.e., the sandwiched concrete and the core concrete, as shown in Fig. 3.39. The behavior of the sandwiched concrete can be considered similar to the concrete core of conventional CFST members.

(a) CHS outer-CHS inner

(b) CHS outer-SHS inner

(c) SHS outer-SHS inner

(d) SHS outer-CHS

inner

FIGURE 3.39 Typical cross sections of DCFST members. (a) CHS outer-CHS inner (b) CHS outer-SHS inner (c) SHS outer-SHS inner (d) SHS outer-CHS inner.

3.2 Stressestrain response of materials

239

For instance, the outer square steel tube can only enhance the ductility of the sandwiched concrete and has a small influence on its compressive strength, whereas the inner steel tube does not affect the behavior of the sandwiched concrete. On the other hand, the concrete core may be affected by the confinement effect provided by both inner and outer steel tubes, depends on the tubes’ cross-section. For example, the concrete core of a DCFST column with CHS as both inner and outer steel tubes, shown in Fig. 3.39a is under the confinement effect provided by both inner and outer steel tubes, whereas the concrete core of a DCFST column with SHS as the outer steel tube and CHC as the inner steel tube, shown in Fig. 3.39d is confined only by the inner circular steel tube. Depending on the outer steel tube’s cross section, the stressestrain response of the sandwiched concrete in DCFST columns can be predicted using models discussed in the previous sections. As an example, the stressestrain response of the sandwiched concrete in the circular DCFST column, depicted in Fig. 3.39a, can be obtained using the model recommended by Liang et al. [51]. Similarly, the stressestrain curve of the concrete core in DCFST columns can be computed using the models described in the last sections. However, depending on the cross section of the outer tube, slight modifications may be required to account for the additional confinement effect provided by the outer steel tube. These modifications are discussed in the following sections. Required modifications for predicting the stressestrain response of the concrete core in DCFST columns with CHS outer and CHS inner steel tubes The

behavior of the sandwiched concrete and the concrete core of DCFST columns with CHS outer and CHS inner tubes can be predicted using the constitutive models discussed in previous sections for circular CFST columns. The size effect can be considered by multiplying the reduction coefficient gc by the compressive strength of unenclosed concrete fc0 (gc fc0 ). As an example, the behavior of the concrete can be predicted using the model shown in Fig. 3.30. The ascending stage of the graph is defined by Eq. (3.56), recommended by Mander et al. [21]. The strain ε0c corresponding to the compressive strength of unconfined concrete is defined using the equation recommended by De Nicolo et al. [45] and represented in Eq. (3.136). The compressive strengths of both the sandwiched concrete and the core concrete fcc0 and their corresponding strains ε0cc are represented using Eqs. (3.58) and (3.59), respectively. The lateral confining pressure fr;o on the sandwiched concrete is similar to the concrete core of a circular CFST column. Therefore, fr;o can be obtained by models developed for predicting the confining pressure of circular CFST columns, such as Eq. (3.166), developed by Liam and Fragomeni [51] or Eq. (3.145) recommended by Hu et al. [46]. The core concrete is confined by both the outer tube and the inner tube. Therefore, the lateral confining pressure fr;i on the concrete core is influenced by the

240

3. Analytical methods

Do to

ratio and the Dti i ratio. Based on Ref. [72], the lateral confining pressure fr;i on the concrete core can be computed by:     Do Di fr;i ¼ 2:2897 þ 0:0066  0:1918 to ti  

  (3.223) Do Di  0:3801 x1  0  0:0585 to ti where x is the confinement factor and is governed by: x¼

Aso fyo þ Asi fyi 0 þ A g f0 Aco gc fc;o ci c c;i

(3.224)

in which Aso and Asi are the cross-sectional areas of the outer and inner steel tubes, respectively, fyo and fyi indicate the yield strengths of the outer and inner steel tubes, respectively, Aco and Aci are the cross-sectional areas 0 of the sandwiched concrete and the concrete core, respectively, and fc;o 0 are the compressive strengths of the sandwiched concrete and the and fc;i concrete core, respectively. The descending stage of the curve for both the sandwiched concrete and core concrete can be defined using Eq. (3.132), developed by Lim and Ozbakkaloglu [41]. The strain corresponding to the inflection point εci can be expressed by: ! ! fres  0 0:12 fres  0 0:47 0 0 þ 10εcc 1  0 (3.225) gc fc gc fc εci ¼ 2:8εcc 0 fcc fcc The residual strength of concrete fres is governed by:   D
 0:0044gc fc0  1:0 0  fres f 0 ¼ 1:2420  0:0029 cc t

(3.226)

Required modifications for predicting the stressestrain response of the concrete core in DCFST columns with SHS outer and CHS inner steel tubes The behavior of the sandwiched concrete and the concrete core

of DCFST columns with SHS outer and CHS inner tubes can be predicted using the constitutive models discussed in previous sections for square and circular CFST columns, respectively. The size effect can be considered by multiplying the reduction coefficient gc by the compressive strength of unenclosed concrete fc0 (gc fc0 ). As an example, the behavior of the concrete can be predicted using the model shown in Fig. 3.30. The ascending stage of the graph is defined by Eq. (3.56), recommended by Mander et al. [21]. The strain ε0c corresponding to the compressive strength of unconfined concrete is defined using the equation recommended by De Nicolo et al. [45] and represented in Eq. (3.136). The sandwiched concrete is confined by the square steel tube. Hence, it can be supposed that the square tube can only enhance the ductility of the

3.2 Stressestrain response of materials

241

sandwiched concrete, and the lateral pressure on the sandwiched  concrete is considered to be zero fr;o ¼ 0 . Therefore, the compressive strength of the sandwiched concrete is fcc0 ¼ gc fc0 . The core concrete is confined by the circular steel tube. Accordingly, the lateral stress between the inner circular steel tube and the concrete core fr;i can be obtained using Eq. (3.166), developed by Liam and Fragomeni [51] or Eq. (3.145) recommended by Hu et al. [46]. The compressive strength of the core concrete fcc0 and its corresponding strains ε0cc are represented using Eqs. (3.129) and (3.130), respectively, as follows: 

fcc0 ¼ gc f 0 c þ 5:2 gc f 0 c

0:91

fro gc f 0 c

!ðgc f 0 cÞ0:06

!1:15 f ro ε0cc ¼ εc þ 0:045 gc f 0 c

(3.227)

(3.228)

in which εc is given by: 

0:225 gc f 0 c εc ¼ 1000

(3.229)

The postpeak region of the curve can be computed by using Eq. (3.132), developed by Lim and Ozbakkaloglu [41]. The strain corresponding to the inflection point εci can be expressed by Eq. (3.221). The residual strength of the concrete core fres;i is expressed by: 8 D > > f0 for i  40 > > ti < cc ! fres;i ¼ (3.230) > fr0:24 Di > 0 0 0 > > : 1:6fcc  0 0:32  fcc  0:15gc f c for 40 < ti  150 gf c c

The residual strength of the sandwiched concrete is taken as fres;o ¼ bc fc0 in which bc is the strength reduction coefficient and is governed by:   8 Bo > >  24 > > B > < 1  to for 24 <  33 15 t bc ¼ >     > 2 > B B B > > : 0:000062 o  0:011225 o þ 0:705288 for 33 <  100 to to t (3.231) Summary of the constitutive stressestrain model for confined concrete

The summary of constitutive models for predicting the stressestrain response of the concrete under triaxial compression and confined concrete in CFST, CFDST, and DCFST members are presented in Tables 3.6e3.11.

TABLE 3.6 Models for the stressestrain curve of the concrete under triaxial compression. Definition

Mander et al. [21]

Stressestrain curve

Equation 


εc ε0  cc l sc ¼
l1þ εc 0 εcc

242

Researcher

fcc0 l

l¼

 Ec  0
Ec  fcc 0 ε cc

in which fcc0 ¼ fc0 þ K1 fr

Stressestrain curve

sc f0

!

ε0cc ¼ ε0c 1 þ

K 2 fl fc0

K1 ¼ 4:1 and K2 ¼ 20:5 2

A:XþB:X ¼ 1þC:XþD:X 2

0  sf0c  1

X ¼ εε0 cX ‡ 0 Constants of the ascending stage Constants of the descending stage

2

A ¼ Efc0ε0 , B ¼ ðA1Þ 0:55  1, C ¼ ðA 2Þ, & D ¼ ðB þ1Þ with either B ‡ 0 or

jBj < A 

 



ε2i Ei i 2i  4εi E2i , B ¼ ðεi ε2i Þ f Efi  f 4E , C ¼ ðA 2Þ, & A ¼ ε2iεε 0 ðf0 fi Þ ðf0 f2i Þ ð 0 i Þ ð 0 f2i Þ D ¼ ðB þ1Þ with B ‡ 0 and either A > 0 or jAj < B. f

f

Ei ¼ εii & E2i ¼ ε2i2i

The secant modulus of concrete

Strain corresponding to the compressive strength of unconfined concrete

qffiffiffiffi 8 1:5 > f 0c for crushed aggregates < 0:043r Ec ¼ qffiffiffiffi  1:5   > r : 3320 f 0 þ 6900 for weaker aggregates c 2320 8 > 0 > > fc 4:24 > pffiffiffiffi Crushed agregates > > < Ec 4 f 0 c εc ¼ 0 > f 3:78 > > c pffiffiffiffi Gravel agregates > > > : Ec 4 fc0

3. Analytical methods

Attard and Setung [25]

Compressive strength and the corresponding strain of confined concrete

Compressive strength and the corresponding strain of confined concrete

f0 fc0 ε0 εc

 ¼

fr ft

þ1

¼1þ



k

17 0:06fc0



! fr fc0

in which !  0 0:21 fr 1 þ0:062 f 0 k ¼ 1:25 fc c

ft ¼ the tensile strength of concrete ¼ 0:9fsp

Stress at the inflection point and the corresponding strain

fsp ¼ fi f0

8  0 0:67 < 0:32 fco

Without silica fume

:

With silica fume

0:410:17 lnð

¼

5:06 εi ε0co

¼

f2i fco0

¼

fr fc0

!0:57

fc0

Þ

þ1

fr fc0

!0:26

fr 0 fco

þ 2 fc0 ‡ 20 MPa

þ1

0:450:25 lnðfco0 Þ 6:35

fc0 ‡ 20 MPa

þ1

0:50:3 lnðfc0 Þ 1:12

The stress f2i corresponding to the axial strain ε2i (see Fig. 3.16)

 0 0:67 0:62 fco

3.2 Stressestrain response of materials

The split cylinder tensile strength

!0:62

þ1

þ1

Continued

243

244

TABLE 3.6 Models for the stressestrain curve of the concrete under triaxial compression.dcont’d Definition

Equation

Binici [26]

Stressestrain curve

8 Ec ε >   > > > ε  ε1e > > r > > > ε0  ε1e < f þ f  f    1e 0 1e ε  ε1e r sc ¼ r1þ > > > ε0  ε1e > > >

 >  >   > : f þ f  f exp  ε  ε0 2 res res 0 a where

ε1e  ε  ε0

ε0  ε

f f

f

Strain corresponding to the compressive strength of unconfined concrete

ε  ε1e

3. Analytical methods

Researcher

0 1e c & Es ¼ εε ε1e ¼ E1ec ; r ¼ EcEE s 1e  εc ¼  0:067fc02 þ29:9fc0 þ1053 106

! f0 fc0

Strain corresponding to the peak stress under confinement effect

ε0 ¼ 5εc

0:8

Stresses f1e , f0 , and fres (see Fig. 3.17)

The amounts of f1e , f0 , and fres are obtained by employing the loading/failure !2 ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

surface defined by: f ¼ fc0 kh where ( kh ¼

0:1 1:0

f

c þ m f r0 ð1 kh Þ c

fr fc0

f

þ fr0 c

at the elastic limit at the peak stress and over the softening stage

( c ¼

0

The ascending stage of the curve

sc f0

Over the softening stage

fc0 ft fc0 ft

m ¼ Samani and Attard [37]

1:0 Over the ascending stage

2

A:XþB:X ¼ 1þðA2Þ:XþðBþ1Þ:X 2

where X ¼ εε0 and 0  ε  ε0 , 0  f  f0 2

A ¼ Efc0ε0 , B ¼ ðA1Þ 0:55  1

Strain corresponding to the compressive strength of unconfined concrete

The descending stage of the curve

ε0 εc

εc ¼

8 > > fc0 4:24 > > pffiffiffiffi > > < Ec 4 f 0 c > fc0 3:78 > > > pffiffiffiffi > > : Ec 4 fc0

f fc0

fi fo εi ε0

¼

 fic fc0

εεc εic εc

0:00367fc0



fr fc0

!½0:3124þ0:002fc0 

Crushed agregates Gravel agregates

2 εc  ε

  ¼ 2:76  0:35 ln fc0 !   f f fic ¼ freso þ 1  freso f0 c

¼

fres f0

 kþ 1

fres fo

  εic εc

2:89 ffiffiffi0 ; k2 ¼ 1:26 þ p fc

Continued

245

The inflection point strain ratio

k1 ¼ 2:9224



εic εc

The inflection point stress ratio

¼



ek1 ;

3.2 Stressestrain response of materials

Strain corresponding to the peak stress under confinement

TABLE 3.6 Models for the stressestrain curve of the concrete under triaxial compression.dcont’d Definition

Equation

The residual stress level ratio

fres f0

246

Researcher

¼1  a

fr fc0

1 ! k3 þ1

where a ¼ 795:7  3:291fc0 !0:694 k3 ¼

The total strain for the concrete of height h

The strain softening behavior The tensile fracture energy Lim and Ozbakkaloglu [41]

The ascending stage of the curve

! þ 1:301

1 pffiffiffi 1 2 pðεic ε0 Þðf0 fres Þ ðf0 fres Þ 2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi þ 2 Ec ¼ 0 lnðf c Þlnðfic Þ  

   8 f  f0 hr hd hr > > ε0 þ ðε0  ε0 Þ hr þ 1  þ ε  h > hd > d < Ec h h h h εh ¼  

   > > f  f0 h hr hr > : ε0 þ ðε0  ε0 Þ r þ 1 þ εd 1  h  hd Ec h h h

Gfcu lc

0:8  2kG f f εd ¼ rð1þkÞft f fo ofres ð oÞ

fres  f  fo

Gft ¼ 0:00097fc0 þ 0:0418 N/mm  
εc ε0  cc l sc ¼
l1þ εc 0 ε fcc0 l

cc

l ¼

 Ec  0
Ec  fcc 0 ε cc

for εc < ε0cc

3. Analytical methods

The compressive fracture energy per unit area under confinement

5:79

fr fc0

The descending stage of the curve

The residual strength

Strain corresponding to compressive strength of unconfined concrete

Strain corresponding to the inflection point The total strain for the concrete of height h

#

2

for εc ‡ ε0cc

þ1

! fres ¼ 1:6fcc0

fr0:24 fc00:32

 fcc0  0:15fc0

ðf 0 Þ d εc ¼ c1000 ks ka where  0:45 kd ¼ 2400 rc  0:1 ks ¼ 152 D  0:13 ka ¼ 2D H 0:225k

fcc0

 0:91 ¼ fc0 þ 5:2 fc0

fr fc0

!ðfc0 Þ0:06

!1:15 ε0cc ¼ εc þ 0:045 εci ¼

2:8ε0cc

fres fcc0

fr fc0

! ! !  0 0:12  0 0:47  r 0:4 f c fc fc þ 10ε0cc 1  fres0 2400 cc

8 if 0  εc  ε0cc > < εc;h  0    εc;h ¼   Hr f  fcc0 Hr ðH  Hr Þ > : c 1 þ ε0cc þ εc  ε0cc þ εd d if εc > ε0cc Ec H H H where Hr ¼ 305 mm & Hd ¼ 2D !  0:5 f 0 f 0 cc c εd ¼ 0:02 fc0 f 0 fres cc

247

The inelastic strain in the damage zone

εc  ε0cc εci  ε0cc

3.2 Stressestrain response of materials

Compressive strength and the corresponding strain of confined concrete

fcc0 fres

sc ¼ fcc0  "

Cross-section

Researcher

Definition

Equation

Circular and square CFSTs

Tao et al. [44]

The ascending stage of the curve

sc A:X þ B:X2 ¼ f0 1 þ ðA  2Þ:X þ ðB þ 1Þ:X2 where X ¼ εε0 and 0  ε  ε0 , 0  f  f0

248

TABLE 3.7 Models for the stressestrain curve of the concrete core in circular and square CFSTs.

2

A ¼ Efc0ε0 , B ¼ ðA1Þ 0:55  1 The descending stage of the curve

 b    cc ε ‡ εcc sc ¼ fres þ fc0 fres exp  εε a

The residual strength

Strain corresponding to the peak stress of the concrete under uniaxial compression The strain at Point B (Fig. 3.21)

( fres ¼ εc0

εcc εc

  0․7 1  e1․38x fc0  0․25fc0

for circular columns

0․1fc0

for rectangular columns qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ¼ 0․00076 þ 0․626f 0c  4․33  107 10 MPa  f 0c  100 MPa

¼ ek1 ;

  k1 ¼ 2:9224 0:00367fc0

fr fc0

!½0:3124þ0:002fc0 

3. Analytical methods

where 8 0:036 > < 0:04  for circular columns 1 þ e6․08x3․49 a ¼ > : 0․005 þ 0․0075x for rectangular columns ( 1:2 for circular columns b ¼ 0:92 for Square columns

The lateral confining pressure

Hu et al. [46]



fc ¼ 1þðRþRE 2Þ

εc ε0cc



Ec εc



ð2R1Þ

εc ε0cc

2

for circular columns

for rectangular columns

 þR

εc ε0cc

3 , 0․5ε0cc  εc  ε0cc 3.2 Stressestrain response of materials

The ascending stage of the curve

8  > D > > > 1 þ 0․027fy ․e0․02 t > > > >   > 4:8 > < 1 þ 1․6e10 : fc0 fr ¼ ffi  0:02pffiffiffiffiffiffiffiffi B2 þD2 > > > t 0․25․ 1 þ 0:027f ․e > y > > >  4:8 > > > 1 þ 1․6e10 ․ fc0 :

where RE ¼

Ec ε0cc fcc0

ðRε 1Þ

Compressive strength and the corresponding strain of confined concrete

fcc0

The lateral confining pressure

Circular cross section   8 > 0:043646  0:000832 D= > < t . fr f ¼   > y > : 0:006241  0:0000357 D =t Square cross section 8   > > < 0:055048  0:001885 D= . t fr f ¼ > y > :0

¼

fc0

& R ¼ RE ðRs 1Þ  R1E 2

þ K1 fr

!

ε0cc ¼ ε0c 1 þ

K2 fr fc0

K1 ¼ 4:1 and K2 ¼ 20:5

21:7  D=  47 t 47  D=t  150 17  D=t  29:2

Continued

249

29:2  D=t  150

TABLE 3.7 Models for the stressestrain curve of the concrete core in circular and square CFSTs.dcont’d Researcher

250

Cross-section

Definition

Equation

The last point of the postpeak stage

s ¼ rfcc0 , εcu ¼ 11εcc

The material degradation coefficient

Circular cross section 8 1 > >
> : 0:0000339 D t  0:010085 D=t þ 1:3491 40  D=t  150

Ellobody et al. [55]

The ascending stage of the curve



fc ¼

1þðRþRE 2Þ

εc ε0cc



Ec εc

ð2R1Þ



εc ε0cc

2

 þR

εc ε0cc

3 , 0․5ε0cc  εc  ε0cc

where RE ¼ Compressive strength and the corresponding strain of confined concrete The lateral confining pressure

fcc0 ε0cc

Ec ε0cc fcc0

¼

fc0

¼

ε0c

& R ¼ RE ðRs 1Þ  R1E 2 ðRε 1Þ

þ K1 f r

!

K f 1 þ f20 r c

K1 ¼ 4:1 and K2 ¼ 20:5

  8 > 0:043646  0:000832 D=t 21:7  D=t  47 > < . ¼ fr   fy > > : 0:006241  0:0000357 D 47  D=t  150 =t

3. Analytical methods

Square cross-section 8    2 > > < 0:000178 D  0:02492 D=t þ 1:2722 17  D=t  70 t K3 ¼ > > : 0:4 70  D=t  150

The last point of the postpeak stage

s ¼ rK3 fcc0 , εcu ¼ 11εcc

Susantha et al. [49]

The ascending stage of the curve

3.2 Stressestrain response of materials

8 1:0 for fc0  30 MPa > > < where r ¼ Linear interpolation for 30 MPa < fc0  100 MPa > > : 0:5 for fc0 > 100 MPa  
fcc0 l εc 0 εcc  l for εc < ε0cc sc ¼
l1þ εc 0 ε cc

l ¼

 Ec  0
Ec  fcc 0 ε cc

Compressive strength and the corresponding strain of confined concrete

fcc0

The ascending stage of the curve

fc ¼ fcc0  Zðε εcc Þ εcc  ε  εcu where circular cross section

ε0cc

¼

fc0

¼

ε0c

þ K1 fr

!

K f 1 þ f20 r c

K1 ¼ 4:1 and K2 ¼ 20:5

Continued

251

Cross-section

Models for the stressestrain curve of the concrete core in circular and square CFSTs.dcont’d Researcher

Definition

252

TABLE 3.7

Equation

Rt ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi 3 1  w2s

fy D Es 2t

Square cross section ! 8 . > 0 >  0:0039 0 for R f > c fy > < Z ¼ ! ! > 0 > . > 0 > 23400R fc :  91:26 for R fc f > 0:0039 y fy qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ffiffiffi ffi q 12ð1w2s Þ fy R ¼ bt Es 4p2

3. Analytical methods

! 8 . > 0 > f  0:006 0 for R > t c f > y > > > > > ! ! > > > . fc0 > 5 0 > >  600 for Rt fc f ‡0:006 and fy  283 MPa 10 Rt > > y fy < Z ¼ ! ! 0 > . > > 0 > 106 Rt fc  600 for R f > t c > fy ‡0:006 and fy ‡336 MPa > fy > > > > > !13:4 > > > fy > > : 283 " ! # ! 0 . fc 5 0 10 Rt  600 for Rt fc  0:006 and 283  fy  336 MPa fy fy

The lateral confining pressure

2t f Circular cross section fr ¼ ðwe  ws Þ D2t y where ws ¼ 0:5 ! ! 0 fc0 0 0 fc þ 4:843ve we ¼ 0:2312 þ 0:3582ve  0:1524 fy fy !2 f0 f0  9:169 c ; 0:04  c  0:2 fy fy  3  2   v0e ¼ 0:881  106 Dt  2:58  104 Dt þ 1:953  102 Dt þ 0:4011

The ultimate strain corresponding to Point B (Fig. 3.24)

Circular cross section f0

εcu ¼ 0:025  εcc þ Zcc Square cross section 8 > > 0:04 > > > > > > > > !2 ! > < f0 f0 εcu ¼ 14:5 R c  2:4R c þ 0:116 > fy fy > > > > > > > > > > : 0:018 εcu  εcc þ

Han et al. [54]

Stressestrain curve

. for R fc0 f y

!  0:042

! . for 0:042 < R fc0 f < 0:073 y ! . 0 for R fc f ‡0:073 y

3.2 Stressestrain response of materials

Square cross section 1:46  1:46 ðf 0 Þ fr ¼ 6:5R c fy þ 0:12 fc0

fcc0 Z

253

  2 8  ε ε ε > >  1 2 > > ε ε ε > 0 0 0 > < ε sc fc0 ¼ > ε ε0 >  > h >1 > > > ε ε ε 0 :b 1 þ 0 ε0 ε0 where   ε0 ¼ εc þ 800x0․2 $106 Continued

Models for the stressestrain curve of the concrete core in circular and square CFSTs.dcont’d Researcher

Definition

Circular CFST

Liang et al. [51]

The ascending stage of the curve

Equation   εc ¼ 1300 þ12․5fc0 $106 8 columns < 2 for circular   h ¼ ε : 1․6 þ 1․5 for Square columns ε0 8  7  5 ½0․25þðx0․5Þ  0 0․5 > ðf c Þ ‡ 0:12 for circular columns > < 0․5 2․36  10 b ¼ 0 0․1 > ðf c Þ > : p ffiffiffiffiffiffiffiffiffiffiffi for Square columns 1․2 1 þ x  
fcc0 l εc 0 ε  cc l for εc < ε0cc sc ¼
l1þ εc 0 ε cc

l ¼

 Ec  0
Ec  fcc 0 ε cc

Compressive strength and the corresponding strain of confined concrete

fcc0

¼ gc

f0

c

þ K1 fc !

ε0cc ¼ ε0c 1 þ

K2 fc fc0

K1 ¼ 4:1 and K2 ¼ 20:5

3. Analytical methods

Cross-section

254

TABLE 3.7

Strength reduction factor Strain corresponding to the compressive strength of unconfined concrete

Lateral confining pressure

gc ¼ 1․85D0․135 , 0․85  gc  1:0 c 8 0․002 for gc f 0c  28 MPa > > > > < g f 0  28 ε0c ¼ 0․002 þ c c for 28 < gc f 0c  82 MPa > 54000 > > > : 0․003 for gc f 0c > 82 MPa

0:04 

fc0  0:2 fy

v0e ¼ 0:881  106

 3 D t

 2:58  104

 2 D t

þ 1:953  102

  D t

þ 0:4011

Aluminum steel tube    2   D D D fr ¼ 0:00249 þ 0:001269ðs0:2 Þ2  0:5329 s0:2  0:002297 t t t

Post-peak behavior

3.2 Stressestrain response of materials

Carbon steel tube 8 2t D > > < 0:7ðwe  ws Þ D  2tfy for t  47   fr ¼  > D > : 0:006241  0:0000357 D= t fy for 47 < t  150 where ws ¼ 0:5 ! ! !2 0 fc0 fc0 0 0 fc þ 4:843ve  9:169 we ¼ 0:2312 þ 0:3582ve  0:1524 ; fy fy fy

 0:7214s0:2 þ 104:24   8   > < bc fcc0 þ εcu  εc fcc0  bc fcc0 for ε0cc < εc  εcu 0 ε  ε cu sc ¼ cc > : 0 bc fcc for εc > εcu

255

Continued

Cross-section

Models for the stressestrain curve of the concrete core in circular and square CFSTs.dcont’d Researcher

Definition Strength reduction coefficient

Square CFST

256

TABLE 3.7

Liang et al. [52]

The ascending stage of the curve

Equation

bc ¼

8 > > < 1․0

   2 > > : 0․0000339 D  0․010085 D þ 1․3491 t t

Post-peak behavior

for 40
> > > < g f 0  28 ε0c ¼ 0․002 þ c c for 28 < gc f 0c  82 MPa > 54000 > > > : 0․003 for gc f 0c > 82 MPa

8 0 for ε0cc < εc  0:005 f > > < cc   sc ¼ bc fcc0 þ 100ð0:015  εc Þ fcc0  bc fcc0 for 0:005 < εc  0:015 > > : 0 for εc > 0:015 bc fcc

3. Analytical methods

 
εc ε0  cc l sc ¼
l1þ εc 0 εcc fcc0 l

l ¼

Compressive strength and the corresponding strain of confined concrete

for

Strength reduction coefficient

Slender circular CFSTs under load eccentricity

Portoles et al. [56]

8 B > > for  24 > 1․0 > > t > > < 1 B B bc ¼ 1:5  for 24 <  48 > 48 t t > > > > > B > : 0:5 for > 48 t

Stressestrain curve

for 0 < εc  εco

for εco < εc  εc1 where   >   ε  ε > c c1 > > for εc1 < εc  εc2 fc0 þ fres  fc0 > > εc2  εc1 > > > : fres for εc2 < εc f0

f0

n ¼ 0:8 þ 17c , X ¼ εεc1c , εco ¼ Ecc εc2 ¼ 0:015

n ðn1Þ;

εc1 ¼ 0:005, &

3.2 Stressestrain response of materials

sc ¼

8 X:n > > fc0 > > > n  1 þ Xn:k > > > > 0 < fc

257

Cross-section

Researcher

Definition

Round-ended rectangular CFSTs

Ahmed and Liang [59]

The ascending stage of the curve

Equation  
fcc0 l εc 0 ε  cc l sc ¼
l1þ εc 0 ε

258

TABLE 3.8 Models for the stressestrain curve of the concrete core in round-ended rectangular and elliptical CFSTs.

for εc < ε0cc

cc

l ¼

 Ec  0
Ec  fcc 0 ε cc

Strain corresponding to the compressive strength of unconfined concrete The descending stage of the curve

fcc0

¼ gc

ε0cc

ε0c

¼

f0

c

þ K1 f r ! K2 fr fc0

1þ

K1 ¼ 4:1 and K2 ¼ 20:5 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi    0:626 gc f 0c  4:33  107 þ 0:00076 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     ε0c ¼ 0․00076 þ 0․626 gc f 0c  4․33  107 ε0c ¼

10 MPa  f 0c  100 MPa

sc ¼ fcc0  "

fcc0 fres

εc  ε0cc εci  ε0cc

εci ¼ 0:007

2

# where þ1

3. Analytical methods

Compressive strength and the corresponding strain of confined concrete

The lateral confining pressure

The residual strength

Patel [60]

> > : 0:9

for 2 

B  4:2 D

fres ¼ bc fcc0

bc ¼

  8 > > 1:436  0:512 B > = < D þ 0:015x > > > : 0:4

for 1:2  for 2
> 10:288  0:928x  3:115 B  0:048 B for 1:2  < 2 < =D =D D

B  4:2 D

Semicircular sections The ascending stage of the curve

 
εc 0 ε  cc l sc ¼
l1þ εc 0 ε fcc0 l

for εc < ε0cc

cc

l ¼

 Ec  0
Ec  fcc 0 ε cc

Compressive strength and the corresponding strain of confined concrete

fcc0 ¼ gc f 0 c þ K1 fc ! ε0cc ¼ ε0c 1 þ

K2 fc fc0

K1 ¼ 4:1 and K2 ¼ 20:5

259

Continued

Cross-section

Researcher

Equation

Lateral confining pressure

8 2t D > for  47 > < 0:7ðwe  ws Þ D  2tfy t   fr ¼  > D > : 0:006241  0:0000357 D= fy for 47 <  150 t t where ws ¼ 0:5 ! ! f0 f0 we ¼ 0:2312 þ 0:3582v0e  0:1524 c þ 4:843v0e c fy fy !2 f0 f0  9:169 c ; 0:04  c  0:2 fy fy  3  2   v0e ¼ 0:881  106 Dt  2:58  104 Dt þ 1:953  102 Dt þ 0:4011

Strain corresponding to the compressive strength of unconfined concrete

8 0․002 > > > > < g f 0  28 ε0c ¼ 0․002 þ c c > 54000 > > > : 0․003

for gc f 0c  28 MPa for 28 < gc f 0c  82 MPa for gc f 0c > 82 MPa

3. Analytical methods

Definition

260

TABLE 3.8 Models for the stressestrain curve of the concrete core in round-ended rectangular and elliptical CFSTs.dcont’d

The descending stage of the curve Strength reduction coefficient

  8   > < bc fcc0 þ εcu  εc fcc0  bc fcc0 for ε0cc < εc  εcu 0 εcu  εcc sc ¼ > : 0 bc fcc for εc > εcu

bc ¼

8 > > < 1․0

for

D  40 t

   2 > > : 0․0000339 D  0․010085 D þ 1․349 for 40 < D  150 t t t 3.2 Stressestrain response of materials

Rectangular section The ascending stage of the curve

 
fcc0 l εc 0 εcc  l sc ¼
l1þ εc 0 ε

for εc < ε0cc

cc

l ¼

 Ec  0
Ec  fcc 0 ε cc

Compressive strength and the corresponding strain of confined concrete The descending stage of the curve

fcc0

f0

¼ gc c 8 0․002 > > > > < g f 0  28 ε0cc ¼ 0․002 þ c c > 54000 > > > : 0․003

for gc f 0c  28 MPa for 28 < gc f 0c  82 MPa for gc f 0c > 82 MPa

Continued

261

8 0 for ε0cc < εc  0:005 f > > < cc  0  0 0 sc ¼ bc fcc þ 100ð0:015  εc Þ fcc  bc fcc for 0:005 < εc  0:015 > > : 0 for εc > 0:015 bc fcc

Cross-section

Researcher

Definition Strength reduction coefficient

Elliptical CFST

The ascending stage of the curve

 
εc ε0  cc l sc ¼
l1þ εc 0 ε fcc0 l

B  24 t B for 24 <  48 t B for > 48 t for

for εc < ε0cc

cc

l ¼

 Ec  0
Ec  fcc 0 ε cc

Compressive strength and the corresponding strain of confined concrete

fcc0

Strength reduction factor

, 0․85  gc  1:0 gc ¼ 1․85D0․135 c where Dc ¼ ð2a 2tÞ

¼ gc

f0

c

þ K1 f c !

ε0cc ¼ ε0c 1 þ

K2 fc fc0

K1 ¼ 4:1 and K2 ¼ 20:5

3. Analytical methods

Ahmed and Liang [61]

Equation 8 > > > 1․0 > > > > < 1 B bc ¼ 1:5  > 48 t > > > > > > : 0:5

262

TABLE 3.8 Models for the stressestrain curve of the concrete core in round-ended rectangular and elliptical CFSTs.dcont’d

The descending stage of the curve

sc ¼ fcc0  "

Residual strength Strength reduction coefficient The ascending stage of the curve

fcc0 fres

εc  ε0cc εci  ε0cc

#

2

þ1

3.2 Stressestrain response of materials

ε0c ¼

Lateral confining pressure

Dai and Lam [62]

q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi    0:626 gc f 0c  4:33  107 þ 0:00076

Strain corresponding to the compressive strength of unconfined concrete

where εci ¼ 0:007 8 a > < 1:065x þ 3:449 for 1:5  < 2 b fr ¼ > : 1:019x þ 0:304 for a‡2 b fres ¼ bc fcc0

bc ¼

8 > < 0:5576x þ 0:225

for 1:5 

a

: 0:0033x3  0:0683x2 þ 0:4575x for a‡2 b 

fc ¼

1þðRþRE 2Þ

εc ε0cc



Ecc εc



ð2R1Þ

εc ε0cc

2

 þR

εc ε0cc

3 0․5ε0cc  εc  ε0cc

where  R1ε RE ¼ Efcc0 εc & R ¼ RE ðRs 1Þ 2 cc

Continued

263

ðRε 1Þ

Cross-section

Researcher

Definition

Equation

Compressive strength and the corresponding strain of confined concrete

fcc0 ¼ fc0 þ K1 fr

Lateral confining pressure

!

ε0cc ¼ ε0c 1 þ

K2 fr fc0

 K1 ¼ 6:770  2:645 ab & K2 ¼ 20:5 s ¼ rK3 fcc0 , εcu ¼ 11εcc where 8 > 1:0 for fu  30 MPa > < r ¼ Linear interpolation for 30 MPa < fu  100 MPa > > : 0:5 for fu > 100 MPa    fr ¼ fy ðv1 v2 Þ  ðaþbÞ t where 8 > > < 0:043646 for 21:7  ða þ bÞ=  47 t v1 ¼ > > : 0:006241 for 47  ða þ bÞ=t  150 8 > > < 0:000832 for 21:7  ða þ bÞ=  47 t v2 ¼ > 0:0000357 for 47  ða þ bÞ=  150 > : t

3. Analytical methods

The last point of the postpeak stage

264

TABLE 3.8 Models for the stressestrain curve of the concrete core in round-ended rectangular and elliptical CFSTs.dcont’d

The descending stage of the curve

The ascending stage of the curve



fc ¼

1þðRþRE 2Þ

εc ε0cc



Ec εc



ð2R1Þ

εc ε0cc

2

 þR

εc ε0cc

3 0․5ε0cc  εc  ε0cc

3.2 Stressestrain response of materials

Patel et al. [63]

  fe ¼ v3 fcc0  fu þ fu at εe ¼ 10εc where v3 ¼ 0:3 for 15  ða þ bÞ=t  30 and a=b ¼ 2. fu ¼ v4 fcc0 ‡ 30 MPa at εe ¼ 30εc where 8 0:7 for fc0 ¼ 30 MPa > > < v4 ¼ Linear interpolation for 30 MPa < fc0  100 MPa > > : 0:3 for fc0 > 100 MPa

where RE ¼ Compressive strength and the corresponding strain of confined concrete Strain corresponding to the compressive strength of unconfined concrete

Ec ε0cc fcc0

& R ¼ RE ðRs 1Þ  R1E 2 ðRε 1Þ

fcc0 ¼ fc0 þ K1 fr

!

ε0cc ¼ ε0c 1 þ

K2 fr fc0

 k1 ¼ 6:7  2:6 ba & K2 ¼ 20:5 8 0․002 > > > > < g f 0  28 ε0c ¼ 0․002 þ c c > 54000 > > > : 0․003

for gc f 0c  28 MPa for 28 < gc f 0c  82 MPa for gc f 0c > 82 MPa

265

Continued

Cross-section

Researcher

Definition Lateral confining pressure The descending stage of the curve

Equation    fy fr ¼ 0:037 0:00755 ðaþbÞ t

266

TABLE 3.8 Models for the stressestrain curve of the concrete core in round-ended rectangular and elliptical CFSTs.dcont’d

17  ðaþbÞ t  29

3. Analytical methods

  8   εc  ε0cc > 0 0 > for ε0cc < εc  εe > fcc þ fe  fcc 0 < εe  εcc sc ¼   >   εc  εe > > : fe þ fcu  fe for εe < εc  εu εcu  εe where fe ¼ ac fcc0 fcu ¼ bc fcc0 ac ¼ 0:889  0:004gc fc0 bc ¼ 0:841429  0:005714gc fc0

TABLE 3.9 Models for the stressestrain curve of the concrete core in hexagonal and octagonal CFSTs. Cross-section

Researcher

Definition

Hexagonal CFST

Hassanein et al. [66]

The ascending stage of the curve

Equation  
fcc0 l εc 0 ε  cc l sc ¼
l1þ εc 0 εcc  Ec 
fcc0 ε0

l ¼

Ec 

Lateral confining pressure

The descending stage of the curve

Strength reduction coefficient Octagonal CFST

Patel et al. [63]

The ascending stage of the curve

¼

fc0

cc

þ K1 fr

! 3.2 Stressestrain response of materials

Compressive strength and the corresponding strain of confined concrete

fcc0

for εc < ε0cc

ε0cc ¼ ε0c 1 þ

K2 fr fc0

K1 ¼ 4:1 and K2 ¼ 20:5  8  BþD BþD > fy for 17  < 63 0:0491703  0:0007943 > < 2t 2t fr ¼   > > : 0:0065311  0:0000044 B þ D fy for 63  B þ D  103 2t 2t   8   > < bc fcc0 þ εcu  εc fcc0  bc fcc0 for ε0cc < εc  εcu εcu  ε0cc sc ¼ > : 0 bc fcc for εc > εcu bc ¼ 0:8726  0:006fc0  
εc ε0  cc l sc ¼
l1þ εc 0 ε fcc0 l

for 47  BþD 2t  103

for εc < ε0cc

267

cc

l ¼

 Ec  0
Ec  fcc 0 ε cc

Continued

TABLE 3.9 Models for the stressestrain curve of the concrete core in hexagonal and octagonal CFSTs.dcont’d Researcher

268

Cross-section

Definition

Equation

Compressive strength and strain of confined concrete

fcc0 ¼ fc0 þ K1 fr

Lateral confining pressure

The descending stage of the curve

K2 fr fc0

K1 ¼ 4:1 and K2 ¼ 20:5 8 0․002 > > > > < g f 0  28 ε0c ¼ 0․002 þ c c > 54000 > > > : 0․003

for gc f 0c  28 MPa for 28 < gc f 0c  82 MPa for gc f 0c > 82 MPa

 8 D D > fy for 17  < 47 0:02508  0:0003977 > > > t t > > > 

t t > > > > > > D : 0:0 ‡150 t   8   εc  ε0cc > 0 0 > for ε0cc < εc  εe > < fcc þ fe  fcc εe  ε0 cc sc ¼   >   εc  εe > > : fe þ fcu  fe for εe < εc  εu εcu  εe where fe ¼ ac fcc0 fcu ¼ bc fcc0 ac ¼ 0:9729  0:0091gc fc0 bc ¼ 0:9987  0:0107gc fc0

3. Analytical methods

Strain corresponding to the compressive strength of unconfined concrete

!

ε0cc ¼ ε0c 1 þ

Susantha et al. [49]

The ascending stage of the curve

 
εc ε0  cc l sc ¼
l1þ εc 0 ε fcc0 l

for εc < ε0cc

cc

l ¼

 Ec  0
Ec  fcc 0 ε cc

Compressive strength and the corresponding strain of confined concrete

fcc0

The ascending stage of the curve

fc ¼ fcc0  Zðε εcc Þ εcc  ε  εcu where ! 8 . > 0 > for R fc f  0:018 > >0 y < Z ¼ ! ! > > . f0 > > : 2:85  104 R c  513 for R fc0 fy > 0:018 fy qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi qffiffiffiffi 12ð1w2s Þ fy R ¼ bt Es 4p2

The lateral confining pressure

fr ¼ 35R

The ultimate strain corresponding to Point B (Fig. 3.24)

(

¼

fc0

þ K1 fr

!

ε0cc ¼ ε0c 1 þ

K2 fr fc0

ðfc0 Þ

1:35

fy

3.2 Stressestrain response of materials

K1 ¼ 4:1 and K2 ¼ 20:5

 1:02 þ 0:22 fc0

εcu ¼ . 0:035 for R fc0 f y fcc0 Z

 0:03  0:566R

fc0 fy

!

! . þ 0:052 for Rt fc0 f > 0:03 y

269

εcu  εcc þ

!

TABLE 3.10 Models for the stressestrain curve of the sandwiched concrete in circular and square CFDSTs. Researcher

Definition

Equation

Circular CFDST

Hu and Su [67]

The ascending stage of the curve

fc ¼

 1þðRþRE 2Þ

εc ε0cc



Ec εc



ð2R1Þ

εc ε0cc

2

 þR

εc ε0cc

3 0․5ε0cc  εc  ε0cc

270

Cross-section

where  R1ε RE ¼ Efc0εc & R ¼ RE ðRs 1Þ 2 ðRε 1Þ

cc

Compressive strength and the corresponding strain of confined concrete

The lateral confining pressure

!

ε0cc ¼ ε0c 1 þ

K2 fr fc0

K1 ¼ 4:1 & K2 ¼ 20:5 s ¼ rK3 fcc0 , εcu ¼ 11εcc where 8 1:0 for fc0  30 MPa > > < r ¼ Linear interpolation for 30 MPa < fc0  100 MPa > > : 0:5 for fc0 > 100 MPa    2   D0 Di D0  0․00897 þ 0․00125 t0 ti t0  2     D0 Di Di   0․00550 þ0․00246 ‡0 t0 ti ti    2   fr2 D0 Di D0 þ 0․00040 þ 0․00001 ¼ 0․01844  0․00055 fyi t0 ti t0  2     D0 Di Di   0․00002 þ0․00001 ‡0 t0 ti ti    2   fr3 D0 Di D0  0․00013 þ 0․00001 ¼ 0․01791  0․00036 fyo t0 ti t0  2     D0 Di Di   0․00002 þ0․00001 ‡0 t0 ti ti fr1 ¼ 8․525  0․166

3. Analytical methods

The last point of the post-peak stage

fcc0 ¼ fc0 þ K1 fr

Descending coefficient

Liang et al. [68]

The ascending stage of the curve

   2   D0 Di D0  0․04731  0․00036 0 < K3 ¼ 1․73916  0․00862 t0 ti t0  2     D0 Di Di   0․00058 þ0․00134  1:0 t0 ti ti  
fcc0 l εc 0 εcc  l for εc < ε0cc sc ¼
l1þ εc 0 ε cc

l ¼

The lateral confining pressure

Strain corresponding to the compressive strength of unconfined concrete The descending stage of the curve

3.2 Stressestrain response of materials

Compressive strength and the corresponding strain of confined concrete

 Ec  0
Ec  fcc 0 εcc

fcc0 ¼ gc f 0 c þ K1 fr ! ε0cc ¼ ε0c 1 þ

K2 fr fc0

K1 ¼ 4:1 and K2 ¼ 20:5

271

   2   D0 Di D0  0․00897 þ 0․00125 fr ¼ 8․525  0․166 t0 ti t0  2     D0 Di Di   0․00550 þ0․00246 ‡0 t0 ti ti 8 0․002 for gc f 0c  28 MPa > > > > < g f 0  28 ε0c ¼ 0․002 þ c c for 28 < gc f 0c  82 MPa > 54000 > > > : 0․003 for gc f 0c > 82 MPa   8   > < bc fcc0 þ εcu  εc fcc0  bc fcc0 for ε0cc < εc  εcu 0 εcu  εcc sc ¼ > : 0 bc fcc for εc > εcu

Continued

TABLE 3.10 Models for the stressestrain curve of the sandwiched concrete in circular and square CFDSTs.dcont’d Cross-section

Researcher

Definition

Strength reduction coefficient

The ascending stage of the curve

l ¼

 Ec  0
Ec  fcc 0 ε cc

Compressive strength and the corresponding strain of confined concrete Strain corresponding to the compressive strength of unconfined concrete

fcc0

¼ gc

f0

c

þ K1 fr !

ε0cc ¼ ε0c 1 þ

K2 fr fc0

K1 ¼ 4:1 and K2 ¼ 20:5 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi    0:626 gc f 0c  4:33  107 þ 0:00076 ε0c ¼

3. Analytical methods

Ahmed et al. [70]

8 D0 > >  40 for > 1․0 > t0 > > > > < D0 for > 40 bc ¼ K3 t0 > > > >  2   > > D0 D0 > > : 0․0000339 þ 1․349 for K3 < 0  0․010085 t0 t0  
fcc0 l εc 0 ε  cc l for εc < ε0cc sc ¼
l1þ εc 0 εcc

272

The ultimate strain corresponding to Point B (Fig. 3.25)

Equation 8 D0 > >  60 0․03 for > > t0 > > > >   <
D0 D0 ð0․03  0․023Þ ð100  60Þ for 60 <  100 εcu ¼ 0․023 þ 100  > t0 t0 > > > > > D0 > > 0․02 : > 100 for t0

The descending stage of the curve

fcc0 fres

sc ¼ fcc0  "

εc  ε0cc εci  ε0cc

2

# þ1

where εci ¼ 0:007 Lateral confining pressure

!

  to Do

fr ¼ 23:471 



fyo gc fc0

 0:096x2  0:067c2 þ 0:411

in which Aso fyo þAsi fyi Ac gc fc0 i ¼ DoD 2to

x ¼ c

Zhao et al. [71]

fres ¼ bc fcc0

Strength reduction coefficient

bc ¼ 2:828 

Stress-strain curve

Compressive strength and the corresponding strain of confined concrete



 

fyo gc fc0

!

  0:03x2 þ 0:302c2 þ 0:01 



to Do

 ε rm 0 εcc  sc ¼ r ε m rm 1þ 0 εcc where rm ¼ Ec 0 fcc Ec  ε0cc !#   8 " fyo t0 > 0 > 1 þ 4:9 for circular cross  section f > > D0 < c fc0 0 fcc ¼ " !#   > > fyo t0 > 0 > : fc 1 þ for square cross  section B0 fc0 " !# fcc0

fcc0 fc0

1

  ε0c ¼ 0:002 þ fc0 20  125  107

  fyi  0:155

273

ε0cc ¼ ε0c 1 þ5 Strain corresponding to the compressive strength of unconfined concrete

to Do

3.2 Stressestrain response of materials

Circular and Square CFDSTs

Residual strength

TABLE 3.11 Models for the stress-strain curve of the sandwiched concrete and the concrete core in circular and square DCFSTs. Researcher

CHS outer-CHS inner

Ahmed et al. [72]

Definition

Equation Confined concrete

The ascending branch of the curve

0 fcc; il

. εc; i ε0 cc; i

l1þ

. εc; i ε0 cc; i

sc; i ¼

E;

l ¼ E; ic 

Strain corresponding to the compressive strength of unconfined concrete The lateral confining pressure on the confined concrete

. 0 fcc; i ε0 cc; i

!l

for εc; i < ε0cc; i

!

0 0 fcc; i ¼ gc f ci þ K1 fl;i !

ε0cc; i ¼ ε0ci 1 þ f 02 l;i K f

co; i

K1 ¼ 4:1 and K2 ¼ 20:5 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi

   0:626 gc f 0 ci  4:33  107 þ 0:00076 ε0c;i ¼

 

      fr;i ¼ 2:2897 þ 0:0066 Dtoo  0:1918 Dtii  0:0585 Dtoo 0:3801 Dti i x1 ‡ 0

The descending branch of the curve

0 " sc; i ¼ fcc;i

Strain corresponding to the inflection point

εci ¼

Residual strength

!

3. Analytical methods

Compressive strength and the corresponding strain of confined concrete

ic

274

Cross-section

2:8ε0cc;i

0 fcc;i fres;i !2 # εc;i  ε0cc;i þ1 εci  ε0cc;i ! 0:12  fres;i 0 gc fc;i þ10ε0cc;i f0 cc;i

! f

1  fres;i 0 cc;i

0:47  0 gc fc;i

  . 0  1:0 0  fres;i 0 ¼ 1:2420  0:0029 Dt  0:0044gc fc;i fcc;i Sandwiched concrete

!

The stressestrain response of the sandwiched concrete is modeled by using the model of Liang et al. for circular CFST members (Table 3.7). SHS outer-CHS inner

Ahmed et al. [73]

Confined concrete !

0 fcc; il

. εc; i ε0 cc; i

l1þ

. εc; i ε0 cc; i

The ascending branch of the curve sc; i ¼

E;

l ¼

Compressive strength and the corresponding strain of confined concrete

. ε0cc; i

for εc; i < ε0cc; i

!

0 fcc; i

3.2 Stressestrain response of materials

E; ic 

ic

!l

0:06

0 fcc; i

0 gc fc;i

¼

0  0:91  ðgc fc;i Þ fr;i 0 þ 5:2 gc fc;i 0 gf c c;i

 f ε0cc; i ¼ εc;i þ 0:045 g r;if 0

1:15

c c;i

Strain corresponding to the compressive strength of unconfined concrete

ε0c;i ¼

ð

Þ

0 0:225 gc fc;i

1000

Continued

275

TABLE 3.11 Models for the stress-strain curve of the sandwiched concrete and the concrete core in circular and square DCFSTs.dcont’d Researcher

Definition The lateral confining pressure on the confined concrete

Equation 8 2ti > > > < 0:7ðwe  ws Þ Di  2ti fy;i fr;i ¼    > > > fy;i : 0:006241  0:0000357 Di=t i

where ws ¼ 0:5 we ¼0:2312 þ 0:3582v0e 0:04 

0 gc fc;i

0 sc; i ¼ fcc;i "

 3 Di ti

0 fcc;i fres;i

εc;i  ε0cc;i !

εci ¼

Residual strength fres;i ¼

2:8ε0cc;i

for 47
> f0 > > < cc > fr0:24 > 0 > > : 1:6fcc  0 0:32 gc f c

cc;i

for !  fcc0  0:15gc f 0c

Di  40 ti

for 40
30MPa  0:23 ε0c ¼ 9:405  104 fc0 0:05

(3.353) (3.354) (3.355)

The term m is used to reflect the concrete strength. The parameter LS represents the impact of the external confinement, LS2 is taken as 0.6466 for CFST columns and LS1 is equal to 0.6650 for CFST columns with external confinement. S the center-to-center spacing of external confinement and is taken as the diameter of external confinement d, H is the height of the column, and ε0c refers to the strain corresponding to the unconfined compressive strength. The average hoop strains εss;E of the external confinement are determined by considering two boundary conditions: 1. For FRP-confined CFST columns having continuous fiber sheet surrounding the steel tube (S ¼ d), εss;E is taken as εsq [94].

3.7 An analytical axial stressestrain model for circular CFST columns

313

2. For CFST columns that are confined by rings at the top only (S z H), εss;E is taken as 0:1εsq [92] The average hoop strains εss;E of the external confinement can be expressed by using the following quadratic function [92]: # " 0:9ðH  SÞ2 εss;E ¼ εsq þ 0:1 (3.356) ðH  dÞ2 The axial loadestrain responses of CFST columns can be defined with the stressestrain response of concrete (based on the model of Attard and Setunge [25]), the stressestrain response of steel tube (based on Eqs. 3.332e3.341), and the interaction of the steel tube and the concrete core (based on Eqs. (3.352)e(3.355), Eqs. (3.342)e(3.351), and Eq. (3.356)). The axial loads carried by the steel tube Ns and the concrete core Nc are obtained by multiplying the axial stresses of the steel tube ssz and the concrete core fcc0 by their respective cross-sectional areas, As and Ac . Therefore, Ns , Nc , and the total axial load of the CFST column Nt are expressed as: Ns ¼ ssz As

(3.357)

Nc ¼ fcc0 Ac

(3.358)

Nt ¼ Ns þ Nc

(3.359)

An iterative process is required to generate the axial loadestrain curves of CFST columns. Fig. 3.46 illustrates the flow chart of the iterative process required for defining the axial loadestrain curves of CFST columns. The can be updated depending on total hoop strain of the following step εiþ1 q i the total hoop strain of the current step εq by a small increment of hoop strain dεsq (0.5mε in elastic stage and 5 mε beyond elastic stage). Next,ssz , ssq , ssr , and fr are determined by assuming the total axial strain iþ1 of the next step εiþ1 z1 . Then, another total axial strain εz2 can be deteriþ1 iþ1 mined by knowing fr and εiþ1 q . If the difference between εz1 and εz2 are less than 0.1%, εiþ1 and εiþ1 q z1 converge. Otherwise, an iterative procedure according to the secant method must be conducted beginning from the second step. By achieving the convergence between εiþ1 and εiþ1 q z1 , the 0 magnitude of fcc , Nc , Ns , and Nt can be determined and a point on the axial loadestrain curve is acquired. Finally, steps (1) to (4) must be repeated until the required total axial strain is reached.

314

3. Analytical methods

FIGURE 3.46 Flow chart of the iterative procedure for defining the stressestrain curves of CFST columns. Lai MH, Ho JCM. A theoretical axial stress-strain model for circular concretefilledsteel-tube columns. Eng Struct 2016;125:124e43.

3.8 Path dependent stressestrain model for CFST columns The models presented in the previous sections assume that the steeleconcrete interface is completely bonded, ignoring the debonding impact that may occur because of the differing lateral dilation characteristics of steel tube and core concrete. However, debonding has been

3.8 Path dependent stressestrain model for CFST columns

315

shown to reduce the strength and stiffness of steeleconcrete composite columns, particularly for steel tubes filled with high-strength concrete. The confining stress, and therefore the strength of columns, may be overestimated if the debonding effect is ignored. On the other hand, the previous models follow the path-independence assumption. In other words, for a given set of axial strain and confining stress, the models assume that the axial stress of confined concrete in the CFST column is similar to the actively confined concrete stress. This is the same as assuming that the confined concrete stress at a particular axial strain is unaffected by the confining stress loading path, or, in other words, the hoop strain history. This hypothesis neglects the progressive growth of tensile splitting cracks in confined concrete because active confining stress applied in the laboratory test is hydraulically generated by an external source and can remain constant throughout the test. In contrast, activation of passive confining needs the dilation of the concrete core because of the axial compression and the development of splitting cracks. Since tensile splitting cracks grow continuously as axial strain increases, the extent of the splitting crack at a given confining stress in passively confined concrete must be greater, resulting in more axial deformation or strain. It means that passively confined concrete should have lower axial stress at the same axial strain and confining stress than actively confined concrete [95]. Assuming the ideal bond condition between the steel tube and the infilled concrete leads to Eqs. (3.228) and (3.229). During the initial loading stage and when the constituent components have linear behavior, the concrete core’s lateral dilation is smaller than that of the steel tube due to the difference between their Poisson’s ratios. This leads to the compressive hoop stress of the tube and negative confining stress if the bonding between the steel tube and the concrete core is assumed to be intact. Exceeding the negative confining stress from the interface bonding stress makes the steel tube debond from the concrete core. The bonding stress is affected by the geometry of columns (Dt ratio and H), concrete cylinder strength (fc0 ), steeleconcrete interface roughness, concrete compaction technique, and concrete age, among other factors. The bond shear stress sb increases with increasing the concrete compressive strength fc0 or reducing the Dt ratio. Based on Ref. [95], sb can be expressed as:    0:1 D exp 0:02 (3.360) sb ¼ 1:32 fc0 t

316

3. Analytical methods

The bond stress fb between the steel tube and the concrete core can be achieved using the Coulomb friction model as follows: mfb  sb

(3.361)

in which the term m is the coefficient of friction between the steeleconcrete interface and can be taken as 0.6 [54]. The magnitude of fb must be less than the ultimate tensile stress of concrete fct . fct can be predicted by EC2 [96]. 8  2 0 3 > > fc0  60 < 0:3 fc  8   (3.362) fb  fct ¼ fc0 > > fc0 > 60 : 2:12 ln 1 þ 10   If fb  fr , no interaction occurs between the constituent components. By reaching the elastoplastic behavior, the concrete’s lateral expansion becomes greater than the steel tube, leading to the development of the composite action. This is the moment when the steel tube confines the concrete core. Equations for calculating the confining stress fr can be found in the previous section. The hoop strains in the column can be derived using Eqs. (3.352)e (3.355). The axial stressestrain relationship of passively confined concrete is commonly accepted to be similar to that of actively confined concrete. This assumes that the confining stress that develops is unaffected by the confinement stress route. The stressestrain relationship, on the other hand, could be impacted by the stress path of confinement. For CFST columns exposed to monotonic axial load, the axial stressestrain relationship is path dependent [97]. This is due to the fact that the confining stress in actively confined concrete remains constant during the loading process, whereas the confining stress in passively confined concrete develops only as the lateral or axial deformation increases. Therefore, due to the larger axial stiffness and slower expansion of tensile splitting crack in actively confined concrete than passively confined concrete at given axial stress, the axial deformation is smaller in actively confined concrete than passively confined concrete [87]. Alternatively, the axial stress will be lower for a given axial strain and confining stress in passively confined concrete. The stressestrain response of actively confined concrete is determined using the model of Samani and Setung [25] as follows:  2   εz εz A: 0 þ B: 0 sc εcc εcc ¼ (3.363)  2   fcc0 εz εz 1 þ C: 0 þ D: 0 εcc εcc

317

3.8 Path dependent stressestrain model for CFST columns

in which fcc0 and ε0cc are the compressive strength and the corresponding axial strain of confined concrete, respectively, and A and B represent curve-shaped parameters and can be found in Section 3.2.3. fcc0 and ε0cc are given by: !1:25



1þ0:062 fr0 fc00:21 f

c fr qffiffiffiffi 0:56 f 0c " #   0 0 0 fr εcc ¼ εc 1 þ z1  0:06fc 0 fc

fcc0 ¼ fc0 1 þ

(3.364)

(3.365)

By comparing the experimental axial stressestrain curves for confined concrete of CFST columns with the actively confined concrete stresse strain curves employing Eqs. (3.363)e(3.365) with the same fr , the impact of stress path dependency can be examined. Fig. 3.47 shows a comparison of confined concrete stressestrain curves in CFST and actively confined columns in which the solid line represents the experimental stressestrain curve of confined concrete, and dotted lines present the actively confined concrete derived at various fr , captured at 0.003 axial strain interval of the CFST columns. The CFST specimen has a circular steel tube with a diameter of 139 mm and a thickness of 10 mm filled with the concrete with the compressive strength of 90 MPa [98]. It is worth mentioning that the experimental confined concrete stressestrain curve can be obtained using the following process: (1) The three-dimensional stress state of the

Confining stress (MPa)

250 Experiment fr=1.16 Mpa fr=9.95 Mpa fr=15.80 Mpa fr=19.60 Mpa fr=22.28 Mpa

200

150

100

50

0 0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

Axial strain

FIGURE 3.47 columns [95].

Stressestrain curves of confined concrete in CFST and actively confined

318

3. Analytical methods

steel tube can be estimated using Eqs. (3.332)e(3.341), utilizing axial and hoop strain readings in the strain gauges on the tube. (2) The confining stress fr can be computed using Eq. (3.342) as the hoop stress ssq is known. (3) The loadestrain curves of confined concrete can be defined as the difference between the calculated axial loadestrain curves of columns and those of steel tubes. The axial loadestrain curves of steel tubes can be calculated by multiplying the axial stress ssz in the steel tube area. (4) After acquiring the loadestrain curves, the experimental concrete stressestrain curves can be determined. The experimental confined concrete curves intersect those of actively confined concrete with confining stresses in a sequential manner, as shown in Fig. 3.47. At these axial strain intervals, the ratio of passively to actively confined concrete stresses  ! fcc;pas fcc;act was calculated, where fcc;pas and fcc;act represent the D¼ fcc;act passively and actively confined concrete stresses, respectively. Based on the acquired results of D, the experimental confined concrete stresses (i.e., passive stresses) are always less than the corresponding stresses calculated from the actively confined concrete stressestrain curve at the same axial strain. The value of fcc0 should be reduced to account for the lesser effectiveness of passive confinement. Based on Ref. [95], fcc0 in passively confined concrete can be obtained using the strength reduction factor b. b is given by: 8 fr > > 1:0 < 0:085 > > fc0 < !0:34 # !0:34 (3.366) b¼ " > > fr fr fr > > 0:085  0  0:611 : 0:71  0:12 0 fc fc0 fc Therefore, fcc0 is expressed as:

  fcc0 ¼ b fc0 þ 4:1fr

(3.367)

The three-dimensional stress state of the steel tube can be calculated using Eqs. (3.332)e(3.341). It is noteworthy that early buckling is the most common cause of failure for very thin-walled steel tubes. As a result, the yield stress ssy cannot be fully developed, necessitating the use of the elastic buckling stress (ssy;b ) specified in EC3 [1]. The buckling resistance pffiffiffi of a CFST column may be enhanced to 3 times that of a steel hollow tube column due to the concrete core’s supportive effect. Hence, the elastic

3.8 Path dependent stressestrain model for CFST columns

319

local buckling stress of the tube ssy;b can be obtained using the following equations: 0 1 ! pffiffiffi pffiffiffi 235 D B90 3 C (3.368) / ssy;b  235@  90 3 A D t ssy;b t ssy;b  ssy

(3.369)

The total axial load of the column Nt is governed by Eqs. (3.357)e (3.359). Like the previous model, determining the full range axial loadestrain curves of CFST columns need an iterative process. Fig. 3.48 illustrates the

FIGURE 3.48 The iterative procedure for defining the path-dependent stressestrain curve of CFST columns. Lai MH, Song W, Ou XL, Chen MT, Wang Q, Ho JCM. A path dependent stress-strain model for concrete-filled-steel-tube column. Eng Struct 2020;211:110312.

320

3. Analytical methods

required iterative procedure for defining the stressestrain curve of CFST columns using the path-dependent model. (1) Eqs. (3.360)e(3.362) are used to determine the average steeleconcrete interface bond stress, fb . (2) can be renewed with a minor The hoop strain of the following step εiþ1 q increase dεq . (3) The values of fr and fb must be compared. The steel tube separates from the concrete core, if fr  fb, causing the core concrete and steel tube to function independently. Because there is no interaction between the steel tube and the concrete core, the uniaxial stressestrain relationship is used for the steel tube, with the elastic buckling resistance pffiffiffi ssy;b equal to that of hollow steel tube (i.e., Eq. (3.368) without 3) because the buckling resistance is reduced. The axial stressestrain response and hoopeaxial strain response of infilled concrete are assumed to follow Eqs. (3.363)e(3.365) by taking fr as zero. At this time, no composite action is being developed. The CFST column’s axial capacity is the arithmetic sum of the concrete core and steel tube axial capacities at that axial strain. When the lateral dilation of concrete surpasses that of steel tube at a given axial strain, or when the axial strain of steel tube exceeds that of concrete iþ1 (i.e., εiþ1 sz ‡εcz ) at a certain hoop strain (lateral dilation), the composite action between concrete and steel tube is reactivated (i.e., rebonding). (4) The axial strain of the following step εiþ1 is assumed if fr < fb. (5) siþ1 z sz , iþ1 iþ1 iþ1 ssq , ssr , and fr can be obtained using Eqs. (3.332)e(3.342), (3.368), and (3.369). (6) Eqs. (3.352)e(3.355) are used to derive another axial strain, εiþ1 z2 . iþ1 The consecutive strains are converged if εiþ1 and ε are sufficiently z1 z2 closed, and any of them can represent the ultimate axial strain. Otherwise, using the secant approach, εiþ1 z1 is assumed again, and the iterative procedure from steps (4), (5), and (6) is repeated until the axial strain converges. (7) If the axial strain converges, Eqs. (3.363), (3.365)e(3.367), and (3.359) can be used to calculate fcciþ1 , Nciþ1 , Nsiþ1 , and Ntiþ1 , and a point in the axial loadestrain and hoopeaxial strain curves can be found. (8) Steps (1) to (7) must be repeated until the required total axial strain is reached. Fig. 3.49 shows the influences of debonding on the overall performance of a circular CFST stub column with D=t ¼ 165 and fc0 ¼ 73:4 MPa [99]. As shown in Fig. 3.49a, debonding leads to an almost 60% reduction in the magnitude of the confining stress. Comparing the longitudinal loade strain curves in Fig. 3.49b indicates that debonding can affect the column’s ductility and increase the slope of the postpeak region (decline the ductility). Besides, the accuracy of the predicted results is increased by considering the debonding, as depicted in Fig. 3.49b. It is worth mentioning that debonding in a thin-walled steel tube column filled with high-strength concrete makes the steel tube more susceptible to early buckling, lowering the steel tube’s elastic buckling stress. The steel tube

321

3.9 Elastic-plastic model for the stressestrain response of CFST columns 3.5

2000 1600

2.5 2 1.5 Including debonding

1 0.5

Without debonding

0 -0.5

0

0.005

0.01

0.015

Confining stress (MPa)

Confining stress (MPa)

3

1200 800 Including debonding Without debonding Experimental result

400 0 0

Axial strain

(a) Confining stress-axial strain curves

0.005

0.01 Axial strain

0.015

(b) Axial load-strain curves

FIGURE 3.49

Effects of debonding of steeleconcrete interface on the behavior of the CFST column [95]. (a) Confining stresseaxial strain curves (b) Axial loadestrain curves.

could not achieve its yield strength once buckling had occurred. As a result, the influence of debonding in this sort of composite column will affect both the prepeak stage of the stressestrain curve and the postpeak stage.

3.9 Elastic-plastic model for the stressestrain response of CFST columns This section discusses a continuum method for predicting the fullrange elastoplastic analysis of circular concrete-filled steel tube columns under axial compression [100]. An elastoplastic analysis based on continuum mechanics can be used to understand the load pattern. Besides, the triaxial stressestrain developments (axial, radial, and perimeter directions) can be anticipated in a full elastoplastic range. The mechanical model is developed according to the following assumptions: 1. No slip occurs between the steel tube and the concrete core. 2. The steel tube and the concrete core are in full contact with each other. 3. No local buckling occurs in the stub column. First, the constitutive models of concrete core and steel tube are defined. The compressive strength of the confined concrete fcc0 is governed by: fcc0 ¼ fc0 þ 3:4sr

(3.370)

The above equation is valid for concrete strength fc0 ranging from 30 to 120 MPa.

322

3. Analytical methods

Based on Ref. [101], the secant modulus at the peak stress in the axial direction Ef for the concrete under triaxial loading state is expressed by: Ef ¼ pffiffiffi !

(3.371)

f0

J2 fc0

in which x ¼

E0 1 þ 4ðA1  1Þx

 p1ffiffi, E0 ¼ εc0 , J2 is the second stress invariants, and 3

c

f 49

A1 ¼ 9:1fcu is the variable corresponding to the ascending stage of the curve in which fcu denotes the cubic compressive strength of concrete [102]. The relationship between fc0 and fcu is as follows: 7

6 fc0 ¼ 0:4fcu

(3.372)

Based on Ref. [103], Eq. (3.371) for the confined concrete under axial compression can be simplified as follows: Ef ¼

E0 sr 1 þ 4:8ðA1  1Þ 0 fc

!0:5

(3.373)

The strain ε0cc corresponding to fcc0 can be predicted by the following equation: !" !0:5 # sr sr 0 0 εcc ¼ εc 1 þ 3:4 0 1 þ 4:8ðA1  1Þ 0 (3.374) fc fc The stressestrain response of the concrete core under axial compression can be expressed as: 8 A2 x þ ðB2  1Þx2 > > for x  1 > < 1 þ ðA2  2Þx þ B2 x2 (3.375) y¼ > x > > for x > 1 : 0:15ðx  1Þ2 þ x in which y ¼ sf 0c , x ¼ εε0c . cc

cc

The terms A2 and B2 are the curve shape parameters. A2 controls the ascending stage of the axial stressestrain curve, and B2 is a parameter to define elastic modulus reduction over the ascending stage of the curve. Parameter A2 is given by: " !0:5 # sr (3.376) A2 ¼ A1 1 þ 4:8ðA1  1Þ 0 fc

3.9 Elastic-plastic model for the stressestrain response of CFST columns

323

Based on the elastic modulus description, when the axial stress of the concrete during the prepeak region is less than 0:4fc0 , the curve can be modeled by a straight line (it can be assumed that the concrete has a linear 0:4f 0

behavior). Therefore, when x ¼ f 0 Ac2 and x ¼ cc

0:4fc0 fcc0 ,

the term B2 can be

defined as: B2 ¼

y þ ðA2  2Þxy  ðA2  xÞx ð1  yÞx2

(3.377)

During the elastic stage, concrete Poisson’s ratio wc under uniaxial compression is in the range of 0.15 and 0.22. By increasing the axial stress on the concrete, wc increases gradually. Poisson’s ratio of the concrete wc can be obtained by: 8w for x  1; y  ya 0 > > sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > >   <  y  ya 2 (3.378) wc ¼ wf  wf  w0 1  for x  1; ya < y  1 > 1  ya > > > : for x > 1; y < 1 wf     in which wf ¼ 1  0:0025 fcu 20 and ya ¼ 0:3 þ 0:0025 fcu 20 , based on Ref. [101]. For the sake of simplicity, the amount of wf can be taken as 0.5 due to the presence of the confinement effect on the concrete core. It is noteworthy that the concrete core’s stressestrain response can also be represented by using the models discussed in Section (3.2.3). The stressestrain relationship for the steel tube can be defined using the models presented in Section (3.2.1), or it can be defined as follows: 8 Eε > for εs  εy > > s s > > > > for εy < εs  12εy < fy (3.379) ss ¼  1  > > f þ  12ε for 12ε < ε  ε E ε ε y s s y y y s u > > 216 > > > :f for εs > εu u 0:5f

in which fu is taken as 1:5fy , εu ¼ 12εy þ 1 y . Poisson’s ratio of steel ws is Es 216 governed by: 8 0:285 for εs  0:8εy > > > ! > < ss (3.380) ws ¼ 1:075  0:8 þ 0:285 for 0:8εy < εs  εy > fy > > > : 0:5 for εs > εy

324

3. Analytical methods

• Stressestrain response of CFST stub columns during the elastic stage: When the axial deformation is small, the axial strains of the steel tube and the concrete core are the same (εcz ¼ εsz ¼ εz ). In this case, the simplification of the axial symmetric plane strain is possible. Based on the Airy stress function, presented in Eq. (3.381), the following expressions can be defined: G ¼ C1 ln r þ C2 r2 ln r þ C3 r2 þ C4

(3.381)

where C1 , C2 , C3 , and C4 are coefficients. 

 1. For the concrete core 0 < r  D2 t : Stress

Strain

scr ¼ scq ¼ 2C3

(3.382)

scz ¼ Ec εz þ 4wc C3

(3.383)

 
εcr ¼ εcq ¼ 2C3 1  wc  2w2c E  w ε c c z

(3.384)

Displacement

  

2C3 1  wc  2w2c ucr ¼ r  w c εz Ec



2. For the steel tube

ucz ¼ Lεz  t  r  D2 : 

D 2

(3.385) (3.386)

Stress ssr ¼

c01 þ 2c03 r2

ssq ¼ 

Strain

c01 þ 2c03 r2

(3.387) (3.388)

ssz ¼ Es εz þ 4ws c03

(3.389)

  8 1 þ ws c01 2c03 1  ws  2w2s > > > þ  w s εz < εsr ¼ E Es r2 s   0 2 > 0 > > : εsq ¼ 1 þ ws c1 þ 2c3 1  ws  2ws  ws εz Es r 2 Es

(3.390)

3.9 Elastic-plastic model for the stressestrain response of CFST columns

Displacement "  #  8 0 2 0 > < usr ¼ r 2c3 1  ws  2ws  ws εz  1 þ ws c1 Es Es r > : usz ¼ Lεz

325

(3.391)

By assuming full contact between the steel tube and the concrete core, the stress condition at midheight of the stub column can be obtained based on Saint-Venant’s principle, as follows: scz Ac þ ssz As ¼ N

(3.392)

Taking fsc ¼ ANsc where Asc ¼ As þ Ac , the following expression can be achieved: fsc ¼ ð1  rÞscz þ rssz

(3.393)

In which r ¼ AAscs z 4t D. The boundary conditions of the steel tube and the concrete core in the midheight region can be expressed as follows: 1. Stress boundary condition sr jr¼D ¼ 0

(3.394)

2

2. Stress and displacement compatible condition sr j  r¼

ur j  r¼

D t 2

D t 2

 ¼ sr j 

þ

(3.395)

 ¼ ur j 

þ

(3.396)

r¼

r¼

D t 2

D t 2

Based on Eqs. (3.382)e(3.391) and Eqs. (3.394)e(3.396), coefficients c01 , and c3 can be defined as follows: 8 D2 > 0 > > c ðwc  ws Þð1  rÞQ:Es εz ¼ > 1 > 4 > > < 1 (3.397) c03 ¼  ðwc  ws Þð1  rÞQ:Es εz > 2 > > > > > > : c3 ¼ 1 ðwc  ws ÞrQ:Es εz 2

   1 in which Q ¼ 1  wc  2w2c nr þ 2  2w2s  r þ rws þ 2rw2s and n ¼ Es Ec .

c03 ,

326

3. Analytical methods

Substituting Eq. (3.397) for Eqs. (3.382)e(3.384) and Eqs. (3.387)e (3.390), the following expressions can be achieved: ( scr ¼ scq ¼ rðwc  ws ÞQ:Es εz (3.398) scz ¼ ½Ec þ 2rwc ðwc  ws ÞQ:Es εz  

 εcr ¼ εcq ¼ r 1  wc  2w2c ðwc  ws ÞQn  wc εz (3.399)

 2  8 D > > s ¼  1 ðwc  ws Þð1  rÞQ:Es εz > sr > > 2r >
ssq ¼   1 ðwc  ws Þð1  rÞQ:Es εz > > > 2r > > : ssz ¼ ½1  2ws ðwc  ws Þð1  rÞQEs εz

 2  8   D > 2 > εsr ¼ ð1 þ w Þ  1  w  2w > s c c :ð1  rÞðwc  ws ÞQεz  ws εz < 2r

 2  > >   > : εsq ¼  D ð1 þ ws Þ  1  wc  2w2 :ð1  rÞðwc  ws ÞQεz  ws εz c 2r (3.401) Replacing the equation of scz in Eq. (3.398) and ssz in Eq. (3.400) for Eq. (3.393), the composite stressestrain response of the stub column during the elastic stage can be expressed as: fsc ¼ Esc εz

(3.402) 2

Esc ¼ ð1  rÞEc þ rEs þ 2ðwc  ws Þ ð1  rÞrQEs

(3.403)

• Stressestrain response of CFST stub columns during the plastic stage: For the steel tube under lateral pressure and axial compressive force, yielding of the internal section (where r ¼ D2  t) occurs first, and the plastic region develops quickly to the whole section of the steel tube considering that t D. The radial stress of the steel tube is neglected while evaluating the stress of the steel tube in the plastic stage. Based on Eq. (3.400), the stresses of the tube after yield can be expresses as follows: ( ssq ¼ 2ðwc  ws Þð1  rÞQt :Ets εz (3.404) ssz ¼ ½1  2ws ðwc  ws Þð1  rÞQt Ets εz in which Qt ¼



  1 1  wc  2w2c nt r þ 2  2w2s  r þ rws þ 2rw2s

(3.405)

3.9 Elastic-plastic model for the stressestrain response of CFST columns

nt ¼

Ets Etc

Etc ¼ Et  2rws ðwc  ws ÞQt Ets

327 (3.406) (3.407)

in which the term Ets denotes the secant modulus of the steel tube and is obtained by Eq. (3.379), and Et is the secant modulus of the concrete achieved by Eq. (3.375). After yielding of the steel tube, the strain at its midheight is expressed by: 3 2 8 7 6ð1 þ ws Þ  > εsr ¼ 4  1  ws  2w2s 5$ð1  rÞðwc  ws ÞQt :εz  ws εz > r > > > 1 < 4 2 3 > > > >  ð1 þ ws Þ  > 2 7 : εsq ¼ 6 4 r þ 1  ws  2ws 5$ð1  rÞðwc  ws ÞQt :εz  ws εz 1 4 (3.408) The strains at the steel tube’s outer surface can be computed by Eq. (3.401) by substituting the term Q with Qt . The behavior of the concrete core turns into the plastic phase by increasing the axial load. The expressions for the stresses and strains of concrete core in the nonlinear phase have similar forms with those of the elastic stage. However, the terms Ec , Es , and Q should be replaced by Etc, Ets , and Qt , respectively. Consequently, the stressestrain response of the CFST stub column during the nonlinear behavior can be expressed as follows: fsc ¼ Etsc εz

(3.409)

Etsc ¼ ð1  rÞ Etc þ rEts þ 2ðwc  ws Þ2 ð1  rÞrQt Ets

(3.410)

Hence, the full elastoplastic composite stressestrain response of the CFST stub column can be written as: ( Esc εz for εz  εz;p (3.411) fsc ¼ Etsc εz for εz > εz;p • Simplified model A typical axial stressestrain response of CFST stub columns subjected to axial compression predicted using the numerical elastoplastic analysis

328

3. Analytical methods

is illustrated in Fig. 3.50 in which Nu is the ultimate capacity, fsc;u is the ultimate strength, Esc denotes the composite modulus of elasticity, fsc;p represents the nominal axial limit stress of elasticity and εz;p is the corresponding strain, εsc;0 is the strain corresponding to the ultimate strength, Ny is the residual capacity, and fsc;y is the residual strength.
The ultimate capacity is achieved when dfsc dε ¼ 0 in the stressestrain z relationship. The ultimate capacity of the stub column might be reached when the concrete core reaches its ultimate strength in the axial direction, based on the elastoplastic model. Meanwhile, the steel tube is at the elastic-perfectly plastic or strain-hardening stage. Considering that the tube is at its perfectly plastic state and under the von Mises yield condition, the compressive strength of the confined concrete is governed by the following linear expression: fcc0 ¼ fc0 þ kscr

(3.412)

in which the term k is a constant. The relationship between scr and ssq in the inelastic phase can be stated using Eqs. (3.398) and (3.404): r (3.413) ssq scr ¼ 2ð1  rÞ Based on the von Mises yield criterion, the following can be expressed for the steel tube: s2sz þ ssz ssq þ s2sq ¼ fy2

(3.414)

High

Axial stress fsc

Middle Low

Axial strain εL

FIGURE 3.50 Typical axial stressestrain relationship for CFST stub columns.

3.9 Elastic-plastic model for the stressestrain response of CFST columns

329

By replacing Eqs. (3.412) and (3.413) with Eq. (3.414), ssz can be defined as:

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 u ssz u 3 scr 1 scr t  ¼ 1 2 0 fy x fc0 x fc

(3.415)

in which the term x is the confinement factor and is expressed as: x¼

As fy r fy ¼ Ac fc0 1  r fc0

(3.416)

The ultimate capacity of the column at its mid-height region is governed by: Nu ¼ fcc0 Ac þ ssz As

(3.417)

By replacing Eqs. (3.415) and (3.412) with Eq. (3.416), the resulting ultimate capacity of the column Nu;1 can be written as: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " !2 # u u s s cr cr t Nu;1 ¼ fc0 Ac 1 þ x2  3 0 (3.418) þ ðk  1Þ 0 fc fc Hence, the following expression can be derived: dNu;1 scr rðk  1Þ ¼ 0 / 0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dscr fc 9 þ 3ðk  1Þ2

(3.419)

The maximum ultimate capacity of the column is achieved by replacing Eq. (3.419) with Eq. (3.418), as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # " 3þðk  1Þ2 (3.420) Nu;2 ¼ fc0 Ac 1 þ x 3 If the load is applied only on the concrete core, the ultimate capacity of the column can be computed by:   kx (3.421) Nu;3 ¼ fc0 Ac 1 þ 2 The term k is a constant. If k ¼ 3:4, it can be clarified by the elastoplastic analysis that Nu;1 is normally less than Nu;3 . For a low-strength concrete N

core, it can be recognized that the amount of the Nu;1 ratio is nearly equal to u;3 N

1. The amount of the Nu;1 ratio reduces and reaches to 0.9 by increasing the u;3 strength of concrete and streel ratio. However, the strain-hardening of steel in the plastic region and its effect on enhancing the confinement

330

3. Analytical methods

effect may prevent such a reduction in the experimental investigation. From this viewpoint, the CFST stub column’s ultimate capacity can be predicted using both the simplified limit equilibrium analysis and the full elastoplastic analysis scheme. Hence, the expression for predicting the ultimate capacity Nu and ultimate strength fsc;u of the CFST stub column can be defined as follows: Nu ¼ fc0 Ac ð1 þ 1:7xÞ

(3.422)

fsc;u ¼ ð1  rÞð1 þ 1:7xÞfc0

(3.423)

Based on the numerical results [100], the part 2ðwc  ws Þ2 ð1 rÞrQEs  in Eq. (3.403) has a quite small value 0 < 2ðwc  ws Þ2 ð1  rÞ rQEs 

ð1 rÞEc þrEs . Therefore, the composite modulus of elasticity Esc given in Eq. (3.403) can be simplified as: Esc ¼ ð1  rÞEc þ rEs

(3.424)

The composite stiffness Esc Asc is governed by: Esc Asc ¼ Ac Ec þ As Es

(3.425)

The concrete core attains its ultimate strength by reaching the ultimate capacity of the CFST column, and εsc;0 becomes ε0cc . Therefore: !" !0:5 # s s cr cr εsc;0 ¼ ε0cc ¼ ε0c 1 þ 3:4 0 1 þ 4:8ðA1  1Þ 0 (3.426) fc fc According to Eqs. (3.412) and (3.415), εsc;0 can be written as: pffiffiffiffiffiffi εsc;0 ¼ ð1 þ 1:7xfÞ½1 þ 3:4 xf ðA1  1Þε0c in which f ¼

ssq fy

(3.427)

is a function of the concrete core’s strength and can be

0 . taken as 0:9  0:005fcu The nominal limit capacity of elasticity Np and the corresponding strain εz;p can be expressed as:

Np ¼ Esc εz;p Asc

(3.428)

fsc;p ¼ Esc εz;p

(3.429)

εz;p ¼

qfy Es

(3.430)

where the term q is governed by: q ¼ 0:48 þ 3:91  r1:62 þ

0:52  3:91r1:62 !2:6 f y 5:5  103 0 þ1 fc

(3.431)

3.9 Elastic-plastic model for the stressestrain response of CFST columns

331

The residual capacity Ny can be expressed as: Ny ¼ ð0:3 þ 2:2rÞfc0 Ac

(3.432)

fsc;y ¼ ð1  rÞð0:3 þ 2:2rÞfc0

(3.433)

or Ny ¼

0:3 þ 2:2r Nu 1 þ 1:7r

(3.444)

fsc;y ¼

0:3 þ 2:2r fsc;u 1 þ 1:7r

(3.445)

It can be observed from Fig. 3.50 that the axial loadestrain response of the CFST stub column can be divided into different stages, as follows: 1. when εz ¼ 0, fsc ¼ 0; dfsc dεz

2. c0  εz < εz;p ,

3. cεz;p  εz < εsc;0 ,

¼ Esc ; d2 fsc dε2z

< 0; and the slope of the curve reduces

gradually and no inflection point occurs; 4. when εz ¼ εsc;0 ; fsc ¼ fsc;u ;

dfsc dεz

¼ 0; at εi ‡εz ;

d2 fsc dε2z

¼ 0, an inflection

point results; 5. at εz /N; fsc /fsc;y ;

dfsc dεz /0.

Hence, the axial loadestrain response of CFST stub columns under concentric axial compression can be defined by the nondimensional mathematical functions, as follows: 8 A x for x  xb > > 3qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi >   > < b3 fsc N a23  ð1  xÞ2  b3 þ 1 for xb  x  1 (3.446) ¼ ¼ a 3 > fsc;u Nu > > > 1 : 3 x a3 ðx  1Þ2 þ b3 ðx  1Þ3 þ x3 for x > 1 ε

;p z in which x ¼ εεsc;0 , xb ¼ εsc;0 , and the term A3 represents the ratio of the composite modulus of elasticity Esc to the secant modulus at the ultimate

f

). strength (Esc;p ¼ εsc;u sc;0 A3 ¼ A1

pffiffiffiffiffiffi ½ð1  rÞnrð1 þ 1:7xfÞ½1 þ 3:4 xf ðA1  1Þ ð1  rÞð1 þ 1:7fÞ

(3.447)

332

3. Analytical methods

Parameters a3 , b3 , a3 , and b3 are given by: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  xb a3 ¼ ðA3  1Þ A3 ðA3 xb þ A3  2Þ

(3.448)

in which ðA3 xb þA3 2Þ must be greater than 0. b3 ¼

A3 xb þ A3  A23 xb  1 A3 xb þ A3  2

(3.449)

0:7  0:5x 0:3 þ 2:2x

(3.450)

b3 ¼

8 > < 0:5 a3 ¼ x4 > : 0:5

for x  1

(3.451)

for x > 1

3.10 Strength enhancement induced during cold forming

800

900

700

800

600

700 Stress (MPa)

Stress (MPa)

Cold forming of steel sheets for forming steel square or rectangular hollow sections can significantly enhance the strength at the corner regions of the cold-formed section. The strength enhancement due to cold forming is more prominent in stainless steel than carbon steel. Compared with the flat regions, the strength enhancement at the corner regions of stainless steel is usually 20% and 100% greater than the 0.2% proof strengths. Fig. 3.51 compares the stressestrain curves of the flat region

500 400 300

600 500 400 300

200

Flat region

200

Flat region

100

Corner region

100

Corner region

0

0 0

0.01

0.02 Strain

0.03

(a) Initial loading stage

0.04

0

0.2

0.4

0.6

Strain (b) Full loading

FIGURE 3.51 The effects of cold forming on the stressestrain curve of stainless steel. (a) Initial loading (b) Full loading. Wang, X.-Q., Tao, Z., Song, T.-Y., & Han, L.-H. (2014). Stress-strain model of austenitic stainless steel after exposure to elevated temperatures. Journal of Constructional Steel Research, 99, 129e139. doi: https://doi.org/10.1016/j.jcsr.2014.04.020.

3.10 Strength enhancement induced during cold forming

333

with the corner region of austenitic stainless steel. It can be observed from the figure that the cold forming of the material increases the yield strength and ultimate strength of the steel. By contrast, the corner region shows less ultimate strain and strain hardening than the flat region throughout the tensile deformation. Due to the combined impacts of the deformation-induced dislocations and martensitic-phase transformation, corner regions of cold-formed austenitic stainless sections exhibit notable strength enhancement. By contrast, duplex and ferritic stainless steels demonstrate less significant deformation hardening, which is attributed to the lack of martensite transformation in the alloys. Accordingly, various models have been recommended for different grades of corner stainless steels for precise analysis of the nonlinear performance of CFST/CFDST. The strength enhancement of the corner materials depends on the production route. The most straightforward forming method is press braking, where single folds are formed in the steel sheet between a tool and die, as shown in Fig. 3.52a. This method of production is controlled manually and is useful for manufacturing a limited number of sections. Another method is cold rolling, which is a more automated process. In this process, the sheet material is straightened and flattened before being formed into the desired section shape by a series of rollers. Cold rolling is a helpful method for producing large quantities of similar sections. For the fabrication of a box section, a circular steel tube is first welded closed and then crushed into a box section using a Turks head die, as shown in Fig. 3.52b.

(a) Press breaking

FIGURE 3.52

(b) Cold rolling of a box section

Cold-forming methods. (a) Press breaking (b) Cold rolling of a box section. Cruise, R. B., & Gardner, L. (2008). Strength enhancements induced during cold forming of stainless steel sections. Journal of Constructional Steel Research, 64(11), 1310e1316. doi: https://doi.org/10. 1016/j.jcsr.2008.04.014.

334

3. Analytical methods

The strength enhancement of the corner materials can be predicted by the following equations. 8 1:673s0:2;f > > > for press  braked sections < r 0:126 i (3.452) sy;c ¼ t > > > : 0:83s for cold  rolled box sections u;f where ri and t are the internal corner radius and the thickness of the cross section, respectively, s0:2;f is 0.2% proof stress of the virgin material, and su;f is the ultimate strength of the virgin material. The revised Rasmussen model can be used to predict the stressestrain curves of corner regions of rectangular austenitic and duplex stainless steel tubes, as follows:  nc 8 s s > þ 0:002 s  sy;c > < E0 E0 ε¼   mc > > : 0:002 þ sy;c þ s  sy;c þ εu;c s  sy;c sy;c < s  su;c E0 Ey;c su;c  sy;c (3.453) where Ey;c is the tangent modulus of the corner material at the yield stress, sy;c and su;c denote the yield stress and the ultimate strength of the corner material, respectively, εu;c presents the ultimate strain corresponding to su;c , and nc and mc are nonlinear parameters for the corner material, respectively. E0   nc E0 1 þ 0:002 sy;c  900  s0:2;f s0:2;f sy;c ¼ 1 þ 0:05 s0:2;f E0 s su;c  u;f ¼ 0:56s0:226 0:2;f  1:4 sy;c s0:2;f Ey;c ¼

εu;c ¼ 1 

sy;c su;c

mc ¼ 0:04s0:2;f  8  m   s0:2;f 0:3n nc ¼ 0:9n2 E0

(3.454)

(3.455) (3.456) (3.457) (3.458) (3.459)

References

335

For defining the stressestrain curves of corner regions of rectangular ferritic stainless steel tubes, Eq. (3.453) can be used. However, the corresponding yield stress sy;c and the ultimate strength su;c must be calculated using the following equations: ! su;f sy;c E0 ¼ 0:25 þ (3.460) s0:2;f 500s0:2;f s0:2;f ! su;c E0 ¼ 1:55  1 (3.461) sy;c 840s0:2;f in which su;f in Eq. (3.460) must be determined by Eq. (3.462) developed particularly for ferritic stainless steel. 8 s0:2;f s0:2;f > > 0:104 þ 360 for 0:00125   0:00235 s0:2;f < E0 E0 ¼ (3.462) s0:2;f > su;f > : 0:95 for 0:00235   0:00275 E0 The mechanical properties of the corner material of different types of stainless steel can be captured with acceptable accuracy with the proposed stressestrain models. Applying these models increases the accuracy of the analytical analysis of cold-formed stainless steel structures.

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C H A P T E R

4 Numerical methods O U T L I N E 4.1 Introduction

343

4.2 Numerical modeling of confined concrete 4.2.1 Concrete plasticity model: DruckerePrager type plasticity model 4.2.1.1 Yield criterion 4.2.1.2 Strain hardening and softening 4.2.1.3 Flow rule 4.2.1.4 Assessment of the presented DruckerePrager type plasticity model 4.2.1.5 DruckerePrager type plasticity model modifications 4.2.1.6 Implementing the modified DruckerePrager type plasticity model in ABAQUS 4.2.2 Concrete plasticity model: concrete damage plasticity model 4.2.2.1 Damage 4.2.2.2 Yield criterion 4.2.2.3 Hardening/softening rule 4.2.2.4 Flow rule 4.2.2.5 CDP model modifications 4.2.3 Effects of different parameters on the numerical results 4.2.3.1 Effects of stressestrain model of steel on the FE results 4.2.3.2 Effects of stressestrain model of concrete on the FE results 4.2.3.3 Effects of the ratio of the second stress invariant on the tensile meridian to that on the compressive meridian on the FE results 4.2.3.4 Effects of the dilation angle on the FE results

343 344 345 349 349

4.3 Investigating the behavior of composite members through numerical analysis 4.3.1 Short columns

Single Skin and Double Skin Concrete Filled Tubular Structures https://doi.org/10.1016/B978-0-323-85596-9.00006-8

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351 354 356 359 360 363 365 366 367 375 376 376 378 380

384 385

© 2022 Elsevier Inc. All rights reserved.

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4.3.1.1 4.3.1.2 4.3.1.3 4.3.1.4 4.3.1.5 4.3.1.6 4.3.1.7

Effects of the outer tube depth-to-thickness ratio Effects of the concrete compressive strength Effects of the outer steel tube yield strength Effects of the inner steel tube yield strength Effects of the hollow ratio Effects of the inner steel tube depth-to-thickness ratio Efficiency of increasing material strength on the loadbearing capacity of composite members 4.3.1.8 Simultaneous effects of the depth-to-thickness ratio of the steel tube and the concrete compressive strength on the ultimate axial strength of the CFDST column 4.3.1.9 Effects of local buckling 4.3.1.10 Effects of section shapes

4.3.2 Mechanism of interaction of steel tubes and concrete 4.3.2.1 Development of interaction between steel tubes and the concrete core 4.3.2.2 Influence of geometric and material properties on the confinement effect 4.3.3 Tapered and inclined composite members 4.3.3.1 Tapered composite members 4.3.3.2 Inclined composite members 4.3.3.3 Effect of the tapered and inclined angles on the ultimate axial capacity of composite members 4.3.4 Slender columns 4.3.4.1 Failure modes 4.3.4.2 Loadestrain curve 4.3.4.3 Effects of column slenderness ratio 4.3.4.4 Effects of width-to-thickness ratio 4.3.4.5 Effects of concrete compressive strength 4.3.4.6 Effects of the hollow ratio 4.3.4.7 Effects of the load eccentricity ratio 4.3.4.8 Load distribution in constituent components of slender composite members 4.3.4.9 Ultimate axial strengths of slender beam columns 4.3.4.10 Effects of concrete confinement 4.3.4.11 Pure bending strengths of slender beam columns 4.3.4.12 Residual strength 4.3.4.13 Behavior of concrete-filled corrugated steel tubular stub column Preloading of composite columns References

385 387 388 391 392 395 403 406 410 412

419

419 424 427 427 434 435 441 443 444 453 453 457 460 463 465 472 475 478 481 481 528

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343

4.1 Introduction Undoubtedly, much information regarding the structural behavior of members can be achieved through experimental studies. However, conducting parametric studies to investigate a wide range of material and geometric properties on structural members’ performance may be time consuming and costly. Besides, an accurate understanding of what occurs inside the elements throughout the loading history is complicated and sometimes impossible via experimental investigations. For instance, observing the interaction pressure between the steel tube and the concrete core of a composite column cannot be achieved through experimental studies. Due to advances in computer performance and engineering software development like ANSYS and ABAQUS, civil engineers and researchers have become increasingly interested in finite element (FE) approaches and fiber-based simulation methods over the past few decades. With the help of FE analysis, researchers can directly simulate the composite action between the steel and concrete elements. Besides, FE simulation enables researchers to integrate a variety of factors into their investigations, such as material imperfections, residual stresses caused by welding, and various loading and boundary conditions. The accuracy of FE findings, on the other hand, is determined by the input parameters, such as constitutive models for characterizing the behavior of steel and confined concrete components, mesh size, and interaction modeling between steel and confined concrete surfaces. This section discusses the influence of various parameters on the FE findings. The numerical methods for simulating the behavior of concrete confined by a steel tube are described in this chapter. In addition, the impacts of parameters that might influence the outputs of nonlinear FE analysis are explored. The basic behavior of short and long CFST and CFDST columns and beam columns is also studied, as are the effects of geometric and material characteristics on composite member performance.

4.2 Numerical modeling of confined concrete Concrete is a nonhomogeneous, anisotropic material with a nonlinear behavior even at low stress levels. Furthermore, under tension and compression stresses, concrete behaves differently. Compression loading causes the concrete’s response to hardening until the maximum stress is reached. As the degree of lateral confinement increases, so does the peak stress. There is a direct correlation between postpeak behavior and the degree of lateral confinement. Postpeak softening occurs at low confinement. When confinement stresses increase, concrete’s response shifts from brittle to ductile hardening. Concrete cracks and ultimately loses its

344

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entire strength when subjected to tensile loads. Confinement can postpone the loss of stiffness and strength while increasing the deformability of concrete. Since the confined concrete has larger internal stress and more excellent deformability, unstable fracture propagation will take longer to begin. Confinement provides the lateral kinematic constraint required to avoid volumetric dilatation and hold the concrete particles together long enough to postpone failure. Under extremely high confining pressures, concrete dilatation can be completely prevented. Concrete’s response to triaxial compression is dramatically influenced by microcrack development and expansion. Microcrack evolution controls concrete brittleness, ductility, dilatancy, and failure mechanisms. All of these phenomena are affected by the triaxial condition of stress imparted to the concrete. Vertical tension splitting causes compression test failure under small lateral confinement, where a sudden loss of residual strength can be expected. Cracking and damage become more evenly distributed as confinement increases, and concrete becomes ductile with minimal or no postpeak deterioration. A precise constitutive model is required for reliable FE modeling of concrete. Researchers have presented different constitutive models for analytical and FE modeling of steel-confined concrete sections, including plasticity models and plastic-damage models. Some constitutive models include damaged elasticity. However, a concrete plasticity model is common to all of them. This section effort to introduce different methods for numerical modeling of the plastic behavior of the concrete core. Various yield functions have been recommended for concrete material. Models based on damage or plasticity or both can be used to predict concrete’s nonlinear behaviour under compression. Plasticity models and plastic-damage models are the most popular approaches for determining the nonlinear behavior of concrete.

4.2.1 Concrete plasticity model: DruckerePrager type plasticity model Plasticity is commonly described as irreversible deformation after all loads have been removed. Models of concrete plasticity are generally based on the same framework as metal plasticity theory, although adjustments are required to reflect concrete’s unique features. The critical aspects of a plasticity model are as follows: 1. The yield surface: 1.1 The initial yield surface 1.2. The subsequent yield surface 2. The flow rule 3. The hardening/softening rule

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4.2 Numerical modeling of confined concrete

The outset of plastic deformation is specified by the initial yield surface, the flow rule specifies the plastic deformation direction, and the development of the yield surface with plastic deformation is determined by the hardening/softening rule. Various yield functions are available for concrete. The number of parameters in these functions varies from one (for example, the Von Mises criteria initially designed for metals) to five. The DruckerePrager (DeP) criteria have been extensively used to model confined concrete, making them one of the most practical methods between the available yield functions. The reason is that the DeP yield criterion is quite simple and requires only two parameters. Additionally, the increase in the shear strength because of the increase in hydrostatic pressure as a unique behavior of the confined concrete can be well captured by this model. Fig. 4.1 compares the DeP yield criterion with the Von Mises criterion as a pressure-insensitive criterion. It can be observed from the figure that the difference between these models is evident in the pffiffiffiffi I1
J2 plane of the stress space. The DeP yield criterion is an inclined line, whereas the Von Mises yield criterion is parallel to the abscissa. 4.2.1.1 Yield criterion The general form of the DeP yield criterion is as follows: pffiffiffiffi F ¼ J2
qI1
k

(4.1)

in which parameters q and k are the frictional parameter and the hardening/softening function and should be determined, and I1 and J2 are the first stress invariant and the second deviatoric stress invariant, respectively. The parameter k is referred to as the hardening/softening function when it is not a constant and is connected to plastic deformation (Eq. 4.2).

Von Mises

pffiffiffiffi

FIGURE 4.1 DruckerePrager yield surface in I1
J2 plane.

346

4. Numerical methods

In this case, Eq. (4.1) represents the initial yield surface as well as the following yield surface.  Z   (4.2) de εp k¼k e εp ¼ k d eεp ¼ Cp

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p p dε j dε j

(4.3)

in which parameter d eεp represents the equivalent plastic strain increment. When the concrete is subjected to triaxial compression, the applied stresses include principal lateral stresses that are equal and the principal stress in the axial direction. In this case, Cp can be expressed as 1 pffiffiffi
q 3 Cp ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3q2 þ 0:5

(4.4)

As a particular case of a DeP type model, taking q ¼ 0 in Eq. (4.4) leads rffiffiffi 2 , which is actually the same as the classical metal plasticity to Cp ¼ 3 model. From Eq. (4.1), it can be found that if the magnitude of q is constant for a considered material, the initial and the following yield surfaces will have an identical form in the stress space. Fig. 4.2 shows the initial and pffiffiffiffi subsequence yield surfaces in the I1
J2 plane. Besides, the surfaces in the deviatoric plane are presented in Fig. 4.3. It can be observed from the figure that the subsequent yield surfaces are formed by a series of straight pffiffiffiffi lines in the I1
J2 plane, and they are all parallel to the initial yield surface. The difference between the yield surfaces is their intersection pffiffiffiffi point with the J2 axis. The intersection point depends on the magnitude of k. The failure surface has the maximum magnitude of k and lies

Subsequent yield surface

Initial yield surface

pffiffiffiffi

FIGURE 4.2 Yield surfaces in I1
J2 plane.

347

4.2 Numerical modeling of confined concrete

Failure surface including the effect of the third deviatoric stress invariant

10

D-P yield criterion

8

6 4 2 -5

0 -2 -4

Triaxial compression 5

10

Non-uniform confinement

-6 -8 -10

Equal biaxial compression

FIGURE 4.3 Failure surfaces in the deviatoric plane.

between the subsequent yield surfaces. From Eq. (4.1), it can be clarified that the DeP yield surface in the deviatoric plane has a circular form, as displayed in Fig. 4.3. The frictional parameter q can be calculated using empirical formulae for concrete under triaxial compression. As discussed in Chapter 3, the compressive strength of confined concrete under triaxial stress state is given by the following: fcc0 ¼ fc0 þ 4:1fr Therefore, I1 ¼ fc0 þ 6:1fr and pffiffiffiffi fc0 þ 3:1fr J2 ¼ pffiffiffi 3 The magnitude of F when the concrete stress is at the peak is equal to 0. Therefore, from Eq. (4.1), F¼ and

pffiffiffiffi   f 0 þ 3:1fr J2
qI1
k ¼ c pffiffiffi
q fc0 þ 6:1fr
k ¼ 0 3     1 3:1 0 pffiffiffi
q fc
6:1q
pffiffiffi fr
k ¼ 0 3 3

348

4. Numerical methods

The value of q is constant and does not depend on the stress state; therefore, 3:1 6:1q
pffiffiffi ¼ 0 3 and hence q ¼ 0:2934 A similar value has been found for q by Karabinis et al [1]. Using the same procedure and taking K1 as 3.5, Teng et al. [2] have found the value of q as 0.2624. The shear strength of concrete subjected to equal biaxial compression (concrete under the same stresses in two principal directions and a zero stress in the third principal direction) is different from the concrete under triaxial compression. This is true even if the first stress invariant of the two stress states is the same. As shown in Fig. 4.3 and according to the theory of plasticity, the stress states of concrete under equal biaxial compression and triaxial compression correspond to various circumferential locations on the deviatoric plane. The shear strength ratio between equal biaxial compression state and triaxial compression state can be deprived using experimental findings or empirical formulae for concrete strengths under equal biaxial compression and triaxial compression. The peak stress fcc0 pffiffiffiffi and the corresponding stress invariants (I1 and J2 ) for concrete subjected to triaxial compression can be obtained using the equations discussed above. For concrete subjected to equal biaxial compression, I1 ¼ 2:32f 0 c pffiffiffiffi 1:16f 0 and J2 ¼ pffiffi c . Applying the condition that the amounts of I1 for the 3

concrete under triaxial compression (I1 ¼ fc0 þ 6:1fr ) and the concrete under biaxial compression cases are the same (I1 ¼ 2:32f 0 c ) leads to fr ¼ 0:22fc0 . Accordingly, the shear strength ratio of concrete subjected to equal biaxial compression to that subjected to triaxial compression is 1:16f 0 c fc0 þ3:10:22fc0

¼ 0:69. A similar value has been found for the shear strength

ratio of concrete by Karabinis et al [1]. Using the same procedure and taking K1 as 3.5, Teng et al. have found the shear strength ratio of 0.725. It can be seen that the shear strength ratio is remarkably smaller than one as designated by the circular failure curve. Consequently, a failure surface that aims to indicate the experimental behavior of concrete should consider the effect of the third deviatoric stress invariant and utilize a noncircular failure curve in the deviatoric plane. Fig. 4.3 depicts one conceivable form for such a failure surface.

4.2 Numerical modeling of confined concrete

349

4.2.1.2 Strain hardening and softening The hardening/softening law specifies the form and position of the loading surface and its development after initial yielding. The hardening law defines the prepeak behavior as the elastic area finishes, and the softening law defines the postpeak behavior during plastic flow. The hardening/softening function (Eq. 4.2) in classical metal plasticity models is solely a function of the equivalent plastic strain. The strain hardening/ softening rule can be calculated using a single uniaxial stressestrain curve of concrete, with or without confinement, if this idea is combined with the DeP yield criteria. The following approach can be used to determine the hardening/softening function k based on a uniaxial stressestrain curve: 1. Calculate the axial stresseplastic strain curve utilizing the stress sl estrain εl relationship and p

εl ¼ εl


sl Ec

(4.5)

2. Specify the relationship between the plastic strain and J2 and I1 according to the axial stresseplastic strain curve achieved in the previous step. 3. Specify the relationship between the hardening/softening function k p and the plastic strain ε l according to Eq. (4.1) and the results of the second step. Ignoring the confinement effect in determining the hardening/softening function k can lead to the results in which the stressestrain curve of confined concrete is underestimated. 4.2.1.3 Flow rule The flow potential and the yield function are the same in the case of adopting the associated flow rule. Therefore, the plastic strain increments are governed by Ref. [3]: p

dε ij ¼ l

vG vF ¼l vsij vsij

(4.6)

in which the term F represents the yield function and l denotes the scalar hardening parameter. It should be noted that the magnitude of l can change during the straining process.

350

4. Numerical methods

When the concrete is under triaxial compression, Eq. (4.6) can be rewritten as   1 2s1
s2
s3 p
q (4.7a) dε 1 ¼ l pffiffiffiffi 3 2 J2   1 2s2
s3
s1 p
q (4.7b) dε 2 ¼ l pffiffiffiffi 3 2 J2   1 2s3
s1
s2 p
q (4.7c) dε 3 ¼ l pffiffiffiffi 3 2 J2 From Eq. (4.7), q can be expressed as follows: pffiffiffi p p p 1 dI1 1 3 dε 1 þ 2dε 2 qffiffiffiffiffiffi ¼ a q¼
p p ¼
0p 6 6 6 dε 1
dε 2 d J2

(4.8)

p

in which the term a denotes the dilation rate and I1 is as follows: p

p

p

p

I 1 ¼ ε1 þ ε2 þ ε 3 and 0p J2 ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i  p  p 1 h p p 2 p 2 p 2 ε1
ε2 þ ε2
ε3 þ ε3
ε1 6

(4.9)

(4.10)

It can be deduced from Eq. (4.8) that how the flow rule defines the plastic deformation direction. The flow rule in the concrete subjected to triaxial compression defines the lateral plastic strain to the axial plastic strain ratio. A flow potential G should be used for computing the plastic strain increments when the nonassociated flow rule is employed. The flow potential G is governed by the following: p

dε ij ¼ l

vG ; vsij

G¼

pffiffiffiffi J2
bI1

(4.11)

in which the term b denotes the potential function parameter and is given by the following: pffiffiffi p p p 3 dε 1 þ 2dε 2 1 dI 1 1 qffiffiffiffiffiffi ¼ a (4.12) b¼
p p ¼
0p 6 dε 1
dε 2 6 6 d J2 b > 0 represents volume dilation, whereas b < 0 represents volume compaction.

351

4.2 Numerical modeling of confined concrete

4.2.1.4 Assessment of the presented DruckerePrager type plasticity model Regarding the yield criterion given by Eq. (4.1), this model cannot accurately determine the peak stress of concrete subjected to equal biaxial compression or nonuniform confinement. According to Fig. 4.3, in the deviatoric plane, the stress state of concrete under nonuniform confinement corresponds to a circumferential position between triaxial and equal pffiffiffiffi biaxial compression. Based on Refs. [4,5], the term J2 in Eq. (4.1) can be pffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi replaced with J2 4 1
C3 Sr in which Sr ¼ J3 3 represents a function of the ðJ2 Þ2

second J2 and the third J3 deviatoric stress invariants, and parameter C3 is a constant. The yield surface in the deviatoric plane takes on a noncircular form as a result of this replacement. According to Refs. [4,5], C3 can be pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi taken as 2.3. Therefore, for triaxial compression, the term 4 1
C3 Sr equals 1.17, but for equal biaxial compression, it equals 0.58. Concrete’s compressive behavior cannot be accurately modeled using an elastic perfectly plastic model. Thus, a plasticity model that incorporates strain hardening and softening is required to fully explain the experiment’s findings. In contrast to the conventional hardening/ softening rule, which solely considers plastic deformation, new research shows that this approach cannot accurately anticipate the increased ductility of confined concrete without taking the confining pressure into account. Fig. 4.4 compares the axial stressestrain curve of unconfined concrete with the compressive strength of 40 MPa with the counterpart concrete under constant active confining pressures of 5 and 10 MPa, respectively, obtained by the FE analysis using the presented DeP model. The strain hardening/softening function was achieved using the process described in Section 4.2.1.2. It can be found from the figure that the 90

Axial stress (MPa)

80

70 60 50 40

Un-confined concrete

30 20 10 0

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

Axial strain

FIGURE 4.4 Predicted axial stressestrain curves of unconfined and confined concrete. Yu T, Teng JG, Wong YL, Dong SL. Finite element modeling of confined concrete-I: Drucker–Prager type plasticity model. Eng Struct 2010;32(3):665e79.

352

4. Numerical methods

proposed model is able to consider the increase in the compressive strength of the concrete due to the confinement effect. However, the postpeak regions of the curves are similar that contradicts the real behavior of the confined concrete. According to experiments, the postpeak area of the stressestrain curve of actively confined concrete has a lower slope than unconfined concrete. This is because the confining pressure must be considered in determining the strain hardening/ softening function k. Hence, it can be deduced that the strain hardening/ softening function k should be linked to the confining pressure so that confined concrete can be predicted precisely. The associated flow rule described above may lead to the overestimated prediction of the confined concrete expansion [6]. The axiale lateral strain curve of concrete with the compressive strength of 41.9 MPa under confining pressure of 12 MPa from the experimental study of Candappa et al. [7] was compared with the corresponding FE results obtained by using DeP model with the associated flow rule and the results are shown in Fig. 4.5 [8]. It can be observed from the figure that the numerical model leads to the overestimated predicted results. Flow rule that specifies a constant potential function parameter b equal to a large positive frictional parameter q causes this overestimation. Fig. 4.6 shows the dilation properties of unconfined concrete, actively confined concrete with different confining pressures, and passively confined concrete [2] in which the compressive strain and tensile strain are taken as positive and negative, respectively. The passively confined concrete curve is obtained using two-ply FRP jackets [9], and the actively confined concrete curves are collected from the works of Candappa et al. [7] with fc0 ¼ 32:8 MPa and Sfer et al. [10] with fc0 ¼ 41:9 MPa. The axial εc and lateral εl strains are normalized by the axial strain of unconfined 0 -0.002 Lateral strain

-0.004 -0.006 -0.008 -0.01 -0.012 FE according to an associated flow rule Test results

-0.014 -0.016 0

0.005

0.01 0.015 Axial strain

0.02

0.025

FIGURE 4.5 Comparison of experimental and numerical axialelateral strain curves of actively confined concrete. Yu T, Teng JG, Wong YL, Dong SL. Finite element modeling of confined concrete-I: Drucker–Prager type plasticity model. Eng Struct 2010;32(3):665e79.

353

4.2 Numerical modeling of confined concrete

Normalized lateral strain εl /εco

0 -1 -2 -3

Unconfined concrete Actively confined σl=1.5MPa

-4

Actively confined σl=4MPa Actively confined σl=12MPa

-5

Passively confined -6 0

1

2

3

4

5

6

7

8

9

Normalized axial strain εc /εco

FIGURE 4.6 Lateral straineaxial strain curves of unconfined and confined concrete. Yu T, Teng JG, Wong YL, Dong SL. Finite element modeling of confined concrete-I: Drucker–Prager type plasticity model. Eng Struct 2010;32(3):665e79.

concrete corresponding to the peak stress εco . It can be observed from the figure that the lateral straineaxial strain curve of passively confined concrete is different from that of actively confined concrete. However, test results proved that the value of axial strain at a specific lateral strain is affected by the current confining pressure sr to that of the compressive strength of unconfined concrete fc0 [2]. This shows that at the intersection points depicted in Fig. 4.6, the magnitude of the confinement ratio sf 0r for c

both actively and passively confined concrete is similar. Another point that can be deduced from Fig. 4.6 is the difference between the tangent Poisson’s ratio (i.e., the increase in lateral strain divided by the increase in axial strain) of actively and passively confined concrete at the interception point despite similar axial strain and the confining pressure. The same observations can be expected for the axiallateral plastic strain curves of actively and passively confined concrete [2]. As a result, a plasticity model that simply connects the flow rule to the confining pressure and plastic strain may accurately predict actively confined concrete behavior. However, the model is likely to be inaccurate for predicting the behavior of FRP-confined concrete. FRP-confined concrete lateral dilatation is typically overestimated by plasticity models, which in turn overestimates the axial stressestrain behavior [8]. As discussed in Chapter 2, the confining pressure in the steel-confined concrete is passive, which is similar to the FRP-confined concrete. Therefore, it can be deduced from the above discussion that the discussed plasticity model can capture the behavior of actively confined concrete with acceptable accuracy. However, it may lead to unreliable predictions for passively confined concrete. The reason is that when concrete is actively confined, the confining pressure remains constant regardless of the deformation of concrete. In contrast, the confinement level changes

354

4. Numerical methods

with the concrete deformation in passively confined concrete. Therefore, for the accurate prediction of the behavior of passively confined concrete, two issues must be considered in the flow rule and the potential function parameter: 1. the confining pressure 2. the rate of confinement increment due to the lateral strain It is noteworthy that the rate of confinement increment in FRP-confined concrete and steel tubeeconfined concrete depends on the stiffness of the FRP jacket and the steel tube, respectively. For instance, the potential function parameter of the concrete confined by the stronger jacket is smaller than its counterpart confined by the weaker jacket. 4.2.1.5 DruckerePrager type plasticity model modifications It is clear from the preceding discussion that a DeP type plasticity model has to be adjusted to have the following three properties if it is to offer substantially accurate predictions of the behavior of both actively and passively confined concrete: 1. The influence of the third deviatoric stress invariant must be reflected by a yield criterion. 2. A strain hardening/softening function that depends on the confinement effect. 3. A nonassociated flow rule that depends on the confinement effect and the confining pressure, in which the confining pressure and confinement increment due to the concrete dilation must be considered in the modified potential function parameter. By performing the first adjustment, the resistance of concrete under active nonuniform confinement can be precisely predicted. Implementing the second modification can increase the accuracy of the predicted axial stressestrain curve of actively confined concrete. Finally, the lateral deformation of both actively and passively confined concrete can be predicted accurately if the third modification is performed. The modified DeP plasticity model can be defined as follows [2]: 1. Yield criterion containing the third deviatoric stress invariant [3,4] F ¼ FðI1 ; J2 ; J3 Þ

(4.13)

2. The strain hardening/softening function that depends on the confining pressure [11]   (4.14) k ¼ k sr ; eεp

355

4.2 Numerical modeling of confined concrete

Based on Ref. [11], the strain hardening/softening function can be expressed in terms of the strain trajectory bε : Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T bε ¼ (4.15) dε p dε p where ε p denotes the plastic strain, and the subscript T represents the transpose of a matrix. The strain hardening/softening function k can be expressed as follows: 0 1 B B k ¼ qB @

ε Ep þ

ε 0 fu
fy0

C C
Rðs3 ÞKε þ fy0 C A

(4.16)

where q is governed by the following: 1 q ¼ pffiffiffi
q 3

(4.17)

ε is given by the following: ε¼

bε 0:5ða
1Þ

(4.18)

in which a is the dilatation of concrete under unconfined compression: p qffiffiffiffiffiffiffi dI1 p p a ¼ slope of the I1
I 2D diagram ¼ qffiffiffiffiffiffiffi (4.19) p d I 2D p

p

where I1 is the first invariant of the plastic strain, I2D is the second invariant of the plastic deviatoric strain, Ep is the plastic deformation modulus, fy0 is the uniaxial yield stress of concrete, fu0 is the ultimate strength of concrete and can be expressed as follows:   Ep fu0 ¼ fy0 þ fc0
fy0 pffiffiffiffiffi pffiffiffiffi 2 (4.20) Ep
K where K denotes the ultimate slope of the unconfined compression stressestrain curve in the postpeak stage and Rðs3 Þ is the ductility function that controls the magnitude of softening as a function of confinement s3 . Rðs3 Þ can be determined using the following expression: Rðs3 Þ ¼ ecs3

(4.21)

356

4. Numerical methods

where the constant c can be expressed as follows: h c ¼ s3 ln K

(4.22)

in which h is the ultimate softening slope. 3. Nonassociated flow rule that depends on confining pressure Eq. (4.11) defines a nonassociated flow rule with a flow potential different from the yield function. The modified potential function parameter b should be the function of plastic deformation eεp , the l confining pressure sl , and increment rate of Ds Dεl in which εl is the lateral strain. Therefore,   Ds (4.23) b ¼ b sl ; l ; eεp Dεl In an actively confined concrete, the magnitude of confining pressure is l constant. Therefore, Ds Dεl ¼ 0 and   b ¼ b sl ; eεp (4.24) When the concrete is confined by a linear elastic confining device such sl l as the FRP layer, Ds Dεl can be taken as εl . Additionally, sl can be defined based on eεp and εl . Therefore, sl would no longer be an independent parameter, and Eq. (4.23) can be rewritten as follows:   s (4.25) b ¼ b l ; eεp εl 4.2.1.6 Implementing the modified DruckerePrager type plasticity model in ABAQUS By modifying the Extended DeP Model and utilizing the facility of user-defined solution-dependent field variables (SDFV), the model presented above can be implemented in the general-purpose FE software ABAQUS. ABAQUS uses an Extended DeP Model which is a DeP type plasticity model with a modified yield criterion. This modification on the yield criterion is done using the extra parameter K to consider the influence of the third deviatoric stress invariant. The parameter K denotes the shear strength of equal biaxial compression to triaxial compression ratio. The yield criterion in the Extended DeP model has the following form: F ¼ s
qI1
k

(4.26)

4.2 Numerical modeling of confined concrete

in which

357

2

0 13 3 pffiffiffi qffiffiffiffi   3  3 J3 C 7 1 pffiffiffiffi6 1 1 B 6 B pffiffiffiffi 2 C 7 J2 61 þ þ 1
s¼ A 7 4 5 2 K K @ J2

(4.27)

where the term J3 represents the third deviatoric stress invariant. pffiffiffiffi Replacing J2 in Eq. (4.1) with s in Eq. (4.26) changes the circular shape of the yield surface in the deviatoric plane. When the concrete is subjected to pffiffi pffiffiffi 3 3 J3 pffiffiffiffi the triaxial compression, pffiffiffi 2 is equal to 1 and s ¼ J2 . When the J2 pffiffi pffiffiffi 3 3 J3 concrete is subjected to the equal biaxial compression, pffiffiffi 2 is equal J2 pffiffiffiffi 1 to
1 and s ¼ K J2 . Consequently, Eq. (4.26) shows the same failure curve in the plane as Eq. (4.1) for the case of triaxial compression if appropriate parameters are specified. The failure surface of the concrete subjected to equal biaxial compression can be expressed as follows: pffiffiffiffi (4.28) F ¼ J2
KqI1
Kk Fig. 4.7 presents the modified yield surface in the deviatoric plane and pffiffiffiffi the I1
J2 plane. In fact, K represents a material parameter to consider the stress path with the deviation of shear strength subjected to the hydrostatic pressure. To assure the convexity of the yield surface, the K value in the ABAQUS Extended DeP Model is restricted between 0.778 and 1.0. The lower limit of K is greater than the strength ratio of 0.7 obtained from the experimental results (see Section 4.2.1.1). Therefore, this model can lead to overestimated strength prediction of concrete under equal biaxial compression. However, if the amount of the frictional parameter q for the triaxial compression state is calibrated from empirical formulas, the predicted results for the triaxial loaded concrete are not affected by the magnitude of K. Using the Extended DeP Model, the user may also provide the strain hardening/softening function as well as a potential function parameter that can be the same as or different from the frictional parameter, depending on whether the associated or the nonassociated flow rule should be utilized. The modified DeP type plasticity model can be implemented in ABAQUS through the following procedure: 1. Yield criterion The extended DeP yield criterion is directly adopted by taking the frictional parameter q as 0.2624 as discussed in Section 4.2.1.1 and K as

358

4. Numerical methods

(a)

Triaxial compression

Biaxial compression

(b)

FIGURE 4.7 Failure surface of the Extended DeP Model in ABAQUS. (A) Failure surface pffiffiffiffi in the deviatoric plane. (B) Failure surface in I1


J2 plane.

0.78. The reason for using 0.78 instead of 0.725 (see Section 4.2.1.1) for K is due to the limitations of ABAQUS0 Extended DeP Model as discussed above. 2. Strain hardening/softening rule The modified hardening/softening rule that depends on the confining pressure can be implemented in ABAQUS using the following steps: First, it is required to determine axial stressestrain responses of actively confined concrete by considering different active confining

359

4.2 Numerical modeling of confined concrete

pressures and using the constitutive models described in Chapter 3. Next, the corresponding axial stresseplastic strain responses of concrete subjected to various confining pressure should be obtained according to the initial concrete material properties, i.e., the initial elastic modulus and Poisson’s ratio, and the following equation [12]: p

ε c ¼ εc


sc s þ 2 l wc Ec Ec

(4.29)

in which Ec and wc are elastic modulus and Poisson’s ratio of the concrete, respectively. Later, the obtained axial stresseplastic strain responses should be defined as the input in ABAQUS. It should be noted that the SDFV option available in ABAQUS can be used to define the relationship between the stressestrain response and the confining pressure. An SDFV is a field variable that alters during the solution process through which additional characteristics can be defined for the material. 3. Flow rule The modified flow rule that depends on the confining pressure can be implemented in ABAQUS using the following steps: First, it is required to determine axial stressestrain responses of confined concrete by considering different confining pressures and using the constitutive models described in Chapter 3. Next, a series of axial strainelateral strain responses of CFST columns for different confining pressures must be obtained using the methods discussed in Chapter 3. Later, the potential function parameters according to the axial straine lateral strain relationships achieved from the previous stages, Eqs. (4.12) and (4.29), and the following equation must be calculated [12]: p

ε l ¼ εl


sl sc ð1
wc Þ þ wc Ec Ec

(4.30)

4.2.2 Concrete plasticity model: concrete damage plasticity model In theory, the damage is typically defined as a decrease in elastic constants. Fig. 4.8 shows a typical stressestrain curve of the concrete subjected to compression in which the reduction of the unloading stiffness and unrecoverable deformation can be observed. Hence, it can be concluded that the concept of plasticity should be combined with the concept of damage to represent concrete’s nonlinear behavior accurately. The concrete damage plasticity (CDP) model, which was first developed by Lubliner et al. [13], is a widely used model for describing the

360

4. Numerical methods

Axial stress

Loading curve

Unloading curve with a declined elastic modulus Plastic strain

Axial strain

FIGURE 4.8 Typical axial stressestrain curve of concrete subjected to compression.

nonlinear behavior of concrete using the concepts of plasticity and damage. For concrete compressive behavior, the essential features of this model are as follows: 1. 2. 3. 4.

The The The The

damage variable yield criterion hardening/softening rule flow rule

The yield function and the potential function are employed to define the plasticity. The yield surface depends on the confining pressure. The magnitude and direction of plastic strains are defined by the flow rule. In the CDP model, a nonassociative flow rule for the yield function and potential function is assumed. 4.2.2.1 Damage According to the traditional elasto-plasticity theory, the total strain (ε) can be divided into two components, including the elastic strain (εel ) and plastic strain (εpl ) to reflect the concrete nonlinearity and unrecoverable deformation. ε ¼ εel þ εpl

(4.31)

Numerous test findings, however, have demonstrated that concrete nonlinearity can be attributable to damage or plasticity separately or to a mixture of both, whereas unloading stiffness deterioration is mostly related to damage progression. As a result, it is preferable to properly separate the effect of damage from that of plasticity throughout the numerical simulation.

361

4.2 Numerical modeling of confined concrete

The CDP model assumes that cracking of the concrete due to tension and crushing of the concrete due to compression are the main two failure modes of the concrete material. The schematic stressestrain responses of the concrete under uniaxial tension and compression are shown in Fig. 4.9. It is supposed that the concrete under tension demonstrates a linear response up to the peak tensile stress sto , and the microcracks develop in the concrete by reaching the peak point. The postpeak region of the curve has a nonlinear softening behavior. When exposed to uniaxial compression, the concrete behaves linearly until the onset of initial yield 7

ε 0

2

(a) 10 9

8 7

ε

(b)

FIGURE 4.9 Axial stressestrain responses of concrete subjected to uniaxial loading. (A) Tensile stressestrain curve. (B) Compressive stressestrain curve.

362

4. Numerical methods

stress sco . The concrete generally exhibits stress hardening from that point until the peak stress scu . Beyond the peak point, the concrete demonstrates a strain-softening response. It can be observed from Fig. 4.9 that the elastic stiffness of the concrete during the strain-softening branch is reduced and damaged when the material is unloaded. CDP model provides a generic capacity for modeling progressive material damage, in which a scalar damage variable 0  d  1:0 and uniaxial compressive damage variable dc and tensile damage variable dt are introduced. The scalar damaged elasticity equation is expressed as follows:   p (4.32) sij ¼ ð1
dÞDeijkl εij
ε ij p

in which sij is the tensor of stress, εij and εij are tensors of strain and plastic strain, respectively, Deijkl is the undamaged elasticity matrix, and d is the damage variable. Eq. (4.21) for the concrete subjected to uniaxial monotonic compression and tension can be simplified to the following:   pl (4.33) sc ¼ ð1
dc ÞEc εc
eε c   pl st ¼ ð1
dt ÞEc εt
eε t (4.34) Damage variables vary in the range of 0 and 1. d ¼ 0 means that the concrete is undamaged, whereas d ¼ 1 means that the concrete is completely lost its strength. The effective compressive and tensile stresses, sc and st , describe the size of the yield surface and are expressed as follows:   sc pl ¼ Ec εc
eε c sc ¼ (4.35) ð1
dc Þ   st pl ¼ Ec εt
eε t st ¼ (4.36) ð1
dt Þ The first effective stress invariant I 1 and the second deviatoric stress invariant J 2 can be defined in terms of the tensor of the effective stress as follows: I 1 ¼ sij

(4.37)

1 J 2 ¼ Sij Sij 2

(4.38)

in which the term Sij represents the effective deviatoric stress tensor.

4.2 Numerical modeling of confined concrete

363

4.2.2.2 Yield criterion The yield surface describes the critical stress state beyond which the plastic deformation is expected to begin. Lubliner et al. [13] proposed a yield function which was then modified by Lee and Fenves [14] to consider the different development of strength under tension and compression. The yield condition adopted by the plastic damage concrete model is based on the yield function of Lubliner et al. [13] and Lee and Fenves [14]. In terms of effective stresses, the yield function takes the following form: F¼

1  1
A

qffiffiffiffiffiffiffi      pl 3J 2
AI 1 þ B eε pl h
smin i
Ch
smin i
scn eε c ¼ 0 (4.39)

where fb0 f 0
1  0:5 0  A ¼ 0 co 2 fb f 0
1 co   pl scn eε c B ¼  pl  ð1
AÞ
ð1 þ AÞ stn eε c C¼

3ð1
KÞ 2K
1

(4.40)

(4.41)

(4.42)

in which the term smin represents the minimum principal effective stress, fb0 is the strength of the concrete subjected to equal biaxial compression, scn is the effective compressive cohesion stress, stn is the effective tensile cohesion stress, and K is the strength ratio of concrete subjected to equal biaxial compression to triaxial compression. According to experimental test results [15], the value of K varies in the range of 0.5e1.0. Fig. 4.10A illustrates the yield surface in the deviatoric plane for K ¼ 1 and 23. The fb0 f 0 ratio can be expressed as follows [12]: co

 0 
0:075 fb0 f 0 ¼ 1:5 fco co

(4.43)

The value of K can be determined using the following equation [8,16]: K¼

5:5fb0  
0:075 0 5 þ 2 fco

(4.44)

364

4. Numerical methods

(a)

(b) FIGURE 4.10

Yield surfaces of the CDP model. (A) Yield surfaces in the deviatoric plane. qffiffiffiffi (B) Yield surfaces in the I 1
J2 plane for triaxial compression.

When the concrete is subjected to triaxial compression, Eq. (4.39) can be replaced by the DeP yield condition as follows:  qffiffiffiffiffiffiffi 1 ðC þ 3AÞ I 1 ¼ ð1
AÞscn (4.45) Cþ1 3J 2
3 3

4.2 Numerical modeling of confined concrete

365

When the concrete is subjected to triaxial compression with smin > 0, pffiffiffiffi the yield surface can be described in the I1
J2 plane by a linear curve, as shown in Fig. 4.8B and in the deviatoric plane by a noncircular curve, as displayed in Fig. 4.8A. 4.2.2.3 Hardening/softening rule In the CDP model, the strain-hardening/softening function for the concrete subjected to uniaxial monotonic compression can be defined by the following:   scn ¼ scn eε pl (4.46) As shown in Fig. 4.11, an isotropic hardening is assumed in ABAQUS, and the evolution law is driven by the equivalent plastic strain, as shown in Eq. (4.39). However, for ease of use, the law is described in a tabular form where the yield stress versus inelastic strain relation is provided. Taking the uniaxial tensile and compressive behaviors represented in Fig. 4.9, the relationship between the yield stress and inelastic strain can be defined as follows: sc Ec st εck t ¼ εt
Ec

εin c ¼ εc


(4.47) (4.48)

ck where εin c represents the compressive inelastic strain and ε t denotes the tensile inelastic strain. However, it should be noted that the CDP model

FIGURE 4.11

Isotropic hardening in deviatoric plane [17].

366

4. Numerical methods

receives unloading data in the form of damage curves, and the inelastic strain values will be automatically converted to plastic strain values by ABAQUS software, meaning that pl

ε c ¼ εin c
pl

ε t ¼ εck t


dc sc ð1
dc Þ Ec

(4.49)

dt st ð1
dt Þ Ec

(4.50)

If the computed plastic strain values are negative or reduced with rising inelastic strain, the ABAQUS software will provide an error notice, pl

pl

indicating incorrect damage curves. If no damage occurs, ε c and ε t are ck taken as εin c and εt , respectively. 4.2.2.4 Flow rule The CDP model assumes a nonassociated flow rule, and the adopted flow potential ðGÞ has a DeP type hyperbolic function: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vG I1 2 p (4.51) ; G ¼ ðɜsto tan jÞ2 þ 3J 2
tan j dε ij ¼ l 3 vsij p

in which dεij denotes the value of the plastic component of strain, l is the plastic multiplier coefficient, ɜ is the flow potential eccentricity, sto shows the uniaxial tensile stress at failure, j is the dilation angle which is qffiffiffiffiffiffiffi I1 measured in the 3
3J 2 plane, and I 1 and J 2 are the effective hydrostatic stress and the second deviatoric stress invariant, respectively. ɜ describes the rate at which the hyperbolic flow potential approaches its asymptote. Typically, the flow potential eccentricity ɜ is taken as 0.1. By taking the flow potential eccentricity ɜ as 0.1, the material’s dilation angle over a wide range of confining pressure stresses will be the same. It can be seen from Eq. (4.51) that the flow potential function G is a function of the dilation angle j. Different values were considered for j by  various researchers. Lai et al. [18] supposed a constant amount of j ¼ 15 for FE analysis of circular and square CFST beam columns. Han et al. [19]  utilized a constant value of j ¼ 30 for FE analysis of circular and square  CFST beam columns. A constant value of j ¼ 40 was considered for FE analysis of rectangular CFST columns by Tao et al. [16]. Besides the following expression for calculating the dilation angle j of circular CFST columns was recommended by Tao et al. [16]: 8 > < 56:3ð1
xÞ for x  0:5 (4.52) j¼ 7:4 > : 6:672e4:64þx for x > 0:5 in which x is the confinement coefficient.

4.2 Numerical modeling of confined concrete

367

When the concrete is subjected to nonuniform confinement, restraint of the concrete in the lateral direction may alter across the section; hence, assuming a constant value for j may reduce the accuracy of results. For the sake of simplicity, the dilation angle j can be considered as a constant value for the concrete subjected to uniform confining pressure. However, in reality, the dilation of concrete declines with increasing the confining pressure on the concrete. 4.2.2.5 CDP model modifications The CDP model can accurately predict the improvement in concrete’s peak strength for various degrees of confinement using an unconfined uniaxial stressestrain curve as the input base model for the concrete [20]. While confined concrete has a higher peak strength, the CDP model, however, underestimates the increase in the strain at that point compared to experimental data [21]. The degree of the increase in axial strain at peak strength due to confinement may be approximately 5 times that of the equivalent rise in concrete compressive strength [22]. Furthermore, the CDP model may not precisely represent the shift in the postpeak slope of the confined concrete’s stressestrain curve. With increased confining pressure, the postpeak strain softening becomes slower. Increasing compressive strength and concrete confinement increase the inaccuracies in postpeak strain prediction as well as the postpeak slope of the stressestrain curve. The peak strength of concrete may be overestimated if the confined uniaxial stressestrain curve is used as the input base model to account for the CDP model’s inability to capture the increase in strain at peak strength and the change in postpeak slope caused by confinement. Accordingly, employing the confined stressestrain curve as the input base model for modeling concrete using the CDP model may be wrong [14,23]. The modifications of the DeP plasticity model discussed in the last section have been developed based on the FRP-confined concrete [8]. As a confining material, FRP acts linearly until rupture, while steel tube exhibits nonlinear force deformation behavior after yielding. This is the most significant difference between an FRP jacketed column and a CFT column. As a result, in order to correctly calculate the strain ductility of CFT columns, an FE modeling approach capable of efficiently reproducing the postpeak stressestrain behavior of concrete of CFT columns for different confining pressure is required. Instead of predefining the confined concrete stressestrain curve, the FE model could identify the appropriate confined concrete base curve throughout the simulation based on the confinement level computed from the interaction of steel tube and in-filled concrete. This would make the model more versatile and robust. This approach would be more advantageous for columns with nonuniform confinement since it is challenging

368

4. Numerical methods

to predefine a confined concrete stressestrain curve that correctly captures the section behavior. With the help of a solution-dependent user-defined field variable (SDFV), a user-defined subroutine (VUSDFLD) is generated in FORTRAN and is integrated with the CDP model in ABAQUS software. As mentioned before, SDFV is a variable that makes the properties of the material to be dependent on the solution and can differ during the solution steps. Further properties of the material are included in the default CDP model available in ABAQUS using SDFV [24]. Modifications are implemented on the default CDP model to make it dependent on the following: 1. The dilation angle (SDFV1) 2. Confining pressure (SDFV2) The considered modifications are performed through two steps: 1. In the first step, compression hardening and damage parameters are determined for various magnitudes of dilation angle and confining pressures. 2. In the second step, the effective confining pressure and dilation angle at each analysis stage are estimated. For each analysis step, the stresses and strains of concrete elements are first imported to VUSDFLD. The stresses and strains are converted to principal stresses and strains. Then, the dilation angle and effective confining pressure are computed inside the subroutine. The dilation angle and confinement pressure values are then returned to ABAQUS in the form of SDFV1 and SDFV2 via the VUSDFLD. The correct predefined concrete material base curve is then used by ABAQUS in accordance with SDFV1 and SDFV2. The flow charts displayed in Figs. 4.12 and 4.13 present the algorithms for the two steps discussed above. The uniaxial stressestrain curve for the actively confined concrete is generated using the confinement model of Teng [2]. This model is based on the model of Mander [22], with the following adjustments to calculate the peak stress and the corresponding strain of confined concrete:   scc sl (4.53) ¼ 1 þ 3:5 scu scu   εcc sl (4.54) ¼ 1 þ 17:5 0 εco scu where scc and εcc denote the peak stress of confined concrete and the corresponding strain, respectively, and scu and ε 0co are the peak stress of unconfined concrete and the corresponding strain, respectively. ε 0co can be

369

4.2 Numerical modeling of confined concrete

Start

Input

Input

,

&

,

& No Stop Yes

Calculate

and

,

No Calculate:

Yes Calculate

No

&

& Yes Calculate: &

FIGURE 4.12 Flow chart of developing the input compression data for FE analysis. Bhartiya R, Sahoo DR, Verma A. Modified damaged plasticity and variable confinement modelling of rectangular CFT columns. J Constr Steel Res 2021;176:106426.

calculated using the models introduced in Chapter 3 such as the model of Nicolo et al. [25] as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (4.55) ε0co ¼ 0:00076 þ ð0:626scu
4:33Þ  10
7 It is worth noting that Eq. (4.54) may overestimate the value of strain for actively confined concrete, especially for weakly confined concrete [26]. As a result, the following equation may be used to determine the strain at peak stress for confined concrete.  1:2 εcc sl ¼ 1 þ 17:5 (4.56) 0 εco scu Based on Ref. [24], Eq. (4.54) can be used when the confining pressure is low, i.e., sscul  0:1, and Eq. (4.56) can be adopted when the confining pressure is high, i.e., sscul > 0:1.

370

4. Numerical methods

Start ABAQUS

At the end of each increment call from ABAQUS stresses and strains for each concrete element into user subroutine VUSDFLD

SDFV1= SDFV2=

Input:

No

Yes

Dilation angle calculation

Convert stresses and strains into principal stresses and strains and

Compute:

Yes No Calculate:

Calculate: Calculate:

FIGURE 4.13

Flow chart of the user subroutine integrated with FE program. Bhartiya R, Sahoo DR, Verma A. Modified damaged plasticity and variable confinement modelling of rectangular CFT columns. J Constr Steel Res 2021;176:106426.

The stressestrain curve of the concrete can be predicted using the constitutive models introduced in Chapter 3, such as the model of Popovics [27], as follows:   scc l εc ε0cc (4.57) sc ¼  l l
1 þ εc ε0cc in which the term l refers to the concrete brittleness constant. Based on Ref. [28], l is governed by the following: l¼

E  c0  Ec
fcc ε0

(4.58)

cc

Fig. 4.14A depicts a set of uniaxial stressestrain curves for actively confined concrete. The adjusted stress values (sac ) over the prepeak region of the curve are kept the same as those of unconfined concrete. The strain values of confined concrete, on the other hand, are adjusted for the prepeak region of the curve. sac ¼ suc

for εuc  εcu

(4.59)

371

4.2 Numerical modeling of confined concrete

6

+1

+∆

5 4

=0

+1

(a)

+∆

=0

1

4

+1

(b)

FIGURE 4.14 Stressestrain curves of confined concrete for FE analysis using the modified CDP model. (A) Uniaxial stressestrain curves of confined concrete. (B) Adjusted uniaxial stressestrain curves. Bhartiya R, Sahoo DR, Verma A. Modified damaged plasticity and variable confinement modelling of rectangular CFT columns. J Constr Steel Res 2021;176:106426.

εac ¼ εuc þ

εcc
εcu u εc εcu

for εuc  εcu

(4.60)

The adjusted stress values of confined concrete are calculated using Eq. (4.61). The compressive strength of confined concrete during the postpeak region is scaled down to meet the peak stress of unconfined concrete while maintaining the same postpeak slope as the softening region of confined concrete. Because the CDP constitutive model incorporates the influence of confinement on strength, any improvement in strength due to confinement is considered automatically in the simulation. The corrected amount of confined concrete stress and strain in the descending stage is given by the following: sac ¼ sc
ðscc
scu Þ for εc > εcc

(4.61)

372

4. Numerical methods

εac ¼ εc

for εuc > εcu

(4.62)

Fig. 4.14B shows a set of scaled uniaxial stressestrain curves for actively confined concrete. Using the above details and the procedure given by Fig. 4.12, the required input for defining the compressive behavior of confined concrete can be generated. These data can be implemented as an input in the ABAQUS software before starting the analysis. The following confining pressures increments can be used to generate a series of axial stresse strain curves of confined concrete: 8 sl > > < 2:5%scu for s  0:1 cu (4.63) Dsl ¼ s > > : 10%scu for l > 0:1 scu The following equation is used in the CDP model to define damage (d) after peak stress, characterized by a decrease in stiffness. sc (4.64) d¼1
scc in which sc denotes the axial stress in concrete. In the suggested model [24], the damage is determined by substituting sc and scc in Eq. (4.64) with their modified values as follows: d¼1


sac scu

(4.65)

To apply the modified stressestrain curve in ABAQUS software, the effective confinement pressure must first be calculated. For the concrete stressestrain relationship, the confinement pressure may be used to calculate the dilation angle. The minimal principal stress in the transverse direction can be considered to be nearly equal to the concrete confinement pressure in order to capture the weakly confined concrete’s postpeak descending branch [29,30]. Based on Ref. [24], the following expression can be adopted to predict the effective lateral confining pressure sleff in the concrete core subjected to nonuniform confinement: sleff ¼

2ðs1c þ 0:039scu Þðs2c þ 0:039scu Þ
0:039scu ðs1c þ s2c þ 0:078scu Þ

(4.66)

in which s1c and s2c denote the principal stresses in the lateral direction. The rate of change of confinement pressure for concrete contained by a linear elastic confinement device, such as FRP jackets, is constant and can sl l be approximated based on the stiffness of the device as Ds Dεl ¼ εl . After that, the dilation angle is determined as a function of the stiffness of the elastic device [31]. Concrete confined by steel tubes and reinforcing ties has a different load deflection behavior than FRP confined concrete. When concrete is passively contained by the FRP jacket, the confining pressure

4.2 Numerical modeling of confined concrete

373

increases until the FRP ruptures. In the case of concrete confined by steel ties, steel tubes, or both, the confining device only acts elastically in the primary ascending stage of the load deflection curve until the steel achieves its yield strain. When the steel yields, the confining pressure becomes constant. Therefore, the in-filled concrete is subjected to a passive state of confinement until the steel yields, after which it behaves similarly to actively confined concrete [32,33]. As a result, the suggested model considers a constant confinement pressure after steel yielding for calculating the dilation angle of concrete. It is worth mentioning that any increase in stress because of the steel strain hardening is ignored only for measuring the dilation angle of confined concrete. Steel hardening is still used in the constitutive model of steel for calculating strength and stiffness. The partial derivative of Eq. (4.57) with respect to εl gives the following formula for a constant confining pressure sl (i.e., for constant scc and εcc ): (

    l
1  l ) εc l vεc εc εc 1 vεc l l
l
1þ εcc εcc εcc εcc vεl εcc vεl 1 vsc ¼ (4.67) ( )2   scc vεl εc l l
1þ εcc Eq. (4.67) can be rewritten as follows: (  l ) εc ðl
1Þ 1
ε cc vsc lscc vεc ¼ (  l )2 vεl vεl εcc εc l
1þ εcc

(4.68)

Based on Ref. [2], for confined concrete, the relationship between axial and lateral strain can be expressed as follows:  "

 0:7 

  # εc sl
εl
εl 1 þ 0:75 (4.69) ¼ 0:85 1 þ 8
exp
7 ε0co fco εco εco in which εl is taken as negative for tension and positive for compression. The partial derivation of Eq. (4.69) with respect to εl when the concrete is subjected to constant confining pressure sl is given by the following: 

 0:3 

  vεc sl
εl
εl
0:525 1 þ 0:75 ¼ 0:85 þ 6:8
7 exp
7 vεl fco εco εco (4.70)

374

4. Numerical methods

The elastic component of axial strain εEc of an element subjected to uniform and constant confining pressure can be obtained based on Hooke’s Law as follows: sc s (4.71) εEc ¼
2 l nc Ec Ec p

The plastic component of axial strain ε c of the element can be determined using additive decomposition as follows: p

εc ¼ εc
εEc

(4.72)

Substituting Eq. (4.71) into Eq. (4.72) leads to the following expression: sc s p ε c ¼ εc
þ 2 l nc (4.73) Ec Ec p

The partial derivation of ε c with respect to εl for the case of constant confining pressure is as follows: p

vε c vεc 1 vsc ¼
vεl vεl Ec vεl

(4.74)

The elastic component of lateral strain εEl of an element subjected to uniform and constant confining pressure is given by the following: s sc εEl ¼ l ð1
nc Þ
nc (4.75) Ec Ec p

and the plastic component of lateral strain ε l of the element is given by the following: p

εl ¼ εl
εEl

(4.76)

Substituting Eq. (4.75) into Eq. (4.76) leads to the following expression: s sc p (4.77) ε l ¼ εl
l ð1
nc Þ þ nc Ec Ec p

The partial derivation of εl with respect to εl for the case of constant confining pressure is as follows: p

vεl vεl nc vsc ¼ þ vεl vεl Ec vεl

(4.78)

Integrating Eq. (4.74) into Eq. (4.78) leads to the following: nc vsc Ec vεl p¼ vεc vεc
1 vsc vεl Ec vεl p

vεl

1þ

(4.79)

4.2 Numerical modeling of confined concrete

375

Based on Ref. [34], the dilation angle j can be calculated using the following expression: ! p vε l 1þ2 p vε c 3 ! tan j ¼
(4.80) p 2 vε l 1
p vε c where s1c and s2c are the principal stresses in the lateral directions. The initial dilation angle of the concrete jo during the initial loading stage can be expressed as follows [35]: jo ¼ 0:117fco þ 28:965 ðDegreeÞ

(4.81)

The in-filled concrete of rectangular CFST columns is in a condition of nonuniform confinement, resulting in various primary stresses in lateral directions. The (absolute) maximum magnitude of primary stresses in lateral directions is used to calculate the dilation angle of concrete in the CDP model since it is recorded under high confining pressure in the qffiffiffiffiffiffiffi I1
3J 2 plane. 3 sl ¼ maxðs1c ; s2c Þ

(4.82)

where s1c and s2c are the principal stresses in the lateral directions. Due to concrete dilatation, the high lateral confining pressure correlates to a lower lateral strain in the concrete. As a result, the (absolute) lowest value of principal lateral strains is used to estimate the dilation angle of nonuniformly confined concrete. εl ¼ minðε1c ; ε2c Þ

(4.83)

where ε1c and ε2c are the principal strains in the lateral directions.

4.2.3 Effects of different parameters on the numerical results Before beginning an FE analysis of steeleconcrete composite members, it is vital to define the behavior of materials, namely steel and concrete. This step is sensitive in performing FE analysis since it can highly affect the final outputs of the analysis. In general, the behavior of the material is defined by its stressestrain relationships. The constitutive models for determining the behavior of steel and concrete are comprehensively discussed in Chapter 3. Additionally, the final results of the analysis can be affected by the parameters used to define the concrete plasticity behavior. This section explores the impact of the factors mentioned above on composite member nonlinear FE analysis results.

376 TABLE 4.1

4. Numerical methods

Geometric and material properties of examined CFDST columns. EPP

BL Esh [ 1%Es

BL Esh [ 1%Es

IML

BPN

Name

Do 3 to (mm)

Di 3 ti (mm)

fsyo (MPa)

fsyi (MPa)

NFE;1 (kN)

NFE;2 (kN)

NFE;3 (kN)

NFE;4 (kN)

NFE;5 (kN)

S2

219  6

102  6

312.5

321.7

3010

3022

3042

3055

3056

C2S1

114:4  4:66

40:14  1:8

416

492

1272

1282

1288

1270

1273

Note: BL, Bilinear; BPN, Bilinear plus nonlinear hardening; EPP, Elastic perfectly plastic; IML, Idealized multilinear; L, Liang model; S, Sakino model.

4.2.3.1 Effects of stressestrain model of steel on the FE results The stressestrain response of steel tubes in composite components has been studied extensively, as mentioned in Chapter 3. A sensitivity analysis was performed to assess the effects of using different steel models on the projected axial stressestrain response of composite members under axial compression. The numerical predictions are compared to experimental results of a circular CFDST stub column tested by Elchalakani et al. [36] and a square CFDST stub column tested by Fang and Lin [37]. Table 4.1 displays the geometry and material properties of examined specimens, in which Nexp represents the experimental compressive capacity and NFE denotes the predicted compressive capacity resulting from the nonlinear FE analysis. Fig. 4.15 depicts a comparison of the experimental and predicted axial loadestrain curves of CFDST stub columns using different steel material models. As displayed in Table 4.1 and Fig. 4.15, steel material models have an insignificant influence on the predicted prepeak region and the ultimate axial strength of columns. By contrast, the effect of the steel material models becomes evident in the predicted postpeak region of the curve, particularly in the square specimen with remarkable strain-hardening characteristics, as shown in Fig. 4.15B. The curve’s postpeak stage deviates from the experimental data when predicted using the elastic perfectly plastic steel material model, as illustrated in Fig. 4.15B. For stresses in the range of general structural interest, the steel material model has negligible influence on the predicted numerical results of composite members. It should be noted that composite members fabricated with compact sections, high strength steel material, or low strength concrete core usually do not have a softening response or may show a softening response at very large deformation. In this case, the choice of steel material model may become of importance. 4.2.3.2 Effects of stressestrain model of concrete on the FE results Compared to steel, the stressestrain model of in-filled concrete has a significant impact on the outcomes of nonlinear FE analysis. This is because the concrete contributes substantially more to the load-bearing

377

4.2 Numerical modeling of confined concrete

S

L

NFE;6 (kN)

NFE;7 (kN)

3034 1299

Nexp (kN)

NFE;1 Nexp

NFE;2 Nexp

NFE;3 Nexp

NFE;4 Nexp

NFE;5 Nexp

2950

1.02

1.03

1.04

1.04

1.05

1295

0.98

0.99

0.99

0.98

0.98

NFE;6 Nexp 1.03

1.00

1400

Axial load (kN)

1200 1000 Experimental 800

EPP BL with Esh=1%Es

600

BL with Esh=2%Es 400

IML BPN

200

L 0 0

0.01

0.02

0.03

0.04

0.05

Axial strain

(a) 4500 4000

Axial load (kN)

3500 3000 Experimental

2500

EPP

2000

BL with Esh=1%Es

1500

BL with Esh=2%Es

1000

IML BPN

500

S 0 0

0.01

0.02

0.03

0.04

NFE;7 Nexp

0.05

Axial strain

(b)

FIGURE 4.15 Comparison of the experimental axial forceestrain curves of CFDST columns with numerical ones predicted using different steel material models. (A) Circular specimen C2S1. (B) Square specimen S2.

378

4. Numerical methods

capacity of composite components than the steel tube, as explained in Chapter 2. A sensitivity analysis was performed to examine the effects of concrete constitutive relationships on the predicted axial stressestrain response of composite members under axial compression. To this end, nonlinear FE analysis was performed on the circular CFST stub column CU-70 tested by Huang et al. [38] using the concrete models of Liang and Fragomeni [39] for circular CFSTs, Liang [40] for square CFSTs, Hassanein et al. [41] for hexagonal CFSTs, and Patel et al. [42] for octagonal CFSTs. Table 4.2 displays the geometry and material properties of the examined specimen, in which Nexp represents the experimental compressive capacity, and NFE denotes the predicted compressive capacity obtained from the nonlinear FE analysis. Fig. 4.16 depicts a comparison of the experimental axial loadestrain curve of the CFST stub column with the predicted ones using different concrete material models [43]. The figure shows that the initial axial stiffness and the ascending branch of the projected curves are in good agreement with the experimental data. As a consequence, it can be inferred that the concrete constitutive models have a negligible influence on the predicted results during the prepeak stage. In contrast, the model used has a significant impact on the anticipated ultimate axial strength and postpeak portion of the curve. The reason lies in the fact that the behavior of the concrete during the initial stage of loading is mostly linear, and no interaction occurs at this stage. Besides, the initial axial stiffness of concrete during the ascending stage is related to the linear behavior of concrete. Accordingly, the concrete model has no remarkable influence on the behavior of the composite column at this stage. By contrast, the peak point and the postpeak region of the curve are highly dependent on the confinement mechanism of the composite column. 4.2.3.3 Effects of the ratio of the second stress invariant on the tensile meridian to that on the compressive meridian on the FE results The ratio of the second stress invariant on the tensile meridian to that on the compressive meridian k should be specified to determine the yield surface of the concrete plasticity model. A sensitivity analysis was performed to assess the effects of the k value on the predicted axial stressestrain response of composite members under axial compression. The experimental results of circular CFST stub column 3HN and square CFST stub column 3MN reported by Tomii et al. [44] are compared with the predicted ones captured by the numerical analysis. Table 4.3 displays the geometry and material properties of the examined specimens, in which Nexp represents the experimental compressive capacity, and NFE denotes the predicted compressive capacity obtained from the nonlinear FE analysis.

Concrete model Circular

Square

Hexagonal

Name

D3t (mm)

fsy (MPa)

fc0 (MPa)

NFE;1 (kN)

NFE;2 (kN)

NFE;3 (kN)

CU-70

280  4

272.6

31.15

2983.5

2875.2

2892.3

Octagonal

NFE;4 (kN)

Nexp (kN)

NFE;1 Nexp

NFE;2 Nexp

NFE;3 Nexp

NFE;4 Nexp

2950.3

3025.2

0.99

0.95

0.96

0.98

4.2 Numerical modeling of confined concrete

TABLE 4.2 Geometric and material properties of the examined CFST column.

379

380

4. Numerical methods

3500 3000

N/kN

2500 2000 1500

Experiment (CU-70) Circular model Square model Hexagonal model Octagonal model

1000 500 0 0

0.005

0.01

0.015 Axial strain

0.02

0.025

0.03

FIGURE 4.16 Comparison of loadeaxial strain response of the circular specimen CU-70 with numerical results predicted using different constitutive models of the concrete core [43].

The comparison between the experimental axial loadestrain curves of CFST stub columns with the predicted ones employing different values of k is illustrated in Fig. 4.17. It can be noticed from the results that the magnitude of k does not influence the curve’s ascending stage when the columns have elastic behavior. The value of k, on the other hand, has a substantial impact on the ultimate axial strength and the postpeak stage of the curve. The influence of k is more evident on the behavior of circular columns than in square columns. Additionally, the predicted compressive strength and postpeak residual strength are increased by reducing k. 4.2.3.4 Effects of the dilation angle on the FE results For defining the plastic flow potential, it is required to determine the dilation angle j. Various constant values of j have been utilized in the past by different researchers. For confined concrete, most researchers used a value of 20 or 30 . The effect of the dilation angle j on the axial loadestrain responses of circular CFST stub column 3HN and square CFST stub column 3MN tested by Tomii et al. [44] is displayed in Fig. 4.18, according to sensitivity analysis. Table 4.4 displays the geometry and material properties of the examined specimens, in which Nexp represents the experimental compressive capacity, and NFE denotes the predicted compressive capacity obtained from the nonlinear FE analysis. As shown in Fig. 4.18, increasing the dilation angle j leads to an increase in the axial  stiffness of the member, and the column with j ¼ 40 shows a stiffer response than other columns. Another point is that the predicted ultimate axial strength increases with an increase in j. The reason is that the value of j affects the dilation rate of concrete. As a general trend, the dilation

Geometric and material properties of examined CFST columns. k k [ 0:6

k [ 2=3

k [ 0:725

k [ 0:8 NFE;1 Nexp

NFE;2 Nexp

NFE;3 Nexp

NFE;4 Nexp

Name

D 3 t (mm)

fsy (MPa)

fc0 (MPa)

NFE;1 (kN)

NFE;2 (kN)

NFE;3 (kN)

NFE;4 (kN)

Nexp (kN)

3HN

150  3:2

287.4

28.7

962

1022

1106

1272

959

0.99

1.07

1.15

1.33

3MN

150  3:2

300.1

27.8

1060

1062

1119

1166

1065

0.99

1.0

1.05

1.09

4.2 Numerical modeling of confined concrete

TABLE 4.3

381

382

TABLE 4.4

Geometric and material properties of examined CFST columns. 





j ¼ 0:01

j ¼ 20

j ¼ 30

j ¼ 40

Name

D3t (mm)

fsy (MPa)

fc0 (MPa)

NFE;1 (kN)

NFE;2 (kN)

NFE;3 (kN)

NFE;4 (kN)

Nexp (kN)

NFE;1 Nexp

NFE;2 Nexp

NFE;3 Nexp

NFE;4 Nexp

3HN

150  3:2

287.4

28.7

922

982

1106

1131

959

0.96

1.02

1.15

1.18

3MN

150  3:2

300.1

27.8

1101

1112

1119

1121

1065

1.03

1.04

1.05

1.05

4. Numerical methods

j 

383

4.2 Numerical modeling of confined concrete

1500

Axial load (kN)

1200

900

600

k=0.6 k=2/3

300

k=0.725 k=0.8

0 0

0.01

0.02

0.03

0.04

Axial strain

(a) 1500

Axial load (kN)

1200

900

600

k=0.6 k=2/3

300

k=0.725 k=0.8

0 0

0.01

0.02

0.03

0.04

Axial strain

(b) FIGURE 4.17 Effect of k on the axial loadestrain curves of composite members. (A) Circular CFST 3HN. (B) Square CFST 3MN.

rate of concrete increases with increasing j. Hence, as the dilation angle increases j, so does the interaction between the steel tube and the concrete core. Consequently, the confining pressure increases with increasing j, leading to a higher ultimate axial strength. Compared with the square composite columns, the influence of the dilation angle j is more prominent on the behavior of circular columns.

384

4. Numerical methods

1500

Axial force (kN)

1200 900 ψ=0.01

600

ψ=20 ψ=30

300

ψ=40 0 0

0.01

0.02

0.03

0.04

Axial strain

(a) 1500

ψ=0.01 ψ=20

1200

Axial force (kN)

ψ=30 ψ=40

900

600

300

0 0

0.01

0.02

0.03

0.04

Axial strain

(b)

FIGURE 4.18 Effect of j on the axial loadestrain curves of composite members. (A) Circular CFST 3HN. (B) Square CFST 3MN.

4.3 Investigating the behavior of composite members through numerical analysis Valuable information about the fundamental behavior of structural elements can be obtained by performing experimental investigations. However, it may be difficult, if not impossible, to investigate the more sophisticated behavior of members using experimental procedures. Experimenting on complicated local effects like residual strength, initial

385

4.3 Investigating the behavior of composite members

imperfections, and the contact mechanism between constituent components of composite elements, for example, is either impossible or very intricate. Additionally, evaluating the effects of geometric and material properties as well as loading and boundary conditions on the structural performance of composite members via experimental studies needs the fabrication and testing of large numbers of specimens that can be time consuming and expensive. FE analysis of structural members is a viable method that can be used along with experimental studies to ease the understanding of the behavior of structural members. The advancement of FE software like ABAQUS and ANSYS has aided researchers and engineers in assessing the complicated behavior of structural members and speeding up parametric studies. The effects of key parameters on the structural performance of composite members, including columns, beams, and beam columns, are addressed in this section.

4.3.1 Short columns 4.3.1.1 Effects of the outer tube depth-to-thickness ratio The effects of the diameter-to-thickness ratio D=t of steel tubes on the performance of CFST stub members were investigated using the nonlinear FE analysis program. The geometric and material properties of the examined circular columns are listed in Table 4.5. Nu and DI in Table 4.5 represent the ultimate axial strength and the ductility index of specimens. Fig. 4.19 illustrates the axial loadestrain responses of CFST stub columns with different D=t ratios. It can be observed from the figure that the initial axial stiffness and the ultimate compressive strength of columns are declined by increasing the D=t ratio. In the postpeak stage of the curve, the column with the largest D=t ratio exhibits the steepest softening response compared with other columns, indicating that the ductility of the column reduces with an increase in the D=t ratio, as shown in Table 4.5. The effects of the D=t ratio on the axial loademoment interaction diagram are presented in Fig. 4.20, in which the ultimate axial strength Pu and the ultimate bending strength Mu are scaled to the pure axial strength TABLE 4.5 Details of the tested specimens for investigating the effects of the D= t ratio. fsy (MPa)

fc0 (MPa)

Nu (kN)

DI

450

45

19,863

14.8

70

12,665

7.2

100

10,836

4.8

Name

D 3 t (mm)

1

500  12:5

40

2

500  7:15

3

500  5

D t

386

4. Numerical methods

25000

Axial load (kN)

20000

15000

10000

D/t=40 5000

D/t=70 D/t=100

0 0

0.01

0.02

0.03

0.04

0.05

Axial strain

FIGURE 4.19 Axial loadestrain curves of CFST columns with various D=t ratios. 1.2 D/t=40 1

D/t=70 D/t=100

Pu / Po

0.8 0.6 0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

1.2

Mu / Mo FIGURE 4.20

Axial loademoment interaction diagrams of CFST beam columns with various D=t ratios.

Po and ultimate pure bending strength Mo . As shown in Fig. 4.20, both Pu and Mu decrease with increasing the D=t ratio, and the effect of the D= t ratio on the magnitude of Mu is slightly more than Pu . These observations can be justified in two aspects. Firstly, the crosssectional area of the steel tube reduces with reducing the thickness of the tube. Secondly, reducing the tube thickness makes it susceptible to

387

4.3 Investigating the behavior of composite members

local buckling. As a result, the confinement effect is reduced by reducing the tube thickness. It should be noted that the same results can be expected for the influences of the D=t ratio on the overall performance of CFSTs and CFDSTs with any cross-section. 4.3.1.2 Effects of the concrete compressive strength In this section, the effects of the concrete compressive strength fc0 on the performance of CFST stub columns are evaluated. The geometric and material properties of the examined square columns are listed in Table 4.6. Fig. 4.21 illustrates the axial loadestrain curves of CFST stub columns with different fc0 . As shown in Fig. 4.21 and Table 4.6, the ultimate axial strength of CFSTs is increased by increasing the concrete strength fc0 . However, the concrete strength fc0 has negligible influence on the ascending stage and initial axial stiffness of columns. Another important influence of the concrete strength fc0 is on the ductility of composite columns. From the axial loadestrain responses, it is clear that increasing the concrete strength fc0 has a negative impact on ductility. This issue can also be noticed from the values of DI, given in Table 4.6. As a result, TABLE 4.6 Details of the tested specimens for investigating the effects of fc0 . Name

D 3 t (mm)

D t

fsy (MPa)

fc0 (MPa)

4

315  6

52.5

350

Nu (kN)

DI

33

7179

4.40

5

70

12,885

3.16

6

90

15,827

2.80

20000

f'c=33MPa

Axial force (kN)

16000

f'c=70MPa

f'c=90MPa 12000

8000

4000

0 0

0.01

0.02

0.03

0.04

0.05

Axial strain

FIGURE 4.21

Axial loadestrain curves of CFST columns with various fc0 .

388

4. Numerical methods

1.2 f'c=33MPa 1 f'c=70MPa

Pu / Po

0.8

f'c=90MPa

0.6 0.4 0.2 0 0

0.5

1

1.5

2

Mu / Mo

FIGURE 4.22 various fc0 .

Axial loademoment interaction diagrams of CFST beam columns with

increasing the concrete strength fc0 can be regarded a viable method for improving the ultimate axial strength of composite members. The postpeak region of composite columns filled with HSC, on the other hand, often exhibits a sudden drop, suggesting brittle behavior, which is not desirable. Fig. 4.22 shows the axial loademoment interaction curves of square CFST columns having different concrete strength fc0 . It can be observed from the figure that the effect of concrete strength fc0 is more prominent on the axial strength of composite columns rather than their bending capacity. Besides, the influence of the concrete strength fc0 increases with a reduction in the bending moment. The reason lies in the fact that concrete is a material that is inherently resistant to compression, whereas its flexural strength when it is subjected to bending is weak. Overall, composite members fabricated with HSC and UHSC should be used for bearing compression load rather than bending moments. It should be noted that the same results can be expected for the influences of the concrete compressive strength fc0 on the overall performance of CFSTs and CFDSTs with any cross-section. 4.3.1.3 Effects of the outer steel tube yield strength The effects of the steel tube yield strength fy on the performance of circular CFST members were investigated using the nonlinear FE analysis program. The geometric and material properties of the examined columns are listed in Table 4.7. Fig. 4.23 illustrates the axial loadestrain curves of

389

4.3 Investigating the behavior of composite members

TABLE 4.7 Details of the tested specimens for investigating the effects of fy . Name

D 3 t (mm)

D t

fy (MPa)

fc0 (MPa)

7

200  6

33.3

235

51

Nu (kN)

DI

8210

22.9

8

355

9194

21.6

9

550

10,657

17.7

12000

Axial force (kN)

10000 8000 6000 fy=235MPa 4000

fy=355MPa fy=550MPa

2000 0 0

0.01

0.02

0.03

0.04

0.05

Axial strain

FIGURE 4.23 Axial loadestrain curves of CFST columns with variousfy .

CFST stub columns with different fy . It can be observed from the figure that the initial axial stiffness of composite members is not affected by the steel yield strength. However, the ultimate axial strength of columns is increased remarkably by increasing the steel tube strength. Regarding the postpeak region of the curves, it can be noted from Table 4.7 and Fig. 4.23 that the column’s ductility reduces as the steel yield strength increases. To further investigate the effect of steel tube strength on the ductility of composite members under axial compression, the axial loadedisplacement curves of circular CFDST columns with inner square steel tube fabricated by mild carbon steel tubes and high strength steel tubes are compared in Fig. 4.24 [45]. The outer circular steel tube section is 219  8:2 mm, and the inner square steel tube section is 89  89  6 mm. The compressive strength of the sandwiched concrete is fc0 ¼ 60 MPa. It can be observed from Fig. 4.24 that increasing the steel tube strength leads to a steeper postpeak region. The column made of high-strength steel tubes fy ¼ 1400 MPa shows a brittle behavior compared with the specimen made of the normal strength steel tube fy ¼ 350 MPa. As a consequence, it can be argued that the usage of high-strength steel can

390

4. Numerical methods

15000 f'y = 350 MPa f'y = 700 MPa f'y = 1050 MPa f'y = 1400 MPa

Axial load (kN)

12000 9000 6000 3000 0 0

5

10 15 20 Axial shortening (mm)

25

30

FIGURE 4.24 Axial loadestrain curves of CFDST columns with various fyo [45]. 1.2 fy=235MPa

1

fy=355MPa fy=550MPa

Pu / Po

0.8 0.6 0.4 0.2 0 0

FIGURE 4.25

0.2

0.4

0.6 0.8 Mu / Mo

1

1.2

1.4

Axial loademoment interaction diagrams of CFST beam columns with

various fy .

result in a good outcome in terms of the column’s ultimate axial strength. The structural engineer, on the other hand, may find it difficult to design a ductile composite column utilizing a high-strength steel tube. Fig. 4.25 shows the axial loademoment interaction diagrams of CFSTs made of different steel tube strength fy . It can be observed from the figure that the compressive capacity and bending strength of columns are enhanced by increasing the steel tube strength fy . However, the effect of fy

391

4.3 Investigating the behavior of composite members

is more prominent on the ultimate pure moment capacity than the ultimate pure axial strength. This can be justified by the fact unlike concrete, the steel tube is susceptible to compression and buckling. Accordingly, for resisting large bending moments, the use of high-strength steel tube can be considered as an efficient solution. It should be noted that the same results can be expected for the influences of the steel tube yield strength fy on the overall performance of CFSTs and CFDSTs with any cross-section. 4.3.1.4 Effects of the inner steel tube yield strength Nonlinear FE analysis was performed on the square CFDST stub columns with the inner circular steel tube to evaluate the effects of the inner tube steel yield strength fyi on the overall behavior of the column. The geometric and material properties of tested CFDST stub columns are given in Table 4.8. The axial loadedisplacement curves of columns having various fyi are depicted in Fig. 4.26. As illustrated in the figure, the effect of the inner tube yield strength fyi on the column’s initial axial stiffness is TABLE 4.8 Details of the tested specimens for investigating the effects of fyi . Name

Bo 3 to (mm)

di 3 ti (mm)

fyo (MPa)

fyi (MPa)

fc0 (MPa)

Nu (kN)

DI

10

300  10

150  7

400

250

40

7539

34

11

350

7853

36

12

450

8166

36.5

9000 8000

Axial force (kN)

7000 6000 5000

fyi=250 MPa

4000 fyi=350 MPa

3000 2000

fyi=450 MPa 1000 0 0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Axial displacement

FIGURE 4.26 Axial loadestrain curves of CFDST columns with various fyi .

392

4. Numerical methods

1.2 fyi=250 Mpa 1 fyi=350 Mpa

Pu / Po

0.8

fyi=450 MPa

0.6 0.4 0.2 0 0

0.25

0.5

0.75 Mu / Mo

1

1.25

1.5

FIGURE 4.27 Axial loademoment interaction diagrams of CFDST beam-columns with various fyi .

negligible. Increasing the inner tube yield strength fyi can improve the ultimate axial strength of the column. However, the inner tube’s effect on the load-bearing capacity of the column is remarkably less than the outer tube and the sandwiched concrete. Similar to the initial axial stiffness, the inner tube yield strength fyi has no apparent influence on the member’s ductility and residual strength, as shown in Table 4.8 and Fig. 4.26. The influence of the inner tube yield strength fyi on the axial loademoment interaction diagram is presented in Fig. 4.27. It is evident that the effect of the inner tube yield strength on the axial loademoment interaction diagram is not significant. 4.3.1.5 Effects of the hollow ratio The hollow ratio c, defined as

di , ðDo
2to Þ

is an important characteristic

that influences the structural performance of CFDST members. The effects of the hollow ratio c on the behavior of square CFDST stub columns with inner circular steel tubes are assessed herein. The geometric and material properties of the examined columns are listed in Table 4.9. Fig. 4.28 illustrates the axial loadestrain curves of CFDST stub columns with different c [46]. As shown in Fig. 4.28 and Table 4.9, the ultimate axial strength of the column increases slightly by increasing the hollow ratio c from 0.1 to 0.3. This enhancement, however, is negligible. Consequently, employing CFDST columns with a hollow ratio c of less than 0.3 is not advised since they can increase the total weight of the structure without a

393

4.3 Investigating the behavior of composite members

TABLE 4.9 Details of the tested specimens for investigating the effects of the c. Name

Bo 3 to (mm) di 3 ti

c

fy (MPa)

fc0 (MPa)

Nu (kN)

DI

13

220  8

20:5  4

0.1

350

40

4131

15.1

14

40  4

0.2

4160

17.8

15

60  4

0.3

4168

18.1

16

102  4

0.5

4035

22.3

17

143  4

0.7

3780

33.4

18

163  4

0.8

3580

39.3

4500

Axial force (kN)

4000 3500 3000 2500

χ=0.10 χ=0.20 χ=0.30 χ=0.50 χ=0.70 χ=0.80

2000 1500 1000 500 0 0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Axial strain

FIGURE 4.28

Axial loadestrain curves of CFDST columns with various c.

significant increase in the columns’ resistance. Increasing the hollow ratio from 0.3 to a higher number, on the other hand, diminishes the column’s ultimate axial strength. The reason is that enlarging the hollow ratio c reduces the cross-sectional area of the sandwiched concrete, which has the most significant contribution to the column’s compressive capacity. Contrary to the ultimate axial strength of the column, a positive correlation between the hollow ratio c and ductility can be expected. This is because increasing the hollow ratio c reduces the area of the sandwiched concrete as a brittle material. The axial loademoment interaction diagrams of CFDST stub columns having different hollow ratios c are presented in Fig. 4.29. As illustrated in Fig. 4.29, axial loademoment interaction curves for columns with hollow ratios c of 0.1e0.3 are nearly identical. This again reinforces the fact that when the hollow ratio c is smaller than 0.3, the impacts of the hollow ratio

394

4. Numerical methods

1.2 χ=0.10 χ=0.20 1

χ=0.30 χ=0.50

0.8

χ=0.70

Pu / Po

χ=0.80 0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

Mu / Mo

FIGURE 4.29

Axial loademoment interaction diagrams of CFDST beam columns with

various c.

c on the overall performance of composite members are negligible. By contrast, the interaction curves shift to the left when the hollow ratio c becomes more than 0.3. The hollow ratio c has a more impact on the column ultimate axial strength than the bending strength because the concrete resistance to compression is more excellent than the bending moment. Fig. 4.30   illustrates the effect of the hollow ratio c on the strength-tos w

Strength-to-weight ratio (s/w)

weight

ratio of CFDST columns under axial compression. It can be 47 46 45 44 43 42 41 40 39 38 37 0.1

FIGURE 4.30 columns.

0.2

0.3 0.5 Hollow ratio χ

0.7

0.8

Effect of the hollow c ratio on the strength to weight ws ratio of CFDST stub

395

4.3 Investigating the behavior of composite members

observed from the figure that the ws ratio enhances with an increase in the hollow ratio c. When the hollow ratio c is increased from 0.1 to 0.8, the ws ratio improves by nearly 15%. 4.3.1.6 Effects of the inner steel tube depth-to-thickness ratio The effects of the depth-to-thickness dtii ratio of the inner steel tube in CFDST members were assessed by performing the FE simulation method. The geometric and material properties of the examined square CFDST columns with inner circular steel tubes are listed in Table 4.10. Fig. 4.31 illustrates the axial forceestrain curves of CFDST stub columns with different dtii ratio [46]. It can be observed from Fig. 4.31 that the thickness of the inner tube does not influence the axial stiffness of composite columns. A comparison of ultimate axial strength of columns, shown in Table 4.10 TABLE 4.10 Details of the tested specimens for investigating the effects of the ratio.

di ti

fy (MPa)

fc0 (MPa)

Nu (kN)

DI

350

40

4272

24

Name

Bo 3 to (mm) di 3 ti

di ti

19

220  8

90  6

15

20

90  6

18

4184

22.5

21

60  5

22.5

4098

21

22

90  4

30

4052

20

23

90  3

45

3802

10.3

4500 4000

Axial load (kN)

3500 3000

di/ti=15

2500 2000

di/ti=18

1500

di/ti=22.5

1000

di/ti=30 di/ti=42.5

500 0 0

0.01

0.02

0.03

0.04

Axial strain

FIGURE 4.31 Axial loadestrain curves of CFDST columns with various dtii ratios.

396

4. Numerical methods

1.2

di/ti=15 di/ti=18

1

di/ti=22.5 Pu / Po

0.8

di/ti=30 di/ti=42.5

0.6 0.4 0.2 0 0

0.25

0.5

0.75

1

1.25

1.5

Mu / Mo

FIGURE 4.32 Axial loademoment interaction diagrams of CFDST beam-columns with

various dtii ratios.

and Fig. 4.31, indicates that with the increase of the dtii ratio from 15 to 18, 22.5, and 30, the ultimate axial strength is reduced by 2%, 4%, 5%, respectively. Accordingly, the thickness of the inner steel tube has a slight effect on the compressive capacity of CFDST columns. Similarly, the effects of the dtii ratio when it ranges from 15 to 30 on the postpeak region and the ductility of columns are negligible. By contrast, the effects of the dtii ratio on the behavior of the CFDST column with ti ¼ 3 mm are evident. The compressive strength of the column with dtii ¼ 45 is 11% smaller than the column with dtii ¼ 15. Besides, a remarkable reduction in the DI value of the column with dtii ¼ 45 compared with other columns can be observed from Table 4.10. Fig. 4.32 shows the axial loademoment interaction diagrams of CFDST stub columns having different dtii ratios. As shown in Fig. 4.32, except for the specimen with dtii ¼ 45, the effects of the dtii ratio on the axial loade moment interaction curve are insignificant. It can be concluded that the thickness of the inner steel tube has the most pronounced influence on the column’s compressive strength. As discussed above, the effects of the inner tube thickness on the overall performance of CFDST members are generally insignificant. This observation can be justified by the fact that compared with other constituent components of CFDST members, the cross-sectional area of the

397

4.3 Investigating the behavior of composite members

inner steel tube, and therefore, its contribution to the load-bearing capacity of the member, is remarkably smaller. The behavior of the column with dtii ¼ 45, on the other hand, contradicts this conclusion when compared to the other

di ti

ratios.

The difference in the effects of the

di ti

ratio on the response of the

di ti

specimen with ¼ 45 compared to other specimens may be explained by comparing the inner tube thickness to the required critical thickness that ensures the prevention of inner tube failure before yielding of the outer tube. Designing a CFDST column with an optimum cross-section requires ensuring that the outer tube yields before the inner tube fails [47]. The inner steel tube’s premature failure can be prevented by determining its minimum thickness to prevent its yield failure (ty:i ) or buckling failure (tbkl:i ). The equilibrium of forces acting on the column’s free body diagram can be used to determine the thicknesses ty:i and tbkl:i , as shown in Fig. 4.33 [46]. Based on Fig. 4.33A, the relationships between the stresses acting on the outer tube (so ), inner tube (si ), and the sandwiched concrete (sl ) are as follows: 2so to ¼ sl tc þ 2si ti

(4.84)

in which the term tc ¼ Bo
Di refers to the thickness of the sandwiched concrete to which lateral limiting pressure is imposed. From Fig. 4.33B, the following expression can be obtained: 2si ti ¼ sl Di

(4.85)

Therefore, si is governed by the following: si t i ¼

(a)

FIGURE 4.33

sl Di 2ti

(4.86)

(b)

(c)

Internal forces in a square CFDST column [46]. (A) Half-section (B) Inner tube (C) Outer tube.

398

4. Numerical methods

The relation between si and so can be determined using Eq. (4.87), derived by integrating Eq. (4.86) into Eq. (4.84).   Di to so  (4.87) si ¼ Bo ti Suppose that when the outer tube reaches its yield stress (so ¼ syo ), the magnitude of stress acting on the inner steel tube is less than its yield stress (si < syi ). In this case, the inner tube has larger yield stress than the outer tube (syi > syo ). Hence,   Di to si ¼  (4.88) syo < syi B o ti Therefore, the minimum thickness tyi is governed by the following: ! Di syo tyi ¼  (4.89) to Bo syi The inner tube is unilaterally restrained, and the sandwiched concrete prevents it from buckling outward. As a result, the buckling strength of the inner tube differs from that of a bilaterally restrained arch or ring. The buckling mechanism of an arch with unilateral boundary conditions is identical to a circular arch when it buckles in the snap-through mechanism. The buckling mechanism of an arch with unilateral boundary conditions is identical to a circular arch when it buckles in the snapthrough mechanism. Hence, it can be assumed that the inner tube’s buckling mechanism is the snap-through behavior of a shallow circular arch [47]. Researchers recommended various buckling load coefficients ðpÞ for the circular arch, which are listed in Table 4.11. Kerr and Soifer [48] developed an equation for predicting the longitudinal buckling resistance of a shallow circular arch, which can be employed for the inner tube of CFDST stub columns as follows: sbk ¼ 2:27

Esi Isi ti r2i

(4.90)

where Esi , Isi , and ri represent the elastic modulus, the moment of inertia, and the radius of the inner tube, respectively. TABLE 4.11 Buckling load coefficients ðpÞ of the circular arch. Buckling load coefficient p Mode

Kerr and Soifer [48]

Haftka et al. [49]

Sun and Natori [50]

Bifurcation

1.91

1.86

1.90

Snap-through

2.27

2.17

2.28

4.3 Investigating the behavior of composite members

399

Early buckling failure of the inner tube can be avoided by taking the inner tube’s buckling strength larger than the confining pressure sl when the outer tube yields. Therefore, it can be considered that the stress acting on the outer tube so is equal to its yield strength syo and: sbk ¼ 2:27

Esi Isi > sl ti r 2i

(4.91)

sl can be determined from Fig. 4.33C as follows: sl ¼

2so to Bo

(4.92)

The minimum thickness that can prevent the premature buckling of the inner steel tube tbkl;i can be obtained by replacing Eq. (4.92) into Eq. (4.91) as follows: !0:5 2:64 d2i syo to tbkl;i ¼ (4.93) Bo Esi The failure mechanism of the inner tube can be anticipated by knowing its critical thicknesses. The first conceivable failure mode is yielding the inner tube before the   outer tube fails. In failure mode 1, si > slim ¼ Min ssyi ; sbk and ssyi < sbk . The second failure mode occurs

when the failure of the outer tube happens after buckling of the inner tube. In failure mode 2, si > and ssyi > sbk . The third failure mode happens when si < slim . In failure mode 3, the inner tube fails before yielding the outer tube. The fourth failure mode occurs when si ¼ slim . In failure mode 4, inner and outer tubes fail simultaneously. The algorithm depicted in Fig. 4.34 describes the analysis procedure for determining the failure mechanism of the inner tube. Table 4.12 summarizes the critical thicknesses of the examined CFDST stub columns. Besides, the strength index (SI) of the columns is given in Table 4.12 to assess the effect of inner tube thickness on the confinement behavior of CFDST columns. SI is given by the following: SI ¼

Nu Aso fyo þ 0:85Aso fc0 þ Asi fyi

(4.94)

The denominator in Eq. (4.94) is the modified design model specified by ACI 318 (2011) for predicting the compressive capacity of CFDST stub columns that ignores the confinement effect. For preventing the premature failure of the inner tube before yielding the outer tube, the thickness of the inner tube  should  be set as the maximum value of ty:i and tbkl:i

treq:i ¼ Max ty:i : tbkl:i

. As shown in

400

4. Numerical methods

start

Geometric and material properties

Compute

.

Calculate

.

Compute

.

.

No

>

.




.

Yes

No

No Failure mode 1

Failure mode 2

Failure mode 3

Failure mode 4

End FIGURE 4.34 Calculation procedure for predicting the failure mechanism of CFDST stub columns steel tubes. Ayough P, Ramli Sulong NH, Ibrahim Z, Hsiao P-C. Nonlinear analysis of square concrete-filled double-skin steel tubular columns under axial compression. Eng Struct 2020;216:110678.

Table 4.12, the inner tube thicknesses in columns 16e18 are larger than treq:i . Hence, the inner tubes fail after yielding the outers tube. The inner tube thicknesses in columns 22 and 23 are smaller than treq:i . Therefore, in these columns, the inner tubes fail prior to the yielding of the outer tube. The inner tubes’ premature failure reduces the concrete core’s confining

Predicting the failure mechanism of CFDST stub columns based on the inner tube wall thickness.

Name

ti (mm)

ty:i (mm)

tbkl:i (mm)

treq:i (mm)

ssyi (MPa)

sbk (MPa)

SI

Failure mechanism

19

6

3.27

1.17

3.27

350

672.6

1.060

Yielding of the outer tube slim > si , treq:i < ti

20

5

467.1

1.053

Yielding of the outer tube slim > si , treq:i < ti

21

4

298.9

1.050

Yielding of the outer tube slim > si , treq:i < ti

22

3

168.2

1.039

Failure of the inner steel tube slim < si , treq:i > ti

23

2

74.7

0.98

Failure of the inner steel tube slim < si , treq:i > ti

4.3 Investigating the behavior of composite members

TABLE 4.12

401

402

4. Numerical methods

Indent buckling

FIGURE 4.35 Failure modes of steel tubes in CFDST columns [46]. (A) dtii ¼ 15 (B) dtii ¼ 18 (C) dtii ¼ 22:5 (D) dtii ¼ 30 (E) dtii ¼ 45.

pressure, specifically in specimen 23. This conclusion can be deduced from the values of SI. The SI magnitude declines by increasing the dtii ratio. The value of SI for Column 23, with the thinner inner steel tube, compared with other columns, is less than unity because of the early failure of the inner tube. As a result, in these columns, the inner tubes fail before the outer tube yields. The failure mechanisms of the steel tubes in CFDST stub columns are depicted in Fig. 4.35. The figure shows that the inner tube thickness has no effect on the outer tube’s failure mechanism and merely alters the outward buckling location. This observation indicates that compared with the thickness of the inner tube, the outer tube thickness has the most pronounced influence on the failure mechanism of CFDSTs. No apparent local buckling can be observed in specimens 19e22, whereas the inner tube of specimen 23 demonstrates indent local buckling, indicating that the inner tube’s failure mode depends on the dtii ratio. Fig. 4.36 shows the effect of the dtii ratio on the ws ratio of CFDSTs. It can be seen from the figure that the ws ratio reduces with increasing the dtii ratio. However, the effect of the inner tube thickness on the ws ratio is moderate, especially in columns 19e22. Hence, it can be noted that the inner tube’s contribution to the load-bearing capacity of CFDST columns is negligible, and the least possible thickness should be considered for it to meet the treq:i and prevent its premature failure. Employing the inner tube with ti ¼ treq:i can lead to an optimal design of CFDST members based on the costsafety balanced criterion.

403

4.3 Investigating the behavior of composite members

44

Strength-to-weight ratio (s/w)

43 42 41 40 39 38 37 36 35 15

18

22.5

30

42.5

di /ti

FIGURE 4.36

Effect of the dtii ratio on the

s w

ratio of CFDST stub columns.

4.3.1.7 Efficiency of increasing material strength on the loadbearing capacity of composite members As previously noted, the material strength of the constituent components directly impacts the load-bearing capacity of composite columns. However, the efficiency of increasing the strength of the outer tube, inner tube, and in-filled concrete is not the same in enhancing the strength of composite members. This section evaluates the effectiveness of increasing the yield strength of the tubes and the compressive strength of concrete in enhancing the load-bearing capacity of composite members. To this end, the efficiency of increasing the material strength on the compressive strength of the square CFDST stub column, evaluated in the previous sections, is given in Fig. 4.37. u The DN Dfyo ratio is determined to assess the efficiency of increasing the outer tube yield strength fyo in enhancing the compressive strength of CFDST columns (Ca and Cb ), as follows [46]: DNu Nu;b
Nu;a ¼ Dfyo fyo;b
fyo;a

(4.95)

where DNu and Dfyo are the increase in the ultimate axial strength and the outer tube yield strength between two columns (Ca and Cb ), respectively. u For determining the DN Dfyo ratio, the column with fyo ¼ 250 MPa is regarded as the benchmark specimen (specimen Ca ). As shown in Fig. 4.37A, when the outer tube yield strength is increased from 250 to 450 MPa, the column’s compressive strength is enhanced by only 23.20% that compared with an 80% increase in fyo is relatively inconsequential. The average efficiency of the outer tube yield strength in the examined CFDST stub column u is DN Dfyo ¼ 0:30.

404

4. Numerical methods

Δfyo Δfy f o

ΔNu

ΔNu/Δfyo ΔNu/Δfy f o 0.35

0.31

0.30

0.29

0.228 0.28

0.3

60 60

70 0.25 0.2 40

50 40

11.84

6.117 6.17

20 10

17.39 17.339

30

23.20

0.15

Efficiency

60

200

Increase in strenght %

80

80

90

0.1 0.05

0

0 fyo=300 ffy o=300

fyo=350 ffy o=350

fyo=400 ffy o=400

fyo=450 ffy o=450

(a) Δfyi Δfy f i

ΔNu

ΔNu/Δfyi 0.12

0.11

0.10

0.10

80

80

90 0.10

60

400

50 0.06

40

Efficiency

0.08

60

0.04

30

6.23 3

4.17

10

2.14

20

8.32 8.332

20

Increase in strenght %

0.1 70

0

0.02

0 fyi=300

fyi=350

fyi=400

fyi=450

(b) FIGURE 4.37 Assessing the efficiency of increasing the material strength in enhancing the ultimate axial strength of composite members [46]. (A) Efficiency of increasing the outer tube yield strength. (B) Efficiency of increasing the inner tube yield strength. (C) Efficiency of increasing the concrete compressive strength.

405

4.3 Investigating the behavior of composite members

0.27

0.27

ΔNu/Δf 'c 0.27

183.33 18 83.33

0.15

49.93

336.49 6.49

22.84 22.8 84

50

0.2

63.799

150

0.3

0.25

133.33 1333.33

200

83.33 83 3.33

Increase in strenght %

250

100

0.27 233.33 23 33.333

300

ΔNu

Efficiency

Δf 'c

0.1

0.05

0

0 f 'c=55MPa

f 'c=70MPa

f 'c=85MPa

f 'c=100MPa

(c) FIGURE 4.37 cont’d.

The

DNu Dfyi

ratio is determined to assess the efficiency of increasing the

inner tube yield strength fyi in enhancing the compressive strength of CFDST columns (Ca and Cb ), as follows: DNu Nu;b
Nu;a ¼ Dfyi fyi;b
fyi;a

(4.96)

where DNu and Dfyi denote the increase in the ultimate axial strength and the inner tube yield strength between two columns (Ca and Cb ), respecu tively. For determining the DN Dfyi ratio, the column with fyi ¼ 250 MPa is regarded as the benchmark specimen (specimen Ca ). As shown in Fig. 4.37B, when the inner tube yield strength is increased from 250 to 450 MPa, the column’s compressive strength is enhanced by only 8.32% that compared with an 80% increase in fyi is negligible. The average efficiency of the inner tube yield strength in the examined CFDST stub u column is DN Dfyi ¼ 0:10. The

DNu Dfc0

ratio is determined to assess the efficiency of increasing the

concrete compressive strength fc0 in enhancing the compressive strength of CFDST columns (Ca and Cb ), as follows: DNu Nu;b
Nu;a ¼ 0 0 Dfc0 fc;b
fc;a

(4.97)

406

4. Numerical methods

where DNu and Dfc0 represent the increase in the ultimate axial strength and the concrete compressive strength between two columns (Ca and Cb ), 0 u respectively. For determining the DN Df 0 ratio, the column with fc ¼ 30 MPa is c

taken as the benchmark specimen (specimen Ca ). As shown in Fig. 4.37C, when the concrete strength is increased from 30 to 100 MPa, the compressive strength of the column is increased by almost 64% that compared with more than 233% increase in fc0 is relatively insignificant. The average efficiency of the concrete compressive strength in the u examined CFDST stub column is DN Df 0 ¼ 0:30. c

DNu u To conclude, the average efficiency ratios of DN Dfyo ¼ 0:30, Dfyi ¼ 0:10, and

DNu Dfc0

¼ 0:30 show that increasing the yield strength of the inner tube for

improving the compressive strength of composite members has the least efficiency compared with the outer tube and the concrete core. 4.3.1.8 Simultaneous effects of the depth-to-thickness ratio of the steel tube and the concrete compressive strength on the ultimate axial strength of the CFDST column The influences of the steel tube depth-to-thickness ratio and the concrete compressive strength on the load-bearing capacity of CFDST stub columns are discussed individually in previous sections. Simultaneous assessment of the effects of these two parameters can help for the optimized design of the outer-tube cross-section. To this end, the change in the cross-sectional areas of the outer tube and the sandwiched concrete, as well as the ultimate axial strength of square CFDST stub columns with Bo 0 to ranging from 44 to 220, filled with the NSC having fc ¼ 40 MPa and 0 HSC having fc ¼ 100 MPa are compared. The results for the examined columns are summarized in Table 4.13 in which DAso:ba is the change in the outer tube area, DAc:ba is the change in the area of the sandwiched concrete, and DNu:ba is the change in the ultimate axial strength of columns. DAso:ba , DAc:ba , and DNu:ba are determined as follows: Aso:b
Aso:a Aso:a

(4.98)

DAc:ba ¼

Ac:b
Ac:a Ac:a

(4.99)

DNu:ba ¼

Nu:b
Nu:a Nu:a

(4.100)

DAso:ba ¼

Columns C5 and C10 are taken as the benchmark specimens ðCa Þ. It can be observed from Table 4.13 that increasing the thickness of the outer tube to (reducing the Bo=t ratio) increases the cross-sectional area of the steel o

tube and the ultimate axial strength of columns, although the change in the cross-sectional area of the sandwiched concrete is minor. By

TABLE 4.13

Simultaneous influences of the Bo=t ratio and fc0 on the ultimate axial strength of CFDST columns. o

Cb

Ca

Bo to

1

C5

C5

220

C4

2

fc0 40

110

Aso (mm2 )

Ac (mm2 )

Nu (kN)

DAso:ba

876

41,165.5

2465

0.00

0.00

0.00

1744

40,297.5

2786

0.50


0.02

0.12

DAc:ba

DNu;ba

C3

73.33

2604

39,437.5

3108

0.66


0.04

0.21

C2

55

3456

38,585.5

3475

0.75


0.07

0.29

C1

44

4300

37,741.5

3710

0.80


0.09

0.34

876

41,165.5

5014

0.00

0.00

0.00

1744

40,297.5

5297

0.50


0.02

0.05

C10 C9

C10

220 110

100

C8

73.33

2604

39,437.5

5487

0.66


0.04

0.09

C7

55

3456

38,585.5

5755

0.75


0.07

0.13

C6

44

4300

37,741.5

6035

0.80


0.09

0.17

4.3 Investigating the behavior of composite members

Group

407

408

4. Numerical methods

comparing the change in the steel tube area and the compressive strength of columns, it can be noticed that DAso:ba increases at a higher rate than DNu:ba . The same result can be observed for both groups of columns filled with NSC and HSC. Regarding the concrete type (i.e., NSC or HSC), it can be noticed from the results that the enhancement in the compressive capacity by increasing to (decreasing the Bo=t ratio) reduces when the HSC o is employed compared with the improvement associated with utilizing the NSC. Increasing the outer tube thickness to for enhancing the ultimate axial strength of columns is more efficient when NSC is used. For example, the change in the cross-sectional area of the steel tube is DAso:ba ¼ 0:80 when the Bo=t ratio is reduced from 220 to 44. However, the ultimate axial o

strength improvement when NSC and HSC are used are DNu;ba ¼ 0:29 and 0.17, respectively. This indicates that filling the hollow steel section by HSC is more economical when a thin-walled steel tube (a large Bo=t ratio) is o

employed. By doing so, the overall cost of the column decreases. The usage of UHSC in CFST columns decreases the cross-sectional area, providing considerable cost benefits [51]. The calculation model recommended by Liang and Fragomeni [39] is used to examine this recommendation. Based on Ref. [39], the ultimate axial strength of circular CFST stub columns under axial compression can be predicted using the following equations:   (4.101) Nu ¼ gc fc0 þ 4:1 frp Ac þ gs fy As in which frp represents the confining pressure and is given by the following: 8  > D D > > 0:02663
0:0002504 fy for 17  < 47 > < t t fr ¼  (4.102)  > D D > > > : 0:01588
0:0000149 t fy for 47  t  221 gc is the concrete strength reduction factor to account the size effect on the ultimate axial strength of column and is expressed as follows: 0:85  gc ¼ 1:85D
0:135  1:0 c

(4.103)

gs is the steel strength factor to incorporate the influences of hoop stresses, strain hardening, geometric flaws, and residual stresses into prediction:  
0:1 D  1:1 (4.104) 0:9  gs ¼ 1:458 t

409

4.3 Investigating the behavior of composite members

The strength prediction of CFST/CFDST members is discussed in detail in Chapter 5. According to Eq. (4.101), a circular CFST stub column with a diameter of 500 mm, tube wall thickness of 25 mm, steel yield strength of 420 MPa, and concrete compressive strength of 190 MPa has an ultimate axial strength of 48,502 MPa. Fig. 4.38A shows the diameter of the column as a function of concrete compressive strength. When UHSC is utilized instead of NSC, the diameter of CFST columns can be reduced in half. For instance, the diameter of the CFST column made of steel with a yield strength of 1200 fy=300 Mpa fy=420 Mpa

Column diameter D (mm)

1000 800 600 400 200 0 0

40

80 120 160 Concrete compressive strength (MPa)

200

(a) 700

Column diameter D (mm)

650 600 550 500 fy=300 Mpa 450

fy=420 Mpa

400 0

40

80

120 D/t

160

200

240

(b)

FIGURE 4.38

Effect of column size on the ultimate axial strength of CFST stub columns [52]. (A) Concrete compressive strength. (B) The Dt ratio. (C) Steel yield strength.

410

4. Numerical methods

800

Column diameter D (mm)

700 600 500 400 D/t=200

300

D/t=20 200 150

190

230

270

310 350 D/t

390

430

470

(c)

FIGURE 4.38 cont’d.

420 MPa and the Dt ratio of 20 can be decreased from 802 to 498 mm by increasing the concrete compressive strength from 20 to 190 MPa. As seen in Fig. 4.38B, the Dt ratio has no substantial impact on the column diameter. Fig. 4.38C displays the diameter of the column as a function of steel yield stress. As can be seen from the figure, the diameter of the column shows a slight reduction with an increase in the steel yield strength. According to the results, concrete compressive strength should be increased rather than D ratio and steel yield strength to lower column size. For ductility, however, t a Dt ratio of less than 30 is required. In order to promote ductility, highstrength steel can be employed as a confining pressure increase. 4.3.1.9 Effects of local buckling A nonlinear FE analysis was performed to assess the effects of the local buckling on the behavior of the square CFDST stub column. The geometric and material properties of the examined specimen are given in Table 4.14. Fig. 4.39 shows the influence of the local buckling on the axial TABLE 4.14 Details of the tested specimens for investigating the effects of the c. Name

Bo 3 to (mm)

di 3 ti (mm)

fy (MPa)

fc0 (MPa)

Nu (kN)

DI

With local buckling

220  8

102  4

350

40

4035

20.1

4453

22.3

Without local buckling

411

4.3 Investigating the behavior of composite members 5000

Axial load (kN)

4000

3000

2000

With local buckling Without local buckling

1000

0 0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Axial strain

FIGURE 4.39 Effects of the local buckling on the axial loadestrain curve of the CFDST stub column.

loadestrain curve of the column. It can be observed from the figure that local buckling has no apparent effect during the initial loading stage. The stiffness of the column begins to deteriorate when the axial load approaches 2200 kN due to the progressive local buckling of the steel tube. Ignoring local buckling leads to a 10.35% overestimation of the ultimate axial strength of the column. Besides, the ductility of the column in which the local buckling is considered is lower than the counterpart without the effect of local buckling, as shown in Table 4.14 and Fig. 4.39. The axial loademoment interaction curves of the column are shown in Fig. 4.40. As shown in Fig. 4.40, the effect of local buckling is more prominent on the ultimate axial strength of the column than its bending 1.2

1

Pu / Po

0.8

0.6

0.4

With local buckling Without local buckling

0.2

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

Mu / Mo

FIGURE 4.40 Effects of the local buckling on the axial load-moment interaction curve of the CFDST stub column.

412

4. Numerical methods

moment capacity. If the local buckling is neglected in the analysis, the pure ultimate moment capacity is overestimated by only 3.33% which is remarkably less than 10.35% overestimation in the ultimate axial strength of the column. 4.3.1.10 Effects of section shapes This section discusses the impact of cross-section shape on composite member performance with the same perimeter, length, steel yield strength, and concrete compressive strength. To this end, the behavior of circular, square, hexagonal, and octagonal CFST stub columns under axial compression is compared. The geometric and material properties of the tested specimens are given in Table 4.15. The axial loadestrain responses of examined CFSTs, along with the loads carried by the steel tube and the concrete core, are illustrated in Fig. 4.41. Four characteristic points are defined on the curves, as shown in Fig. 4.41. The steel proportional limit, beyond which the steel tube enters into the elasticeplastic behavior, is denoted by point A. Yielding or buckling of the steel tube occurs at Point B. The ultimate axial strength is reached at Point C, and Point D is TABLE 4.15 Details of the examined CFST stub columns [43]. Sectional shape

P (mm)

t (mm)

H (mm)

fsy (MPa)

fc0 (MPa)

Nu;FE (kN)

1256.64

6.00

1200.00

345.0

51.0

1661.9

345.0

51.0

995.8

345.0

51.0

4730.3

345.0

51.0

930.6

Circular

Square

Hexagonal 1256.64

Octagonal

6.00

1200.00

413

4.3 Investigating the behavior of composite members = 5937

C B

8000

D

Concrete core

A 6000 ,

4000

8000

CFST

= 2010 Steel tube

2000

= 3035 B

Axial force (kN)

Axial force (kN)

10000

C

6000 A 4000 Steel tube 2000 ,

Concrete core

= 2380

0

0 0

0.01

0.02 0.03 Axial strain

0

0.04

0.01

0.02 0.03 Axial strain

(a) = 4447

C B

8000

D

CFST

A 6000

Concrete core

4000 Steel tube 2000

,

0.04

(b) 10000

Axial force (kN)

10000

Axial force (kN)

CFST

D

= 5085

C

8000 B A 6000

Concrete core

4000 Steel tube

2000

= 2071

CFST

D

,

= 2194

0

0 0

0.01

0.02 0.03 Axial strain

(c)

0.04

0

0.01

0.02 0.03 Axial strain

0.04

(d)

FIGURE 4.41

Axial loadestrain responses of CFSTs, concrete cores and steel tubes [43]. (A) Circular (B) Square (C) Hexagonal (D) Octagonal.

described as ε ¼ 0:02. The longitudinal stress distribution over the crosssection of in-filled concrete (sc;l ) at their mid-height at each characteristic point is shown in Fig. 4.42. As shown in Fig. 4.41, the axial loadestrain curves can be divided into four phases, as follows: 1. Step O-A: Elastic performance. During this phase, CFST columns in any shape have the same performance, and the steel tubes and the concrete cores sustain the axial load independently. Except for the square CFST, the longitudinal stress distributions over the concrete cores’ cross-sections of other shapes are uniform, as depicted in Fig. 4.42. However, due to the negligible difference between the stress values over the square concrete core cross-section, the stress distribution in the square CFST column can be considered uniform. 2. Step A-B: Step A-B: Elasticeplastic performance. By reaching Point A, columns begin to display inelastic behavior. Due to the formation of cracks in the concrete core, the stiffness of the curves is lower than in the previous stage. The longitudinal stress distribution across the circular concrete core is still uniform, as seen in Fig. 4.42. Polygonal CFST concrete cores, on the other hand, have nonuniform longitudinal stress distributions, with stress concentrations appearing at corners. This proves that the circular steel tube can uniformly confine the concrete core, whereas the largest

414

4. Numerical methods

(1) Point A

(2) Point B

(3) Point C

(4) Point D

(a)

(1) Point A

(2) Point B

(3) Point C

(4) Point D

(b)

(1) Point A

(2) Point B

(3) Point C

(4) Point D

(c)

(1) Point A

(2) Point B

(3) Point C

(4) Point D

(d)

FIGURE 4.42 Longitudinal stress distributions of infilled concrete at four characteristic points (Unit: Pa) [43]. (A) Circular. (B) Square. (C) Hexagonal. (D) Octagonal.

longitudinal stress over the perimeter of the concrete core confined by polygonal steel tubes occurs at corners. At Point B, steel tubes reach their maximum load-bearing capacities. It can be observed from Fig. 4.41 that the values of strain corresponding to Point B for circular, square, hexagonal, and octagonal specimens are 2010, 2380, 2071, and 2194 mε, respectively. Comparing the axial loadestrain responses of steel tubes and concrete cores indicates that the tubes attain their maximum capacities earlier than the concrete cores. 3. Step BeC: Plastic performance. During this stage, the concrete cores’ longitudinal stresses keep increasing. The longitudinal stresses in the concrete core when columns reach the peak Point C exceed the concrete compressive strength fc0 but with varying degrees, as shown in Fig. 4.42. The comparison of results shows that the increase in longitudinal stresses over the cross-sections of circular, octagonal, and hexagonal concrete cores is more noticeable than that of the

4.3 Investigating the behavior of composite members

415

square concrete core. In the case of polygonal CFSTs, the magnitude of longitudinal stresses around the flat parts of the steel tubes is less than corners. However, hexagonal and octagonal steel tubes can confine the concrete core so that longitudinal stresses in the vicinity of flat sections surpass the concrete cylinder strength. As shown in Fig. 4.42, the magnitudes of sc;l around the flat parts of hexagonal and octagonal steel tubes are 7.45% and 10% higher than the concrete cylinder strength, respectively. By contrast, the amount of sc;l adjacent to the flat areas of the square steel tube is less than the cylinder strength of the concrete. The difference between the longitudinal axial stresses of the concrete cores is due to the difference in susceptibility of plat parts of the steel tubes to local buckling and inner angles of steel tubes. The square, hexagonal, and octagonal steel tubes have widths of 314, 209, and 157 mm, respectively. As a result, the square steel tube is more prone to local buckling than hexagonal and octagonal tubes. Fig. 4.43 shows the cross-section of a polygonal CFST column. As shown in Fig. 4.43, the concrete core can be split into enhanced and nonenhanced sections in which the boundary between these two sections has a parabola shape. The internal angle between the intersection of the tube and the tangent of the boundary is 45 , as shown in Fig. 4.43. The unconfined parabola cross-section area is Auncon ¼ 2 3s2 tan2 W, and the concrete core area is Ac ¼ ns2 tan a where the term n denotes the number of sides, a ¼ pn, and s is the radius of the polygonal section. For assessing the ability of polygonal steel tubes in confining the concrete core, polygon confinement effectiveness coefficient kn is defined as follows: =

kn ¼

FIGURE 4.43

Aeff ¼ Ac

Ac


n P i¼1

Ac

Auncon (4.105)

Different zones of concrete core in a polygonal CFST column [43].

416

4. Numerical methods

kn for the examined square, hexagonal, and octagonal CFSTs are 0.33, 0.57, and 0.71, respectively. Besides, the area of each unconfined parabolas in square, hexagonal, and octagonal sections are 0:1675, 0:072, and 0:036 Ac , respectively. It can be seen that octagonal and hexagonal CFST columns have smaller unconfined concrete areas than the square CFST column, respectively. The strains corresponding to peak point for circular, square, hexagonal, and octagonal columns are 5937, 3035, 4477, and 5085 mε, respectively. The circular cross-section shows the largest εcu compared with other specimens. Besides, as the number of flat sides of the steel tube increases, so does the value of εcu . This shows that hexagonal and octagonal steel tubes show better local stability than the square steel tube, resulting in better ductility. 4. Step C-D: Plastic behavior. Due to the buckling of steel tubes, all columns show strain-softening behavior at this point. However, the square column loses strength at a faster pace than the others. It can be observed in Figs. 4.41A,C, and D that the hardening behavior of circular, hexagonal, and octagonal steel tubes compensates for the strain-softening behavior of columns and improves the columns’ strength. This characteristic, however, is not evident in the square CFST column because of the weaker buckling strength and confinement effect. The steel tubes’ out-of-plane deformations along their longitudinal middle line corresponding to the strain when the axial load is dropped to 90% of the ultimate axial strength in the descending stage are shown in Fig. 4.44. The figure illustrates that the largest out-of-plane deformation occurs in the square steel tube, whereas the least deformation belongs to the circular steel tube. Compared with the square tube, octagonal and hexagonal tubes display smaller deformations. The difference in the out-of-plane deformations of polygonal steel tubes can be explained by the difference in their buckling strength. The joints of flat plates in

Outward deflection (mm)

7

Circular Square Hexagonal Octagonal

6 5 4 3 2 1 0 0

0.2

0.4

0.6

0.8

Normalized length FIGURE 4.44

Out-of-plane deformation of steel tubes [43].

1

417

4.3 Investigating the behavior of composite members

polygonal tubes perform as support. Compared to hexagonal and octagonal tubes, the square steel tube has a larger flat area and hence has a larger unsupported area. Accordingly, the square steel tube’s buckling strength is lower than the hexagonal and octagonal tubes. The nonuniform longitudinal stress distribution over the concrete cores cross-sections of polygonal CFSTs is evident at Point D, as shown in Fig. 4.42. Compared with the previous stage, the magnitudes of sc;l are smaller at point D. The square column has the most significant reduction in the sc;l , whereas the circular column exhibits the slightest reduction in longitudinal stress at this stage. Between polygonal CFSTs, octagonal and hexagonal concrete cores show lower longitudinal stress reduction. This observation is in line with the strain-softening behavior of columns and reinforces the fact that hexagonal and octagonal steel tubes can better confine the concrete core than the square steel tube. The ratios of the maximum longitudinal stress to the average amount are used herein to compare the rate of the longitudinal stress sc;max s c;ave

of the unevenness of the longitudinal stresses over the concrete cross ratios against the corresponding axial strains ε sections. The sc;max s c;ave

are shown in Fig. 4.45. It can be observed from the figure that the square ratio during the loading CFST column shows the largest sc;max s c;ave

procedure, whereas the circular CFST column has the lowest. The ratio is nearly equal to unity for the circular column. sc;max s c;ave

Compared with the square CFST column, hexagonal and octagonal have ratios during the loading history. This proves that the smaller sc;max s c;ave

provided confinement effect by the hexagonal and octagonal steel tubes is more uniform than that of the square steel tube. At Point D, the 4

Square

σc, max / σc, ave

3

Hexagonal Octagonal

2

l Circular 1 0 0

0.005

0.01

0.015

0.02

Axial strain FIGURE 4.45 columns [43].

Comparison of the sc;max s
ε histories between different CFST stub c;ave

418

4. Numerical methods

25

10

D

8

Point 1

20

Points 1-4

6

p (MPa)

p (MPa)

D C

4

15 C

10

Point 2 B

2

5 A

0

B

0

0.01

0.02 Axial strain

0.03

Points 3 & 4

A

0

0

0.04

0.01

(a)

0.02 Axial strain

0.03

(b)

40

30

p ( MPa)

p (MPa)

20 C

Point 2

10 A 0 0

Point 3 0.01

0.02 Axial strain

(c)

D

20 C 10

Point 4 B

Point 1

Point 1

D

30

0.04

Point 2 A

0 0.03

0.04

0

Point 4

B 0.01

0.02

Point 3 0.03

0.04

Axial strain

(d)

Confining stress p
axial strain ε histories of CFST stub columns [43]. (A) Circular (B) Square (C) Hexagonal (D) Octagonal.

FIGURE 4.46

sc;max s ratios for the circular, square, hexagonal, and octagonal CFSTs c;ave are found to be 1.15, 3.8, 2.42, and 1.78, respectively. The interaction stresses p are investigated from the magnitude and distribution perspectives to further study the confinement mechanism in composite columns of diverse forms. The interaction stresses are collected from the mid-height of the columns at various positions of the crosssections, as illustrated in Fig. 4.46. The interaction stresseaxial strain histories of CFSTs are shown in Fig. 4.46. It can be observed from the figure that there is no trace of interaction between the steel tube and the concrete core during the initial loading stage. The reason is that the Poisson’s ratio of steel tube during the elastic phase is ws ¼ 0:3 which is larger than the concrete core wc ¼ 0:15w0:2. As a result, the lateral expansion of the tube exceeds that of the concrete core. By increasing the axial loading, the concrete core cracks, causing the volume of the concrete to surpass the steel tube. This results in contact between the concrete core and the steel tube. The described behavior applies to composite columns of any cross-sectional shape. The amplitude of the stress p along the perimeter of the circular concrete core is uniform, as shown in Fig. 4.46, indicating that the confinement effect generated by the circular tube is uniform. In contrast, the largest confining stress occurs at corner point 1 in polygonal CFST columns.

419

4.3 Investigating the behavior of composite members

35

Strain at peak load (Square, Hexagonal, and Octagonal)

Hexagonal

30

Octagonal

p1 (MPa)

25 Square

20 15

Strain at peak load (Circular)

10

Circular

5 0 0

FIGURE 4.47

0.01

0.02 Axial strain

0.03

0.04

Comparison between p1
ε responses of CFST stub columns [43].

Fig. 4.47 shows the p1
ε histories of CFSTs in which p1 is the confining stress at point 1. As illustrated in Fig. 4.47, p1
ε responses have different trends because of the difference in the cross-section shape. p1
ε curves of polygonal CFST columns show almost the same initial stiffness, whereas the circular CFST column has a smaller initial stiffness. The magnitudes of p1 corresponding to the ultimate axial strain in polygonal CFST columns are almost the same and are more significant than the corresponding confining stress in the circular CFST column. The rate of growth in the p1
ε curve of the circular CFST column is lower when compared to polygonal CFSTs. Furthermore, the circular column has a more consistent p1
ε response than polygonal columns. Polygonal CFST columns present approximately similar p1
ε curves before reaching the ultimate axial strain. From then onward, their p1
ε curves begin to deviate. This disparity in the p1
ε response of CFSTs is caused by the difference in the buckling resistance of steel tubes. In general, the composite mechanism of hexagonal and octagonal CFST columns is nearly identical to that of the square CFST column.

4.3.2 Mechanism of interaction of steel tubes and concrete 4.3.2.1 Development of interaction between steel tubes and the concrete core The preceding section examined the confinement impact of CFST stub columns under axial compression, as well as the effect of section shape on the confining pressure. To further understand the confinement mechanism of composite columns, the confinement effect generated by the outer square steel tube and the inner circular steel tube in a CFDST stub column is investigated in this section. The considered CFDST stub column has a square outer steel tube with a dimension of 220  5 mm and a circular inner steel tube with a dimension of 90  6 mm. The steel tubes with a yield strength of 350 MPa are filled with concrete with a compressive

420

4. Numerical methods

FIGURE 4.48 The examined interaction points. Ayough P, Ramli Sulong NH, Ibrahim Z, Hsiao P-C. Nonlinear analysis of square concrete-filled double-skin steel tubular columns under axial compression. Eng Struct 2020;216:110678. 4000

B A

3000

Axial force (kN)

C D

2000

1000

0 0

5000

10000

15000

20000

25000

30000

Axial strain ε (με)

FIGURE 4.49 Axial loadestrain history of the examined CFDST stub column. Ayough P, Ramli Sulong NH, Ibrahim Z, Hsiao P-C. Nonlinear analysis of square concrete-filled double-skin steel tubular columns under axial compression. Eng Struct 2020;216:110678.

strength of 40 MPa. First, the interaction stresselongitudinal strain responses across the perimeter of the sandwiched concrete are investigated. The considered positions are illustrated in Fig. 4.48, where the parameter x specifies the distance of the studied point from the cornet of the section. Later, the influences of material and geometric properties on the confining pressure are evaluated. Fig. 4.49 shows the axial loadestrain curve of the examined CFDST stub column. The p
ε responses corresponding to the investigated points are represented in Fig. 4.50, where po denote the confining stress between the sandwiched concrete and the outer tube, and pi is the confining stress between the sandwiched concrete and the inner tube. The longitudinal stress (sl ) and transverse stress (sq ) development in the outer steel tube during the loading history provides valuable information about the confinement mechanism. Fig. 4.51 depicts the sl
ε curves and st
ε curves at various points along the circumference of the outer steel tube.

421

4.3 Investigating the behavior of composite members 70

6 5

B

C

40

pi (MPa)

A

50

po (MPa)

Point 5

Point 1

60

Point 2

30 20

3

Point 8

2

Point 7

Point 6

Point 3&4

10

4

1

0

0 0

5000

10000

15000

20000

0

5000

Axial strain ε (με) (a)

10000

15000

20000

Axial strain ε (με) (b)

FIGURE 4.50 Interaction stress over the cross-section of the examined CFDST stub col umn. (A) Interaction stress between the outer steel tube and the in-filled concrete (po . (B) Interaction stress between the inner steel tube and the in-filled concrete (pi ). Ayough P, Ramli Sulong NH, Ibrahim Z, Hsiao P-C. Nonlinear analysis of square concrete-filled double-skin steel tubular columns under axial compression. Eng Struct 2020;216:110678.

As shown in Fig. 4.50, the magnitude of confining stresses po and pi during the initial loading stage is zero. The outer steel tube starts to induce confining pressure po on the sandwiched concrete by reaching Point A. However, no interaction occurs between the inner tube and the sandwiched concrete before reaching Point C, as shown in Fig. 4.50B. The confining pressure of the inner steel tube is noticeably lower than that of the outer steel tube. This demonstrates that the outer tube has a far more critical function in confining the in-filled concrete than the inner tube. It can be observed from Fig. 4.50A that the maximum po is developed at point 1 with x ¼ 0 and the confining stress po is reduced by increasing the distance x. The middle part of the steel tube with x > B6o cannot induce any confining pressure on the sandwiched concrete. Contrary to the square outer tube, the confining pressure provided by the inner circular steel tube is almost uniform, as depicted in Fig. 4.50B. The multiaxial stress state of the outer tube is shown in Fig. 4.51. As shown in Fig. 4.51A, the outer tube is under axial stress sl , hoop stress sq , and confining stress po . The histories of sq , sl , and the corresponding von Mises stress sMises are given in Fig. 4.51B. To better understand the development of stresses on the tube during the elastic and elasticeplastic stages, the von Mises yield criterion is also depicted in Fig. 4.51C, in which the compression side is considered as positive and vice versa. Assuming that the steel tube exhibits fully plastic behavior, the yield stress of the tube subjected to the plane stress can be stated using the Mises criterion as follows: ss ¼ s2l
sl sq þ s2q
s2y

(4.106)

422

4. Numerical methods

(a) Constitutive behaviour 400

Point 1

Point 2

Point 3

Steel stress σs (MPa)

B

1

300

= 9500

2

= 17500

A

200

100

0 0

10000

20000

30000

40000

Steel strain ε (με) (b) / 1.5 Von Mises yield criterion

B (0.39,0.82)

1

Outer square steel tube subjected to axial loading

0.5

-1.5

-1 -1

-0.5 -0.5

=√

A

[(

−

)2 + 2

2

+

2]

O

0 0

0.5 0.5

1

1

1.5/

-0.5 -1

(c)

FIGURE 4.51 Composite action of the examined CFDST stub column. (A) Stress state of the outer square steel tube. (B) Stressestrain responses over different points of the outer square steel tube. (C) Biaxial stress condition of the outer steel tube. Ayough P, Ramli Sulong NH, Ibrahim Z, Hsiao P-C. Nonlinear analysis of square concrete-filled double-skin steel tubular columns under axial compression. Eng Struct 2020;216:110678.

4.3 Investigating the behavior of composite members

423

When the steel material has an elastic behavior, stresses sq and sl develop linearly, and the steel stress is ss < 0. By entering the stresses sq and sl into the plastic stage, the steel stress is ss ¼ 0. The following expression can be used to calculate the increase of stresses and the corresponding strain: 8 2 1 30 > 1 w dε > s l E > s > 4 A 5@ > if ss < 0 > ! > 1
w2s ws 1 > dε < q dsl ¼ 8 2 3 90 1 3 2 > 2 dsq > = dεl < E 1 w S S S s > q l l 1 > s > 4 5 @ A if ss ¼ 0 5
4 > > > ; dε S 2 : :1
w2s ws 1 Sl Sq Sq q (4.107) in which Sl ¼

Es ðs 1
w2s l

þws sq Þ, Sq ¼

Es ðs 1
w2s q

þws sl Þ, and S ¼ sl Sl þ sq Sq , Es

is the Young’s modulus of steel tube, and ws is the steel Poisson’s ratio. sq and sl represent the stress deviators of circumferential and longitudinal directions, respectively. As shown in Fig. 4.51B and C, during the elastic behavior, the steel tube is subjected to the uniaxial stress condition, and no hoop stress sq is developed in the tube ðsq ¼ 0Þ. At Point A, the magnitude of longitudinal stress is sl;A ¼ 0:55syo and the corresponding strain is εA ¼ 950 mε. By reaching Point A, the sandwiched concrete starts to crack and expand laterally. Therefore, the tube and the concrete interaction happens, inducing the hoop stress sq on the steel tube. As depicted in Fig. 4.51B, during the elasticeplastic stage and owing to the von Mises yield criterion, the stiffness of the sl
ε response is gradually decreased until reaching Point B, where the steel tube yields. The stress and strain corresponding to the yield point are sMises ¼ 350 MPa and εB ¼ 2286 mε, respectively. At the yielding point B, sl and st are 0.82syo and 0.39syo , respectively, as depicted in Fig. 4.51B and C. By reaching the yield point, sl shows strain-softening behavior. The rate of reduction in the stress sl increases with increasing the distance x. This is because the middle part of the steel tube (point 3) is more vulnerable to local buckling. The stiffness of the sq
ε curve at point 1 when the column has an elasticeplastic behavior is more than points 2 and 3. Furthermore, the hoop stress sq at point 1 is higher than that at points 2 and 3 during the loading history. This observation shows that the steel tube corners can efficiently confine the concrete core, and the confinement effect reduces with increasing the distance x. The strains corresponding to the intersection points of the sl
ε curve with the sq
ε curve at point 1 and 2 are εint:1 z 9500 mε and εint:2 z 17500 mε, respectively. Contrary to points 1 and 2, the sl
ε curve and the sq
ε curve corresponding to point 3 do not intersect, as shown in Fig. 4.51B. Assessing the intensity of the confinement effect and the development of sq
ε and sl
ε curves at different points shows that the confinement effect is more efficient at points where the related sq
ε

424

4. Numerical methods

FIGURE 4.52 Longitudinal stresses distribution over the section of CFDST sandwiched concrete [54].

and sl
ε curves intersect [46,53]. As depicted in Fig. 4.51B, εint:1 z 9500 mε < εint:2 z 17500 mε, therefore po;1 > po;2 . Besides, no intersection occurs between sq
ε and sl
ε curves at point 3, and the magnitude of po;3 is less significant than points 1 and 2. As shown in the preceding discussion, the sandwiched concrete is predominantly constrained by the outer steel tube than the inner steel tube. However, if the hollow ratio c is large, the conclusion that the outer steel tube plays a more prominent role in confining the sandwiched concrete may not be correct. The cross-sectional stress distribution of sandwiched concrete of a CFDST column with a large hollow ratio of c ¼ 0:82 and a height of 1465 mm during the loading history is presented in Fig. 4.52 in which point A denotes the steel proportional limit, point B denotes the ultimate axial strength, and point C denotes to the time when the load reduces to 85% of the ultimate axial strength at the descending stage of the curve. The Dtoo ratio and dtii ratio are 113.33 and 109.6, respectively, and the yield strength of outer and inner tubes are 326.67 MPa and 336.67 MPa, respectively, and the concrete compressive strength is 37.65 MPa. As can be seen, before attaining ultimate axial strength, the concrete stress distribution is not uniform, and the stress magnitude around the inner tube is greater than that of the outer tube. By reaching the ultimate axial strength (point B), the concrete longitudinal stress around the inner steel tube exceeds fc0 , whereas the stress values near the outer tube are smaller than fc0 . This shows that the confinement effect provided by the outer steel tube in CFDST columns with large hollow ratios is limited. However, because the inner tube’s dtii ratio is smaller than that of the outer tube, the inner tube’s out-of-plane stiffness and stability are greater than those of the outer tube. As a result, the inner tube can impose lateral pressure on the sandwiched concrete, enhancing the compressive stress of surrounding concrete [54]. 4.3.2.2 Influence of geometric and material properties on the confinement effect A parametric study is performed on the square CFDST stub column defined in the previous section to show the effects of material and geometric properties on the confinement effect. Fig. 4.53 shows the full

4.3 Investigating the behavior of composite members

425

FIGURE 4.53 Effects of geometric and material properties on the Po
ε response.

(A) Effect of fyo (B) Effect of fyi (C) Effect of Bo (D) Effect of to (E) Effect of c (F) Effect of Di =ti (G) Effect of fc0 . Ayough P, Ramli Sulong NH, Ibrahim Z, Hsiao P-C. Nonlinear analysis of square concrete-filled double-skin steel tubular columns under axial compression. Eng Struct 2020;216:110678.

426

4. Numerical methods

FIGURE 4.53 cont’d.

history of po
ε curves in which po is the average value of the confining pressure around the perimeter of the sandwiched concrete. As shown in Fig. 4.53A, the overall trend of the po
ε curve is not sensitive to the outer tube yield strength. However, the magnitude of the confining pressure po is increased with an increase in the outer tube strength. Contrary to the outer tube, the influence of the inner tube strength on the po
ε curve is little, as depicted in Fig. 4.53B. Fig. 4.53C and D illustrate the influences of the outer tube dimension on the po
ε curve. It can be observed from Fig. 4.53C that the confining pressure enhances with increasing the thickness of the outer tube to (reducing the Btoo ratio). On the other hand, increasing the width of the outer tube Bo declines the confining pressure (increasing the Btoo ratio), as shown in Fig. 4.53D. It is noteworthy that the overall trend of the curve becomes unpredictable as the slenderness of the outer tube increases. For instance, the po
ε curve corresponding to the Btoo ¼ 220 fluctuates widely after reaching the pick point, whereas the po
ε curve corresponding to the Btoo ¼ 44 shows almost a steady behavior, as depicted in Fig. 4.53C. This is due to the susceptibility of a thin steel tube to local buckling. The effects of the inner tube dimension on the po
ε curve are shown in Fig. 4.53E and F. As shown in Fig. 4.53E, the interaction pressure po reduces with increasing the hollow ratio c. This issue becomes evident in the column with c > 0:3. Compared with the width of the inner tube, the effect of the inner tube thickness ti on the po
ε curve is negligible, as depicted in Fig. 4.53F. Fig. 4.53G presents the effects of the concrete compressive strength fc0 on the po
ε curve. It can be observed from the figure that the strain associated with the onset of confining pressure increases with an increase in the concrete strength fc0 . This is because increasing fc0 enhances concrete

4.3 Investigating the behavior of composite members

427

stiffness. Therefore, the concrete’s lateral expansion with a higher fc0 is less than the one with a smaller fc0. The delay in the onset of the confining pressure development reduces the steel tube’s efficiency in confining the sandwiched concrete. Consequently, the confining pressure po is reduced by increasing the concrete strength fc0 . The effects of the concrete strength, dimension, and the yield strength of the outer tube on the confinement effect can be discussed mathematif

cally by the confinement coefficient x ¼ an fsyo0 . Increasing the outer tube ck

thickness or reducing its width increases the steel nominal ratio an , resulting in an increase in the confinement coefficient value. Similarly, increasing the outer steel tube yield stress fsyo enhances the magnitude of the confinement coefficient. By contrast, the confinement coefficient de0 . clines by increasing the concrete compressive strength fck

4.3.3 Tapered and inclined composite members The use of tapered or inclined composite members may be required due to aesthetic or unique architectural purposes. The inclined column can provide the audience with a unique space experience in spatial constructions or help to deal with the uneven architecture style found in highrise buildings. Furthermore, it may be utilized to create an efficient load transmission system in the structure. Tapered members with changing sections along the longitudinal direction might be employed to adapt to the architecture need. The structural behavior of tapered and inclined composite members under axial compression is discussed in this section. The tapered angle q can be expressed as follows:   Dob
Dot (4.108) q ¼ arctan 2H where Dob and Dot are the diameters of the outer tube at the bottom, and top sections, respectively, and H represents the height of the column. 4.3.3.1 Tapered composite members A nonlinear FE analysis was performed on the tapered CFDST stub column under axial compression to investigate the effect of tapered angle q on the behavior of the member. The behavior of the tapered composite member is compared to that of its straight CFDST counterpart to acquire a better understanding of its behavior. The geometric and material properties of the examined columns are given in Table 4.16. The relative stress distribution of the sandwiched concrete over column height when ultimate axial strength is achieved is presented in Fig. 4.54, where sc is the concrete stress, fc0 is the concrete cylinder strength, h is the height of the column, and S11 and S22 are the transverse and longitudinal stresses, respectively. It should be noted that the negative values denote that the section is under compression. As shown in Fig. 4.54, S11 is under

428

Do 3 to (mm) 

di 3 ti (mm)

Top

Bottom

Top

Bottom

fyo (MPa)

fyi (MPa)

fc0 (MPa)

Nu (kN)

0

350  3:8

350  3:8

231  2:9

231  2:9

439

397

42

5298

1.14

308  3:8

350  3:8

189  2:9

231  2:9

Name

q

Straight Tapered

4410

4. Numerical methods

Geometric and material properties of examined tapered and straight CFDST columns.

TABLE 4.16

429

4.3 Investigating the behavior of composite members

1 0.9 h

0.8

h/H

0.7 0.6 O

0.5 0.4 0.3 S22-Straight

0.2

S22-Tapered S11-Straight

0.1

S11-Tapered

0 -1.5

-1

-0.5

0

Relative stress σc /fc'

FIGURE 4.54

Longitudinal and transverse stress distributions over the height of the sandwiched concrete.

compression due to the confining pressure induced by the steel tube. The distribution of S22 over the straight column’s height is almost uniform, with an average value of 1:22fc0 . This demonstrates that the confinement effect of the circular steel tube enhances the concrete strength by 22%. Similarly, the distribution of S11 along the column’s height is nearly uniform, except at 0.2H adjacent to the column’s ends. Concerning the tapered CFDST column, S22 progressively increases over the column’s height, with the largest S22 occurring at the top part of the column with a magnitude of 1:33fc0 , whereas S22 at the bottom of the column is almost 1:12fc0 . The difference between the values of S22 at the top and bottom parts of tapered columns increases with an increase in the tapered angle q. It is noteworthy that the hollow ratio c has little impact on the concrete stress distribution along the height axis of the tapered CFDST column. Similar to the longitudinal stress, the maximum S11 occurs at the top section of the column, and it is reduced by increasing the section of the column. Hence, it can be concluded that the confinement effect in the tapered composite member is not uniform and varies over the column height. The maximum confinement effect is developed at the top part of the tapered composite column. The relative stress distribution of the inner and outer steel tubes over column height when ultimate axial strength is achieved is presented in Fig. 4.55, where ss is the steel stress, fy is the steel tube yield stress, h is the height of the column, and S11 and S22 are the transverse and longitudinal stresses, respectively. As shown in Fig. 4.55A, the magnitude of S22 over

430

4. Numerical methods

1 0.9 0.8

h

h/H

0.7 0.6 O

0.5 0.4 S22-Straight

0.3

S22-Tapered S11-Straight

0.2

S11-Tapered

0.1

0 -1.5

-1

-0.5

0

0.5

0

0.5

Relative stress σsi /fyi

(a) 1 0.9 0.8

h

h/H

0.7 0.6 O

0.5 0.4 0.3

S22-Straight S22-Tapered

0.2

S11-Straight

0.1

S11-Tapered

0 -1.5

-1

-0.5

Relative stress σso /fyo

(b)

FIGURE 4.55 Longitudinal and transverse stress distributions over the height of the inner and outer steel tubes. (A) Inner steel tube. (B) Outer steel tube.

4.3 Investigating the behavior of composite members

431

the tapered CFDST column height increases by increasing the column height, and the maximum S22 occurs at the top section of the column where the steel tube has the least cross-sectional area. S11 of the tapered column’s inner steel tube is negative (compression), indicating that the inner tube restricts the lateral expansion of the sandwiched concrete. As illustrated in Fig. 4.55B, the developed S11 in the outer steel tube is under tension for both straight and tapered CFDST columns due to restricting the lateral expansion of the sandwiched concrete. Concerning the straight CFDST column, the distribution of S11 over the column height is uniform. By contrast, for the tapered CFDST column, S11 progressively increases as the column height increases. This again reinforces the fact that the distribution of confinement effect over the height of the tapered composite column is not uniform, and the maximum confinement effect occurs at the top part of the column where the cross-sectional area is the least. It is noteworthy that the hollow ratio c has a negligible effect on the stress distribution of steel tubes along the longitudinal axis. Fig. 4.56 shows the development of the longitudinal strain of the outer steel tubes in two circular tapered CFDST columns as well as their straight CFDST counterpart at their ultimate axial strengths [54]. The dimensions of the outer and inner steel tubes at the bottom section are Do  to ¼ 398  3 mm and di  ti ¼ 332  2:5 mm, respectively. The yield strengths of outer and inner steel tubes are 327 and 337 MPa, respectively, and the sandwiched concrete has a compressive strength of 37.65 MPa. The first column has a tapered angle of q ¼ 1:13 , whereas the tapered angle of the second column is q ¼ 1:92 . It can be observed from the figure that the longitudinal strain at the top areas of the tapered column is more significant than those at the lower areas. The plastic strain is mainly

FIGURE 4.56 The longitudinal plastic strain development of the outer tube of the tapered CFDST column [54]. (A) q ¼ 1:13 (B) q ¼ 1:92 (C) q ¼ 0 .

432

4. Numerical methods

concentrated in the top section of the column, whereas the steel tube in the bottom half is mostly elastic. Local buckling is more likely to occur at the part around the top with higher plastic strain, which corresponds to the failure position of the outer tube in the experiments (see Chapter 2). In contrast to tapered columns, the plasticity of the straight column develops symmetrically over the whole column height except for the boundary section, and the strain gradient is less than that of tapered columns. The same trend can be expected for the plasticity development over the height of inner steel tubes. f

The confinement coefficient x ¼ an fsyo0 can be used to evaluate the ck

confinement effect in the tapered composite column. The cross-sectional areas of the steel tube As and the in-filled concrete Ace of the tapered composite member are reduced with the increase of the column height. However, the reduction in the Ace outweighs the decrease in the As . Therefore, the steel nominal ratio an ¼ AAces increases over the column height. The schematic view of the confinement coefficient distribution over the column height is depicted in Fig. 4.57. The figure illustrates that the confinement coefficient along the height of the straight column remains constant. By contrast, the confinement coefficient’s distribution over the longitudinal direction of the tapered column is not uniform and increases with the increase in the column height, indicating that the top section of the tapered steel tube with a smaller area provides a more substantial confinement effect to the sandwiched concrete. The distribution of confining stress over the columns’ height at the ultimate axial strength is shown in Fig. 4.58, in which p1 and p2 are the confining stresses induced on the in-filled concrete by the outer and inner steel tubes, respectively. It can be observed from the figure that the H Straight column

Tapered column

O0

FIGURE 4.57 Schematic view of the distribution of confinement coefficient x over the column height.

433

4.3 Investigating the behavior of composite members

1 0.9 0.8

h/H

0.7 0.6 0.5 0.4 Straight-P1

0.3

Tapered-P1

0.2

Straight-P2

0.1

Tapered-P2

0 0

1

2

3

4

5

Stress (MPa) FIGURE 4.58

Confining pressure between the steel tubes and the sandwiched concrete.

confining stress along the longitudinal direction of the straight column is almost uniform. By contrast, the magnitude of the confining pressure in the tapered CFDST column increases with increasing the column height. This is because the confinement effect is improved by reducing the column section. The schematic view of the confining pressure in the straight and tapered composite columns under axial compression is illustrated in Fig. 4.59 based on the results provided in Fig. 4.58.

FIGURE 4.59 Schematic view of the confining pressure in the straight and tapered CFDST columns under axial compression.

434

4. Numerical methods

4.3.3.2 Inclined composite members Contrary to the tapered composite members, the section of the inclined composite members is non-sway along the longitudinal axis. Assuming fixed boundary conditions for an inclined composite column, combined forces, i.e., bending, compression, and shear, are developed at column ends, as shown in Fig. 4.60. Because of the bending moment, the confinement effect across the column height is nonuniform and differs from that of a straight composite column under axial compression. However, the nonuniform confinement effect in inclined composite members under axial compression has not been studied until now, and there are no theoretical models for considering the inclined angle reduction influence on the confinement effect. The induced vertical load Nv can be decomposed into the longitudinal and horizontal Nh forces, as depicted in Fig. 4.60C. A nonlinear FE analysis was performed on the inclined CFDST stub column under axial compression to investigate the effect of inclined angle q on the behavior of the member. The behavior of the tapered composite column is compared to that of its straight CFDST counterpart to better understand its behavior. The geometric and material properties of the examined columns are presented in Table 4.17. The vertical loade displacement curves of the examined columns are shown in Fig. 4.61. It can be observed from the figure that the inclined angle has a moderate effect on the overall shape of the curve. The comparison of the ultimate axial strengths in Table 4.17 shows that, while the compressive strength of the inclined column is less than that of the equivalent with the vertical longitudinal axis, the difference is negligible.

=0 =0

=0 =0 =0

(a)

(b)

(c)

FIGURE 4.60 Free body diagram of an inclined column. (A) Boundary condition (B) Supports reactions (C) Decomposition of the vertical force.

435

4.3 Investigating the behavior of composite members

TABLE 4.17 Geometric and material properties of examined CFDST columns. Name

q

Do 3 t o (mm)

di 3 ti (mm)

fyo (MPa)

fyi (MPa)

fc0 (MPa)

Nu (kN)

Straight

0

220  3:6

159  3:7

439

397

51.4

2535

Inclined

4

2463

9

2317



3000

Axial force (kN)

2500 2000 1500 1000

Strainght Inclined-θ=4°

500

Inclined-θ=9° 0 0

5

10

15

20

25

30

Axial displacement (mm)

FIGURE 4.61 Axial loadedisplacement curves of straight and inclined CFDST stub columns.

4.3.3.3 Effect of the tapered and inclined angles on the ultimate axial capacity of composite members The FE modeling program was employed to assess the effects of the taped and inclined angles on the behavior of composite members under axial compression. The geometric and material properties of the examined tapered and inclined members collected from the performed experimental works are given in Table 4.18 and Table 4.19, respectively. The straight counterpart column of each specimen has the vertical longitudinal axis with a cross-section similar to the bottom section of the tapered column. Fig. 4.62 shows the influence of the in-filled concrete on increasing the ultimate axial strength of members by comparing the compressive strength of the columns Ncomposite with the axial strength of the corresponding hollow steel sections NHollow . As shown in Fig. 4.62, the

Geometric and material properties of examined tapered composite columns. Outer tube D0 3 B0 3 t0 ðmm 3 mm 3 mmÞ

References

Section

q

[55]

SHS-CHS

CHSeCHS

[55]

Round-endrectangular

fsyi (MPa)

fc0 (MPa)

ð106
159Þ ð106
159Þ  3․72

319.6

380.6

51.4

ð83
136  ð83
136Þ  3․72

ð106
159Þ  ð106
159Þ  3․72

319.6

380.6

51.4

ð250
350Þ  ð250
350Þ  3․82

ð150
231Þ ð150
231Þ  2․92

ð165
231Þ ð165
231Þ  2․92

439.3

396.5

52.5

215  135  3․62

240  160  3․62

ð117
161Þ  ð37
81Þ  3․72

ð142
186Þ  ð62
106Þ  3․72

319.6

380.6

51.4

215  135  3․62

240  160  3․62

ð117
161Þ  ð37
81Þ  3․72

ð142
186Þ  ð62
106Þ  3․72

319.6

380.6

52.5

Section B-B

Section A-A

Section B-B

1

197  197 3․62

220  220 3․22

ð83
136Þ  ð83
136Þ  3․72

1

197  197 3․62

220  220  3․22

0.57e1.14

ð235
350Þ  ð235
350  3․82

1

Elliptical

[57]

fsyo (MPa)

Section A-A

SHS

0e4

ð116
158Þ  ð116
158  3․75

200  200 3․75

410.1

51.4

CHS

0e4

ð116
200Þ ð116
200  3․75

200  200 3․75

410.1

51.4

4. Numerical methods

[56]



Inner tube D0 3 B0 3 t0 ðmm 3 mm 3 mmÞ

436

TABLE 4.18

Geometric and material properties of examined inclined composite columns.

References

Section

q

Outer tube D0 3 B0 3 t0 ðmm 3 mm 3 mmÞ

[55]

SHS-CHS

9

220  220  3․62

ð106
159Þ  ð106
159Þ  3․72

319.6

380.6

51.4

CHS-CHS

9

220  220  3․62

ð106
159Þ  ð106
159Þ  3․72

319.6

380.6

51.4

Round-endrectangular

9

240  160  3․62

ð142
186Þ  ð62
106Þ  3․72

319.6

380.6

51.4

Elliptical

9

240  160  3․62

ð142
186Þ  ð62
106Þ  3․72

319.6

380.6

52.5

SHS

9

200  200  3․75

[57]



Inner tube D0 3 B0 3 t0 ðmm 3 mm 3 mmÞ

fsyo (MPa)

fsyi (MPa)

fc0 (MPa)

410.1

51.4

4.3 Investigating the behavior of composite members

TABLE 4.19

437

438

4. Numerical methods

Circular

Square

Round-end rectangular

Elliptical

3

Ncomposite/Nhollow

2.5 2 1.5 1 0.5 0 Straight

Inclined

Tapered

FIGURE 4.62

Effect of the in-filled concrete on increasing the compressive strength of hollow steel sections.

Ncomposite NHollow ratios of the straight circular, square, round-ended rectangular, and elliptical members are 1.66, 2.44, 1.81, and 1.39, respec tively. The Ncomposite NHollow ratios of the inclined circular, square, roundended rectangular, and elliptical members are 1.68, 2.78, 1.95, and 1.62, respectively. The Ncomposite NHollow ratios of the tapered circular, square, round-ended rectangular, and elliptical members are 1.72, 2.32, 1.68, and 1.52, respectively. The comparison of the Ncomposite NHollow ratios reveals that the effect of in-filled concrete is the most pronounced on the ultimate axial strength of the square cross-section. The reason is the hollow square steel tube’s susceptibility to inward buckling that is restricted by the infilled concrete. The simultaneous effects of the hollow ratio c and the inclined and tapered angles on the ultimate axial strength of CFDST columns can be assessed using the strength index SI that is governed by the following: SI ¼

Nu Nu; straight

(4.109)

where Nu and Nu; straight are the ultimate axial strengths of tapered or inclined column and the straight counterpart, respectively. Fig. 4.63 shows the strength index SI against the hollow ratio c for straight, inclined, and tapered CFDST stub columns having different shapes. As demonstrated in Fig. 4.63A, increasing the hollow ratio from 0.5 to 0.75 reduces the strength index SI of the circular, square, round-ended

4.3 Investigating the behavior of composite members Circular

Square

Round-end rectangular

439

Elliptical

1.6 1.4 1.2

SI

1 0.8 0.6 0.4 0.2 0 χ=0.5

χ=0.75

(a) Circular

Square

Round-end rectangular

Elliptical

1.6 1.4 1.2

SI

1 0.8 0.6 0.4 0.2 0 χ=0.5

χ=0.75

(b) Circular

Square

Round-end rectangular

Elliptical

1.4 1.2 1

SI

0.8 0.6 0.4 0.2 0 χ=0.5

χ=0.75

(c)

FIGURE 4.63 Relationship of the strength index SI with the hollow ratio c. (A) Straight. (B) Inclined. (C) Tapered.

440

4. Numerical methods

rectangular, and elliptical straight columns by 22%, 13.5%, 18.1%, and 21%, respectively. As shown in Fig. 4.63B, increasing the hollow ratio from 0.5 to 0.75 reduces the strength index SI of the circular, square, roundended rectangular, and elliptical-inclined CFDST columns by 28.5%, 18.18%, 21.54%, and 20.6%, respectively. As shown in Fig. 4.63C, increasing the hollow ratio from 0.5 to 0.75 reduces the strength index SI of the circular, square, round-ended rectangular, and elliptical tapered CFDST columns by 19.5%, 15.42%, 22%, and 14.2%, respectively. The strength index versus the tapered or inclined angle is depicted in Fig. 4.64. The strength of the tapered stub column is determined by the

Circular

Square

Round-end rectangular

Elliptical

1.2 1

SI

0.8

0.6 0.4 0.2 0 0

1

2

3

4

5

Tapered angle

(a) Circular

Square

Round-end rectangular

Elliptical

1.1 1

SI

0.9 0.8

0.7 0.6 0.5 0

2

4

6

8

10

Inclined angle

(b)

FIGURE 4.64 Strength index SI against the tapered or inclined angle q. (A) Tapered column. (B) Inclined column.

4.3 Investigating the behavior of composite members

441

smallest section of the column where failure is most likely to occur. It can be observed from the figure that the strength index SI is reduced by increasing the tapered or inclined angle. However, the influence of the inclined angle on SI is relatively moderate. The effect of the tapered angle on the column capacity is due to the change in the column’s cross-section along its longitudinal axis and, therefore, the change in the concrete confinement effect over the height of the column. Contrary to the tapered column, the cross-section of the inclined column is constant along the longitudinal axis. However, the imposed compressive load is not parallel to the inclined longitudinal axis, as depicted in Fig. 4.60. The generated bending moment in the inclined column is increased by increasing the inclined angle, leading to the reduction in the confinement effect and the compressive strength of the member. It is noteworthy that the tapered and inclined angles have the most pronounced influence on the strength index SI of the square cross-section.

4.3.4 Slender columns Slender composite members are among the most popular structural members in high-rise structures for bearing vertical loads and lateral loads induced by wind and earthquakes [58]. As an approximate criterion, the columns with a slenderness ratio (L=r) greater than 22 are considered slender columns. More information on the definitions for the classification of columns in terms of stub, medium, and slender can be found in Chapter 2. As discussed in sections above and Chapter 2, the resistance of stub composite members is governed by the geometric and material properties of the section. The resistance of slender composite members, on the other hand, is often determined by their overall stability. Increasing the length of the composite member reduces its stability and, as a result, its bearing capacity. As a result, the stability effects must be considered in the nonlinear analysis and design of slender columns. The typical failure mode of a slender composite column with pin-ended boundary conditions is shown in Fig. 4.65. As illustrated in the figure, the column fails in a half-sine wave shape. Taking the deflection at the mid-height of the column as um , the deflection u over the column height at any point (y, z) is governed by the following: u ¼ um sin

pz L

(4.110)

The momentecurvature modeling approach is used to compute the internal section moment. An equation to anticipate the mid-height curvature as a function of deflection is required to calculate the internal

442

4. Numerical methods

Undeformed shape

FIGURE 4.65

Buckled shape

Typical failure mode of composite slender column [59,60].

moment. This equation can be determined numerically by conducting the double differentiation of Eq. (4.110) as 4¼

v 2 u p2 pz ¼ 2 um sin 2 L vz L

(4.111)

Eq. (4.111) is used to determine the mid-height column curvature as a function of mid-height deflection um and column length L as follows: 4m ¼ um

p2 L2

(4.112)

The column initial geometric imperfection uoy can be determined using the following equation: uoy ¼ uo sin

pz L

(4.113)

in which the term uo denotes the initial geometric imperfection at the midheight of the column. With this approach, loading eccentricity, initial member imperfection, and second-order effects on slender column nonlinear responses are all considered. The second-order effect amplifies the magnitude of the bending over the length of the column. The external bending moment at the mid-height of the column subjected to the axial compression P can be

4.3 Investigating the behavior of composite members

443

expressed using the following equation in which the effects of the load eccentricity e, geometric imperfection uo , and the second-order effect are considered. Mme ¼ Pðuo þ um þ eÞ

(4.114)

4.3.4.1 Failure modes The composite beam columns slenderness ratio is a critical geometric property that can remarkably influence the structural performance and failure mechanism of composite members. Fig. 4.66 shows the typical ultimate axial strength Pu against the slenderness ratio l curve for columns subjected to axial compression. The term slenderness ratio l is expressed as follows: Le l ¼ qffiffiffiffiffiffiffi I=A

(4.115)

where Le represents the effective length of the column, and I and A are the second moment of area and the cross-sectional area of the composite member, respectively. The curve is split into three failure stages: plastic, elasticeplastic, and elastic buckling. Therefore, three types of failures are possible based on the column slenderness ratio. For short columns with l < lp , members fail in plastic buckling failure mode. By contrast, long columns with l > lr fail in elastic buckling failure mode. If lp < l < lr , the column is classified as intermediate length, which typically experiences elasticeplastic buckling failure mode. The failure mode of the intermediate-length column is actually the interaction between the local

Plastic stage

Elastic-plastic stage

Elastic stage

FIGURE 4.66

Strengtheslenderness ratio relationship of columns.

444

4. Numerical methods

buckling and the overall buckling of the column. Design codes recommend the limits for lp and lr of columns, based on their cross-sectional shapes. More details regarding the values of lp and lr can be found in Chapter 5. For discussing the effect of the slenderness ratio l on the column failure modes, the nonlinear FE analysis was performed on a series of circular CFST columns with the dimension of D ¼ 400 mm and the tube thickness ranging from 6.78 to 20 mm. The steel yield stress is 520 MPa, and the concrete compressive strength is in the range of 25 and 100 MPa. More details of the examined columns are listed in Table 4.20, in which εlc is the mid-height longitudinal strain at the compression side of the column and εy is the yield strain of the steel tube. The strength versus slenderness ratios relationship for Group 1 specimens are given in Fig. 4.67, where the ultimate axial strength Pu is normalized to the plastic strength Ppl ¼ As fy þ Ac fc0 . The slenderness limits lr and lp are taken as 0:09 Efys and 0:19 Efys based on AISC360-16 (see Chapter 5). It can be observed from the figure that the ultimate axial strength of short columns with l < lp exceeds the corresponding plastic strength Ppl and the PPplu ratio reduces with increasing

the columns slenderness ratio l. The difference between the ultimate axial strength and the plastic strength in long members with a large slenderness ratio l is remarkable. The reason is that the susceptibility to overall buckling is increased with increasing the slenderness ratio l. Therefore, the confinement effect provided by the steel tube is declined with increasing the slenderness ratio l. Fig. 4.68 shows the εεlcy ratio versus the slenderness ratio l of Group 1 columns. The εεlcy ratio exceeds unity for columns with a PPplu ratio larger than

1, and vice versa. Therefore, the εεlcy ratio for short and intermediate-length

columns that fail by plastic buckling is greater than one, whereas, for long columns that fail by elasticeplastic buckling or elastic buckling, the ratio of εεlcy is less than one. 4.3.4.2 Loadestrain curve The axial force P versus longitudinal strain εl responses captured from the mid-height of columns CFST6, CFST8, and CFST17 are presented in Fig. 4.69, in which the compression side is considered negative and vice versa. It can be observed from Table 4.20 that the εεlcy ratios of columns CFST6, CFST8, and CFST17 are 1.29, 0.75, and 0.32, respectively. As shown in Fig. 4.69, the whole section of column CFST6 is under compression before reaching the ultimate axial strength, and the strain corresponding to the ultimate axial strength is higher than the steel yield strain. After the ultimate axial strength, part of the section is reversed from compression to

TABLE 4.20

Geometric and material properties of the examined CFST columns [59].

10

12,856

4.00

25

20

12,697

3.18

40

25

30

11,841

2.98

10

40

25

40

9785

1.80

400

10

40

25

50

7962

1.12

6000

400

10

40

25

60

6932

1.29

CFST7

7000

400

10

40

25

70

6012

0.86

CFST8

8000

400

10

40

25

80

5140

0.75

CFST9

9000

400

10

40

25

90

4677

0.57

CFST10

10,000

400

10

40

25

100

4270

0.48

CFST11

11,000

400

10

40

25

110

3847

0.43

CFST12

12,000

400

10

40

25

120

3318

0.33

CFST13

13,000

400

10

40

25

130

3050

0.37

CFST14

14,000

400

10

40

25

140

2729

0.35

CFST15

15,000

400

10

40

25

150

2389

0.33

CFST16

16,000

400

10

40

25

160

2147

0.33

CFST17

17,000

400

10

40

25

170

1895

0.32

CFST18

18,000

400

10

40

25

180

1702

0.32

CFST19

19,000

400

10

40

25

190

1562

0.30

CFST20

20,000

400

10

40

25

200

1385

0.31

G1

CFST1

L (mm)

fc0 (MPa)

D (mm)

t (mm)

D t

1000

400

10

40

25

CFST2

2000

400

10

40

CFST3

3000

400

10

CFST4

4000

400

CFST5

5000

CFST6

l

Continued

445

εlc εy

Column

4.3 Investigating the behavior of composite members

Pu (MPa)

Group

TABLE 4.20

Geometric and material properties of the examined CFST columns [59].dcont’d

60

10,504

1.12

25

60

8371

1.24

40

25

60

6932

1.29

8.00

50

25

60

5865

0.92

400

6.78

59

25

60

5389

0.98

15,000

400

20.00

20

25

150

3761

0.29

CFST26

15,000

400

13.33

30

25

150

2924

0.32

CFST15

15,000

400

10.00

40

25

150

2389

0.33

CFST27

15,000

400

8.00

50

25

150

2030

0.30

CFST28

15,000

400

6.78

59

25

150

1825

0.31

CFST6

6000

400

10.00

40

25

60

6932

1.29

CFST29

6000

400

10.00

40

40

60

7653

1.07

CFST30

6000

400

10.00

40

60

60

8621

1.14

CFST31

6000

400

10.00

40

80

60

9219

0.99

CFST32

6000

400

10.00

40

100

60

9856

1.28

CFST15

15,000

400

10.00

40

25

150

2389

0.33

CFST33

15,000

400

10.00

40

40

150

2503

0.30

CFST34

15,000

400

10.00

40

60

150

2626

0.29

CFST35

15,000

400

10.00

40

80

150

2651

0.29

CFST36

15,000

400

10.00

40

100

150

2768

0.33

L (mm)

G2

CFST21

G3

G4

G5

fc0 (MPa)

D (mm)

t (mm)

D t

6000

400

20.00

20

25

CFST22

6000

400

13.33

30

CFST6

6000

400

10.00

CFST23

6000

400

CFST24

6000

CFST25

l

4. Numerical methods

εlc εy

Column

446

Pu (MPa)

Group

1.6 1.4 1.2

0.8 0.6

= 35

0.4 0.2

= 73

Pu/Ppl

1

0 0

50

100

150

200

λ

FIGURE 4.67

The PPus ratio versus l relationship.

4.5 4 3.5 εlc/εy

3

> 1.0

2.5

Plastic buckling

2 1.5 1

< 1.0 Elastic buckling

0.5 0 0

50

100

150

200

λ FIGURE 4.68 The

εlc εy

ratio versus l relationship.

P (kN) 8000 7000

Comp-CFST17

6000

Tens-CFST17

5000

Comp-CFST8

3000

Tens-CFST8

Yield strain

Yield strain

4000

2000 1000

Comp-CFST6 Tens-CFST6

0 -20000

FIGURE 4.69

-10000

0

10000

20000

Axial load versus longitudinal strain relationships of CFST slender columns. Hassanein MF, Elchalakani M, Patel VI. Overall buckling behaviour of circular concrete-filled dual steel tubular columns with stainless steel external tubes. Thin-Walled Struct 2017;115:336e48; Hassanein MF, Kharoob OF. Analysis of circular concrete-filled double skin tubular slender columns with external stainless steel tubes. Thin-Walled Struct 2014;79:23e37.

448

4. Numerical methods

tension. The loadestrain response of column CFST8 is almost similar to that of column CFST17. However, its ultimate axial strength is attained in the elastic stage. The reason is that the εεlcy ratio in column CFST8 is 0.75, which is smaller than 1.0. Contrary to columns CFST6 and CFST8, part of column CFST17 is under tension at the ultimate axial strength, and it buckles in the elastic range. The effects of the slenderness ratio on the axial load versus longitudinal strain responses of outer and inner steel tubes of CFDST columns are shown in Fig. 4.70. Table 4.21 shows the geometric and material properties of the examined circular CFDST columns. Similar to the CFST columns, 12000

P (kN)

10000 8000 Comp-Outer tube

6000

-20000

Yield strain

Yield strain

Comp-Inner tube 4000 2000 0

-10000

0

Tens-Outer tube Tens-Inner tube

10000

20000

(a) 4500 4000 3500 3000 2500

-5000

Comp-Outer tube

1500

Comp-Inner tube

1000

Tens-Outer tube Tens-Inner tube

500 0 -2500

0

2500

Yield strain

Yield strain

2000

5000

(b)

FIGURE 4.70 Axial load versus longitudinal strain relationships of CFST slender columns. (A) CFDST 1. (B) CFDST 2. Hassanein MF, Elchalakani M, Patel VI. Overall buckling behaviour of circular concrete-filled dual steel tubular columns with stainless steel external tubes. Thin-Walled Struct 2017;115:336e48; Hassanein MF, Kharoob OF. Analysis of circular concretefilled double skin tubular slender columns with external stainless steel tubes. Thin-Walled Struct 2014;79:23e37.

Geometric and material properties of the examined CDFST columns [61]. Outer tube

Column

L (mm)

Inner tube

Do (mm)

to (mm)

fyo (MPa)

di (mm)

ti (mm)

fyi (MPa)

fc0 (MPa)

l

Pu (kN)

εlc εy

CFDST1

5500

500

10

530

250

10

25

25

40

11,200

1.20

CFDST2

17,500

500

10

530

250

10

25

25

127

3951

0.23

4.3 Investigating the behavior of composite members

TABLE 4.21

449

450

4. Numerical methods

CFDST columns with columns with εεlcy

εlc εy

> 1:0 fails in the plastic stage, whereas CFDST

< 1:0 experience elastic buckling. As shown in Fig. 4.70A,

both external and internal steel tubes of column CFDST1 are subjected to compression before reaching the ultimate axial strength. By reaching the peak point, part of the outer steel tube is subjected to tension, whereas the inner steel tube remains under compression. Furthermore, the longitudinal compressive stresses of both tubes are larger than the yield strains of respective materials at the ultimate axial strength of column CFDST1. However, once the maximum load is achieved, the stress in a portion of the cross-section of the outer tube of CFDST1 changed from compression to tension, while the internal tube remained under compression in the same loading range. Contrary to the specimen CFDST1, part of the outer steel tube of the specimen CFDST2 is under tension before the ultimate axial strength is reached, as depicted in Fig. 4.70B. It is also worth noting that the longitudinal compressive strains in both CFDST2 tubes are smaller than the yield strains in their respective materials. Additionally, due to the elastic buckling associated with such a long column, the strains of both sides of the steel tubes differ from each other from the early loading stage. For evaluating the confinement effect in the slender composite columns, the axial load ratio PPu -strain ratio εεlch curves of the specimens CFST6 as an intermediate-length column and CFST17 as a long column are drawn in Fig. 4.71A in which the terms εlc and εh represent the hoop and longitudinal strains of the steel tube in the compression zone, as measured at the mid-height section of each column. As shown in Fig. 4.71A, the εεlch ratio of the long column CFST17 is almost constant and is equal to 0.3 prior to reaching the peak point. By contrast, the εεlch ratio of the intermediate-length column CFST6 starts to increase by reaching the axial load ratio of PPu ¼ 0:67, and it continues to increase until achieving the peak point. Fig. 4.71B shows the axial load ratio PPu -strain ratio εεlch curves of the specimens CFDST1 as an intermediate-length column and CFDST2 as a long column. Similar to the long CFST column, the εεlch ratio for the outer steel tube of the long CFDST column is 0.3 prior to reaching the peak point. A similar trend can be observed for the inner steel tube. Contrary to the long specimen CFDST2, the εεlch ratios of the outer and inner steel tubes of the intermediate-length column CFDST1 start to increase by reaching the load ratio PPu ¼ 0:72 and PPu ¼ 0:56, respectively. By reaching the ultimate axial strength PPu ¼ 1, the εεlch ratios of steel tubes for both long- and intermediate-length columns start to reduce.

451

4.3 Investigating the behavior of composite members

1.2 1

P/Pu

0.8 0.6

0.4

CFST6

0.2

CFST17

0 0

0.2

0.4

0.6

0.8

1

εh /εlc (a) 1.2 1

P/Pu

0.8 0.6 CFDST1-Outer tube

0.4

CFDST1-Inner tube CFDST2-Outer tube

0.2

CFDST2-Inner tube

0 0

0.2

0.4

0.6

0.8

1

εh /εlc (b) The relationships between the normalized load PPu and strain ratio εεlch of composite columns. (A) CFST columns. (B) CFDST columns. Hassanein MF, Elchalakani M, Patel VI. Overall buckling behaviour of circular concrete-filled dual steel tubular columns with stainless steel external tubes. Thin-Walled Struct 2017;115:336e48; Hassanein MF, Kharoob OF. Analysis of circular concrete-filled double skin tubular slender columns with external stainless steel tubes. Thin-Walled Struct 2014;79:23e37.

FIGURE 4.71

The following conclusions can be derived about the confinement effect of long- and intermediate-length composite columns based on the preceding discussions:   1. During the elastic stage, the Poisson effect exists εεlch ¼ 0:3 . 2. During the elastic stage, the amount of the εεlch ratio for the   intermediate-length column is constant εεlch ¼ 0:3 . Therefore, no interaction occurs between the steel tube and the in-filled concrete

452

4. Numerical methods

during this stage. However, the εεlch ratio for the intermediate-length

column starts to increase at the load ratio PPu < 1:0, whereas the εεlch ratio for the long column has a constant amount of 0.3 before reaching the peak point. This shows that only the Poisson effect occurs in the steel tube and the in-filled concrete in the long column has an elastic behavior before reaching the peak point. Consequently, the induced circumferential stress on the steel tube due to the dilation of the infilled concrete is small and the provided confinement effect by the long composite column’s steel tube during the prepeak region can be ignored. By contrast, the lateral expansion of in-filled concrete in the intermediate-length column exceeds the steel tube when the load ratio is PPu < 1:0. Therefore, the confinement effect in the intermediatelength column occurs at a specific limit during the ascending stage and increases before reaching the peak load. The effects of the slenderness ratio on the axial load-lateral deflection at the mid-height of CFST columns are shown in Fig. 4.72. It can be observed from the figure that the initial axial stiffness, as well as the ultimate axial strength, is diminished by increasing the slenderness ratio. However, the postpeak region of the curve becomes softer with an increase in the slenderness ratio. It should be noted that the mid-height deflection corresponding to the ultimate axial strength increases with increasing the slenderness ratio. This shows that the influence of the second-order effect is more prominent on long composite columns than the intermediatelength ones. In general, no discernible difference could be seen between intermediate-length and slender CFST columns. A similar trend is true for the effects of the slenderness ratio on the axial load-lateral deflection at the mid-height of CFDST columns. 12000 10000

CFST4

CFST5

Axial force (kN)

8000

CFST3 CFST7

6000

CFST9 4000

CFST13 CFST16

2000

CFST20

0 0

50

100

150

200

Mid-height deflection (mm)

FIGURE 4.72 Axial loademid-height deflection curves of slender CFST columns. Hassanein MF, Elchalakani M, Patel VI. Overall buckling behaviour of circular concrete-filled dual steel tubular columns with stainless steel external tubes. Thin-Walled Struct 2017;115:336–48; Hassanein MF, Kharoob OF. Analysis of circular concrete-filled double skin tubular slender columns with external stainless steel tubes. Thin-Walled Struct 2014;79:23e37.

453

4.3 Investigating the behavior of composite members

4.3.4.3 Effects of column slenderness ratio The effects of the slenderness ratio l on the axial loadestrain response and the axial loademoment interaction of composite beam columns are investigated in this section. Fig. 4.73 shows the axial loadestrain curves of slender CFST columns, given in Table 4.20. It can be observed from Fig. 4.73 that, in general, the initial axial stiffness is not affected by the slenderness ratio l. By contrast, the ultimate axial strength and the corresponding axial strain are declined by increasing the slenderness ratio l. Another distinguishable difference between the long- and intermediatelength columns is the transition from the prepeak region to the postpeak region. As shown in Fig. 4.73, intermediate-length columns typically have a rounded descending response (for instance, column CFST4), whereas long columns typically show a sharp postpeak region (for instance, column CFST20). The strength envelopes of columns CFST4, CFST8, and CFST17 are shown in Fig. 4.74. As illustrated in the figure, the pure ultimate axial loads and ultimate bending capacities of composite columns subjected to the same axial load level are declined by increasing the slenderness ratio l. However, the slenderness ratio does not affect the pure moment capacity. 4.3.4.4 Effects of width-to-thickness ratio Groups 2 and 3, presented in Table 4.20, are used to discuss the influences of the width-to-thickness Dt ratio of the steel tube on the performance of slender composite columns. It can be noticed from Table 4.20

12000 CFST4 CFST5

10000

Axial force (kN)

CFST6 8000

CFST7 CFST8

6000

CFST9 CFST12

4000

CFST13 CFST16

2000

CFST20 0 0

2000

4000

6000

8000

10000

Axial strain ε(με)

FIGURE 4.73 Effect of the slenderness ratio on the axial loadestrain responses of composite columns. Hassanein MF, Elchalakani M, Patel VI. Overall buckling behaviour of circular concrete-filled dual steel tubular columns with stainless steel external tubes. Thin-Walled Struct 2017;115:336e48; Hassanein MF, Kharoob OF. Analysis of circular concrete-filled double skin tubular slender columns with external stainless steel tubes. Thin-Walled Struct 2014;79:23e37.

454

4. Numerical methods

1.2

CFST6 1

CFST8 CFST17

Pu/Po

0.8 0.6 0.4 0.2 0 0

0.25

0.5

0.75

1

1.25

1.5

Mu/Mo FIGURE 4.74 Effect of the slenderness ratio on the axial loademoment interaction curves of composite columns.

that increasing the Dt ratio reduces the compressive capacity of slender composite columns. However, the effect of the Dt ratio on the ultimate axial strength is the most prominent on the intermediate-length columns. The relationship between the axial load PPplu ratio and the Dt ratio is shown in Fig. 4.75. As shown in the figure, the Dt ratio has a negligible influence on the PPplu ratio of intermediate-length and long composite columns. 0.8 0.7

Pu /Ppl

0.6 0.5

G2

0.4

G3

0.3 0.2 0.1 0 10

20

30

40

50

60

70

D/t FIGURE 4.75 The effect of the Dt ratio on the ultimate axial strength of slender composite

columns.

455

4.3 Investigating the behavior of composite members

The effect of the Dt ratio on the confinement effect of slender composite columns can be assessed through the axial load P-εεlch ratio responses of columns, displayed in Fig. 4.76. For intermediate-length columns, the confinement effect is declined by increasing the Dt ratio, resulting in lower ultimate axial strengths. As shown in Fig. 4.76A, the confinement effect provided by the steel tube is negligible when P < 20% Pu . The magnitude of the confinement effect remains constant until the axial load reaches a particular load level (60%w70% Pu ). From then onwards, the confinement effect increases remarkably until reaching the peak load Pu . Contrary to the intermediate-length columns, the Dt ratio has no substantial impact on the axial load P-εεlch ratio responses of long columns, as shown in Fig. 4.76B. 12000 C21 CFST21

8000

C22 CFST22

6000

C6CFST6 CFST23 C23

Axial load (kN)

10000

4000

CFST24 C24

2000 0 0.10

0.20

0.30 εh /εlc

0.40

0.50

(a)

Axial load (kN)

4000 3500

C25 CFST25

3000

C26 CFST26

2500 2000 1500 1000

C15 CFST15 C27 CFST27 CFST28 C28

500 0 0.10

0.20

0.30 εh /εlc

0.40

0.50

(b) The effect of the ratio on the axial load versus strain ratio εεlch responses of slender composite columns. (A) G2. (B) G3. Hassanein MF, Elchalakani M, Patel VI. Overall buckling behaviour of circular concrete-filled dual steel tubular columns with stainless steel external tubes. Thin-Walled Struct 2017;115:336e48; Hassanein MF, Kharoob OF. Analysis of circular concrete-filled double skin tubular slender columns with external stainless steel tubes. Thin-Walled Struct 2014;79:23e37.

FIGURE 4.76

D t

456

4. Numerical methods

This demonstrates that the confinement effect offered by steel tubes in long composite columns may be disregarded. The axial loadestrain responses of Groups 2 and 3 specimens are depicted in Fig. 4.77. Regardless of the slenderness ratio, the ultimate axial strength, as well as the initial axial stiffness of columns, is reduced with increasing the Dt ratio. A comparison of the descending region of curves shows that the intermediate-length columns have a smooth and roundended postpeak stage, whereas the descending stage of long columns is sharp. 12000 10000

Axial force (kN)

CFST21 C21

8000

CFST22 C22 CFST6 C6

6000

CFST23 C23

4000

CFST24 C24

2000 0 0

2000

4000

6000

8000

Axial strain ε(με) (a) 4000 3500

Axial force (kN)

3000

CFST25

2500

CFST26

2000

CFST15

1500

CFST27

1000

CFST28

500 0 0

2000

4000 6000 Axial strain ε(με)

8000

(b) Effect of the Dt ratio on the axial loadestrain responses of slender composite columns. (A) G2. (B) G3. Hassanein MF, Elchalakani M, Patel VI. Overall buckling behaviour of circular concrete-filled dual steel tubular columns with stainless steel external tubes. Thin-Walled Struct 2017;115:336e48; Hassanein MF, Kharoob OF. Analysis of circular concrete-filled double skin tubular slender columns with external stainless steel tubes. Thin-Walled Struct 2014;79:23e37.

FIGURE 4.77

457

4.3 Investigating the behavior of composite members

1.2 C21 1

C22 C6

Pu /Po

0.8

C23 C24

0.6

0.4

0.2

0

0

0.2

0.4

0.6

0.8

1

1.2

Mu /Mo FIGURE 4.78 Effect of the Dt ratio on the axial loademoment interaction curves of composite columns.

The effect of the Dt ratio on the strength envelopes of slender composite columns is given in Fig. 4.78. As shown in the figure, the axial loade moment interaction curves of the columns are enlarged by increasing the D ratio. t 4.3.4.5 Effects of concrete compressive strength Groups 4 and 5 specimens, presented in Table 4.20, are employed to investigate the effects of the concrete compressive strength fc0 on the performance of slender composite columns. As shown in Table 4.20, the slenderness ratios of Groups 4 and 5 specimens are 60 and 150, respectively, and a constant Dt ratio of 40 is considered for both Groups of specimens. The considered fc0 is in the range of 25 and 100 MPa to cover NSC, HSC, and UHSC. Fig. 4.79 shows the effect of the concrete compressive strength fc0 on the ultimate axial strength of slender composite columns. As shown in Table 4.20 and Fig. 4.79A, the ultimate axial strength of intermediate-length columns is enhanced by increasing the concrete strength fc0 . The same issue can be observed for long composite columns. However, as compared to intermediate-length columns, the effect of fc0 on long columns compressive capacity is insignificant. As shown in Table 4.20, the magnitudes of the εεlcy ratios for Group 4 specimens are greater than unity, whereas the opposite is true for Group 5 specimens.

458

4. Numerical methods

12000 10000

Pu (kN)

8000 G4 G5

6000 4000 2000 0 0

20

40

60 f c'

80

100

120

(a) 1.00

Pu /Ppl

0.80

0.60 G4 G5

0.40 0.20 0.00 10

30

50

70

90

110

f c' (b)

FIGURE 4.79

The effect of fc0

Ultimate axial strength. (B) The

on the load-bearing capacity of slender CFDST columns. (A) ratio.

Pu Ppl

Therefore, long composite columns fail by the elastic buckling. As a result, the resistance of long composite columns is dominated by the flexural stiffness of the cross-section rather than the material strength. Increasing fc0 results in a modest increase in flexural stiffness, and hence in the compressive strength of long columns. It may be concluded that using HSC and UHSC to fill long composite columns is not an efficient technique for improving their load-bearing capacities since they raise manufacturing costs without providing a significant increase in long column resistance.

459

4.3 Investigating the behavior of composite members

Fig. 4.79B depicts the relationships between the axial load PPplu ratio and fc0 . It can be noticed from the figure that the PPplu ratios of Group 4 columns that fail in the elasticeplastic range are more sensitive to the concrete strength fc0 than Group 5 columns, which fail by elastic buckling. The axial loadestrain responses of Groups 4 and 5 specimens are shown in Fig. 4.80. It can be observed from the figure that the influence of the concrete strength fc0 on the initial axial stiffness of columns is insignificant. By contrast, the ultimate axial strength of columns is remarkably increased by increasing the concrete strength fc0 . However, the effect of increasing the concrete strength fc0 on enhancing the ultimate axial strength of columns is declined by increasing the column slenderness.

Axial force (kN)

12000 10000

CFST6

8000

CFST29

6000

CFST30

4000

CFST31

2000

CFST32

0 0

2000

4000 6000 Axial strain ε(με)

8000

(a)

Axial force (kN)

3000 2500

CFST15

2000

CFST33

1500

CFST34

1000

CFST35

500

CFST36

0 0

2000

4000 6000 Axial strain ε(με)

8000

(b) fc0

FIGURE 4.80 The effect of on the axial loadestrain responses of slender composite columns. (A) G4. (B) G5. Hassanein MF, Elchalakani M, Patel VI. Overall buckling behaviour of circular concrete-filled dual steel tubular columns with stainless steel external tubes. Thin-Walled Struct 2017;115:336e48; Hassanein MF, Kharoob OF. Analysis of circular concrete-filled double skin tubular slender columns with external stainless steel tubes. Thin-Walled Struct 2014;79:23e37.

460

4. Numerical methods

1.2 1

Pu /Po

0.8 0.6

CFST6 CFST29

0.4

CFST30 CFST31

0.2

CFST32 0 0

0.2

0.4

0.6

0.8

1

1.2

Mu /Mo FIGURE 4.81 Effect of fc0 on the axial loademoment interaction curves of slender composite columns.

Fig. 4.81 shows the effect of the concrete compressive strength fc0 on the strength envelope of composite columns. It can be observed from the figure that the effect of the concrete strength fc0 is more prominent on the axial strength rather than the moment capacity of slender composite columns. This clarifies that contrary to the ultimate axial strength of slender composite columns, increasing the concrete strength fc0 for enhancing their moment capacity is neither a practical or cost-effective solution. 4.3.4.6 Effects of the hollow ratio The effects of the hollow ratio c on the behavior of slender CFDST columns are investigated in this section. The geometric and material properties of the examined specimens are given in Table 4.22. Two groups of specimens are considered so that the influence of the hollow ratio c on the behavior of intermediate-length columns and long columns can be assessed. Fig. 4.82 compares the ultimate axial strength of slender CFDST columns having different hollow ratios c. It can be seen from Fig. 4.82A that increasing the hollow ratio c has no noticeable impact on the compressive strength of intermediate-length columns. By contrast, the ultimate axial strength of long columns may be increased by increasing the hollow ratio c. This observation contradicts the influence of hollow ratio c on the ultimate axial strength of short CFDST columns. As mentioned in Section 4.3.1.5,

TABLE 4.22

Geometric and material properties of slender CFDST columns for investigating the effects of the hollow ratio c.

Column

G1

CFDST1

G2

L (mm) 5500

Do (mm)

to (mm)

fyo (MPa)

di (mm)

ti (mm)

fyi (MPa)

c

PFE (kN)

500

10

530

125

10

235

0.26

43

11,237

l

CFDST2

175

0.36

42

11,226

CFDST3

250

0.52

40

11,200

CFDST4

300

0.63

38

11,158

CFDST5

375

0.78

36

11,130

125

0.26

114

4652

CFDST7

175

0.36

111

4939

CFDST8

250

0.52

105

5014

CFDST9

300

0.63

101

5145

CFDST10

375

0.78

95

5253

CFDST6

14,500

4.3 Investigating the behavior of composite members

Group

461

462

4. Numerical methods

12000 10000 G1

8000 Pu

G2 6000 4000 2000 0 0

0.2

0.4

0.6

0.8

1

0.6

0.8

1

(a) 1 0.8

Pu/Ppl

0.6

G1 G2

0.4 0.2 0 0

0.2

0.4 (b)

FIGURE 4.82 Effect of the hollow ratio c on the load-bearing capacity of slender CFDST columns. (A) Ultimate axial strength. (B) The PPplu ratio.

increasing the hollow ratio c reduces the load-bearing capacity of CFDST stub columns because enlarging the hollow ratio c reduces the crosssectional area of the sandwiched concrete, which bears the majority of the external load in short columns. In the case of the intermediate-length column, the cross-sectional area of the sandwiched concrete decreases as the hollow ratio c increases. Increasing the hollow ratio c, on the other hand, increases the distance between the inner steel tube and the centroid of the column, resulting in an increase in the section’s flexural stiffness by increasing the contribution of the inner tube to the moment of inertia. As a result, the decrease in ultimate axial strength caused by the reduced

463

4.3 Investigating the behavior of composite members

cross-sectional area of the sandwiched concrete is offset by the increase in flexural rigidity acquired by the inner steel structure. As mentioned in Section 4.3.4.4, the influence of in-filled concrete on the performance of composite members is declined by increasing the column slenderness. Besides, it is known that the ultimate axial strength of long composite columns that fail by elastic buckling is dominated by flexural rigidity. Therefore, the increase in the moment of inertia in long columns due to the increase in their hollow ratio c enhances their ultimate axial strength. The effect of the hollow ratio c on the PPplu ratio is given in Fig. 4.82B. It can be seen that increasing the hollow ratio c increases the PPplu ratio in both intermediate-length and long CFDST columns. This is due to the fact that when the hollow ratio c enlarges, the Pu values tend to remain constant. However, the plastic strength Ppl of CFDST columns is reduced by increasing the hollow ratio c. 4.3.4.7 Effects of the load eccentricity ratio The nonlinear FE analysis was performed on a series of slender CFST columns to investigate the influence of load eccentricity ratio (e= r) on the behavior of slender composite columns. The geometric and material properties of the tested specimens are given in Table 4.23. The axial loadedeflection responses of slender CFST columns subjected to eccentric axial compression load are depicted in Fig. 4.83. It can be observed from the figure that increasing the load eccentricity ratio leads to a remarkable decline in the initial axial stiffness as well as the ultimate axial strength of columns. The reduction effect of the load eccentricity ratio is because increasing the e=r ratio increases the column’s end moments and diminishes the confining pressures on the concrete. Another point is that the postpeak region of a column with a greater e=r ratio has a smoother TABLE 4.23 Geometric and material properties of the slender CFST columns for investigating the effects of the load eccentricity ratio. Column

L (mm)

D (mm)

t (mm)

D t

fc0 (MPa)

fy (MPa)

e=r

Pu (MPa)

CFST37

8000

530

12

44.2

55

320

0.1

13,883

CFST38

0.2

11,792

CFST39

0.3

9273

CFST40

0.4

7382

CFST41

0.5

5222

464

4. Numerical methods

16000

14000

CFST37

Axial force (kN)

12000 CFST38

10000 8000

CFST39

6000

CFST40

4000

CFST41

2000 0

0

25

50 75 100 Mid-height deflection (mm)

125

FIGURE 4.83 Effect of the load eccentricity ratio on the axial loadedeflection curves of slender composite columns.

0.8 0.7 0.6

Pn /Po

0.5 0.4 0.3 0.2 0.1 0 0

0.5

1

1.5

2

2.5

3

e/D FIGURE 4.84 Effect of the load eccentricity ratio on the normalized compressive strength of slender composite columns.

response than the column with a smaller e=r ratio. Fig. 4.84 shows the relationship between the normalized ultimate axial strength PPno and the load eccentricity e=D ratio in which Po is the pure ultimate axial strength. As illustrated in the figure, when the e=D ratios are large, the decrease in the ultimate axial strength of the column lessens as the e=D increases. This is mainly owing to the fact that a large load eccentricity can remarkably decrease the confinement effect provided by the tube.

465

4.3 Investigating the behavior of composite members

4.3.4.8 Load distribution in constituent components of slender composite members The contribution of composite beam columns constituent components in load-bearing capacity can be assessed using the contribution ratios as follows: Pu;os Pu

(4.116)

CCR ¼

Pu;c Pu

(4.117)

ISCR ¼

Pu;is Pu

(4.118)

OSCR ¼

where the terms OSCR; CCR, and ISCR denote contribution ratios of the outer steel tube, concrete core, and inner steel tube, respectively, Pu;os , Pu;c , and Pu;is are the ultimate axial loads sustained by the outer steel tube, concrete core, and inner steel tube, respectively, and Pu is the ultimate axial strength of the composite member. The load distribution between the constituent components of slender CFST and slender CFDST beam columns is investigated in this section using the nonlinear FE analysis. Load distribution in constituent components of slender CFST beam columns

Nonlinear FE analysis was performed on slender CFST beam columns   to assess the influence of the slenderness ratio Lr on the load distribution. The geometric and material properties of the tested specimens are given in Table 4.24. Fig. 4.85 shows the axial loadedeflection curves of the examined slender CFST columns. The load carried by the steel tube and the concrete core are also presented in Fig. 4.85. The figure shows that the concrete core carries the most majority of the external load, regardless of the slenderness ratio Lr. This is similar to the behavior of short composite columns. During the loading history, the concrete core and steel tube of TABLE 4.24 Geometric and material properties of the slender CFST columns for investigating the influence of the slenderness ratio Lr on the load distribution. Column

D (mm)

t (mm)

D t

CFST42

600

10

60

L r

30

fc0 (MPa)

fy (MPa)

e=D

Pu (MPa)

80

320

0.1

13,352

CFST43

60

9156

CFST44

80

6612

CFST45

100

5098

466

4. Numerical methods

10000

16000 14000

8000 Axial force (kN)

Axial force (kN)

12000 10000 8000 CFST

6000

Concrete core 4000

6000 4000

Steel tube

0

2000

-2000

0 0

50 Mid-heghit deflection (mm)

0

100

8000

8000

6000

6000 Axial force (kN)

Axial force (kN)

CFST Concrete core Steel tube

2000

4000 CFST Concrete core Steel tube

2000 0

100 200 300 Mid-heghit deflection (mm)

400

4000 CFST

2000

Concrete core Steel tube

0

-2000

-2000 0

100 200 300 Mid-heghit deflection (mm)

400

0

100 200 300 Mid-heghit deflection (mm)

400

FIGURE 4.85 Axial loadedeflection responses of slender CFST columns and the corresponding constituent components with different slenderness ratios. (A) Lr ¼ 30 (B) Lr ¼ 60 (C) Lr ¼ 80 (D) Lr ¼ 100.

 the intermediate-length column

L¼ r

 30 are under compression. On the

other hand, steel tubes in long columns (Lr  60) are exposed to tension during the postpeak region of the curve. This is because increasing the slenderness ratio Lr increases the lateral deflection and hence the bending moments at the column’s mid-height. Fig. 4.86 depicts the relationships between the load-bearing contribution ratios of slender CFSTs constituent components and the slenderness ratio Lr. It is evident from the figure that SCR increases with increasing the slenderness ratio Lr. By contrast, increasing the slenderness ratio Lr leads to

467

4.3 Investigating the behavior of composite members

Contribution ratio

1 0.8 0.6

CCR SCR

0.4 0.2 0 0

20

40

60

80

100

120

L/r FIGURE 4.86

Effect of the slenderness ratio on the load-bearing contribution ratios of slender CFSTs constituent components.

a reduction in CCR. This again reinforces the fact that increasing the concrete compressive strength fc0 is not a viable way to enhance the ultimate axial strength of long composite columns (columns with a large L ratio). r For investigating the effect of the load eccentricity ratio e= D on the load distribution in slender CFST columns, a circular specimen with a constant diameter of 700 mm, a steel tube wall thickness of 10 mm, and a slenderness ratio of 50 was analyzed at various load eccentricity ratios ranging from 0.1 to 0.4. The geometric and material properties of the tested slender CFSTs are given in Table 4.25. Fig. 4.87 presents the axial loadedeflection responses of the examined slender CFST columns. The load carried by the steel tube and the concrete core are also presented in Fig. 4.87. The figure shows that the concrete core and steel tube are both under compression when the load eccentricity ratio is set at De ¼ 0:1. By contrast, imposing the load eccentricity ratio larger than 0.1 makes the TABLE 4.25 Geometric and material properties of the slender CFST columns for investigating the influence of the load eccentricity ratio on the load distribution. Column

D (mm)

t (mm)

D t

L r

fc0 (MPa)

fy (MPa)

e=D

Pu (MPa)

CFST46

700

10

70

50

80

320

0.1

16,850

CFST47

0.2

12,274

CFST48

0.3

9302

CFST49

0.4

7455

468

4. Numerical methods

18000

14000 12000

15000 Axial force (kN)

Axial force (kN)

10000 12000 9000

CFST Concrete core

6000

8000 6000

CFST Concrete core Steel tube

4000 2000

Steel tube 3000

0

0

-2000 0

100 200 Mid-heghit deflection (mm)

300

0

10000

8000

8000

6000

6000 4000

CFST Concrete core Steel tube

2000

400

(b)

0

Axial force (kN)

Axial force (kN)

(a)

100 200 300 Mid-heghit deflection (mm)

4000 CFST Concrete core Steel tube

2000 0 -2000

-2000 -4000

-4000 0

100 200 300 Mid-heghit deflection (mm)

(c)

400

0

100 200 300 Mid-heghit deflection (mm)

400

(d)

FIGURE 4.87 Axial loadedeflection responses of slender CFST columns and the corresponding constituent components under different load eccentricity ratios. (A) De ¼ 0:1 (B) e e e D ¼ 0:2 (C) D ¼ 0:3 (D) D ¼ 0:4.

steel tube go under tension when the columns are under a large lateral deflection. It is noteworthy that the intersection point of the axial loade deflection response of the steel tube with the horizontal axis shifts to the left with increasing the load eccentricity ratio. Therefore, increasing the load eccentricity ratio causes the steel tube to be under tension at a smaller lateral deflection. Fig. 4.88 depicts the relationship between the concrete core and steel tube load contribution ratios and the load eccentricity ratio. As illustrated in the figure, the concrete core’s contribution in bearing the ultimate axial strength is increased by increasing the er ratio, whereas the opposite is true for the steel tube.

469

4.3 Investigating the behavior of composite members

1.2 Contribution ratio

1 0.8 CCR SCR

0.6 0.4 0.2 0 0

0.1

0.2

0.3

0.4

0.5

e/D

FIGURE 4.88 Effect of the load eccentricity ratio on the load-bearing contribution ratios of slender CFSTs constituent components. Load distribution in constituent components of slender CFDST beam columns

Nonlinear FE analysis was performed on slender CFDST beam   columns to assess the influence of the slenderness ratio Lr on the load distribution. The geometric and material properties of the tested specimens are given in Table 4.26. Fig. 4.89 shows the axial loadedeflection responses of the examined slender CFDST columns. The load carried by the steel tubes and the sandwiched concrete are also presented in Fig. 4.89. It can be observed from the figure that regardless of the slenderness ratio L, the most majority of the axial load is carried by the sandwiched conr crete, which is similar to CFSTs. On the other hand, the role of the inner steel tube in the load-bearing contribution is the least. The constituent components of the CFDST column with the slenderness ratio of Lr ¼ 30 are under axial compression during the loading history. By contrast, as a result of the enormous deflections and bending moments experienced at the mid-height of very slender CFDST columns (Lr  60), steel tubes undergo tensile force during the postpeak region. Fig. 4.90 shows the relationships between the load-bearing contribution ratios and the slenderness ratio Lr. It can be observed from the figure that increasing the slenderness ratio Lr leads to a reduction in the CCR and ISCR, whereas OSCR is increased by increasing the slenderness ratio Lr. For investigating the effect of the load eccentricity ratio e= Do on the load distribution of slender CFDST columns, a circular specimen with the constant material and geometric properties was analyzed under various load eccentricity ratios e=Do ranging from 0.1 to 0.4. The geometric and material properties of the tested slender CFDST column are given in Table 4.27. Fig. 4.91 presents the axial loadedeflection responses of the

470

TABLE 4.26

Geometric and material properties of the slender CFDST columns for investigating the influence of the slenderness ratio on the load distribution. Do (mm)

to (mm)

fyo (MPa)

di (mm)

ti (mm)

fyi (MPa)

fc0 (MPa)

e=Do

CFDST11

600

10

300

300

10

300

70

0.1

L=r

Pu (MPa)

30

13,993

CFDST12

60

9258

CFDST13

80

6897

CFDST13

100

5237

4. Numerical methods

Column

471

4.3 Investigating the behavior of composite members

10000

16000 14000

8000

10000 8000 CFDST Concrete core Outer steel tube Inner steel tube

6000 4000 2000 0

Axial force (kN)

Axial force (kN)

12000 6000 CFDST Concrete core Outer steel tube Inner steel tube

4000 2000 0 -2000

0

50 100 150 Mid-heghit deflection (mm)

0

200

8000

100 200 Mid-heghit deflection (mm)

300

6000

7000

5000

5000 4000 CFDST Concrete core Outer steel tube Inner steel tube

3000 2000 1000

Axial force (kN)

Axial force (kN)

6000

4000 3000

CFDST Concrete core Outer steel tube Inner steel tube

2000 1000 0

0 -1000

-1000 0

100 200 300 Mid-heghit deflection (mm)

400

0

100 200 300 400 Mid-heghit deflection (mm)

500

FIGURE 4.89 Axial loadedeflection responses of slender CFDST columns and the corresponding constituent components with different slenderness ratios. (A) Lr ¼ 30 (B) Lr ¼ 60 (C) ¼ 80 (D) Lr ¼ 100.

L r

examined slender CFDST columns. The load carried by the steel tubes and the sandwiched concrete are also presented in Fig. 4.91. It can be observed from the figure that increasing the load eccentricity ratio leads to the reduction of the ultimate axial strength. Components of slender CFDST columns under load eccentricity ratios of Deo < 0:4 are under compression, as opposed to the slender CFST columns discussed in the preceding section. Contrary to the slender CFST columns examined in the previous section, constituent components of slender CFDST columns under the load eccentricities of Deo < 0:4 are under compression. This can be due to the fact that the flexural rigidity a CFDST column is more than its CFST counterpart.

472

4. Numerical methods

1

Contribution ratio

0.8 0.6

CCR OSCR ISCR

0.4 0.2 0 0

20

40

60 L/r

80

100

120

FIGURE 4.90

Effect of the slenderness ratio on the load-bearing contribution ratios of slender CFDSTs constituent components.

Fig. 4.92 shows the effect of the load eccentricity ratio on the load contribution ratios of the sandwiched concrete, outer steel tube, and inner steel tube. As illustrated in the figure, the contribution of the inner steel tube and the sandwiched concrete in bearing the ultimate axial strength are both increased by increasing the er ratio, whereas the opposite is true for the steel tube. Generally, it can be concluded that the behavior of slender CFST and slender CFDST columns is similar. 4.3.4.9 Ultimate axial strengths of slender beam columns In this section, the influences of different parameters, namely the     slenderness ratio Lr , depth-to-thickness ratio Dt , load eccentricity ratio   ðDe Þ, and the compressive strength of the concrete fc0 on the ultimate axial strength of slender composite beam columns are discussed. To this end, rectangular CFST beam columns fabricated with the steel tube with B  D ¼ 400  500 mm and the yield strength of fy ¼ 390 MPa are investigated. Fig. 4.93 shows the influences of the parameters mentioned above on the ultimate axial strength of columns. It can be seen from the figure that the ultimate axial strength of beam column is declined by increasing the Bt ratio and the Be ratio. This is because increasing the Bt ratio reduces the buckling strength and the contribution of the steel tube to the loadbearing capacity. The influence of the Be ratio on the ultimate axial strength can be explained by the increase in the primary moment P e and the secondary moment P  um caused by the increase in the Be ratio [62]. Besides, the ultimate axial strength of rectangular CFST slender beam columns is diminished by increasing the slenderness ratio Lr. By contrast,

Column

Do (mm)

to (mm)

fyo (MPa)

di (mm)

ti (mm)

fyi (MPa)

fc0 (MPa)

e=Do

L=r

Pu (MPa)

CFDST14

700

10

300

320

8

300

70

0.1

50

16,499

CFDST15

0.2

13,715

CFDST16

0.3

11,489

CFDST17

0.4

11,783

4.3 Investigating the behavior of composite members

TABLE 4.27 Geometric and material properties of the slender CFDST columns for investigating the influence of the load eccentricity ratio on the load distribution.

473

474

4. Numerical methods

18000

16000 14000

15000 Axial force (kN)

Axial force (kN)

12000 12000 9000 CFDST Concrete core Outer steel tube Inner steel tube

6000 3000

10000 8000 CFDST Concrete core Outer steel tube Inner steel tube

6000 4000 2000

0

0 0

50 100 150 Mid-heghit deflection (mm)

200

0

50 100 150 200 Mid-heghit deflection (mm)

(b)

14000

14000

12000

12000 10000

10000

Axial force (kN)

Axial force (kN)

(a)

250

8000 6000 CFDST Concrete core Outer steel tube Inner steel tube

4000 2000

8000 CFDST Concrete core Outer steel tube Inner steel tube

6000 4000

2000 0

0

-2000 0

50 100 150 200 Mid-heghit deflection (mm)

0

250

50 100 150 200 Mid-heghit deflection (mm)

(c)

250

(d)

FIGURE 4.91 Axial loademid-height deflection responses of slender CFDST columns and the corresponding constituent components under different load eccentricity ratios. (A) e e e e Do ¼ 0:1 (B) Do ¼ 0:2 (C) Do ¼ 0:3 (D) Do ¼ 0:4.

Contribution ratio

0.8 0.6 CCR OSCR ISCR

0.4 0.2 0 0

FIGURE 4.92

20

40

60 e/Do

80

100

120

Effect of the slenderness ratio on the load-bearing contribution ratios of slender CFDSTs constituent components.

475

4.3 Investigating the behavior of composite members

14000 f 'c=100MPa, L/r=50

15000

f 'c=40MPa, L/r=50 f 'c=100MPa, L/r=100

12000

f 'c=40MPa, L/r=100

9000 6000 3000

Ultimate axial strength (kN)

Ultimate axial strength (kN)

18000

f 'c=100MPa, L/r=50

12000

f 'c=40MPa, L/r=50

10000

f 'c=100MPa, L/r=100 f 'c=40MPa, L/r=100

8000 6000 4000 2000 0

0 0

0.5 1 1.5 2 2.5 Load eccentricity ratio (e/B) (a)

0

0.5 1 1.5 2 2.5 Load eccentricity ratio (e/B) (b)

FIGURE 4.93

Effects of the geometric properties and the concrete compressive strength on the ultimate axial strength of slender beam columns. (A) Bt ¼ 50 (B) Bt ¼ 100.

increasing the concrete compressive strength fc0 leads to an increase in the ultimate axial strength. It can be observed from the results that the effects of the geometric and material properties on the ultimate axial strength of columns with the load eccentricity ratio of Be > 2:0 are insignificant. Compared to the primary moment P  e, the effect of the mid-length deflection of eccentrically loaded CFST/CFDST stub columns is insignificant. Axial strength decreases as column length increases because an increase in mid-length deflection results in a rise in the mid-length secondary moment P  um . The total mid-length bending moment for slender columns equals the sum of the primary moment P  e and secondary moment P  um . The secondary moment P  um increases dramatically as the column length increases due to the mid-length deflection um increase. The secondary moment has a considerable influence on slender columns. Instead of failing under compression, the slender column bends fail by bending due to the secondary moment. The instability of the column causes this behavior. 4.3.4.10 Effects of concrete confinement Experimental studies showed that circular composite slender beam columns having pin-ended boundary conditions typically failed at their mid-height. Therefore, the strength at the middle of the beam column height dominates the resistance of the member. The concrete confinement provided by the steel tubes can positively enhance the strength at the mid-height of slender beam columns. To assess the influence of the confinement effect on improving the load-bearing capacity of composite

476

4. Numerical methods

Pu/Ppl

1.6 1.4

Confinement considered

1.2

Confinement ignored

1 0.8 0.6 0.4 0.2 0 0

50

100

150

200

λ FIGURE 4.94

Effect of the concrete confinement on the relative compressive capacity of circular composite columns.

columns, the ultimate axial strength of circular CFDST columns with the slenderness ratios l ranging from 7 to 182 obtained from the nonlinear analysis were compared with the case when the confinement effect was ignored. All the examined specimens have a circular outer steel tube with a diameter of 500 mm and a thickness of 10 mm and a circular inner steel tube with a diameter of 250 mm and a thickness of 10 mm. The yield strengths of the outer and inner tubes are 530 and 235 MPa, respectively, and the compressive strength of the sandwiched concrete is 25 MPa. Fig. 4.94 shows the PPus ratio as a function of the slenderness ratio l. It can be seen from the figure that the effect of confinement on enhancing the compressive capacity of CFDST columns is reduced by increasing the slenderness ratio l. The reason is that the lateral deformation of the column before failure increases the secondary bending moment, leading to the reduction of the mean compressive strain in the in-filled concrete. For composite columns with the slenderness ratio l  64, the PPus ratio when the confinement effect is considered is almost equal to the case in which the confinement effect is ignored. The compressive strength enhancement because of the confinement effect as a function of the slenderness ratio l is depicted in Fig. 4.95. It can be observed from the figure that for the column with l ¼ 7, the confinement effect improves the ultimate axial strength by more than 8%. However, the strength enhancement in the column with l ¼ 62 is only 0.85%. Therefore, it can be stated that the confinement effect has no influence in long composite columns with l  64 and can be ignored. Similar results can be expected for circular CFST columns. The concrete confinement effect on the axial loademoment strength interaction diagrams of the CFDST slender column with a slenderness ratio of l ¼ 29 is shown in Fig. 4.96. It is evident that the compressive

477

4.3 Investigating the behavior of composite members

Compressive strength enhancement (%)

10 8 6 4 2 0 0

20

40

60

λ FIGURE 4.95 Compressive strength enhancement due to the confinement effect in circular CFDST columns with different slenderness ratios.

15000

Axial load Pn (kN)

12000 9000 6000 Confinement considered 3000

Confinement igonored

0 0

300

600 900 1200 1500 Bending moment Mn (kNm)

1800

FIGURE 4.96 Effects of concrete confinement on the loademoment strength interaction curves of circular CFDST beam columns.

capacities of the beam column are underestimated if the confinement effect provided by the steel tubes is ignored in the analysis. The effect of confinement becomes more prominent with increasing the axial load or reducing the end bending moment. Another point is that the influence of the concrete confinement on the pure compressive strength is more than the pure bending strength, as depicted in Fig. 4.96. Fig. 4.97 demonstrates the influence of the concrete confinement on the ultimate axial strength of the CFDST slender beam column with the

478

4. Numerical methods

16000

Axial load Pn (kN)

14000 12000

Confinement considered

10000

Confinement igonored

8000 6000 4000 2000 0 0

1

2

3

4

e/Do FIGURE 4.97

Effects of concrete confinement on the compressive capacity of the circular CFDST slender beam column subjected to different load eccentricity ratios.

slenderness ratio of l ¼ 29 under various load eccentricity ratios. As shown in the figure, when the amount of load eccentricity ratio is 0.05, the confinement effect enhances the compressive strength of the column by more than 7%. However, when the eccentricity ratio is 2.0, the strength enhancement due to the confinement effect is less than 1%. Therefore, it can be stated that the confinement effect in slender beam columns subjected to the load eccentricity greater than 0.2 can be neglected. 4.3.4.11 Pure bending strengths of slender beam columns In this section, the influences of different parameters, namely the     slenderness ratio Lr , depth-to-thickness ratio Dt , load eccentricity ratio   e B , steel yield strength fy , and the compressive strength of the concrete   core fc0 on the ultimate pure bending strengths ðMo Þ of slender beam columns are discussed. To this end, the theoretical model recommended by Liang and Fragomeni is employed [63]. The benchmark square CFST beam column has the steel tube with the cross-section of B  D ¼ 350  350 mm and steel yield strength of 340 MPa filled with the concrete with a compressive strength of 80 MPa. Fig. 4.98 shows the ultimate pure bending strength Mo as a function of the Dt ratio in which the term Ze represents the elastic modulus of the section. As illustrated in the figure, the magnitude of Mo reduces with increasing the Dt ratio. The reason is that the area and the steel tube’s buckling strength are declined by increasing the Dt ratio (reducing the wall thickness). A similar trend is

479

4.3 Investigating the behavior of composite members

1.2

Mo/(Ze fy )

1 0.8 0.6 0.4 0.2 0 0

30

60

90

120

150

D/t FIGURE 4.98

Effect of the

D t

ratio on the pure bending strength Mo .

true for circular beam columns. Based on Refs. [63,64], the inverse relationship between the Dt ratio and the pure bending strength of beam columns can be considered using the factor lm, defined as follows:    2 8 t t > > 0:0087 þ 12:3 for circular beam
columns with 10  D=t  120
36 > < D D lm ¼    2 > > > : 0:0187 þ 24:2 t
61:1 t for rectangular beam
columns with 10  D=t  120 D D

ð4:119Þ

fc0

The effect of the concrete compressive strength on the pure bending strengths Mo of beam columns is illustrated in Fig. 4.99, in which fc0 varies 1.2

Mo /(Ze λm fy )

1 0.8 0.6 0.4 0.2 0 0

30

60

90

120

150

f c' FIGURE 4.99 Effect of fc0 on the pure bending strength Mo .

480

4. Numerical methods

in the range of 20 and 120 MPa and lm is the function of the Dt ratio. It is evident from Fig. 4.99 that the moment ratio ZeMlmo fy is increased with

increasing the concrete strength fc0 . To consider the direct relationship between fc0 and the pure bending strength of beam columns Mo , the factor ac is defined as follows [63,64]: ac ¼

8   < 0:774 fc0 0:075 :

 0:0668 0:7266 fc0

for circular CFST beam
columns with 30  fc0  120 MPa for rectangular CFST beam
columns with 20  fc0  120 MPa

ð4:120Þ The effect of the steel yield strength fy on the pure bending strengths Mo of beam columns is illustrated in Fig. 4.100, in which ac is the function of the concrete compressive strength fc0 . It can be observed from the figure that the Ze lMm ao c fy ratio is reduced with increasing the steel yield strength. To consider the direct relationship between fy and the pure bending strength of beam columns Mo , the factor as is defined as follows [63,64]:

as ¼

8 21:147 4202 0:882 þ þ  2 > > fy > > fy
262:62 28119 > > > : 0:471 þ fy
 2 fy

for rectangular CFST beam
columns with 250  fy  690 MPa

ð4:121Þ

The prediction of steel-concrete composite beam column ultimate pure bending strength Mo is based on factors lm , ac , and as and will be explored further in Chapter 5. 1.2

Mo /(Ze λmαc fy )

1 0.8 0.6 0.4 0.2 0 0

200

400

600

800

fy FIGURE 4.100

Effect of fy on the pure bending strength fy .

4.3 Investigating the behavior of composite members

481

4.3.4.12 Residual strength The residual strength ratio PPur is a useful measure for assessing the ductility of CFST/CFDST columns under axial compression, especially when they have been fabricated using high-strength and ultrahighstrength materials [52]. The residual strength Pr is determined from the 1:5f L

loadestrain response at the shortening limit 4lim , which is taken as Esy [51] in which the term L represents the column length. In order to keep the columns ductile, the residual strength ratio must be higher than 0.7 [51]. The residual strength is generally influenced by three parameters, i.e., the steel contribution ratio, concrete compressive strength, and the depth-tothickness ratio. A series of parametric tests on 99 circular CFST stub columns were done using nonlinear FE analysis to evaluate the influence of these factors on Pr [52]. Steel yield strength fy ranged between 250 and 450 MPa, concrete compressive strength fc0 ranged between 60 and 190 MPa, and the Dt ranged between 20 and 50. The specimens were all 600 mm in diameter and 1800 mm in length. Fig. 4.101 depicts the effects of steel contribution, concrete compressive strength fc0 , and steel yield strength fy on the residual strength ratio. Fig. 4.101A shows that the residual strength increases when the steel contribution ratio increases, but it slightly reduces as concrete compressive strength fc0 increases, as shown in Fig. 4.101B. Reducing the Dt ratio, which corresponds to an increase in the steel contribution ratio, while keeping the concrete strength fc0 constant can lead to greater ductility in concrete, as displayed in Fig. 4.101A. As illustrated in Fig. 4.102, increasing steel yield strength fy increases both ultimate axial strength Pu and residual strength Pr . As a result, as seen in Fig. 4.101C, the steel yield strength fy does not affect the residual strength ratio. However, highstrength steel must be used to achieve the required UHSC’s confining pressure and keep composite columns ductile enough. Generally speaking, a smaller Dt (less than 30) combined with a larger steel contribution ratio (more than 0.3) for ultrahigh-strength composite columns can result in appropriate ductility, according to Ref. [52]. It should be noted that AS/NZS 5100.6:2017 [65] specifies that the steel contribution ratio for CFST columns with HSC must be between 0.2 and 0.9. It is recommended that the steel contribution ratio be raised to 0.3 in order to make practical use of UHSC in CFST columns [52]. 4.3.4.13 Behavior of concrete-filled corrugated steel tubular stub column CFST and CFDST columns have been widely employed in industrial and civil buildings, bridges, and underground works because of their exceptional load-bearing capacity and seismic ductility [66,67]. Nonetheless,

1 0.9 0.8

0.7

D/t=40

D/t=190

0.6 Pu/Pr

D/t=17

D/t=20 D/t=30

0.5

D/t=50

0.4

D/t=60 D/t=70

0.3

D/t=130

0.2 0.1 0 0

0.1

0.2 Steel contribution ratio

0.3

0.4

(a) 1 0.9 0.8 0.7 Pu/Pr

0.6 0.5

D/t=20

0.4

D/t=30

0.3

D/t=40

0.2

D/t=50

0.1 0 0

50 100 150 200 Concrete compressive strength (MPa)

250

(b)

1 0.9 0.8

Pu/Pr

0.7

0.6 0.5

f'c=60 Mpa

0.4

f'c=120 Mpa

0.3

f'c=190 Mpa

0.2 0.1 0 200

250

300 350 400 Steel yiled strength (MPa)

450

500

(c)

FIGURE 4.101

Residual strength ratio of CFST columns made with ultrahigh-strength concrete. (A) Effect of steel contribution ratio. (B) Effect of concrete compressive strength. (C) Effect of steel yield strength. Patel VI, Hassanein MF, Thai H-T, Al Abadi H, Elchalakani M, Bai Y. Ultra-high strength circular short CFST columns: axisymmetric analysis, behaviour and design. Eng Struct 2019;179:268e83.

483

4.3 Investigating the behavior of composite members

60000 50000

Pr (kN)

40000 30000 f'c=60 Mpa 20000

f'c=120 Mpa

10000

f'c=190 Mpa

0 0

100

200 300 Steel yield strength (MPa)

400

500

400

500

(a) 80000 70000

Pu (kN)

60000 50000 40000 f'c=60 Mpa

30000

f'c=120 Mpa

20000

f'c=190 Mpa

10000 0 0

100

200 300 Steel yield strength (MPa) (b)

FIGURE 4.102 Effects of steel yield strength on residual and ultimate axial strengths of CFST columns made with ultrahigh-strength concrete. (A) Residual strength. (B) Ultimate axial strength. Patel VI, Hassanein MF, Thai H-T, Al Abadi H, Elchalakani M, Bai Y. Ultra-high strength circular short CFST columns: axisymmetric analysis, behaviour and design. Eng Struct 2019;179:268e83.

corrosion of steel tubes is becoming a major concern in practice since the environment is increasingly complicated with ever-expanding applications, such as being exposed to the industrial atmosphere, seawaters, and saline soil environment. Between anticorrosive materials (such as stainlesssteel tubes and FRP tubes), galvanized corrugated steel pipe (CSP) has been well developed and used in civil engineering for more than a century [68e71]. Cold pressing and rolling are used to create the CSP. A spiral lockseam is created by corrugating a flat steel sheet first and then folding both of its edges together to connect. Cold pressing and rolling are used to create the CSP. A spiral lock-seam is created by corrugating a flat steel sheet first and then folding both of its edges together to connect, as shown in Fig. 4.103. It is then rolled up to turn into a circular tube. Structure corrosion can be effectively addressed by using a zinc coating on CSP to provide

484

4. Numerical methods

FIGURE 4.103

Corrugated steel pipe/tube [76].

protection. In North America and Canada, CSP culverts have reportedly been in use for decades without any maintenance [72e75]. There are various examples of CSP and concrete being used together in the early 21st century, including bridge piers, pile foundations, highway bridges, and pier rehabilitation [77], as displayed in Fig. 4.104. Thinwalled CSP may be used as a concrete formwork because of the significant stiffness of the corrugation [78], and research on CFST columns with vertical corrugations has shown that the local stability of fabricated columns can be improved as a result [79,80]. When a circular tube is corrugated, it has a larger capacity for absorbing energy under lateral loading [81,82]. According to the Handbook of Steel Drainage and Highway Construction Products [69], nearly any needed service life for CSP buildings may be achieved by selecting suitable coatings. Zinc oxide layer formation on the surface and cathodic protection of the iron benefit the structures [83,84]. There is also no easy way to remove the cladding layer, reducing maintenance costs and increasing CSP structural life. As a result, concrete-filled steel tubular (CCFT) should have better corrosion resistance than conventional CFT structures. When flat sheets are deformed to become corrugated, the cross-sectional inertia moment and strength both rise noticeably, improving the mechanical performance of CFCST members, as well as various other potential benefits. (a) The construction may be effectively improved by employing CSP as a concrete form, which benefits from CSP’s ease of assembly; (b) Because of its low axial stiffness, CSP may not fully transfer longitudinal load when compressed, which is advantageous for confining the concrete core. A CFCST column’s behavior may be comparable to that of a tubed-reinforced concrete (TRC) or a CSFT column. As a result, CFCST columns may perform well when subjected to compression and lateral cyclic loads; (c) Because of the

485

4.3 Investigating the behavior of composite members

(a)

(b)

(c)

(d)

FIGURE 4.104

Practice case of CSPeconcrete composite structures[76]. (A) Pile foundation (B) Rehabilitation. (C) Wind turbine pile foundation (D) Old bridge over the La Have River.

corrugation of CSP, there is a robust interlocking effect between CSP and concrete, which prevents the outer tube from being separated from the concrete core; (d) During transportation and construction, the initial imperfections of the tubes can be reduced or avoided. Because CSPeconcrete constructions are still a relatively novel application, limited is known about the combined structural effects of CSP and concrete. The CSP is frequently utilized just as a corrosioneprotection template for the concrete. This section investigates the structural performance of CFCST columns. The confinement effect given by CSP substantially adds to the structural benefits of CFCST, not only under concentrically axial compression but also under eccentric compression. Thorough knowledge and precise quantification of the confinement effect is a crucial issue that must be addressed is. Therefore, the nonuniform confinement effect in CFCST members is also discussed herein. The symbols shown in Fig. 4.103, as well as other symbols used in the following section, are introduced in Table 4.28. Based on the standards and manufacturers’ data [68e71], the geometric properties of CSP are presented in Table 4.29.

486

4. Numerical methods

TABLE 4.28 Nomenclature of symbols used to investigate the behavior of CFCST members. Ac

Cross-sectional area of concrete

As

Effective cross-sectional area of tube

B

Width of corrugated belt

B0

Width of flat unprocessed sheet

Din

Inner diameter

Dout

Outer diameter

D0

Nominal diameter

L

Effective length of specimen

M

Bending moment

Mu

Ultimate bending moment

N

Load

Nu

Ultimate load-bearing capacity

R

Arc radius of corrugation

Z

Relative height to trough

TL

Length of flat segment of corrugation

D

Lateral deflection/displacement

Du

Lateral deflection/displacement on the peak load

b

Width of lock seam

e

Eccentricity of load

fy-cr

Yield strength of crest

fy-mid

Yield strength of middle

fy-tr

Yield strength of trough

fr

Confining stress at peak load

fr0

Confining stress at peak load of concentrically axial compression

h

Wave height of corrugation

l

Wave length of corrugation

r

radius (polar coordinates)

r0

Nominal radius

t

Thickness of tube

wcr

Arc length of crest

wmid

Arc length of middle

4.3 Investigating the behavior of composite members

487

TABLE 4.28 Nomenclature of symbols used to investigate the behavior of CFCST members.dcont’d wtr

Arc length of trough

x

Cross-sectional depth/coordinates

z

Vertical coordinates/height

a

Steel ratio of tube

b

Radius of curvature

ε

Strain

εcc

Compressive strain of confined concrete corresponding to the strength

εh

Horizontal strain/hoop strain

εv

Vertical strain

h

amplification coefficient for steel ratio

lcr

Weighted coefficient of crest

lmid

Weighted coefficient of middle

ltr

Weighted coefficient of trough

q

Helical angle of corrugation

s

Stress

sc

Compressive stress of concrete

se

Mises stress

sh

Hoop stress

sh-cr

Hoop stress at crest

sh-tr

Hoop stress at trough

ss

Stress of steel

st

Tensile stress of concrete

sv

Vertical stress

shu

Average hoop stress at the ultimate eccentric load

sh-cs

Average hoop stress on the extreme compression side

sh-ts

Average hoop stress on the extreme tension side

sh0

Average hoop stress at the ultimate axial load

u

Offset angle in the cross-section

x

Confinement factor

488

4. Numerical methods

TABLE 4.29 Geometric parameters of CSP. l 3 h ðmmÞ

38 3 6.5

68 3 13

75 3 25

125 3 25

qs;max ð Þ

68e69

62e63

43e45

56e57

R0 ðmmÞ

7.1

17.5

14.3

40

wcr ðmmÞ

5.4

17.2

23.0

47.3

wmid ðmmÞ

28.7

36.0

42.5

39.1

wtr ðmmÞ

5.4

17.5

23.0

47.3

ws ðmmÞ

39.5

70.4

88.5

133.7



Like conventional CFST/CFDST members, geometric and material properties affect the structural performance of CFCST members. For instance, the steel ratio a can highly influence the load-bearing capacity of CFCST columns, which is similar to CFST/CFDST members. However, due to the corrugation in CSP, the calculating procedure for CFCST is different from that for circular CFST members. A cold-formed steel sheet is used to make CSP, crimped to clamp along the spiral helix, as displayed in Fig. 4.105. The distance between each part of adjacent crimps B is displayed in Fig. 4.105. The width of the virgin (flat unprocessed) sheet is represented by Bo. The steel consumption of CST is BBo times that of a flat tube with the same diameter and thickness. Accordingly, a of CFCST can be expressed as follows: a¼h

FIGURE 4.105

4ts Do

Width of sheet and corrugated belt [85].

(4.122)

4.3 Investigating the behavior of composite members

489

in which h ¼ BBo ¼ wl s (ws is the arc length of one corrugation period and l is the wavelength, shown in Fig. 4.103) refers to the corrugation and crimping amplitude factor, ts and Do are the wall-thickness and the nominal diameter of CSP, respectively. Do is taken as the average of the outer diameter and inner diameter. The steel consumption of CSP is larger than that of a flat tube. Hence, the amplification coefficient h is used to describe that. In the case of flat tubes, h ¼ 1 due to ws ¼ l. Another parameter that can affect the performance of CFCST is the steel yield stress. In general, the yield strength of virgin steel fy
vir is directly employed to design conservatively in CSP constructions. However, the mechanical properties of virgin steel are substantially altered by cold-roll forming and cold bending [86]. Crimps are cold-formed in practical manufacturing due to cold bending, rolling, and pressing, as shown in Fig. 4.106. The crest and trough endured bending and rolling, with the crest area suffering more cold rolling than the trough due to its outermost location. The middle area is only strengthened in the final rolling if cold bending is not conducted. Finally, the procedure results in increased strength. Consequently, CSP does not have a uniform material property. As a result, different areas, such as the crest, middle, and trough, must be tested for steel properties, as shown in Fig. 4.107. Based on design codes [87,88], the horizontal coupons must be cut at each original location, as displayed in Fig. 4.107A. As seen in Fig. 4.107B, the strength of CSP is improved significantly, but its plasticity is decreased when compared to its base sheet. It exhibited typical cold-form steel tensile performance with no evident yielding plateau. Fig. 4.107C and D display the longitudinal

FIGURE 4.106 Manufacture of CSP [85].

490

4. Numerical methods

FIGURE 4.107 Test of coupons and hollow CSP [76]. (A) Horizontal coupons. (B) Comparison of horizontal coupons. (C) Tensile corrugated coupon. (D) Tensile lock-seam coupon. (E) Hollow CSP and corrugatedcoupons. (F) Enlarged view. (G) Tensile hollow CSP. (H) Compressive hollow CSP.

491

4.3 Investigating the behavior of composite members

coupons. The longitudinal strain condition is found to be reverse on both sides of the corrugation, corresponding to the inner and outer surfaces of CSP. This observation suggests that the strain distribution is not uniform across the thickness of the steel skin. Wang et al. [85] discovered that the yield strength of the crimps region is 40% more than that of the virgin sheet, 37% greater than that of the crest region, 17% greater than that of the trough region, and 11% greater than that of the middle zone. There are two basic degrees of strength based on the four previously described cold-formed places. The crest and crimps at the upper level have a 20% stronger strength than the remainder of the structure, which includes the trough and intermediate levels. When importing the yield strength into the steel constitutive model, this large discrepancy is also taken into account in the subsequent stress analysis. The yield strength of CPS is represented by weighted average yield strength, defined by the following: fy ¼ lcr fy
cr þ lmid fy
mid þ ltr fy
tr

(4.123)

The weighting coefficients “l”s for distinct parts show their arc length proportion of arc length in each corrugation. The subscripts “cr,” “mid,” and “tr” denote the corrugation’s crest, middle, and trough, respectively. The weighting coefficients are presented in Table 4.30, based on the standards and manufacturers’ data [68e71]. The confinement factor and ultimate load-bearing capacity of CFCST can be calculated using the average yield strength fy. However, the corresponding strengths (i.e., fy
cr , fy
mid , and fy
tr , respectively) should be used in the steel constitutive model to accurately identify the local stress behavior while modeling the local stress of CSP and its confinement mechanism. Due to the existence of corrugation and lock seams over the height, corrugated coupons and lock-seam coupons must be examined under tension (see Fig. 4.107C and D). Because compressive testing is complex, the compressive corrugation is studied using a finite-element model. The whole deformation of the corrugated coupons under tension is a combination of flatting and stretching of the material. The smaller they deformed, the flatting effect accounted for a greater share of the TABLE 4.30 The weighting coefficients. l 3 h ðmmÞ

38 3 6.5

68 3 13

75 3 25

125 3 25

lcr

13.7%

24.5%

26.0%

35.0%

lmid

72.6%

51.0%

48.0%

30.0%

ltr

13.7%

24.5%

26.0%

35.0%

492

4. Numerical methods

deformation. In Fig. 4.107E, there are two steps for the flatting procedure: OA and AB. The yield of crest and trough causes the decrease of rigidity in stage AB. The strength can approach the ultimate strength of steel after flatting. However, a very small tensile strength (near to 0.1fy ) may be obtained during testing the lock seam coupons, as displayed in Fig. 4.107F. This is due to the fact that it opens preferentially throughout corrugation deformation. As a result, it is unable to fully utilize the steel material’s strength. Even though large stresses on the surfaces have been achieved, the rigidity and strength of corrugation are both modest when compressed (about 0.1fy and 0.03 Es As , respectively). This is due to the bending action at the crest and trough, which causes longitudinal stress offset from the outer to the inner surface. The lock seam determines the behavior of the hollow CSP under tension, as shown in Fig. 4.107G and H, where its stiffness and ultimate strength are near the lock-seam coupons, suggesting that such a clamped hollow tube is not appropriate for bearing tensile load. On the contrary, the lock seam adds to the compressive behavior of CSP owing to its stiffening effect, together with that after encircling to be a tube, resulting in a 50% increase in compressive strength when compared to corrugation alone. Despite this, the value is just 60 N/mm2, or 0.17fy . In summary, the hollow CSP is not appropriate for bearing axial load independently because of the negative influence of corrugation bending effect, stability difficulty under compression, and opening of lock seam under tension. However, the performance of CSP is fundamentally altered after filling with concrete. Concrete significantly eliminates the local buckling of corrugation. Besides, longitudinal bars can be added to limit the opening of the lock seam. Following then, the CSP switches to lateral confinement rather than carrying the axial stress. Comparison of CFCST, CFST, and RC

A nonlinear FE analysis was performed to compare the behavior of a CFCST stub column with its conventional CFST and reinforced concrete (RC) column with almost the same steel ratio and uniform concrete strength. The corrugated circular steel tube has a dimension of Do  ts  L ¼ 305  1:42  840 mm and yield strength of 297.8 MPa filled with the concrete with a compressive strength of 47.2 MPa. Fig. 4.108 shows the axial load-strain histories of CFCST, CFST, and RC columns. Comparison of results shows that the ultimate axial strength of CFCST (a ¼ 2.3%, x ¼ 0.19) is 5% higher than the CCFT column (a ¼ 2.2%, x ¼ 0.16) and 28% higher than the RC column (a ¼ 2.3%). As a result, the ultimate axial strength of the CFCST column is marginally greater than that of the CCFT column and significantly higher than that of the RC column. Additionally,

4.3 Investigating the behavior of composite members

FIGURE 4.108

493

Comparison of the CFCST column with the conventional CFST and RC

columns [85].

the strain εcc corresponding to the ultimate axial strength of the CFCST column is remarkably greater than CFST and RC columns. Compared to CFST and RC columns, the value of εcc for CFCST is increased by 47% and 92%, respectively. Hence, the deformation corresponding to the ultimate axial strength of the CFCST column is significantly delayed compared to the CCFT and RC columns. In general, CFCST columns have a slightly higher strength index SI and ductility than typical CFST ones. For the same steel ratio and dimensions, CFCST columns have a much higher strength index SI and ductility than RC columns. Influence of geometries

The uniqueness of CFCST lies in the outer tube’s corrugated profile, which has the potential to affect how such members behave. According to the requirements for spiral CSP concluded from Refs. [68e71], five parameters influence the profile and its size, which are described in Table 4.31 and Fig. 4.109, including the corrugation type l  h, the crimps spacing B, the nominal diameter Do, the helical angle q, and the thickness ts . The effects of these parameters are investigated using an FE analysis performed on a CFCST stub column with a dimension of Do  L ¼ 600  1800 mm. • Multiple-factor analysis According to the practical matching of parameters, four types of corrugation are first examined based on the geometry displayed in

494

4. Numerical methods

TABLE 4.31 Demand for CSP [85]. D0 ðmmÞ

q ð Þ

l 3 h ðmm 3 mmÞ

B ðmmÞ

38  6.5

152

150e600

5e18

1.2

1.6

2.0

e

51  13

153

150e900

3e18

e

1.6

3.2

4.2

68  13

544

300e2400

4e30

e

1.6

3.2

4.2

75  25

525

900e3600

3e11

e

1.6

3.2

4.2

FIGURE 4.109

ts ðmmÞ

Corrugation profile [85].

Fig. 4.109 inferred from GB/T 34,567 [71]. Despite that the length l and depth h of corrugation are utilized in pairs, the h=l ratios of these mixed patterns are distinct. In such a situation, the h=l ratio is used to denote various forms of corrugation. Once D0 and l  h are known, the B and q are uniquely defined by fixed manufacturing techniques. The first two types are appropriate for a 185 mm-wide virgin sheet, leading to four periods of waves within every two adjacent crimps for 38 mm  6.5 mm (B ¼ 4l) and three periods for 51 mm  13 mm (B ¼ 3l). While the latter two types are appropriate for the 692 mm-wide virgin sheets, they result in eight periods of waves for 68 mm  13 mm (B ¼ 8l) and seven periods for 75 mm  25 mm (B ¼ 7l) between every two crimps. As can be seen in Fig. 4.110, there is no monotonic rise or reduction as a result of the combined parameters like h=l, B, and q that have been matched with each other. Nevertheless, the following discussion points are provided: (a) The h=l ratio indicates the virgin sheet’s folding extent and, in turn, the steel consumption (or steel ratio a). A hypothetical conclusion is that as the h=l ratio increases, the load-bearing capacity Nu enhances because of the increasing a. However, the Nu of “75 mm  25 mm” is placed second with the biggest h=l ratio, which is smaller than “51 mm  13 mm.” It is worth noting that the latter’s crimps are the most intensive (B ¼ 3l), and it can raise the steel ratio and provide local reinforcement for the concrete, implying that crimps spacing has a more significant impact on Nu than h=l.

4.3 Investigating the behavior of composite members

FIGURE 4.110

495

Effect of practical types [85].

(b) Because the diameter is 600 mm, the crimps spacing B of “38 mm  6.5 mm” is 4l and B of “51 mm  13 mm2 is 3l, resulting in a slight helical angle q of just 5 . The hoop stress increases as the helical angle is reduced, improving the confined concrete strength fcc0 and corresponding strain εcc . The cross-sectional hoop stress shy of these two types is higher than the other two, as seen in Fig. 4.110, possibly due to the smaller q. As a result, a small helical might help to achieve a high level of confinement. As a result, the dense crimps and narrow helix angle may be advantageous to CFCST behavior. However, because of the varied matching patterns of the CSP parameters in practice, the impact of the h= l ratio cannot be determined directly. There may be an optimal matching of these parameters for CFCSTs in order to obtain good behavior. Independent parameters are studied separately in the following discussion to give an acceptable type of CSP for future design and fabrication of such a member. • Single-factor analysis (a) Corrugation profile h=l. In Fig. 4.111, the impact of corrugation profile is examined by specifying specific parameters, such as diameter (D0 ¼ 600 mm), helical angle (q ¼ 0 ), and crimps spacing B equal to the length of column L (i.e., no crimp). As shown in Fig. 4.111A, the h=l ratio has only a modest effect on Nu since the corrugation is so little compared to the overall member size,

496

4. Numerical methods

FIGURE 4.111 Effect of corrugation profile [85]. (A) On Nu. (B) On average crosssectional stress

but it had a substantial effect on the stress components. As previously stated, the h=l ratio virtually depicts the folded shape’s extent, which might impact vertical stress. The average vertical stress svy of “75 mm  25 mm” is found to be tensile because of the most folded form, which has robust bowstring-like tension, as shown in Fig. 4.111B. This issue will be discussed further in the following section. Due to the flattest corrugation, the svy of “38 mm  6.5 mm” is determined to be compressive stress with a maximum amount of 0:24fy . As a result, with increasing folding extent (i.e., h=l), the svy is displayed as a decreased compressive value or a growing tensile value. The average cross-sectional stress shy for the type “68 mm 13 mm” is the highest, indicating the most substantial confinement effect. In addition, as shown in Fig. 4.109, this type has one of the lowest ws =l values, implying that steel consumption is reduced indirectly. Based on this scenario, the “68 mm 13 mm” is an optimum corrugation between these four varieties since it has the most substantial confinement effect and uses the least amount of steel. (b) Crimps spacing B. As previously indicated, the reinforcement provided by crimps would most likely strengthen the local part and limit lateral expansion. The crimps (together with the corrugation) serve to restrict the local maximum principal strain of CSP, and reducing the crimps spacing reduces the shy . While decreasing the space between crimps improves the steel ratio, leading in an increase in Nu . Accordingly, the improved Nu by crimps is not because of the strengthening confining stress but to the increase of transverse steel consumption, as displayed in Fig. 4.112C. Widening crimps spacing when the steel ratio is constant does not change the Nu , whereas Nu is improved with an increase in a. The improvement of Nu is found to be smaller than 5% in the typical range (4le8l), and it tends to be

4.3 Investigating the behavior of composite members

497

FIGURE 4.112 Effect of crimps [85]. (A) On deformed shape and principal strain. (B) On Nu and shy . (C) With a uniform steel ratio

uniform with widening spacing. Therefore, it is pointless to enhance the Nu by narrowing crimps, which diminish the plasticity of steel and the ductility of the member. Increasing the thickness may be a more efficient method to improve Nu . In this way, the plasticity of steel can be preserved, and the confinement effect can be enhanced. Consequently, in the case of CFCSTs, wide rolling belts (large-distance crimps) should be used. (c) Helical angle q. In practice, the helical angle is often between 0 and 20 , and hardly reaching 30 , for instance, when the diameter is small and crimps distance is large. According to this ultimate angle, the helical angle in FE analysis was taken as 0 e30 to examine its effect on Nu and stress. The corrugation profile had a dimension of 68 mm  13 mm and B ¼ L (i.e., no crimp). It can be observed from Fig. 4.113 that the main effect of q is not on Nu but the components of stress, particularly when q is greater than 10 . The average cross-sectional vertical compressive stress svy is reduced with reducing the helical angle, while the hoop stress shy increases with it.

498

4. Numerical methods

FIGURE 4.113

Effect of helical angle [85]. (A) On load-bearing capacity. (B) On crosssectional average yield stress

It was explained in the multiple-factor analysis that increasing shy is advantageous for improving fcc and Nu, whereas the decreased svy indicates less contribution of CSP to direct loading. These two counteractions and the small common range of 0 e20 lead to a slight improvement of Nu with the reduction of q. However, strong confinement and adequate ductility can be achieved using a small helical angle. (d) Thickness t s . It can be observed from Fig. 4.114 that increasing ts can remarkably improve the Nu because of increased steel consumption. (a) Steel ratio a.

FIGURE 4.114

Effect of thickness [85].

4.3 Investigating the behavior of composite members

FIGURE 4.115

499

Effect of steel ratio [85].

The steel ratio a is a comprehensive parameter that can highly affect the geometries of CSP. Except for the independent parameter q, all other geometric parameters are included in a. A uniform tendency with changing a regardless of how geometric parameters alter, including variable h=l (0.17e0.33), B (2e10l), and ts (3.2e6.0 mm), can be found by comparing the results shown in Fig. 4.115. As a result, the effect of the CSP geometries on Nu should be expressed by a. Confinement mechanism

The CFCST can be categorized as confined concrete. The convex crest of CSP interrupts the compressive load distribution, which is the primary reason. Another possibility is that thin-walled CSP carries minimal compression load and only serves to confine the concrete. CSP crest likely moves outward under compression and functions flexibly without any external support, causing tensile behavior at the crest and physically preventing axial load transfer. Fig. 4.116 presents the local strain distribution of a CFCST column with a circular steel tube with a dimension of Do  ts  L ¼ 308  1:68  851 mm and yield strength of 416.1 MPa filled with the concrete with a compressive strength of 47.2 MPa. The trough of CSP tends to move inward when a CFCST column is under compression. However, the inward movement of tends is restricted by the in-filled concrete because of the frictional force among CSP and the compressed concrete, causing a vertical compressive strain at the trough, as shown in Fig. 4.116A. The CSP tends to withstand the lateral expansion of the concrete core as the plastic dilatation of the concrete increases. As a result, a hoop strain may

500

FIGURE 4.116

4. Numerical methods

Hoop and vertical strains [85]. (A) Strain at trough. (B) Strain at crest

develop. Because the CSP may transfer compressive load at the trough when supported by concrete inside, the vertical strain at the trough is always compressive, while the hoop strain is always tensile, which acts similarly to the outer tube of CFT columns. Because of the lateral expansion of concrete, the hoop strain should be tensile at the crest, as indicated in Fig. 4.116B. Because of the combined impact of compression and bowstring-like tension, it is hard to precisely determine the property of vertical strain at the crest. Under compression, the crest tends to move outward and acts flexibly without any external support, resulting in a vertical tensile strain. The vertical strain at the crest (Fig. 4.116B) first exhibited tension but then shifted to compression as the axial load approached the maximum capacity Nu. The fact that the vertical strain changed from tension to compression revealed that, for the crest of CSP, bowstring-like pugging impacts the vertical strain during the early loading stage, whereas compression has a growing effect on the vertical strain during loading. When the load reached the ultimate Nu , the latter influence eventually overtook the former, producing a slight vertical compressive strain. In conclusion, even if the CSP may carry the load at the trough during the compression, the fraction of the trough is less than half of the crosssection, resulting in only a negligible contribution to bearing. Meanwhile, because the vertical strain is usually low, the vertical load can barely cross the crest. The axial load-bearing capabilities of CSP may be neglected based on these two factors, demonstrating that the CFCST is a tube confined concrete member. • Strain of CSP At various places, the confinement effect is variable. The transverse deformation coefficient εεhv is determined by dividing the hoop strain εh by the vertical strain εv , and it is employed to describe the confinement level at different areas of the CSP. At crests, there are two opposing effects

4.3 Investigating the behavior of composite members

FIGURE 4.117

εh εv

501

ratio of CSP [85].

(tension at first, compression afterward during the residual stage), causing the value of vertical strain at the crest (εv
crest ) to grow steadily. As a result, the εεhv ratio at crest increases quickly with load, as demonstrated by the absolute value in Fig. 4.117, indicating a rapidly expanding confinement at the crest. The vertical strain tends to transition from a tensile strain to a compressive strain as the load approaches the peak, resulting in a value near zero and an accelerated εεhv . The εεhv of the trough, on the other hand, grows much more slowly and is always smaller than 1.0. The confinement in the trough zone of CSP is weak, while it is significant in the crest region. When compared to conventional tube confined concrete, this nonuniformity may result in different overall confinement. Fig. 4.118 compares the strain of CSP with that of the circular tubes on the conventional CFST column. Both the compressive vertical strain and the tensile hoop strain of CSP and the flat tube of CFST increase in opposite directions, i.e., the compressive vertical strain due to column shortening and the tensile hoop strain due to lateral expansion, which follows the typical deformation of CFST columns. The distinction is that the CSP hoop strain is always greater than the flat tube hoop strain. After the load is increased to the ultimate amount (NNu ¼ 1:0), the bigger hoop strain of CSP can be seen at the crest, where it is much larger than the circular tube of CFST. This outcome indicates that CSP confinement is more substantial than flat tube confinement, particularly after the peak load, which is represented in ductility, as the ductility of CFCST is more than that of CFST, as discussed before. • Stress of CSP and the in-filled concrete

502

FIGURE 4.118

4. Numerical methods

Strain of CSP and flat tube [85]. (A) Trough and flat tube. (B) Crest and flat

tube

When the load reach to the ultimate amount (NNu ¼ 1:0) (see Fig. 4.119), the vertical strain of CSP was shown to be consistently compressive at the trough and tensile at the crest, implying that there must be a transition point between the trough and the crest where the vertical stress equals zero. In such a situation, a wave’s period may be split into two halves, resulting in separating two independent ideal bodies by the cutting plain

FIGURE 4.119

Local mechanics [85].

503

4.3 Investigating the behavior of composite members

where the transition point is placed, called the “trough area” and “crest region,” as shown in Fig. 4.119. For each body, two reference planes, i.e., RP1 and RP2, are placed on the trough and crest points, respectively. There is a pair of frictional forces f between the interior surface of CSP and concrete in each reference plane. The horizontal component of frictional force f in RP1 counteracts the normal force n at the trough, based on the mechanicalR analysis of these two bodies with respect to each reference plane (i.e., ½f cos l
n sin ldX, where l ¼ arctan Y0 ðXÞ). The horizontal R superposition of f and n in RP2 (i.e., ½f cos l þn sin ldX) raises the hoop stress sh and the effective confining stress fr in the crest area. Accordingly, CSP’s crest may have a greater hoop confining effect than the trough. The horizontal and vertical stress components are estimated using an elasticeplastic analysis with shear taken into account for validation. Based on two classified levels (see Fig. 4.107B), the crest and trough yield strength are imported into the constitutive equations of steel. The stress components of a CFCST specimen (see Table 4.32) are used as examples, including hoop stress sh , vertical stress sv , shear stress shv , and equivalent stress se (or Mises stress) at the trough and crest of CSP, as shown in Fig. 4.120. Eq. (4.124) determines the equivalent stress of CSP: se ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2h þ s2v
sh sv þ 3s2hv

(4.124)

Vertical stress at the trough is continuously compressive with supporting normal force and a pair of frictional forces, as indicated in Fig. 4.120A. This is similar to the stress distribution shown in Fig. 4.119. However, the pulling effect along the convex of corrugation induced a tensile behavior at the crest, as shown in Fig. 4.119. Due to the formation of vertical cracks, the lateral expansion is accelerated toward the peak load as the load increases, inducing an increasingly tight normal contact and vertical frictional force subjected to compression. When the load reaches the peak, the increasing frictional force causes a reduction in vertical tensile stress at the crest. The hoop stress increases quickly, whereas it grows slowly at the trough, as shown in Fig. 4.120B. Increasing the load and dilation of infilled concrete accelerates the rise of hoop stress, particularly by TABLE 4.32 Geometric and material properties of examined CFCST stub columns [85]. Specimen

Do (mm)

ts (mm)

L (mm)

fy (MPa)

fc0 (MPa)

a

x

CFCST 1

308

1.68

851

416.2

47.2

2.7%

0.31

504

4. Numerical methods

FIGURE 4.120

Stresses of CFCST specimens [85]. (A) Vertical stress. (B) Hoop stress. (C) Shear stress. (D) Equivalent stress. (E) Vertical stress of concrete

approaching the peak load. This discrepancy implies that the crest’s confinement effect is stronger than the trough’s. As displayed in Fig. 4.120A, the vertical stress at the trough is compressive with a magnitude of
345.5 MPa, whereas the stress at the crest is tensile with a value of þ169.9 MPa. Therefore, the cross-sectional

505

4.3 Investigating the behavior of composite members

TABLE 4.33 Geometric and material properties of examined CFCST stub columns [85]. Specimen

Do (mm)

ts (mm)

L (mm)

fy (MPa)

fc0 (MPa)

a

x

CFCST 2

306

1.20

850

411.5

47.2

2.0%

0.21

CFCST 3

305

1.42

843

297.8

47.2

2.3%

0.19

CFCST 4

308

1.68

851

416.1

47.2

2.7%

0.31

CFCST 5

307

1.68

850

344.7

47.2

2.7%

0.24

vertical stress svy is on average 87.8 MPa, which is 21% of the yield strength fy of the CFCST specimen, resulting in a low level of the directly load-bearing capacity of CSP. The load-bearing proportion of CSP in four CFCST stub columns under axial compression is given in Fig. 4.121. The geometric and material propeorties of the examined columns are given in Table 4.33. It can be observed from Fig. 4.121 that the contribution of CSPs to capacity of the column is less than 5%   the load-bearing Ncsp sv A s N ¼ N
> > > shy ðuÞdu > > > 0 > < shy ¼ 2p (4.128) Z 2p > > > > svy ðuÞdu > > > > svy ¼ 0 : 2p Based on the experimental tests performed by Wang et al. [85], the average amount of shy is 0:77fy (tensile), and the average value of svy is
0:25 (compressive). The hoop yield stresses of circular TRC columns and CFST columns are 0:54 fy [89] and 0:41fy [85]. This indicates that the CSP’s hoop confinement is more robust than that of the flat tube. Due to the helical corrugation, the cross-section of CFCST is not a perfect circle; hence, a simplification is performed to attain a consistent value of fr . A single period of corrugation is considered as the unit object, as indicated in Fig. 4.125. The fr is calculated using the circumferential and vertical integrals of hoop stress as follows: Z l 2 shy ðXÞdX:ts   ws 2ts shy ¼ (4.129) fr ¼ 0 lDo l Do

4.3 Investigating the behavior of composite members

FIGURE 4.125

509

Internal stresses of the CPS [85].

where ws denotes the path length of one period of corrugation, l represents the length of the wavem, and shy is the cross-sectional hoop stress at the peak load. It can be observed from Eq. (4.129) that fr is a function of the wl s ratio. The wl s ratio for the flat tube is 1.0, whereas the wl s ratio for CSP is always bigger than 1.0, indicating that the confinement effect provided by the CSP is more robust than that of the flat steel tube. As shown in Eq. (4.129), shy influences the amount of the confining pressure fr . As discussed before, shy is affected by the h=l ratio, B, ts , and q (see Figs. 4.110e4.114). It was also argued that q has the most significant influence on shy compared to other parameters. Even though the q has a minor impact on Nu , it has a significant effect on the contribution of stress components and cannot be overlooked when predicting hoop stress. Other f

parameters can be expressed using the confinement coefficient x ¼ a fy0 to c

make it easier to reflect the of shy . These two independent parameters have a specific amount as long as the geometric and material properties of CSP are known. Based on Ref. [85], shy can be expressed as follows: pffiffi shy ¼ 0:78ðcos qÞe0:24
0:4 x fy (4.130) in which q is calculated based on Fig. 4.123. Behavior of CFCST beam columns

The main focus of the previous section was to investigate the confinement mechanism of CSP. Therefore, no steel reinforcement was used in the concrete of CFCST columns. However, longitudinal bars or

510

4. Numerical methods

structural steel should be positioned to resist the bending moment in practical engineering. Accordingly, reinforcements are used in the investigation of CFCST beam columns. Fig. 4.126 shows a schematic view of the fabrication process of CFCST beam columns with reinforcements. The embedded reinforcement ratios of the longitudinal bars and the stirrups are taken as 2.2% and 0.2%, respectively. The bending deflection of CFCST columns is discussed in this section. For convenience, the columns are labeled as shown in Fig. 4.127. Fig, 4.128 shows the bending deflection of nine CFCST stub columns under eccentric compression. The geometric and material properties of specimens are represented in Table 4.34. Fig. 4.128A compares the deformed shape of columns at the ultimate with the half-sine curve. Besides, the deformation during the loading history is also compared in Fig. 4.128B, where the

FIGURE 4.126

Fabrications of CFCST beam columns [76].

FIGURE 4.127 Nomenclature of columns.

TABLE 4.34

Geometric and material properties of examined CFCST stub columns under eccentric compression [76].

Specimens

Din

Dout

D0

t (mm)

a (%)

x

e (mm)

fy (N/mm2)

fco (N/mm2)

C-1.45-25-1

255.7

272.0

263.9

1.452

2.29

0.24

25

316.7

30.4

C-1.45-25-2

255.5

272.4

264.0

1.447

2.28

0.24

C-1.45-25-3

255.5

272.1

263.8

1.451

2.29

0.24

C-1.45-50-1

255.7

272.4

264.1

1.449

2.28

0.24

50

316.7

30.4

C-1.45-50-2

255.5

272.4

264.0

1.450

2.28

0.24

C-1.45-50-3

255.6

272.2

263.9

1.444

2.28

0.24

C-1.45-75-1

255.6

272.1

263.9

1.450

2.29

0.24

75

316.7

30.4

C-1.45-75-2

255.6

269.9

262.8

1.450

2.30

0.24

C-1.45-75-3

255.5

272.0

263.8

1.451

2.29

0.24

4.3 Investigating the behavior of composite members

Diameter (mm)

511

512

4. Numerical methods

FIGURE 4.128 Bending deflection of CFCST beam columns [76]. (A) Deformed shape at ultimate load. (B) Typical deformed shape during loading

column C-1.45-50-2 is taken as an example. As shown in Fig. 4.128, the columns deformed symmetrically due to the bending moment, which matches the half-sine wave. As the loading progressed, the agreements tended to improve. Based on the results, the sine function can describe the deformed shape of the CFCST member exposed to bending moment, which is a basic assumption of the numerical fiber-beam element model for CFST and TRC beam columns [40,90]. The load-bearing capacity Nu and ultimate bending moment Mu are affected by two variables, namely the eccentricity ratio e=ro and the confinement coefficient x, in which ro ¼ D2o denotes the nominal radius of the CSP. The effect of the eccentricity ratio e=ro on the performance of specimen C-145 is presented in Fig. 4.129 in which the e=ro ratio is in the range of 0e75. As shown in Fig. 4.129A, the loadelateral deflection response is highly affected by the e=ro ratio. Nu is decreased significantly by increasing the e=ro ratio from 0 to around 0.57, as displayed in Fig. 4.129A and B. For instance, when e=ro is increased from 0 to 0.19, Nu is diminished by 20%e30%. However, when e=ro is more than about 0.57, its effect on Nu is decreased. As seen in Fig. 4.129C, as the e=ro ratio increases from 0 to the value of pure bending columns, the deflection at ultimate load Du continues to increase. Besides, the value of Du is increased by increasing x from 0.22 to 0.87. The reason is that the stronger confinement leads to a more considerable compressive strain of concrete in the compressive zone, and therefore, greater curvatures and larger Du. The ultimate bending moments Mu and the ultimate axial compressive strengths Nu are determined in pairs by examining columns with similar characteristics (i.e., cross-sectional dimensions, height, reinforcements, and material properties subjected to axial compression with varied eccentricity ratios), as shown in Fig. 4.130. CFCST beam columns subjected to small load eccentricities (notably when reo < 0:2) can resist the compression effectively, despite Nu diminishes with an increase in Mu . When reo is larger than 0.2, Nu reduces substantially, while the Mu increases

4.3 Investigating the behavior of composite members

513

FIGURE 4.129 Effect of load eccentricity on CFCST beam columns [76]. (A) On average N-D curve of specimens. (B) On Nu. (C) On Du

FIGURE 4.130

Nu
Mu envelopes of CFCST beam columns [76]. (A) C-1.45. (B) C-1.65

with less intensity. Under high eccentricity, the tendency is the opposite, and CFCST members are at a disadvantage in terms of axial capacity. The CSP curves easily, much like the bellows of an accordion. In this situation, CSP can barely support the axial load and bending force, but it mainly

514

4. Numerical methods

confines the concrete, comparable to FRP-confined concrete beam columns [91]. The Nu
Mu curve also has a balanced failure point corresponding to the critical value of the eccentricity ratio, which is similar to the behavior of CFT members when compressed eccentrically. The Mu achieves its maximum amount at the balanced failure point. For instance, the critical eccentricity ratios corresponding to the confinement coefficients x of 0.24 and 0.39 are 0.61 and 0.69, respectively. Performance mechanism of CFCST columns under compression-bending combined loading

As discussed before, the geometry of CFCST is distinctive because of its helical corrugations, which makes the working mechanism different from conventional CFST and TRC members. The stresses in CFCST beam columns are non uniform in three directions: through-thickness, height, and cross-sectional depth. There is a local bending effect at the crest and trough due to the curved shape, resulting in a nonuniform distribution of strain and stress across the thickness of CSP. Moreover, unequal stress values are caused by the opposite curved directions from crest to trough, resulting in nonuniform stress distribution over the height. Because of the eccentric compression load, the stress varies over the cross-sectional depth. Nonuniformities in these three directions eventually influence the confinement and cross-sectional behavior. Stress of CSP over the thickness Because of the local bending effect, the longitudinal stress state in Fig. 4.131 is opposite from the internal to the

FIGURE 4.131 Longitudinal stress at ultimate eccentric load [76]. (A) Compressive zone. (B) Tensile zone

4.3 Investigating the behavior of composite members

FIGURE 4.132

515

Hoop stress at ultimate eccentric [76].

external surface, which corresponds to the results shown in Fig. 4.107C and D. For instance, in a compression zone, the crest longitudinal stress is tensile on the external surface and compressive on the internal surface. Because the corrugation becomes compressed, the outer and inner fibers of the crest become longer or shorter, respectively. Similarly, the tendency is opposite at the inward concave trough (valley). Instead, the CSP tends to be flat in the tension area. Thus, the stress state of the CSP skin is the inverse of that of the compressive zone. As a result of this, the hoop stress is likewise nonuniform, as seen in Fig. 4.132. According to the overall deformation state, sv acts compressive in the compression region and tension in the tension region, as illustrated in Fig. 4.132. The net value of CSP longitudinal stress is substantially lower as compared to the tube of TRC beam columns, but the hoop stress value is significantly larger. For instance, in the compression region of CFCST and TRC [91] under the load eccentricity ratio reo of 0.1, the value of sv at the ultimate strength of CSP is in the range of
0:15 and
0:25 fy , as shown in Fig. 4.133, while it ranges from
0:7 to
0:8 fy in the flat steel tube. In the tension area, the former varies only in the range of þ 0:22 to þ 0:28 fy , as displayed in Fig. 4.134, while the latter can reach þ 1:0fy . The value of sh of CSP at the ultimate load is in the range of þ0:63 and þ 1:2 fy , while for flat tube, it is in the range of þ0:33 and þ0:82 fy over the section. Accordingly, it can be concluded that the contribution of CSP to the axial load is small, but it can efficiently confine the concrete core. Stress of CSP over the height Aside from nonuniformity in thickness, the aforementioned local bending action also causes nonuniformity in height, as displayed in Figs. 4.133 and 4.134. For instance, in the compressive area, the amount of sv is greatest at the trough and the lowest at the crest. If the equivalent stress (Mises stress) is provided in a plane stress

516

4. Numerical methods

FIGURE 4.133

CSP net longitudinal stress over the height [76].

FIGURE 4.134

CSP net hoop stress over the height [76].

condition, the stress components might affect each other, and so the hoop stress of CSP is nonuniform throughout the height due to the nonuniformity of longitudinal stress, as displayed in Fig. 4.134. Due to the described local bending effect, the measured outer-surface strain can partially illustrate such nonuniformity across height, as shown in Fig. 4.135, even though its law is not totally consistent with net stresses throughout the thickness. Fig. 4.135 shows the strain on the outer surface of specimen C-1.45-50-1 obtained through nonlinear FE analysis and experimental test results [76] in which MP denotes the location of strain gauges for measuring the strain values. Aside from nonuniformity, the stresses also have periodicity, which is reflected in height. As shown in Figs. 4.133 and 4.134, both stress

4.3 Investigating the behavior of composite members

517

components alter periodically with the corrugation, except for those at the ends of columns because of the restraint at the ends. The stress components can be expressed using cosine functions, as follows:  z sv
cr þ sv
tr sv
cr
sv
tr þ cos 2p sv ðzÞ ¼ (4.131) 2 2 l  z s þ sh
tr sh
cr
sh
tr þ cos 2p sh ðzÞ ¼ h
cr (4.132) 2 2 l in which z and l represent the vertical coordinates value and wave length, respectively. The stress before averaging is nonuniform within a period. The total level is uniform when taking into account the periodicity over the height of the column. Accordingly, the nominal longitudinal stress sv and the nominal hoop stress sh can be achieved by integral averaging over one period to reflect their whole extent, as expressed by Eqs. (4.133) and (4.134), in which the net longitudinal stress sv is over the surface in a local coordinate system, while the nominal longitudinal stress sv is axial in a global coordinate system. Parameters z and b in Eqs. (4.133) and (4.134)

(a) FIGURE 4.135

Strain on the outer-surface of specimen C-1.45-50-1 [76]. (A) Measurement points. (B) Axial load-strain histories.

518

4. Numerical methods

(b) FIGURE 4.135

cont’d.

4.3 Investigating the behavior of composite members

519

represent the height and surface sloping angle, respectively, as shown in Fig. 4.103. Z l sv cos bdz (4.133) sv ¼ 0 l Z l sh dz (4.134) sh ¼ 0 l By taking the average of stresses over the thickness and the height, the stresses of CSP can be comparable to those of the flat steel tube. These simplifications can be used in the quantitative analysis of load-bearing contribution, the evaluation of rigidity of CSP, and the defining of the equivalent confining stress. However, the unsimplified sv and sh are still used in the theoretical study of the change of confinement mechanism with height. Stress of CSP over the cross-section

• Nominal longitudinal stress The stresses in CFCST beam columns fluctuate over the section because of the combined compression-bending action, unlike CFCSTs under concentric axial compression. The nominal longitudinal stress at the peak load svu is obtained using Eq. (4.133) and presented in Fig. 4.136. With an increasing eccentricity ratio e=r0 , the neutral axis advances toward the compressive side. svu is similar to that under axial compression at severe compression, i.e., re0 ¼ 0. svu becomes tensile in the tension zone and rises as e=r0 grows.

FIGURE 4.136

Nominal longitudinal stress of CSP at ultimate load [76]. (A) Influenced by eccentricity ratio. (B) Influenced by helical angle

520

4. Numerical methods

The svu over the cross-section of CSP varies in the range of
0.4 to 0:2 fy (the negative is compressive and corresponding to the compressive area). Compared with TRC beam columns, the values of svu are remarkably less, leading to a negligible load-bearing contribution of CSP. Based on Eqs. (4.135) and (4.136), the direct contribution of CSP to the ultimate axial load and to the corresponding bending moment is less than 5%   Nu
csp Nu

< 5% and

Mu
csp Mu

< 5% . This can also be attributed to the low

rigidity of CSP. The local bending effect may cause such slight rigidity, leading to opposite longitudinal stress states from the inner to the external surface and the average value of stress reduction. Consequently, the CSP is unable to provide much resistance to compression and bending loads and instead effectively restrict the concrete longitudinally, as previously stated. Z2p Nu
CSP ¼

ts r0 svu ðuÞ du

(4.135)

ts r20 svu ðuÞsin u du

(4.136)

0

Z2p Mu
CSP ¼ 0

• Nominal hoop stress The nonuniformity of nominal hoop stress at the peak load over the cross-sectional depth x, i.e., shu ðxÞ, is displayed in Fig. 4.137A. It can be observed from the figure that shu ðxÞ reduces by moving from the extreme compression fiber to tension fiber, where x=r0 is in the range of
1.0 and þ1.0. shu ðxÞ is reduced linearly over the depth of the cross-section, as expressed by Eq. (4.137). Eq. (4.137) can be applied for defining the

FIGURE 4.137 Nominal hoop stress of CSP at ultimate load [76]. (A) Across crosssectional depth. (B) On extreme compressive and tensile side

4.3 Investigating the behavior of composite members

521

equivalent confining stress. This issue is discussed in detail in the next section. shu ðxÞ ¼

sthu þ schu sthu
schu þ x 2 2D0

(4.137)

where sthu and schu represent the nominal hoop stress on extreme compression and tension side at the peak load, respectively, which are taken as the two limits of the linear function. Increasing the e=r0 ratio also diminishes the shu . The boundary amounts of schu and sthu corresponding to the maximum and minimum values of the hoop stress are plotted in Fig. 4.137B to further explain the  effect of the e=r0 ratio, i.e., sthu x ¼ r0 . It is evident that the eccentricity ratio of re0 < 1:0 leads to a considerable shu even if the increase in e= r0 is against the shu . After then, shu continues to reduce until it is below 70% of its value when compressed uniformly. Local confinement and theoretical analysis

• Local confinement Due to the confinement, the majority of the in-filled concrete is in triaxial compression. Contrary to this, the stress of concrete encased in corrugation is unique because of the varied interactions between CSP and concrete over the height, resulting in nonuniform local confinement from crest to trough. Furthermore, the local confinement characteristics change from compression to tension zones. Confinement in the compressive zone is mostly caused by the trough area, which has the highest radial confining stress of concrete, as shown in Fig. 4.138. The reason for this is that the trough interaction tends to be tighter, whereas the crest of CSP separates from the in-filled concrete.

FIGURE 4.138

Local confining stress at ultimate load [76].

522

FIGURE 4.139

4. Numerical methods

Typical failure mode of CFCST stub under eccentric compression [76].

Without sufficient confinement, the concrete hoop stress might even become tensile at the crest, resulting in vertical cracks, as shown in Fig. 4.139. During loading, the CSP in the tension zone becomes flat. As previously stated, the concrete tends to contact CSP more firmly at the crest while separating in the trough where horizontal cracks emerge, as displayed in Fig. 4.139. As a result, the confinement in the tension zone comes mainly from the crest region. These areas are displayed in dark color in Fig. 4.138. • Theoretical analysis The nonuniform confinement effect over the height of CFCST stub columns under eccentric compression is explained via stresses. One period of corrugation distant from the ends is chosen as a sample unit based on stress and geometrical periodicities, as illustrated in Fig. 4.140A. The geometry of spiral CSP is described in a cylindrical coordinate system (i.e., global coordinate system), defined over the axial centerline where the radius r varies along vertical coordinate z and alters with cross-sectional central angle q0 because of the helical corrugation. The origin of vertical coordinate can be set at trough around the circumference of CSP (at any q0 ) to exclude the influence of helix. The corrugations consist of circular arcs, including crest and trough and flat (or middle) segments, as seen in Fig. 4.103. The following continuous cosine function can be used to describe the geometry of CSP:  z h rðzÞ ¼ r0
cos 2p (4.138) 2 l

4.3 Investigating the behavior of composite members

FIGURE 4.140

523

Mechanics of CFCST beam columns [76]. (A) Vertical mechanics. (B) Mi-

cro unit analysis

where r0, h and l represent nominal radius, wave height, and length, respectively. The sloping angle qs and meridian curvature radius R can be defined according to Eq. (4.138), as follows: h  z qs ðzÞ ¼ arctan½r0 ðzÞ z 0:8p sin 2p (4.139) l l (   )32 ph 2pz 2 sin 1þ  3=2 l l 1 þ r0 ðzÞ2 l2   (4.140) ¼
RðzÞ ¼
00 2pz r ðzÞ h 2 2p cos l A microunit of CSP is used in Fig. 4.140 to develop the confining stress equation. Because the directions of stresses in an arbitrary unit are unknown, the following are the specifications for positive values and directions for geometric parameters and stresses. For qs, sloping outward: þ. For R, incurve (arc center is inside of CSP): þ. For hoop and longitudinal

524

4. Numerical methods

stresses, tensile: þ. For confining stress, pressure: þ. Three local orthogonal axes can be constructed on the surface of a microunit along its tangential and normal directions, i.e., local coordinate axes X1, Y1, and Z1, respectively. The confining stress n can be defined along the Z1-axis (normal direction) according to the fundamental equilibrium equation of revolving-shell membrane stresses [92]. As expressed by Eq. (4.141), it is generated by hoop stress and longitudinal stress, respectively, i.e., nh and nv , which is distinct from the flat tube, whose confining stress is solely developed by the hoop stress due to the meridian curvature radius R ¼ N. The longitudinal frictional stress fv can be calculated using Eq. (4.142) and illustrated in Fig. 4.140B over the Y1-axis as one of the tangential directions. It can be observed that fv is periodically up-downreverse, leading to the accumulation of longitudinal stress sv alternately increased or reduced. nðz; uÞ ¼ nh þ nv ¼

ts sh ðz; uÞ ts sv ðz; uÞ cos½qs ðzÞ þ 0 rðzÞ RðzÞ

fv ðz; uÞ ¼

ts sh ðz; uÞ ts s0 ðz; uÞ sin½qs ðzÞ þ v 0 rðzÞ RðzÞqs ðzÞ

(4.141) (4.142)

in which ts , r, R, z, and qs represent thickness of CSP, radius, meridian curvature radius, vertical coordinate, and sloping angle, respectively. sh and sv are net hoop stress and net longitudinal stress of CSP, respectively. By replacing Eqs. (4.131, 4.132, 4.138, 4.139, 4.140) into Eq. (4.141), the confining stress n can be defined as plotted in Fig. 4.141A. The most significant value of n is at the trough in the compressive area, while it turns to be at crest in the tensile area. In Fig. 4.141A and B, the contributions of the hoop stress and longitudinal stress to confining stress are compared. Because the meridian curvature radius R is significantly less than the radius r z r0 , nv has a more significant impact at the crest and trough than nh . Because of the opposite curving direction from crest to

FIGURE 4.141

Theoretical confining stress [76]. (A) In compressive zone of specimen C1.45-50. (B) In compressive zone of specimens C-1.45

4.3 Investigating the behavior of composite members

525

trough, nv is positive-negative-reverse over the height, as shown in detail in Fig. 4.140A. The CSP is detached from the concrete at the crest of the compressive area, where local confinement is invalid, the concrete hoop stress is tensile, as shown in Fig. 4.132, and vertical cracks emerge in the wrapped concrete, as displayed in Fig. 4.139. The separation of CSP can be included in the derivation by limiting the value of confining stress n. Because of the possible negative contribution of nv , the computed value of n might be negative without the limit, as shown in Fig. 4.141, representing the bonding force between CSP and concrete resisting debonding failure and separation. Theoretically, the debonding failure occurs, and CSP separates from the concrete when the negative amount of n exceeds the ultimate adhesive strength. However, the adhesive force can be ignored due to the smooth inner surface of galvanized CSP and the complexity in defining the actual cohesive strength. Consequently, n must be taken as 0 in the debonding segments (Figs. 4.138, 4.140 and 4.141), i.e., n  in Eq. (4.141). Stress redistribution can occur as a result of such a separating (debonding) impact. As a result, in the theoretical derivation, n must be determined from the hoop and longitudinal stresses after separation, which are commonly derived in the modeling based on the frictional contact while taking into account the debonding behavior of steel tube and concrete [89,90,93]. • Equivalent confining stress Confinement can highly affect the cross-sectional behavior of CFST and TRC members. The strength and plasticity of confined core concrete will improve as the confining stress n increases. As a result, the confining stress has a direct impact on the stressestrain response of confined concrete. The nonuniform confinement can be precisely defined by the proposed n. However, from a design standpoint, it is inconvenient. By flatting uniformization of CFCSTs into the ideal circular TRC members, which is derived by the integral average of the horizontal projection of n, an entirely radial and uniformly distributed confining stress is obtained, i.e., equivalent confining stress f r , which can be a better solution. Usually, the confining stress at the peak load is used to measure the stressestrain constitutive relationship. Similar to the uniformization method introduced in the previous section for defining the confining stress, if axially loaded CFCSTs, the value of n at the peak load can be utilized for defining f r of CFCST beam columns. The confining stress f r can be defined using the local mechanic analysis scheme or the simplified approach. A local unit cross-sectional depth within one period of corrugation is displayed in Fig. 4.142, in which the stress value alters over the height. The lateral confining force Fr is provided by the resulting counterforce of n and fv on

526

4. Numerical methods

FIGURE 4.142

Mechanics of debonding and confining segments [76].

the surface of in-filled concrete. The counterforce of Fr is balanced with the hoop stress shu and longitudinal stresses svu;deb in the confining section of CSP (point 1 in Fig. 4.142), as defined by Eq. (4.143). Zle Fr ¼

shu ðzÞ 0

ts dz þ 2ts svu;deb sin qs;deb rðzÞ cos qs ðzÞ

(4.143)

where z, ts , ro , le , qs , shu , and svu;deb represent vertical coordinate, thickness, nominal radius, vertical length of effective confining segment, meridional sloping angle of surface (Eq. 4.139), hoop stress (Eq. 4.132), and longitudinal stress at the transition cross-section of each debonding segment, respectively. The subscripts “u” and “deb” represent “at ultimate load” and “at transition point of debonding region to confining region,” respectively. The mechanics of debonding segment of CSP, shown by point 2 in Fig. 4.142, are straightforward since there are only two pairs of stresses over the edges of CSP belt, namely shu, and svu;deb . The equilibrium equation of shu , and svu;deb is governed by the follow: Zl 2ts svu;deb sin qs;deb ¼

shu le

ts dz ro cos qs

(4.144)

A simple equation for the resultant confining force Fr can be derived by replacing Eq. (4.144) into Eq. (4.143), as follows: ts Fr ¼ ro

Zle 0

dz ts shu þ cos qs ro

Zl le

dz ts shu ¼ cos qs ro

Zl shu 0

dz cos qs

(4.145)

4.3 Investigating the behavior of composite members

FIGURE 4.143

527

Uniformization of confining stress [76].

Therefore, the equivalent confining stress f r can be expressed as follows: Fr ts f r ðxÞ ¼ ¼ l ro l

Zl shu 0

dz ws ts z s ðxÞ l ro hu cos qs

(4.146)

in which x, ws , and shu denote the cross-sectional depth, wave arc length, and nominal hoop stress (given by Eq. 4.137), respectively. The other approach is according to the equivalent scheme, as displayed in Fig. 4.143. One period of corrugation is employed as the unit belt, including one of each complete crest, middle, and trough segments, because of the periodicity of stresses. n and fv are only balanced with hoop stress over the radial direction since the periodical longitudinal stresses on the top and bottom edges have balanced themselves in pairs. Eq. (4.146) can also be used to derive a formula for f r based on the equilibrium equation of the equivalent unit belt. The equivalent thickness is defined as teq ¼ ts wl s to maintain the same equation form as TRC members. To describe the equivalent circular tube more clearly, Eq. (4.146) can be rewritten as follows: f r ðxÞ ¼

2teq s ðxÞ Do hu

(4.147)

528

4. Numerical methods

FIGURE 4.144

Simulated equivalent confining stress [76].

f r ðxÞ is determined by the nonuniformity of shu . As shown in Fig. 4.144, the magnitude of f r ðxÞ alters over the cross-sectional depth. Preloading of composite columns Typically, hollow steel tubes are placed and later filled with concrete to construct composite members in practice. This causes the steel columns to be under preload induced from the constructional dead load, self-weight of tubes, and wet concrete weight. Preloading of steel tubes causes prestresses and additional deformation in the steel tubes and can influence the composite member’s load-bearing capacity. In this section, the effects of preloading of outer and inner steel tubes on the structural behavior of CFDST columns are discussed using nonlinear FE analysis. Table 4.35 shows the geometric and material properties of the examined CFDST columns in which the term hp represents the preload ratio and is governed by the following: hp ¼

Np Nus

(4.148)

where Np is the preload imposed on the steel tube and Nus is the ultimate axial strength of the steel tube. Two stages were taken into account in the computer program to consider the preloading effects:

Geometric and material properties of the CFDST columns for investigating the influence of preloading.

Column

Do (mm)

to (mm)

fyo (MPa)

di (mm)

ti (mm)

fyi (MPa)

fc0 (MPa)

hp

L (mm)

Pu (MPa)

CFDST18

400

9

345

190

6

345

50

0

2200

10,300

CFDST19

0.8

9880

4.3 Investigating the behavior of composite members

TABLE 4.35

529

530

4. Numerical methods

12000

Axial force (kN)

10000 C

D

8000 B 6000 4000

A

O

0 0

FIGURE 4.145

Without preload With preload

2000

1500

3000 4500 6000 Axial strain ε(με)

7500

9000

Axial loadestrain responses of the CFDST column with and without

preload.

First step: Applying the specified preload only on steel tubes and deactivating the concrete elements. This step causes the initial deformations to be developed in the steel tubes and imposes prestresses. Second step: Activating the concrete elements and imposing the vertical load on the top of the column. At this step, all the constituent components are loaded simultaneously until the failure of the column. It is worth noting that the preload applied in the previous step remains constant at this point. The axial loadestrain responses of the examined CFDST columns are given in Fig. 4.145. It can be observed from the figure that preloading the steel tubes results in a reduction in the column’s initial axial stiffness and ultimate axial strength. Based on Fig. 4.145, the axial loadestrain response of the preloaded CFDST column can be divided into four steps, as follows: Step 1: Preload stage (Points O-A): In this phase, the preload is imposed on steel tubes only. Preloading steel tubes develop deformation and longitudinal stress in them. Step 2: Elastic stage (Points A-B): The whole cross-section of the column is compressed axially during this phase. The constituent components of the column carry the load independently. Fig. 4.145 shows that the rigidity of the preloaded counterpart is significantly lower when compared to the column without preload. Step 3: Elastic-plastic stage (Points B-C): At this phase, the interaction between the steel tubes and the sandwiched concrete occurs owing to the different Poisson’s ratio of two materials. The magnitude of the confinement effect provided by the outer steel tube and the axial load increases with the column’s deformation. Compared with the CFDST column

4.3 Investigating the behavior of composite members

531

without preload, the ultimate axial strength of the preloaded counterpart is smaller. Besides, preloading steel tubes increases the axial strain corresponding to the peak load. The reason is that preloading the steel tube of composite columns reduces the confinement effect. Step 4: Hardening or softening stage (Points C-D): At this phase, the axial load will either increase or reduce, based on the confinement effect provided by the outer tube. The stressestrain responses at the mid-height of the steel tubes and sandwiched concrete of the preloaded CFDST column were compared to those of the unloaded counterpart, and the results are presented in Fig. 4.146, where sl and sq are the longitudinal and transverse stresses, respectively. The negative values represent that the member is under compression and vice versa. When there is no preload on tubes, the longitudinal stresses in the outer and inner tubes grow proportionally from the outset of loading, as shown in the figure. By reaching the Mises yield stress for the outer tube, the magnitude of the longitudinal stress reduces, whereas transverse stress keeps increasing. Contrary to the outer tube, no significant transverse stress is developed in the inner tube during the loading history, due to the moderate confinement effect of the inner tube. Due to the confinement effect produced by the steel tubes, transverse stresses are formed in sandwiched concrete, and the magnitude of the longitudinal stress reaches about 1:32 f 0c over the loading history. Concerning the preloaded CFDST column, almost no transverse stress is developed in the outer and inner steel tubes during the preloading stage. This shows that no interaction occurs between the tubes and the sandwiched concrete in this step. The transverse stress increases continuously when the tubes and sandwiched concrete are loaded simultaneously, but the longitudinal stress rises initially and then subsides. Compared with the column without preload, the increase of transverse stress in the preloaded counterpart is less. Another point is that preloading the steel tubes increases the axial strain corresponding to the peak point of the longitudinal stress in the sandwiched concrete. Fig. 4.147 shows the interaction stressestrain histories of the CFDST column with and without preload in which Po and Pi are the interaction stresses between the concrete and outer and inner steel tubes, respectively. As illustrated in the figure, the magnitude of Po during the initial loading stage of the column without preload is almost zero due to the difference in the larger lateral expansion of the steel tube than the sandwiched concrete. At this stage, a small interaction occurs between the inner tube and the concrete because of the greater Poisson’s ratio of the inner tube than the sandwiched concrete. When the sandwiched concrete turns into the elasticeplastic stage; however, this impact becomes insignificant.

532

4. Numerical methods

1.5 Mises stress 1

σs/fy

0.5 0 -0.5 -1

Without preload With preload

-1.5 0

5000

10000

15000

20000

Axial strain ε (με) (a) 1.5

Mises stress

1 Without preload

0.5 σs/fy

With preload 0 -0.5 -1 -1.5 0

5000

10000 Axial strain ε (με)

15000

20000

(b)

1.5 Without preload

1

With preload

σs/fy

0.5 0

-0.5 -1 -1.5 0

5000

10000 Axial strain ε (με)

15000

20000

(c)

FIGURE 4.146

Stressestrain histories of constituent components of the preloaded CFDST column. (A) Outer steel tube. (B) Inner steel tube. (C) Sandwiched concrete.

533

References

Interaction stress P (MPa)

8 Without preload-Po

7

Without preload-Pi

6

With preload-Po

5

With preload-Pi

4 3 2 1 0 0

2000

4000

6000

8000

10000

12000

Strain ε(με)

FIGURE 4.147

Interaction stressestrain histories of the CFDSTcolumn with and without

preload.

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C H A P T E R

5 Design rules and standards O U T L I N E 5.1 Limitations of design regulations on the strength of materials and section slenderness 5.2 Compressive design strength of composite members based on design guidelines 5.2.1 Compressive strength of CFST columns subjected to axial compression 5.2.1.1 AS5100 5.2.1.2 BS5400 5.2.1.3 DBJ13-51 5.2.1.4 Eurocode 4 5.2.1.5 AISC 360-16 5.2.2 Buckling curves 5.2.2.1 AS5100 5.2.2.2 BS5400-part 5 5.2.2.3 DBJ13-51 5.2.2.4 EC4 5.2.3 Design capacity of CFST columns subjected to axial compression 5.2.3.1 AS5100 5.2.3.2 BS5400 5.2.3.3 DBJ13-51 5.2.3.4 EC4 5.2.3.5 AISC360-16 5.2.4 Examples 5.2.4.1 Example one 5.2.4.2 Example two 5.2.4.3 Example three

Single Skin and Double Skin Concrete Filled Tubular Structures https://doi.org/10.1016/B978-0-323-85596-9.00004-4

539

542 546 546 546 548 551 552 553 555 555 555 556 556 556 556 557 558 558 558 559 559 568 575

© 2022 Elsevier Inc. All rights reserved.

540

5. Design rules and standards

5.2.5 Design strength of CFDST columns subjected to axial compression

577

5.3 Moment design strength of composite members based on design guidelines 5.3.1 Moment strength of CFST beams 5.3.1.1 Plastic moment capacity of CFST beams 5.3.2 Moment capacity of CFST beams based on design codes 5.3.2.1 AS5100 5.3.2.2 BS5400 5.3.2.3 DBJ13-51 5.3.2.4 EC4 5.3.2.5 AISC360-16 5.3.3 Examples 5.3.3.1 Example four 5.3.3.2 Example five 5.3.3.3 Example six 5.3.4 Moment capacity of CFDST beams 5.3.4.1 Example seven

577 577 580 586 586 587 588 589 589 592 592 599 602 604 607

5.4 CFST members under combined axial loading and bending 5.4.1 Determining the resistance of CFST members under combined axial loading and bending based on design rules 5.4.1.1 BS5400 5.4.1.2 DBJ13-51 5.4.1.3 EC4 5.4.1.4 AISC360-16 5.4.2 Second-order effect 5.4.2.1 EC4 5.4.2.2 AISC360-16 5.4.3 Example eight 5.4.3.1 Solution:

609

5.5 Strength of composite members based on research works 5.5.1 CFST members under axial compression 5.5.1.1 Compressive stiffness of CFST short columns 5.5.1.2 Ultimate axial strain of CFST short columns 5.5.2 CFDST members under axial compression 5.5.3 CFST members under bending 5.5.3.1 Simplified moment capacity model of Elchalakani et al. 5.5.3.2 Simplified moment capacity model of Han 5.5.3.3 Simplified moment capacity model of Liang et al. 5.5.3.4 Flexural stiffness of CFST beams 5.5.3.5 Moment M-curvature f response of CFST beams

635 635 654 654 655 655

611 611 615 618 619 623 623 626 626 626

655 670 671 671 672

5. Design rules and standards

541

5.5.4 CFDST members under bending 5.5.4.1 Example nine 5.5.5 CFDST members under combined axial loading and bending

673 674 676

5.6 Local buckling of steel plates 5.6.1 Elastic local buckling of steel plates 5.6.1.1 Steel hollow sections 5.6.1.2 Steel hollow sections filled with the concrete 5.6.2 Postlocal buckling of steel plates 5.6.2.1 Steel hollow sections 5.6.2.2 Square steel hollow sections filled with the concrete 5.6.2.3 Circular steel hollow sections filled with the concrete 5.6.2.4 Example ten 5.6.2.5 Example eleven

678 678 678 682 685 685 689

5.7 Further discussion on local buckling of steel plates in rectangular CFST columns under axial compression 5.7.1 Local buckling of steel plates in rectangular CFST columns with binding bars under axial compression 5.7.2 Elastoplastic local buckling of steel plates in rectangular CFST columns under axial compression 5.7.3 A refined model for local buckling of steel plates in rectangular CFST columns under axial compression 5.7.4 Effects of different parameters on the local buckling strength of rectangular steel plates in CFST columns with binding bars 5.7.4.1 Effect of spacing between binding bars 5.7.4.2 Effect of the diameter of binding bars 5.7.4.3 Effect of the D=B ratio 5.7.4.4 Further discussion on the effect of the b=t ratio and the D=B ratio on the buckling strength of rectangular steel plates in CFST columns with binding bars 5.7.5 Design recommendation of steel plates in rectangular CFST columns with binding bars 5.8 Compressive strength of CFST stub columns with stiffeners 5.8.1 Square CFST stub columns with PBL 5.8.2 Square CFST stub columns with inclined stiffener ribs 5.8.3 Square CFST stub columns with binding bars 5.8.4 Rectangular CFST stub columns with binding bars 5.8.5 Square CFST stub columns with spiral tension bars 5.8.6 Circular CFST stub columns with tie bars 5.8.7 Circular CFST stub columns with tension bars 5.8.8 Circular CFST stub columns with external ring bar stiffeners

694 695 697

706 712 713 717 721 721 722 724

726

727 732 732 736 737 741 746 749 752 754

542

5. Design rules and standards

5.8.9 5.8.10 5.8.11 5.8.12

Circular CFST stub columns with jacket strip stiffeners Circular CFST stub columns with spiral bar stiffeners L-shaped CFST stub columns with binding bars T-shaped CFST stub columns with binding bars

5.9 Compressive strength of CFST stub columns with local corrosions 5.9.1 Circular CFST stub columns with local corrosions 5.9.2 Square CFST stub columns with local corrosions

757 758 759 767 776 776 784

5.10 Strain compatibility between the steel tube and the concrete core

785

References

786

5.1 Limitations of design regulations on the strength of materials and section slenderness Design codes consider limitations on the strength of materials and slenderness of steel tubes in the construction of concrete-filled steel tubular (CFST) members. Recent advances in the construction and materials industries have made it feasible to use high-strength materials such as concrete and steel. For instance, rectangular CFST members have been employed in the Latitude Building in Sydney, where steel rectangular hollow sections with a yield strength of 690 MPa have been filled with 80 MPa strength concrete. However, it can be seen from Table 5.1 that the

TABLE 5.1 Ranges of allowable material strengths for the use in composite columns. Design code

Steel strength (MPa)

Concrete compressive strength (MPa)

AISC 360-16

fy
525

21
fc0
69

AS 2327

fy
690

20
fc0
100

AS 5100

230
fy
400

25
fc0
65

DBJ 13-51-2010

235
fy
420

24
fc0
70

EC4

235
fy
460

20
fc0
50

5.1 Limitations of design regulations on the strength

543

use of high-strength and ultra-high-strength materials has been neglected by almost all design codes. Employing high-strength materials in steelconcrete composite structures can decrease the size of the structural members, leading to providing more architectural space and diminishing construction expenses. Besides, the confinement effect provided by steel tubes can be enhanced by using high-strength steel materials due to their better sustainability and buckling resistance. Accordingly, it is crucial to examine the accuracy of current design codes in predicting the capacities of steel-concrete composite members constructed with high-strength materials. Table 5.2 summarizes the slenderness limits of the steel sections of CFST/concrete-filled double skin steel tubular (CFDST) members subjected to axial compression and bending. Contrary to AS 2327 and AISC 360-16, EC4 recommends the same maximum slenderness ratios for members subjected to axial compression and flexure. The slenderness limits recommended in AISC 360-16 are similar to those given in AS 2327. However, there is a slight discrepancy between the American and Australian codes in the definition of slenderness ratio. Both AISC 360-16 and AS 2327 allow the occurrence of local buckling in steel sections, whereas EC 4 does not permit the design of columns made of steel tubes with a slenderness ratio beyond the recommended limits. The filling of hollow steel sections (HSSs) with concrete changes the behavior of members. As discussed in Chapter 2, HSS members under axial compression are susceptible to local buckling. However, the infilled concrete in CFST/CFDST members under compressive loading can prevent the inward buckling of the steel tube wall. Therefore, CFST/CFDST members usually present a more ductile postbuckling behavior compared with equivalent HSS members. This is because of the more extensive wavelength of the buckling mode, spreading of plastic deformation, and a slight improvement in the moment of inertia of the tube. Therefore, the slenderness limits of CFST/CFDST members are larger than that of equivalent HSS members. For instance, the slenderness of lr ¼ 0:11Es =fy is defined as the limit for the occurrence of elastic local buckling of circular hollow sections under axial compression. The use of infilled concrete in the circular hollow section increases the critical local buckling stress by 1.73 times. As shown in Table 5.2, the noncompact/slender limit lr for circular CFST/CFDST members recommended by AISC 360-16 is equal to lr ¼ 0:19Es =fy .

Loading condition

Cross section

Design code

Slenderness ratio l

Compact/ noncompact

Noncompact/ slender

Permitted limit

Axial compression

Circular

EC4

D t

e

e

90 235 fy

AS 2327

D fy t 250

e

e

0:19 Efys

AISC 360-16

D t

lp  0:09 Efys

lr  0:19 Efys

BS5400

D t

e

e

0:31 Efys qffiffiffiffiffiffi

DBJ13-51

D t

e

e

EC4

B t

e

e

Flexure

Circular

Rectangular

qffiffiffiffiffiffi

150 235 fy qffiffiffiffiffi ffi 52 235 fy qffiffiffiffi 5 Efys qffiffiffiffi 5 Efys qffiffiffiffiffiffi

AS 2327

B t

AISC 360-16

B t

qffiffiffiffi lcp  2:26 Efys qffiffiffiffi lp  2:26 Efys

BS5400

B t

e

e

DBJ13-51

B t

e

e

qffiffiffiffiffiffi 60 235 fy

EC4

D t

e

e

90 235 fy

AS 2327

D fy t 250

e

e

0:31 Efys

AISC 360-16

D t

lp  0:09 Efys

lr  0:31 Efys

BS5400

D t

e

e

0:31 Efys qffiffiffiffiffiffi

DBJ13-51

D t

e

e

EC4

B t

e

e

fy 250

qffiffiffiffi ley  3 Efys qffiffiffiffi lr  3 Efys

8Es fy

3Es fy

8Es fy

150 235 fy qffiffiffiffiffi ffi 52 235 fy

5. Design rules and standards

Rectangular

544

TABLE 5.2 Slenderness limits of steel sections of composite members subjected to axial compression and bending.

qffiffiffiffiffiffi fy

e

e

e

e

qffiffiffiffi lr  3 Efys

qffiffiffiffi 3 Efys qffiffiffiffi 5:7 Efys qffiffiffiffi 5 Efys

qffiffiffiffi lr  5:7 Efys

qffiffiffiffi 5:7 Efys

B t

AS 2327 (web)

h t

AISC 360-16 (flange)

B t

qffiffiffiffi lp  2:26 Efys

AISC 360-16 (web)

h t

lp  3

BS5400

B t

e

e

DBJ13-51

B t

e

e

250 qffiffiffiffiffi ffi fy 250

qffiffiffiffi Es fy

qffiffiffiffiffiffi 3Es fy

qffiffiffiffiffiffi 60 235 fy

5.1 Limitations of design regulations on the strength

AS 2327 (flange)

545

546

5. Design rules and standards

5.2 Compressive design strength of composite members based on design guidelines 5.2.1 Compressive strength of CFST columns subjected to axial compression 5.2.1.1 AS5100 • Rectangular cross section In Ref. [1], the design plastic capacity of rectangular CFST columns to axial compressive force is defined by superimposing constituent components’ contributions, and the confinement effect is neglected. AS5100 ¼ fAs fy þ fc Ac fc0 Npl;Rd

(5.1)

in which As and fy are the cross-sectional area and yield stress of the steel tube, respectively, and Ac and fc0 are the cross-sectional area and the characteristic compressive strength of the concrete core, respectively. The terms 4 and 4c are the capacity factors of the steel and the concrete, respectively. According to AS5100, 4 is taken as 0.9, and 4c is taken as 0.6. The nominal design plastic capacity of rectangular CFST columns to axial compressive force can be calculated by ignoring the capacity factors in Eq. (5.1) as follows: AS5100 ¼ As fy þ Ac fc0 Npl;Rd;nom

(5.2)

• Circular cross section AS5100 considers the increase in the concrete strength due to the confinement effect provided by the circular steel tube if the following requirements are met: 1. The relative slenderness l is smaller than 0.5. D. 2. The load eccentricity ratio e is less than 10 The relative slenderness l is defined by: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ns (5.3) l¼ AS5100 Ncr in which Ns ¼ As fy þ Ac fc0 AS5100 ¼ Ncr

p2 ðEIÞAS5100 eff ðKe LÞ2

(5.4.a) (5.4.b)

where L is the length of the column, Ke represents the effective length denotes the effective elastic flexural factor, and the term ðEIÞAS5100 eff stiffness of the column.

547

5.2 Compressive design strength of composite members

TABLE 5.3 Effective length factors for braced and swayed frame members [1]. Braced member

Ke ¼ 0:7

Ke ¼ 0:85

Swayed member

Ke ¼ 1:0

Ke ¼ 1:2

Ke ¼ 2:2

Ke ¼ 2:2

Table 5.3 defines the magnitude of the effective length factor Ke for columns having idealized end conditions. The effective length factor of frame members is determined based on the compression member stiffness to the end restraint stiffness ratio. AS5100 [1] has provided charts for the effective length factor of members in frames. The effective elastic flexural stiffness ðEIÞe is obtained by the following expression: ¼ fEs Is þ fc EAS5100 Ic ðEIÞAS5100 c eff

(5.5)

in which Es and EAS5100 are Young’s modulus of the steel tube and the c concrete core, respectively, and Is and Ic are the second area moments of is governed by: the steel tube and the concrete core, respectively. EAS5100 c qffiffiffiffi (5.6) EAS5100 ¼ 0:5r1:5  0:043 f 0c c where r represents the density of concrete. The design plastic capacity of circular CFST columns to axial compressive force is governed by: " # t fy AS5100 0 Npl;Rd ¼ fAs h2 fy þ fc Ac fc 1 þ h1 (5.7) D fc0

548

5. Design rules and standards

where t and D represent the thickness and the diameter of the tube, respectively. Similar to rectangular CFST columns, the nominal design plastic capacity of circular CFST columns to axial compressive force can be achieved by taking the capacity factors 4 ¼ 4c ¼ 1 as follows: " # t fy AS5100 0 Npl; Rd; nom ¼ As h2 fy þ Ac fc 1 þ h1 (5.8) D fc0 Coefficients h1 and h2 depend on the load eccentricity ratio e. The load eccentricity e is given by: e¼

MEd NEd

(5.9)

in which MEd denotes the maximum design bending moment, and NEd is the axial design force. When the column is under concentric compression ðe ¼ 0Þ, h1 and h2 are named h10 and h20 , respectively, and are expressed as: h1 ¼ h10 ¼ 4:9  18:5l þ 17l2  0   h2 ¼ h20 ¼ 0:25 3 þ 2l
1:0

(5.10.a) (5.10.b)

It can be seen from Eq. (5.10) that the magnitude of h2 is less than unity. This is to consider the influence of the lateral expansion of the infilled concrete on reducing the steel tube yield stress.   D For the load eccentricities ranging from 0 to 10

are governed by:

  10e h1 ¼ h10 1  D h2 ¼ h20 þ ð1  h20 Þ

10e D

D , h and h 0 < e
10 1 2

(5.11.a) (5.11.b)

5.2.1.2 BS5400 • Rectangular cross section In Ref. [2], the design plastic capacity of rectangular CFST columns to axial compressive force is defined by superimposing constituent components’ contributions, and the confinement effect is neglected. BS5400 0 ¼ fAs fy þ fc Ac fcu Npl;Rd

(5.12)

in which 4 ¼ 0:95 and 4c ¼ 0:45 are, respectively, the capacity factors of 0 represents the characteristic cube the steel and the concrete, and fcu compressive strength of concrete.

5.2 Compressive design strength of composite members

549

According to Ref. [2], the concrete contribution ratio ac for rectangular CFST members is given by: ac ¼

0 0:45Ac fcu BS5400 Npl;Rd

(5.13)

The concrete contribution ratio must be between 0.1 and 0.8 ð0:1 < ac < 0:8Þ. The nominal design plastic capacity of rectangular CFST columns to axial compressive force can be achieved by taking the capacity factors 4 ¼ 4c ¼ 1 as follows: BS5400 0 ¼ As fy þ Ac fcu Npl;Rd;nom

(5.14)

• Circular cross section The design plastic capacity of circular CFST columns to axial compressive force is governed by: BS5400 ¼ 4As fy0 þ 4c Ac fcc0 Npl;Rd

(5.15)

in which 4 and 4c are taken as 0.95 and 0.45, respectively, and fcc0 and fy0 are given by: t 0 þ C 1 fy fcc0 ¼ fcu D

(5.16)

fy0 ¼ C2 fy

(5.17)

where the terms C1 and C2 are constants and depend on the effective le . The effective length of column le depends length to the diameter ratio D on the end conditions and can be obtained from Table 5.4. The values of C1 and C2 can be obtained from Table 5.5 or Fig. 5.1. It can be observed from Table 5.5 or Fig. 5.1 that the magnitude of C2 is less than unity.

TABLE 5.4 Effective length le based on the end conditions. End conditions

Effective length le

Fixed-fixed

0:7L

Fixed-pinned

0:85L

Pinned-pinned

L

Fixed-slide

1:5L

Fixed-free or pinned-slide

2:0L

550

5. Design rules and standards

TABLE 5.5 The magnitudes of coefficients C1 and C2 . le D

C1

C2

0

9.47

0.76

5

6.40

0.80

10

3.81

0.85

15

1.80

0.90

20

0.48

0.95

25

0

1.0

10 9 8 7 C1

6 5 4 3 2 1 0 0

2

4

6

8

10

12 14 le/D

16

18

20

22

24

26

16

18

20

22

24

26

(a) Coefficient 1 0.9

C2

0.8 0.7 0.6 0.5 0

2

4

6

8

10

12 14 le/D

(b) Coefficient

FIGURE 5.1 cient C2 .

Variations of C1 and C2 against the

le D

ratio. (a) Coefficient C1 . (b) Coeffi-

5.2 Compressive design strength of composite members

551

This is to consider the influence of the lateral expansion of the infilled concrete on decreasing the steel tube yield stress. According to Ref. [2], the concrete contribution ratio ac for circular CFST members is given by: ac ¼

0:45Ac fcc0 BS5400 Npl;Rd

(5.18)

The concrete contribution ratio must be in the range of 0.1 and 0.8 ð0:1 < ac < 0:8Þ. The nominal design plastic capacity of circular CFST columns to axial compressive force can be achieved by taking the capacity factors f ¼ fc ¼ 1 as follows: BS5400 ¼ As fy0 þ Ac fcc0 Npl;Rd;nom

(5.19)

5.2.1.3 DBJ13-51 The design plastic capacity of CFST columns to axial compressive force in Ref. [3] is governed by: DBJ1351 ¼ fsc Asc Npl;Rd

(5.20)

Asc ¼ As þ Ac

(5.21)

where ( fsc ¼

ð1:18 þ 0:85x0 Þfc0

for the square section

ð1:14 þ 1:02x0 Þfc0

for the circular section

(5.22)

in which the term x0 is the confinement factor and is governed by: x0 ¼

As f Ac fc0

(5.23)

where the term f represents the design yield stress of the tube, and fc0 is the f

compressive design strength of infilled concrete. f and fc0 can be taken as gy fck0 gc ,

s

0 fck

and respectively, in which is the characteristic amount of the cylinder compressive strength of concrete at 28 days, and gc ¼ 1:4 and gs ¼ 1:12 are the material property factors. The nominal design plastic capacity of CFST columns to axial DBJ1351 compressive force Npl;Rd;nom can be achieved by taking the capacity factors gs ¼ gc ¼ 1:0 in Eq. (5.20).

552

5. Design rules and standards

5.2.1.4 Eurocode 4 • Rectangular cross section The design plastic capacity of rectangular CFST columns to axial compressive force is defined by superimposing constituent components’ contributions, and the confinement effect is neglected. EC4 0 Npl; Rd ¼ As fyd þ Ac fcd

(5.24) f0

f

0 ¼ ck is the where fyd ¼ gy is the design yield stress of the tube and fcd gc s design compressive strength of infilled concrete in which gs ¼ 1:5 and gc ¼ 1:0 are the material property factors. Eurocode 4 defines a limit for the steel contribution ratio ðdÞ, as follows: As fyd (5.25) 0:2 < d ¼ EC4 < 0:9 Npl;Rd

The steel contribution ratio d is a basic parameter for evaluating the composite action of the section. If d is greater than 0.9, the behavior of the section is dominated by the steel tube, and the role of the concrete core must be neglected. Therefore, the member should be designed as a steel member. The role of the concrete core increases with reducing d. For members with d less than 0.2, the section’s behavior is dominated by the concrete core, and the member must be considered a concrete structure. The nominal design plastic capacity of CFST columns to axial comEC4 pressive force Npl; Rd; nom can be achieved by taking the capacity factors

gs ¼ gc ¼ 1:0 in Eq. (5.24). • Circular cross section Ref. [4] considers the increase in the concrete strength due to the confinement effect provided by the circular steel tube if the following requirements are satisfied: 1. The relative slenderness l is smaller than 0.5. 2. The load eccentricity ratio e is less than 0.1. The relative slenderness l is defined by: sffiffiffiffiffiffiffiffiffiffiffiffiffi Npl; Rk (5.26) l¼ N EC4 cr 0 Npl; Rk ¼ As fy þ Ac fck EC4 Ncr ¼

p2 ðEIÞEC4 eff ðKe LÞ2

(5.27.a) (5.27.b)

5.2 Compressive design strength of composite members

553

in which L is the length of the column, Ke represents the effective length factor, and the term ðEIÞEC4 eff denotes the effective elastic flexural stiffness of the column. Table 5.3 can be used to define the magnitude of Ke . ðEIÞEC4 eff is obtained by the following equation: EC4 ðEIÞEC4 eff ¼ Es Is þ 0:6Ec Ic

(5.28)

in which EEC4 is the elastic modulus of the concrete and is given by: c  0 0:3 f EC4 (5.29) Ec ¼ 22000 c 10 The reduction factor 0.6 in Eq. (5.28) is to consider the effects of cracking the concrete core on the elastic flexural stiffness of the column. The design plastic capacity of circular CFST columns to axial compressive force is given by: " # t fy EC4 0 (5.30) Npl; Rd ¼ As hs fyd þ Ac fcd 1 þ hc 0 D fck The coefficients hs and hc depend on the load eccentricity ratio e. The magnitude of e is given by Eq. (5.9). If the axial compressive load is imposed concentrically, hs and hc are named hs0 and hc0 , respectively, and are governed by: hc ¼ hc0 ¼ 4:9  18:5l þ 17l2  0   hs ¼ hs0 ¼ 0:25 3 þ 2l
1:0

(5.31.a) (5.31.b)

For the load eccentricities ranging from 0 to 0:1 ð0 < e
0:1Þ, hs and hc are governed by:   10e (5.32.a) h1 ¼ hc0 1  D h2 ¼ hs0 þ ð1  hs0 Þ

10e D

(5.32.b)

If e > 0:1, EC4 ignores the confinement effect, and therefore, hs ¼ 1:0 and hc ¼ 0. 5.2.1.5 AISC 360-16 Ref. [5] recommends design equations of CFST columns based on the steel tube’s slenderness. It is supposed that CFST columns with compact steel tubes can reach plastic strength. Therefore, the nominal plastic

554

5. Design rules and standards

capacity columns with compact sections to axial compressive   of CFST force l  lp is expressed by: AISC Nnom ¼ As fy þ Ac C2 fc0

(5.33)

in which the coefficient C2 depends on the shape of the cross section and is taken as 0.85 for rectangular columns and 0.95 for circular columns. It can be seen from Eq. (5.33) that Ref. [5] considers that the steel tube can achieve its yield stress before the occurrence of local buckling, whereas the safety factor C2 < 1:0 is considered for the concrete section. Additionally, Ref. [5] defines the following condition for the ratio of the steel tube and the concrete core cross-sectional areas: As  0:01 As þ Ac

(5.34)

The nominal compressive capacity of CFST columns with noncompact   steel tubes lp < l
lr is defined by the quadratic interpolation between Np and Ny , as follows: 2 Np  Ny  AISC ¼ Np   Nnom 2 l  lp lr  lp

(5.35)

Ny ¼ As fy þ Ac 0:7fc0

(5.36)

in which

l, lr , and lp can be found in Table 5.2. Similar to the compact section, it is supposed that the noncompact steel tube can reach its yield stress. In contrast, it is supposed that due to the lower confinement effect provided by the noncompact tube than the compact one, the infilled concrete experiences remarkable volumetric dilation after the longitudinal stress of the concrete core reaches 0:7fc0 . Contrary to the compact and noncompact sections, Ref. [5] assumes that elastic local buckling occurs in the slender steel tubes of CFST columns under compression. Therefore, the steel tube cannot reach its yield stress, and the compressive strength of CFST columns with slender steel tube ðlr < lÞ is governed by: AISC Nnom ¼ As fcr þ Ac 0:7fc0

(5.37)

in which

fcr ¼

8 9Es > > > b2 > > > > > < t

for rectangular section

0:72fy > > > !0:2 > > > Df > y > : Es t

(5.38) for circular section

5.2 Compressive design strength of composite members

555

5.2.2 Buckling curves The effects of local and global buckling on the section capacity of composite columns must be considered in predicting their compressive strength. The bending stiffness of the member influences the load-bearing capacity of long columns. Hence, a remarkable elastic buckling resistance is expected for composite members due to the increased bending stiffness. The infilled concrete postpones or hinders the local buckling of long columns. However, local and overall buckling interaction must be applied in predicting the member strength having an intermediate length. Short columns’ member strength can be taken as the maximum strength, discussed in the previous section. The member strength of slender composite columns is taken as the design strength, and a slenderness reduction factor of the member must be incorporated. Buckling curves can be used for calculating the member strength of CFST columns subjected to axial compression. 5.2.2.1 AS5100 Depends onthe slendernessratio ofthe memberln, AS5100 employs three buckling curves for CFST members. The slenderness ratio ln is given by: ln ¼ 90lr where

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u AS5100 uN pl; Rd lr ¼ t AS5100 N cr

(5.39)

(5.40)

AS5100 is governed by Eq. (5.1) for rectangular columns, and Eq. in which Npl; Rd AS5100 is given by Eq. (5.4.b) in which the (5.7) for circular columns and Ncr magnitude of 4 and 4c must be taken as 1.0.

5.2.2.2 BS5400-part 5 Ref. [2] defines four buckling curves for CFST members. The overall trend of the curves is similar to steel columns recommended by BS5400Part 3. The nondimensional slenderness l description, however, is different. Based on Ref. [2], l is governed by: l¼

le lE

(5.41)

where the term le represents the effective length of the column, presented in Table 5.3, and the length of the column lE is obtained by taking the BS5400 equal to the Euler load. Therefore: squash load Npl; Rd sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:45Ec Ic þ 0:95Es Is (5.42) lE ¼ p N BS5400 pl; Rd

556

5. Design rules and standards

BS5400 can be calculated using Eq. (5.12) for rectangular columns where Npl; Rd

and Eq. (5.15) for circular columns. 5.2.2.3 DBJ13-51 Instead of the term slenderness reduction factor, Ref. [3] uses the term column stability factor f, which is a function of steel tube yield stress, the compressive strength of the concrete, steel ratio, and slenderness of the member l. l can be calculated based on the recommendation of Ref. [3], as follows: 8 Lo > > for circular section

> :2 3 for rectangular section B in which Lo represents the calculated length of the column and can be achieved from Clause 5.3 of GB50017-2013. 5.2.2.4 EC4 Ref. [4] employs two buckling curves, i.e., curve a and curve b. Curve a is used for CFST columns with the ratio AAsrc
3%, and curve b is applied for CFST columns with 3% < AAsrc
6% in which the term Asr represents the cross-sectional area of the steel rebar.

5.2.3 Design capacity of CFST columns subjected to axial compression 5.2.3.1 AS5100 AS5100 recommends the following expression for calculating the design capacity of CFST columns: AS5100 AS5100 Nu; design ¼ ac Npl; Rd

(5.44)

AS5100 is the design strength of the member defined in Eq. (5.1) in which Npl; Rd

for rectangular CFST columns and Eq. (5.7) for circular CFST columns, and ac is the slenderness reduction factor of the member and can be achieved from the buckling curves, recommended by AS5100 or using the following expression: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2

90 (5.45) ac ¼ x 1  1  xl

557

5.2 Compressive design strength of composite members

where  x¼

l 90

2

þ1þh  2 l 2 90

(5.46.a)

l ¼ ln þ aa $ab

(5.46.b)

h ¼ 0:00326$ðl  13:5Þ  0

(5.46.c)

ln ¼ 90lr

(5.46.d)

aa ¼

2100$ðln  13:5Þ l2n  15:3ln þ 2050

(5.46.e)

where lr can be obtained by using Eq. (5.40), and the term ab represents the compression member section constant and depends on the fabrication method of the HSS, as displayed in Table 5.6. TABLE 5.6 The magnitude of ab based on Table 10.3.3 of AS5100-part 6. ab

Steel hollow section

1.0

Hot-formed circular and rectangular hollow sections having form factor kf ¼ 1:0.

1.0

Stress relieved cold-formed circular and rectangular hollow sections having form factor kf ¼ 1:0.

0.5

Nonestress relieved cold-formed circular and rectangular hollow sections having form factor kf ¼ 1:0.

0.5

Hot-formed circular and rectangular hollow sections having form factor kf < 1:0.

0.5

Cold-formed circular and rectangular hollow sections having form factor kf < 1:0.

0

Welded box sections

5.2.3.2 BS5400 Ref. [2] recommends the following expression for calculating the design capacity of CFST columns: BS5400 BS5400 Nu; design ¼ 0:85K1 Npl; Rd

(5.47)

BS5400 is defined by Eq. (5.12) for rectangular CFST columns and where Npl; Rd

Eq. (5.15) for circular CFST columns, and K1 is the slenderness reduction

558

5. Design rules and standards

factor of the member and can be achieved by the buckling curves, recommended by Ref. [2] or using the following expression: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 



 1þh 1þh 2 4 1þ 2 (5.48)   2 K1 ¼ 0:5 1 þ 2 l l l where l can be calculated using Eq. (5.41) and h is governed by: ( 0 for l
0:2 h¼ 75:5aðl  0:2Þ for l > 0:2

(5.49)

in which the magnitude of a depends on the buckling curves, recommended by Ref. [2], and is taken as 0.0025, 0.0045, 0.0062, and 0.0083 for the buckling curves of A, B, C, and D, respectively. 5.2.3.3 DBJ13-51 Ref. [3] recommends the following expression for calculating the design capacity of CFST columns: DBJ1351 DBJ1351 Nu; design ¼ 4Npl; Rd

(5.50)

DBJ1351 where Npl; is defined by Eq. (5.20), and f is the slenderness reduction Rd

factor and can be achieved from the buckling curves, defined by Ref. [3]. 5.2.3.4 EC4 The design capacity of CFST columns based on Ref. [4] is governed by the following equation: EC4 EC4 Nu; design ¼ cNpl; Rd

(5.51)

EC4 is defined by Eq. (5.24) for rectangular CFST columns and in which Npl; Rd

Eq. (5.30) for circular CFST columns, and the term c is the slenderness reduction factor and is governed by: c¼

1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f þ f2  l2

i h   f ¼ 0:5 1 þ 0:21 l  0:2 þ l2

(5.52.a)

(5.52.b)

5.2.3.5 AISC360-16 The design capacity of CFST columns in Ref. [5] is governed by: AISC AISC Nu; design ¼ fc Nn

(5.53)

5.2 Compressive design strength of composite members

559

in which fc is the resistance factor and is taken as 0.75 and NnAISC is given by:   8 AISC Nnom NAISC > AISC Ncr > N 0:658 if nom
2:25 > nom AISC > < Ncr NnAISC ¼ (5.54) > AISC > N > > 0:877NAISC if nom > 2:25 : cr AISC Ncr AISC is discussed in Section 5.2.1.5, and N AISC reprein which the term Nnom cr sents the buckling strength of the member and is given by: AISC ¼ Ncr

p2 ðEIÞAISC eff ðKe LÞ2

AISC Ic ðEIÞAISC eff ¼ Es Is þ C3 Ec

As
0:9 As þ Ac qffiffiffiffi ¼ 0:043r1:5 f 0c EAISC c

C3 ¼ 0:45 þ 3

(5.55) (5.56.a) (5.56.b) (5.56.c)

5.2.4 Examples 5.2.4.1 Example one Determine the design compressive strength of a circular hollow steel tube with a dimension of 500  15 mm2 filled with normal strength concrete with the compressive cylinder strength of 50 MPa and cubic strength of 60 MPa under axial compression. The modulus of elasticity and the yield stress of the tube are 2:05  105 MPa and 345 MPa, respectively, and the length of the column is 4200 mm. Solution:

The geometric and material properties of the member are as follows: D ¼ 500 mm t ¼ 15 mm As ¼

pD2 pðD  2tÞ2 p  5002 p  ð500  2  15Þ2  ¼  ¼ 22855:07 mm2 4 4 4 4 Ac ¼

pðD  2tÞ2 p  ð500  2  7Þ2 ¼ ¼ 173494:31 mm2 4 4

560 Is ¼

5. Design rules and standards

pD2 pðD  2tÞ2 p  5002 p  ð500  2  7Þ2  ¼  ¼ 672653198 mm4 64 64 64 64 Ic ¼

pðD  2tÞ2 p  ð500  2  7Þ2 ¼ ¼ 2395305786 mm4 64 64 0 fcu ¼ 60 MPa

fc0 ¼ 50 MPa 0 ¼ 50 ¼ 33:3 MPa Based on EC4: fcd 1:5 ( 0 fck ¼ 38:5 MPa Based on GB50010 [6]: fc0 ¼ 27:5 MPa Based on AS5100: qffiffiffiffi pffiffiffiffiffi AS5100 Ec ¼ 0:5r1:5  0:043 f 0c ¼ 0:5  24001:5  0:043 50 ¼ 17875 MPa 0 ¼ 450  60 ¼ 27000 MPa Based on BS5400: EBS5400 ¼ 450fcu c  0 0:3  0:3 fc 50 Based on EC4: EEC4 ¼ 22000 ¼ 22000 ¼ 35654 MPa c 10 10 Based on AISC 360-16: qffiffiffiffi pffiffiffiffiffi AISC Ec ¼ 0:043r1:5 f 0c ¼ 0:0043  24001:5 50 ¼ 35749:5 MPa

fy ¼ 345 MPa Es ¼ 205000 MPa Based on GB50017 [7]: f ¼ 295 MPa 345 Based on EC4: fyd ¼ ¼ 345 MPa 1:0 • AS5100 To predict the member’s compressive strength, first, it is required to calculate the relative slenderness l and control the existence of the confinement effect. l is calculated by Eqs. (5.3) and (5.4), as follows: ¼ fEs Is þ fc EAS5100 Ic ¼ 0:9  205000  672653198 þ 0:6  17875 ðEIÞAS5100 c eff  2395305786 ¼ 149794  109 N:mm2 AS5100 Ncr ¼

p2 ðEIÞAS5100 eff ðKe LÞ2

¼

p2  149794  109 ¼ 83810:25 kN 42002

Ns ¼ As fy þ Ac fc0 ¼ 22855:07:3  345 þ 173494:31  50 ¼ 16559:71 kN

5.2 Compressive design strength of composite members

561

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ns 16559:71 ¼ l¼ ¼ 0:444 < 0:5 AS5100 83810:25 N cr It can be observed that l < 0:5. Therefore, the circular steel tube can provide the confinement effect. Coefficients h1 and h2 are calculated based on Eq. (5.10) as: h1 ¼ h10 ¼ 4:9  18:5l þ 17l2 ¼ 4:9  18:5  0:444 þ 17  0:4442 ¼ 0:036  0

  h2 ¼ h20 ¼ 0:25 3 þ 2l ¼ 0:25  ð3 þ 2  0:444Þ ¼ 0:97
1:0

From Eq. (5.7), the design plastic capacity of the column to axial compressive force is: " # t fy AS5100 0 ¼ 0:9  22855:07  0:97  345 Npl; Rd ¼ fAs h2 fy þ fc Ac fc 1 þ h1 D fc0   15 345  ¼ 12143 kN þ 0:6  173494:31  50  1 þ 0:036  500 50 and the nominal design plastic capacity of the column is obtained by taking the capacity factors 4 ¼ 4c ¼ 1 as follows: Es Is þ EAS5100 Ic ¼ 205000  672653198 þ 17875  2395305786 ðEIÞAS5100 c eff ¼ 180709  109 N:mm2 p2 ðEIÞAS5100 eff

p2  180709  109 ¼ 101107:65 kN 42002 ðKe LÞ2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ns 16559:71 ¼ l¼ ¼ 0:40 < 0:5 AS5100 101107:65 N cr

AS5100 Ncr ¼

¼

It can be observed that l < 0:5. Therefore, the circular steel tube can provide the confinement effect. Coefficients h1 and h2 are calculated based on Eq. (5.10) as: 2

h1 ¼ h10 ¼ 4:9  18:5l þ 17l ¼ 4:9  18:5  0:40 þ 17  0:402 ¼ 0:2  0   h2 ¼ h20 ¼ 0:25 3 þ 2l ¼ 0:25  ð3 þ 2  0:40Þ ¼ 0:95
1:0 " # t fy AS5100 0 ¼ 22855:07  0:95  345 Npl; Rd; nom ¼ As h2 fy þ Ac fc 1 þ h1 D fc0   15 345  ¼ 16538:35 kN þ 173494:31  50  1 þ 0:2  500 50

562

5. Design rules and standards

The slenderness ratio ln is obtained by Eq. (5.39): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N AS5100 16538:35 pl; Rd; nom ¼ lr ¼ ¼ 0:444 Ncr 83810:25 ln ¼ 90lr ¼ 90  0:44 ¼ 40 From Eqs. (5.45) and (5.46), the reduction slenderness factor of the member is: aa ¼

2100$ðln  13:5Þ 2100  ð40  13:5Þ ¼ 2 ¼ 18:31 2 40  15:3  40 þ 2050 ln  15:3ln þ 2050

ab is taken as 0, based on Table 5.6. l ¼ ln þ aa $ab ¼ 40 h ¼ 0:00326:ðl  13:5Þ ¼ 0:00326  ð40  13:5Þ ¼ 0:086  0  2  2 l 40 þ1þh þ 1 þ 0:086 90 90 x¼ ¼ ¼ 3:25  2  2 l 40 2 2 90 90 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2   2

90 90 ¼ 3:25  1  1  ¼ 0:90 ac ¼ x 1  1  xl 3:25  40 Therefore, from Eq. (5.44), the design strength of the member under axial compressive load is: AS5100 AS5100 Nu; design ¼ ac Npl; Rd ¼ 0:90  12143 ¼ 10988 kN

and the nominal design strength of the member under axial compressive load is: AS5100 AS5100 Nu; nom; design ¼ ac Npl;Rd;nom ¼ 0:90  16538:35 ¼ 14855:68 kN

• BS5400 The first step in calculating the compressive strength of circular CFST columns based on BS5400 is to define C1 and C2 . The magnitude of le 4250 D ¼ 500 ¼ 8:4. Therefore, based on Fig. 5.1, C1 and C2 are taken as 4.4 and 0.84, respectively. fcc0 and fy0 are calculated using Eqs. (5.16) and (5.17), respectively, as follows: t 15 0  345 ¼ 105:54 MPa þ C1 fy ¼ 60 þ 4:4  fcc0 ¼ fcu D 500

5.2 Compressive design strength of composite members

563

fy0 ¼ C2 fy ¼ 0:84  345 ¼ 289:8 MPa The design plastic capacity of the column to axial compressive force is governed by Eq. (5.15) as: BS5400 0 0 Npl; Rd ¼ fAs fy þ fc Ac fcc ¼ 0:95  22855:07:3  289:8 þ 0:45  173494:31

 105:54 ¼ 14532 kN The concrete contribution ratio is calculated by Eq. (5.13) as follows: ac ¼

0 0:45Ac fcu 0:45  173494:31  60 ¼ ¼ 0:32 BS5400 14532  103 Npl; Rd

Therefore, the limits 0:1 < ac < 0:8 are satisfied. The nominal compressive strength of the section is calculated based on Eq. (5.19) as follows: BS5400 0 Npl; Rd; nom ¼ As fy þ Ac fcu ¼ 22855:07:3  289:8 þ 173494:31  105:54

¼ 18294:66 kN For determining the design strength of the member, first, the buckling curve must be defined. The selection of the buckling curve depends on the r y ratio in which r and y are governed by: D 500 ¼ ¼ 250 2 2 rffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Is 672653198 ¼ r¼ ¼ 171:6 mm As 22855:070 y¼

Hence: r 171:6 ¼ ¼ 0:69 < 0:7 y 250 According to Ref. [2], the buckling curve A must be used. From Eqs. (5.41) and (5.42), the nondimensional slenderness l is obtained as follows: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:45Ec Ic þ 0:95Es Is lE ¼ p Nu rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 0:45  27000  2395305786 þ 0:95  205000  672653198 ¼ 3:1416 13939:11 ¼ 10427:7 mm

564

5. Design rules and standards

l¼

le 4200 ¼ ¼ 0:40 > 0:2 lE 10427:7

l is greater than 0.2. Hence, the term h is obtained by Eq. (5.49) in which for buckling curve B, a should be taken as 0.0045: h ¼ 75:5aðl  0:2Þ ¼ 75:5  0:0045  ð0:40  0:2Þ ¼ 0:038 Therefore, the slenderness reduction factor of the member K1 is calculated based on Eq. (5.48) as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 



 1þh 1þh 2 4 1þ 2   2 K1 ¼ 0:5 1 þ 2 l l l sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 



 1 þ 0:038 1 þ 0:038 2 4 ¼ 0:5  1 þ  1þ ¼ 0:96  0:42 0:42 0:42 Finally, the design capacity of the member is given by Eq. (5.47) as: BS5400 BS5400 Nu; design ¼ 0:85K1 Npl; Rd ¼ 0:85  0:96  14532 ¼ 11816:83 kN

and the nominal design capacity of the member is: BS5400 BS5400 Nu; nom; design ¼ 0:85K1 Npl; Rd; nom ¼ 0:85  0:96  18294:66 ¼ 14876:5 kN

• DBJ13-51 From Eqs. (5.21)e(5.23), the composite section properties are: Asc ¼ As þ Ac ¼ 196349:4 mm2 x0 ¼

As f 22855:07  295 ¼ 1:41 ¼ Ac fc0 173494:31  27:5

fsc ¼ ð1:14 þ 1:02x0 Þfc0 ¼ ð1:14 þ 1:02  1:41Þ  27:5 ¼ 71 MPa Therefore, the design plastic capacity of the column to axial compressive force is calculated based on Eq. (5.20) as follows: DBJ1351 ¼ fsc Asc ¼ 71  196349:4 ¼ 13938:6 kN Npl; Rd

The nominal ultimate axial strength is calculated by taking gs ¼ gc ¼ 1:0, therefore: x0 ¼

As fy 22855:07  345 0 ¼ 173494:31  38:5 ¼ 1:18 Ac fck

fsc ¼ ð1:14 þ 1:02x0 Þfc0 ¼ ð1:14 þ 1:02  1:18Þ  38:5 ¼ 90:25 MPa

5.2 Compressive design strength of composite members

565

DBJ1351 Npl; Rd; nom ¼ fsc Asc ¼ 90:25  196349:4 ¼ 17720 MPa

The stability factor is the function of the steel ratio as and the slenderness of the member l. as ¼

As 22855:07 ¼ 0:13 ¼ Ac 173494:31

l¼4

Lo 4200 ¼4 ¼ 33:6 B 500

The stability factor f is taken as 0.86, based on the buckling curves recommended by DBJ13-51. Therefore, the design strength of the section is achieved from Eq. (5.50) as follows: DBJ1351 DBJ1351 ¼ 0:86  19489 ¼ 12126:57 kN Nu; design ¼ 4Npl; Rd

and the nominal design capacity of the member is: DBJ1351 DBJ1351 Nu; ¼ 0:86  24662 ¼ 15416:37 kN nom; design ¼ 4Nu; nom

• EC4 To predict the member’s compressive strength, it is first required to calculate the relative slenderness l and control the existence of the confinement effect. l is calculated by Eqs. (5.26)e(5.28), as follows: EC4 ðEIÞEC4 eff ¼ Es Is þ 0:6Ec Ic ¼ 205000  672653198 þ 0:6  35654:45

 2395305786 ¼ 1:89136  1014 N:mm2 EC4 Ncr ¼

p2 ðEIÞEC4 eff ðKe LÞ

2

¼

p2  1:89136  1014 ¼ 105821:6 kN 42002

0 ¼ 22855:07  345 þ 173494:31  50 ¼ 16559:71 kN Npl; Rk ¼ As fy þ Ac fck sffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Npl; Rk 16559:71 ¼ l¼ ¼ 0:395 < 0:5 105821:6 NEC4 cr

It can be observed that l < 0:5. Therefore, the circular steel tube can provide the confinement effect. Coefficients h1 and h2 are calculated based on Eq. (5.31) as:   hs ¼ hs0 ¼ 0:25 3 þ 2l ¼ 0:25  ð3 þ 2  0:395Þ ¼ 0:95 hc ¼ hc0 ¼ 4:9  18:5l þ 17l2 ¼ 4:9  18:5  0:395 þ 17  0:3952 ¼ 0:24

566

5. Design rules and standards

From Eq. (5.30), the design plastic capacity of the column to axial compressive force is: " # t fy EC4 0 Npl; Rd ¼ As hs fyd þ Ac fcd 1 þ hc 0 ¼ 22855:07  0:95  345 D fck   15 345  ¼ 13546:14 kN þ 173494:31  33:3  1 þ 0:24 500 50 and the nominal design plastic capacity of the column is obtained by 0 with f and f 0 , respectively, as follows: replacing fyd and fcd y ck " # t fy EC4 0 Npl; Rd; nom ¼ As hs fy þ Ac fck 1 þ hc 0 ¼ 22855:07  0:97  345 þ 183984:1 D fck   15 345  ¼ 16582:55 kN  50  1 þ 0:24 500 50 The steel contribution ratio is calculated by Eq. (5.25) as follows: d¼

As fyd EC4 Npl; Rd

¼

22855:07  345 ¼ 0:48 13546:14

Therefore, the limits 0:2 < ac < 0:9 are satisfied. From Eq. (5.52), the slenderness reduction factor is: h i    2 f ¼ 0:5 1 þ 0:21 l  0:2 þ l ¼ f ¼ 0:5  1 þ 0:21ð0:395  0:2Þ þ 0:3952 ¼ 0:60 c¼

1 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ c ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0:95 2 0:60 þ 0:602  0:3952 f þ f2  l

Therefore, the design strength of the section is achieved from Eq. (5.51) as follows: EC4 EC4 Nu; design ¼ cNpl; Rd ¼ 0:95  13546:14 ¼ 12922:26 kN

and the nominal ultimate axial strength is calculated by taking gs ¼ gc ¼ 1:0, therefore: EC4 EC4 Nu; nom; design ¼ cNpl;Rd; nom ¼ 0:95  16582:55 ¼ 15818:82 kN

• AISC360-16

5.2 Compressive design strength of composite members

567

The slenderness of the section must be determined according to Table 5.2: D 500 ¼ ¼ 33:33 t 15 sffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Es 205000 ¼ 0:09 lp ¼ 0:09 ¼ 54:8 fy 345 l¼

It can be seen that l < lp . Therefore, the steel tube is classified as a compact section. The nominal plastic capacity of the column with compact sections to axial compressive force is calculated based on Eq. (5.33) as: AISC Nnom ¼ As fy þ Ac C2 fc0 ¼ 22855:07  345 þ 173494:31  0:95  50

¼ 16126 kN From Eq. (5.34) the cross-sectional ratio of the steel tube and the infilled concrete is: As 22855:07 ¼ 0:12  0:01 ¼ As þ Ac 22855:07 þ 173494:31 Therefore, the requirement is satisfied. The next step is to calculate the design strength of the column. The effective flexural stiffness of the section is governed by Eq. (5.56) as follows: C3 ¼ 0:45 þ 3

As 22855:07 ¼ 0:8
0:9 ¼ 0:45 þ 3 22855:07 þ 173494:31 As þ Ac

AISC ðEIÞAISC Ic ¼ 250000  672653198 þ 0:8  35749:5 eff ¼ Es Is þ C3 Ec

 2395305789 ¼ 202967  109 N:mm2 From Eq. (5.55), the elastic critical load is: AISC ¼ Ncr

The ratio of

p2 ðEIÞAISC eff

AISC Nnom NcrAISC

ðKe LÞ2

¼

ð4200Þ2

¼ 113560:8 kN

16126 ¼ 0:14
2:25. Therefore, according to Eq. ¼ 113560:8

(5.54): AISC NnAISC ¼ Nnom

p2  202967  109

0:658

AISC Nnom AISC Ncr

!

  ¼ 16126  0:6580:14 ¼ 15195:45 kN

568

5. Design rules and standards

and the design capacity of the column is: AISC AISC ¼ 0:75  15195:45 ¼ 11396:6 kN Nu; design ¼ fc Nn

5.2.4.2 Example two Determine the design compressive strength of a square hollow steel tube with a dimension of 500  500  20 mm3 filled with normal strength concrete with the compressive cylinder strength of 50 MPa and cubic strength of 60 MPa under axial compression. The modulus of elasticity and the yield stress of the tube are 2:05  105 MPa and 345 MPa, respectively, and the length of the column is 4200 mm. Solution:

The geometric and material properties of the member are as follows: B ¼ H ¼ 500 mm t ¼ 20 mm AS ¼ B$H  ðB  2tÞðH  2tÞ ¼ 500  500  ð500  2  25Þ  ð500  2  25Þ ¼ 38400 mm2 Ac ¼ ðB  2tÞðH  2tÞ ¼ ð500  2  25Þ  ð500  2  25Þ ¼ 211600 mm2 Is ¼

BH3 ðB  2tÞðH  2tÞ3 500  5003 ð500  2  20Þð500  2  20Þ3  ¼  12 12 12 12 4 ¼ 1477120000 mm

Ic ¼

ðB  2tÞðH  2tÞ3 500  5003 ð500  2  20Þð500  2  20Þ3 ¼  12 12 12 ¼ 3731213333 mm4 0 fcu ¼ 60 MPa

fc0 ¼ 50 MPa 0 ¼ 50 ¼ 33:3 MPa Based on EC4: fcd 1:5 ( 0 fck ¼ 38:5 MPa Based on GB50010 [6]: fc0 ¼ 27:5 MPa

qffiffiffiffi 1:5  0:043 f 0 ¼ 0:5  24001:5  ¼ 0:5r Based on AS5100: EAS5100 c c pffiffiffiffiffi 0:043 50 ¼ 17875 MPa 0 ¼ 450  60 ¼ 27000 MPa Based on BS5400: EBS5400 ¼ 450fcu c

5.2 Compressive design strength of composite members

569

0:3  0:3 fc0 50 Based on EC4: ¼ 22000 ¼ 22000 ¼ 35654 MPa 10 10 Based on AISC 360-16: qffiffiffiffi pffiffiffiffiffi EAISC ¼ 0:043r1:5 f 0c ¼ 0:0043  24001:5 50 ¼ 35749:5 MPa c 

EEC4 c

fy ¼ 345 MPa Based on GB50017 [7]: f ¼ 295 MPa Based on EC4: fyd ¼ 345 1:0 ¼ 345 MPa • AS5100 The design plastic capacity of the column to axial compressive force is defined by Eq. (5.1) as follows: AS5100 0 Npl; Rd ¼ fAs fy þ fc Ac fc ¼ 0:9  38400  345 þ 0:6  211600  50

¼ 18271:2 kN and the nominal design plastic capacity of the column to axial compressive force is defined by Eq. (5.2) as follows: AS5100 0 Npl; Rd; nom ¼ As fy þ Ac fc ¼ 38400  345 þ 211600  50 ¼ 23828 kN

The effective flexural stiffness of the section is governed by Eq. (5.5) as follows: ¼ fEs Is þ fc EAS5100 Ic ¼ 0:9  205000  1477120000 þ 0:6 ðEIÞAS5100 c eff  17875  3731213333 ¼ 312545  109 N:mm2 The elastic critical load of the column is obtained by Eq. (5.4.b) as: AS5100 ¼ Ncr

p2 ðEIÞAS5100 eff ðKe LÞ

2

¼

p2  312545  109 ¼ 174869:27 kN 42002

The slenderness ratio ln is given by Eqs. (5.39) and (5.40) as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ns 23828 ¼ ¼ 0:37 lr ¼ AS5100 174869:27 N cr ln ¼ 90lr ¼ 90  0:37 ¼ 33:22 According to Eqs. (5.45) and (5.46), the slenderness reduction factor of the member is: aa ¼

2100$ðln  13:5Þ 2100  ð33:22  13:5Þ ¼ 15:66 ¼ l2n  15:3ln þ 2050 33:222  15:3  33:22 þ 2050

570

5. Design rules and standards

ab should be taken as 0, based on Table 5.6. l ¼ ln þ aa $ab ¼ 33:22 h ¼ 0:00326$ðl  13:5Þ ¼ 0:00326  ð33:22  13:5Þ ¼ 0:064  0  2   l 33:22 2 þ1þh þ 1 þ 0:064 90 90 x¼ ¼ ¼ 4:40   2  l 33:22 2 2 2 90 90 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2     2

90 90 ¼ 4:40  1  1  ¼ 0:93 ac ¼ x 1  1  xl 4:40  33:22 Therefore, the design capacity of the column is: AS5100 AS5100 Nu; design ¼ ac Npl; Rd ¼ 0:93  18271:2 ¼ 17018 kN

and the nominal design capacity of the column is: AS5100 AS5100 Nu; nom; design ¼ ac Npl;Rd; nom ¼ 0:93  23828 ¼ 22193:64 kN

• BS5400 The design plastic capacity of the column to axial compressive force is defined by Eq. (5.12) as follows: BS5400 0 Npl; Rd ¼ fAs fy þ fc Ac fcu ¼ 0:95  345  38400 þ 0:45  60  211600

¼ 18299 kN The concrete contribution ratio is calculated by Eq. (5.13) as follows: ac ¼

0 0:45Ac fcu 0:45  211600  60 ¼ ¼ 0:31 BS5400 18299  103 Npl; Rd

Therefore, the limits 0:1 < ac < 0:8 are satisfied. The nominal compressive strength of the section is calculated based on Eq. (5.14) as follows: BS5400 0 Npl; Rd; nom ¼ As fy þ Ac fcu ¼ 345  38400 þ 60  211600 ¼ 25944 kN

For determining the design strength of the member, first, the buckling curve must be defined. Selection of the buckling curve depends on the yr ratio in which r and y are governed by: y¼

D 500 ¼ ¼ 250 2 2

5.2 Compressive design strength of composite members

571

rffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Is 1477120000 ¼ 196:13 mm r¼ ¼ As 38400 Hence: r 196:13 ¼ ¼ 0:78 > 0:7 y 250

According to Ref. [2], the buckling curve B must be used. From Eqs. (5.41) and (5.42), the nondimensional slenderness l is obtained as follows: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:45Ec Ic þ 0:95Es Is lE ¼ p Nu rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:45  27000  3731213333 þ 0:95  205000  1477120000 ¼ 3:1416 18299 ¼ 13402 mm l¼

le 4200 ¼ ¼ 0:31 lE 13402

l is greater than 0.2. Hence, the term h is obtained by Eq. (5.49) in which for the buckling curve B, a should be taken as 0.0045: h ¼ 75:5aðl  0:2Þ ¼ 75:5  0:0045  ð0:31  0:2Þ ¼ 0:039 Therefore, the slenderness reduction factor of the member K1 is calculated based on Eq. (5.48) as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 



 1þh 1þh 2 4 1þ 2   2 K1 ¼ 0:5 1 þ 2 l l l sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 



 1 þ 0:039 1 þ 0:039 2 4 ¼ 0:5  1 þ  1þ ¼ 0:96  0:312 0:312 0:312 Finally, the design capacity of the member is given by Eq. (5.47) as: BS5400 BS5400 Nu; design ¼ 0:85K1 Npl; Rd ¼ 0:85  0:96  18299 ¼ 14919:44 kN

and the nominal design capacity of the member is: BS5400 BS5400 Nu; nom; design ¼ 0:85K1 Npl;Rd; nom ¼ 0:85  0:96  25944 ¼ 21152:75 kN

• DBJ13-51 From Eqs. (5.21)e(5.23), the composite section properties are: Asc ¼ As þ Ac ¼ 250000 mm2

572

5. Design rules and standards

x0 ¼

As f 38400  295 ¼ ¼ 1:95 0 Ac fc 211600  27:5

fsc ¼ ð1:18 þ 0:85x0 Þfc0 ¼ ð1:18 þ 0:85  1:95Þ  27:5 ¼ 77:95 MPa Therefore, the design plastic capacity of the column to axial compressive force is calculated based on Eq. (5.20) as follows: DBJ1351 ¼ fsc Asc ¼ 77:95  250000 ¼ 19487:5 kN Npl; Rd

The nominal ultimate axial strength is calculated by taking gs ¼ gc ¼ 1:0, therefore: x0 ¼

As fy 38400  345 0 ¼ 211600  38:5 ¼ 1:63 Ac fck

0 fsc ¼ ð1:18 þ 0:85x0 Þfck ¼ ð1:18 þ 0:85  165Þ  38:5 ¼ 98:65 MPa DBJ1351 Npl;Rd; nom ¼ fsc Asc ¼ 98:5  250000 ¼ 24662 MPa

The stability factor is the function of the steel ratio as and the slenderness of the member l. As 38400 ¼ ¼ 0:18 Ac 211600 pffiffiffi Lo pffiffiffi 4200 l¼2 3 ¼2 3 ¼ 29:1 B 500 as ¼

The stability factor f is taken as 0.91, based on the buckling curves recommended by DBJ13-51. Therefore, the design strength of the section is achieved from Eq. (5.50) as follows: DBJ1351 DBJ1351 ¼ 0:91  19487:5 ¼ 17734:7 kN Nu; design ¼ 4Npl; Rd

and the nominal design capacity of the member is: DBJ1351 DBJ1351 Nu; ¼ 0:91  24662 ¼ 22442:3 kN nom; design ¼ 4Nu; nom

• EC4 From Eq. (5.24), the design plastic capacity of the column to axial compressive force is: EC4 0 Npl; Rd ¼ As fyd þ Ac fcd ¼ 38400  345 þ 211600  33:3 ¼ 20301:33 kN

5.2 Compressive design strength of composite members

573

and the nominal design plastic capacity of the column is obtained by 0 with f and f 0 , respectively, as follows: replacing fyd and fcd y ck EC4 Npl;Rd;nom ¼ As fy þ Ac fc0 ¼ 38400  345 þ 211600  50 ¼ 23828 kN

The steel contribution ratio is calculated by Eq. (5.25) as follows: d¼

As fyd NuEC4

¼

38400  345 ¼ 0:65 20301:33

Therefore, the limits 0:2 < ac < 0:9 are satisfied. The effective flexural stiffness of the section is governed by Eq. (5.28) as follows: EC4 ðEIÞEC4 eff ¼ Es Is þ 0:6Ec Ic ¼ 205000  1477120000 þ 0:6  35654:45

 3731213333 ¼ 38263  1010 N:mm2 The elastic critical load of the column is obtained by Eq. (5.27.b) as: EC4 ¼ Ncr

p2 ðEIÞEC4 eff ðKe LÞ2

¼

p2  38263  1010 ð4200Þ2

¼ 214083 kN

The relative slenderness l is given by Eq. (5.26) as: sffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Npl; Rk 23828 ¼ l¼ ¼ 0:33 EC4 214083 N cr From Eq. (5.52), the slenderness reduction factor is h  2i   f ¼ 0:5 1þ 0:21 l  0:2 þl ¼ f ¼ 0:5  1 þ 0:21ð0:33 0:2Þ þ 0:332 ¼ 0:57 c¼

1 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ c ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0:97 2  0:332 2 0:57 þ 0:57 f þ f2  l

Therefore, the design strength of the section is achieved from Eq. (5.51) as follows: EC4 EC4 Nu; design ¼ cNpl; Rd ¼ 0:97  20301:33 ¼ 19682:24 kN

and the nominal ultimate axial strength is calculated by taking gs ¼ gc ¼ 1:0, therefore: EC4 EC4 Nu; nom; design ¼ cNpl; Rd; nom ¼ 0:97  23828 ¼ 23101:36 kN

574

5. Design rules and standards

• AISC360-16 The slenderness of the section must be determined according to Table 5.2: B 500 ¼ ¼ 25 t 20 sffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Es 205000 ¼ 2:26 lp ¼ 2:26 ¼ 55:1 fy 345 l¼

It can be seen that l < lp . Therefore, the steel tube is classified as a compact section. The nominal plastic capacity of the column with compact sections to axial compressive force is calculated based on Eq. (5.33) as: AISC Nnom ¼ As fy þ Ac C2 fc0 ¼ 38400  345 þ 211600  0:85  50 ¼ 22241 kN

From Eq. (5.34) the cross-sectional ratio of the steel tube and the infilled concrete is: As 38400 ¼ ¼ 0:15  0:01 As þ Ac 38400 þ 211600 Therefore, the requirement is satisfied. The next step is to calculate the design strength of the column. The effective flexural stiffness of the section is governed by Eq. (5.56) as follows: C3 ¼ 0:45 þ 3

As 38400 ¼ 0:91 > 0:9 ¼ 0:45 þ 3 As þ Ac 38400 þ 211600

Therefore the coefficient C3 is taken as 0.9. AISC ðEIÞAISC Ic ¼ 205000  1477120000 þ 0:9  35749:5 eff ¼ Es Is þ C3 Ec

 3731213333 ¼ 422859806224:53 kN:mm2 From Eq. (5.55), the elastic critical load is: AISC ¼ Ncr

p2 ðEIÞAISC eff ðKe LÞ2

¼

p2  422859806224:53 ð4200Þ2

¼ 236590:65 kN

N AISC

22241 The ratio of Nnom AISC ¼ 236590:65 ¼ 0:094
2:25. Therefore, according to Eq. cr

(5.54): AISC NnAISC ¼ Nnom

0:658

AISC Nnom AISC Ncr

!

  ¼ 22241  0:6580:094 ¼ 21383 kN

5.2 Compressive design strength of composite members

575

and the design capacity of the column is: AISC AISC ¼ 0:75  21383 ¼ 16037:25 kN Nu; design ¼ fc Nn

5.2.4.3 Example three Determine the ultimate axial strength of an elliptical CFST column with the dimension of 400  200  12:5 mm and the length of 4 m according to EC4. The yield strength of the steel tube is 355 MPa and the concrete compressive strength is 30 MPa. Solution:

As shown in Fig. 5.2, the dimensions of the column are as follows: 2a ¼ 400 mm 2b ¼ 200 mm t ¼ 12:5 mm L ¼ 4000 mm The cross-sectional area of the concrete core Ac is: p p Ac ¼ ð2a  2tÞð2b  2tÞ ¼ ð400  2  12:5Þð200  2  12:5Þ ¼ 51542 mm2 4 4 The cross-sectional area of the steel tube As is [8]:   hm ffiffiffiffiffiffiffiffiffiffiffiffiffiffi p t As ¼ Pm t ¼ pðam þ bm Þ 1 þ 3 10 þ 4  hm in which Pm is the mean perimeter and: am ¼

ð2a  tÞ ð400  12:5Þ ¼ ¼ 193:75 mm 2 2

bm ¼ hm ¼

ð2b  tÞ ð200  12:5Þ ¼ ¼ 93:75 mm 2 2

ðam  bm Þ2

ð193:75  93:75Þ2

ðam þ bm Þ

ð193:75 þ 93:75Þ2

FIGURE 5.2

¼ 2

¼ 0:121

The cross section of the elliptical CFST column.

576

5. Design rules and standards

Therefore

 As ¼ pð193:75 þ 93:75Þ 1 þ 3

 0:121 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12:5 ¼ 11632 mm2 10 þ 4  0:121

The second-moment areas of the concrete core Ic and the steel tube Is are: p p Ic ¼ ð2a  2tÞð2b  2tÞ3 ¼ ð400  2  12:5Þð200  2  12:5Þ3 4 4 ¼ 1:578  109 mm4 p p Is ¼ ð2aÞð2bÞ3  Ic ¼ ð400Þð200Þ3  1:578  109 mm4 4 4 ¼ 9:348  108 mm4 Based on EC4, the effective material properties of the section are: fyd ¼

fy 355 ¼ ¼ 355 MPa gs 1:0

Es ¼ 210000 MPa 0 fcd ¼

0 fck 30 ¼ ¼ 20 MPa gc 1:5

Ec ¼ 33000 MPa To predict the compressive strength of the member, first, it is required to calculate the relative slenderness l and control the existence of the confinement effect. l is calculated by Eqs. (5.26)e(5.28), as follows: EC4 8 ðEIÞEC4 eff ¼ Es Is þ 0:6Ec Ic ¼ 210000  9:348  10

þ 0:6  33000  1:578  109 ¼ 2:276  1014 N:mm2 EC4 Ncr ¼

p2 ðEIÞEC4 eff ðKe LÞ2

¼

p2  2:276  1014 ¼ 14040 kN 400002

0 ¼ 11632  355 þ 51542  20 ¼ 5160 kN Npl; Rk ¼ As fy þ Ac fck sffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffi Npl; Rk 5160 ¼ l¼ ¼ 0:61 EC4 14040 N cr

It can be observed that l > 0:5. Therefore, the steel tube cannot provide the confinement effect.

5.3 Moment design strength of composite members

577

The design plastic capacity of the column to axial compressive force is: EC4 0 Npl; Rd ¼ As fyd þ Ac fcd ¼ 11632  355 þ 51542  20 ¼ 5160 kN

No steel reinforcement is considered in the column. Therefore, the buckling curve b should be used and a ¼ 0:34. From Eq. (5.52), the slenderness reduction factor is: i h    f ¼ 0:5 1 þ 0:21 l  0:2 þ l2 ¼ f ¼ 0:5  1 þ 0:21ð0:61  0:2Þ þ 0:612 ¼ 0:756 c¼

1 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ c ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0:832 0:756 þ 0:7562  0:612 f þ f2  l2

Therefore, the design strength of the section is achieved from Eq. (5.51) as follows: EC4 EC4 Nu; design ¼ cNpl; Rd ¼ 0:832  5160 ¼ 4293 kN

5.2.5 Design strength of CFDST columns subjected to axial compression None of the design regulations provide guidelines for designing CFDST members. Consequently, structural engineers are required to employ the modified equations of CFST columns to design CFDST columns. Usually, the inner steel tube’s load-bearing capacity is added to the design code’s equations based on the strength superposition principle. However, the confinement effect provided by the inner tube is ignored. Table 5.7 summarizes the design strength equations of CFDST columns based on the modified recommendations of design codes in which Aso and fyo are, respectively, the area and yield strength of the outer tube, and Asi and fyi are, respectively, the area and yield strength of the outer tube.

5.3 Moment design strength of composite members based on design guidelines 5.3.1 Moment strength of CFST beams Before discussing the given equations in design codes for predicting the composite beam’s moment capacity, it is necessary to understand the stress distribution in composite members under pure bending. This allows reaching a model for predicting the plastic moment capacity of composite beams. Figs. 5.3e5.5 illustrate the typical stress distributions of rectangular without rounded corners, with rounded corners, and circular CFST members under bending, respectively.

578

5. Design rules and standards

TABLE 5.7 Design strength equations of CFDST columns according to design codes. Design guideline

Shape of the cross section

AS5100

Rectangular Circular

Design equation AS5100 ¼ 4A f 0 Npl; so yo þ 4c Ac fc þ 4Asi fyi Rd " AS5100 ¼ 4A h f 0 t Npl; so 2 yo þ 4c Ac fc 1 þh1 D Rd

# fy fc0

þ 4Asi h2 fyi

ðEIÞAS5100 ¼ 4Eso Iso þ 4c EAS5100 Ic þ 4Esi Isi c eff Ns ¼ Aso fyo þ Ac fc0 þ Asi fyi p2 ðEIÞAS5100 eff ðKe LÞ2

AS5100 ¼ Ncr qffiffiffiffiffiffiffiffiffiffiffiffi Ns l ¼ NAS5100 cr

BS5400

0 þ fA f ¼ fAso fyo þ fc Ac fcu si yi

Rectangular

BS5400 Npl;Rd

Circular

BS5400 ¼ fA f 0 þ f A f 0 þ fA f Npl; so yo si yi c c cc Rd 0 þC t f fcc0 ¼ fcu 1 D yo 0 fy ¼ C2 fyo

DBJ13-51

Rectangular and circular

DBJ1351 Npl; ¼ fsc Asc þ Asi fyi Rd

Asc ¼ Aso þ Ac ( ð1:18 þ 0:85x0 Þfc0 fsc ¼ ð1:14 þ 1:02x0 Þfc0 x0 ¼

for the square section for the circular section

Aso f Ac fc0

f

f ¼ gyo s

EC4

Rectangular Circular

EC4 ¼ Npl; Rd f fyod ¼ gyo s

0 þA f Aso fyod þ Ac fcd si yi

"

 EC4 ¼ h A f 0 1 þh to þA f f Npl;Rd þ A so yod c cd si yi s c Do

# fyo fck0

EC4 I þ E I ðEIÞEC4 c si si eff ¼ Eso Iso þ 0:6Ec 0 þA f Npl; Rk ¼ Aso fyo þ Ac fck si yi p2 ðEIÞEC4

EC4 ¼ eff Ncr ðK LÞ2 rffiffiffiffiffiffiffiffiffiffi e N l ¼ Npl;EC4Rk cr

AISC36016

Rectangular and circular

8A f þ A C f0 þ A f so yo c 2 c si yi Compact > > > > > < 2 Np  Ny  AISC ¼ Noncompact Np   2 l  lp Nnom > lr  lp > > > > : Aso fcr þ Ac 0:7fc0 þ Asi fyi Slender Ny ¼ Aso fyo þ Ac 0:7fc0 þ Asi fyi 8 9Eso > >  2 for rectangular section > > > bo > > > > < to fcr ¼ 0:72fyo > > > !0:2 for circular section > > > D > o fyo > > : E t so o

5.3 Moment design strength of composite members

FIGURE 5.3

579

Stress distribution of rectangular CFST members under pure bending.

FIGURE 5.4 Stress distribution of rectangular CFST members with round-ended corners under pure bending [9].

580

5. Design rules and standards

FIGURE 5.5 Stress distribution of circular CFST members with round-ended corners under pure bending [10].

The main assumption of all the design codes in developing expressions for predicting the flexural strength of composite members is that the compression region of the concrete core and both the compression and tension regions of the steel tube participate in providing the flexural resistance of the member. In other words, the load-bearing capacity of part of the concrete, which is under tension, is ignored because of the rapid crack development at the initial loading stage. Compared with HSSs under bending, the postbuckling behavior of the composite members is more stable. This is because the infilled concrete prevents the inward buckling of the steel tube. Three mechanisms govern the moment capacity of composite members, including: 1. Yielding of the steel tube tensile area 2. Buckling of the steel tube compressive area 3. Crushing of the concrete In this section, first, expressions for predicting the plastic moment capacity of circular and rectangular CFST beams are defined. Then, recommendations of design regulations for calculating the moment capacity of CFST beams are discussed. 5.3.1.1 Plastic moment capacity of CFST beams The method for defining the plastic moment capacity of CFST beams is based on the plastic stress distribution method. According to this method, both the steel tube and the concrete core have rigid-plastic behavior. The stresses of the steel tube under tension and compression reach fy , and the

5.3 Moment design strength of composite members

581

concrete core’s compressive stress reaches fc0 . By contrast, the tension stress of the concrete core is assumed to be zero. Rectangular section with flat steel plates

When a composite member is under bending, part of the concrete below the neutral axis is under tension. Therefore, the strength of the concrete section under the neutral axis can be omitted. As shown in Fig. 5.3, the neutral axis position can be defined based on the mechanical equilibrium of the internal forces acting on the concrete core and the steel tube as: C1 þ C2 þ C3 ¼ T1 þ T2

(5.57)

where C1 and C2 are the compression forces acting on the widths B and H, respectively, C3 is the compression force on the concrete core, and T1 and T2 are the tension forces acting on the widths B and H, respectively. From Fig. 5.3, the magnitudes of internal forces can be expressed as: C1 ¼ Btfy

(5.58.a)

C2 ¼ 2dn tfy

(5.58.b)

C3 ¼ ðB  2tÞdn fc0

(5.58.c)

T1 ¼ 2ðH  2t  dn Þtfy

(5.58.d)

T2 ¼ Btfy

(5.58.e)

Therefore, the position of the neutral axis position is defined as:   H  2t (5.59) fRHS dn ¼ 2 in which 1 1 z (5.60) 0 0:25fc B  2t 0:25fc0 B $ $ 1þ 1þ fy fy t t

 The magnitude of fRHS for the HSS is 1.0 fy ¼ fc0 ¼ 0 . In this case, Eq. fRHS ¼

(5.59) gives the position of the neutral axis of the HSS. According to the internal forces acting on the section, shown in Fig. 5.3, the plastic moment capacity is governed by: Mpl ¼ C1 dc1 þ C2 dc2 þ C3 dc3 þ T1 dT1 þ T2 dT2

(5.61)

where dc1 ¼ dn þ

t 2

(5.62.a)

582

5. Design rules and standards

dn 2 dn dc3 ¼ 2 dc2 ¼

dT1 ¼

(5.62.b) (5.62.c)

ðH  2t  dn Þ 2

dT2 ¼ H  2t  dn þ

t 2

(5.62.d) (5.62.e)

Hence,   2 Mpl ¼ fy t BðH  tÞ þ 0:5ðH  2tÞ2 þ 0:5fy tðH  2tÞ2 1  fRHS þ 0:5ðB  2tÞd2n fc0

(5.63)

in which the first term of the equation is the plastic moment capacity of an unfilled rectangular steel hollow section:  (5.64) Mpl; RHS ¼ fy t BðH  tÞ þ 0:5ðH  2tÞ2 The plastic design strength of rectangular CFST members with flat steel tubes can also be determined as follows:     (5.65) Mpl ¼ Wpa  Wpa;n fy þ 0:5 Wpc  Wpc;n fc0 in which Wpa and Wpc represent the plastic section modulus of the steel tube and the concrete core, respectively, and Wpa;n and Wpc;n are the plastic section modulus of the steel tube and the concrete core in the 2dn region, respectively. ðB  2tÞðH  2tÞ2 4

(5.66.a)

BH2  Wpc 4

(5.66.b)

Wpc;n ¼ ðB  2tÞd2n

(5.66.c)

Wpa;n ¼ Bd2n  Wpc;n

(5.66.d)

Wpc ¼

Wpa ¼

The position of the neutral axis dn is given by: dn ¼

Ac fc0

 2Bfc0 þ 4t 2fy  fc0

(5.67)

Rectangular section with round-ended corners

The overall trend for defining the plastic moment capacity of rectangular CFST beams with round-ended corners is similar to those with flat steel plates. The only difference is that the distribution of stresses at

583

5.3 Moment design strength of composite members

rounded corners must be included in calculations. Fig. 5.4 shows the stress distribution of rectangular CFST beams with round-ended corners. The neutral axis position can be defined based on the mechanical equilibrium of the internal forces acting on the concrete core and the steel tube as: 6 X

Ci ¼

i¼1

6 X

Ti

(5.68)

i¼1

Therefore, the neutral axis position is governed by Ref. [9]: ! 0:5fc0 rint  p  ðH  2tÞ þ 2rext  2t  rint fy t 2 ! dn ¼  0:5fc0 B  2t 2þ fy t

(5.69)

in which the terms rext and rint ¼ rext  t are the external and internal radiuses of the member, respectively. rext can be taken as 2t for steel hollow sections with t  3 mm and 2:5t for HSSs with t < 3 mm, respectively [11]. Therefore, Eq. (5.69) can be rewritten as: ! !   8 fc0 t fc0 t > 1 þ 0:242 1 þ 0:242 >     > > fy fy H  2t H > H  2t H  2t > > ! !  >  z > 2 2 > 0:25fc0 B  2t 0:25fc0 B > > > 1þ 1þ > > fy fy t t < dn ¼ ! !     > > > f0 t f0 t > > 1 þ 0:108 c 1 þ 0:108 c >     > f f H  2t H > y y H  2t H  2t > > ! !  >  z > > 2 2 > 0:25fc0 B  2t 0:25fc0 B > : 1þ 1þ fy fy t t

for t  3 mm

for t < 3 mm

ð5:70Þ

According to Fig. 5.4, the plastic moment capacity of the section can be defined as follows [9]: Mpl ¼

3 X

Ti dTi þ

6 X

Ci dCi

(5.71)

T1 ¼ C1 ¼ fsu tðB  2rext Þ   t T2 ¼ C2 ¼ fsu tp rint þ 2

(5.72.a)

i¼1

i¼1

where

T3 ¼ fsu t2ðH  2t  dn  rint Þ

(5.72.b) (5.72.c)

584

5. Design rules and standards

C3 ¼ fsu t2ðdn  rint Þ

(5.72.d)

C4 ¼ fc0 ðB  2rext Þ p C5 ¼ fc0 r2int 2

(5.72.e)

C6 ¼ fc0 ðB  2tÞðdn  rint Þ

(5.72.g)

dT1 ¼ H  2t  dn þ dT2 ¼ ðH  2t  dn  rint Þ þ

(5.72.f)

t 2 

p t rint þ 2 2

ðH  2t  dn  rint Þ 2 t dC1 ¼ dn þ 2   p t rint þ dC2 ¼ ðdn  rint Þ þ 2 2 dT3 ¼

ðdn  rint Þ 2 rint dC4 ¼ dn  2 4 dC5 ¼ ðdn  rint Þ þ rint 3p dC3 ¼ dC6 ¼

(5.72.h)  (5.72.i) (5.72.j) (5.72.k) (5.72.L) (5.72.m) (5.72.n) (5.72.o)

The plastic design strength of rectangular CFST members with roundended corners can also be derived from Eq. (5.65), in which the terms Wpa , Wpa;n , Wpc , and Wpc;n are governed by:   ðB  2tÞðH  2tÞ2 2 2 H 2  rint  rint ð4  pÞ (5.73.a)  t  rint Wpc ¼ 4 3 2   BH2 2 H 3 2 Wpa ¼  ðrint þ tÞ  ðrint þ tÞ ð4  pÞ  t  rint  Wpc (5.73.b) 4 3 2 Wpc;n ¼ ðB  2tÞd2n

(5.73.c)

Wpa;n ¼ Bd2n  Wpc;n

(5.73.d)

The position of the neutral axis dn from the compression face of the steel tube is given by Eq. (5.67). Circular section

The overall trend for defining the plastic moment capacity of circular CFST beams is similar to the rectangular section. Fig. 5.5 depicts the stress

585

5.3 Moment design strength of composite members

distribution of circular CFST beams in which go represents the angular position of the plastic neutral axis and is obtained by using the equilibrium principle, as follows: FSC þ FCC ¼ Fst

(5.74)

The magnitudes of the internal forces are governed by Ref. [10]: FSC ¼ fy trm ðp  2go Þ   1 0 2 p  go  sinð2go Þ FCC ¼ fc ri 2 2 Fst ¼ fy trm ðp þ 2go Þ

(5.75.a) (5.75.b) (5.75.c)

in which dt 2 d  2t ri ¼ 2 rm ¼

(5.76.a) (5.76.b)

Based on Eq. (5.74), go can be defined using an iterative approach. By assuming sin go ¼ go : p go ¼ FCHS 2 in which

! !   d  2t 1 fc0 d t 8 fy t ! !  FCHS ¼ z  1 fc0 d  2t 1 fc0 d 1þ 1þ 4 fy t 4 fy t 1 fc0 8 fy

(5.77)

(5.78)

The magnitude of FCHS for the hollow section is equal to 0, which coincides with the plastic neutral axis’s angular position in the circular hollow section ðgo ¼ 0Þ. The plastic moment capacity of the member is governed by: Mpl ¼ Msc þ Mcc þ Mst

(5.79)

in which Mcc is the plastic moment strength of the concrete core under compression, and Mst and Msc are the plastic moment strengths of the steel tube under tension and compression, respectively, and are given by Ref. [10]: Msc ¼ Mst ¼ 2fy tr2m cos go

(5.80.a)

586

5. Design rules and standards

2 Mcc ¼ fc0 r3i cos3 go 3

(5.80.b)

Therefore, Eq. (5.79) can be rewritten as follows: 2 Mpl ¼ 4fy tr2m cos go þ fc0 r3i cos3 go 3

(5.81)

The plastic moment resistance of circular CFST beams can also be defined by Eq. (5.65), in which the terms Wpa , Wpa;n , Wpc , and Wpc;n are governed by: Wpc ¼

ðD  2tÞ3 6

(5.82.a)

Wpa ¼

D3  Wpc 6

(5.82.b)

Wpc;n ¼ ðD  2tÞd2n

(5.82.c)

Wpa;n ¼ Dd2n  Wpc;n

(5.82.d)

The position of the neutral axis dn from the compression face of the steel tube is given by: dn ¼

Ac fc0

 2Dfc0 þ 4t 2fy  fc0

(5.83)

5.3.2 Moment capacity of CFST beams based on design codes 5.3.2.1 AS5100 AS5100 (2017) does not provide design equations for predicting the moment capacity of CFST beams and only suggests using the plastic theory for determining the bending moment capacity. Hence, equations developed in Section 5.3.1.1 can be used for predicting the nominal moment capacity ðMn Þ of CFST beams. The design moment strength MAS5100 u; design is then achieved by multiplying the capacity factor 4 ¼ 0:9 into

 Mn MAS5100 u; design ¼ 4Mn . The principal hypothesis for predicting the plastic moment capacity of CFST beams is that no buckling occurs in circular or rectangular steel tubes. The slenderness limits for the circular and rectangular steel tubes under pure bending to prevent local buckling qffiffiffiffiffiffi are 90ε02 and 52ε0 , respectively, in which ε0 ¼ 235 fy .

5.3 Moment design strength of composite members

587

5.3.2.2 BS5400 • Rectangular cross section The ultimate design moment strength of rectangular CFST beams is governed by:   h  dc BS5400 þ btðt þ dc Þ (5.84) Mu; design ¼ 0:95fy As 2 where the term h represents the concrete depth, and dc is the distance between the inner surface of the steel tube and the neutral axis, similar to dn , depicted in Fig. 5.3. dc can be expressed as: dc ¼

As  2bt ðb  2tÞr þ 4t

(5.85)

0 0:4fcu 0:95fy

(5.86)

where r¼

• Circular cross section The ultimate design moment strength of circular CFST beams is given by: MBS5400 u; design ¼ 0:95Sfy ð1 þ 0:01mÞ

(5.87)

where the parameter m is for incorporating the influence of the concrete core on the moment capacity of the column and can be achieved from graph A.2 given by standard BS5400-Part 5 [2], and the term S represents the plastic section modulus of the steel section and is governed:  2 D 1 (5.88) S ¼ t3 t According to Ref. [2], the wall thickness of rectangular and circular steel tubes in CFST members under bending must be greater than qffiffiffiffiffiffi qffiffiffiffiffiffi qffiffiffiffiffiffi f f f t  b 3Ey s and t  D 8Ey s , respectively. Based on Ref. [2], t  b 3Ey s qffiffiffiffiffiffi qffiffiffiffiffiffi qffiffiffiffiffiffi 3Es 3Es 275 and  2 ¼ $ε  2 in which ε ¼
can be rewritten as b2t 275 t fy fy qffiffiffiffiffiffi qffiffiffiffiffiffi fy D s Es ¼ 202; 000 MPa. Similarly, t  D 8Es can be rewritten as t
8E fy ¼

588

5. Design rules and standards

pffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffi 8Es fy 275 8Es fy ¼ 275 ε2 where ε2 ¼ 275 275 $ fy fy and Es ¼ 202; 000 MPa. The local buckling of steel tubes is no longer required to be considered if the thickness of the tubes is larger than the limits. The nominal moment strength can be achieved by replacing 0:95fy with 0 with f 0 in the above equations. fy and 0:4fcu cu 5.3.2.3 DBJ13-51 Ref. [3] recommends the following equation for predicting the ultimate design moment strength of CFST beams: MDBJ1351 u; design ¼ gm Wsc fsc

(5.89)

in which the terms gm , Wsc ; and fsc depend on the cross section of the beam. For rectangular CFST beams, the variables are governed by: gm ¼ 1:04 þ 0:48 lnðx þ 0:1Þ

Wsc ¼

(5.90.a)

BH2 6

(5.90.b)

fsc ¼ ð1:18 þ 0:85x0 Þfc0

(5.90.c)

For circular CFST beams, the variables are governed by: gm ¼ 1:1 þ 0:48 lnðx þ 0:1Þ Wsc ¼

(5.91.a)

pD3 32

(5.91.b)

fsc ¼ ð1:14 þ 1:02x0 Þfc0

(5.91.c)

where x is the confinement factor and x0 is the design confinement factor and is defined by: x0 ¼

As f Ac fc0

(5.92)

In the above equations, B, H, and D are defined in Figs. 5.3 and 5.5, f represents the design yield stress of the tube and fc0 is the design f

f0

compressive strength of infilled concrete. f and fc0 can be taken as gy and gck , s c 0 represents the characteristic amount of the cylrespectively, in which fck inder compressive strength of concrete at 28 days, gc ¼ 1:4 and gs ¼ 1:12 are the material properties factors. The nominal moment strength can be achieved by taking gs ¼ gc ¼ 1:0. MDBJ1351 u; nom

589

5.3 Moment design strength of composite members

The slenderness limits for preventing the local buckling in circular and rectangular steel tubes under pure bending are 150ε02 and 60ε0 , respecqffiffiffiffiffiffi tively, in which ε0 ¼ 235 fy . 5.3.2.4 EC4 Like AS5100, Ref. [2] does not recommend design equations for predicting the moment capacity of CFST beams and only suggests using the plastic theory. Hence, equations developed in Section 5.3.1.1 can be used 

for predicting the nominal moment capacity MEC4 u; nom

of CFST beams.

The design moment strength MEC4 u; design can be achieved by replacing fy f

f0

0 with y and c , respectively. The slenderness limits for the circular and fck gs gc and rectangular steel tubes for preventing the local buckling under pure qffiffiffiffiffiffi bending are 90ε02 and 52ε0 , respectively, in which ε0 ¼ 235 fy .

5.3.2.5 AISC360-16 Based on Ref. [5], the moment capacity of CFST beams depends on the slenderness of the steel tube. Ref. [5] assumes that the moment capacity of CFST beams with compact steel tube under bending can reach the plastic moment capacity of the section Mp defined based on the plastic stress distribution method. Therefore, for compact circular and rectangular CFST beams: MAISC nom ¼ Mp

(5.93)

in which Mp is the plastic moment resistance of the section and can be obtained based on Section 5.3.1.1. It should be noted that for compact composite members under bending, Ref. [5] considers that the tension and compression regions of the steel tube fully reach the stress fy , whereas the concrete core of circular and rectangular sections can obtain 0:95fc0 and 0:85fc0 , respectively. In addition to the method discussed in Section 5.3.1.1, the stress blocks of compact rectangular CFST members, shown in Fig. 5.6, can be used for defining the plastic moment resistance Mp . The location of the neutral axis ap from the compression face is given by: ap ¼

2fy Htw þ 0:85fc0 btf 4tw fy þ 0:85fc0 b

and the plastic moment resistance Mp is expressed as [12]:    

a  tf tf p þ fy btf H  ap  þ fy ap 2tw Mp ¼ fy btf ap  2 2 2  

 ap  t    H  ap f þ 0:85fc0 ap  tf b þ fy H  ap 2tw 2 2

(5.94)

(5.95)

590

5. Design rules and standards

FIGURE 5.6 Stress blocks of compact section for defining Mp .

  For noncompact sections lp < l < lr , the nominal moment strength of the composite beam can be obtained by linear interpolation between the plastic moment strength Mp and yield moment strength My of the section and tube slenderness as follows:     l  lp (5.96) ¼ M  M  M MAISC p p y nom lr  lp in which l, lp , and lr are listed in Table 5.2, and My is the yield moment of the section and is determined based on the elastic stress distribution on the section. The moment strength of noncompact and slender sections is obtained based on the lower bound theory of plasticity. For noncompact sections, it is assumed that the tension and compression regions of the steel tube have elastic-plastic behavior and elastic behavior up to the stress fy , respectively, and the compression area of the concrete core has an elastic behavior up to 0:7fc0 . Besides, the contribution of the tension area of the concrete core is ignored. Fig. 5.7 shows the stress blocks of

FIGURE 5.7

Stress blocks of noncompact section for defining My .

5.3 Moment design strength of composite members

591

noncompact CFT members for defining My . It can be observed from the figure that the maximum compressive strength of the concrete core is 0:7fc0 . This assumption is because of the low stability of the noncompact steel tube after local buckling. Therefore, a noncompact steel tube cannot fully confine the concrete core. Besides, the concrete core may present remarkable expansion in the lateral direction after reaching the compressive stress of 0:7fc0 . The volumetric dilatation intensifies the steel tube’s local buckling and its disability in confining the concrete core. From Fig. 5.7 and the use of the equilibrium equation, the location of the neutral axis ay from the compression face is given by: ay ¼

2fy Htw þ 0:35fc0 btf 4tw fy þ 0:35fc0 b

(5.97)

and the elastic moment resistance My is expressed as:       tf tf 2ay þ fy btf H  ay  þ 0:5fy ay 2tw My ¼ fy btf ay  2 2 3      

 2 ay  tf    2ay H þ fy H  ap 2tw þ 0:5fy ay 2tw þ 0:35fc0 ap  tf b 3 3 2 (5.98) The stress blocks of slender sections are depicted in Fig. 5.8. It can be observed from the figure that both tension and compression regions of the steel tube have elastic behavior. The maximum stress of the steel tube’s compression region is limited to the local buckling stress Fcr , and the tension region of the curve can achieve the yield stress fy . Besides, the concrete core’s compression area has an elastic behavior up to 0:7fc0 , and

FIGURE 5.8 Stress blocks of slender section for defining first yield moment Mcr .

592

5. Design rules and standards

the contribution of the tension area of the concrete core is ignored. The local buckling stress Fcr is expressed by: 8 9Es > for rectangular section > > b2 > > > > > < t (5.99) Fcr ¼ 0:75fy > > for circular section > " # >   0:2 > > > D fy > : t Es From Fig. 5.8 and the use of equilibrium equation, the location of the neutral axis acr from the compression face is given by:

 fy Htw þ 0:35fc0 þ fy  Fcr btf

 (5.100) acr ¼ tw fy þ Fcr þ 0:35fc0 b The moment strength of CFST members with slender steel tubes is given by:       tf tf 2acr AISC þ fy btf H  acr  þ 0:5Fcr acr 2tw Mnom ¼ Mcr ¼ Fcr btf acr  2 2 3    

 2 acr  tf  2ðH  acr Þ 0 þ 0:5fy ðH  acr Þ2tw þ 0:35fc acr  tf b 3 3 (5.101)

5.3.3 Examples 5.3.3.1 Example four Determine the moment strength of a square hollow steel tube with a dimension of 500  500  20 mm3 filled with normal strength concrete with the compressive cylinder strength of 50 MPa and cubic strength of 60 MPa under pure bending. The modulus of elasticity and the yield stress of the tube are 2:05  105 MPa and 345 MPa, respectively, and the length of the column is 4200 mm. The geometric and material properties of the member are as follows: B ¼ H ¼ 500 mm t ¼ 20 mm AS ¼ 38400 mm2 Ac ¼ 211600 mm2 0 fcu ¼ 60 MPa

5.3 Moment design strength of composite members

593

fc0 ¼ 50 MPa 0 ¼ Based on EC4: fcd

Based on GB50010½6:

50 ¼ 33:3 MPa 1:5 ( 0 fck ¼ 38:5 MPa fc0 ¼ 27:5 MPa

¼ 17875 MPa Based on AS5100: EAS5100 c ¼ 27000 MPa Based on BS5400 : EBS5400 c ¼ 35654 MPa Based on EC4: EEC4 c ¼ 35749:5 MPa Based on AISC 360  16: EAISC c fy ¼ 345 MPa Based on GB50017½7: f ¼ 295 MPa Based on EC4: fyd ¼

345 ¼ 345 MPa 1:0

• AS5100 The slenderness of the section is: rffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffi fy B 500 345 ¼ ¼ 29:37 le ¼ 20 250 20 250 From Table 5.2, the slenderness limit for preventing the local buckling of the tube under bending is: sffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffi   B 235 235 0 ¼ 42:9 ¼ 52ε ¼ 52 ¼ 52 t limit fy 345 It can be observed that le
: fy;is Se for slender flange

for noncompact flange

ð5:104Þ in which tf is the thickness of the flange, S is the elastic section modulus about the bending axis, and Se is the effective section modulus based on the effective width of the section be . be is expressed as:

be ¼

8 > 1:92tf > > > > > > >
> > sffiffiffiffiffiffiffi sffiffiffiffiffiffiffi ! > > > E 0:34 E > > : 1:92tf 1 
b for rectangular section without round ended corners fy;is fy;is b tf

ð5:105Þ Mw is

The moment strength of the noncompact web is governed by: ! rffiffiffiffiffiffiffi

 h fy;is Mw 0:305  0:78
Mpl;is (5.106) is ¼ Mpl;is  Mpl;is  fy;is S E tw in which h and tw are the depth and thickness of the web, respectively.

606

5. Design rules and standards

The moment strength of the slender web Mw is is defined based on yielding or local buckling of the compression flange, as follows: ( Rpg fy;is S compresion flange yielding w Mis ¼ (5.107) Rpg fcr Sxc compresion flange buckling in which aw hc Rpg ¼ 1   5:7 1200 þ 300aw tw aw ¼

sffiffiffiffiffiffiffi ! E
1:0 fy;is

hc tw
10 bfc tfc

Fcr ¼

(5.108.a) (5.108.b)

9Es kc !2 b tf

(5.108.c)

where bfc and tfc are the width and thickness of the compression flange, respectively, tw is the thickness of the web. The height hc is taken as h  2t, and the constant kc is taken as 4.0. If Fcr > fy , the member strength is governed by Rpg fy;is S. Mis for circular hollow sections is expressed by: For compact sections: Mis ¼ Mpl;is ¼ fy;is Zis For noncompact sections:

2

3

60:021E 7 Mis ¼ 4   þ fy;is 5Sis D t For slender sections:

(5.109)

2

(5.110)

3

60:33E7 Mis ¼ 4   5Sis D t

(5.111)

in which the term Sis represents the elastic moment of inertia of the inner steel tube. It is noteworthy that the minimum cross section must be considered for predicting the flexural strength of tapered CFDST members.

5.3 Moment design strength of composite members

607

5.3.4.1 Example seven Determine the moment strength of a CFDST beam with a rectangular steel tube with round-ended corners bo  do  to as the outer steel tube and a rectangular steel tube with bu  di  ti as the inner steel tube. The compressive strength of the concrete is fc0 and the yield stress of the steel tubes are fy;os ¼ fy;is ¼ fy . Suppose that the inner steel tube has a compact section. Solution:

The inner steel tube has a compact section. Therefore, the moment strength of the inner tube Mis is governed by: Mis ¼ Mpl;is ¼ fy;is Zis The stress distribution of the section is shown in Fig. 5.9. Fig. 5.9b shows the stress distribution of the concrete core. The position of the neutral axis can be obtained based on the equilibrium principle, as follows: 3 X i¼1

Tos ¼

3 X i¼1

Cos þ

3 X

Cc

i¼1

in which Tos and Cos are the tensile and compressive forces in the outer tube, respectively, and Cc is the compressive force in the sandwiched concrete. The neutral axis position can be defined using the above equation. Two cases can occur. The first one is that the neutral axis locates above the inner steel tube. In this case, the whole sections of the inner steel tube are

FIGURE 5.9 Stress distribution of the CFDST beam [65].

608

5. Design rules and standards

under tension. The other case occurs when part of the inner steel tube is under compression. Therefore: ! fc0 ðdo  2to Þ þ 0:215 to fy;os do  2to  di ! if dn < dn ¼  2 f0 bo  2to 2 þ 0:5 c fy;os to !     fc0 do  2to  di bi ðdo  2to Þ þ to 0:215  0:25 fy;os to to ! dn ¼  f0 bo  2to  bi 2 þ 0:5 c fy;os to if dn 

do  2to  di 2

The moment strength Mosc is the total of the moments generated by all the internal forces: Mosc ¼

3 X i¼1

Tsoi dTsoi þ

3 X

Csoi dCsoi þ

i¼1

3 X

Cci dCci

i¼1

The forces Tsoi , Csoi , and Cci and distances dTsoi , dCsoi , and dCci when dn  do 2t2o di are expressed as: Tso1 ¼ Cso1 ¼ fy;os to ðbo  2rext Þ

p   Tso2 ¼ Cso2 ¼ fy;os r2  r2int 2 ext Tso3 ¼ fy;os to ðdo  2to  dn  rint Þ Cso1 ¼ fy;os to 2ðdn  rint Þ Cc1 ¼ fc0 ðbo  2rext Þrint

p Cc2 ¼ fc0 r2 2 int Cc3 ¼ fc0 ðbo  2to Þð0:5ðdo  2to  di Þ  rint Þ 

 do  2to  di 0 Cc4 ¼ fc ðbo  2to  bi Þ dn  2 dTso1 ¼ do  2to  dn þ

to 2

to 2 dTso2 ¼ dCso1 þ do  2to  2dn dCso1 ¼ dn þ

5.4 CFST members under combined axial

609

  to 2 rint þ 2 dCso2 ¼ dn  rint þ p dTso3 ¼ 0:5ðdo  2to  dn  rint Þ dCso3 ¼ 0:5ðdn  rint Þ dCc1 ¼ dn  0:5rint dCc2 ¼ dn  rint þ

4rint 3p

dCc3 ¼ dCc1  0:25ðdo  2to  di Þ dCc4 ¼ 0:5dn  0:25ðdo  2to  di Þ Except forces Cc3 and Cc4 and the distance dCc3 , rest of the above equations are valid for the case dn < do 2t2o di . Cc3 and dCc3 are governed by the following equations when the neutral axis is above the inner steel tube cross section: Cc3 ¼ fc0 ðbo  2to Þðdn  rint Þ Cc4 ¼ 0 dCc3 ¼ 0:5ðdn  rint Þ

5.4 CFST members under combined axial loading and bending Developing the axial loadebending moment ðN MÞ interaction curve based on the plastic stress distribution can help predict composite sections’ resistance under combined axial loading and bending. Figs. 5.10 and 5.11 show the schematic view of the stress distribution and N M interaction curve for the circular and square CFST members, respectively. The level of the axial compression load against the bending moment affects the position of the neutral axis. Therefore, four states can occur depending on the neutral axis position: At state A, the section is under pure compression, and there is no trace of bending. State B relates to the pure bending when no axial load is induced on the section, and the neutral axis locates above the centroid of the section. At state C, the member is under both axial compression and bending simultaneously. At this point, the magnitude of the axial force is equal to the load capacity of the concrete core alone, and the bending capacity is equal to the bending strength in state B. The location of the neutral axis at this state is under the centroid of the section. At state D, the neutral axis locates at the

610

5. Design rules and standards

Point A:

Point C:

Point D:

Point B:

FIGURE 5.10 Stress distribution of CFST members under combined loading conditions.

centroid of the section. Compared with state C, the axial load is half, whereas the maximum bending capacity occurs at this state. The compressive strength and the flexural strength of the member can be obtained using the methods discussed in Sections 5.2 and 5.3, respectively. The magnitude of Mmax can be determined from the stress distribution given in Fig. 5.10. At point D, the neutral axis locates at the centroid

611

5.4 CFST members under combined axial 12

A 10 8

C 6

D 2 0

B 0

5

10

15

20

FIGURE 5.11 Axial loadebending moment ðNeMÞ interaction curve of CFST members.

of the section. Therefore, for the rectangular CFST member, the location of the neutral axis at point D is governed by: dn ¼

ðH  2tÞ 2

(5.112)

Mmax for rectangular CFST members can be achieved using Eq. (5.63) and taking the term fRHS as 1. Therefore:    H  2t 0 Mmax ¼ fy t BðH  tÞ þ 0:5ðH  2tÞ2 þ 0:5ðB  2tÞ fc (5.113) 2 For calculating the magnitude of Mmax of circular CFST members, the term go in Eq. (5.81) must be taken as 0. Therefore: 2 Mmax ¼ 4fy tr2m þ fc0 r2i 3

(5.114)

in which the terms rm and ri are expressed by Eqs. (5.76.a) and (5.76.b), respectively.

5.4.1 Determining the resistance of CFST members under combined axial loading and bending based on design rules 5.4.1.1 BS5400 The schematic view of the NeM interaction curve recommended by Ref. [2] is illustrated in Fig. 5.12. It can be observed from the figure that Ref. [2] ignores the state D and only considers three states A, C, and D, for the NeM interaction curve.

612

5. Design rules and standards

A

C

6

B

FIGURE 5.12 Axial loadebending moment ðNeMÞ interaction curve based on BS5400.

The design code [2] considers four cases of combined loading conditions, as follows: • Uniaxial bending about the minor axis " N
Ny ¼ Nu

  #  My My 2  4K3 K1y  K1y  K2y  4K3 Muy Muy 

(5.115)

in which the subscript y denotes the minor axis. Based on Ref. [2], it is required that: My
Muy My  N  0:03b, where b represents the smallest lateral dimension of the member. Nu is the column’s axial compressive capacity and is calculated based on Ref. [2], discussed in Section 5.2.1. Muy denotes the design flexural strength of the member about the minor axis. Muy for the circular cross section can be achieved by Eq. (5.87), whereas Muy for the rectangular cross section is expressed as follows:    ðb  dc Þ Muy ¼ 0:95fy As þ h þ 2tf tf tf þ dc 2

(5.116)

in which h is the depth of the infilled concrete, b denotes the clear width of the rectangular steel tube, r is governed by Eq. (5.86), and dc is expressed as follows: dc ¼

 As  2 h þ 2tf tf hr þ 4tf

(5.117)

613

5.4 CFST members under combined axial

K1y is defined using Eq. (5.48), in which the radius of gyration and effective length should be considered based on the minor axis. K2y is calculated as:

 " # 115  30 2by  1 ð1:8  ac Þ  100ly  8 2 

for circular section 0:9ac þ 0:2 > > > > 50 2:1  by < K2y ¼

 > " # > > 90  25 2b  1 ð1:8  ac Þ  C4 ly > y   : 2 

0:9ac þ 0:2 for square section 30 2:5  by

(5.118)

0
K2y
0:75 K3 is expressed as:

   8 2 0:5b þ 0:4 a  0:5 þ 0:15 ly > y c a > < 0:04  c þ 3 15 K3y ¼ 1 þ ly > > : 0 for square section

for circular section

(5.119)

ac is governed by Eqs. (5.13) and (5.18) for the rectangular and circular CFST members, respectively, ly is determined based on Eq. (5.41) and the effective length of the member corresponding to the minor axis. by depends on the loading condition. If the member is under end bending 2 moments only, by is taken as M M1
1:0; ðjM1 j  jM2 jÞ, while in the case of the existence of transverse loads, by can be taken as 1.0. The constant C4 depends on the buckling curve recommended by Ref. [2] and is equal to 100, 120, and 140 for the buckling curves of A, B, and C, respectively. • Uniaxial bending about the major axis restrained from failure about the minor axis     Mx Mx 2  4K3 N
Nx ¼ Nu K1x  ðK1x  K2x  4K3 Þ Mux Mux

(5.120)

in which the subscript x denotes the major axis. Based on Ref. [2], it is required that: Mx
Mux Mx  N  0:03b, where b represents the smallest lateral dimension of the member. Nu is the column’s axial compressive capacity and is calculated based on Ref. [2], discussed in Section 5.2.1. Mux denotes the design flexural strength of the member about the major axis. Mux for the rectangular and

614

5. Design rules and standards

circular cross sections can be achieved by Eqs. (5.84) and (5.87), respectively. K1x is defined using Eq. (5.48). K2x is calculated as:

K2x ¼

  8  115  30ð2bx  1Þð1:8  ac Þ  100lx 2 > > 0:9a þ 0:2 for circular section > c < 50ð2:1  b Þ x

  >   90  25ð2bx  1Þð1:8  ac Þ  C4 lx > > : 0:9a2c þ 0:2 30ð2:5  bx Þ

for square section (5.121)

0
K2x
0:75 K3 is expressed as:     8 ð0:5bx þ 0:4Þ a2c  0:5 þ 0:15 lx a >

: 0 for square section (5.122) ac is governed by Eqs. (5.13) and (5.18) for the rectangular and circular CFST members, respectively, lx is determined based on Eq. (5.41). by depends on the loading condition. If the member is under end bending 2 moments only, by is taken as M M1
1:0; ðjM1 j  jM2 jÞ, while in the case of existence transverse loads, by can be taken as 1.0. The constant C4 depends on the buckling curve recommended by Ref. [2] and is equal to 100, 120, and 140 for the buckling curves of A, B, and C, respectively. • Uniaxial bending about the major axis unrestrained from failure about the minor axis If the column is prone to fail about the minor axis, the condition N
Nx , presented in Eq. (5.120), should be satisfied. Besides: N
Nxy ¼

1 1 1 1 þ  Nx Ny Nax

(5.123)

where Nx and Ny are defined by Eqs. (5.120) and (5.115), respectively, by taking My ¼ N  0:03b, and Nax is governed by: Nax ¼ K1x Nu

(5.124)

in which K1x is defined using Eq. (5.48) and Nu is calculated based on Ref. [2], discussed in Section 5.2.1. • Biaxial bending

5.4 CFST members under combined axial

615

If the column is under biaxial bending condition, i.e., N, Mx , and My , the following requirements should be satisfied: Mx
Mux

(5.125.a)

My
Muy

(5.125.b)

Mx  N  0:03b

(5.125.c)

My  N  0:03b

(5.125.d)

N
Nxy ¼

1 1 1 1 þ  Nx Ny Nax

(5.125.e)

5.4.1.2 DBJ13-51 The schematic view of the NeM interaction curve recommended by Ref. [3] is shown in Fig. 5.13. The interaction curve for calculating the section capacity of the CFST member subjected to combined axial load and uniaxial bending around the minor or major axis is defined as follows: N abm M þ
1:0 Nu Mu

if

bN 2 cM bm M  þ
1:0 Nu Mu Nu2

N  2h0 Nu if

N < 2h0 Nu

(5.126.a) (5.126.b)

where a ¼ 1  2h0

(5.127.a)

FIGURE 5.13 Axial loadebending moment ðNeMÞ interaction curve based on DBJ13-51.

616

5. Design rules and standards

1  xo h2o

(5.127.b)

2ðxo  1Þ h0

(5.127.c)

b¼ c¼

xo and h0 for circular CFST members are expressed as: ( h0 ¼

xo ¼ 1 þ 0:18x1:15 0:5  0:2445x for x
0:4 0:1 þ 0:14x0:84 for x > 0:4

(5.128.a) (5.128.b)

xo and h0 for rectangular CFST members are expressed as: ( h0 ¼

xo ¼ 1 þ 0:14x1:3 0:5  0:3175x 0:1 þ 0:13x

for x
0:4

0:81

for x > 0:4

(5.129.a) (5.129.b)

in which Nu is the column’s ultimate axial strength and is calculated based on Ref. [3], discussed in Section 5.2.1. Mu is the ultimate flexural strength of the column and is calculated based on Ref. [3], discussed in Section 5.3.2. The term bm denotes the equivalent moment coefficient. If the members are under the end bending moments only, bm can be expressed as follows: M2 M1

bm ¼ 0:65 þ 0:35

(5.130)

in which M1 and M2 are end moments ðjM1 j  jM2 jÞ. In the case of the existence of transverse loads, bm can be taken as 1.0. The interaction curve for defining the capacity of the member is expressed as follows: N ab M þ m
1:0 4Nu dm Mu

if

N  24h0 Nu

bN2 cM bm M N  þ
1:0 if < 243 h0 2 N u dm M u Nu Nu

(5.131.a) (5.131.b)

where a ¼ 1  242 h0 b¼

1  xo 43 h2o

(5.132.a) (5.132.b)

5.4 CFST members under combined axial

c¼

2ðxo  1Þ h0

617 (5.132.c)

in which Nu is the axial compressive capacity of the column and is calculated based on Ref. [3], discussed in Section 5.2.1. Mu is the column’s ultimate flexural strength and is calculated based on Ref. [3], discussed in Section 5.3.2. The term f is the slenderness reduction factor and can be achieved by buckling curves, defined by Ref. [3]. bm is introduced above. xo and h0 are governed by Eqs. (5.128) and (5.129) for circular and rectangular members, respectively. The term dm is defined to take into account the second-order effect and is governed by: 8 N > > for circular section > < 1  0:4 N E dm ¼ (5.133) > N > > for square section : 1  0:25 NE in which l is given by Eq. (5.43), and NE is the elastic buckling load and is expressed as: p2 Esc Asc l2 Asc ¼ As þ Ac

NE ¼

(5.134) (5.135)

Esc is given by: Esc ¼

fscp εscp

(5.136)

fscp and εscp depend on the cross section of the member. For circular members, fscp and εscp are given by: " ! # fy (5.137.a) fscp ¼ 0:192 þ 0:488 fscy 235 fscy ¼ ð1:14 þ 1:02xÞfc0

(5.137.b)

εscp ¼ 3:25  106 fy

(5.137.c)

For rectangular members: ! " # fy 20 fscp ¼ 0:263 þ 0:365 0 þ 0:104 fscy 235 fc

(5.138.a)

fscy ¼ ð1:18 þ 0:85xÞfc0

(5.138.b)

εscp ¼ 3:01  106 fy

(5.138.c)

618

5. Design rules and standards

If the column is subjected to uniaxial bending about the major axis while it is prone to fail about the minor axis, the following requirement must be controlled: N b M þ m
1:0 4Nu 1:4Mu

(5.139)

For members subjected to biaxial bending, the following condition must be satisfied:     My 1:8 Mx 1:8 þ
1:0 (5.140) Mux Muy in which Mx and My are the maximum bending moment about the major and minor axes, respectively; Mux and Muy denote the design flexural strength of the composite member about the major and minor axes, respectively, and are calculated based on Ref. [3], discussed in Section 5.3.2.3. It should be noted that for defining Muy , the term Wsc is given by: Wsc ¼

HB6 6

(5.141)

5.4.1.3 EC4 The interaction curve in Ref. [2] is simplified by a polygon ACBD, as shown in Fig. 5.14. At Point A, the member is under pure compression,

FIGURE 5.14 Axial loadebending moment ðNeMÞ interaction curve based on EC4.

5.4 CFST members under combined axial

619

and the corresponding axial load is equal to the plastic resistance of the member Npl; Rd. Npl; Rd for the rectangular and circular cross sections are given by Eqs. (5.24) and (5.30), respectively. At Point B, the member is under pure bending, and M  B is equal to the plastic flexural strength of the section

MB ¼ Mpl; Rd . Mpl; Rd can be

obtained from Section 5.3.1.

At Point C, the bending moment is equal to that of Point B MC ¼ MB ¼ Mpl; Rd . The magnitude of the axial load at Point C is given

by: NC ¼ Npm;Rd ¼ Ac fc0

(5.142)

of the axial load is half of that at Point C   At Point D, the magnitude ND ¼ 0:5NC ¼ 0:5Ac fc0 . The magnitude of the bending moment is equal to the maximum flexural strength of the section MMax . MMax can be derived from Eqs. (5.113) and (5.114) for rectangular and circular members, respectively, or by the following simplified expression: MD ¼ MMax; Rd ¼ Wpa fy þ 0:5Wpc fc0

(5.143)

where Wpa and Wpc can be derived from Section 5.3, depending on the cross-sectional shape of the member. 5.4.1.4 AISC360-16 Ref. [5] recommends two methods for predicting the strength of CFST members under combined loading conditions. The first method

In the first method, the interaction curve is developed using a bilinear model, as shown in Fig. 5.15. This method is applicable for compact, noncompact, and slender sections. However, modification is required for the bilinear interaction curve of noncompact and slender sections. Generally, this method leads to conservative results, and the intensity of conservatism depends on the contribution of the concrete core in the member’s load-bearing capacity. In general, increasing the concrete compressive strength fc0 increases the degree of conservatism. The bilinear interaction curve of compact sections is defined by:   Mry Nr 8 Mrx Pr
1:0 if þ þ  0:2 (5.144.a) fc Nn 9 fb Mnx fb Mny fc Pn   Mry Nr Mrx Nr
1:0 if þ þ < 0:2 (5.144.b) 2fc Nn fb Mnx fb Mny fc Nn

620

5. Design rules and standards

FIGURE 5.15 Axial loadebending moment ðNeMÞ interaction curve based on AISC36016-first method.

in which Nr and Mr and is the required axial strength and flexural strength, respectively, 4c Nn is the design axial strength, 4b Mn is the design flexural capacity, 4c ¼ 0:75 and 4b ¼ 0:9 are resistance factors for compression and bending, respectively, and subscripts x and y denote the major axis and minor axis, respectively. For the noncompact and slender sections, the modified bilinear interaction curve is expressed by:   1  cp Nr Mr Nr
1:0 if þ  cp (5.145.a) 4c N n cm 4b M n 4c Nn   Nr 1  cp Mr Nr þ
1:0 if < cp (5.145.b) fc Nn cm f b Mn fc Nn in which 4c ¼ 0:75 and 4b ¼ 0:9 are resistance factors for compression and bending, respectively, and the constants cp and cm are defined from Table 5.8. csr in Table 5.8 is governed by: csr ¼

As fy Ac fc0

(5.146)

The second method

This method is based on the plastic stress distribution scheme. The schematic view of the interaction curve developed based on the second

621

5.4 CFST members under combined axial

TABLE 5.8 Coefficients cp and cm . cm Cross section

cp

csr ‡0:5

csr < 1:065x þ 3:449 for 1:5
< 2 b fr ¼ > : 1:019x þ 0:304 for a  2 b Af 0
x ¼ Ac gs yf 0
9:0 c c 

Nu;Uenaka ¼ 1:46 As fy þAc fc0  EC4 ¼ A h f þ A f 0 1 þh t Nu;Jamaluddin ¼ Npl;Rd s s yd c cd c De

fy fck0

 where parameters hs

5.5 Strength of composite members

(5.192)

for 17
ðaþbÞ t
29

ad hc should be computed according to the equivalent inscribed circular section and based on equations introduced in Section 5.2.1.5. De is the equivalent diameter and can be obtained using equations recommended by Ruiz-Teran and Gardner [37] as:

 De ¼ 2a ba

647

TABLE 5.14

Design strengths of round-ended rectangular CFST short columns.

Steel material

Vertical axis status

Loading

Carbon steel jacket

Short straight column

Fully loaded columns

Number of design equation

Cross section References

Round-ended rectangular

(5.195)

[38]

Nu; Diang ¼ fc0 Ac ð1 þKxÞ Where: x ¼

f y As fc0 Ac

K ¼ 0:8 þ 0:9 D B

(5.196)

[39]

 Nu; Hassanein ¼ gs fy As; RE þ gc fc0 þ4:1frp Ac;RE þ fy As; RP; eff þ gc fc0 Ac;RP Where: As;RE ¼ round ends steel tube cross  sectional area Ac;RE ¼ round ends concrete core cross  sectional area As; RP; eff ¼ effective cross  sectional area of steel tube in the rectangular part based on EN 1993-1-4 [40] Ac;RP ¼ cross  sectional area of the concrete in the middle rectangular parts 8 2t D > > < 0:7ðwe  ws Þ D  2tfy for t
47   fr ¼  > D > : 0:006241  0:0000357 D= t fy for 47 < t
150 ! ! !2 f0 f0 f0 we ¼ 0:2312 þ 0:3582v0e  0:1524 c þ 4:843v0e c  9:169 c fy fy fy  3  2   v0e ¼ 0:881  106 Dt  2:58  104 Dt þ 1:953  102 Dt þ 0:4011 0:85
gc ¼ 1:85D0:135
1:0 c  0:1
1:1 0:9
gs ¼ 1:458 Dt

(5.197)

[41]

Nu; Ahmed ¼ Ase fy þ Ac fcc0 fcc0 ¼ gc fc0 þ 4:1fr     8 B D B > > 10:288  0:928x  3:115  0:045 for 1:2
< 2 < D t D fr ¼ > > : 0:9 for 2
B
4:2 D As f y x ¼ Ac gc fc0

(5.198)

[42]

Nu;Wang ¼ Nu;c þ Nu;r where Nu;c and Nu;r are the ultimate axial strengths of the semi-circular and rectangular sections, respectively. Nu;c is governed by the model recommended by EC4 for circular cross section and Nu;r ¼ Asr fy þ 0:85Acr fc0 in which Asr and Acr are the cross-sectional areas of the steel tube and the concrete core in rectangular part, respectively.

(5.199)

[42]

Nu; Wang ¼ Asc fscy where    8 B B > > 1:14 þ 0:27x þ 0:75 x fck for 1

1:44 < D D Asc ¼ As þ Ac fscy ¼ > > B : ð1:18 þ 0:85xÞf
4 ck for 1:44 < D

TABLE 5.15 Steel material Austenitic

Design strengths of circular CFST short columns with stainless steel tube. Vertical axis status

Loading

Short straight column

Fully loaded columns

Number of design equation

Cross section References

Circular

(5.200)

[43]

Based on GB  50936-2014 [44]:  Nup; Dai ¼ 1:09 þ2:02x þ0:0059x2 fck Asc Based on EC4 [40]: EC4 Nup; Dai ¼ Npl;Rd Where: EC4 is defined by Eq. (5.30) Npl;Rd

hc is by Eq. (5.31.a) ( 4:924  26:155l for l
0:132 hs ¼ 1:472 for l > 0:132 l is defined by Eq. (5.26) Duplex

(5.201)

[43]

Based on GB 50936-2014p[44]: ffiffiffi Nup; Dai ¼ ð0:23 þ 2:56 x þ 0:56xÞfck Asc Based on EC4 [40]: EC4 Nup; Dai ¼ Npl;Rd Where: EC4 is defined by Eq. (5.30) Npl;Rd

hc is defined by Eq. (5.31.a) hs ¼ 2:358  6:409l l is defined by Eq. (5.26) Stainless steel

(5.202)

[45]

Nu; Patel ¼ gs As s0:2 þ Ac fcc0 where fcc0 ¼ gc fc0 þ 4:1fr 8 2t D > > < 0:7ðwe  ws Þ D  2ts0:2 for t
47   fr ¼  > D > : 0:006241  0:0000357 D= t s0:2 for 47 < t
150

 0   0 2  0  f fc f  9:169 c we ¼ 0:2312 þ 0:3582v0e  0:1524 c þ 4:843v0e s0:2 s0:2 s0:2  3  2   v0e ¼ 0:881  106 Dt  2:58  104 Dt þ 1:953  102 Dt þ 0:4011 8  0:1 D > >3 D for < 80 > < t t gs ¼ > D0:1 > D > :2 for  80 t t

Partially loaded

¼ Ac fc0 þ 2As s0:2

(5.203)

[46]

Nu;

(5.204)

[47]

 Nu; Zhao ¼ fc0 Ab 1 þnc Dt

Ding

s0:2 fc0

 kn where

nc is determined by Eq. (5.31.a)  0:689 Ac kn ¼ 1:05x0:275 þ 0:291 Ab Ab ¼ cross  sectional area of the load  bearing plate

TABLE 5.16 Steel material Stainless steel

Design strengths of square CFST short columns with stainless steel tube. Vertical axis status

Loading

Short straight column

Fully loaded columns

Number of design equation

Cross section References

Circular

(5.205)

[48]

Nu; Tao ¼ kAs s0:2;a þ Ac fc0 Where: k¼ Coefficient representing the effect of nonlinearity and strain hardening of stainless steel ¼ 0:946 þ 0:021x s0:2;a ¼ The average nominal yield strength of the steel tube ¼ Cs0:2;c þ ð1 CÞs0:2;V in which 1:637s

(5.206)

[49,50]

0:2;V s0:2;c ¼ 0:2% proof stress of the corner material ¼ r 0:126 s0:2;V ¼ i t 0:2% proof stress of the virgin material ri ¼ Internal corner radius   Nu; Ellobody ¼ C Ae Fn þ0:85Ac fc0 Where: 8 H > > < 60 < 1:0 for 16
t For unstiffened columns: C ¼  > > : 1:14 for 60
H
158 t 8 H > > < 45:5 < 1:0 for 26:5
t For stiffened columns: C ¼  > > : 1:10 for 45:5
H
74 t H ¼ maxðH; BÞ p2 Et Fn ¼  2 KL r E o fy Et ¼ !n1 Fn fy þ 0:002nEo fy

Ae ¼ effective cross  sectional area of steel tube based on EN 1993-91-4 [40] Eo ¼ initial elastic modulus n ¼ RambergeOsgoodparamet strain  hardening coefficient ¼ 4:67 Fn ¼ flexural buckling strength Duplex

(5.207)

[51]

Nu; Lam ¼ As s0:1 þ 1:1Ac fc0 where s0:1 ¼The steel strength at 1.0% strain

654

5. Design rules and standards

5.5.1.1 Compressive stiffness of CFST short columns In general, the behavior of the stressestrain curve for the steel material before achieving the elastic limit is linear. At the same time, the expansion of microcracks in the concrete reduces the secant modulus. The modulus of elasticity for concrete ðEc Þ can be taken as the secant modulus corresponding to 0:4fc0 . Since the concrete core has the most prominent role in the load-bearing capacity of composite members, cracking of the concrete with increasing compressive load leads to a decrease in the secant compressive stiffness of composite columns. As a result, the compressive stiffness of composite members ðEAÞ can be taken as the secant stiffness corresponding to 0.4 of the compressive capacity of the column ð0:4Nu Þ [52]. The compressive stiffness of composite members EA can be predicted using the superposition principle as follows: EA ¼ Es As þ EC AC

(5.208)

Eq. (5.208) can be used for predicting EA when the confinement is not tfy strong. As a criterion, if the 0 ratio for circular CFSTs is less than 0.3 or Dfc tfy the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ratio for rectangular CFSTs is less than 0.35, it can be B2 þ D2 fc0 expected that Eq. (5.208) leads to an acceptable prediction of EA. Numerical analysis showed that the steel tube usually has an elastic behavior at the axial load 0:4Nu , whereas the longitudinal stress over the concrete cross section exceeds 0:4fc0 because of the confinement effect provided by the steel tube. In the case of imposing strong confinement, a remarkable decrease in concrete stiffness can be expected. Therefore, a modification factor kc
1:0 is required for the concrete stiffness and EA can be predicted using the following equation: EA ¼ Es As þ kc EC AC

(5.209)

where

kc ¼

8  0:004 D > > > þ > > < t

tfy Dfc0

!1:5 

> 0:003 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > B2 þ D2 > : þ t

 0:004  D 56:8  56:3 t

tfy pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B2 þ D2 fc0

for circular columns

!2:3 " 103:12  102:26

0:003 # pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B2 þ D 2 for rectangular t columns

ð5:210Þ 5.5.1.2 Ultimate axial strain of CFST short columns The axial strain εu corresponding to the ultimate axial strength of composite columns is to some extent indicative of the member’s ductility, which can be a useful parameter for the designers. As a general trend, εu is

5.5 Strength of composite members

655

increased by increasing the steel yield strength fy . By contrast, increasing the concrete compressive strength fc0 or reducing the tube wall thickness t leads to a reduction of εu . This is due to the influences of the material and geometric properties of the composite column on the confinement effect. The strain εu increases with increasing the confinement effect. The following expressions can be employed for predicting the value of εu [21].   1:5  8 D 1:4 0 1:2 > >
10000mε for circular columns > 3000  10:4fy f c 0:73  3785:8 t > < εu ¼ !2 > h i > tfy > > 2300 þ 31:2f 0 0:7 þ 2:32  104  3:88  106 f 0 1:8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : for rectangular columns c c B2 þ D2 fc0

ð5:211Þ

5.5.2 CFDST members under axial compression The available design models for predicting the ultimate axial strength of CFDST columns are discussed in this section. Various configurations of steel tubes, including CHS-CHS, CHS-SHS, SHS-CHS, SHS-SHS, RHSRHS, elliptical, round-ended rectangular, octagonal, and dodecagonal, are considered through Tables 5.17e5.27. The cross section of CFDST columns under axial partial loading over ring bearing plates is shown in Fig. 5.20. Definition of symbols of the CFDST columns with SHS inner and CHS outer tubes, and SHS inner and SHS outer tubes are given in Figs. 5.21 and 5.22, respectively. The relationship between the stability ratio 4o and the slenderness ratio l based on the Chinese code GBJ17-88 is presented in Fig. 5.23.

5.5.3 CFST members under bending In addition to the design equations recommended by international design codes for predicting the bending moment capacities of CFST beams, researchers have also tried to develop simplified models for calculating the ultimate pure bending strengths of CFST/CFDST beams, which are discussed in this section. 5.5.3.1 Simplified moment capacity model of Elchalakani et al. The model developed by Elchalakani et al. [10] is based on the experimental test results of eight compact circular CFST beams with D= t
40 and four slender circular CFST beams with 74 < D=t
110 under pure bending. Elchalakani et al. recommended Eq. (5.81) for calculating the ultimate pure bending strengths Mu of circular CFST beams.

TABLE 5.17 Design strengths of CFDST short columns with CHS outer and CHS inner tubes.

Steel material Carbon steel jacket

Vertical axis status

Straight short columns

Loading

Fully loaded columns

Number of design equation

Cross section

References

CHSeCHS

(5.212)

[53]

Nu; Uenaka ¼

(5.213)

[54]

(5.214)

[55]

Nu; Tao ¼ Nosc;u þ Ni;u Where: Nosc;u ¼ fscy Asco with Asco ¼ Aso þ Asc fscy ¼ C1 c2 s0:2 þ C2 ð1:14 þ1:02zÞfck ( fscy and fck [N/mm2]) Aso fsyo a 1 þ an Aso Aso ,a ¼ , an ¼ ,z¼ and fck ¼ 0:67fcu ¼ 0:84fc0 C1 ¼ , C2 ¼ 1þa Asc Ac;no min al Ac;no min al fck 1þa Ni;u ¼ Asi fsyi    8 > 1 þ 0:3 z U fsyo Aso þ fcc0 Asc þ fsyi Asi : D to
150 > < 1þz Nu; Hassanein ¼ > . > : fsyo Aso þ 0:85fc0 Asc þ fsyi Asi : D to > 150

  2:86 2:59 Dd fsyo Aso þ fsyi Asi þ Asc fc0 with 0:2 < Dd < 0:7

Where:

. z ¼ afsy fck , a ¼ Aso =Asc and  2  p D  2t U ¼ Asc o 4 Partially loaded

(5.215)

[56]

Nu; Yang ¼ kbc NTao Where: NTao is the strength fully loaded composite sections according to Tao et al. [54] kbc represents the    þ 0:52 0:9 þ 1:28c  2:16c2 0:12t0:4  a bearing capacity factor. kbc ¼ 0:5 0:28b þ 0:44 Asc b¼ Ap Ap is the partial bearing area of the compressive load and ta is the top endplate thickness (not the ring bearing plate).

Tapered columns Stainless steel jacket

Straight short columns

Fully loaded columns

(5.216)

[57]

The formula for the prediction of the sectional strength of straight CFDST section proposed by Tao et al. [54] has been adopted to design the tapered CFDST columns. That formula is to be used by considering the smallest section of the column in the calculations.

(5.217)

[58]

(5.218)

[59]

The same method of design of circular CFDST column with inner CHS and with outer carbon steels [54] was suggested to predict the strength for circular CFDST column with outer stainless steels.

 0 Asc Nu; Hassanein2 ¼ gse fsye Ase þ gsi fsyi Asi þ gsc fc0 þ4:1frp:se 0 Where: fcc0 ¼ gc fc0 þ 4:1frp;se 8 2t D > > < 0:7ðwe  ws Þ D  2tfy for t
47 0   frp;se ¼  > D > : 0:006241  0:0000357 D= t fy for 47 < t
150 The Poisson ratios (ne and ns ) are given as follows: ns ¼ 0:5 ! ! !2 fc0 fc0 fc0 0 0 þ 4:843ve  9:169 we ¼ 0:2312 þ 0:3582ve  0:1524 fy fy fy  3  2   D D D v0e ¼ 0:881  106  2:58  104 þ 1:953  102 þ 0:4011 t t t The factors gse , gsi and gc are given as follows:  0:1  0:1 D d gse ¼ 1:458 and gsi ¼ 1:458 ð0:9
gs
1:1Þ te ti 0:135 with Dc ¼ D  2te and ð0:85
gc
1:0Þ gc ¼ 1:85Dc

Design strengths of CFDST short columns with CHS outer and SHS inner tubes.

Steel material

Vertical axis status

Loading

Carbon steel jacket

Straight short columns

Fully loaded columns

Cross section References

CHS-SHS

(5.219)

[60]

Nu; Elchalakani ¼ Pso þ Pc þ Psi Where:   Pso ¼ p4 fyo D2e D2ei   Psi ¼ Pcorner þ Pflat ¼ fyi;corner p r2exti r2inti þ 4fyi;flat ðbi 2rexti Þti

 Pc ¼ 0:85fc0 Ac ¼ 0:85fc0 p4 D2i þr2ext ð4 pÞ B2 8 > le
82

: le > 82 :D lso 8 > < Dei : le
82 Dei ¼ ley > : le > 82 : Dei lso   fyo Dei ¼ D  2to and le ¼ D to 250 as defined by AS4100 [61]. 8 > le
40 < bo  2rexto : beo ¼ and ley > : le > 40 : ðbo  2rexto Þ le qffiffiffiffiffiffi  fyo exto le ¼ bo 2r to 250 as defined by AS4100 [61].

5. Design rules and standards

Number of design equation

658

TABLE 5.18

TABLE 5.19 Steel material Carbon steel jacket

Vertical axis

Loading

Straight short columns

Fully loaded columns

Cross section

No. of equation

References

SHS-CHS

(5.220)

[62]

Nu; Han ¼ Nosc;u þ Ni;u Where: Nosc;u ¼ fscy Asco with Asco ¼ Aso þ Asc !  "  # fsyo f þ0:7646 z þ 0:0727 ck þ0:0216 z2 fck fscy ¼ 1:212 þ 0:138 235 20

(5.221)

[63]

(5.222)

[58]

(fscy and fck [N/mm2]) Aso fsyo and fck ¼ 0:67fcu ¼ 0:84fc0 z ¼ Ac;no min al fck Ni;u ¼ Asi fsyi   to Nu; Ayough ¼ 0:76fyo Aso þ Ac fc0 þ1:7612 fyo þ fyi Asi Bo It was proposed to use the same method of design of rectangular CFDST column (inner RHS and outer RHS) [64].

5.5 Strength of composite members

Stainless steel jacket

Design strengths of CFDST short columns with SHS outer and CHS inner tubes.

659

660

TABLE 5.20

Design strengths of CFDST short columns with SHS outer and SHS inner tubes. Vertical axis

Loading

Carbon steel jacket

Straight short columns

Cross section

No. of equation

References

SHS-SHS

Fully loaded columns

(5.223)

[65]

Nu; Zhao ¼ Pso þ Pc þ Psi Where:   Pso ¼ Pcorner þ Pflat ¼ fyo;corner p r2exto r2into þ 4fyo;flat beo to 2  Psi ¼ Pcorner þ Pflat ¼ fyi;corner p rexti r2inti þ 4fyi;flat ðbi 2rexti Þti Pc ¼ 0:85fc0 Ac 8 > le
40 < bo  2rexto : Where: beo ¼ ley > : le
40 : ðbo  2rexto Þ le rffiffiffiffiffiffiffiffi  fyo bo  2rexto le ¼ as defined by AS4100 [61]. to 250

Partially loaded columns

(5.224)

[56]

Nu; Yang ¼ kbc NTao&Han Where: PTao&Han is the strength of the corresponding fully loaded composite ð0:94þ1:12c2c2 Þ$ð0:12t0:4 a þ0:2Þ sections according Tao and Han [64]. kbc ¼ ð0:04b0:5 þ0:84Þ

5. Design rules and standards

Steel material

TABLE 5.21

Design strengths of CFDST short columns with RHS outer and RHS inner tubes.

Vertical axis

Loading

Carbon steel jacket

Straight short columns

Fully loaded columns

References

SHS-SHS

(5.225)

[64]

NTao&Han ¼ Nosc;u þ Ni;u Where: Nosc;u ¼ fscy Asco with Asco ¼ Aso þ Asc fscy ¼ C1 c2 fsyo þ C2 ð1:18 þ0:85zÞfck (fscy and a , C ¼ 1þan , a ¼ Aso , fck [N/mm2]) C1 ¼ 1þa 2 1þa Asc A f

an ¼ Ac;noAsomin al , z ¼ Ac;no sominsyoal fck and

fck ¼ 0:67fcu ¼ 0:84fc0 Ni;u ¼ Asi fsyi

5.5 Strength of composite members

Steel material

Cross section

No. of equation

661

662

TABLE 5.22

Design strengths of CFDST short columns with elliptical or round-ended rectangular steel tubes.

Vertical axis

Loading

Stainless steel jacket

Straight short columns

Fully loaded columns

References

Elliptical or round-ended rectangular

(5.226)

[58]

The same method of design of circular CFDST column (inner CHS and outer CHS) with outer carbon steels was proposed to predict the strengths of the round-ended rectangular and elliptical sections. But the hollow ratio of the CFDST section, for the round-ended rectangular and elliptical sections, is to be taken as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c ¼ ðB2to bd ÞðD2to Þ

5. Design rules and standards

Steel material

Cross section

No. of equation

TABLE 5.23 Design strengths of octagonal CFDST short columns. Steel material

Vertical axis

Loading

Carbon steel jacket

Straight short columns

Fully loaded columns

Cross section

No. of equation

References

Octagonal

(5.227)

[66]

¼ Asc fscy þ Asi fyi  Where: fscy ¼ 1:13 þ0:9x þ0:45 CCso1 so2 Nu;

Yang

an 2 ar x

 fck

x¼

5.5 Strength of composite members

Asc ¼ Aso þ Ac Cso1 ¼ 4b Cso2 ¼ 4a Aso fyo Ac fck

ar ¼ real steel ratio a ¼ vertical length of octagon side b ¼ horizontal length of hypotenuse side of octagon (5.228)

[67]

Nu; Yuan ¼ Asc fscy þ Asi fyi Where:

 pd2 pD2 Asc ¼ Aso þ Ac ¼ a2 o ðao 1Þ þ 4c2i 1 ceq o eq   fscy ¼ 1:1329 þ0:7656x þ0:1631x2 fcu x¼

Aso fuo Ac fcu

¼

4ðao 1ÞD2o c2eq fuo

ð1c2eq Þa2o d2i fcu

Do to

ao ¼ ceq ¼ hollow ratio of equivalent circular section fuo ¼ ultimat strength of outer steel tube (5.229)

[68]

663

Nu; Ge ¼ Asc fscy þ Asi fyi Where: Asc ¼ Aso þ Ac so fscy ¼ g1 fcp þ fzgo A Ace fcp ¼ longitudinal compressive strength of core concrete ¼ Continued

Steel material

Vertical axis

Loading

No. of equation

664

TABLE 5.23 Design strengths of octagonal CFDST short columns.dcont’d Cross section References

Octagonal qffiffiffiffiffiffiffi p g 2p g fc0 1 þ1:5 1f 0 2 þ f10 2 c

!

c

p1 ¼ uequivalent circular outer tube0 s plastic bearing   1 þ bo ro þ to capacity under circumferential pressure ¼ 2fyo ln 2 þ bo ro

p1 g2 r2o ðro þ to Þ2  r2o ro ¼ equivalent radius of octagon bo ¼ parameter of intermediate principal stress (5.230)

[69]

The same method of design of polygonal CFST columns recommended by Yu et al. [24] was suggested to predict the strength of octagonal CFDST columns. Additional term for adding the ultimate axial strength of inner steel tube was included in the model as follows:    x Ac fck þAso fyo þ Asi fyi Nu; Alqawzai ¼ 1 þ0:5kn 1þx Where: Af

x ¼ Acsfcky kn ¼ polygon confinement effectiveness coefficien ¼ n ¼ side number ¼ 8

n2  4 n2 þ 20

5. Design rules and standards

fzgo ¼ equivalent cylindrical steel tube longitudinal compressive strength ¼

TABLE 5.24 Design strengths of dodecagonal CFDST short columns. Steel material

Vertical axis

Loading

Carbon steel jacket

Straight short columns

Fully loaded columns

Cross section

No. of equation

References

Dodecagonal

(5.231)

[70]

Nu; Wang ¼ 4sc

    x 1 þ0:3kn ð1 jÞ 1þx gc Ac fck þgs Aso fyo þgs Asi fyi


1:0 0:85
gc ¼ 1:85D0:135 c sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  

2 l2sc þ kc lsc þ 1  4l2sc 4sc ¼ stability factor ¼ 12 l2sc þkc lsc þ1  2lsc

kc ¼ equivalent initial imperfection coefficient ¼ 0:25  0:09Kn ð1 jÞ n2  4 kn ¼ polygon confinement effectiveness coefficient ¼ 2 rffiffiffiffiffiffiffiffiffiffiffin þ 20 N0 l lsc ¼ non  dimensional slenderness ratio ¼ p Esc Isc l ¼ rLsc Esc Isc ¼ composite bending rigidity ¼ Ec Ac þ Es As qffiffiffiffiffi rsc ¼ gyration radius ¼ AIscsc

5.5 Strength of composite members

Where: c j ¼ AcAþA h Ah ¼ cross  section area of the hollow part  0:1
1:1 0:9
gs ¼ 1:458 Dt

Asc ¼ As þ Ac

665

TABLE 5.25 Design strengths of preloaded CFDST short columns with CHS inner and CHS outer steel tubes. Vertical axis

Carbon steel jacket

Straight short columns

Cross section References

CHSeCHS

Preload on outer steel tube

(5.232)

[71]

Nu; Li ¼ kp NTao Where: NTao is the strength of the corresponding fully loaded composite sections according to Tao et al. [54]. The strength index (kp ) was proposed as: kp ¼ 1  f ðlo Þf ðe =rÞhp
1:0 Where For axially loaded CFDST columns with both CHSs, f ðe =rÞ is 0.9. f ðlo Þ is a function accounting for the influence of slenderness ratio (l), where lo ¼ l=80 and l ¼ 4l=D. l ( : lo
1:0 0:17lo  0:02 is the column length. f ðlo Þ ¼ 0:13l2o þ 0:35lo  0:07 : lo > 1:0 Np Np hp ¼ ¼ Nus fo fsyo Aso where Np is the preload applied on the outer steel hollow section; Nus is the ultimate strength of the outer steel tubular column. fo is the stability ratio (Fig. 5.19 according to GBJ17-88 [72] The validity limits of this proposed strength are hp ¼ 0  0:8; c ¼ 0:25  0:75; an ¼ 0:04  0:2; l ¼ 10  80; fck ¼ 20  60 MPa and fsyo ¼ 200  420 MPa.

Preload on both steel tubes

(5.233)

[71]

The design strength is typical to the design model of preload on the outer steel tube, with a modification in kp , as follows: kp ¼ 1  f ðlo Þhp
1:0 f ðlo Þ ¼ 0:40lo  0:01 lo ¼ l=100 4l l ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D2 þ ðd  2tsi Þ2 Np Np hp ¼ ¼ Nus fo fsyo Aso þ fo fsyi Asi The validity limits of this proposed strength are hp ¼ 0  0:8; c ¼ 0:25  0:75; an ¼ 0:04  0:2; l ¼ 10  60; fck ¼ 20  70 MPa; fsyi ¼ 235  420 MPa and fsyo ¼ 235  420 MPa.

Loading

5. Design rules and standards

No. of equation

666

Steel material

TABLE 5.26 Design strengths of preloaded CFDST short columns with SHS inner and CHS outer steel tubes. Steel material

Vertical axis

Loading

Carbon steel jacket

Straight short columns

Preload on outer steel tube

Cross section

No. of equation

References

SHS-CHS

(5.234)

[71]

Nu; Li ¼ kp NHan Where: NHan is the strength of the corresponding fully loaded   composite sections according to Han et al. [62]. The strength index kp was proposed as: kp ¼ 1  f ðlo Þf ðe =rÞhp
1:0 For axially loaded CFDST columns with inner circular and outer square tubes, f ðe =rÞ is 0.9. f ðlo Þ is a function accounting for the influence of slenderness ratio

hp ¼

Np Np ¼ Nus fo fsyo Aso

where Np is the preload applied on the outer steel hollow section; Nus is the ultimate strength of the outer steel tubular column. The stability ratio ðfo Þ can be calculated from Fig. 5.19 according to GBJ17-88 [72]. The validity limits of this proposed strength are hp ¼ 0  0:8; c ¼ 0:25  0:75; an ¼ 0:04  0:2; l ¼ 10  80; fck ¼ 20  60 MPa and fsyo ¼ 200  420 MPa. Preload on both steel tubes

(5.235)

[71]

667

The design strength is typical to the design model of preload on the outer steel tube, with a modification in kp , as follows: kp ¼ 1  f ðlo Þhp
1:0 f ðlo Þ ¼ 0:33l2o þ 0:051lo þ 0:03 lo ¼ l=100 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4l 12D2  3pðd  2tsi Þ2 l ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . 16D4  3pðd  2tsi Þ4 Np Np ¼ hp ¼ Nus fo fsyo Aso þ fo fsyi Asi The validity limits of this proposed strength are hp ¼ 0  0:8; c ¼ 0:25  0:75; an ¼ 0:04  0:2; l ¼ 10  60; fck ¼ 20  70 MPa; fsyi ¼ 235  420 MPa and fsyo ¼ 235  420 MPa.

5.5 Strength of composite members

ðlÞ, where lo ¼ l=80 and l ¼ 4l=D. l is the column length. ( lo
1:0 0:14lo þ 0:02 : f ðlo Þ ¼ 2 0:15lo þ 0:42lo  0:11 : lo > 1:0

668

TABLE 5.27 Design strengths of CFDST long columns with CHS inner and CHS outer steel tubes. Steel material

Vertical axis

Loading

Stainless steel jacket

Straight long columns

Fully loaded columns

Cross section References

CHSeCHS

(5.236)

[73]

NHassanein ¼ cNpl;Rd Where: 8 

< g Ase s0:2 þ g f 0 þ 4:1f 0 Asc þ g A fy ss c c s si rp Npl;Rd ¼ :A s þ A f0 þ A f se 0:2 sc c si y In which l is determined using Eq. (5.26) as: qffiffiffiffiffiffiffiffiffi Npl;Rk l ¼ Ncr

: l
0:5 : l > 0:5

0 Npl; Rk ¼ As fy þ Ac fck

Ncr ¼

p2 ðEIÞEC4 eff ðKe LÞ2

The reduction factor ðcÞ is to be calculated using the European strut curves as: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . 2 c¼1 4 þ 42  l
1:0

 2  4 ¼ 0:5 1 þa l l0 þl lo is taken as 0.2 following the carbon steel predictions; see EC3 [37]. Hassanein and kharoob [12] suggested to use a value 0.49, as given by EC3 [44] for welded hollow sections.

5. Design rules and standards

No. of equation

5.5 Strength of composite members

Ring bearing plate Sandwiched concrete

669

Ring bearing plate Sandwiched concrete

(a) Circular CFDST with CHS outer and CHS inner tubes

(b) Square CFDST with SHS outer and SHS inner tubes

FIGURE 5.20 Axial partial loading over ring bearing plates. (a) Circular CFDST with CHS outer (b) Square CFDST with SHS outer and and CHS inner tubes SHS inner tubes.

FIGURE 5.21

Definition of symbols of the CFDST (SHS inner and CHS outer).

rint o rint i

FIGURE 5.22

Definition of symbols of the CFDST (SHS inner and SHS outer).

670

5. Design rules and standards

1.2 1.0 0.8

Mo

0.6 0.4 0.2 0.0 0.0

40

80

120

160

200

O FIGURE 5.23 fo versus l relationship of pure steel tubular column.

5.5.3.2 Simplified moment capacity model of Han The model of Han [74] is applicable for predicting the ultimate pure bending strengths Mu of circular and rectangular CFST beams. This model is governed by: Mu ¼ gm Wscm fscy

(5.237)

where the term gm denotes the flexural strength index, Wscm represents the section modulus of the CFST beam, and fscy is the nominal yield strength of the section and is similar to the recommendation of Ref. [3] presented by Eq. (5.22). gm is governed by: ( 1:1 þ 0:48 lnðx þ 0:1Þ for circular CFST beams gm ¼ 1:04 þ 0:48 lnðx þ 0:1Þ for rectangular and square CFST beams (5.238) Wscm depends on the cross section of the CFST beam and is determined as follows: 8 pD3 > > > for circular CFST beams > > 32 > > > > > > B3 > > for square CFST beams < 6 Wscm ¼ > > BD2 > > for rectangular CFST beams about the major axis ðx  xÞ > > > 6 > > > > B2 D > > : for rectangular CFST beams about the minor axis ðy  yÞ 6 (5.239)

671

5.5 Strength of composite members

The validity limits of this model are 100 mm
D
2000 mm, 0:04
a
0:2, 200 MPa
fy
500 MPa, and 20 MPa
fck
80 MPa. 5.5.3.3 Simplified moment capacity model of Liang et al. Liang et al. [75,76] recommended equations for predicting the ultimate pure bending strengths Mu of circular and rectangular CFST beams. The model of Liang et al. is expressed as: Mu ¼ lm ac as Ze fy

(5.240)

in which the term Ze represents the elastic section modulus, and parameters lm , ac , and as are introduced to take into account the effects of the Dt ratio, compressive strength of the concrete, and steel yield strength on the bending capacity of the composite beam. lm and ac are introduced in Section 4.3.4.10. However, they are provided in this section for ease of understanding.    2 8 t t > > for circular CFST beam  columns with 10
D=t
120  36 >0:0087 þ 12:3 < D D lm ¼     2 > > > :0:0187 þ 24:2 t  61:1 t for rectangular CFST beam  columns with 10
D=t
120 D D (5.241) 8  0 0:075 0 < 0:774 fc for circular CFST beam  columns with 30
fc
120 MPa ac ¼ (5.242)  0 0:0668 : for rectangular CFST beam  columns with 20
fc0
120 MPa 0:7266 fc

as ¼

8 21:147 4202 > þ 2 > > 0:883 þ f > fy y < > > 262:62 28119 > >  2 : 0:471 þ f fy y

for circular CFST beam  columns with 250
fy
690 MPa for rectangular CFST beam  columns with 250
fy
690 MPa (5.243)

5.5.3.4 Flexural stiffness of CFST beams A typical moment M-curvature f response of composite beams is displayed in Fig. 5.24, in which Ki and Ks are the initial flexural stiffness and the serviceability-level section flexural stiffness, respectively. As shown in Fig. 5.24, the behavior of the beam before reaching 20% of the bending moment capacity is elastic. Hence, Ki can be taken as the secant stiffness corresponding to the moment of 0:2Mu . The initial flexural stiffness of CFST beams is given by Ref. [77]: Ki ¼

0:2Mu 4e

(5.244)

672

5. Design rules and standards

FIGURE 5.24

Typical moment M-curvature f graph of composite beams.

where Mu represents the bending moment capacity of the section and can be defined by models presented in Section 5.5.3. 4e is the elastic curvature of the beam at Point A (see Fig. 5.24) and is expressed as follows: 8 >  > > ½ð4:25bc þ 100:14Þ þ ð15:8bc þ 3:65Þxb0:82 for circular CFST beams > s > ðEs DÞ > > > > < 0:82 4e ½ð10:64bc þ 91:18Þ þ ð8:66bc þ 5:93Þxbs ðE DÞ for rectangular or square CFST beams about the major axisðx  xÞ s > > > > > >  > for rectangular for square CFST beams about the minor axisðy  yÞ > ½ð10:64bc þ 91:18Þ þ ð8:66bc þ 5:93Þxb0:82 s > : ðEs BÞ

ð5:245Þ The initial serviceability-level flexural stiffness of CFST beams is governed by:

0:6Mu (5.246) 40:6 where f0:6 is the curvature of the beam at Point B (see Fig. 5.24) and is expressed as follows: Ki ¼

8 > 0:82 > > > ½ð41:48bc þ 343:43Þ þ ð17:32bc þ 30:39Þxbs ðE DÞ for circular CFST beams > > s > > > < 0:82 4e ½ð38:9bc þ 319:11Þ þ ð12:61bc þ 23:1Þxbs ðE DÞ for rectangular or square CFST beams about the major axisðx  xÞ s > > > > > > 0:82 > for rectangular or square CFST beams about the minor axisðy  yÞ > ½ð38:9bc þ 319:11Þ þ ð12:61bc þ 23:1Þxbs > ðEs BÞ :

ð5:247Þ In the above equations, bc and bs should be taken as respectively.

fcu 30

and

fy 345,

5.5.3.5 Moment M-curvature f response of CFST beams The moment M-curvature f curve of CFST beams, shown in Fig. 5.24, can be determined as follows [77]:

5.5 Strength of composite members

673

(1) Elastic step (Points O-A): At this step, the beam has a linear behavior and 0 < f
fe . The elastic stage of the curve can be determined using the following expression. M¼

0:2Mu 4 4e

(5.248)

(2) Elastic-plastic step (Points A-B): By reaching Point A, the beam enters into the elastic-plastic behavior and fe < f
fo . The elastic-plastic stage of the curve can be given as:  2  8   4 4 > > Mu for circular CFST beams > < 1:8 4  0:9 4 o o M¼     2  > > > : 1:9 4  0:95 4 Mu for square and rectangular CFST beams 4o 4o (5.249) where fo is the curvature of the beam at Point B and is given by: ( 8:484e for circular CFST beams (5.250) 4o ¼ 8:934e for square and rectangular CFST beams 3) Hardening step (Points BeC): The hardening stage starts when f > fo and can be defined using the following expression:   8 4 > Mu for circular CFST beams > < 0:882 þ 0:018 4 o M¼    > > : 0:94 þ 0:0 4 Mu for square and rectangular CFST beams 4o (5.251)

5.5.4 CFDST members under bending In addition to the modified equations of design codes for predicting the flexural strength of CFDST beams, the bending moment capacity of CFDSTs can be predicted using the method developed by Han et al. [78], as follows: (5.252) Mu ¼ gm1 Wscm fscy þ gm2 Wsi fsyi

674

5. Design rules and standards

in which the terms gm1 and gm2 represent the flexural capacity coefficients, Wscm is the compound flexural modulus of the outer steel tube and infilled concrete, and Wsi is the flexural modulus of the inner steel tube. (

gm1 ¼

  0:48 lnðx þ 0:1Þ  0:85c2 þ 0:06c þ 1 þ 1:1 for circular section   0:48 lnðx þ 0:1Þ  0:85c2 þ 0:06c þ 1 þ 1:04  0:3c2 for square section

( gm2 ¼

(5.253.a)

0:02c2:76 ln x þ 1:04c0:67 0:04c

2:5

for circular section

0:8

ln x þ 1:04c for square section

 8 p D 4  d4 o i > > for CHS outer  CHS inner > > > 35D > > > ! > > > < B4o pd4i  Wscm ¼ 6 32 > > > for SHS outer  CHS inner > > Bo > > > > > > : B3o b4  i for SHS outer  SHS inner 6 6Bo

8 < C c2 fsyo þ C ð1:14 þ 1:02xÞf 1 2 ck fscy ¼ : C1 c2 fsyo þ C2 ð1:18 þ 0:85xÞfck

for circular section for square section

(5.253.b)

(5.253.c)

(5.253.d)

a , C ¼ 1þan , a ¼ Aso , and a ¼ Aso . in which C1 ¼ 1þa n 2 1þa Ac Ace

5.5.4.1 Example nine Determine the flexural capacity of a CFDST beam with the geometric and material properties, given in Table 5.28.

TABLE 5.28 Geometric and material properties of the section beam for predicting the flexural capacity. Steel tubes Outer steel tube SHS 120  120  3 (mm)

fsyo ¼ 275:9 MPa Eso ¼ 200000 MPa

Inner steel tube CHS 58  3 (mm)

Concrete core

fsyo ¼ 374:5 MPa

0 ¼ 46:8 fcu

Esi ¼ 200000 MPa

Ec ¼ 33300 MPa

5.5 Strength of composite members

675

Solution:

The cross-sectional areas of the steel tubes and the sandwiched concrete are: Aso ¼ B2o  ðBo  2to Þ2 ¼ 1202  ð120  2  3Þ2 ¼ 1404 mm2 Asi ¼

pd2i pðdi  2ti Þ2 p  582 pð58  2  3Þ2  ¼  ¼ 518:36 mm2 4 4 4 4

Ac ¼ ðBo  2to Þ2 

pd2i p  582 ¼ ð120  2  3Þ2  ¼ 10353:9 mm2 4 4

Ace ¼ ðBo  2to Þ2 ¼ ð120  2  3Þ2 ¼ 12996 mm2 The flexural strength of the section is calculated as: a¼

Aso 1404 ¼ 0:136 ¼ Ac 10353:9

an ¼ C1 ¼ C2 ¼ c¼ x¼

Aso 1404 ¼ ¼ 0:108 Ace 12996

a 0:136 ¼ ¼ 0:12 1 þ a 1 þ 0:136

1 þ an 1 þ 0:108 ¼ ¼ 0:975 1þa 1 þ 0:136

di 58 ¼ 0:51 ¼ ðBo  2to Þ ð120  2  3Þ

Aso fsyo 1404  275:9 ¼ ¼ 0:95 Ace fck 12996  ð0:67  46:8Þ

fscy ¼ C1 c2 fsyo þ C2 ð1:14 þ 0:85xÞfck ¼ 0:12  0:512  275:9 þ 0:975  ð1:14 þ 0:85  0:95Þ  ð0:67  46:8Þ ¼ 69:68 MPa  gm1 ¼ 0:48 lnðx þ 0:1Þ  0:85c2 þ 0:06c þ 1 þ 1:04  0:3c2   ¼ 0:48  lnð0:95 þ 0:1Þ  0:85  0:512 þ 0:06  0:51 þ 1 

þ 1:04  0:3  0:512 ¼ 1:11 gm2 ¼  0:04c2:5 ln x þ 1:04c0:8 ¼ 0:04  0:512:5  ln 0:95 þ 1:04  0:510:8 ¼ 1:08

676

5. Design rules and standards

Wscm ¼ Wsi ¼

B4o pd4i  6 32

!

Bo

 p d4i  ðdi  2ti Þ4 32di

  1204 p  584  6 32 ¼ ¼ 2:79  105 mm3 120   p 584  ð58  2  3Þ4 ¼ ¼ 6:78  103 mm3 32  58

Mu ¼ gm1 Wscm fscy þ gm2 Wsi fsyi ¼ 1:11  2:79  105  69:68 þ 1:08  6:78  103  374:5 ¼ 24:32 kNm

5.5.5 CFDST members under combined axial loading and bending The axial loadebending moment interaction diagram of CFDST members can be determined by Ref. [78]:   8 1 N a2 M N > > for  243 ho > < 4 N u þ d2 M u ¼ 1 Nu (5.254)       > N N 1 M N > > : b2  c2 þ ¼ 1 for < 243 ho Nu Nu d2 M u Nu where

8 > 1:0 > > > < 2 4 ¼ a1 l þ b1 l þ c1 > > > d 0:23c2 þ 1 > : 1 ðl þ 35Þ2 a1 ¼

8" > > > 13000 þ 4657 ln > > < d1 ¼ " > > > > 13500 þ 4810 ln > :

for l
lo for lo < l
lp

(5.255.a)

for l > lp

  1 þ 35 þ 2lp  lo e1  2 lp  lo

(5.255.b)

b1 ¼ e1  2a1 lp

(5.255.c)

c1 ¼ 1  a1 l2o  b1 lo

(5.255.d)

!# 0:3  235 25 an 0:05 for circular CFDST beam  columns fyo fck þ 5 0:1 !# 0:3  235 25 an 0:05 for square CFDST beam  columns fyo fck þ 5 0:1 (5.255.e)

677

5.5 Strength of composite members

e1 ¼ 

lp ¼

d1 lp þ 35

2

(5.255.f)

8 1743 > > > pffiffiffiffiffiffi > < fyo

for circular CFDST beam  columns

> 1811 > > ffi > : pffiffiffiffiffi f

for square CFDST beam  columns

(5.255.g)

yo

lo ¼

8 p > rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > . ffi > > > ð420x þ 550Þ f > > < scy

for circular CFDST beam  columns

p > > rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > . ffi > > ð220x þ 450Þ > : fscy

for square CFDST beam  columns

(5.255.h)

fscy is governed by Eq. (5.253.d) a2 ¼ 1  242 ho 1  zo 43 h2o

(5.255.j)

2ðzo  1Þ ho

(5.255.k)

b2 ¼ c2 ¼

(5.255.i)

circular CFDST-beam columns: (   ð0:5  0:245xÞ  1:8c2 þ 0:7c þ 1 ho ¼    0:1 þ 0:14x0:84  1:8c2 þ 0:7c þ 1

for x
0:4 for x > 0:4

(5.255.l)

square CFDST-beam columns: (   ð0:5  0:318xÞ  0:8c2 þ 0:2c þ 1 for x
0:4 ho ¼  (5.255.m)   0:84 2 0:1 þ 0:13x  0:8c þ 0:2c þ 1 for x > 0:4   8 N > > for circular CFDST beam  columns > < 1  0:4 NE (5.255.n) d2 ¼   > N > > : 1  0:25 for square CFDST beam  columns NE p2 Escm ðAsoþ Acþ Asi Þ l2 Eso Iso þ Ec Ic þ Esi Isi Escm ¼ Iso þ Ic þ Isi NE ¼

(5.255.o) (5.255.p)

678

5. Design rules and standards

5.6 Local buckling of steel plates Steel tubes used in composite members are typically fabricated by thinwalled steel plates formed by different methods like hot rolling, welding, or cold forming. The main weakness of thin steel plates is their susceptibility to local buckling since this phenomenon reduces the resistance and stiffness of composite members. Hence, it is crucial to understand the local buckling behavior of thin-walled steel tubes and consider the effects of local buckling in the design of composite members. The use of rational analysis for assessing the local buckling of steel tubes can lead to an economical design of steel casings. Different parameters affect the elastic local buckling mechanism of thin steel plates, i.e., the steel plate’s width-to-thickness ratio, loading conditions, boundary conditions, material properties, and geometric imperfections. By the occurrence of elastic local buckling in a thin steel plate, the plate does not fail and still can resist loads due to the effective postlocal buckling reverse of strength. Similar to the elastic buckling strength, the postlocal buckling strength is influenced by the material properties and slenderness of the steel plate. Besides, residual stresses developed during the fabrication process can also affect the postlocal buckling strength. Compared with a HSS, the existence of the infilled concrete highly increases the buckling strength of the steel tube since it restricts the inward buckling of the plates.

5.6.1 Elastic local buckling of steel plates 5.6.1.1 Steel hollow sections Steel hollow columns subjected to uniform compressions are likely to undergo local buckling. The buckling mechanism of a square steel hollow stub column subjected to uniform compression is shown in Fig. 5.25. The figure shows that two sides of the column present outward local buckling, whereas the other two show inward local buckling. According to the failure mechanism shown in Fig. 5.25, it can be supposed that steel plates have hinged boundary conditions. Hence, they can rotate about edges. As a result, flanges and webs of the rectangular or square steel hollow sections under axial compression can be idealized by simply supported on the edges. Fig. 5.26 presents a simply supported thin steel plate in which two opposite edges are subjected to uniform edge compression, where the terms L, b, and t represent the length, width, and thickness of the plate.

5.6 Local buckling of steel plates

679

FIGURE 5.25 Buckled shape of a square steel hollow short column under uniform compression [79].

FIGURE 5.26 A steel plate subjected to uniform edge compression with simple supports.

Imposing the compressive load equal to the plate’s elastic local buckling strength makes the plate experience local buckling by presenting outof-plane deformation. Considering a steel plate with simple supports at four edges subjected to uniform compression, shown in Fig. 5.26, the local buckling displacements of the plate is governed by:

npx mpy sin (5.256) u ¼ um sin L b where um denotes the undetermined deflection at the center of the plate, and n and m are the numbers of half-waves parallel to the applied compressive load direction and number of half-waves across the width, respectively.

680

5. Design rules and standards

The elastic buckling force of the steel plate is expressed as follows [80]:   p2 L2 bDr n2 m2 Pcr ¼ (5.257) þ n2 L2 b2 in which the term Dr denotes the flexural rigidity of the steel plate and is governed by: Dr ¼

Es t 3  12 1  w2s 

(5.258)

where Es and ws are Young’s modulus and Poisson’s ratio of the steel and t is the plate thickness. The minimum amount of Pcr is achieved when m is equal to unity. In this case, one half-wave buckled area is generated across the width of the steel plate, whereas several half-waves are formed along the applied load direction. The elastic buckling stress of the steel plate is [80,81]: scr ¼

k b p2 Es    b 2  12 1  w2s t

(5.259)

in which kb is the elastic buckling coefficient and depends on the boundary condition of the steel plate. For a simply supported plate, kb is expressed as:   nb L 2 þ (5.260) kb ¼ L nb According to Eq. (5.259), the initial local buckling stress of the plate reduces with increasing the bt ratio. Therefore, increasing the thickness of the plate can increase the elastic local buckling stress of the plate. However, when the elastic local buckling strength scr is greater than yield strength sy (scr > sy ), the difference between the critical buckling stresses of stocky steel plates may be negligible since the plates yield under the critical buckling load. It can be observed from Eq. (5.260) that the elastic buckling coefficient of a simply supported steel plate depends on n and the Lb ratio. The variation of the buckling coefficient of a simply supported steel plate under axial compression with n and the Lb ratio is given in Fig. 5.27. The least amount of kb for any half wave number is 4. Another point is that the overall shape of the curve flatters by increasing the number of half-waves n. The limiting width-to-thickness ratio of the steel plate for preventing the early elastic local buckling before reaching the yield strength can be achieved by taking scr ¼ sy .

681

5.6 Local buckling of steel plates

10 9

Buckling coefficient kb

8 7 6 5 4 3 2 1 0 0

1

2

3

4

5

6

L/b

FIGURE 5.27 Variation of buckling coefficient of the simply supported plate under compression with n and L=b.

In addition to the width-to-thickness ratio of the plate, the elastic local buckling strength is affected by initial geometric imperfections, yield stress, and residual stresses. Initial imperfections can highly reduce the initial local buckling strength of the steel plate. However, their effects are not considered in Eq. (5.259) since the equation is basically developed based on the elastic buckling behavior of straight steel plates with ideal behavior without any initial imperfections. Therefore, using Eq. (5.259) for predicting the initial elastic local buckling strength of real steel plates can lead to overestimated results. Bleich [81] applied the reduction coefficient h into Eq. (5.259) to consider the effects of initial imperfections and material inelasticity in predicting the initial elastic local buckling strength of steel plates as follows: scr ¼

in which h¼

kb p2 hEs     b 2 12 1  w2s t

(5.261)

8 2 < lp

if lp < ley

: l2  1  6l  0  4217 p p

if lp  ley

(5.262)

682

5. Design rules and standards

where ley is the yield slenderness limit and can be taken as 0.42 [82], and lp is the slenderness ratio of the steel plate and can be defined as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ffi rffiffiffiffiffiffi sy b 12 1  w2s sy ¼ (5.263) lp ¼ scr t k b p2 Es 5.6.1.2 Steel hollow sections filled with the concrete When a steel plate is in contact with the concrete, the concrete restrains all the edges of the plate. Hence, it can be assumed that the steel plate in contact with the concrete has a clamped boundary condition, as shown in Fig. 5.28. As discussed in Chapter 2, the existence of the concrete core prevents the inward buckling of the steel tube and hence, increases the buckling strength of the plate. Fig. 5.29 shows the buckling shape of a

FIGURE 5.28 Boundary condition of a flat steel plate in contact with the concrete.

FIGURE 5.29 Buckled shape of a square steel tube short column filled with the concrete under uniform compression [79].

5.6 Local buckling of steel plates

683

square steel tube filled with concrete. Depending on the existence of initial imperfections, the elastic critical local buckling stress of the steel plates of steel-concrete composite members under concentric axial compression (uniform compressive stresses) can be predicted using Eq. (5.259) or Eq. (5.261) by taking kb ¼ 10:3. Stress gradients may occur when a composite beam-column is subjected to combined loading conditions such as axial compression and biaxial bending. In this case, the steel plates of the square tube are no longer under uniform compression; two steel plates are subjected to nonuniform compressive stresses while the other steel plates are subjected to in-plane bending stresses, as shown in Fig. 5.30. Similar to the column under uniform compressive stresses, the elastic local buckling stress of clamped steel plates in composite columns subjected to compressive stress gradients can be obtained by Eq. (5.259) or Eq. (5.261). However, the elastic buckling coefficient kb should be computed using the following equation [83]: kb ¼ 18:89  14:38a þ 5:3a2

(5.264)

in which the term a represents the stress gradient coefficient and is defined as the ratio of the minimum edge stress (s2 ) to the maximum edge stress (s1 ) acting on the plate. It can be observed from Eq. (5.264) that the buckling coefficient kb is reduced by increasing the stress gradient coefficient a. For a clamped steel plate subjected to the uniform compression, the magnitude of a is equal to 1.0, leading to kb ¼ 9:81, which is slightly smaller than the value of 10.3 recommended by Uy [84]. During the fabrication process of HSSs, initial imperfections may be developed in the steel plate. In general, two types of initial imperfections may occur: 1. Initial out-of-plane deflections 2. Residual stresses

FIGURE 5.30 Clamped steel plates subjected to compressive stress gradients.

684

5. Design rules and standards

The main effects of initial imperfections are on reducing the strength and stiffness of the steel plates. For the numerical studies, the maximum magnitude of the initial geometric imperfections at a plate center can be taken as 0:1t [85]. Welding of steel plates leads to a complicated distribution of temperatures and stresses that are time-dependent. However, the residual stress pattern can be idealized for the sake of simplicity, as shown in Fig. 5.31. As shown in Fig. 5.31, the tensile residual stress generated at the welded edges can reach the steel yield stress, while the compressive residual stress typically varies in the range of 25% and 30% steel yield stress. The tensile and compressive residual stresses balance each other [85]. The initial local buckling stress of steel plates with initial imperfections scr is the function of the plate’s width-to-thickness ratio, steel yield stress, and stress gradient coefficient [83]. The initial local buckling stress of steel plates with 30 < b=t < 100 subjected to linearly varying edge compression can be expressed as:    2  3 scr b b b ¼ a1 þ a2 þ a4 (5.265) þ a3 fy t t t

FIGURE 5.31 Idealized residual stress pattern in welded composite members.

685

5.6 Local buckling of steel plates

TABLE 5.29 a

Constant coefficients for determining the initial local buckling stresses of the plate under stress gradients [83].

a1

a2

a3

a4  104

1:718  106

0.0

0.6925

0.02394

4:408

0.2

0.8293

0.01118

2:427  104

8:164  107

0.4

0.6921

0.01223

2:488  104

8:676  107

0.6

0.4028

0.02152

3:742  104

1:446  106

0.8

0.5096

0.0112

2:11  104

7:092  107

1.0

0.5507

0.005132

9:869  105

1:198  107

Where b is the clear width of the steel plate, and a1 , a2 , a3 , and a4 are constant coefficients and depend on the magnitude of the stress gradient coefficient a and are listed in Table 5.29.

5.6.2 Postlocal buckling of steel plates 5.6.2.1 Steel hollow sections The axial stiffness of the highly buckled regions of the steel plate is reduced by the occurrence of the initial local buckling. Therefore, the buckled areas tend to unload, and the stress starts to redistribute from the buckled area to the intact parts, particularly the areas adjacent to the supported edges that are almost unaffected by the local buckling of the plate. As a result, the steel plate can still bear increasing loads and present postlocal buckling reserve strength without failure, as shown in Fig. 5.32. From Fig. 5.32, the postlocal buckling reserve strength sp of a steel member is the difference between the critical local buckling strength scr and the ultimate strength of the steel member su. sp ¼ su  scr

(5.266)

When compression stresses are induced on the edges of a thin steel plate, its postlocal buckling behavior can be described by the steel plate deflections in the transverse direction and the in-plane redistribution of stress through the buckled region. The in-plane redistribution of stress depends on the in-plane boundary conditions of the plate [86]. The deflections of the steel plate in transverse directions and the axial strain lead

686

5. Design rules and standards

Ultimate strength

σu Axial stress σ

Initial stiffness Post-buckling

σc Local buckling

Axial strain ε FIGURE 5.32

General axial stressestrain curve of a steel plate.

to a continued axial shortening of boundary lines at the loaded edges. The axial shortening caused by the plate’s deflections in the transverse direction is not constant and alters across the plate. The maximum axial shortening occurs at the center, whereas the minimum axial shortening occurs at the unloaded edges. However, the axial shortening induced by the axial strain compensates for this variation, where the maximum and minimum axial shortenings due to the axial strain occur at the unloaded edges and the center, respectively. The in-plane distribution of stress at the steel plate’s buckled region in the direction of loading should be the same as that of the axial strain. This denotes that the buckled area of the plate bears smaller stresses than the loaded strips of the edges. Thin steel plates in steel-concrete composite columns usually have a great postlocal buckling strength. Consequently, the postlocal buckling reserve strengths of thin steel plates should be considered in the design of composite members. The postlocal buckling of thin steel plates is generally determined by the effective width concept. The in-plane distribution of ultimate stress in a thin steel plate with simply supported boundary conditions subjected to uniform edge compression is shown in Fig. 5.33. The actual distribution of ultimate stress can be replaced by an idealized stress distribution in which it is assumed that the central buckled regions do not carry any stresses. In contrast, the effective width of the plate be bears the stress equal to the steel yield stress, as shown in Fig. 5.33b.

5.6 Local buckling of steel plates

(a) Ultimate stress distribution

687

(b) Effective width

FIGURE 5.33 The effective width theory for simply support plates. (a) Ultimate stress distribution (b) Effective width.

According to the effective width theory, the effective width of the steel plate be is defined as: be su ¼ b fy

(5.267)

where su and fy represent the average ultimate stress acting on the plate and yield stress of the plate, respectively. von Karman et al. [87] recommended the following equation for calculating the effective width be of a simply supported thin steel plate under uniform edge compression: rffiffiffiffiffiffi be scr (5.268) ¼ b fy Generally, the initial curvatures in steel plates as an initial imperfection generated by hot rolling or welding of steel plates diminishes the stiffness and strength of the plates. Therefore, the Australian design guide AS 4100 applies a reduction factor a to Eq. (5.268) for predicting the effective width be of HSSs with initial imperfections as follows: rffiffiffiffiffiffi be scr (5.269) ¼a b fy The influences of initial curvatures and residual stresses are incorporated into the reduction coefficient a. Residual stresses can lead to premature failure of steel plates. Based on AS 4100, the magnitude of a for

688

5. Design rules and standards

hot-rolled steel plates can be taken as 0.65. It is noteworthy that the effects of initial curvatures are more prominent on the behavior of steel plates with intermediate slenderness ratios than the stocky ones. The effective width of steel plates with initial curvatures in coldformed members can be defined as [88]: rffiffiffiffiffiffi rffiffiffiffiffiffi ! be scr scr ¼ 1  0:22 (5.270) b fy fy Eq. (5.270), known as the Winter equation, is the basis of equations recommended by design codes such as Eurocode 3 and AISC 360-16. For determining the effective width of the steel plates in rectangular steel hollow sections, AISC 360-16 recommends the following expressions: sffiffiffiffiffiffi 8 fy > > b for l
lr > > < scr (5.271) be ¼ sffiffiffiffiffiffi ffiffiffiffi s s ffiffiffiffi ! > > > fy fel fel > : b 1  0:20 for l > lr fy fy scr in which l is the width-to-thickness ratio of the member and lr is the limiting width-to-thickness ratio, defined in Table 5.2. fel denotes the elastic local buckling stress and is expressed as:   lr 2 fy fel ¼ 1:382 l

(5.272)

For circular steel hollow sections, AISC360-16 recommends the following equations for determining the effective area of the section Ae :

Ae ¼

8 Ag > > > > >

27 60:038E s > > 4   þ 5Ag > > D 3 : fy t

D Es
0:11 fy t

Es D Es for < < 0:45 fy fy t

(5.273)

For calculating the effective width of the steel plates in rectangular steel hollow sections, EC3 recommends the following expressions: 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > b for lp
0:5 þ 0:085  0:055a > > < be ¼ lp  0:055ð3 þ aÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (5.274) > >
1:0 for l
0:5 þ 0:085  0:055a p > 2 : lp

5.6 Local buckling of steel plates

689

where a is the stress gradient coefficient and lp is the slenderness parameter and is expressed as: sffiffiffiffiffiffi fy ¼ lp ¼ scr

in which ks is given by:

b t vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u pffiffiffiffiffi 235 u  ks 28:4u  t fy N=mm2

(5.275)

8 >
: 1:0 for a ¼ 1

(5.276)

AS 4100 recommends a simple method for predicting the effective width of rectangular hollow section be and the effective diameter of circular hollow sections De . be and De in AS 4100 are functions of the slenderness of the steel plate le and the yield slenderness limit of the steel plate ley . The magnitude of the yield slenderness limit of the steel plate ley depends on the boundary condition, type of steel plate, stress distribution, and the level of residual stress and can be found in Table 6.2.4 of AS 4100. The effective width be of flat steel plates is governed by:   ley be ¼ b
b (5.277) le The effective diameter De of circular hollow sections is governed by: " sffiffiffiffiffiffi   # ley ley 2 De ¼ min D ;D 3
D (5.278) le le 5.6.2.2 Square steel hollow sections filled with the concrete The use of postbuckling strength in the design of bare steel sections is permitted by international design codes. By contrast, no design codes allow the use of this advantageous behavior of steel sections when it comes to design of composite members. Instead, they define width-tothickness limits for steel plates of composite members to guarantee that they yield before the occurrence of local buckling. Liang et al. [83]

690

5. Design rules and standards

suggested an equation for predicting the postlocal buckling strength of steel plates in contact with the concrete core as follows:    2  3 su b b b ¼ c1 þ c2 þ c4 (5.279) þ c3 fy t t t where c1 , c2 , c3 , and c4 are constant coefficients and depend on the magnitude of the stress gradient coefficient a and are listed in Table 5.30. Steel plate ultimate strength having stress gradient coefficient a > 0 can be approximately determined by the following expression [83]: s1u su ¼ ð1 þ 0:54s Þ (2.280) fy fy in which 0
fs ¼ 1  a
1:0 and su is the ultimate stress of steel plates subjected to uniform compression and is governed by Eq. (5.279) by taking a as 1.0. Different models are available for calculating the effective width of steel plates in composite members under uniform compression. Table 5.31 summarizes the models recommended by researchers for predicting the magnitude of bbe for the steel plates of rectangular CFST columns. For calculating the effective width of steel plates subjected to stress gradients in composite beam-columns, presented in Fig. 5.34, the following expression can be applied [83]:  2   be1 b 4 b ¼ 0:2777 þ 0:01019 þ 9:605  1:972  10 b t t (5.281.a)  3 7 b  10 for a > 0:0 t

TABLE 5.30 Constant coefficients for determining the ultimate strength of plate under stress gradients [83]. a

c1

c2

c3

c4 104

1:407  106

0.0

1.257

0.006184

1:608 

0.2

0.6855

0.02894

4:89  104

2:134  106

0.4

0.6538

0.02888

5:215  104

2:424  106

0.6

0.7468

0.01925

3:689  104

1:677  106

0.8

0.6474

0.02088

4:171  104

2:058  106

1.0

0.5554

0.02038

3:944  104

1:921  106

0.2

1.48

0.01584

2:868  104

1:742  106

TABLE 5.31 Effective width models for rectangular CFST columns under uniform compression. Researcher

Equations number (5.21)

Winter [88]

(5.22)

Nakai et al. [89]

(5.23)

Ge and Usami [90]

(5.24)

Liang and Uy [91]

(5.25)

Song et al. [82]

(5.26)

be g ¼ b lp 8 lp > > <  175 g ¼ 0  9 lp  0  42 > > : lp  lp

Elastic local buckling, kb 10.3

10.3 9.81 4 9.81

5.6 Local buckling of steel plates

Von Karman et al. [87]

Effective width formula be rffiffiffiffiffiffi  1 be su scr ¼ ¼ lp ¼
1 b sy sy  1 be su 0:25 ¼ ¼ lp   2
1 b sy lp 2    be su ¼ ¼ 0:433 lp  0:5  0:831 lp 0:5 þ 1 b sy be su 1:2 0:3 ¼ ¼  2
1 b sy lp lp  1 8 scr 3 > > 0  675 for scr
sy > < sy be su ¼ ¼ > b sy > 1  > : scr 3 for s > s 0  915 cr y scr þ sy

10.3 if lp < 0  42 if lp > 0  42

691

692

5. Design rules and standards

FIGURE 5.34

Determining effective widths of rectangular steel hollow sections filled with the concrete under stress gradients.

 2  3  be1 b 5 b 7 b ¼ 0:4186  0:002047  4:685 10 for a ¼ 0:0 þ 5:355  10 b t t t (5.281.b) be2 be1 ¼ ð1 þ 4s Þ b b

(5.281.c)

where the effective widths be1 and be2 are depicted in Fig. 5.34. If ðbe1 þbe2 Þib, the whole width of the plate is effective in bearing loads, and therefore, the ultimate strength of the plate can be calculated using Eqs. (5.279) and (5.280). It should be noted that except for the models recommended by Song et al. [82], the rest of them are developed based on the normal strength steel material. The accuracy of models for predicting the steel plates ultimate strength of CFST built with normal and high strength steel material with fy ranging from 235 to 960 MPa was assessed by Ayough et al. [92] based on the numerical study of Song et al. [82], and the results are presented in Fig. 5.35. It is evident from the figure that the ultimate strengths predicted by Liang and Uy equation [91] are conservative, and the degree of conservatism increases with increasing the steel yield stress. The model recommended by Liang et al. [83] is able to estimate the ultimate strength of normal strength steel plates with reasonable accuracy. In contrast, it underestimates the results for high-strength steel plates. Considering the accuracy of the Ge and Usami model [90], it can be observed from the

693

1.2

1.2

1

1

0.8

0.8

σu /σy (be /b)

σu /σy (be /b)

5.6 Local buckling of steel plates

0.6 0.4

0.4

fy=235 Mpa Liang et al. Liang and Uy Ge and Usami Nakai et al.

0.2

0.6 fy=460 Mpa Liang et al. Liang and Uy Ge and Usami Nakai et al.

0.2

0

0 0

0.5

1

1.5

0

0.5

1

λp

1.2

1.2

1

1

0.8

0.8

0.6

0.6 0.4

fy=690 Mpa Liang et al. Liang and Uy Ge and Usami Nakai et al.

0.2

1

fy=960 Mpa Liang et al. Liang and Uy Ge and Usami Nakai et al.

0.2 0

0 0

2

0

1

2

3

λp

λp

(c)

2

(b)

σu /σy (be /b)

σu /σy (be /b)

(a)

0.4

1.5

λp

(d)

FIGURE 5.35 Comparisons of effective width (ultimate stress) between FE analyses and theoretical formulas. (a) fy ¼ 235MPa (b) fy ¼ 460 MPa. (c) fy ¼ 690 MPa (d) fy ¼ 960 MPa. Ayough P, Sulong NHR, Ibrahim Z. Analysis and review of concrete-filled double skin steel tubes under compression. Thin-Walled Struct 2020;148:106495.

figure that their equation can predict the ultimate strength of low-strength steel plates with reasonable accuracy. However, for high-strength thin steel plates, their model underestimates the ultimate strength. Out of all considered equations, the model of Nakai et al. [89] leads to the most accurate result for all ranges of steel yield strength. The effective width curves recommended by Song et al. [82], von Karman et al. [87], and Winter [88] are compared with the numerical test results by Ayough et al. [92] based on the numerical study of Song et al. [82], and the results are presented in Fig. 5.36. Additionally, the local buckling stress curves calculated by using Eqs. (5.261)e(5.263) are shown in Fig. 5.36. It can be observed from the figure that, in general, Karman’s model [87] overestimates the ultimate strength of steel plates, especially for compact steel plates. Winter’s model [88] overestimates the ultimate strength of thick steel plates, whereas it underestimates the ultimate

694

5. Design rules and standards

1.2 fy=235 Mpa fy=460 Mpa fy=690 Mpa fy=960 Mpa Song et al. Karman Winter Elastic buckling strength

1

σu /σy (be /b)

0.8

0.6

0.4

Post-buckling strength

0.2

0 0

0.5

1

1.5

2

2.5

3

λp FIGURE 5.36 Comparisons of effective width (ultimate stress) predicted by Song et al., Karman, and Winter models with the measured ones. Ayough P, Sulong NHR, Ibrahim Z. Analysis and review of concrete-filled double skin steel tubes under compression. Thin-Walled Struct 2020;148:106495.

strength of thin steel plates. The model recommended by Song et al. [82], however, leads to conservative results for the whole given range of lp . Comparison of the elastic buckling curve with the effective width curves indicates that the postbuckling strength can remarkably enhance su the strength of the plate. For instance, the ratio for a steel plate with sy lp ¼ 1:5 is 0.189 when the postbuckling strength is ignored, whereas the su ratio for the same slenderness lp ¼ 1:5 when the postbuckling strength sy is considered is more than 0.565. Again, this reinforces the fact that filling the HSS and considering the postbuckling strength of steel plates leads to a remarkable strength enhancement, especially for thin steel plates with large values of lp . 5.6.2.3 Circular steel hollow sections filled with the concrete Uy and Bradford [93] recommended a model for predicting the effective diameter de of circular steel tubes filled with concrete. To this end, they adopted the von Karman theory of postlocal buckling. Based on Uy and Bradford’s model, the effective diameter is obtained using Eq. (5.267), in which the effective width be is replaced by the effective diameter de. de su ¼ d fy

(5.282)

5.6 Local buckling of steel plates

695

The magnitude of the load that causes local buckling over the effective cross section of the circular steel tube is governed by:   2Es t ðpde tÞ (5.283) Ncr; eff ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  w2 de   2E s ffi t p ffiffiffiffiffiffiffiffi is the minimum elastic buckling stress. in which sel ¼ 2 de 1w

Therefore, Eq. (5.283) can be rewritten as:   d ðpde tÞ Ncr; eff ¼ sel;eff de

(5.284)

The magnitude of the load that causes yielding of the circular steel tube over its effective cross section is expressed as: Ny; eff ¼ ðpde tÞfy

(5.285)

Therefore, Eq. (5.282) can be rewritten as: su de sel ¼ ¼ fy d fy

(5.286)

5.6.2.4 Example ten The cross section ð600 500 8 mmÞ of a rectangular CFST column subjected to biaxial bending is given in Fig. 5.37. The web and the flange are under compressive stress gradients, as shown in the figure. Determine the effective area of the steel tube.

FIGURE 5.37 Cross section of the rectangular CFST column subjected to compressive stress gradient.

696

5. Design rules and standards

Solution:

Since the member is under compressive stress gradients, Eq. (5.281) is applied to determine the effective width of the cross section. • The effective width of the flange The clear width of the flange is: b ¼ 500  2  8 ¼ 484 mm 30 ¼ 0:8 > 0:0. Therefore, the The stress gradient coefficient a ¼ ss21 ¼ 37:5 effective width of the flange is calculated using Eq. (5.281.a) as:

 2  3   be1 b 4 b 7 b ¼ 0:2777 þ 0:01019 þ 9:605  10  1:972  10 b t t t     2  484 484 484 3 ¼ 0:2777 þ 0:01019 þ 9:605  107  1:972  104 8 8 8 ¼ 0:385 Hence: be1 ¼ 0:385  484 ¼ 186:34 mm The effective width of be2 is calculated using Eq. (5.281.c) as: 4s ¼ 1  a ¼ 1  0:8 ¼ 0:2 be2 be1 ¼ ð1 þ 4s Þ b b be2 ¼ ð1 þ 4s Þbe1 ¼ ð1 þ 0:2Þ  186:34 ¼ 223:61 mm The total effective width of the flange is be1 þ be2 ¼ 186:34 þ 223:61 ¼ 409:95, which is less than the clear width of the flange b ¼ 484. • The effective width of the web The clear width of the web is: b ¼ 600  2  8 ¼ 584 mm 42 ¼ 0:452 > 0:0. Therefore, The stress gradient coefficient a ¼ ss21 ¼ 92:92 the effective width of the flange is calculated using Eq. (5.281.a) as:

 2  3   be1 b 4 b 7 b ¼ 0:2777 þ 0:01019 þ 9:605  10  1:972  10 b t t t     2  584 584 584 3 ¼ 0:2777 þ 0:01019 þ 9:605  107  1:972  104 8 8 8 ¼ 0:344

5.6 Local buckling of steel plates

697

Hence: be1 ¼ 0:344  584 ¼ 200:9 mm The effective width of be2 is calculated using Eq. (5.281.c) as: 4s ¼ 1  a ¼ 1  0:452 ¼ 0:548 be2 be1 ¼ ð1 þ 4s Þ b b be2 ¼ ð1 þ 4s Þbe1 ¼ ð1 þ 0:548Þ  200:9 ¼ 311 mm The total effective width of the flange is be1 þ be2 ¼ 200:9 þ 311 ¼ 511:9, which is less than the clear width of the web b ¼ 584. • The effective cross-sectional area of the tube As shown in Fig. 5.37, only one flange and one web of the steel tube are subjected to compressive stress. Therefore, it can be assumed that local buckling does not occur in other areas of the tube. The ineffective crosssectional areas of the flange Aineff; f and web Aineff; w are: Aineff; f ¼ ðb  be Þt ¼ ð484  409:95Þ  8 ¼ 592:4 mm2 Aineff; w ¼ ðb  be Þt ¼ ð584  511:9Þ  8 ¼ 576:8 mm2 The gross cross-sectional area of the steel tube Ag is: Ag ¼ ðb  hÞ  ðb  2tÞðh  2tÞ ¼ ð500  600Þ  ð500  2  8Þð600  2  8Þ ¼ 17344 mm2 The effective cross-sectional area of the steel tube is therefore: Aeff ¼ Ag  Aineff; f  Aineff; w ¼ 17344  592:4  576:8 ¼ 16174:8 mm2 Fig. 5.38 illustrates the effective area of the steel tube. 5.6.2.5 Example eleven The cross section ð600 500 8 mmÞ of a rectangular CFST column is subjected to uniform compression, in which the steel tube is fabricated by welding four flat steel plates. Determine the effective area of the steel tube. The yield stress and the modulus of elasticity of the steel tube are fy ¼ 235 MPa and Es ¼ 205000 MPa, respectively.

698

5. Design rules and standards

FIGURE 5.38

Effective and ineffective areas of steel tube of CFST column subjected to compressive stress gradient.

Solution:

The effective cross-sectional area of the given column is determined using models listed in Table (5.31), and the results are compared. The clear width of the flange and width are as follows: b ¼ 500  2  8 ¼ 484 mm b ¼ 600  2  5 ¼ 590 mm • von Karman’s model First, the effective width of the flange is determined. From Eq. (5.263), the slenderness ratio of the flange lp is: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ffi   rffiffiffiffiffiffi sy b 12 1  w2s sy 484 12 1  0:32  235 ¼ ¼ 0:67 lp ¼ ¼ scr t 8 10:3  p2  205000 Kp2 Es According to von Karman’s equation, the effective width of the flange is: be  1  1 ¼ lp ¼ lp ¼ ð0:67Þ1 ¼ 1:49 > 1:0 b

5.6 Local buckling of steel plates

699

Therefore bbe ¼ 1:0 and the overall width of the flange is effective under the uniform compression. The slenderness ratio of the web lp is: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi    rffiffiffiffiffiffi sy b 12 1  w2s sy 590 12 1  0:32  235 ¼ ¼ 1:19 lp ¼ ¼ scr t 5 10:3  p2  205000 Kp2 Es According to von Karman’s equation, the effective width of the web is: be  1  1 ¼ lp ¼ lp ¼ ð1:19Þ1 ¼ 0:84 < 1:0 b Therefore, be ¼ 0:84b ¼ 0:84  590 ¼ 495:6 mm be1 ¼ be2 ¼

be 495:6 ¼ ¼ 247:8 mm 2 2

The effective cross-sectional area of the steel tube is calculated as: Aineff ; f ¼ 0 mm2 Aineff ; w ¼ ðb  be Þt ¼ ð590  495:6Þ  5 ¼ 472 mm2 The gross cross-sectional area of the steel tube Ag is: Ag ¼ ðb  hÞ  ðb  2tÞðh  2tÞ ¼ ð500  600Þ  ð500  2  8Þð600  2  5Þ ¼ 14440 mm2 The effective cross-sectional area of the steel tube is therefore: Aeff ¼ Ag  Aineff ; f  Aineff ; w ¼ 14440  0  472 ¼ 13968 mm2 • Winter’s model First, the effective width of the flange is determined. From Eq. (5.263), the slenderness ratio of the flange lp is: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ffi   rffiffiffiffiffiffi sy b 12 1  w2s sy 484 12 1  0:32  235 ¼ ¼ ¼ 0:67 lp ¼ scr t 8 10:3  p2  205000 kb p2 Es According to Winter’s equation, the effective width of the flange is: be su  1 0:25 0:25 ¼ ¼ lp   2 ¼ ð0:67Þ1  ¼ 0:94 b sy ð0:67Þ2 lp

700

5. Design rules and standards

Therefore, be ¼ 0:94b ¼ 0:94  484 ¼ 455 mm be1 ¼ be2 ¼

be 455 ¼ ¼ 227:5 mm 2 2

The slenderness ratio of the web lp is: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ffi   rffiffiffiffiffiffi sy b 12 1  w2s sy 590 12 1  0:32  235 ¼ ¼ ¼ 1:19 lp ¼ scr t 5 10:3  p2  205000 k b p2 Es According to Winter’s equation, the effective width of the web is: be su  1 0:25 0:25 ¼ ¼ lp   2 ¼ ð1:19 Þ1  ¼ 0:66 b sy ð1:19 Þ2 lp Therefore, be ¼ 0:66b ¼ 0:66  590 ¼ 389:4 mm be1 ¼ be2 ¼

be 389:4 ¼ ¼ 194:7 mm 2 2

The effective cross-sectional area of the steel tube is calculated as: Aineff ; f ¼ ðb  be Þt ¼ ð484  455Þ  8 ¼ 232 mm2 Aineff ; w ¼ ðb  be Þt ¼ ð590  389:4Þ  5 ¼ 1003 mm2 The gross cross-sectional area of the steel tube Ag is: Ag ¼ ðb  hÞ  ðb  2tÞðh  2tÞ ¼ ð500  600Þ  ð500  2  8Þð600  2  5Þ ¼ 14440 mm2 The effective cross-sectional area of the steel tube is therefore: Aeff ¼ Ag  Aineff; f  Aineff; w ¼ 14440  232  1003 ¼ 13205 mm2 • Nakai’s model First, the effective width of the flange is determined. From Eq. (5.263), the slenderness ratio of the flange lp is: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ffi   rffiffiffiffiffiffi sy b 12 1  w2s sy 484 12 1  0:32  235 ¼ ¼ ¼ 0:688 lp ¼ scr t 8 9:81  p2  205000 k b p2 Es

5.6 Local buckling of steel plates

701

According to Nakai’s equation, the effective width of the flange is:  2   be su ¼ ¼ 0:433 lp  0:5  0:831 lp  0:5 þ 1 b sy ¼ 0:433  ð0:688  0:5Þ2  0:831  ð0:688  0:5Þ þ 1 ¼ 0:86 Therefore, be ¼ 0:86b ¼ 0:86  484 ¼ 416:24 mm be1 ¼ be2 ¼

be 416:24 ¼ ¼ 208:12 mm 2 2

The slenderness ratio of the web lp is: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ffi   rffiffiffiffiffiffi sy b 12 1  w2s sy 590 12 1  0:32  235 ¼ ¼ ¼ 1:34 lp ¼ scr t 5 9:81  p2  205000 kb p2 Es According to Nakai’s equation, the effective width of the web is:  2   be su ¼ ¼ 0:433 lp  0:5  0:831 lp  0:5 þ 1 b sy ¼ 0:433  ð1:34  0:5Þ2  0:831  ð1:34  0:5Þ þ 1 ¼ 0:61 Therefore, be ¼ 0:61b ¼ 0:61  590 ¼ 360 mm be1 ¼ be2 ¼

be 360 ¼ ¼ 180 mm 2 2

The effective cross-sectional area of the steel tube is calculated as: Aineff; f ¼ ðb  be Þt ¼ ð484  416:24Þ  8 ¼ 542:1 mm2 Aineff; w ¼ ðb  be Þt ¼ ð590  360Þ  5 ¼ 1150 mm2 The gross cross-sectional area of the steel tube Ag is: Ag ¼ ðb  hÞ  ðb  2tÞðh  2tÞ ¼ ð500  600Þ  ð500  2  8Þð600  2  5Þ ¼ 14440 mm2 The effective cross-sectional area of the steel tube is therefore: Aeff ¼ Ag  Aineff; f  Aineff; w ¼ 14440  542:1  1150 ¼ 12747:9 mm2 • Ge’s model

702

5. Design rules and standards

First, the effective width of the flange is determined. From Eq. (5.263), the slenderness ratio of the flange lp is: rffiffiffiffiffiffi sy b ¼ lp ¼ scr t

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ffi   12 1  w2s sy 484 12 1  0:32  235 ¼ ¼ 1:077 8 4  p2  205000 k b p2 Es

According to Ge’s equation, the effective width of the flange is: be su 1:2 0:3 1:2 0:3 ¼ ¼   2 ¼ ¼ 0:86 2 b sy lp 1:077 1:077 lp Therefore, be ¼ 0:86b ¼ 0:86  484 ¼ 416:24 mm be 416:24 ¼ ¼ 208:12 mm 2 2 The slenderness ratio of the web lp is: be1 ¼ be2 ¼

rffiffiffiffiffiffi sy b ¼ lp ¼ scr t

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ffi   12 1  w2s sy 590 12 1  0:32  235 ¼ ¼ 2:1 5 4  p2  205000 k b p2 Es

According to Winter’s equation, the effective width of the web is: be su 1:2 0:3 1:2 0:3 ¼ ¼   2 ¼ ¼ 0:5 b sy lp 2:1 2:12 lp Therefore, be ¼ 0:5b ¼ 0:5  590 ¼ 295 mm be 295 ¼ ¼ 147:5 mm 2 2 The effective cross-sectional area of the steel tube is calculated as: be1 ¼ be2 ¼

Aineff; f ¼ ðb  be Þt ¼ ð484  416:24Þ  8 ¼ 542:1 mm2 Aineff; w ¼ ðb  be Þt ¼ ð590  295Þ  5 ¼ 1475 mm2 The gross cross-sectional area of the steel tube Ag is: Ag ¼ ðb  hÞ  ðb  2tÞðh  2tÞ ¼ ð500  600Þ  ð500  2  8Þð600  2  5Þ ¼ 14440 mm2 The effective cross-sectional area of the steel tube is therefore: Aeff ¼ Ag  Aineff; f  Aineff; w ¼ 14440  542:1  1475 ¼ 12422:9 mm2

5.6 Local buckling of steel plates

703

• Liang and Uy’s model The elastic critical buckling stress scr of the flange is calculated by Eq. (5.259) as: scr ¼

k b p 2 Es 9:81  p2  205000  ¼ 496:58 MPa  2 ¼     b  484 2 2 2 12  1  0:3 12 1  w t 8

scr ¼ 496:58 MPa > sy ¼ 235 MPa. Therefore, according to the Liang and Uy’s equation, the effective width of the flange is:   1 1 3 3 be su scr 496:58 ¼ ¼ 0:915 ¼ 0:915 ¼ 0:80 b sy scr þ sy 496:58 þ 235 Hence: be ¼ 0:8b ¼ 0:8  484 ¼ 387:2 mm be1 ¼ be2 ¼

be 387:2 ¼ ¼ 193:6 mm 2 2

The elastic critical buckling stress scr of the flange is calculated by Eq. (5.259) as: scr ¼

k b p 2 Es 9:81  p2  205000  ¼ 130:54 MPa  2 ¼     b  590 2 2 12  1  0:32 12 1  w t 5

scr ¼ 130:58 MPa < sy ¼ 235 MPa. Therefore, according to the Liang and Uy’s equation, the effective width of the web is:  1  1 be su scr 3 130:54 3 ¼ ¼ 0:675 ¼ 0:675 ¼ 0:55 b sy sy 235 Therefore: be ¼ 0:55b ¼ 0:55  590 ¼ 324:5 mm be1 ¼ be2 ¼

be 324:5 ¼ ¼ 162:25 mm 2 2

The effective cross-sectional area of the steel tube is calculated as: Aineff; f ¼ ðb  be Þt ¼ ð484  387:2Þ  8 ¼ 744:4 mm2

704

5. Design rules and standards

Aineff; w ¼ ðb  be Þt ¼ ð590  324:5Þ  5 ¼ 1327:5 mm2 The gross cross-sectional area of the steel tube Ag is: Ag ¼ ðb  hÞ  ðb  2tÞðh  2tÞ ¼ ð500  600Þ  ð500  2  8Þð600  2  5Þ ¼ 14440 mm2 The effective cross-sectional area of the steel tube is therefore: Aeff ¼ Ag  Aineff; f  Aineff; w ¼ 14440  744:4  1327:5 ¼ 12368:1 mm2 • Song’s model First, the effective width of the flange is determined. From Eq. (5.263), the slenderness ratio of the flange lp is: rffiffiffiffiffiffi sy b ¼ lp ¼ scr t

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi    12 1  w2s sy 484 12 1  0:32  235 ¼ ¼ 0:67 8 10:3  p2  205000 k b p2 Es

lp ¼ 0:67 > 0:42. Therefore, based on the Song’s model, g is calculated as:  1:75 0:9 lp  0:42 0:9  ð0:67  0:42Þ1:75 ¼ 0:55 g ¼ lp  ¼ 0:67  lp 0:67 and be g 0:55 ¼ ¼ ¼ 0:82 b lp 0:67 Therefore: be ¼ 0:82b ¼ 0:82  484 ¼ 396:9 mm be 396:9 ¼ ¼ 198:45 mm 2 2 The slenderness ratio of the web lp is: be1 ¼ be2 ¼

rffiffiffiffiffiffi sy b ¼ lp ¼ scr t

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ffi   12 1  w2s sy 590 12 1  0:32  235 ¼ ¼ 1:19 5 k b p2 Es 10:3  p2  205000

lp ¼ 1:19 > 0:42. Therefore, based on the Song’s model, g is calculated as:  1:75 0:9 lp  0:42 0:9  ð1:19  0:42Þ1:75 ¼ 0:71 g ¼ lp  ¼ 1:19  lp 1:19

705

5.6 Local buckling of steel plates

and be g 0:71 ¼ ¼ ¼ 0:60 b lp 1:19 Therefore: be ¼ 0:6b ¼ 0:6  590 ¼ 354 mm be1 ¼ be2 ¼

be 354 ¼ ¼ 177 mm 2 2

The effective cross-sectional area of the steel tube is calculated as: Aineff; f ¼ ðb  be Þt ¼ ð484  396:9Þ  8 ¼ 696:8 mm2 Aineff; w ¼ ðb  be Þt ¼ ð590  354Þ  5 ¼ 1180 mm2 The gross cross-sectional area of the steel tube Ag is: Ag ¼ ðb  hÞ  ðb  2tÞðh  2tÞ ¼ ð500  600Þ  ð500  2  8Þð600  2  5Þ ¼ 14440 mm2 The effective cross-sectional area of the steel tube is therefore: Aeff ¼ Ag  Aineff; f  Aineff; w ¼ 14440  696:8  1180 ¼ 12563:2 mm2 The summary of the results obtained by different models is given in Table 5.32. It can be seen from the table that the model of von Karman et al. leads to the largest effective area. In contrast, the models of Nakai et al., Ge and Usami, Lian and Uy, and Song et al. almost lead to the same results, and they are all smaller than the effective area predicted by the

TABLE 5.32 Summary of the results predicted by different models. Flange

Web

Researcher

be

be

Effective area

von Karman et al.

484

495.6

13,968

Winter

455

389.4

13,205

Nakai et al.

416.24

360

12,747.9

Ge and Usami

416.24

295

12,422.9

Liang and Uy

387.2

324.5

12,368.1

Song et al.

396.9

354

12,563.2

706

5. Design rules and standards

models of von Karman et al. and Winter. This is because the models of von Karman et al. and Winter are based on the HSSs, whereas the rest of the models are according to the behavior of steel tubes filled with concrete.

5.7 Further discussion on local buckling of steel plates in rectangular CFST columns under axial compression As discussed above, the critical buckling strength model has been developed by assuming that the four edges of steel plates in CFST columns with rectangular or square cross sections are fully clamped because of the restraint afforded by the infilled concrete. In fact, the restraint of the edge depends on the stiffness of adjacent plates fields. It could be discussed that the adjacent plate fields and infilled concrete restrain the unloaded edges of the rectangular steel plate field from rotation. However, the extent of restraint is not perfect because steel plate fields’ stiffness over corners usually is not significant enough to provide a fully clamped boundary condition. Hence, it can be assumed that the unloaded edges of rectangular steel plates are elastically restricted against rotation, while the loaded edges have clamped boundary conditions. The differential equation for the stability analysis of the thin steel plate can be expressed based on the classical theory of elastic stability [80], as follows: ! v4 u v4 u v4 u v2 u v2 u v2 u þ N þ 2 þ þ 2N (5.287) ¼ N Dr x xy y vxvy vx4 vx2 vy2 vx4 vx2 vy2 in which the term Dr is the flexural rigidity of the steel plate and is governed by Eq. (5.258). The energy associated with the thin plate deforming U and the energy associated with the applied load V can be expressed based on the classical theory of elastic stability [80], as follows: Dr U¼ 2

V¼

Z Z (

1 2

Z Z "

v2 u v2 u þ vx2 vy2

!2

"

v2 u v2 u v2 u  2ð1  ns Þ  vxvy vx2 vy2

!2 #)

  2   2 # vw vw vw vw Nx þ Ny þ 2Nxy dxdy vx vx vy vy

dxdy (5.288) (5.289)

The boundary stress of the steel plate for the CFST stub columns subjected to axial compression is sy ¼ sxy : Ny ¼ sy t ¼ Nxy ¼ sxy t ¼ 0

(5.290)

707

5.7 Further discussion on local buckling

Clamped

Elastically restrained against rotation FIGURE 5.39 Local buckling of steel plates in rectangular CFST columns [94].

Nx ¼  px t ¼ sx t

(5.291)

in which px is the uniform compressive stress applied in the x-axis direction, as shown in Fig. 5.39. Fig. 5.39 shows the local buckling of steel plates in rectangular CFST stub columns under axial compression in which it is assumed that the unloaded edges of the steel plate are elastically restricted against rotation, and the loaded edges are clamped. It can be observed from Fig. 5.39 that the failure mechanism of the steel plate is outward buckling since the infilled concrete restricts the inward buckling of the tube. The steel plate is under uniform uniaxial compression in the x-direction in which a half buckling wavelength equal to a is developed in the x-direction. Therefore, the boundary conditions of the steel plate with the given boundary and loading conditions can be expressed as: At x ¼ 0 or x ¼ a: w¼

vw ¼0 vx

(5.292)

At y ¼ 0 or y ¼ b:

My ¼

w¼0 8 v2 u vw > > > Dr 2 ¼ zr > < vy vy > > v2 u vw > > : Dr 2 ¼ þzr vy vy

(5.293) for y ¼ 0 (5.294) for y ¼ b

708

5. Design rules and standards

Besides, w  0. zr in Eq. (5.294) stands for the rotational rigidity of the elastic restraint at the unloaded edges. Suppose that the deformation functions of the steel plate in the x and y directions are a cosine function and a biquadratic functional, respectively, the deformed shape of the steel plate is given by:  

y 2

y

y2 y y y y 2 y2 þ a4 1  þ a3 1 w ¼ C a 1 1  þ a2 1 b b b b b b b b   2px 1  cos a (5.295) Both the boundary conditions and the requirement of compatibility are satisfied by Eq. (5.295). By replacing the derivative function of Eq. (5.295) with respect to y into Eqs. (5.293) and (5.294), it can be proved that coefficient a2 is equal to coefficient a3 . Therefore, Eq. (5.295) can be rewritten as:   

y2 y y y 2 2px 1  cos (5.296) 1 w¼C 1 þm b b b b a in which the coefficient m can be derived by replacing the first-order partial derivative and the second-order partial derivative of Eq. (5.296) with respect to y into Eqs. (5.293) and (5.294), as follows: m¼cþ 1

(5.297)

in which the term c denotes the elastically restraining factor of unloaded edges as is governed by: c¼

zr b 2Dr

Eq. (5.296) can be simplified as:  

y2

y3

y4  y 2px 1  cos þ 42 þ 43 w ¼ C þ 41 b b b b a

(5.298)

(5.299)

in which C is a constant, and coefficients f1 , f2 , and f3 are governed by: 8 > > < 41 ¼ c 42 ¼ 2ðc þ 1Þ (5.300) > > : 43 ¼ c þ 1

5.7 Further discussion on local buckling

709

The total potential energy of the plate P can therefore be represented as: P ¼ U þ UG þ V

(5.301)

in which U and V are respectively governed by Eqs. (5.288) and (5.289). UG represents the energy associated with the elastic restraint over the unloaded edges and can be expressed as:  2 # Z "  2 zr vw vw dG (5.302) þ UG ¼ 2 vy y¼0 vy y¼b G

Eqs. (5.288)e(5.294), and Eqs. (5.299), (5.301), and (5.302) lead to: " # Dr 8p4 C2 b 3C2 a 2p2 C2 4p2 C2 P¼ A3  A4 A1 þ 3 A2 þ 2ð1  ns Þ 2 ab ab a3 2b (5.303)   3C2 zr a 1 þ A25 px tp2 C2 b þ  A1 a 4b2 in which A1 , A2 , A3 , A4 , and A5 are governed by: 8 1 4 42 þ 242 43 þ 41 42 422 þ 241 43 42 43 423 > A1 ¼ þ 1 þ 1 þ þ þ þ > > 5 3 7 4 9 3 2 > > > > > > 144 2 > 2 2 > > > A2 ¼ 441 þ 1242 þ 1241 42 þ 5 43 þ 1641 43 þ 3642 43 > < A3 ¼ 1 þ 341 þ 2421 þ 442 þ 543 þ 541 42 þ 641 43 þ 742 43 þ 3422 þ 4423 > > > > > > > > 242 þ 642 1441 43 þ 6422 12423 > > þ 343 þ 241 42 þ þ 342 43 þ > A 4 ¼ 41 þ 1 > > 3 5 7 > : A5 ¼ 1 þ 241 þ 342 þ 443 (5.304) Based on the minimum potential energy principle: vP ¼0 vC

(5.305)

Eqs. (5.303)e(5.305) lead to: "  # p2 Dr 4 3g2 A2 2ð1  ns ÞA3  2A4 3cg2 1 þ A25 p2 D r þ 4 þ þ ¼k 2 px t ¼ 2 2 2 4 g 2p A1 b 4p A1 p A1 b (5.306)

710

5. Design rules and standards

where the term g ¼ ba represents the half-wave parameter, and k is the elastic local buckling coefficient of the steel plate and can be expressed as:   4 3g2 A2 2ð1  ns ÞA3  2A4 3cg2 1 þ A25 þ (5.307) k¼ 2þ 4 þ g 2p4 A1 4p A1 p 2 A1 The minimum amount of k is shown by kcr and can be derived by vk vk ¼ 0. Replacing Eq. (5.307) into ¼ 0 leads to: taking vg vg " #1 4 A1 gcr ¼ 2p (5.308)   3A2 þ 6c 1 þ A25 Hence, kcr can be derived by replacing Eq. (5.308) into Eq. (5.307). The critical local buckling stress of the steel plates with the clamped boundary conditions at the loaded edges and elastically restrained against rotation at the unloaded edges is: scr ¼

kcr p2 Es    b 2  12 1  w2s t

(5.309)

In steel plates with clamped boundary conditions at the loaded edges and simply supported boundary conditions at the unloaded edges, the rotational rigidity of the elastic restraint zr is 0. Hence, c ¼ 0 and gcr ¼ 1:52 based on Eq. (5.308). Relacing gcr ¼ 1:52 into Eq. (5.307) results in kcr ¼ 5:46. Therefore: scr ¼ 5:46

p2 Es     b 2 2 12 1  ws t

(5.310)

In the steel plates with the clamped boundary conditions at the loaded and unloaded edges, the rotational rigidity of the elastic restraint zr / N. Therefore, c/N and gcr ¼ 1:0 based on Eq. (5.308). Replacing gcr ¼ 1:0 into Eq. (5.307) results in kcr ¼ 10:32 which is almost similar to kcr ¼ 10:31, recommended by Uy and Bradford [84]. Therefore: scr ¼ 10:32

p2 Es     b 2 2 12 1  ws t

(5.311)

711

5.7 Further discussion on local buckling

Based on Ref. [81], the restraining factor c for the steel plates of hollow sections without concrete core is governed by: !3 tw r (5.312) c¼ tf r tf bw tw bf

r¼1 

1 pbw r ¼  tanh 4bf p

!

!2

2 6 6 61 þ 4

(5.313) pbw 2bf sinh

! 3

7 !7 7 pbw 5

(5.314)

2bf

tb

in which tfw bwf
1, tf and bf are the thickness and the width of the calculated steel plate, respectively, and tw and bw are the thickness and the width of the adjacent steel plate, respectively. According to Eqs. (5.313) and (5.312), the values of r and c for the bare steel tube with a uniform thickness are 0. In this case, global buckling happens, and there is no trace of restraint between the adjacent steel plates. However, the previous buckling mechanism does not happen in the steel plates of rectangular CFST columns. Therefore, it is required to modify Eq. (5.312) for the steel plate of rectangular CFST columns as follows: !3 atw r0 (5.315) c¼ tf r r0 ¼ r þ 1

(5.316)

in which a ¼ 0:8 represents the modification factor for calculating the adjacent steel plate thickness, and it is employed to take into account the advantageous influences of restraint provided by the infilled concrete. Cai and Long [95] clarified that the assumption that the unloaded edges of the steel plate are elastically restrained against rotation and the loaded edges have the clamped boundary conditions is more reasonable than considering the fully clamped boundary conditions for all the edges of the steel plates. Therefore, Eq. (5.309) leads to more accurate results than Eqs. (5.310) and (5.311).

712

5. Design rules and standards

5.7.1 Local buckling of steel plates in rectangular CFST columns with binding bars under axial compression As discussed in Chapter 2, the local buckling in steel plates of square CFST columns with binding bars occurs at the spacing within two adjacent rows of level binding bars. Fig. 5.40 shows the typical failure mechanism of steel plates in rectangular CFST columns with binding bars. Also, the lateral displacement at the areas where binding bars are arranged is remarkably less than that at the peak of the half buckling wave. Therefore, it can be assumed that the lateral displacement at the locations where binding bars are arranged equals zero, leading to a half buckling wavelength in the x-direction a equals to the spacing of binding bars in the longitudinal direction bs , that results in: g¼

bs b

(5.317)

The critical local buckling stress of the steel plates in rectangular CFST columns with binding bars can be predicted using Eq. (5.309) in which the critical elastic local buckling coefficient kcr is governed by Eq. (5.307), and the elastically restraining factor of unloaded edges c is given by Eq. (5.315). Fig. 5.41 shows the relationship between k, g, and c. It can be observed from the figure that the local buckling coefficient k is increased remarkably by reducing the half-wave parameter g. The relationship between scr , k, and the b=t ratio is shown in Fig. 5.42. It is evident from the figure that the critical elastic local buckling stress scr is declined significantly by increasing the b=t ratio. It is noteworthy that the values obtained by Eq. (5.36) are the elastic critical local buckling stress. When scr > sy , no elastic buckling happens in the steel tube before it yields. Based on the experimental test results [96,97], local buckling of rectangular CFST columns with binding bars typically happens when the column achieves the

FIGURE 5.40 Local buckling of steel plates in rectangular CFST columns with binding bars [95].

713

5.7 Further discussion on local buckling

50 γ= 1.0

40

γ= 0.7 γ= 0.5

30

k

γ= 0.3 20 10 0 0

20

40

60

80

100

x

FIGURE 5.41

Relationship between k, c, and g.

6000

σcr (MPa)

5000 4000

k=10 k=15

3000

k=20

2000

k=30

1000

k=40

0 0

50

100 150 200 b/t FIGURE 5.42 Relationship between scr , k, and the B=t ratio. Cai J, Long Y-L. Local buckling of steel plates in rectangular CFTcolumns with binding bars. J Constr Steel Res 2009;65(4):965e72.

ultimate axial strength or during the postpeak region. Also, the elastic local buckling strengths scr calculated using the model are typically larger than test results sol , indicating that the rectangular or square steel tubes yield before the occurrence of buckling.

5.7.2 Elastoplastic local buckling of steel plates in rectangular CFST columns under axial compression The model for investigating the elastic local buckling of steel plates [95] can be extended to the elastoplastic phase [94]. The reduction coefficient h recommended by Bleich [81] for the bending stiffness of the steel plate over the loading direction during the elastoplastic phase can be employed to reduce the torsional stiffness in the loading direction. h can be

714

5. Design rules and standards

determined based on the stressestrain relationship model of the steel plate as follows: h¼

Et Es

(5.318)

where Et stands for the tangent modulus of steel plate, and Es denotes the elastic modulus of steel plate. Therefore, the differential equation governing the buckling of a thin steel plate during the elastoplastic step can be expressed as: ! v4 u v4 u v2 u v2 u v2 u pffiffiffi v4 u þ N þ þ 2N (5.319) ¼ N Dr h 4 þ 2 h x xy y vxvy vx vx2 vy2 vx4 vx2 vy2 in which the term Dr is the flexural rigidity of the steel plate and is governed by Eq. (5.258). The energy associated with the thin plate deforming U can be expressed based on the classical theory of elastic stability [80], as follows: !2 !2 Z Z ( Dr v2 u v2 u pffiffiffi v2 u v2 u h þ2 h þ U¼ 2 vxvy vx2 vx2 vy2 (5.320) " !2 #) v2 u v2 u v2 u pffiffiffi  2 h ð1  ns Þ  dxdy vxvy vx2 vy2 The energy associated with the applied load V is given by Eq. (5.289). Besides, the boundary stress of the steel plate is similar to those determined by Eqs. (5.290) and (5.291). Like the previous model, it is supposed that the loaded edges have the clamped boundary conditions, and the unloaded edges of the plate are elastically restricted against rotation, as shown in Fig. 5.39. Hence, Eqs. (5.292)e(5.294) are used to develop the buckling model of the steel plate in the elastoplastic phase. Likewise, the deformed shape of the steel plate can be expressed using Eqs. (5.298) e(5.300). The total potential energy of the plate P is governed by: # " Dr 8p4 C2 b 3C2 a 2p2 C2 pffiffiffi pffiffiffi 4p2 C2 A3  h A4 A1 þ 3 A2 þ 2 h ð1  ns Þ h P¼ 2 ab ab a3 2b   3C2 zr a 1 þ A25 px tp2 C2 b þ  A1 2 a 4b (5.321) in which coefficients A1 , A2 , A3 , A4 , and A5 can be obtained by Eq. (5.304).

5.7 Further discussion on local buckling

715

Based on the minimum potential energy principle vP vC ¼ 0. Using Eqs. vP (5.321) and (5.304) and by taking vC ¼ 0:  # pffiffiffi 2 " hp Dr pffiffiffi 4 3g2 A2 2ð1  ns ÞA3  2A4 3cg2 1 þ A25 þ h 2 þ pffiffiffi 4 þ px t ¼ pffiffiffi g 2 hp4 A1 b2 4 hp A1 p2 A1 pffiffiffi 2 hp Dr ¼k b2 (5.322) in which g ¼ ba represents the half-wave parameter, and k is the elastic local buckling coefficient of the steel plate and can be expressed as:   2ð1  ns ÞA3  2A4 pffiffiffi 4 3g2 A2 þ 6cg2 1 þ A25 þ (5.323) k¼ h 2 þ pffiffiffi g 4p4 A1 h p2 A 1 vk ¼ 0 leads to: vg " #1 4 hA1 gcr ¼ 2p   3A2 þ 6c 1 þ A25

Replacing Eq. (5.323) into

(5.324)

The amount of kcr can be determined by replacing Eq. (5.324) into Eq. (5.323), as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   3A2 þ 6c 1 þ A25 2ð1  ns ÞA3  2A4 pffiffiffiffiffiffi kcr ¼ þ 2 p2 A1 p A1   (5.325) 3A2 þ 6c 1 þ A25 ffi þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    p2 A1 3A2 þ 6c 1 þ A25 The critical local buckling stress of the steel plate can be obtained based on Eqs. (5.258), (5.291), and (5.322), as follows: pffiffiffi kcr p2 hEs (5.326) scr ¼     b 2 2 12 1  ws t

716

5. Design rules and standards

Consequently, the elastoplastic critical local buckling stress of the steel plates in rectangular CFST columns subjected to axial compression can be derived by replacing Eq. (5.325) into Eq. (5.326), as follows:   (qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 3A2 þ 6c 1 þ A25 kcr p2 hEs 2ð1  ns ÞA3  2A4 pffiffiffiffiffiffi scr ¼ þ  2 ¼ 2 A p 2 A1 p   b 1 12 1  w2s t )   pffiffiffi 3A2 þ 6c 1 þ A25 p2 hEs þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     ffi    b 2 2 p2 A1 3A2 þ 6c 1 þ A25 12 1  ws t (5.327) In steel plates with clamped boundary conditions at the loaded edges and simply supported boundary conditions at the unloaded edges, c ¼ 0 and gcr ¼ 1:52. Therefore, the elastoplastic critical buckling stress can be expressed as: pffiffiffi p2 Es h (5.328) scr ¼ 5:46     b 2 2 12 1  ws t In the steel plate with the clamped boundary conditions at the loaded and unloaded edges, c/N and gcr ¼ 1:0. Therefore, the elastoplastic critical buckling stress can be expressed as: pffiffiffi p 2 Es h (5.329) scr ¼ 10:32     b 2 2 12 1  ws t In the elastic phase, the reduction coefficient h equals 1. Therefore, Eq. (5.326) is degraded as:   (qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3A2 þ 6c 1 þ A25 kcr p2 Es 2ð1  ns ÞA3  2A4 pffiffiffiffiffiffi scr ¼ þ  2 ¼ 2 p 2 A1 p A1   b 12 1  w2s t )   3A2 þ 6c 1 þ A25 p2 Es ffi þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi        b 2 p2 A1 3A2 þ 6c 1 þ A25 12 1  w2s t (5.330)

717

5.7 Further discussion on local buckling

5.7.3 A refined model for local buckling of steel plates in rectangular CFST columns under axial compression The model discussed in Sections 5.7 and 5.7.1 for local buckling of steel plates with binding bars only considers the longitudinal spacing of binding bars bs . In addition to the value of bs , the local buckling behavior of steel plates in rectangular CFST columns with binding bars depends on the horizontal spacing as well as the diameter of binding bars. Fig. 5.43 displays the effectively and ineffectively confined areas of the concrete core in rectangular CFST columns with binding bars. Besides, the failure mechanism of the steel plate is shown in Fig. 5.44. As shown in Fig. 5.44, for ease of calculation, the effectively confined area in the plate is taken as a diameter of the binding bar ðds Þ distance from the center of binding bars

Binding bars Effectively confined area Ineffectively confined area Boundary region

FIGURE 5.43 Effectively and ineffectively confined areas of the infilled concrete in rectangular CFST columns with binding bars [94].

718

5. Design rules and standards

Effectively confined area Ineffectively confined area Local buckling mode of effectively confined area Binding bar Local buckling mode of ineffectively confined area Elastically restrain against rotation Clamped

FIGURE 5.44

Local buckling modes of effectively and ineffectively confined areas of the

steel plate [94].

in the horizontal direction. Therefore, the steel plate located in the effectively confined area can only experience outward buckling in the range of bs  2ds , i.e., the local buckling mode of the plate located in the effectively confined areas is different from that of the plate in CFST columns without binding bars. If the horizontal spacing of binding bars as is long enough ðas  2ds Þ, other areas of the steel plate except for the effectively confined areas can be considered as the ineffectively confined area, as depicted in Fig. 5.44. The failure mode of the plate located in the ineffectively confined area is similar to the steel plate in the conventional CFST columns without binding bars. The empirical critical local buckling coefficient kcr is introduced [94] to assess the local buckling behavior of the steel plate in rectangular CFST columns with binding bars by considering the difference between the

5.7 Further discussion on local buckling

719

local buckling modes of effectively and ineffectively confined areas of the steel plate. kcr can be expressed as: Pm Pm1 i¼1 As1i kcr1 þ j¼1 As2j kcr2 þ 2As3 kcr3 kcr ¼ (5.331) As in which As is the cross-sectional area of the steel plate, As1i ad As2j are cross-sectional areas of the effectively confined area and ineffectively confined area of the steel plate, respectively, and As3 is the cross-sectional area of the ineffectively confined areas next to the unloaded edge of the plate (i.e., boundary regions of the plate). kcr1 represents the critical local buckling coefficient of the plate located in the effectively confined area, kcr2 denotes the critical local buckling coefficient of the plate located in the ineffectively confined area, kcr3 is the critical local buckling coefficient of the steel plate in the boundary region. The critical local buckling stress scr can be expressed as: pffiffiffi kcr p2 hEs scr ¼ (5.332)     b 2 2 12 1  ws t As mentioned above, the local buckling happens within the range of bs  2ds in the longitudinal direction of the effectively confined area in the steel plate. The local buckling modes of the ineffectively confined area of the steel plate and the steel plate in CFST columns without binding bars are the same. Additionally, binding bars can highly enhance the stiffness of the steel plate located in the effectively confined area. Therefore, it can be assumed that unloaded edges of steel plates located in the effectively confined area are simply supported, as shown in Fig. 5.45a. This is because the adjacent steel plates (i.e., the steel plate located in the ineffectively confined area) have a relatively small stiffness and are unable to restrict the steel plate located in the effectively confined area. Likewise, it can be assumed that unloaded edges of the steel plate located in the ineffectively confined area have clamped boundary conditions, as shown in Fig. 5.45b. This is because the adjacent steel plates (i.e., the steel plate located in the effectively confined area) have a relatively large stiffness. For the sake of simplicity, it can be supposed that the steel plate located in the ineffectively confined area next to the unloaded edge of the plate is elastically restricted against rotation at unloaded edges [95], as shown in Fig. 5.45c.

720

5. Design rules and standards

Clamped

Binding bar

Clamped

Simply supported

Elastically restrain against rotation

Elastically restrain against rotation

Clamped

Clamped

Clamped

(a) Effectively confined area (b) Ineffectively confined area (c) Boundary region Boundary conditions of the steel plate located in the effectively confined area, ineffectively confined area, and boundary region. (a) Effectively confined area (b) Ineffectively confined area (c) Boundary region.

FIGURE 5.45

The local buckling half-wave parameter g is governed by: a bs  2ds g¼ ¼ b b

(5.333)

The local buckling coefficient kcr1 can be determined using Eqs. (5.323) and (5.333). The critical local buckling coefficient kcr2 of the steel plate located in ineffectively confined areas, presented in Fig. 5.45b, is governed by Eq. (5.325) when c/N. The critical local buckling coefficient kcr3 of the steel plate located in the boundary regions, shown in Fig. 5.45c, is given by Eq. (5.325) and Eqs. (5.314)e(5.316). It must be mentioned that the model for the elastoplastic buckling of steel plates has been developed based on rectangular CFST columns with binding bars of a normal size [94]. Therefore, the model is reasonable for columns with B
1000 mm; D
1000 mm; 1:0
D=B
3:0; t  4 mm; ds  12 mm.

721

5.7 Further discussion on local buckling

5.7.4 Effects of different parameters on the local buckling strength of rectangular steel plates in CFST columns with binding bars 5.7.4.1 Effect of spacing between binding bars The critical longitudinal spacing ðbs Þ and the horizontal spacing ðas Þ of binding bars affect the local buckling stress ðscr Þ of the steel plate in rectangular CFST columns with binding bars, as displayed in Fig. 5.46.

500

(MPa)

400 300 200 R1 (as=D/2, bs=D/2) R2 (as=D/2, bs=D/4) R3 (as=D/2, bs=D/5) R4 (as=D/4, bs=D/2) R5 (as=D/4, bs=D/2)

100 0 0

25

50

75

100 D/t

125

150

175

200

(a) 500

(MPa)

400 300 200

R1 (as=D/2, bs=D/2) R2 (as=D/2, bs=D/4) R3 (as=D/2, bs=D/5) R4 (as=D/4, bs=D/2) R5 (as=D/4, bs=D/2)

100 0 0

25

50

75

100 D/t

125

150

175

200

(b) D Effect of bs and as on scr at different D B ratios (ds ¼ 0:02D). (a) B ¼ 1:0. (b) D ¼ 2:0. (c) D ¼ 3:0. Long Y-L, Zeng L. A refined model for local buckling of rectangular CFST B B columns with binding bars. Thin-Walled Struct 2018;132:431e41; Cai J, Long Y-L. Local buckling of steel plates in rectangular CFTcolumns with binding bars. J Constr Steel Res 2009;65(4):965e72.

FIGURE 5.46

722

5. Design rules and standards

500

(MPa)

400 300 200 R1 (as=D/2, bs=D/2) R2 (as=D/2, bs=D/4) R3 (as=D/2, bs=D/5) R4 (as=D/4, bs=D/2) R5 (as=D/4, bs=D/2)

100 0 0

25

50

75

100 D/t

125

150

175

200

(c)

FIGURE 5.46 continued.

The reduction of bs remarkably enhances the elastic critical local buckling stress. This issue can be found by comparing the critical buckling stresses of columns R1, R2, and R3. Additionally, the effect of bs on the critical local buckling stress is more prominent than that of as . 5.7.4.2 Effect of the diameter of binding bars The effect of binding bar diameter ðds Þ on the steel plate’s elastic critical local buckling stress in rectangular CFST columns with binding bars is shown in Fig. 5.47. It should be noted that the D=B ratio of all columns is 2.0 because the effect of the aspect ratio D=B on the elastic critical local buckling stress of the steel plate in rectangular CFST columns with binding bars is negligible. This issue is discussed in detail in the next section. It can be observed from Fig. 5.47a and b that the influence of ds on the critical local buckling stress when the spacing of binding bars is large can be ignored. By contrast, ds can affect the critical local buckling stress when the binding bar spacing is small, as shown in Fig. 5.47c and d. The increase in the diameter of binding bars when the spacing between them is small modestly enhances the steel plate’s critical local buckling stress. Reducing longitudinal spacing to postpone the occurrence of local buckling of steel plates can be considered an efficient method with priority over other methods.

723

5.7 Further discussion on local buckling

500 ds=0.014D 400

ds=0.016D ds=0.018D ds=0.020D

(MPa)

300 200 100 0 0

50

100

150 D/t

200

250

300

(a) 500 ds=0.014D ds=0.016D

400

(MPa)

ds=0.018D ds=0.020D

300 200 100 0 0

50

100

150 D/t

200

250

300

FIGURE 5.47 Effect of ds on scr at different spacings of binding bars. (a) as ¼ bs ¼ D2 . (b) as ¼ bs ¼ D3 . (c) as ¼ bs ¼ D4 . (d) as ¼ bs ¼ D5 . Long Y-L, Zeng L. A refined model for local buckling of rectangular CFST columns with binding bars. Thin-Walled Struct 2018;132:431e41; Cai J, Long Y-L. Local buckling of steel plates in rectangular CFTcolumns with binding bars. J Constr Steel Res 2009;65(4):965e72.

724

5. Design rules and standards

500 ds=0.014D ds=0.016D

400

(MPa)

ds=0.018D ds=0.020D

300

200

100

0 0

25

50

75 100 125 150 175 200 225 250 275 300 D/t (c)

500 ds=0.014D 400

ds=0.016D

(MPa)

ds=0.018D ds=0.020D

300

200

100

0 0

25

50

75 100 125 150 175 200 225 250 275 300 D/t (d)

FIGURE 5.47 continued.

5.7.4.3 Effect of the D=B ratio Fig. 5.48 shows the effect of the cross-sectional aspect ratios ðD =BÞ on the elastic critical local buckling stress of the steel plate in rectangular CFST columns with binding bars. It can be observed from the figure that the D=B ratio has negligible effects on the elastic critical local buckling stress and can be ignored in the design of rectangular CFST columns with binding bars.

725

5.7 Further discussion on local buckling

500 D/B=1 400

D/B=2 D/B=3

(MPa)

300

200

100

0 0

50

100

150 D/t

200

250

300

(a) 500 D/B=1

(MPa)

400

D/B=2 D/B=3

300

200

100

0 0

50

100

150 D/t

200

250

300

(b)

Effect of the D B ratio on scr at different spacings of binding bars ðas ¼ bs Þ. (a) as ¼ bs ¼ D2 . (b) as ¼ bs ¼ D3 . (c) as ¼ bs ¼ D4 . (d) as ¼ bs ¼ D5 . Long Y-L, Zeng L. A refined model for local buckling of rectangular CFST columns with binding bars. Thin-Walled Struct 2018;132:431e41; Cai J, Long Y-L. Local buckling of steel plates in rectangular CFTcolumns with binding bars. J Constr Steel Res 2009;65(4):965e72.

FIGURE 5.48

726

5. Design rules and standards

500 D/B=1

(MPa)

400

D/B=2 D/B=3

300

200

100

0 0

25

50

75 100 125 150 175 200 225 250 275 300 D/t (c)

500 D/B=1

(MPa)

400

D/B=2 D/B=3

300

200

100

0 0

25

50

75 100 125 150 175 200 225 250 275 300 D/t (d)

FIGURE 5.48 continued.

5.7.4.4 Further discussion on the effect of the b=t ratio and the D=B ratio on the buckling strength of rectangular steel plates in CFST columns with binding bars The broad faces of rectangular CFST/CFDST columns are susceptible to extensive buckling. Consequently, the confinement effect provided by the broad faces is weaker than the narrow faces. Fig. 5.49 shows the relationship between scr , the b=t ratio, and the aspect ratio D=B. As shown in Fig. 5.49, the critical buckling stresses of broad faces with a larger D= B ratio are smaller than those of narrow sides. Hence, the required number of binding bars for improving the buckling strength of broad faces is more

727

5.7 Further discussion on local buckling

2000 Narrow faces Broad faces (D/B=1.5)

1600 σcr (MPa)

Broad faces (D/B=2.0)

1200

Broad faces (D/B=2.5) Broad faces (D/B=3.0)

800 400 0 0

20

40

60

80

100

b/t FIGURE 5.49 Relationship between scr of narrow faces, the B=t ratio, and the D= B ratio for CFST columns without binding bars. Long Y-L, Zeng L. A refined model for local buckling of rectangular CFST columns with binding bars. Thin-Walled Struct 2018;132:431e41; Cai J, Long Y-L. Local buckling of steel plates in rectangular CFTcolumns with binding bars. J Constr Steel Res 2009;65(4):965e72.

than narrow faces. Appropriate values of aspect ratios D=B can be determined by arranging binding bars with reasonable longitudinal spacing bs so that the local buckling strengths of broad and narrow faces are greater than their yield strengths. Binding bars are mostly used on the broad faces to enhance their buckling strength and prevent premature buckling before they yield. The relationship between critical elastic local buckling stresses scr of broad and narrow faces with the B=t ratio corresponding to different aspect ratios D=B is presented in Fig. 5.50. It can be observed from Fig. 5.50 that connecting the broad faces by binding bars remarkably increases their critical buckling stresses, and their critical buckling stresses can even become greater than the critical buckling stresses of narrow faces of steel tubes. Based on Fig. 5.50, the premature buckling of the mild structural steel that is typically employed for the fabrication of composite columns can be prevented by setting appropriate values for the B=t ratio and binding bar longitudinal spacing with respect to the D=B ratio.

5.7.5 Design recommendation of steel plates in rectangular CFST columns with binding bars The local buckling coefficient k is increased by reducing the half-wave parameter g (reducing the longitudinal spacing between binding bars), as shown in Fig. 5.41. However, when the predicted elastic local buckling strength of steel tube scr exceeds the steel tube yield strength sy , the local buckling strength is equal to the steel tube yield strength, leading to the smaller limit of bs . For square CFST columns with binding bars, it can be considered that bw ¼ bf and tw ¼ tf . Hence, from Eqs. (5.314)e(5.316), the restraining factor of unloaded edges c will be equal to 1.38. The relationship between scr and the b=t ratio for square CFST stub columns with

728

5. Design rules and standards

1000 900 800 σcr (MPa)

700 600 500 400

Narrow faces (no binding bars)

300

Broad faces (bs=D/2)

200

Broad faces (bs=D/3)

100

Broad faces (bs=D/4)

0 0

10

20

30

40

50

60

70

80

90

60

70

80

90

b/t (a) 600

σcr (MPa)

500 400 300 Narrow faces (no binding bars)

200

Broad faces (bs=D/2) Broad faces (bs=D/3)

100

Broad faces (bs=D/4)

0 0

10

20

30

40

50 b/t

(b)

FIGURE 5.50 Relationship between scr of broad faces, the B=t ratio, and D= B ratio for D D D CFST columns with binding bars. (a) D B ¼ 1:5. (b) B ¼ 2:0. (c) B ¼ 2:5. (d) B ¼ 3:0. Long Y-L, Zeng L. A refined model for local buckling of rectangular CFST columns with binding bars. Thin-Walled Struct 2018;132:431e41; Cai J, Long Y-L. Local buckling of steel plates in rectangular CFTcolumns with binding bars. J Constr Steel Res 2009;65(4):965e72.

729

5.7 Further discussion on local buckling

600 500

σcr (MPa)

400 300 200

Narrow faces (no binding bars) Broad faces (bs=D/2)

100

Broad faces (bs=D/3) Broad faces (bs=D/4)

0 0

10

20

30

40

50

60

70

80

90

70

80

b/t (c) 600 500

σcr (MPa)

400 300 200

Narrow faces (no binding bars) Broad faces (bs=D/2)

100

Broad faces (bs=D/3) Broad faces (bs=D/4)

0 0

10

20

30

40 b/t

50

60

(d)

FIGURE 5.50 Continued.

binding bars (c ¼ 1:38Þ is given in Fig. 5.51. It can be observed from the graph that the predicted elastic local buckling strength scr is larger than 450 MPa when bs ¼ 3b and bt
120. This issue indicates that the mild structural steel which is typically employed for the fabrication of CFST columns yields prior to buckling. The predicted elastic local buckling strengths are slightly enhanced compared with the case without binding bars when bs ¼ b. Therefore, the lower limit of bs can be taken as bs ¼ 3b,

730

5. Design rules and standards

500 450 400 σcr (MPa)

350 300 250 No binding bars bs=b bs=0.7b bs=0.5b bs=0.33b bs=0.25b

200 150 100 50 0 0

50

100

150

200

b/t FIGURE 5.51 Relationship between scr , bs , and the B=t ratio ðc ¼ 1:38Þ. Long Y-L, Zeng L. A refined model for local buckling of rectangular CFST columns with binding bars. Thin-Walled Struct 2018;132:431e41; Cai J, Long Y-L. Local buckling of steel plates in rectangular CFTcolumns with binding bars. J Constr Steel Res 2009;65(4):965e72.

while the upper limit of bs can be taken as bs. Additionally, it is recommended to take bs as 3b or 2b for the design of square CFT columns with binding bars in practice since the predicted buckling strengths improve remarkably as bs ¼ 3b or bs ¼ 2b compared with the cases without binding bars [95], as shown in Fig. 5.51. Based on the results shown in Fig. 5.51, the maximum b=t ratios for the given bs values to predict the premature buckling of the mild structural steel tube before yielding can be summarized in Table 5.33. The simultaneous local buckling of the steel plate occurs when the elastic critical local buckling stresses of the calculated steel plate scr1 and the adjacent steel plate scr2 are equal ðscr1 ¼ scr2 Þ. Replacing scr1 ¼ scr2 corresponding bs , as , and ds into Eq. (5.326) leads to the rational relationship between the wall thicknesses of the calculated steel plate t1 and the adjacent steel plate t2 . For example, the relationship between t1 and t2 in square CFST columns with binding bars when ds ¼ 0:02D is given in Table 5.34.  TABLE 5.33 Limitations for the B=t ratio based on the bs values fy ¼ 450 MPa). Longitudinal spacing of binding bars bs

The b=t ratio limitation

0:7b

b=t
65

0:5b

b=t
80

b 3 b 4

b=t
125 b=t
160

731

5.7 Further discussion on local buckling

TABLE 5.34 Relationship between t1 and t2 in square CFST columns with binding bars at different bars spacing when ds ¼ 0:02D. bs as

D=2

D=3

D=4

D=5

D= 2

t1 ¼ 0:74t2

t1 ¼ 0:64t2

t1 ¼ 0:55t2

t1 ¼ 0:47t2

D= 3

t1 ¼ 0:70t2

t1 ¼ 0:59t2

t1 ¼ 0:49t2

t1 ¼ 0:42t2

D= 4

t1 ¼ 0:68t2

t1 ¼ 0:55t2

t1 ¼ 0:45t2

t1 ¼ 0:38t2

D= 5

t1 ¼ 0:66t2

t1 ¼ 0:52t2

t1 ¼ 0:42t2

t1 ¼ 0:35t2

Based on Fig. 5.50, the premature buckling of the mild structural steel that is typically employed for the fabrication of composite columns can be prevented by setting appropriate values for the B=t ratio and binding bar longitudinal spacing with respect to the D=B ratio. Table 5.35 summarizes ranges of B=t ratios and bs for different values of D=B ratios. For example, it can be assured that the critical buckling stresses of broad and narrow faces are larger than 380 and 400 MPa, respectively, when D B ¼ 2:5, B= t
55, and bs
D3 , i.e., the mild structural steel tube yields before the occurrence of buckling. Normally, mild structural steel is used for the fabrication of steelconcrete composite members in which the yield stress does not exceed 450 MPa. The premature local buckling of steel plates before they yield can be assured if scr1 and scr2 be greater than fy ¼ 450 MPa. Replacing scr1 ¼ scr2 ¼ 450 MPa corresponding bs , as , and ds into Eq. (5.326) gives B reasonable relationships between the D t1 ratio and the t2 ratio at different longitudinal spacing and horizontal spacing between binding bars. For B example, the relationship between D t1 ratio and the t2 in square CFST columns with binding bars when ds ¼ 0:02D is given in Table 5.36.

TABLE 5.35 Recommendation for ranges of b=t ratios based on the bs values. scr (MPa) D= B

B=t

bs

Wide faces

Narrow faces

1.5


55


D=2

450

400

2.0


55


D=3

550

400

2.5


55


D=3

380

400

3.0


55


D=4

430

400

732

5. Design rules and standards

TABLE 5.36

B Relationship between D t1 ratio and the t2 in square CFST columns with binding bars at different as and bs when ds ¼ 0:02D.

bs as

D=2

D=3

D=4

D=5

D= 2

B=t2
47

B=t2
46

B=t2
45

B=t2
44

D=t1
64

D=t1
71

D=t1
81

D=t1
94

B=t2
47

B=t2
45

B=t2
44

B=t2
44

D=t1
67

D=t1
76

D=t1
90

D=t1
106

B=t2
46

B=t2
45

B=t2
44

B=t2
44

D=t1
68

D=t1
81

D=t1
97

D=t1
117

B=t2
46

B=t2
44

B=t2
44

B=t2
44

D=t1
70

D=t1
85

D=t1
104

D=t1
127

D= 3

D= 4

D= 5

5.8 Compressive strength of CFST stub columns with stiffeners It was discussed in Chapter 2 that a possible solution for improving the buckling strength of steel tubes, and therefore, the confinement effect and the mechanical performance of steel-concrete composite members, is to use stiffeners. This section discusses models for predicting the ultimate axial strength of CFST/CFDST stub columns with stiffeners under axial compression.

5.8.1 Square CFST stub columns with PBL The cross section of the square CFST columns with PBL is presented in Fig. 5.52, in which B and t are the width and the wall thickness of the square steel tube, respectively, ws and ts are the width and the thickness of PBL stiffeners, respectively, and d and y are the diameter and the spacing of the holes on PBL stiffeners, respectively. The concrete core can be divided into enhanced (confined) and nonenhanced (unconfined) parts, as shown in Fig. 5.53. q in Fig. 5.53 is the corner cut and is governed by Ref. [98]: 3:6tfy tan q ¼  2 B t t

(5.334)

733

5.8 Compressive strength of CFST stub columns

FIGURE 5.52 The cross section of square CFST stub columns with PBL.

Unconfined area

θ

Confined area

FIGURE 5.53 Different zones of concrete core in square CFST stub columns with PBL.

As shown in Fig. 5.53, the cross-sectional area of the concrete core consists of the effective area ðAe Þ and ineffective area ðAi Þ. The crosssectional area of the concrete core ðAc Þ is given by: Ac ¼ ðB  2tÞ2  4ws ts

(5.335)

734

5. Design rules and standards

The length of ineffective area is 0:4ðB 2t ts Þ. Hence: Ai ¼ 8

½0:4ðB  2t  ts Þ2 tan q 6 Ae ¼ Ac  Ai

(5.336) (5.337)

The confinement effectiveness coefficient ðke Þ is defined as the ratio of the area of the effectively confined section Ae to the cross-sectional area of the concrete core Ac : ke ¼

1:2tfy ðB  2t  ts Þ2 Ae A ¼1 i ¼1   2 Ac Ac  B t ðB  2tÞ2  4ws ts 2

(5.338)

ke converts the lateral confining pressure uniformly distributed on the surface of the infilled concrete into the effective lateral confining pressure. The free-body diagram of a square steel tube is shown in Fig. 5.54, in which ssh is the hoop stress of the steel tube, fl is the lateral confining stress from the concrete core, and Qt is the force in the longitudinal stiffeners. The equilibrium of forces acting on the steel tube and PBL leads to the following equation: 2ssh tH þ nQt ¼ fl ðB  2t  ts ÞH

(5.339)

in which H is the height of the steel tube, and n denotes the number of qffiffiffiffi holes on PBL stiffeners. Qt can be taken as 1:95A f 0c [99], in which A is the area of the holes on PBL stiffeners. Based on Ref. [100], the width of stiffeners bs and the diameter of the holes d affect the behavior of stiffeners, and the optimum case can be

FIGURE 5.54

Internal forces in a square CFST stub column with PBL stiffeners.

735

5.8 Compressive strength of CFST stub columns

achieved when d ¼ w2s . By defining the impact factor g ¼ 1  jwsw2dj , s Eq. (5.339) can be rewritten as: qffiffiffiffi   g 1:95nA f 0c 2ssh t þ H (5.340) fl ¼ B  2t  ts Consequently, the radial stress acting on the infilled concrete is: fl0 ¼ ke fl

(5.341)

The infilled concrete is under the three-dimensional stress state. As a result, sc;3 < sc;1 ¼ sc;2 < 0, in which sc;1 , sc;2 , sc;3 are principal stresses. Assuming sc;1 ¼ sc;2 ¼ fr0 , the axial compressive stress of the infilled concrete sc;3 can be derived using the unified strength theory as follows: sc;3 ¼ fc0 þ ks1

(5.342)

fÞ in which the coefficient k is given by 1:0
k ¼ ð1þsin
7:0, where the ð1sin fÞ

term f is the friction of angle of concrete. Accordingly, the axial compressive strength of confined concrete (effective area in Fig. 5.53) is governed by: fce ¼ fc0 þ kfl0

(5.343)

Consequently, the axial compressive capacity of effectively confined concrete is governed by:    (5.344) Nce ¼ Ae fce ¼ ke ðB  2tÞ2  4ws ts fc0 þ kfl0 The axial compressive strength of the unconfined area of concrete can be derived by incorporating the strength reduction factor gc to consider reduction of the constraints, as follows: fci ¼ fc0 þ gc kfl0

(5.345)

Models for calculating the strength reduction factor gc can be found in Chapter 2. Therefore, the compressive strength of the unconfined area of the concrete core is given by:    (5.346) Nci ¼ Ai fci ¼ ð1  ke Þ ðB  2tÞ2  4ws ts fc0 þ gc kfl0 Finally, the compressive capacity of square CFST stub columns stiffened with PBLs under axial compression can be derived using the superposition principle [98], as follows:

 (5.347) Nu ¼ Nce þ Nci þ As fy þ fy;PBL

736

5. Design rules and standards

    d ts ws is the total cross-sectional area of where As ¼ 4 t B t þ 1  bs the steel tube and PBL stiffeners, fy is the yield stress of the steel tube, and fy;PBL is the yield stress of PBL stiffeners.

5.8.2 Square CFST stub columns with inclined stiffener ribs The ultimate axial strength of square CFST stub columns with inclined plates can be predicted using the following expression [101,102]: Nu ¼ Asc fscy þ As;s fy;s

(5.348)

in which As;s and fy;s are the cross-sectional area and yield strength of stiffeners, respectively. As;s is governed by: As;s ¼ n

ts ls ws bs

(5.349)

in which n is the number of faces on which the stiffeners are placed, ts is the thickness of stiffeners, ws is the width of stiffeners, and bs is the spacing distance between the stiffeners along the specimen axis direction. Asc in Eq. (5.349) is taken as: Asc ¼ As þ Ac

(5.350)

fscy is the nominal average strength of the steel tube and concrete core and is governed by: fscy ¼ ð1:75 þ 2:85kÞfck

(5.351)

where k represents the coefficient of the axial compressive capacity of stiffened CFST stub column to take into account the effects of confinement coefficient, stiffened method, and effective concrete core area. Based on Ref. [103], k is given by: k ¼ xki ka

(5.352)

in which ki ¼

n0:8 2

ka ¼ 0:1eke

(5.353.a) (5.353.b)

where n denotes the number of stiffened tube faces, and the confinement effectiveness coefficient ke can be derived based on the effective area of the concrete core, shown in Fig. 5.55.

737

5.8 Compressive strength of CFST stub columns

Effective area of the concrete core Effective areas of concrete core in square CFST columns with inclined Stiffeners

FIGURE 5.55 plates.

5.8.3 Square CFST stub columns with binding bars The ultimate axial strength of square CFST stub columns with binding bars under axial compression can be expressed as: Nu ¼ As fa þ Ac fcc0

(5.354)

in which As is the cross-sectional area of steel tube, fa is the strength of steel in the longitudinal direction, Ac is the cross-sectional area of the concrete core, and fcc is the longitudinal strength of the concrete core. For calculating fcc , Cai and He [96] modified the equation developed by Mander et al. [104] as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   f0 7:94f 0l 0 fcc ¼ fck  1:254 þ 2:254 1 þ (5.355) 2 l fck fck in which fl0 is equivalent lateral confined stress and is governed by: fl0 ¼ ke fl

(5.356)

where fl represents the lateral confining stress of the concrete core. As discussed in Chapter 2 and based on the free-body diagram, shown in Fig. 5.56, the following equation can be driven for defining fl : B  as As;s $ fy;s 2t as bs þ fl ¼ B=ð2tÞ  1 B=ð2tÞ  1 ssh

(5.357)

738

FIGURE 5.56

5. Design rules and standards

Internal forces in a square CFST stub column with binding bars.

where As;s is the cross-sectional area of binding bars, as and bs are the transverse and longitudinal spaces of binding bars, respectively, fy;s is the yield stress of the binding bars, and ssh is the steel hoop stress. The confinement effectiveness coefficient ke consists of the level confinement effectiveness coefficient ðke1 Þ and the longitudinal confinement coefficient ðke2 Þ, as follows: ke ¼ ke1 $ke2

(5.358)

Based on Ref. [104]: ke1 ¼

Ae1 Ac1

(5.359.a)

ke2 ¼

Ae2 Ac2

(5.359.b)

in which Ae1 represents the level area of an effectively confined concrete core, Ae2 is the longitudinal area of an effectively confined concrete core, Ac1 is the level area of the concrete core, and Ac2 is the longitudinal area of the concrete core. Effectively and ineffectively confined concrete core areas are presented in Fig. 5.57. The hoop reinforcement can only effectively induce the maximum transverse pressure on the part of the infilled concrete where the confining stress has fully developed owing to arching action. Based on Ref. [104], it is assumed that the arching action occurs in the form of a second-degree parabola having an initial tangent angle q at

5.8 Compressive strength of CFST stub columns

739

Ineffective area of the concrete core

θ

Effective area of the concrete core

FIGURE 5.57 Different zones of concrete core in square CFST stub columns with binding bars. Cai J, He Z-Q. Axial load behavior of square CFT stub column with binding bars. J Constr Steel Res 2006;62(5):472e83.

the cross section of the concrete core. Therefore, Ae1 and Ac1 can be expressed as follows: # "   2 tan q ðB  2tÞ B=a s n X (5.360) Ae1 ¼ ðB  2tÞ2  6 i¼1 Ac1 ¼ ðB  2tÞ2 Replacing Eqs. (5.360) and (5.361) into Eq. (5.359.a) leads to: " # n X ðas =BÞ2 tan q ke1 ¼ 1  6 i¼1

(5.361)

(5.362)

Likewise, it can be concluded that:   bs tan q 2 ke2 ¼ 1  2ðb  2tÞ

(5.363)

Consequently, the confinement effectiveness coefficient ke can be expressed as: " #  n X ðas =BÞ2 tan q bs tan q 2 (5.364) ke ¼ 1  1 6 2ðB  2tÞ i¼1

740

5. Design rules and standards

The magnitude of the initial tangent angle q depends on the level spacing of binding bars. Cai and He [96] recommended Eq. (5.365) to predict q:   p 9:2as ðradÞ. (5.365) 13 þ q¼ 100 180 As discussed in Chapter 2, square CFST stub columns with binding bars under axial compression typically fail in postlocal buckling mode. Therefore, the steel materials reach their yield stress by reaching the ultimate axial strength of the column. Based on the von Mises yield criteria: fa2  fa ssh þ s2sh ¼ s2y

(5.366)

The local buckling strength of a steel plate ðsb Þ in square CFST stub columns can be predicted using the model recommended by Ge and Usami [90] as follows: sb 1:2 0:3  2
1:0 ¼ sy R R

(5.367)

in which the parameter R is the width-to-thickness ratio governed by Eq. (5.263) by taking the elastic local buckling coefficient k ¼ 4:0. It should be noted that R must be greater than 0.85 to ensure that the requirement sb
sy is satisfied (see Eq. 5.367). According to Eq. (5.367), two scenarios can be determined for the stress states of square steel tubes as follows: If R  0:85: Taking sa ¼ sb in Eqs. (5.366) and (5.367) leads to:   1:2 0:3  2 sy ðin compressionÞ (5.368.a) fa ¼ R R qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fa  4s2y  3f 2a ðin tensionÞ (5.368.b) ssh ¼ 2 The effects of local buckling can be ignored if R < 0:85 [90]. Sakino et al. [105] assumed that the hoop stress ssh and longitudinal stress fa in the steel tube at the ultimate axial strength are less than sy . Therefore: ssh ¼ au sy

(5.369.a)

fa ¼ buc sy

(5.369.b)

in which au ¼ 0:19 and buc ¼ 0:89 are stress coefficients and were defined according to experimental results [105]. It is assumed that stress

741

5.8 Compressive strength of CFST stub columns

coefficients are independent of the geometric and material properties of columns. Therefore, Eq. (5.369) can be rewritten as follows: ssh ¼ 0:19sy fa ¼ 0:89sy

ðin tensionÞ

(5.370.a)

ðin compressionÞ

(5.370.b)

In addition to the model presented above, a simplified model can also be used for predicting the compressive strength of square CFST stub columns with binding bars under axial compression. Eq. (5.354) can be expressed as: Nu ¼ 4s As fa þ 4c Ac fck

(5.371)

in which fs and fc represent the strength coefficients of steel and concrete, respectively, and are given by: 4s ¼

fa fy

(5.372.a)

4c ¼

fcc fck

(5.372.b)

According to the regression analysis performed by Cai and He [96], fs can be expressed as: ( 0:89 R < 0:85 4s ¼ (5.373) ð0:7407Þ 0:897R R  0:85 The uniaxial compressive capacity of infilled concrete in square CFST stub columns with binding bars depends on the constraining coefficient of binding bars ðzÞ and the width-to-thickness ratio parameter ðRÞ [96]. f A

Therefore, fc is a function of z ¼ fckyas bs s and R, and can be expressed as: 4c ¼ 1:039Rð0:7407Þ ð7:3836z þ 1:0588Þ

(5.374)

5.8.4 Rectangular CFST stub columns with binding bars In square CFST stub columns, the binding bars are typically designed uniformly over the four faces of the steel tube. Therefore, the lateral confining pressure at all faces is the same. In rectangular CFST columns, however, binding bars are typically arranged to span the narrow direction, as shown in Fig. 5.58. Consequently, the lateral confining pressure provided on narrow faces is different from that provided on broad faces. The model for predicting the ultimate axial strength of rectangular CFST short columns with binding bars can be derived by modifying the method

742

5. Design rules and standards

FIGURE 5.58

Cross section of a rectangular CFST column with binding bars.

discussed in the previous section for predicting the compressive capacity of square CFST short columns with binding bars [97]. Fig. 5.59 shows the free-body diagram of a rectangular CFST column with binding bars. Suppose that a length column equal to the longitudinal direction is equivalent to the spacing of binding bars in the longitudinal direction ðbs Þ. Hence, the equilibrium of forces according to the free-body diagram shown in Fig. 5.59a leads to: fl1 bs ðB  2tÞ  2ssh1 bs t ¼ 0

(5.375)

Therefore, the lateral confining stress acting on the narrow faces fl1 is governed by: fl1 ¼

(a) Forces acting on the width

2ssh1 B 2 t

(5.376)

(b) Forces acting on the length

FIGURE 5.59 Internal forces in a rectangular CFST column with binding bars. (a) Forces acting on the width (b) Forces acting on the length.

5.8 Compressive strength of CFST stub columns

From Fig. 5.59b, it can be concluded that:   D  1 Fs ¼ 0 fl2 bs ðD  2tÞ  2ssh2 bs t  as

743

(5.377)

The stress in binding bars Fs is governed by: Fs ¼ Es;b εs;b
fy;s

(5.378)

in which Es;b and εs;b are Young’s modulus and tension strain of binding bars, respectively. The lateral confining stress acting on the wide direction can be derived by replacing Eq. (5.378) into Eq. (5.377) leads to: fl2 ¼

2ssh2 þ Es;b εs;b

As;b ðD  as Þ t as bs

D 2 t

(5.379)

It can be assumed that by reaching the ultimate axial strength, binding bars yield. Therefore the term Es;b εs;b in Eq. (5.378) can be replaced by the

 yield strength of binding bars fy;s . It can be concluded from Eq. (5.379) that reducing the horizontal spacing as and vertical spacing bs of binding bars enhances the contribution of binding bars to lateral confining pressure on the infilled concrete and the lateral confining pressure acting on the wide faces fl2 . Finally, the ultimate axial strength of rectangular CFST stub columns with binding bars can be expressed as: Nu ¼ As fa þ Ac fcc0 ¼ As1 fa1 þ As2 fa2 þ Ac fcc0

(5.380)

where fa1 is the longitudinal strength of the steel of the broad faces of the tube, fa2 is the longitudinal strength of steel of the narrow faces of the tube. The stress state of infilled concrete at the ultimate axial strength is the true triaxial compressive stress state. The compressive strength of confined concrete fcc0 is determined based on the failure criterion for concrete subjected to triaxial compression [106] as follows:   0:09  s0 0:9297 s0 ¼ 6:9638 (5.381) c  s0 c ¼ 12:2445ðcos 1:5 aÞ1:5 þ 7:3319ðsin 1:5 aÞ2 soct s0 ¼ fck soct s0 ¼ fck

(5.382) (5.383) (5.384)

744

5. Design rules and standards

fl10 þ fl20 þ fcc0 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   0  2  2 2 f l1  f 0l2 þ f 0l2  fcc þ f 0cc  f 0l1 soct ¼

soct ¼

3 cos a ¼

2fl10  fl20  fcc0 pffiffiffi 3 2soct

(5.385)

(5.386) (5.387)

where soct represents the normal stress, soct is the shear stress, a is the rotational variable that defines the direction of the deviatoric component on the octahedral plane, fl10 is the effective lateral confining pressure on the narrow faces, and fl20 is the effective lateral confining pressure on the broad faces. fl10 and fl20 are determined based on the method used by Mander et al. [104]. fl10 and fl20 are, respectively, governed by: fl10 ¼ ke fl1

(5.388)

fl20 ¼ ke fl2

(5.389)

in which fl1 and fl2 can be defined using Eqs. (5.376) and (5.379), respectively, and ke is given by Eqs. (5.358), (5.359a) and (5.359b). Ae1 in Eq. (5.359.a) is the horizontal area of an effectively confined concrete core, whereas Ae2 in Eq. (5.359.b) is the longitudinal area of an effectively confined concrete core. Ac1 in Eq. (5.359.a) is the horizontal area of a concrete core, while Ac2 in Eq. (5.359.b) is the longitudinal area of a concrete core. Similar to square CFST columns with binding bars, it can be supposed that the arching action occurs in the form of a second-degree parabola having an initial tangent angle q at the cross section of the rectangular concrete core, as shown in Fig. 5.60.

θ θ

Ineffectively confined concrete

θ

Effectively confined concrete

FIGURE 5.60 Different zones of concrete core in rectangular CFST stub columns with binding bars.

745

5.8 Compressive strength of CFST stub columns

Based on Fig. 5.60, Ae1 and Ac1 can be expressed as: Ae1 ¼ ðB  2tÞðD  2tÞ 

n X

AI 

i¼1

m X

AII

(5.390)

i¼1

(5.391) Ac1 ¼ ðB  2tÞðD  2tÞ in which AI and AII represent the parabolas areas, shown in Fig. 5.60 and are governed by: ðB  2tÞ2 tan q 6 2 32

AI ¼

(5.392)

6ðD  2tÞ7 6   7 tan q 4 5 D as

AII ¼

6

(5.393)

Therefore, ke1 can be expressed as: ke1 ¼ 1 

n ðB  2tÞ X i¼1

a 2 s

tan q

D 6ðD  2tÞ

a 2 s m ðD  2tÞ tan q X D  6ðB  2tÞ i¼1

(5.394)

When the binding bars are both designed to span the narrow and broad directions, ke2 is given by:    bs tan q bs tan q ke2 ¼ 1  1 (5.395) 2ðD  2tÞ 2ðB  2tÞ When the binding bars are only designed to span the narrow direction, ke2 is governed by bs tan q (5.396) 2ðB  2tÞ The initial tangent angle q can be calculated using Eq. (5.365). However, the experimental test results [97] showed that reducing the horizontal spacing between bonding bars has a negligible influence on the ultimate axial strength of rectangular CFST stub columns. Therefore, q can be taken as 12 degrees [97]. Like square CFST stub columns with binding bars, rectangular CFST stub columns with binding bars under axial compression usually fail in postultimate local buckling. Hence, the steel materials reach their yield stress by reaching the ultimate axial strength of the column. Based on the von Mises yield criteria: ke2 ¼ 1 

2  fa1 ssh1 þ s2sh2 ¼ s2y fa1

(5.397)

746

5. Design rules and standards

The local buckling strength of a steel plate ðsb Þ in rectangular CFST stub columns can be predicted using the model recommended by Ge and Usami [90] as follows: sb 1:2 0:3 ¼ 
1:0 sy R1 R21

(5.398)

in which the parameter R1 is the depth-to-thickness ratio governed by: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffisffiffiffiffiffi 2 fy D 12 1  ws (5.399) R1 ¼ Es t 4p2 It should be noted that R1 must be greater than 0.85 to ensure that the requirement sb
sy is satisfied. According to Eq. (5.398), two scenarios can be determined for the stress states of rectangular steel tubes as follows: If R1  0:85: Taking fa ¼ sb in Eqs. (5.397) and (5.398) leads to: ! 1:2 0:3  (5.400.a) sy ðin compressionÞ fa1 ¼ R1 R21 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fa  4s2y  3f 2a1 ssh ¼ ðin tensionÞ (5.401.b) 2 The effects of local buckling can be ignored if R1 < 0:85 [90]. Therefore, similar to the square CFST stub column with binding bars, ssh1 and fa1 can be expressed as: ssh1 ¼ 0:19sy

ðin tensionÞ

fa1 ¼ 0:89sy ðin compressionÞ

(5.402.a) (5.402.b)

ssh2 and fa2 can be derived using the same process discussed above.

5.8.5 Square CFST stub columns with spiral tension bars The stress distribution of square CFST stub columns with spiral tension bars is shown in Fig. 5.61 [28]. This method is according to the stress distribution and the superposition principle of the core concrete at the ultimate strength. As shown in Fig. 5.61, the cross section of the concrete core is split into different regions, including Ae1 , the area constrained by both square steel tube and annular stirrups, Ae2 the area constrained by annular stirrups only, Ae3 is area constrained by square steel tube only,

747

5.8 Compressive strength of CFST stub columns

(a) Cross-section

(b) Free-body diagram

FIGURE 5.61 Different zones of concrete core in square CFST stub columns with spiral tension bars. (a) Cross section (b) Free-body diagram.

and Ae4 is the unconstrained concrete area. The length of the unconfined area is 0:4B, as depicted in Fig. 5.61. Consequently, the relationship between different areas of infilled concrete can be expressed as: 8 > > < Ae1 þ Ae2 þ Ae3 þ Ae4 ¼ Ac Ae1 þ Ae2 ¼ Acor (5.403) > > : Ae1 þ Ae3 ¼ 0:75Ac  2

in which Ac ¼ ðB  2tÞ and Acor ¼ p

B 2

td

2 where d is the diameter

of the stirrup. For the convenience of calculation, the spiral tension bars are transformed into a circular steel tube with equivalent wall thickness. The inner diameter of the equivalent steel tube is taken as d so that the concrete area Acor confined by stirrups does not alter. Besides, the equivalent steel tube has the same steel ratio as that of the spiral tension bars. Therefore, the area of the equivalent steel tube Ass0 is given by: Ass0 ¼

p2 d2 ðB  2t  2dÞ 4bs

(5.404)

where bs is the spacing distance between the stirrups along the specimen axis direction.

748 

5. Design rules and standards

The relationship between the concrete stress in the radial  direction  sr;c1 induced by stirrups and yield stress of stirrups fy;s at the ulti-

mate state is governed by: sr;c1 ¼

fy;s As;s 2Acor

(5.405)

According to the free-body diagram shown in Fig. 5.61, the relationship between  concrete stress in the radial direction developed by the steel  the tube sr;c2 and the hoop stress in the steel tube ðssh Þ at the ultimate state can be expressed as: 2tssh (5.406) sr;c2 ¼ B Hence, the axial compressive stress of concrete core in different zones by considering the lateral confining stress can be expressed as: 8   fl1 ¼ fc0 þ k sr;c1 þ sr;c2 > > > > > < fl2 ¼ f 0 þ ksr;c1 c (5.407) 0 > f ¼ f þ ksr;c3 > l3 c > > > : fl4 ¼ fc0 According to the finite element (FE) analysis performed by Ding et al. [28] on square CFST stub columns with spiral tension bars, when columns achieve their compressive capacities, the amount of the ratio of the   mean axial compressive stress of steel tube sL;s to its yield stress fy is: sL;s ¼ 0:78 fy

(5.408)

Based on von Mises yield criterion, the tensile hoop stress of the steel tube ðsh Þ can be expressed as: ssh ¼ 0:33fy

(5.409)

The ultimate axial strength of square CFST stub columns with spiral stirrups can be expressed as: (5.410) Nu ¼ fl1 Ae1 þ fl2 Ae2 þ fl3 Ae3 þ fc0 Ai þ fy As Substituting Eqs. (5.403)e(5.409) into Eq. (5.410) leads to: Nu ¼ fc0 Ac þ 1:7fy;s Ass0 þ 1:2fy As

(5.411)

5.8 Compressive strength of CFST stub columns

749

The ultimate axial strength of square CFST stub columns without spiral stirrups can be derived from Eq. (5.411) by removing the term 1:7fy;s Ass0 as follows: Nu ¼ fc0 Ac þ 1:2fy As

(5.412)

The comparison between Eqs. (5.411) and (5.412) indicates that the increase in the ultimate axial strength of square CFST stub columns with spiral stirrups is because of the increase in the ultimate axial strength of the concrete core confined by stirrups. Besides, the coefficient 1.7 in Eq. (5.411) shows that the confinement effect provided by stirrups is more remarkable than that of the steel tube.

5.8.6 Circular CFST stub columns with tie bars Fig. 5.62 presents the free-body diagram of a circular CFST column with tie bars. Based on the equilibrium of forces, the relationship between the lateral confining stress of the concrete core fl , steel hoop stress sh , and the stress of tie bars Fs can be expressed as: p fl ðD  2t  ds ÞH ¼ 2ssh tH þ Fs d2s n (5.413) 4 in which D and t are the diameter and the thickness of the steel tube, respectively, H shows the height of the column, ds is the diameter of tie bars, n is the total layers of tie bars, and Fs is given by Eq. (5.378). From Eq. (5.413), fl can be determined as: fl ¼

2ssh t Fs pd2s n þ D  2t  ds 4HðD  2t  ds Þ

Tie bar

FIGURE 5.62

Internal forces in a circular CFST stub column with tie bars.

(5.414)

750

5. Design rules and standards

It can be assumed that tie bars yield at the ultimate state. Therefore, Fs can be taken as fy;s , and Eq. (5.414) can be rewritten as follows: fl ¼

fy;s pd2s 2ssh t Fs pd2s n 2ssh t ¼ þ þ D  2t  ds 4HðD  2t  ds Þ D  2t  ds 4bs ðD  2t  ds Þ (5.415)

where bs stands for the center-to-center spacing of the tie bars in the longitudinal direction. It can be seen in Eq. (5.415) that for determining the lateral confining stress, the magnitude of hoop stress in the steel tube ssh should be determined. Based on the research performed by Lai and Ho [107], the hoop stress ssh is a function of steel tube yield stress sy and the confinement coefficient x and can be expressed as follows: 8 1 > > 0 0
x< > > 75 > < 1 ssh ¼ (5.416)
x < 6:68 sy ð0:15x  0:002Þ > > > 75 > > : x  6:68 sy Typically, circular steel tubes show strain hardening. Therefore, sy in

 Eq. (5.416) can be increased by 10% s0y ¼ 1:1sy . It can be observed from   1 Eq. (5.416) that when the confinement coefficient is small x < 75 , the developed confinement effect is weak and can be ignored. Therefore, no hoop stress occurs in the steel tube  ðssh ¼ 0Þ, and the longitudinal stress in the steel tube fa is equal to fy fa ¼ sy . In this case, the steel tube is similar to longitudinal steel reinforcement bars and does not provide any confinement effect. By contrast, when the confinement coefficient is large ðx  6:68Þ, the steel tube can strongly confine the infilled concrete, similar to confining reinforcement bars. Hence, it can be assumed ssh ¼  sy , while the tube has no contribution to the axial resistance fa ¼ 0 . The compressive strength of infilled concrete fcc is expressed as: fcc0 ¼ fc0 þ 4:1fl

(5.417)

Based on the superposition principle, the ultimate axial strength of circular CFST stub columns with tie bars under axial compression is governed by: Nu ¼ fcc0 Ac þ fy;s As;s þ fa As

(5.418)

5.8 Compressive strength of CFST stub columns

751

in which pD 2 d 2 bs s

(5.419)

pd2 t p 2 D  ðD  2tÞ2  s 2 bs

(5.420)

p p D  2t 2 Ac ¼ ðD  2tÞ2  ds 4 2 bs

(5.421)

As;s ¼ As ¼

The axial stress fa in steel tube subjected to biaxial state can be determined using von Mises yield criterion. The model presented above has been developed based on the circular CFST stub columns with normal strength concrete [108]. Therefore, it may underestimate the ultimate axial strength of circular CFST columns made with high-strength concrete (HSC) since the relationship between the confinement coefficient and the concrete compressive strength fc0 is no Af

longer linear. Therefore, using the equation x ¼ Asc fy0 will underestimate the c

effectiveness of the steel tube and tie bars in providing the confinement effect. A possible solution to solve the drawback of the presented model is to convert the high-strength circular CFST stub columns with fc0  60 MPa and x
1 into an equivalent section of normal-strength circular CFST stub columns. To this end, the following equations can be used to convert the areas of the steel and concrete:

s  y A0s ¼ As (5.422) 355  0 f (5.423) A0c ¼ Ac c 30 A0st ¼ Ast rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4  0 0 0 D ¼ AA p s c rffiffiffiffiffiffiffiffiffi 4 0 0 Dc ¼ A p c b0s ¼

bs D0 D

(5.424) (5.425) (5.426) (5.427)

in which A0s is the equivalent area of the circular tube, A0c is the equivalent area of the concrete core, A0st is the equivalent area of the tie bars, D0 is the diameter of the equivalent area including concrete core and steel tube, and b0s is the equivalent spacing of the tie bars. The ultimate axial strength

752

5. Design rules and standards

of high-strength circular CFST stub columns with tie bars can be achieved by replacing As by A0s, Ac by A0c, bs by b0s, D by D0, and ðD 2tÞ by D0c in the above equations. For applying Eq. (5.416), the hardening stress for highstrength CFST stub columns can be considered by taking s0y ¼ 1:2sy . In practice, the cross-sectional area of columns is usually much bigger

 than the area of tie bars dDs  50 . Therefore, the area of tie bars in predicting the ultimate axial strength using the above model can be ignored. The model of Ho and Lai [108] leads to a reasonable prediction of ultimate axial strengths of circular CFST stub columns with tie bars when x < 3:0. By contrast, when x > 3:0, the model overestimates the confinement effect and thus the predicted ultimate axial strength of columns. Achieving the upper boundary condition defined in Eq. (5.416) is almost impossible in practice because of the existence of friction force between the steel tube and infilled concrete. Therefore, Eq. (5.416) can be modified [109] as follows: 8 > 0
x < 0:03 >

> : 0:244sy x  2:5

5.8.7 Circular CFST stub columns with tension bars Fig. 5.63 shows the simplified stress distribution of circular CFST stub column with stirrups. For the convenience of design, stirrups are converted into a circular steel tube employing an equivalent wall thickness to match the steel ratio. The area of the equivalent steel tube Ass0 for a

FIGURE 5.63

Internal forces in a circular CFST stub column with stirrups.

5.8 Compressive strength of CFST stub columns

Orthogonal stirrups

Bidirectional stirrups

753

Loop stirrups

(a) Orthogonal stirrups (b) Bidirectional stirrups (c) Loop stirrups Circular CFST columns with stirrups. (a) Orthogonal stirrups (b) Bidirectional stirrups (c) Loop stirrups.

FIGURE 5.64

circular CFST stub column with different types of stirrups, shown in Fig. 5.64, can therefore be expressed as:

Ass0 ¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8  2  2 2 > d 1 1 > > p8 pffiffiffi  ðD  2tÞ ðD  2tÞ > > > 4 2 6 2 2ðD  2tÞd2 p > > ¼ for orthogonal stirrups > > > 8 3bs < ðD  2tÞd2 p > > for bidirectional stirrups > > bs > > > > > 2 2 > > : ðD  2tÞd p for loop stirrups 4bs (5.429)

where d represents the diameter of stirrups and bs is the spacing of stirrups in the longitudinal axis. The relationship between fl and ssh can be obtained from the free-body diagram, shown in Fig. 5.63, as follows: fl ¼

fy;s Ass0 rs s þ 2Ac 2ð1  rs Þ sh

(5.430)

s is the steel ratio of the steel tube. in which rs ¼ AsAþA c According to the FE analysis performed by Ding et al. [22] on circular CFST stub columns with stirrups, when columns achieve their compressive capacities, themean  amount of the ratio of the axial compressive stress of steel tube sL;s to its yield stress fy is:

sL;s ¼ 0:69 fy

(5.431)

754

5. Design rules and standards

Based on von Mises yield criterion, the tensile hoop stress of the steel tube ðssh Þ can be expressed as: ssh ¼ 0:59fy

(5.432)

The axial compressive stress of the concrete core can be expressed using Eq. (5.342) by taking the lateral pressure coefficient k as 3.4 [22]. Based on the superposition principle, the ultimate axial strength of circular CFST stub columns with stirrups under axial compression can be expressed as: Nu ¼ sc;3 Ac þ kfl

(5.433)

Replacing Eq. (5.342) and Eqs. (5.429)e(5.432) into Eq. (5.433) leads to:

 Nu ¼ fc0 Ac þ 1:7 fs As þ fy;s Ass0 (5.434) in which fy;s is the yield strength of stirrups.

5.8.8 Circular CFST stub columns with external ring bar stiffeners • Model of Ho and Lai [108]: The free-body diagram of a circular CFST column with ring bar stiffeners is shown in Fig. 5.65. According to the equilibrium of forces, the relationship between the lateral confining stress of the concrete core fl , steel hoop stress sh , and the stress of ring bars Fs can be expressed as: p fl ðD  2tÞH ¼ 2ssh tH þ 2Fs d2s n 4

(5.435)

(a) Side view (b) Internal forces FIGURE 5.65 Circular CFST columns with external ring bar stiffeners. (a) Side view (b) Internal forces.

5.8 Compressive strength of CFST stub columns

755

in which D and t are the diameter and the thickness of the steel tube, respectively, H shows the height of the column, ds is the diameter of tie bars, n is the total layers of tie bars, and Fs is given by Eq. (5.378). Replacing Eq. (5.378) into Eq. (5.435) leads to: p 2ssh tH þ 2Es;b εs;b d2s n 4 fl ¼ ðD  2tÞH

(5.436)

It can be assumed that ring bars yield at the ultimate state. Therefore, Fs can be taken as fy;s , and Eq. (5.436) can be rewritten as follows: p 2ssh tH þ 2fy;s d2s n A pd2s st 4 ¼ ssh þ fy;s fl ¼ ðD  2tÞH 2Ac 2ðD  2tÞbs

(5.437)

where bs stands for the spacing of ring bars in the longitudinal direction, shown in Fig. 2.65, Ast is the total area of the steel tube and ring bars and can be expressed as: Ast ¼ As þ

p pd2s D fy;s 4 bs fy

(5.438)

For determining the lateral confining stress fl using Eq. (5.437), it is required to calculate the magnitude of hoop stress in the steel tube ssh . ssh can be determined using Eq. (5.428). The compressive strength of infilled concrete fcc is expressed as: fcc0 ¼ fc0 þ 4:1fl

(5.439)

Based on the superposition principle, the ultimate axial strength of circular CFST stub columns with tie bars under axial compression can be expressed as: Nu ¼ fcc0 Ac þ fa Ast

(5.440)

in which ss;L is the axial stress in the steel tube and can be determined using von Mises yield criterion, as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 1h fa ¼ (5.441)  ssh þ 4s2y  3s2sh 2 Typically, circular steel tubes show strain hardening. Therefore, sy in

 von Mises yield criterion can be increased by 10% s0y ¼ 1:1sy . The model presented above has been developed based on the circular CFST stub columns with normal strength concrete [107]. Therefore, it may

756

5. Design rules and standards

underestimate the ultimate axial strength of circular CFST columns made Af

with HSC since the confinement coefficient x ¼ Asc fy0 underestimates the c

confinement effects provided by circular steel tubes and the external ring bars for a specific concrete compressive strength. In circular CFST stub columns with the HSC core, the external confinement effect provided by ring bars improves the confining stress. Therefore, ring bars remarkably enhance the compressive capacity and ductility of the concrete core for a Af

given strength [110]. The equation of x ¼ Asc fy0 is based on CFST columns c

without external stiffeners (without additional confinement effect). Therefore, the equation of x must be modified to take into account the effect of external confinement. Similar to the model introduced for circular CFST stub columns with tie bars, the high-strength circular CFST stub columns can be converted into the section of normal-strength circular CFST stub columns. • Model of Cai and Jiao [111]: In this model, which has been developed based on the limit equilibrium analysis, it is supposed that at the ultimate state, the steel rings achieve their yield strength, and the steel tube is at the yielding stage. Additionally, the infilled concrete reaches its confined concrete strength. The ultimate axial strength of CFST stub columns with external stiffeners can be expressed as: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " !2 !# u 2 k  1 fr pds fy;s u 3 2Ac fr pd2s fy;s t 0 þ þ 1  Nu ¼ 0:8fc Ac 1 þ x x fc0 8tbs fy 4 As fy 4tbs fy (5.442) in which k ¼ 4:1 is the coefficient of confining pressure. It can be seen in Eq. (5.442) that the predicted ultimate axial strength is taken as 80% of the compressive strength of the column. This is because the ultimate axial strength of columns obtained from experimental tests will always be smaller than the values predicted by limit equilibrium analysis. The reason is that reaching the fully efficient confinement effect at the early stages of loading is unlikely due to geometric imperfections of materials. The magnitude of confining stress fl for the maximum amount of Nu can be achieved by taking the partial derivative with respect to fl on both sides and setting it to zero. k1 pd2 As fl ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xfc0 þ s fy;s 8tbs Ac 3ðk  1Þ2 þ 9

(5.443)

5.8 Compressive strength of CFST stub columns

Replacing Eq. (5.443) into Eq. (5.442) leads to: " sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !# k  1 fl pd2s fy;s 3 0 Nu ¼ 0:8fc Ac 1 þ x þ þ 0 x fc 8tbs fy ðk  1Þ2 þ 3

757

(5.444)

5.8.9 Circular CFST stub columns with jacket strip stiffeners Fig. 2.66 shows the free-body diagram of a circular CFST column with jacket strip stiffeners. It can be observed from the figure that the concrete core is confined by the steel tube and the jacket stiffeners. Therefore, the lateral confining stress of the concrete core fl can be expressed as: fl ¼

2t 2nAs;s FE s þ D  2t sh HðD  2tÞ

(5.445)

in which the first term refers to the confining stress from the steel tube, and the second term refers to the confining stress from the steel jackets. n in Eq. (5.445) is the number of steel jackets. It can be assumed the steel jackets yield at the ultimate state. Therefore, Fs can be taken as fy;s , and Eq. (5.445) can be rewritten as follows: fl ¼

2t 2nAs;s fy;s ssh þ HðD  2tÞ D  2t

(5.446)

in which the hoop stress of steel tube ssh is determined using Eq. (5.428) (Fig. 5.66). The compressive strength of infilled concrete fcc0 can be derived using Eqs. (5.417) and (5.446). The cross-sectional area of steel jackets is negligible and can be ignored. Finally, the ultimate axial strength of circular CFST stub columns with steel jackets can be expressed as follows: Nu ¼ fcc0 Ac þ fy As

(5.447)

Jacket strip stiffeners

FIGURE 5.66

Internal forces in a circular CFST column with jacket strip stiffeners.

758

5. Design rules and standards

5.8.10 Circular CFST stub columns with spiral bar stiffeners The free-body diagram of a circular CFST stub column with spiral bar stiffeners is shown in Fig. 5.67. As shown in the figure, spirals contain two parts, including middle spirals and end horizontal hoops. Consequently, the confinement effect provided by spirals is split into the middle part and end part over the height H of the column [112]. From the free-body diagram displayed in Fig. 5.67, the confining stress fl can be determined as follows: p H p fl ðD  2tÞH ¼ 2ssh tH þ 4Fs;h d2s þ 2 Fs;sp d2s 4 bs 4

(5.448)

in which Fs;h stands for the confining stress provided by horizontal hoops and Fs;sp is the confining stress provided by spirals. It can be supposed that when the column achieves its compressive capacity, spiral bars yield. Consequently, Eq. (5.448) can be rewritten as: 2 0 13 fl ¼

2t pd2s bs pd2s 6 B 7 C fy;s cos4arc tan@  ssh þ A5 ds HðD  2tÞ 2bs ðD  2tÞ D  2t p Dþ 2 (5.449)

in which the hoop stress of steel tube ssh is determined using Eq. (5.428).

Horizontal hoop

Middle part spirals

)

FIGURE 5.67 Internal forces.

)

(a) Side view (b) Internal forces Circular CFST columns with external spiral bar stiffeners. (a) Side view (b)

759

5.8 Compressive strength of CFST stub columns

The confinement coefficient x should be taken as

Ast fy Ac fc0

in which Ast is the

total area of the steel tube and spiral bar stiffeners and can be expressed as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !     pd2s d d 2 fy;s þ n b2s þ p2 D þ (5.450) Ast ¼ As þ 2p D þ 4H fy 2 2 in which n denotes the number of spirals. The ultimate axial strength of circular CFST stub columns with spiral ring bars can be expressed as: 13 2 0 p bs 7 6 B C Nu ¼ fcc Ac þ As fa þ 2nfy;s d2s sin4arc tan@  A5 ds 4 p Dþ 2

(5.451)

in which the compressive strength of infilled concrete fcc is governed by Eq. (5.417), and the longitudinal stress of the steel tube fa can be determined using the von Mises yield criterion.

5.8.11 L-shaped CFST stub columns with binding bars L-shaped CFST stub columns with binding bars have the same confinement mechanism as square and rectangular CFTS stub columns with binding bars [96,97]. L-shaped CFST stub columns can be converted to rectangular RC columns with stirrups, where steel tubes serve as longitudinal and stirrups reinforcement bars with close spacing, and binding bars can be considered as tie bars arranged in stirrups [113]. Consequently, the model of Mander et al. [104] for predicting the compressive capacity of the confined concrete core of RC columns can be adopted to assess the load-bearing capacity of axially loaded L-shaped CFST stub columns with binding bars, similar to square and rectangular CFST stub columns [96,97], discussed in previous sections. For estimating the compressive capacity of L-shaped CFST stub columns with binding bar, the column should be split into a square CFST column and two rectangular CFST columns, as displayed in Fig. 5.68. According to the schemes for predicting the compressive capacities of square and rectangle CFST stub columns with binding bars, the model for predicting the ultimate axial strength of L-shaped CFST stub columns with binding bars is developed. For ease of calculation, two rectangular CFST columns are designed with the same dimensions. Besides, binding bars are only provided along the short-side direction for rectangular CFST columns, as indicated in Fig. 5.68a. The following boundary conditions are assumed for the boundary surfaces:

760

5. Design rules and standards

(a) Divisional L-section

(b) Boundary conditions

FIGURE 5.68 The division of the L-shaped CFST stub column and boundary conditions. (a) Divisional L-section (b) Boundary conditions. Zuo Z-L, Cai J, Yang C, Chen Q-J, Sun G. Axial load behavior of L-shaped CFT stub columns with binding bars. Eng Struct 2012;37:88e98.

1. The boundary surfaces have infinite rigid lateral rigidity. However, they can be compressed along the longitudinal direction (see Fig. 5.68b). 2. The boundary surfaces of square and rectangular CFST columns have a uniform longitudinal deformation. The ultimate axial strength of L-shaped CFST stub columns with and without binding bars under axial compression can be expressed as: 1 0 3 X 0 0 faj Asj A þ Ac1 fcc1 þ Ac2 fcc2 (5.452) Nu ¼ As fa þ Ac fcc0 ¼ @2 j¼1 0 fcc1

0 fcc2

and represent the compressive stresses of the confined where concrete core of square and rectangular CFST columns, respectively, fa1 denotes the longitudinal stress of the a1 -side steel plates of the columns, fa2 denotes the longitudinal stress of the short side (b-side) steel plates of the columns, Aci ði ¼ 1 and 2Þ and Asj ð j ¼ 1; 2; and 3Þ are the crosssectional areas of the concrete core and steel plates corresponding to 0 ði ¼ 1 and 2Þ and f ð j ¼ 1; 2; and 3Þ, respectively. fcci aj The stress state of the concrete core at the ultimate axial strength is the triaxial compressive stress state. Similar to rectangular CFST stub columns 0 ði ¼ 1 and 2Þ are deterwith binding bars, the compressive stresses fcci mined using the failure criterion of concrete [106]: socti ¼

0 þ f0 þ f0 fli1 cci li2 3

(5.453)

5.8 Compressive strength of CFST stub columns

soct ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  0 2  2  2 f li1  f 0li2 þ f 0li2  fcci þ fcci  f 0li1 3

socti fck s s0i ¼ octi fck

s0i ¼

cos ai ¼

0  f0  f0 2fli1 cci pffiffiffili2 3 2socti

ci ¼ 12:2445ðcos 1:5ai Þ1:5 þ 7:3319ðsin 1:5ai Þ2   0:09  s0i 0:9297 s0i ¼ 6:9638 ci  s0i

761

(5.454) (5.455) (5.456) (5.457) (5.458) (5.459)

where the subscript i ¼ 1 and 2, socti represents the normal stress, socti is the shear octahedral stress, ai is the rotational variable that defines the 0 and f 0 direction of the deviatoric component on the octahedral plane, fli1 li2 are the effective lateral confining stresses of the infilled concrete, where 0 ¼ f 0 are induced over the sides of the square column, and f 0 ¼ f 0 fl21 l22 l11 l12 are induced over the long sides and short sides of the rectangular columns, respectively. 0 and f 0 are determined based The effective lateral confining stresses fli1 li2 0 and f 0 are, respectively, on the method used by Mander et al. [104]. fli1 li2 governed by: 0 ¼ ke1 fl1 fl11

(5.460)

0 fl12 ¼ ke1 fl2

(5.461)

0 0 fl21 ¼ fl22 ¼ ke2 fl2

(5.462)

where ke1 and ke2 are the confinement effectiveness coefficients corresponding to rectangular and square cross sections, respectively, fl1 is the lateral confining stress of the concrete core induced over the long sides of the rectangular columns, and fl2 is the lateral confining stress of the concrete core induced over the short sides of the rectangular columns or the sides of the square column. The confinement effectiveness coefficients of the rectangular column ke1 and square column ke2 can be expressed as the horizontal confinement effectiveness coefficient keih ði ¼ 1 and 2Þ and the longitudinal confinement effectiveness coefficient keil ði ¼ 1 and 2Þ, as follows: kei ¼ keih keil  0 ði ¼ 1 and 2Þ

(5.463)

762

5. Design rules and standards

wherte keih and keil ði ¼ 1 and 2Þ are governed by: keih ¼

Aeih  0 ði ¼ 1 and 2Þ Aci

(5.464.a)

keil ¼

Aeil  0 ði ¼ 1 and 2Þ Aci

(5.464.b)

in which Aeih and Aeil are the areas of horizontal and longitudinal effectively confined concrete core of the rectangular CFST columns ði ¼ 1Þ or the square CFST column ði ¼ 2Þ, respectively, and Aci represents the areas of the concrete core of the rectangular CFST columns ði ¼ 1Þ or the square CFST column ði ¼ 2Þ. The boundaries between the confined and unconfined areas of the concrete core over the sides of the cross section as well as longitudinal sections of the L-shaped CFST column can be assumed to have secondorder parabola shapes with initial tangent angle q, as shown in Fig. 5.69. It should be noted that the hatched areas in Fig. 5.69 represent the ineffectively confined concrete core. When the column has no binding bars, it can be assumed that the boundaries along the sides of cross sections span over the corner points of the tube, and no ineffective confined infilled concrete areas develop along the longitudinal section. In the case of using binding bars, it is assumed that the binding bars are uniformly arranged in rectangular columns for ease of calculation. Besides, it is assumed that boundaries over the sides of cross sections span between the corner points of the steel tubes and the bars, and the spans of boundaries over the sides of longitudinal sections are specified as spacing of binding bars in the longitudinal section ðbs Þ. For controlling the amount of the confinement effects provided by binding bars and the steel tube, the initial tangent angle q is employed. The stiffness of the steel palates can also affect the magnitude of q. However, when the difference between the spans of different boundaries is not remarkable, i.e., a=b < 3, the same q can be considered for each span to simplify calculations. Ae2h and Ac2 of the square CFST column without binding bars can be expressed as:   ðb  tÞ 2 2 (5.465) Ae2h ¼ ðb  tÞ  2ðb  tÞ tan q 0:5  ð3a  6tÞ Ac2 ¼ ðb  tÞ2

(5.466)

763

5.8 Compressive strength of CFST stub columns

(a) Cross-sections of columns with and without binding bars

Unconfined area

A-A

B-B

C-C

(b) Longitudinal sections with binding bars The division of L-shaped CFST stub columns and boundary conditions. (a) Cross sections of columns with and without binding bars. (b) Longitudinal sections with binding bars. Zuo Z-L, Cai J, Yang C, Chen Q-J, Sun G. Axial load behavior of L-shaped CFT stub columns with binding bars. Eng Struct 2012;37:88e98.

FIGURE 5.69

764

5. Design rules and standards

Hence, ke2h can be determined by replacing Eqs. (5.465) and (5.466) into Eq. (5.464.a), as follows:   A ðb  tÞ (5.467) ke2h ¼ e2h ¼ 1  2 tan q 0:5  Ac2 ð3a  6tÞ Likewise, ke1h and keil ði ¼ 1 and 2Þ for the column without binding bars are governed by: tan q ke1h ¼ 1  6ða  b  tÞðb  2tÞ "

# ða  2tÞ3  ðb  tÞ2 ð3a  2b  4tÞ ða  b  tÞ þ ðb  2tÞ þ a  2t 2

2

(5.468) keil ¼ 1 ði ¼ 1 and 2Þ

(5.469)

In the case of using binding bars, keih and keil ði ¼ 1 and 2Þ can be expressed by Eqs. (5.470)e(5.473), respectively.   ða  b  tÞ b  2t þ tan q (5.470) ke1h ¼ 1  3nðb  2tÞ 6ða  b  tÞ    bs tan q bs tan q 1 (5.471) ke1l ¼ 1  2ðb  2tÞ 4ða  b  tÞ ke2h ¼ 1  tan

q 3

  bs tan q 2 ke2l ¼ 1  4ðb  tÞ

(5.472) (5.473)

Consequently, the confinement effectiveness coefficient kei ði ¼ 1 and 2Þ of rectangular and square columns can be expressed as: For columns without binding bars: tan q ke1 ¼ 1  6ða  b  tÞðb  2tÞ "

# ða  2tÞ3  ðb  tÞ2 ð3a  2b  4tÞ ða  b  tÞ þ ðb  2tÞ þ a  2t 2

2

  ðb  tÞ ke2 ¼ 1  2 tan q 0:5  ð3a  6tÞ

(5.474) (5.475)

5.8 Compressive strength of CFST stub columns

765

For columns with binding bars:  

   ða  b  tÞ b  2t bs tan q bs tan q þ tan q 1  1 ke1 ¼ 1  3nðb  2tÞ 6ða  b  tÞ 2ðb  tÞ 4ða  b  tÞ (5.476)  2  q bs tan q 1 (5.477) ke2 ¼ 1  tan 3 4ðb  tÞ According to Ref. [113], the initial tangent angle q is given by:  p   0:861  0:115A þ 0:961A0:606 ð7:99  0:0317BÞ q¼ 180 (5.478)   p 1:48  0:406C þ 0:0214C2 ð2:23 þ 0:92DÞ
4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi in which A ¼ r2a þ r2b is the combined perimeter of sectional dimension, ra and rb are the width-to-thickness ratios of the a1 -side and b-side steel plates, respectively; B is given by: 8 > < 0:001ða þ bÞ$bs with binding bars s 2 (5.479) B¼ > : 0 without binding bars and C¼

fy fck

(5.480)

a b

(5.481)

D¼

Fig. 5.70 displays the free-body diagram of the L-shaped CFST stub column. Suppose that a length column equal to the longitudinal direction is equivalent to the spacing of binding bars in the longitudinal direction

FIGURE 5.70

Internal forces in an L-shaped CFST stub column with binding bars.

766

5. Design rules and standards

(bs ). According to the static equilibrium equations, fl1 and fl2 can be expressed as: 8 ðnF þ s tb Þ s sh s > > < ½ða  b  tÞbs  with binding bars fl1 ¼ (5.482) > ssh t > : without binding bars ða  b  tÞ fl2 ¼

ðssh1 þ ssh2 Þt ðb  2tÞ

(5.483)

in which n stands for the numbers of binding bars placed over the short pd2 fy;s side direction of the rectangular CFST columns, Fs ¼ s is the force of 4 a binding bar, ssh1 is the hoop stress of the a1 -side steel plates of the Lshaped column, ssh2 is the hoop stress of the short-side (b-side) steel plates of the rectangular columns, ssh3 is the hoop stress of the long-side (a2 -side) steel plates of the rectangular columns. It can beassumed that at the ultimate axial strength, binding bars yield Fs ¼ fy;s .

It is supposed that steel plates are in-plane stress states and yield according to the von Mises yield criteria at the ultimate axial strength. Consequently, the relationships between the longitudinal and hoop stresses of the steel plates can be expressed as: faj2 þ s2shj  faj sshj ¼ fy2

ð j ¼ 1; 2; and 3Þ

(5.484)

As discussed above, steel plates present local buckling at the ultimate axial strength if the width-to-thickness ratio parameter R is greater than 0.85. Otherwise, steel plates yield and no local buckling occurs. The stress states of the steel plates, faj , and sshj ðj ¼ 1; 2; and 3Þ, can be determined according to the value of Rj . Based on Eq. (5.263), Rj can be defined as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffisffiffiffiffiffi 2 fy a 12 1  ws (5.485.a) R1 ¼ Es t 4p2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffisffiffiffiffiffi 2 fy b 12 1  ws R2 ¼ (5.485.b) 2 Es t 4p sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffisffiffiffiffiffi fy a  b 12 1  w2s R3 ¼ (5.485.c) 2 t Es 4p

5.8 Compressive strength of CFST stub columns

767

If Rj < 0:85 ð j ¼ 1; 2; and 3Þ, faj and sshj ð j ¼ 1; 2; and 3Þ are governed by Eqs. (5.369.a) and (5.369.b), respectively, as follows: sshj ¼  0:19sy

(5.486.a)

faj ¼ 0:89sy

(5.486.b)

For Rj  0:85, the values of the faj and sshj can be determined based on Eqs. (5.368.a) and (5.411), as follows: ! 1:2 0:3 faj ¼  sy
0:89sy Rj R2j qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi faj  4s2y  3f 2aj sshj ¼ 2

(5.487.a)

(5.487.b)

5.8.12 T-shaped CFST stub columns with binding bars The method used for developing models for predicting the ultimate axial strength of T-shaped CFST stub columns with and without binding bars is similar to the one employed for L-shaped CFST stub columns with and without binding bars [113] discussed in the previous section. The cross section of a T-shaped CFST column can be divided into square and rectangular CFST columns [114]. Steel tubes serve as longitudinal and stirrups reinforcement bars with close spacing, and binding bars can be considered as tie bars arranged in stirrups. Therefore, the model of Mander et al. [104] for predicting the compressive capacity of the confined concrete core of RC columns can be adopted to assess the loadbearing capacity of axially loaded T-shaped CFST stub columns with binding bars, similar to square and rectangular CFST stub columns [96,97], discussed in previous sections. It is assumed that the T-shaped CFST column consists of four rectangular CFST (RCFST) columns, including a first RCFST, a second RCFST, and two third RCFSTs on the flange legs, as displayed in Fig. 5.71. For ease of calculation, it is supposed that binding bars are uniformly placed over the short side of the second and third RCFSTs. The terms as2 and n2 refer to the spacing of binding bars horizontally and the numbers of binding bars in the second RCFST, respectively. Likewise, as3 and n3 are the spacing of binding bars in the horizontal direction and numbers of binding bars in the third RCFST columns, respectively. The following boundary conditions are assumed for the boundary surfaces:

768

5. Design rules and standards

(a) Divisional T-section

(b) Boundary conditions The division of L-shaped CFST stub columns and boundary conditions. (a) Divisional T-section. (b) Boundary conditions. Zuo Z-L, Cai J, Chen Q-J, Liu X-P, Yang C, Mo T-W. Performance of T-shaped CFST stub columns with binding bars under axial compression. Thin-Walled Struct 2018;129:183e96.

FIGURE 5.71

5.8 Compressive strength of CFST stub columns

769

1. The boundary surfaces have infinite rigid lateral rigidity. However, they can be compressed along the longitudinal direction (see Fig. 5.71b). 2. The boundary surfaces of square and rectangular CFST columns have a uniform longitudinal deformation. The ultimate axial strength of T-shaped CFST stub columns with and without binding bars under axial compression can be expressed as: 1 0 4 X 0 0 0 faj Asj A þ fa5 As5 þ Ac1 fcc1 þ Ac2 fcc2 þ 2Ac3 fcc3 Nu ¼ As fa þ Ac fcc0 ¼ @2 j¼2

(5.488) 0 fcci

where ði ¼ 1 3Þ are the compressive strengths of the concrete core in the first, second, and third RCFST columns, respectively; faj ð j ¼ 1 5Þ are the longitudinal strengths of the steel plates of the a-side, b1 -side, a1 -side, b2 -side, and a2 -side, respectively; Aci ði ¼ 1 3Þ are the areas of the con0 ði ¼ 1 3Þ, crete core corresponding to the compressive strengths of fcci respectively; and Asj ðj ¼ 1 5Þ are the areas of the steel plates corresponding to the longitudinal strengths of faj ð j ¼ 1 5Þ, respectively. Parameters with i ði ¼ 1 3Þ subscripts relate to the first, second, and third RCFST columns, respectively. The stress state of the concrete core at the ultimate axial strength is the 0 ði ¼ 1 3Þ triaxial compressive stress state. The compressive strengths fcci are determined according to the failure criterion of concrete [106] and by using Eqs. (5.453)e(5.459) in which the subscript i is in the range of 1 and 0 and f 0 are the effective lateral confining stresses of the infilled 3. fli1 li2 concrete induced on two sides of RCFST columns and can be determined using the model of Mander et al. [104], as follows: 0 fli1 ¼ kei fli ði ¼ 1  3Þ

(5.489)

0 fli2 ¼ kei fli2 ði ¼ 1  3Þ

(5.490)

where fli1 and fli2 are the confining stresses of the concrete core uniformly induced over two sides of RCFST columns. kei ði ¼ 1 3Þ are confinement effectiveness coefficients can be expressed as the horizontal confinement effectiveness coefficient keih and the longitudinal confinement effectiveness coefficient keil , as follows: kei ¼ keih keil  0 ði ¼ 1  3Þ

(5.491)

in which keih ¼

Aeih  0 ði ¼ 1  3Þ Aci

(5.492.a)

770

5. Design rules and standards

keil ¼

Aeil  0 ði ¼ 1  3Þ Aci

(5.492.b)

in which Aeih refers to the effective confined concrete core areas in horizontal sections with or without binding bars, as displayed in Fig. 5.72a and b, respectively. It can be observed from Fig. 5.72a that the boundaries between confined and unconfined areas of the concrete core in the horizontal sections span between the corner points of the divided RCFST columns and the ends of binding bars. As shown in Fig. 5.72a, there are

(a)

areas for the column with binding bars

(c)

areas for the column with binding bars

(b)

areas for the column without binding bars

(d) side view

FIGURE 5.72 Idealized boundaries between confined and unconfined concrete core areas. (a) Aeih areas for the column with binding bars. (b) Aeih areas for the column without binding bars. (c) Aeil areas for the column with binding bars. (d) side view. Zuo Z-L, Cai J, Chen Q-J, Liu X-P, Yang C, Mo T-W. Performance of T-shaped CFST stub columns with binding bars under axial compression. Thin-Walled Struct 2018;129:183e96.

5.8 Compressive strength of CFST stub columns

771

small gaps between curved corners and their adjacent binding bars. For simplification purposes, these gaps are ignored in the model, and it is assumed that binding bars are uniformly placed in the second and third RCFST columns. For T-shaped columns without binding bars, it is assumed that boundaries span over the corner points of the section, as displayed in Fig. 5.72b. Aeil represents the longitudinal effective confined concrete core areas projected in the horizontal direction, as shown in Fig. 5.72c. Aeil in the divided RCFST columns with binding bars are determined according to the maximum lengths of the boundaries in the longitudinal sections where binding bars are placed, as shown in Fig. 5.72d. bs is the spacing of binding bars in the longitudinal direction, as shown in Fig. 5.72d. In other words, bs is the spread of boundaries in the longitudinal sections. Aeil for columns without binding bars are equal to Aci . It is assumed that the boundaries between the confined and unconfined concrete core areas are formed in 1.5-order parabola curves with the same initial tangent angle q. Consequently, for columns without binding bars keih ði ¼ 1 3Þ and keil ði ¼ 1 3Þ can be expressed correspondingly by: ke1h ¼ 1 

 tan q 0:5 5a2 ða  2tÞ  2a2:5 2 ða  2tÞ 15a2 ðb1  tÞ ke2h ¼ 1 

2ðb2  tÞtan q ða2  2tÞtan q  ða2  2tÞ 5ðb2  tÞ

(5.493) (5.494)

 tan q 0:5 3ða  2tÞ2  5a2 ða  2tÞ þ 2a2:5 2 ða  2tÞ 30ða1  2tÞðb1  2tÞ ða1  tÞtan q ðb1  2tÞtan q   5ðb1  2tÞ 5ða1  tÞ (5.495)

ke3h ¼ 1 

keil ¼ 1 ði ¼ 1  3Þ

(5.496)

keih ði ¼ 1 3Þ and keil ði ¼ 1 3Þ for T-shaped CFST stub columns with binding bars are correspondingly governed by: a2 tan q 5ðb1  tÞ

(5.497)

ke2h ¼ 1 

2ðb2  tÞtan q ða2  2tÞtan q  5ða2  2tÞn2 5ðb2  tÞ

(5.498)

ke3h ¼ 1 

2ða1  tÞtan q ðb1  2tÞtan q  5ðb1  2tÞn3 5ða1  tÞ

(5.499)

ke1h ¼ 1 

772

5. Design rules and standards

bs tan q 3ðb1  tÞ

(5.500)

ke2l ¼ 1 

2bs tan q 3ða2  2tÞ

(5.501)

ke3l ¼ 1 

2bs tan q 3ðb1  2tÞ

(5.502)

ke1l ¼ 1 

According to Ref. [114], the initial tangent angle q is given by:    p  q¼ p11 þ p12 A þ p13 A2 p21 þ p22 B þ p23 B2 p31 þ p32 C þ p33 C2 180   p p41 þ p42 D þ p43 D2
4 (5.503) qffiffiffiffiffiffiffiffiffiffiffiffiffi in which A ¼ ab11 þ ba22 is the combined perimeter of sectional dimension, B is given by: B¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kmax2 $kmax3

(5.504)

where kmax2 and kmax3 represent the width-thickness ratios of steel plates with the largest length in the second and the third RCFST columns, respectively. fy fck pffiffiffiffiffiffiffiffiffiffiffi D ¼ z2 $z3 C¼

(5.505) (5.506)

in which z2 and z3 are the confining coefficients of binding bars for the second and the third RCFST columns, respectively, and are governed by: z2 ¼

As;s fys $ as2 bs fck

(5.507)

z3 ¼

As;s fys $ as3 bs fck

(5.508)

Coefficients p in Eq. (5.503) depend on the binding bars and can be found in Table 5.37. Fig. 5.73 shows the free-body diagram of the T-shaped CFST stub column under axial compression. Based on the static equilibrium equations and by assuming the uniform distribution of the lateral confining stresses

Coefficients p in Eq. (5.503).

Coefficients

p11

p12

p13

p21

p22

p23

p31

p32

p33

p41

p42

p43

With binding bars

0.25

1.97

0.88

41.89

4.54

0.15

24.64

6.53

0.35

0.42

0.31

1.87

Without binding bars

0.20

0.07

0.01

28.12

2.24

0.07

31.01

8.17

0.44







5.8 Compressive strength of CFST stub columns

TABLE 5.37

773

774

5. Design rules and standards

(a) 1st RCFST column

(b) 2nd-RCFST column

rd (c) 3 -RCFST column Internal forces in a T-shaped CFST stub column with binding bars. (a) first RCFST column. (b) second RCFST column. (c) third RCFST column. Zuo Z-L, Cai J, Chen Q-J, Liu X-P, Yang C, Mo T-W. Performance of T-shaped CFST stub columns with binding bars under axial compression. Thin-Walled Struct 2018;129:183e96.

FIGURE 5.73

on the concrete core, the lateral confining stresses of the concrete core fli1 and fli2 ði ¼ 1 3Þ can be expressed as:   n2 Fs þ ðssh1 þ ssh5 Þ bs t fl21 ¼ (5.509) b 2 t 2s (5.510) fl22 ¼ a sh4 2 2 t   n3 Fs þ ssh2 bs t fl31 ¼ (5.511) a 1 2t

5.8 Compressive strength of CFST stub columns

ssh1 þ ssh3 b1 2 t   fl11 ¼ min fl31 ; fl22   fl12 ¼ min fl32 ; fl21 fl32 ¼

775 (5.512)

(5.513) (5.514)

pd2s fy;s is the yielding force of a binding bar, sshj ð j ¼ 1 5Þ 4 are the hoop stresses of the steel plates corresponding to the faj ð j ¼ 1 5Þ, respectively. n2 and n3 should be taken as zero for columns without binding bars. As discussed above, steel plates present local buckling at the ultimate axial strength if the width-to-thickness ratio parameter R is greater than 0.85. Otherwise, steel plates yield and no local buckling occurs. The stress states of the steel plates of T-shaped CFST stub columns under axial compression, faj and sshj ð j ¼ 1 5Þ, can be determined according to the value of Rj . Based on Eq. (5.263), Rj can be defined as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffisffiffiffiffiffi  2 fy 12 1  w a s (5.515.a) R1 ¼ 2 Es t 4p sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffisffiffiffiffiffi  fy b1 12 1  w2s R2 ¼ (5.515.b) 2 t Es 4p in which Fs ¼

R3 and R4 in columns with binding bars: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffisffiffiffiffiffi  2 fy a1 12 1  ws R3 ¼ t Es 4p2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffisffiffiffiffiffi  2 fy b1 12 1  ws R4 ¼ t Es 4p2 R3 and R4 in columns without binding bars: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffisffiffiffiffiffi  fy a1 þ b2 12 1  w2s R3 ¼ R4 ¼ 2 t Es 4p sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ffiffiffiffiffi s   fy a2 12 1  w2s R5 ¼ t Es 4p2

(5.515.c)

(5.515.d)

(5.515.e)

(5.515.f)

In columns without binding bars, a1 -side and b2 -side of steel plates are supposed to deform continuously in the out-of-plane direction at their

776

5. Design rules and standards

Relationship between sshj and faj when Rj  0:85. Zuo Z-L, Cai J, Chen Q-J, Liu X-P, Yang C, Mo T-W. Performance of T-shaped CFST stub columns with binding bars under axial compression. Thin-Walled Struct 2018;129:183e96.

FIGURE 5.74

intersected edge as a continuous steel plate, even though an angle is formed between them. This assumption has been considered in Eq. (5.515.e). If Rj < 0:85 ð j ¼ 1 5Þ, faj and sshj ðj ¼ 1 5Þ are governed by Eqs. (5.369.a) and (5.369.b), respectively. For Rj  0:85, the values of the faj and sshj can be determined based on Eqs. (5.368.a) and (5.411). The relationship betweenfaj and sshj can also be determined according to a simplified stress path before yielding [114], as depicted in Fig. 5.74. It is assumed that the stress path is formed in a second-order parabola curve in which the coordinates of ending points are sshj ¼ 0:19fy and faj ¼ 0:89fy . Therefore: sshj ¼

faj2 4:169fy

(5.516)

5.9 Compressive strength of CFST stub columns with local corrosions 5.9.1 Circular CFST stub columns with local corrosions Corrosion of steel tubes reduces the confinement effect on the infilled concrete. Therefore, the confinement coefficient x should be modified using the adjustment factor 4 considering the less efficient confinement on the infilled concrete provided by the notched steel tube than the same

5.9 Compressive strength of CFST stub columns

777

intact tube [115]. The confinement coefficient x in steel-concrete composite columns with artificial notch is governed by: x¼4

A s fy Ac fc0

(5.517)

Therefore, it is required to determine the f factor for composite stub columns with a slanted notch having the orientation angle q, the length l, the depth d, and the width b. The approach presented here is practical for determining the f factor of any composite column with the notched steel tubes. For instance, by taking q as 0 and 90 degrees, the method considers vertical-direction and horizontal-direction notches, respectively. Additionally, the through-depth notch or the notch with a depth less than the tube thickness can be examined by taking d
t. A steel tube with a notch can be split into inner and outer layers, as shown in Fig. 5.75. The thickness of the inner layer ti and the outer layer to, respectively, are: (5.518) ti ¼ t  d to ¼ d

(5.519)

The f factor for adjusting the confinement effect of infilled concrete from a tube without a notch to that of a notch must be defined. The f factor must combine the effect of the inner and outer layers of the notched steel tube. Therefore: (5.520) 4 ¼ 4i þ 4o
1:0 in which fi and fo consider the effects of the inner and outer layers of the steel tube with a notch on the confinement effect. The fi factor for the inner layer can be expressed as: 4i ¼

fli fl

(5.521)

in which fli is the lateral confining pressure generated by the inner layer of the notched tube, and fl stands for the lateral confining pressure provided by the intact tube. fli and fl are induced on the infilled concrete because of the developed hoop stress in the inner layer of the notched tube and the intact tube, respectively. Based on the equilibrium of forces: fli ¼

2ti sshi Dc

(5.522)

fl ¼

2tssh Dc

(5.523)

where sshi and ssh represent the developed hoop stresses in the inner layer of the steel tube with a notch and the intact steel tube, respectively. According to Ref. [105], the hoop stresses of the tube can be taken as 0.19fy .

778

5. Design rules and standards

Inner layer

Ineffectively confined concrete

Outer layer

Effectively confined concrete

(a) Section B-B Outer layer

Inner layer

Ineffectively confined concrete

Notch Outer layer

(b) Section A-A

(c) Equivalent hoops model

FIGURE 5.75

The cross section of the stub column with a notch depth lower than tube thickness d < t. (a) Section B-B. (b) Section A-A. (c) Equivalent hoops model. Huang H, Guo L, Qu B, Jia C, Elchalakani M. Tests of circular concrete-filled steel tubular stub columns with artificial notches representing local corrosions. Eng Struct 2021;242:112598.

Dc is the diameter of the infilled concrete, as presented in Fig. 5.75, and can be expressed as: Dc ¼ D  2t

(5.524)

Assuming that sshi ¼ ssh as discussed above and replacing Eqs. (5.522) and (5.523) into Eq. (5.21), fi can be expressed as: 4i ¼

ti t

(5.525)

5.9 Compressive strength of CFST stub columns

779

Similarly, fo can be expressed as: 4o ¼

flo0 fl

(5.526)

where flo0 is the effective lateral confining pressure provided by the outer layer in the steel tube with a notch. It should be noted that the existence of the notch in the outer layer of the tube makes the efficiency of the confinement effect provided by the outer layer to be less than the confinement effect provided by the inner layer. flo can be determined based on the notched area of the column, shown in Fig. 5.75. The notched outer layer is unable to develop the hoop stress. Consequently, the confinement effect provided by the outer layer is provided discretely by parts of the tube with the length of s at the ends of the notched area, which are expressed as the equivalent hoops in Fig. 5.75. Consequently, the discontinuous confinement effect afforded by the equivalent hoops can be estimated according to the confinement effect provided by the circular hoop steel reinforcement in a circular reinforced concrete column. flo can be defined based on the equilibrium equation of forces and by supposing that the distribution of flo over the surface of the infilled concrete is uniform. Therefore, flo can be expressed as: 2tssh s  s þ lv 2 Dc =

flo ¼ 

(5.527)

in which ssh is the hoop stress in the equivalent hoop and lv denotes the length of the slanted notch projected to the vertical direction. lv can be calculated as: lv ¼ l cos q

(5.528)

Based on Ref. [105], the maximum hoop stress on intact circular CFST stub columns is 0:19fy . At the limit state, ssh can be taken as the maximum value of 0:19fy , and the value allowed by von Mises yielding criterion. Therefore, ssh can be determined as: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!  sv  4s2y  3s2v    (5.529) ssh ¼ max 0:19fy ;     2 where sv is the longitudinal stress of the steel tube in the equivalent hoop and is associated with yielding of the cross section of the tube with a notch under the longitudinal stress, and is governed by: sv ¼

Asi þ Aso fy As

(5.530)

780

5. Design rules and standards

in which Asi is the cross-sectional area of the inner layer in the notched tube, Aso is the cross-sectional area of the outer layer in the notched tube. Based on Ref. [104], flo0 can be expressed as: flo0 ¼ ke flo

(5.531)

where ke is the confinement effectiveness coefficient and can be calculated as: ke ¼

Ae Ac

(5.532)

in which Ae is the cross-sectional area of the concrete core that is effectively confined by the equivalent hoops. The circular hoop reinforcement can only effectively induce the maximum transverse pressure on the part of the infilled concrete where the confining stress has fully developed owing to arching action. Based on Ref. [104], it is assumed that the arching action is developed in the form of a second-degree parabola with an initial tangent angle of q ¼45 degrees. Therefore, the cross section of the effectively confined infilled concrete happens at the middle of the two equivalent hoops, as shown in Fig. 5.75c. Ae can be defined based on Fig. 5.75c as:   s þ lv 2 Ae ¼ Ac 1  (5.533) 2Dc flo0 can be expressed by replacing Eqs. (5.527), (5.532), and (5.533) into Eq. (5.531) as follows:   s þ lv 2 2tssh s (5.534) flo0 ¼ 1    2Dc s þ lv 2 Dc =

It can be noticed from Eq. (5.334) that flo0 depends on s. The maximum amount of flo0 can be obtained by taking s as: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 3 2lv Dc  l2v  lv 16 4 (5.535) s¼ 2 The infilled concrete over the notched area of the tube is supposed to develop the confined compressive strength fcc . Eq. (5.355) developed by Mander et al. [104] can be used to predict the value of fcc . fl0 in Eq. (5.355) is the effective lateral confining pressure on the infilled concrete and can be expressed as: (5.536) fl0 ¼ fli þ flo0

5.9 Compressive strength of CFST stub columns

781

in which fli and flo0 can be determined using Eqs. (5.527) and (5.534), respectively. In a CFST stub column with the through-thickness notch in the tube, fli must be taken as zero. When the depth of the notch is less than the thickness of the tube, it is supposed that by reaching the limit state, the inner layer of the tube yield under the combination of longitudinal and hoop stresses, whereas the outer layer of the tube yield under the longitudinal stress only over the notched area. It is noteworthy that the interaction between the inner and outer layers of the tube is neglected in this hypothesis. Consequently, the ultimate axial strength of the CFST stub column under axial compression can be expressed as: Nu ¼ fcc0 Ac þ sL Asi þ fy Aso

(5.537)

in which Asi is the cross-sectional area of the inner layer in the notched tube, Aso is the cross-sectional area of the outer layer in the notched tube, sL represents the maximum compressive stress in the inner layer of the steel tube obtained when the inner layer yields under the biaxial stress condition. Based on Ref. [105], sL can be taken as 0:89fy . Fig. 5.76 displays an idealized model for circular CFST columns with vertical or slanted notches in the steel tubes. When the notch depth is equal to the thickness of the tube (through-thickness notch), part of the tube over the notched area does not form the hoop stress. In this case, the

FIGURE 5.76 An idealized model for composite columns with vertical or slanted notches in the steel tubes. Huang H, Guo L, Qu B, Jia C, Elchalakani M. Tests of circular concrete-filled steel tubular stub columns with artificial notches representing local corrosions. Eng Struct 2021;242:112598.

782

5. Design rules and standards

behavior of the notched tube depends on the property of the extremely outer fix-fix tube strip that has a unit width, as displayed in Fig. 5.77. Therefore, the local buckling would happen at the limit state in the areas near the notch (for instance, see Fig. 5.77b), except the length of the strip lv is small enough. For predicting the occurrence of local buckling, parameter lcr is defined as the strip length of the tube associated with the tube strip elastic buckling stress of fy . It can be assumed that the local buckling happens if lv  lcr . It is supposed that by the occurrence of local buckling, each locally buckled region extends along the 45 degree directions on the surface of the tube, as depicted in Fig. 5.76. Therefore, the arc length of each locally buckled zone ðBÞ can be expressed as: B¼

lv  lcr 2

(5.538)

At the limit state, the area outside the locally buckled zones is supposed to reach the longitudinal stress of fy . Additionally, the average longitudinal stress over each locally buckled zone ðsLB Þ is supposed to be: sLB ¼

scr  fy
fy 2

(5.539)

where scr is the critical elastic buckling stress in the extremely outer fix-fix tube strip displayed in Fig. 5.76. Subsequently, the ultimate axial strength of the circular CFST stub column with the through-thickness notch in the tube in which the tube local buckling behavior occurs can be expressed as: Nu ¼ fcc0 Ac þ sLB AsB þ fy AsNB

(5.540)

in which AsB is the cross-sectional area of the locally buckled zones and AsNB is the cross-sectional area of the tube outside of the locally buckled zones over the notched tube cross section. When lv < lcr , no local buckling occurs. In this case, Eq. (5.540) can be used by taking AsB as zero. It was discussed in Chapter 2 that when the projected lengths of a slanted notch and a vertical notch in the vertical direction are the same, they lead to the identical strength in a circular CFST stub column. Accordingly, the slanted notch in a circular CFST stub column can be replaced by a vertical notch with the same projected length in the vertical direction and the same width and depth, as displayed in Fig. 5.75. By doing that, the relevant models determined above for predicting the strength of the stub column with a vertical notch in the steel tube can be then utilized.

(d) With a horizontal notch

FIGURE 5.77 Failure modes of circular CFST stub columns with artificial notches in the steel tubes. (a) Intact column. (b) With a vertical notch. (c) With a slanted notch. (d) With a horizontal notch. Huang H, Guo L, Qu B, Jia C, Elchalakani M. Tests of circular concrete-filled steel tubular stub columns with artificial notches representing local corrosions. Eng Struct 2021;242:112598.

783

(c) With a slanted notch

(b) With a vertical notch

5.9 Compressive strength of CFST stub columns

(a) Intact column

784

5. Design rules and standards

5.9.2 Square CFST stub columns with local corrosions As discussed above, design codes employed the superimposing constituent components’ contributions for predicting the compressive strength of square CFST stub columns under axial compression and ignored the confinement effect. Using the same method, the compressive strength of square CFST stub columns under axial compression can be calculated as the following: Nu ¼ fc0 Ac þ fy As

(5.541)

The loading patch is cut for the square column with a notch in the steel tube because of the notch. Part of the steel plate around the notch cannot transfer the axial load directly. Besides, the confinement effect in the zone close to the notch would reduce. Contrary to the circular CFST stub column, the effect of the notch on the confinement effect of the square CFST column is neglected. This is because the compressive strength of the square CFST stub column can be taken as the summation of compression strength of constituent components since the confinement effect provided by the square tube is weak. According to Ref. [116], the ultimate axial strength of the notched square CFST stub columns can be derived by modifying Eq. (5.541) as follows: Nu ¼ fc0 Ac þ fy Ase

(5.542)

in which Ase represents the effective cross-sectional area of the steel tube. According to Fig. 5.78, Ase can be expressed as: Ase ¼ 4tðB  tÞ  lh d

(5.543)

in which B is the width of the square CFST column with a notch in the steel tube, t is the thickness of the steel tube, lh stands for the length of the slanted notch projected to the horizontal direction, and d is the depth of the notch. lh can be expressed as: lh ¼ l sin q þ b cos q

(5.544)

in whichl and b are the length and width of notch. Accordingly, Ase can be derived by replacing Eq. (5.544) into Eq. (5.543) as: Ase ¼ 4tðB  tÞ  ðl sin q þ b cos qÞd

(5.545)

5.10 Strain compatibility between the steel tube

(a) side view

785

(b) Cross-section A-A

FIGURE 5.78 The notched square CFST stub column. (a) side view (b) cross section A-A. Guo L, Huang H, Jia C, Romanov K. Axial behavior of square CFST with local corrosion simulated by artificial notch. J Constr Steel Res 2020;174:106314.

5.10 Strain compatibility between the steel tube and the concrete core In the design of composite members, it is required to ensure that steel tube yield before the core concrete achieves its maximum stress. This is to guarantee that the entire plastic resistance of the composite section is fully used. This is especially important for composite members made by HSC and ultra-high-strength concrete (UHSC) due to the brittle nature of HSC and UHSC. Consequently, the steel yield strength and the concrete compressive strength must be selected so that the steel yield strain is smaller than the compressive strain of the concrete corresponding to the peak stress. The yield strain of the steel tube εy is given by: fy (5.546) Es According to Ref. [117], the axial strain εcu corresponding to the compressive strength of the concrete is given by: εy ¼

0:31 εcu ¼ 0:7fcm

(5.547)

where the term fcm ðMPaÞ denotes the mean compressive strength of concrete at 28 days. The strain compatibility between the steel tube grade

786 TABLE 5.38

5. Design rules and standards

Compatibility between the steel grade and concrete class for composite columns.

and the concrete core class can be achieved if εy < εcu . Table 5.38 provides the appropriate selection of steel grade and concrete class in which the criterion εy < εcu has been satisfied [118]. As shown in Table 5.38, HSSs up to Grade S550 can be used for confining the concrete class up to C190. It should be noted that the effect of the confinement on increasing εcu is ignored in this table. The following equation can be used for determining the highest steel yield strength based on the characteristic concrete strength with strength class up to C190 [118]:

 0:31  ; 550 (5.548) fy ¼ min 0:7Es fck þ 8

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[52] Huo J, Huang G, Xiao Y. Effects of sustained axial load and cooling phase on post-fire behaviour of concrete-filled steel tubular stub columns. J Constr Steel Res 2009;65(8): 1664e76. [53] Uenaka K, Kitoh H, Sonoda K. Concrete filled double skin circular stub columns under compression. Thin-Walled Struct 2010;48(1):19e24. [54] Tao Z, Han L-H, Zhao X-L. Behaviour of concrete-filled double skin (CHS inner and CHS outer) steel tubular stub columns and beam-columns. J Constr Steel Res 2004; 60(8):1129e58. [55] Hassanein MF, Kharoob OF. Compressive strength of circular concrete-filled double skin tubular short columns. Thin-Walled Struct 2014;77:165e73. [56] Yang Y-F, Han L-H, Sun B-H. Experimental behaviour of partially loaded concrete filled double-skin steel tube (CFDST) sections. J Constr Steel Res 2012;71:63e73. [57] Li W, Ren Q-X, Han L-H, Zhao X-L. Behaviour of tapered concrete-filled double skin steel tubular (CFDST) stub columns. Thin-Walled Struct 2012;57:37e48. [58] Han L-H, Ren Q-X, Li W. Tests on stub stainless steeleconcreteecarbon steel doubleskin tubular (DST) columns. J Constr Steel Res 2011;67(3):437e52. [59] Hassanein MF, Kharoob OF, Liang QQ. Circular concrete-filled double skin tubular short columns with external stainless steel tubes under axial compression. ThinWalled Struct 2013;73:252e63. [60] Elchalakani M, Zhao X-L, Grzebieta R. Tests on concrete filled double-skin (CHS outer and SHS inner) composite short columns under axial compression. Thin-Walled Struct 2002;40(5):415e41. [61] Standards Australia. Steel structures. Sydney: Standards Australia: Australian Standard AS4100; 1998. [62] Han L-H, Tao Z, Huang H, Zhao X-L. Concrete-filled double skin (SHS outer and CHS inner) steel tubular beam-columns. Thin-Walled Struct 2004;42(9):1329e55. [63] Ayough P, Ramli Sulong NH, Ibrahim Z, Hsiao P-C. Nonlinear analysis of square concrete-filled double-skin steel tubular columns under axial compression. Eng Struct 2020;216:110678. [64] Tao Z, Han L-H. Behaviour of concrete-filled double skin rectangular steel tubular beamecolumns. J Constr Steel Res 2006;62(7):631e46. [65] Zhao X-L, Grzebieta R. Strength and ductility of concrete filled double skin (SHS inner and SHS outer) tubes. Thin-Walled Struct 2002;40(2):199e213. [66] Yang J, Xu H, Peng G. Behavior of concrete-filled double skin steel tubular columns with octagon section under axial compression. Front Architect Civ Eng China 2008; 2(3):205e10. [67] Yuan W-B, Yang J-J. Experimental and numerical studies of short concrete-filled double skin composite tube columns under axially compressive loads. J Constr Steel Res 2013;80:23e31. [68] Yang J, Xu H, Peng G. A study on the behavior of concrete-filled double skin steel tubular columns of octagon section under axial compression. Tu Mu Gong Cheng Xue Bao 2007;40(2):33e8. [69] Alqawzai S, Chen K, Shen L, Ding M, Yang B, Elchalakani M. Behavior of octagonal concrete-filled double-skin steel tube stub columns under axial compression. J Constr Steel Res 2020;170:106115. [70] Wang J, Cheng X, Wu C, Hou C-C. Analytical behavior of dodecagonal concrete-filled double skin tubular (CFDST) columns under axial compression. J Constr Steel Res 2019;162:105743. [71] Li W, Han L-H, Zhao X-L. Axial strength of concrete-filled double skin steel tubular (CFDST) columns with preload on steel tubes. Thin-Walled Struct 2012;56:9e20.

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¨ zkan O ¨ . Influence of high temperature on the properties of con_ Siddique R, O [72] Yu¨ksel I, cretes made with industrial by-products as fine aggregate replacement. Construct Build Mater 2011;25(2):967e72. [73] Hassanein MF, Kharoob OF. Analysis of circular concrete-filled double skin tubular slender columns with external stainless steel tubes. Thin-Walled Struct 2014;79:23e37. [74] Han L-H. Flexural behaviour of concrete-filled steel tubes. J Constr Steel Res 2004; 60(2):313e37. [75] Liang QQ, Fragomeni S. Nonlinear analysis of circular concrete-filled steel tubular short columns under eccentric loading. J Constr Steel Res 2010;66(2):159e69. [76] Patel VI, Liang QQ, Hadi MNS. Nonlinear analysis of biaxially loaded rectangular concrete-filled stainless steel tubular slender beam-columns. Eng Struct 2017;140: 120e33. [77] Han L-H, Lu H, Yao G-H, Liao F-Y. Further study on the flexural behaviour of concrete-filled steel tubes. J Constr Steel Res 2006;62(6):554e65. [78] Han L-H, Huang H, Zhao X-L. Analytical behaviour of concrete-filled double skin steel tubular (CFDST) beam-columns under cyclic loading. Thin-Walled Struct 2009; 47(6):668e80. [79] Kazemzadeh Azad S, Uy B. Effect of concrete infill on local buckling capacity of circular tubes. J Constr Steel Res 2020;165:105899. [80] Timoshenko S, Gere JM. Theory of elastic stability. New York: McGraw-Hill; 1961. [81] Bleich F. Buckling strength of metal structures. McGraw-Hill; 1952. [82] Song Y, Li J, Chen Y. Local and post-local buckling of normal/high strength steel sections with concrete infill. Thin-Walled Struct 2019;138:155e69. [83] Liang QQ, Uy B, Richard Liew JY. Local buckling of steel plates in concrete-filled thinwalled steel tubular beamecolumns. J Constr Steel Res 2007;63(3):396e405. [84] Uy B, Bradford MA. Elastic local buckling of steel plates in composite steel-concrete members. Eng Struct 1996;18(3):193e200. [85] Liang QQ, Uy B. Theoretical study on the post-local buckling of steel plates in concrete-filled box columns. Comput Struct 2000;75(5):479e90. [86] Trahair NS, MA B. The behaviour and design of steel structures to AS4100. ed. 3. London: E & FN Spon; 1998. [87] Von Karman T. The strength of thin plates in compression. Trans. ASME 1932;54:53e7. [88] Winter G. Strength of thin steel compression flanges. Trans. ASCE 1947;112:527. [89] Nakai H, Kitada T, Yoshikawa O. A design method of steel plate elements in concrete filled square steel tubular columns. Dob Gakkai Ronbunshu 1985;1985(356):405e13. [90] Ge HB, Usami T. Strength analysis of concrete-filled thin-walled steel box columns. J Constr Steel Res 1994;30(3):259e81. [91] Liang Q, Uy B. Parametric study on the structural behaviour of steel plates in concretefilled fabricated thin-walled box columns. Adv Struct Eng 1998;2:57e71. [92] Ayough P, Sulong NHR, Ibrahim Z. Analysis and review of concrete-filled double skin steel tubes under compression. Thin-Walled Struct 2020;148:106495. [93] Bradford MA, Loh HY, Uy B. Slenderness limits for filled circular steel tubes. J Constr Steel Res 2002;58(2):243e52. [94] Long Y-L, Zeng L. A refined model for local buckling of rectangular CFST columns with binding bars. Thin-Walled Struct 2018;132:431e41. [95] Cai J, Long Y-L. Local buckling of steel plates in rectangular CFTcolumns with binding bars. J Constr Steel Res 2009;65(4):965e72. [96] Cai J, He Z-Q. Axial load behavior of square CFT stub column with binding bars. J Constr Steel Res 2006;62(5):472e83. [97] Cai J, Long Y-L. Axial load behavior of rectangular CFT stub columns with binding bars. Adv Struct Eng 2007;10(5):551e65.

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[98] Ling Y, Feng WY, Zhao JH, Li Y. Study on the ultimate bearing capacity of concrete filled steel square tubular short column with PBL. Adv Mater Res 2014;941e944: 770e5. [99] Zhang ZH, Liu P, He SH, Ji TG. Research on influence factors and formulae for the bearing capacity of twin-PBL shear connectors. Appl Mech Mater 2013;353e356: 3167e71. [100] Zhang JG. Research on the mechanical performance of concrete-filled square steel tube stiffened with PBL under axial compression load. Xi’an: Chang’an University; n.d. [101] Tao Z, Han L-H, Wang Z-B. Experimental behaviour of stiffened concrete-filled thinwalled hollow steel structural (HSS) stub columns. J Constr Steel Res 2005;61(7): 962e83. [102] DBJ1351. Technical specification for concrete-filled steel tubular structures [in Chinese]. Fuzhou: The Construction Department of Fujian Province; 2003. [103] Liang W, Dong JF, Yuan SC, Wang QY. Behavior of self-compacting concrete-filled steel tube columns with inclined stiffener ribs under axial compression. Strength Mater 2017;49(1):125e32. [104] Mander JB, Priestley MJN, Park R. Theoretical stress-strain model for confined concrete. J Struct Eng 1988;114(8):1804e26. [105] Sakino K, Nakahara H, Morino S, Nishiyama I. Behavior of centrally loaded concretefilled steel-tube short columns. J Struct Eng 2004;130(2):180e8. [106] Guo ZH, Wang CZ. Investigation of strength and failure criterion of concrete under multi-axial stresses. China Civ Eng J 1991;24:1e14. [107] Lai MH, Ho JCM. Behaviour of uni-axially loaded concrete-filled-steel-tube columns confined by external rings. Struct Des Tall Special Build 2014;23(6):403e26. [108] Ho JCM, Lai MH. Behaviour of uni-axially loaded CFST columns confined by tie bars. J Constr Steel Res 2013;83:37e50. [109] Lai MH, Ho JCM. Confinement effect of ring-confined concrete-filled-steel-tube columns under uni-axial load. Eng Struct 2014;67:123e41. [110] Zhong ST. Unified theory of CFST: research and application. first edn. Beijing, China: Tsinghua University Press; 2006 [In Chinese]. [111] Cai S, Jiao Z. Behavior and ultimate strength of short concrete-filled steel tubular columns. J Build Struct 1984;5(6):13e29. [112] Lai MH, Ho JCM. Effect of continuous spirals on uni-axial strength and ductility of CFST columns. J Constr Steel Res 2015;104:235e49. [113] Zuo Z-L, Cai J, Yang C, Chen Q-J, Sun G. Axial load behavior of L-shaped CFT stub columns with binding bars. Eng Struct 2012;37:88e98. [114] Zuo Z-L, Cai J, Chen Q-J, Liu X-P, Yang C, Mo T-W. Performance of T-shaped CFST stub columns with binding bars under axial compression. Thin-Walled Struct 2018; 129:183e96. [115] Huang H, Guo L, Qu B, Jia C, Elchalakani M. Tests of circular concrete-filled steel tubular stub columns with artificial notches representing local corrosions. Eng Struct 2021;242:112598. [116] Guo L, Huang H, Jia C, Romanov K. Axial behavior of square CFST with local corrosion simulated by artificial notch. J Constr Steel Res 2020;174:106314. [117] Eurocode C. Design of concrete structures e part 1-1: general rules and rules for buildings, EN1992-1-1. CEN, Brussels: European Committee for Standardization; 2004. [118] Liew JYR, Xiong M, Xiong D. Design of concrete filled tubular beam-columns with high strength steel and concrete. Structures 2016;8:213e26.

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C H A P T E R

6 Future research O U T L I N E 6.1 Introduction

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6.2 Material properties 6.2.1 Carbon steel tube and concrete core 6.2.2 Stainless steel tube

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6.1 Introduction Over the last decades, extensive experimental and numerical studies have been performed on composite members. Burr [4] conducted some of the initial experiments on concrete-filled steel tubular (CFST) columns in 1912, followed by the experimental works of Teichmann et al. [42] on concretefilled double skin steel tubular (CFDST) columns fabricated by aluminum tubes filled with the cellular-cellulose-acetate (CCA) core in 1951.

Single Skin and Double Skin Concrete Filled Tubular Structures https://doi.org/10.1016/B978-0-323-85596-9.00003-2

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The appropriate results of composite members achieved from the previous tests convinced Montague [34] to examine the feasibility of using composite members to construct chambers in deep-sea. The satisfactory results of the studies stated above motivated researchers to further investigate CFST and CFDST members. Since then, various researches have been performed concerning the structural performance of composite members. Various cross-section geometries were examined in the previous works, such as circular, rectangular, elliptical, round-ended rectangular, and polygonal. Recently, the effects of using steel tubes fabricated by corrugated plates were also investigated [18,19,48]. Different configurations of the above cross-sections were used for the outer and inner steel tubes of CFDST members. Concerning loading conditions, experimental and numerical examinations have been performed on composite members that cover compressive, tensile, torsional, bending, and combined loading conditions. The imposed loads were monotonic or cyclic. Besides, the effects of load eccentricity and partial compression have been investigated. Furthermore, the behavior of composite members under fire, blast loads, and impact loads have been studied. In terms of material imperfections, the influences of concrete imperfections, namely circumferential and spherical gaps, and steel imperfections, namely, initial geometric imperfections and corrosion, have been analyzed. Lastly, the effects of preloading of the steel tubes induced by fresh concrete weight, selfweight of the tubes, and other constructional loads have also been discussed. Regarding material properties, different types of steel and concrete material have been used by researchers for investigating the performance of composite members fabricated by different materials. The examined steel tubes consisted of mild carbon steel, high strength steel, stainless steel, high-performance fire-resistant steel, and aluminum. For filling the steel tubes, various types of concrete, including normal strength concrete (NSC), high-strength concrete (HSC), and ultra-high-strength concrete (UHSC), were used. Recently, the use of other concrete types, such as polymer concrete, foam concrete, rubberized concrete, recycled aggregate concrete, and concrete made by oil palm boiler clinker or agricultural solid wastes, has been examined. One of the most popular types of concrete between researchers is the self-consolidating concrete (SCC) since the whole part of the framework can be filled using the own weight of SCC, and there is no need for mechanical compaction. Despite the numerous conducted research on composite members and the achieved valuable information regarding their structural performance, increasing use of CFST and CFDST elements in civil engineering projects, especially in high-rise buildings and bridges, necessitates further studies on these types of composite members. This chapter discusses the current challenges and provides a future research direction.

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6.2 Material properties 6.2.1 Carbon steel tube and concrete core Fig. 6.1 presents the histograms of the steel tube yield strength and concrete compressive strength used in the experimental studies conducted on CFST and CFDST members over the last decades. It should be noted that elliptical composite members are considered as the circular members, and rectangular, square, round-ended rectangular, hexagonal, octagonal, and dodecagonal composite members are considered as the polygonal members. As shown in Fig. 6.1, the majority of studies were performed on composite members built with normal strength steel tubes having fy
460 MPa and NSC having fc0
50 MPa. Regarding circular specimens, almost 85.5% of studies used normal strength steel tubes. Besides, 70.5% of

FIGURE 6.1 The material strength of tested composite members.

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research studies have employed NSC, 17.8% have used HSC 50 MPa < fc0


90 MPa , and only 11.7% utilized UHSC 90 MPa < fc0 . Concerning polygonal specimens, 73% of studies used normal strength steel tubes. 57.75% of research performed on polygonal specimens was based on NSC, whereas the percentage is reduced to 18.5% and 24.05% for HSC and UHSC, respectively. Due to the recent advances in the materials and construction industries, the use of high-strength materials has increased over the past decade. Table 6.1 summarizes some modern structures in which highstrength materials have been employed. Using high-strength materials lets the structural designer achieve a more economical design by reducing the required size of the elements and leading to a significant reduction in the structure’s overall dead load. Besides, the strength-to-weight ratio of members can be enhanced using high-strength materials. Despite the appropriate performance of composite members made of high strength materials and the increasing use of them in modern structures, the design of composite structures with high-strength materials is not allowable by most of the design codes, as shown in Fig. 6.2 based on [53]. It can be observed from Fig. 6.2a that except for the Canadian steel structures design code [5] and the American code ACI [22], other regulations do not TABLE 6.1 The use of high-strength materials in modern structures. Strength of materials

Name

Location

Height

Petronas Twin Towers

Kuala Lumpur, Malaysia

88-Storey, 375 m

Grade 80 HSC

Sail

Marina Bay, Singapore

70-Storey, 245 m

Grade 80 HSC

International Commerce Centre

Hong Kong

110-Storey, 480 m

Grade 90 HSC

World Financial Center

Shanghai, China

101-Storey, 492 m

Grade 450 steel plate

Sky Tree

Tokyo

634 m

Grade 700 steel tube

Techno Station

Japan

780 MPa high strength steel 160 MPA UHSC

Latitude Building

Sydney

51-Storey, 222 m

690 MPa high strength steel 80 MPa HSC

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FIGURE 6.2 The allowable range of material strengths recommended by different design codes. Ayough P, Sulong NHR, Ibrahim Z. Analysis and review of concrete-filled double skin steel tubes under compression. Thin-Walled Struct 2020;148:106495. https://doi.org/10.1016/j.tws.2019. 106495. (a) Steel, (b) Concrete.

allow the use of high-strength steel material or define limitations for it. The maximum steel strength is limited to 525 MPa by AISC 360-16 [41], whereas this design code defines no lower limit. The maximum allowable steel yield strength in Ref. [52] and European design code EC4 [35] is limited to 400 and 460 MPa, respectively, and the recommended lower limit in both design codes is 230 MPa. The lower and higher limits of the steel strength defined by the Chinese design code DBJ 13-51-2010 [8] are 235 and 420 MPa, respectively, which is close to AS 5100 and EC4. Similarly, the Japanese design standard, AIJ [36], only lets the use of steel material with a strength ranging from 235 to 440 MPa. It should be noted that filling hollow steel sections with concrete enhances composite members’ ductility and buckling resistance. Nevertheless, the range of steel strength for the design of composite members permitted by EC4 is more conservative than that defined for bare steel sections by EC3 [7,17].

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Accordingly, further experimental studies are required to assess the application of materials with strength beyond the limits defined by design codes and modify the current limits on material strength.

6.2.2 Stainless steel tube Appropriate stainless steel properties than conventional carbon steel, such as its excellent durability, fire resistance, corrosion resistance, and ease of maintenance, have convinced researchers and structural engineers to consider it as a proper structural material. However, the expensive cost of stainless steel may limit its wide use in the construction industry. A possible solution to reduce the high cost of stainless steel is to employ them in composite members. Over the last decade, extensive experimental and numerical studies were performed on stainless composite members. However, there are still more research areas to understand better the behavior of composite members with stainless steel tubes. Although the fire performance of composite members with stainless steel tubes has attracted increasing interest among researchers, further research work is required to be done about the structural fire design of composite members with stainless steel tubes. It is noteworthy that a performance-based method for the fire design of composite members with stainless steel tubes is recommended. This is because no insulation is expected to be used on the outer surface of stainless steel tubes.

6.3 Geometric properties Fig. 6.3 shows the histograms of the steel tube slenderness ratios investigated in the experimental studies conducted on CFST and CFDST members over the last decades. It should be noted that elliptical composite members are considered as the circular members, and rectangular, square, round-ended rectangular, hexagonal, octagonal, and dodecagonal composite members are considered as the polygonal members. As shown in Fig. 6.3a, most circular specimens were fabricated using compact/ noncompact steel tubes, and slender circular steel tubes consisted of only 2.25% of studies. With respect to polygonal steel tubes, 96% of studies were based on compact/noncompact steel tubes, and only 4% of investigations were according to the use of slender steel tubes. The steel tube thickness can be reduced using high-strength steel. Therefore, more efforts are required to assess the feasibility of using composite members built with high-strength slender steel tubes. Design of plated structural components in bare steel sections based on post-buckling strength design methods such as the effective width method is allowed by design codes like EC3 and AISC 360-16. However,

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FIGURE 6.3 The slenderness ratios of tested composite members. (a) Circular, (b) Polygonal.

the use of the postbuckling behavior in the design of composite sections is generally not permitted by most design codes such as AISC 360-16 and EC4. Design codes apply width to thickness limits to steel plate components of composite members to ensure that the section yields prior to the occurrence of local buckling, as presented in Table 5.2 in Chapter 5. Ignoring the post-buckling strength of plated structural elements in the design of composite members may lead to uneconomic design outcomes. Therefore, it may be required to modify the slenderness limits defined by the design codes by performing more experimental and analytical tests on composite members built with slenderness ratios beyond the design codes’ limitations.

6.4 Nonuniform confinement As discussed in Chapters 2e4, steel tubes of straight CFST and CFDST members with the ideal surface can uniformly confine the infilled concrete. Researchers have well investigated the uniform confinement effect in composite members, and precise numerical models have been developed to predict the uniform confinement effect provided by various steel tube sections. However, in some cases, the provided confinement effect may be nonuniform. The cross-section of tapered and straight-tapered-straight (STS) composite columns alters gradually over the longitudinal axis. Therefore, the steel tube’s width-to-thickness ratio varies along the column’s longitudinal axis, which leads to the nonuniform confinement effect provided by

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the steel tube. Performing the numerical modeling of tapered and STS composite members by incorporating the variation of confinement effect over the column height may be challenging. Therefore, researchers tried to use simplified methods to predict the concrete core’s constitutive behavior in tapered and STS composite members. A possible solution for modeling tapered and STS composite members is to split the column height into very small sections according to the shape of the profile, and later different confinement effect is applied to each section, based on the section change. This method can reflect the confinement effect with reasonable accuracy; however, it can be time-consuming. A more straightforward method is to apply the confinement effect of the weakest part of the composite member to the whole model in the numerical modeling. This is because the weakest section governs the compressive capacity of the column. The section of the inclined composite members over the longitudinal axis does not change. Nevertheless, the horizontal restraints at the column ends generate bending moments that lead to the nonuniform confinement effect in the inclined composite members, which is less than the confinement effect in the straight counterpart. The difference between the confinement effect in the inclined composite member with the straight counterpart depends on the inclined angle. To date, no theoretical formulation has been developed for considering the confinement effect in tapered and STS composite members. Besides, there is no empirical model to consider the loss of the confinement effect due to the column inclination. Therefore, theoretical studies are required to develop material constitutive relationships for confined concrete in tapered and STS composite members, as well as inclined composite members. Another issue that can cause the nonuniform confinement effect is the corrosion of steel tubes. Based on the report presented by the World Corrosion Organization (WCO), the corrosion of structures causes the loss of almost 2.4 trillion dollars each year [45]. According to the steel damage characteristics, two types of corrosions can be developed over the steel surface, i.e., local corrosion and uniform corrosion [3]. Local corrosion means that the corrosion is localized in a specific area of the metal surface. When corrosion is developed uniformly over the metal surface, it is called uniform corrosion. In practical engineering, the chance of the occurrence of local corrosion is remarkably more than that of uniform corrosion. According to the statistics, more than 80% of the whole corrosion damage was due to local corrosion [3,45]. The corroded part of the steel tube cannot confine the concrete core effectively due to the reduction in its wall thickness. As a result, corroded composite members’ performance would degrade compared with that of the undamaged counterpart. Another effect of the corrosion is changing the section centroid position that causes the composite column to be subjected to eccentric load even if the axial

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load is induced on the column’s center. This issue will further deteriorate the structural performance of the corroded composite member. Although the use of composite members in offshore structures and submarine pipeline structures is increasing worldwide, research on the effects of corrosion has been limited to a small number of experimental and analytical studies. Therefore, more efforts are required to fully understand the behavior of corroded composite members, especially the ones that suffer from local corrosion, and develop a design-oriented material law for concrete core under nonuniform confinement developed by corroded steel tubes.

6.5 Fire performance of composite members One of the most severe accidents in engineering structures is fire, which can cause economic losses and, more importantly, casualties. Due to the outstanding performance of composite members and their popularity among structural engineers, the fire resistance of CFST and CFDST members has always been an important research topic between researchers. The experimental test results indicate that the composite members’ fire resistance reduces with increasing the normal concrete strength [6,20,23,38]. By contrast, the use of HSC can enhance the fire resistance of composite members [33,44,49]. It can be observed that there is no consensus regarding the influence of concrete strength on the fire performance of composite members. Hence, the effect of concrete strength, particularly HSC, on composite members’ structural behavior should be further examined. The collapse mechanisms because of the fire damage of composite members can be considered as another research gap. Another aspect is investigating the performance of composite members subjected to multi-hazards, including fire. Concerning the loading condition, more research needs to be performed to further investigate the fire performance of composite members subjected to tensile loading and torsion. Regarding the cross-section shapes of composite members subjected to high temperature, most studies have been conducted on circular and rectangular members. Therefore, the fire resistance of other shapes of composite members can be examined in the future.

6.6 Stiffened composite members The structural performance of composite members having various kinds of stiffening methods has been wildly assessed over the past 2 decades using experimental, numerical, and analytical studies. Different

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types of stiffeners used in composite members are discussed in detail in Chapter 2. From the whole studies concerned the behavior of stiffened composite members, more than 64% of them were performed on the behavior of stiffened composite columns subjected to the axial compressive load. In contrast, other loading conditions, i.e., bending, combined loading, dynamic, and thermal loading, generally consist of only 37% of the conducted studies [1]. Therefore, more research is required to be done on stiffened composite members to fully understand the behavior. The influences of applying PBL ribs stiffeners on the behavior of CFST short columns were investigated by Ling et al. [30]. Zhang et al. [51] performed experimental and numerical tests on the performance of CFST slender columns having PBL ribs stiffeners. It was clarified that PBL ribs stiffeners have an insignificant influence on the tested specimens’ compressive capacity. Nevertheless, they can positively improve the ductility of columns. However, their tests were limited to a small number of experimental results. Given that less amount of steel material is required for PBL ribs stiffeners than the ordinary ribs stiffeners and taking into account that they can enhance the ductility and shear strength of composite columns, more study can be done in the future on the behavior of composite members with PBL ribs stiffeners. Experimental test results [28] clarified that the use of inclined steel plates as stiffeners in rectangular composite columns could improve the overall performance of columns, such as the buckling strength of the steel tube, the ultimate axial strength, and ductility of the column. However, no study has been conducted to compare the performance of the longitudinal plate stiffeners with the inclined plate stiffeners. This issue can be considered as a future research topic. As discussed in Chapter 2, circular steel tubes can be stiffened with welding ring bars, battlement-shaped tie bars with or without longitudinal bars, regular longitudinal ribs, and horizontal plate rings. The performance of these types of stiffeners in circular CFST columns under axial compression was examined by Yang et al. [55]. In general, it was observed that, except for the horizontal plate rings, other types of stiffeners could improve the performance of composite columns. The performance of circular CFST columns with horizontal plate rings stiffeners was weak compared with the nonstiffened counterparts because the fresh concrete could not flow over the entire part of the column and made gaps under the stiffeners. Therefore, further studies require to be conducted on the performance of horizontal plate ring stiffeners by using selfconsolidating concrete (SCC). In general, employing stiffeners can reduce the required tube wall thickness by improving the steel tube’s buckling strength. Nevertheless, no research has been conducted on the slenderness impact of stiffened

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circular composite columns. Besides, the performance of stiffened circular composite columns under eccentric loading has not been investigated. Figs. 6.4e6.5, modified from the study performed by Alatshan et al. [1], give information regarding the number of studies performed on composite members concerning the cross-section shape, position of the stiffeners, and loading conditions. The majority of the studies were performed on rectangular composite members. The popularity of investigating the performance of stiffeners in rectangular composite members compared

FIGURE 6.4 The proportion of studies performed on internal and external stiffeners.

FIGURE 6.5 The number of studies performed on each kind of stiffeners based on crosssection shape and loading condition. Alatshan F, Osman SA, Hamid R, Mashiri F. Stiffened concrete-filled steel tubes: a systematic review. Thin-Walled Struct 2020;148:106590. https://doi. org/10.1016/j.tws.2019.106590. The number of studies performed on each kind of stiffeners based on cross-section shape and loading condition.

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with circular composite members between researchers may be due to the lower buckling strength and the poor distribution of concrete confinement effect in composite members made with flat steel plates. Another point is that limited research was performed on the performance of composite members with external stiffeners, which can be because of their better appearance of internal stiffeners than the external ones in terms of architectural and aesthetic criteria. However, for strengthening and seismic rehabilitation of the constructed composite structures, there may be a need for using external stiffeners. Therefore, there is still a demand for assessing the performance of external stiffeners on improving the seismic performance of composite members. The majority of the conducted tests were focused on the axial performance of the stiffened composite columns, and the available studies regarding the stiffened composite beams are rare. Additionally, the available discussions regarding the effects of stiffeners in composite beams are not comprehensive. Therefore, further investigations need to be focused on the influence of using different types of stiffeners on the behavior of composite beams having different width-to-thickness ratios. Besides, there is no research concerning the influence of employing external stiffeners in composite beams. Experimental research performed on the stiffened rectangular and special-shaped composite members subjected to combined loading conditions indicated that using stiffeners can positively improve members’ buckling strength, capacity, and ductility. Nevertheless, no research focused on the performance of stiffened circular composite members under combined loading conditions. Besides, the effects of using external stiffeners in composite members subjected to combined loading actions were not investigated. Different types of internal stiffeners, such as headed shear studs, were used in composite members under axial compression to investigate the feasibility of using stiffeners in improving the bond action. The push-out test results showed that stiffeners could improve the bond strength between the steel tube and the concrete core when the load was induced on the concrete core. However, there is no information regarding the bond behavior and the influence of stiffeners on bond action in composite members under other types of loadings like bending. Another topic that needs further investigation is the bond behavior of fire exposed composite members. Although using stiffeners can enhance composite members’ bond strength subjected to thermal loading, the performed research on this issue is quite limited. In general, using stiffeners in composite members can enhance the seismic performance of composite members. The experimental test results demonstrated that stiffeners could be placed only at the weak plastic hinge zones to strengthen the seismic resisting of composite members. This issue can prevent a large increase in the utilized amount of steel

6.7 Environmentally sustainable material

805

material. However, no research has been conducted on the seismic rehabilitation of composite members by adding external stiffeners. The research results on the seismic performance of composite members with external stiffeners can become of importance in the seismic rehabilitation of structures located in high-risk seismic zones. As shown in Fig. 6.5, no research has been conducted on the performance of stiffened composite members under tensile loading and torsion. Therefore, further research can be focused on these topics. Composite columns are a popular structural member in the design of high-rise structures due to their advantages in increasing the structure’s overall stiffness and stability. Although the use of composite columns in structures typically leads to smaller designs compared with the conventional RC columns or bare steel columns, the size of composite columns in high-rise buildings may still be a challenge for structural engineers. Therefore, a combination of different types of stiffeners can be employed to assure adequate buckling resistance and composite action in composite members with large cross-sections. For instance, T-shaped rib stiffeners and headed shear studs were used in circular CFST columns in Wuhan Center Tower, in Wuhan, China. In Taipei 101 tower in Taipei, Taiwan, eight large rectangular CFST columns were used at four faces of the tower. The size of columns from the ground floor up to the 62nd floor is 2:4 m  3:0 m and they were constructed using high-performance concrete with 69 MPa compressive strength. For higher floors, rectangular hollow steel sections were used only to reduce the induced dead load on the tower [39]. Three different types of stiffeners were used to design these supercolumns, namely, headed studs, internal longitudinal ribs, and orthogonal tension bars. Despite the potential of using the combination of stiffeners in practice, no research has been performed to evaluate the performance of large size composite columns with a combination of stiffeners. Lastly, choosing the proper kind of stiffeners is financially important, especially in the design of high-rise structures. As a result, a full costsafety balanced analysis of employing various stiffeners in composite members can be done in the future.

6.7 Environmentally sustainable material Environmental problems caused by the construction industry nowadays are one of the main concerns of modern societies. Excessive extraction of natural resources, emission of greenhouse gas, pollutants release, and the volume of produced waste are only some of the concerns regarding the devastating effects of the construction industry. Therefore, there is an urgent need to develop sustainable construction to protect the

806

6. Future research

ecosystem, preserve natural resources, and enhance the living organisms’ habitat conditions. A possible solution for achieving environmentally friendly constructions is to recycle and reuse construction wastes. To this end, researchers have tried to assess the feasibility of processing construction wastes to produce recycled aggregates to construct concrete structures. This type of concrete, called recycled aggregate concrete (RAC), is an efficient approach to construct green concrete structures. Compared with natural aggregate concrete (NAC), the durability, elastic modulus, and compressive and tensile strengths of RAC are lower. By contrast, RAC demonstrates higher shrinkage, creep, and ductility than NAC [31,32,43,46]. Confining RAC by steel tubes can be considered as an efficient method to improve the shortcomings of RAC. Replacing NAC by RAC in composite members for enhancing the mechanical characteristics of RAC was first examined by Konno et al. [24]. Since then, several experiments were conducted to assess the efficient employment of RAC in different kinds of steel tubes, i.e., carbon steel tubes, stainless steel tubes, and carbon steel tubes reinforced with fiber-reinforced polymers. However, there are still some research gaps regarding the performance of RAC composite members. The behavior of RAC composite members subjected to different loading conditions such as axial compression, bending, combined loading condition, and cyclic loading has been well evaluated over the last decades. Nevertheless, the main focus of the performed studies was on composite members having circular and rectangular crosssections. Besides, the number of performed tests on RACFDST members is limited. Moreover, more efforts are needed to investigate the mechanical properties of composite members filled with RAC subjected to fire. Another concern is the seismic performance of beam-to-column connections in composite members with RAC. Therefore, design models are required for the joints of RACFSTs and RACFDSTs. Another environmental problem is waste tires recycling management, especially in developed countries. This is because the durability of waste tires is high, and they required a wide waste dumping landfill area. According to statistics, the United States produces more than 20.5 million tons of tire rubber waste annually, and almost 90% is recycled yearly [14]. In Europe, this number is increased to almost 30 million tons per year, while only less than 70% of them are recycled. The produced tire rubber waste per year in Australia is 50 million [14]. The landfill disposal of waste tires has devastating repercussions on the environment. Due to the chemical compound of tires, the occurrence of a fire accident in a tire landfill can lead to poisonous emissions. Consequently, the management of waste tires is a challenging environmental concern. An alternative solution to reduce the required landfill disposal and protect natural resources is to use waste tires in the construction industry using Rubberized Concrete (RuC). Replacing the natural aggregate of concrete

6.7 Environmentally sustainable material

807

with rubber particles changes the brittle behavior of NAC to a ductile material. Besides, using rubber particles in concrete can enhance the viscous damping and kinetic energy of the concrete [21]. However, similar to RAC, the tensile and compressive strengths of RuC are smaller than NAC. The drawbacks of RuC can be improved by confining it with steel tubes and using RuCFST or RuCFDST members [56]. The axial performance of RuCFST having circular and square cross-sections was examined by Duarte et al. [11,12]. Silva et al. [40] investigated the flexural behavior of circular CFST through experimental tests. Regarding the cyclic performance of RuCFST, Duarte et al. [12] performed an experimental study on a total of 12 circular, rectangular, and square RuCFSTs. Concerning RuCFST members under combined loading conditions, Dong et al. examined the performance of circular and square RuCFSTs columns, beams, and beam-columns [9,10]. The only available test results about the axial performance of RuCFDST columns are from the experimental works of Elchalakani et al. [15,16] conducted on circular and square stub columns. Recently, the fire performance of RuCFST has been investigated through experimental tests by researchers such as Duarte et al. [13]. It is evident that all the performed studies were focused on short columns having circular and square cross-sections. Besides, the available test results regarding the cyclic and fire performance of composite members are limited. Also, there is no research about the bond action of RuC composite members. Hence, further studies must be performed on RuCFST and RuCFDST subjected to cyclic loading, heat loading, tensile loading, torsion, and push-out loading. Exorbitant need for the fine aggregate and freshwater as the raw materials required for the construction of concrete members in recent years has led to excessive exploitation of natural resources worldwide. Hence, experimental tests were performed on seawater and sea sand concrete (SWSSC) behavior to produce green concrete. SWSSC consists of sea sand and seawater and alkali-activated geopolymer [25,26]. The short-term mechanical characteristics of SWSSC and the conventional Portland cement concrete are the same [27]. Therefore, NAC in composite members can be replaced by SWSSC to build a sustainable structural member and decrease the environmental problems caused by using normal material. The main problem with using SWSSC is its large composition of chloride iron and lightweight impurities that can increase the chance of corrosion in the typical carbon steel tube. A practical solution to solve the corrosion problem in the conventional composite members is to use stainless steel tubes due to their outstanding durability and corrosion resistance. Concerning the behavior of CFST and CFDST members with stainless steel tubes and SWSCC, only a limited number of experimental tests were performed on circular and square stub columns by Liao et al. [29] and Li et al. [25,26]. Therefore, further studies should be done on other shapes of

808

6. Future research

composite members with SWSCC having different slenderness ratios subjected to other loading conditions, such as bending, combined loading, tensile, torsion, and cyclic loadings. Besides, the structural behavior of SWSCC composite members subjected to long-term sustained loading should be assessed. Another possible solution for reducing the excessive exploitation of river sand resources for concrete constructions is replacing river sand with dune sand in the concrete mix design, especially in countries with vast desert areas. In China, for instance, almost a quarter of the country’s land area consists of desert areas [47]. It should be noted that the fineness modulus of dune sand is lower in comparison with river sand. Besides, river sand possesses poor gradation. Also, the alkaliesilica reaction must be considered as a possible detrimental effect if dune sand is used [51]. Further investigation is required to investigate the alkaliesilica reaction in dune sand concrete. The experimental test results of Zhang et al. [50] indicated that dune sand concrete could be employed for general engineering construction. Al-Harthy et al. [2] reported that replacing the river sand aggregate with dune sand reduces the workability and compressive strength of the concrete. Full replacement of fine aggregates with dune sand leads to a maximum 25% strength loss. To conclude, dune sand concrete usually possesses large cohesiveness, inadequate workability, and weak fluidity. Besides, the compressive strength and elastic modulus of concrete are reduced by using dune sand in the mixtures [54]. A possible solution for mitigating dune sand’s defects is to confine the concrete with a steel tube. The behavior of CFST beams and columns fabricated with dune sand concrete was first examined by Wang et al. [47], where 10% of fine aggregate was replaced by dune sand. Later, Ren et al. [37] examined the performance of short CFST columns with dune sand as the fine aggregate. The test results showed that using dune sand as the fine aggregate has no detrimental influence on the failure mechanism of CFST members, and dune sand concrete can work efficiently when it is confined by the steel tube. The investigation on dune sand composite members is still limited, and no research has been performed on dune sand CFST and CFDST members under various loading conditions such as cyclic and long-term loadings. Hence, further studies are required to assess the long-term performance of dune sand composite members and study the effects of concrete shrinkage and creep.

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Index Note: ‘Page numbers followed by “f ” indicate figures and “t” indicate tables.’

A ABAQUS software, 2e3, 343, 376e378 implementing modified DeP type plasticity model in, 356e359 Active confinement, 190e191 Age-adjusted effective modulus method, 237 Agricultural solid waste, 794 AIJ (Japanese code), 15 AISC 360e16 (American steel design code), 15, 553e554, 798e799 concrete, 22 design capacity of CFST columns subjected to axial compression, 558e559 moment capacity of CFST beams based on design codes, 587e588 resistance of CFST members under combined axial loading and bending based on design rules, 607e609 first method, 609e635 second method, 611e622 second-order effect, 618e619 steel, 17 Aluminum, 171 American steel design code AISC 360e16, 15 Analytical axial stressestrain model for circular CFST columns, 298e305 Analytical methods axial and lateral stressestrain response of CFST column, 284e298 constitutive model for computing lateral strain of confined concrete, 279e284 creep model, 277e279 elastoplastic model for stressestrain response of CFST columns, 308e313 fundamental step in, 168 path-dependent stressestrain model for CFST columns, 305e308 strength enhancement induced during cold forming, 314e321 stressestrain response of materials, 168e279 Analytical stress-strain models, 22e23

813

ANSYS software, 2e3, 343, 376e378 AS5100 (Australian code), 15 AS5100 part 6, steel, 16 buckling curves, 555 compressive strength of CFST columns subjected to axial compression, 546e548 concrete, 17 design capacity of CFST columns subjected to axial compression, 556e557 determining resistance of CFST members under combined axial loading and bending based on design rules, 602e604 moment capacity of CFST beams based on design codes, 582e584 Associated flow rule, 582 Austenitic stainless steels, 179e180 Axial bending, 24 Axial compression, 24 CFDST members under, 626 CFST members under axial compression, 620e622 compressive stiffness of composite short columns, 623e626, 636te642t ultimate axial strain of composite short columns, 623e625 compressive strength of CFST columns subjected to, 546e554 design capacity of CFST columns subjected to, 556e559 test, 37 Axial loadebending moment interaction curve (NeM interaction curve), 634 Axial loadedisplacement curves for CHSs, 41 Axial stressestrain response of CFST column, 284e298

B Beams, 144e153 Behavior of composite members through numerical analysis, 385e419, 443e444

814

Index

Behavior of composite members through numerical analysis (Continued) inclined composite members, 435e441 mechanism of interaction of steel tubes and concrete, 419e424 development of interaction between steel tubes and concrete core, 424e427 influence of geometric and material properties on confinement effect, 427e441 short columns, 385e387 effects of concrete compressive strength, 388e391 effects of depth-to-thickness ratio of inner steel tube, 403e406 effects of hollow ratio, 395e402 effects of inner steel tube yield strength, 392e395 effects of local buckling, 412e419 effects of outer steel tube yield strength, 391e392 effects of outer tube depth-to-thickness ratio, 387e388 effects of section shapes, 419e427 increasing material strength efficiency on load-bearing capacity of composite members, 406e410 simultaneous effects of depth-tothickness ratio and concrete compressive strength, 410e412 slender columns, 443e444 effect of tapered and inclined angles on ultimate axial capacity of composite members, 441e531 tapered composite members, 434 effects of column slenderness ratio, 453e457 effects of concrete compressive strength, 460e463 effects of concrete confinement, 475e478 effects of hollow ratio, 463e464 effects of load eccentricity ratio, 465e472 effects of width-to-thickness ratio, 457e460 failure modes, 444e452 load distribution between constituent components of slender composite members, 465e468 loadestrain curve, 453 preloading of composite columns, 481e528 pure bending strengths of slender beamcolumns, 481

ultimate axial strengths of slender beamcolumns, 478e480 Bending CFDST members under, 655 CFST members under, 626e635 Bilinear plus nonlinear hardening model, 172 Binding bars, 75 Binici’s model, 201 Boltzmann principle of superposition, 239 Brittle mode, 43 BS 5400 (British bridge code), 15 BS5400 part, steel 5, 16 BS5400-part 5, buckling curves, 555e556 compressive strength of CFST columns subjected to axial compression, 548e551 concrete, 21 design capacity of CFST columns subjected to axial compression, 557e558 moment capacity of CFST beams based on design codes, 584e586 Buckling curves, 555e556 AS5100, 555 BS5400-Part 5, 555e556 DBJ13e51, 556 EC4, 556 Buckling mode, 41

C Cantilever launching techniques, 10e13 Canton Tower in Guangzhou, China, 8, 11f Carbon steel, 31e32, 168e178. See also Stainless steel comparison of experimental stressestrain curves with different material models, 176f engineering stressestrain curve of, 168f existing stressestrain model for hotrolled carbon steel material, 170f idealized piecewise-linear model for structural carbon steel, 169f input parameters for defining stressestrain curve of square steel tubes, 177t material coefficients for CSM material model, 174t series of standard structural steel grades, 177t tube, 795e798 use of high-strength materials, 796t CDP model. See Concrete damage plasticity model (CDP model)

Index

Cellularcellulose-acetate (CCA), 30e31, 793e794 CFDST. See Concrete-filled double skin steel tubular (CFDST) CFST. See Concrete-filled steel tubular (CFST) Circular CFST, 3e4 columns, 12e13 members, 192e225 Circular columns, 43e48 biaxial stress condition, 44f failure mode of LOC+CC, 43f failure mode of SLB+SC, 42f Circular hollow section (CHS), 3e4, 32 Circular hollow steel sections, 17 Circular steel hollow sections filled with concrete, 678e685 Circumferential cap gaps, 135e143 Circumferential stress, 45e46 Circumferential tensile stress, 45e46 Classical elastic-perfectly plastic curve, 170e171 Cold forming, strength enhancement induced during, 314e321, 314f Cold rolling method, 309 Cold-formed process. See Hot-rolled process Collapse mechanisms, 801 Column(s), 143e153 beams, 70e72 under eccentric load, 63e65 with outer stainless steel tube, 66 under partial compression load, 54e58 with rubberized concrete core, 66e70 slenderness ratio effects on load eccentricity, 115e116 effects on steel tube, 453e457 Compaction of concrete core, 129e135 Composite beams, 70e72 circular composite beams filled, 68f failure mechanism of RuCFST under, 67f typical momenterotation response, 67f Composite columns, 805 stiffeners in, 72e89 Composite frame systems, 6e7 Composite member(s), 13 application, 6e13 Canton tower, 11f CFDST columns application in electricity transmission tower, 15f of composite members in bridges, 12f example of submerged tube tunnel, 7f mega CFST column cross section, 11f

815 in power plant workshop, 14f Ruifeng International Commercial Building in Hangzhou, China, 10f schematic view of composite structural system, 8f SEG plaza in Shenzhen, China, 9f in subway stations, 13f Wangchang East River Bridge, China, 12f Zhoushan electricity transmission tower, 14f based on research works, strength of, 619e620 CFDST members under axial compression, 626 under bending, 655 under combined axial loading and bending, 670e671 CFST members under axial compression, 620e622 under bending, 626e635 design strengths of circular CFST short columns, 636te642t of circular CFST short columns with stainless steel tube, 650te651t of elliptical CFST short columns, 647t of hexagonal CFST short columns, 645t of octagonal CFST short columns, 646t of rectangular CFST short columns, 643te644t of round-ended rectangular CFST short columns, 648te649t of square CFST short columns with stainless steel tube, 652te653t failure modes, 43e72 circular columns, 43e48 columns under eccentric load, 63e65 columns under partial compression load, 54e58 columns with outer stainless steel tube, 66 columns with rubberized concrete core, 66e70 composite beams, 70e72 corrugated columns, 50e51 inner steel tube of CFDST columns, 53e54 polygonal columns, 52e53 rectangular/square columns, 48e50 slender columns, 59e63 tapered columns, 58e59

816

Index

Composite member(s) (Continued) future research of carbon steel tube and concrete core, 795e798 environmentally sustainable material, 805e808 fire performance of composite members, 801 geometric properties, 798e799, 799f material properties, 795e798 nonuniform confinement, 799e801 stainless steel tube, 798 stiffened composite members, 801e805 Composite structures, 2 Compressive design strength of composite members based on design guidelines, 546e577 buckling curves, 555e556 compressive strength of CFST columns subjected to axial compression, 546e554 design capacity of CFST columns subjected to axial compression, 556e559 design strength of CFDST columns subjected to axial compression, 568e575, 578t examples, 559e577 Compressive stiffness of composite short columns, 623e626 Compressive strength of concrete core, 463 Concrete, 2, 17e22 AISC 360e16, 22 AS5100, 17 BS5400, 21 confinement, 190e279 effects on composite columns, 475e478 contribution ratio, 568, 688 core, 43, 81, 795e798 beams, 144e153 circumferential and spherical cap gaps, 135e143 columns, 143e153 compaction of, 129e135 imperfections, 128e143 maximum limit for gap ratio, 143e144 DBJ13e51, 21 effects of material properties, 105e112 Eurocode 4, 21 filling inner steel tube with, 120e125 made by oil palm boiler clinker, 794 material properties of, 23t mechanism of interaction of steel tubes and, 419e424

tensile region, 61e62 compressive strength, 21, 795e796 effects of, 388e391, 460e463 effects on load eccentricity, 117e118 on ultimate axial strength of CFDST column, 410e412 Concrete damage plasticity model (CDP model), 23e24, 360e362 modifications, 375e383 Concrete plasticity model, 23e24, 344e359 assessment of presented DeP type plasticity model, 351e354 DeP type plasticity model modifications, 354e356 flow rule, 349e350 implementing modified DeP type plasticity model in ABAQUS, 356e359 strain hardening and softening, 349 yield criterion, 345e348 Concrete-filled double skin steel tubular (CFDST), 2, 30e31 beams, 70e71 columns, 4e5, 793e794 design strengths of CFDST long columns, 668t design strengths of CFDST short columns with CHS outer and CHS inner tubes, 656te657t with CHS outer and SHS inner tubes, 658t with elliptical or round-ended rectangular steel tubes, 662t with RHS outer and RHS inner tubes, 661t with SHS outer and CHS inner tubes, 659t with SHS outer and SHS inner tubes, 660t design strengths of dodecagonal CFDST short columns, 665t design strengths of preloaded CFDST short columns with CHS inner and CHS outer steel tubes, 666t with SHS inner and CHS outer steel tubes, 667t erection of CFDST columns, 5e6, 5f shapes of composite members, 6f members, 4e5, 233e238 advantages, 2e5 under axial compression, 626 under bending, 655

Index

under combined axial loading and bending, 670e671 example, 655e673 geometric and material properties of section, 655 stressestrain model of Hu and Su, 233e234 stressestrain model of Liang et al., 234e236 stressestrain model of Zhao et al., 236e237 simultaneous effects of depth-tothickness ratio and concrete compressive strength, 410e412 symbols of CFDST, 669f Word Table 23 design strengths of octagonal CFDST short columns, 663te664t Concrete-filled steel tubular (CFST), 2, 30e31, 542e543 CFST/CFDST columns, 32 members, 31e32 column, 4e5, 793e794 analytical axial stressestrain model for circular, 298e305 axial and lateral stressestrain response of, 284e298 elastoplastic model for stressestrain response of, 308e313 path-dependent stressestrain model for, 305e308 compressive strength of CFST columns subjected to axial compression, 546e554 AISC 360e16, 553e554 AS5100, 546e548 BS5400, 548e551 DBJ13e51, 551 Eurocode 4, 552e553 slenderness limits of steel sections of composite members, 550t design capacity of CFST columns subjected to axial compression, 556e559 AISC360e16, 558e559 AS5100, 556e557 BS5400, 557e558 DBJ13e51, 558 EC4, 558 design strength of CFDST columns subjected to axial compression, 568e575

817

erection of CFST columns, 5e6 members, 5 advantages, 2e5 under bending, 626e635 flexural stiffness of CFST beams, 654 moment M-curvature 4 response of CFST beams, 654e655 simplified moment capacity model of Elchalakani et al., 626e635 simplified moment capacity model of Han, 635e677 simplified moment capacity model of Liang and Fragomeni, 635e655 under axial compression, 620e622 circular and square, 192e225 hexagonal and octagonal CFST, 230e233 round-ended rectangular and elliptical CFST, 225e230 members under combined axial loading and bending, 600e602 example, 619e622 resistance of CFST members under combined axial loading and bending based on design rules, 602e604 second-order effect, 611e615 Concrete-steel contribution ratio (CSCR), 115 Confine concrete, 190e279 constitutive model for computing lateral strain of, 279e284 constitutive stressestrain model for, 191e279 free body diagram for CFDST columns, 190f Confinement, 190e191 factor, 40, 424 effects of, 102e104 ratio, 200e201 Constitutive models, 344 for computing lateral strain of confined concrete, 279e284 Constitutive stressestrain model for confined concrete, 191e279 CFDST members, 233e238 of CFST/CFDST members under longterm sustained loading, 241 circular and square CFST members, 192e225 stressestrain model of Attard and Setung, 193e197 stressestrain model of Binici, 197e202

818

Index

Constitutive stressestrain model for confined concrete (Continued) stressestrain model of Ellobody et al., 223 stressestrain model of Han et al., 222e223 stressestrain model of Hu et al., 213e215 stressestrain model of Liang et al., 219e222 stressestrain model of Lim and Ozbakkaloglu, 206e211 stressestrain model of Mander et al., 192e193 stressestrain model of Portoles et al., 223e225 stressestrain model of Samani and Attard, 202e206 stressestrain model of Susantha et al., 216e219 stressestrain model of Tao et al., 211e213 dual skin concrete-filled steel tube columns, 237e238 hexagonal and octagonal CFST members, 230e233 stressestrain model of Hassanein et al., 230e231 stressestrain model of Patel, 232 stressestrain model of Susantha et al., 232e233 proposed models for stressestrain curve of concrete in circular and square CFDSTs, 270te273t in circular and square CFSTs, 248te257t in circular and square DCFSTs, 274te276t in hexagonal and octagonal CFSTs, 267te269t in round-ended rectangular and elliptical CFSTs, 258te266t under triaxial compression, 242te247t round-ended rectangular and elliptical CFST members, 225e230 stressestrain model of Ahmed and Liang, 225e228 stressestrain model of Dai and Lam, 228e230 stressestrain model of Patel, 226e227, 230 Construction process, 7, 31 Continuous strength method (CSM), 171

Corrugated columns, 50e51 comparison of failure mechanism, 49f variation of corrugated steel plates, 47f Corrugated steel tubes, 48e50 Coupon specimens, 33e36 Cracks’ extensions pattern, 56e57 Creep compliance function, 238 Creep model, 277e279 Creep Poisson’s ratio, 238e239 Cubic strength (fcu), 21 Curling ribs, 75 Cyclic loadings, 808

D DBJ13e51 (Chinese design code), 15, 796e798 buckling curves, 556 compressive strength of CFST columns subjected to axial compression, 551 concrete, 21 design capacity of CFST columns subjected to axial compression, 558 determining resistance of CFST members, 604e609 moment capacity of CFST beams based on design codes, 586e592 steel, 17 DCFST columns. See Dual-skin concretefilled steel tube columns (DCFST columns) Depth-to-thickness ratio, 463 of inner steel tube, 403e406 simultaneous effects on ultimate axial strength of CFDST column, 410e412 Design rules and standards CFST members under combined axial loading and bending, 600e602 compressive design strength of composite members based on design guidelines, 546e577 limitations of design regulations on strength of materials and section slenderness, 542e543 local buckling of steel plates, 671 moment design strength of composite members based on design guidelines, 575e577 strain compatibility between steel tube and concrete core, 685e689 strength of composite members based on research works, 619e620 Dilation angle, 374 effects on FE results, 384e531

Index

Double power-law function, 238 “Double skin” composite construction, 6 DruckerePrager type plasticity model (DeP type plasticity model), 23e24, 344e359 assessment of presented, 351e354 implementing modified DeP type plasticity model in ABAQUS, 356e359 modifications, 354e356 Dual-skin concrete-filled steel tube columns (DCFST columns), 237e238, 310 required modifications for predicting stressestrain response of concrete core in with CHS outer and CHS inner steel tubes, 238e241 with SHS outer and CHS inner steel tubes, 239e240 Ductility index (DI), 103e104 Duplex stainless steels, 179e180 Dynamic modulus, 238

E EC3 (European design codes), 798e799 Eccentric load, columns under, 63e65 composite columns under eccentric compression, 61f eccentrically loaded circular concretefilled double steel tubular columns, 62f eccentrically loaded rectangular concretefilled double steel tubular columns, 63f severe tensile crack, 62f Eccentricity ratio effects on load eccentricity, 116e117 Economic members, 2 Effective elastic flexural stiffness, 548 Effective modulus method, 237 Effective Poisson’s ratio, 238e239 Effective width method, 740, 798e799 Elastic local buckling of steel plates, 671e672. See also Post-local buckling of steel plates steel hollow sections, 672e673 steel hollow sections filled with concrete, 673e676 Elastic modulus of concrete, 225e226 Elastic Poisson’s ratio, 238e239 Elasticelinear hardening model, 171

819

Elasticeperfectly plastic steel material model, 375 Elasticeplastic buckling, 434 Elastoplastic model, 302 for stressestrain response of CFST columns, 308e313 Elephant-foot-type buckling, 41 Elliptical CFST members, 225e230 Elliptical composite members, 798 Environmentally sustainable material, 805e808 Equilibrium principle, 598 Equivalent cylinder strength, 36e37 Erection process, 5e6 Eurocode 4 (EC4), 15, 552e553, 798e799 buckling curves, 556 concrete, 21 design capacity of CFST columns subjected to axial compression, 558 moment capacity of CFST beams based on design codes, 586 resistance of CFST members under combined axial loading and bending based on design rules, 607e609 second-order effect, 615e618 steel, 17 Experimental procedure, typical, 33e38 typical tensile coupon specimen, 36f Experimental tests confinement factor, effects of, 102e104 cross-sectional shapes, 33 straight, inclined, and tapered columns, 35f typical cross sections of CFDSTs, 34f experiments, 33e38 effective parameters, 39e40 typical experimental procedure, 33e38 failure modes, 41e72 composite member failure modes, 43e72 hollow tube failure modes, 41e43 hollow ratio, 100e102 load eccentricity, 113e119 long-term sustained load, effects of, 153e163 material imperfections, 125e153 material properties, effects of, 105e113 concrete, 105e112 steel tube, 112e113 preload, effects of, 148e153 role of inner steel tube, 119e125 size effects, 90e100

820

Index

Experimental tests (Continued) steel ratio effect on concrete damage status, 85f typical axial loadedisplacement curve of CFST/CFDST columns, 84f typical vertical stress vs. strain relationship of circular and square steel tubes, 86f variety of columns’ size in arch bridges, 83f stiffeners in composite columns, 72e89 Extended DeP model, 357 External confining methods, 190e191

F Fabrication of hollow steel tubes, 6 Failure modes, 444e452 Ferritic stainless steels, 179e180 Fiber-reinforced polymers (FRPs), 190e191, 458 Filling inner steel tube with concrete, 120e125 Finite element (FE), 2e3, 168, 343 analysis software, 376e378 dilation angle effects on FE results, 384e531 effects of ratio of second stress invariant on tensile meridian on FE results, 380e383 effects of stressestrain model of concrete material on FE results, 378e380 of steel material on FE results, 376e378 Fire performance of composite members, 801 Flat steel plates, rectangular section with, 577e609 Flexural stiffness index (FSI), 127 “Flip disk” mechanism, 41e43 Flow rule, 349e350, 367e375 concrete plasticity model, 360e362 concrete damage plasticity model modifications, 375e383 damage, 363e365 strain hardening and softening, 366e367 yield criterion, 365e366 different parameters effects on numerical results, 376 dilation angle effects on FE results, 384e531 effects of ratio of second stress invariant on tensile meridian on FE results, 380e383

effects of stressestrain model of concrete material on FE results, 378e380 effects of stressestrain model of steel material on FE results, 376e378 Foam concrete, 794 Framework, 33

G Gardner and Nethercot’s model, 185 Generalized Hooke’s Law, 365 Generalized multistage model, 187e188 Geometric properties of composite members, 798e799, 799f Gyration, 198

H Headed shear studs, 804e805 Hexagonal CFST members, 230e233 High strength concrete (HSC), 15e16, 33, 794 High-strength materials, 542e543, 796e798 in modern structures, 796t Hollow ratio, 39, 100e102 effects of, 395e402 effects on slender CFDST columns, 463e464 Hollow tube failure modes, 41e43 axial loadedisplacement curves for CHSs, 38f elephant-foot-like buckling in CHSs, 36f roof-type mechanism in SHSs, 38f typical failure modes, 42f Hooke’s Law, 170e171, 372e373 Hoopeaxial strain relationships, 412 “Horizontal swing” techniques, 10e12 Hot-rolled process, 6 Hybrid structural systems, 6e7

I Idealized piecewise-linear model, 169e170, 169f Inclined composite members, 435e441 effect on ultimate axial capacity of composite members, 441e531 Inflection point stress ratio, 201e202, 384 Initial geometric imperfections, 2e3 Inner steel tube of CFDST columns, 53e54 failure mechanism of polygon CFDST stub columns, 50f typical failure mode of corrugated circular CFST columns, 49f

Index

effects of inner steel tube yield strength, 392e395 role of, 119e125 effects of inner steel tube, 119e120 filling inner steel tube with concrete, 120e125 Internal stiffeners, 804 Internal transverse reinforcement, 190e191 International design guidelines, 15 Inversion of stressestrain relationship, 189

L Lateral load resisting system, 6e7 Lateral stressestrain response of CFST column, 284e298 Least-squares procedure, 175 Liang and Uy’s model, 754 Liang’s model, 178 Lightweight concrete (LWC), 21, 206e211 Linear variable displacement transducers (LVDTs), 37 Load eccentricity, 113e119, 517 deformed shape of CFST/CFDST members under eccentric load, 90f effects of column slenderness ratio, 115e116 effects of concrete compressive strength, 117e118 effects of steel tube yield stress, 118e119 ratio, 463 effects of, 116e117 effects on slender CFDST columns, 465e472 Load-bearing capacity, 45e46 Loading conditions, 2e3 Local buckling, 798e799 effects on composite members, 412e419 of steel plates, 671 elastic local buckling of steel plates, 671e672 post-local buckling of steel plates, 674e676 Local corrosion, 800e801 Long-term loadings, 808 Lu¨ders strain, 168e169

M Masonry, 2 Material(s), 15e24, 16t behavior, 188 coefficient for CSM material model, 173, 174t

821

concrete, 17e22 imperfections, 31, 125e153 concrete core imperfections, 128e143 steel plate imperfections, 125e128 steel, 16e17 strength efficiency on load-bearing capacity of composite members, 406e410 Maximum limit for gap ratio, 143e144 Mirambell and Real’s model, 183e185 Modified hardening/softening rule, 359 Modified RambergeOsgood model, 183 Modified size effect law (MSEL), 81e82 MohreCoulomb friction law, 200 Moment capacity of CFST beams, 592e599 based on design codes, 581e582 AISC360e16, 587e588 AS5100, 582e584 BS5400, 584e586 DBJ13e51, 586e592 EC4, 586 example, 599e602 Moment design strength of composite members based on design guidelines, 575e577 moment strength of CFST beams, 575e577 examples, 588e589 moment capacity of CFDST beams, 592e599 plastic moment capacity of CFST beams, 577 Multilinear stressestrain curve, 171e172

N Nakai’s model, 703 Natural aggregate concrete (NAC), 806 NeM interaction curve. See Axial loadebending moment interaction curve (NeM interaction curve) Nonassociated flow rule, 354 Nonlinear FE analysis, 386e387, 426, 457, 459 Nonuniform confinement, 799e801 effect, 799e800 Normal concrete (NC), 15e16, 65 Normal strength concrete (NSC), 15e16, 31e32, 794 Normal weight concrete (NWC), 21, 206e211 Normal-strength steel (NSS), 105 Numerical computing, 168 Numerical methods of confined concrete core, 343e383

822

Index

Numerical methods of confined concrete core (Continued) behavior of composite members through numerical analysis, 385e419 concrete plasticity model, 344e359 flow rule, 359e375

O Octagonal CFST members, 230e233 Ordinary longitudinal ribs, 72 Orthogonal tension bars, 75 Outer stainless steel tube, columns with, 66 Outer steel tube yield strength, effects of, 391e392 Outer tube depth-to-thickness ratio effects of steel tubes, 387e388

circular steel hollow sections filled with concrete, 678e685 examples, 678e682 rectangular steel hollow sections filled with concrete, 678e706 Postbuckling behavior, 798e799 strength design methods, 798 PrandtleReuss equations, 326 Preloading of composite columns, 481e528 Proportional limit, 180 Pure bending strengths of slender beamcolumns, 481

Q Quad-linear stressestrain model, 173 Quadratic function, 286

P Partial compression load, columns under, 54e58 CFDST columns under partial compression, 52f partially loaded CFDST columns, 53f Passive confinement, 190e191 Path-dependent stressestrain model for CFST columns, 305e308 PBL ribs stiffeners, 802 Piecewise-linear model, 170 Plastic moment capacity of CFST beams, 577 circular section, 580e586 rectangular section with flat steel plates, 577e609 with round-ended corners, 577e586 Plasticity, 344 Poisson’s ratio, 44e45 of concrete, 291, 458 effective, 238e239 of RuC, 65 of steel, 48, 81 tangent, 628 Polygon confinement effectiveness coefficient, 408 steel tubes, 51 Polygonal columns, 52e53 Polymer concrete, 794 Post-local buckling of steel plates, 674e676. See also Elastic local buckling of steel plates steel hollow sections, 675e676 steel hollow sections filled with concrete, 676e677

R RambergeOsgood model, 180e183 Rankine criterion, 200 Rate of creep method, 237, 389 Rate of flow method, 237 Rectangular columns, 48e50 boundary conditions of flat steel plate, 45f typical failure mechanism, 45f Rectangular hollow steel sections, 17 Rectangular section, 577e609 with flat steel plates, 577e609 with round-ended corners, 577e586 Rectangular steel hollow sections filled with concrete, 678e706 skins, 50 Recycled aggregate concrete (RAC), 794, 806 Reinforced concrete (RC), 3e4 Relative slenderness, 548e549, 554, 584 Residual strength of concrete, 396 Residual stress, 2e3 level ratio, 410 Revised Rasmussen’s model, 310 Round-ended corners, rectangular section with, 577e586 Round-ended rectangular CFST members, 225e230 Rubberized concrete (RuC), 15e16, 65, 103e104, 794, 806e807 columns with RuC core, 66e70 concrete and outer steel tube bonding, 66f

Index

failure mechanism in RuCFDST columns, 65f schematic failure modes, 64f Ruifeng International Commercial Building in Hangzhou, China, 8, 10f

S Samani and Attard model, 204 Scalar damaged elasticity equation, 362 Seawater and sea sand concrete (SWSSC), 807e808 Second-order effect, 434, 611e615 AISC360e16, 618e619 EC4, 615e618 Section shapes effects on composite members, 419e427 SEG plaza in Shenzhen, China, 7, 9f Self consolidating concrete (SCC), 15e16, 33, 794, 802 SHS. See Square hollow section (SHS) SI. See Strength index (SI) SIC. See Strength index of column (SIC) Simplified moment capacity model of Elchalakani et al., 626e635 of Han, 635e677 of Liang and Fragomeni, 635e655 Slender columns composite member failure modes, 59e63 CFDSTcolumns, 56f CFST slender columns, 58f effect of slenderness ratio, 60f strength-slenderness ratio relationship, 59f Slender steel tubes, 798 Slenderness ratio (l), 39e40, 427, 434, 463, 575 effects of column, 453e457 Song’s model, 756 Spherical cap gaps, 135e143 Square CFST members, 3e4, 192e225 Square columns, 48e50 boundary conditions of flat steel plate, 45f typical failure mechanism, 45f Square hollow section (SHS), 3e4, 32 SRI. See Strength lost index (SRI) Stability factor, 503, 570 Stainless steel, 31e32, 171, 179e189. See also Carbon steel Gardner and Nethercot’s model, 185 generalized multistage model, 187e188 grades and chemical compositions of, 181t

823

grades of, 179e180 inversion of stressestrain relationship, 189 Mirambell and Real’s model, 183e185 nominal yield strength and ultimate strength of stainless steels, 182t Quach’s model, 185e187 RambergeOsgood model, 180e183 stressestrain curves of carbon steel and stainless steel, 182f tube, 798 types, 179e180 typical material stressestrain curves, 179f Steel, 2, 16e17 AISC 360e16, 17 AS5100 part 6, 16 BS5400 part 5, 16 contribution ratio, 547 DBJ13e51, 17 Eurocode 4, 17 hollow sections elastic local buckling of steel plates, 672e673 post-local buckling of steel plates, 675e676 material limitation, 17 plates imperfections, 125e128 local buckling of, 671 rectangular section with flat, 577e609 reinforcement bars, 17 steel-concrete composite structural members, 2 stiffeners, 72 yield strength, 17 Steel nominal ratio (an), 40 Steel tube(s), 31, 190e191 effects of material properties, 112e113 fabrication, 15e16 mechanism of interaction of concrete and, 419e424 strength, 802 yield strength, 795e796 stress effects on load eccentricity, 118e119 Stiffened circular composite columns, 802e803 Stiffened composite members, 801e805 number of studies performed on each kind of stiffeners, 803f proportion of studies performed on internal and external stiffeners, 803f

824

Index

Stiffeners circular CFST columns external stiffeners in, 79f tension bar stiffeners in, 78f in composite columns, 72e89 comparison of local buckling, 74f confinement distribution of unstiffened and stiffened square CFST columns, 80f failure modes, 77f internal stiffeners, 69f local buckling mode of CFST columns, 82f longitudinal ribs on buckling mode, 70f schematic view of buckling modes, 73f tension bar stirrup stiffeners, 71f in composite members, 804e805 Stiffening methods, 76e78, 801e802 Straight-tapered-straight (STS), 799e800 Strain compatibility between steel tube and concrete core, 685e689 Strain ductility index, 103e104 Strain hardening, 169e170, 349, 354, 366e367 modulus, 173 Strain softening, 349, 354, 366e367 behavior, 205 Strength enhancement induced during cold forming, 314e321 Strength index (SI), 140e141, 398, 426e427 Strength index of column (SIC), 188 Strength lost index (SRI), 117e118 Strength-to-weight ratio, 5, 796e798 of CFDST columns, 93 Stressestrain behavior of stainless steels, 179 Stressestrain model CFDST members of Hu and Su, 233e234 of Liang et al., 234e236 of Zhao et al., 236e237 circular and square CFST members of Attard and Setung, 193e197 of Binici, 197e202 of Ellobody et al., 223 of Han et al., 222e223 of Hu et al., 213e215 of Liang et al., 219e222 of Lim and Ozbakkaloglu, 206e211 of Mander et al., 192e193 of Portoles et al., 223e225 of Samani and Attard, 202e206 of Susantha et al., 216e219

of Tao et al., 211e213 of concrete material effects on FE results, 378e380 effects of ratio of second stress invariant on tensile meridian on FE results, 380e383 hexagonal and octagonal CFST members of Hassanein et al., 230e231 of Patel, 232 of Susantha et al., 232e233 round-ended rectangular and elliptical CFST members of Ahmed and Liang, 225e228 of Dai and Lam, 228e230 of Patel, 226e227, 230 of steel material effects on finite element results, 376e378 Stressestrain response of materials, 168e279 carbon steel, 168e178 confine concrete, 190e279 stainless steel, 179e189 Structural carbon steel, 171 Structural members, 2 Structural performance of composite members, 801e802 STS. See Straight-tapered-straight (STS) Superposition principle, 714 SWSSC. See Seawater and sea sand concrete (SWSSC) Symmetrical steel sections, 16

T T-shaped rib stiffeners, 805 Tangent Poisson’s ratio, 425 Tapered columns, 58e59 failure modes of concrete core, 54f typical failure modes of straight and tapered CFDST columns, 55f Tapered composite members, 434 effect on ultimate axial capacity of composite members, 441e531 Tensile fracture energy, 204e205 Tensile strength, 17 Tensile stressestrain curve, 168e169 Thin-walled square steel tube, 76 Timber, 2

U Ultimate axial strain of composite short columns, 623e625 Ultimate axial strengths of slender beamcolumns, 478e480

Index

Ultimate tensile stress, 168e169 Ultra-high-strength concrete (UHSC), 15e16, 31e32, 100e101, 794 Uniform compression, 48 Uniform confinement effect, 799 Uniform corrosion, 800e801 User-defined subroutine (VUSDFLD), 489

Width-to-thickness ratio effects on steel tube, 457e460 Winter equation, 741, 759e760 Winter’s model, 703 World Corrosion Organization (WCO), 800e801

Y V “Vertical swing” techniques, 10e12 Von Mises yield criterion, 415, 491, 528 Von Mises’ failure criterion, 309

W Wangchang East River Bridge, China, 10e12, 12f Waste dumping landfill area, 806e807 Waste tires recycling management, 806e807

825

Yield criterion, 345e348, 354, 365e366 in extended DeP model, 357 Yield slenderness limit, 16 Yield stress, 17, 33e36 Young’s modulus, 33e36 of concrete, 217

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