A D V A N C E S IN STEEL S T R U C T U R E S Proceedings of The Second International Conference on Advances in Steel Structures 1517 December 1999, Hong Kong, China
Volume I
A D V A N C E S IN STEEL S T R U C T U R E S Proceedings of The Second International Conference on Advances in Steel Structures 1517 December 1999, Hong Kong, China
Volume I
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A D V A N C E S IN STEEL S T R U C T U R E S Proceedings of The Second International Conference on Advances in Steel Structures 1517 December 1999, Hong Kong, China
Volume I
Edited by S L C h a n and J G T e n g The Hong Kong Polytechnic University
Organised by Department of Civil and Structural Engineering The Hong Kong Polytechnic University
Sponsored by The Hong Kong Institution of Engineers
1999
ELSEVIER AMSTERDAM
9L A U S A N N E
9N E W
YORK
9O X F O R D
9S H A N N O N .
SINGAPORE.
TOKYO
E L S E V I E R S C I E N C E Ltd The Boulevard, Langford Lane Kidlington, Oxford OX5 1GB, U K 9 1999 Elsevier Science Ltd. All rights reserved.
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Preface These two volumes of proceedings contain 9 invited keynote papers and 126 contributed papers presented at the Second International Conference on Advances in Steel Structures held on 1517 December 1999 in Hong Kong. The conference was a sequel to the International Conference on Advances in Steel Structures held in Hong Kong in December 1996. The conference provided a forum for discussion and dissemination by researchers and designers of recent advances in the analysis, behaviour, design and construction of steel structures. The papers presented at the conference cover a wide spectrum of topics and were contributed from over 15 countries around the world. They report the current stateoftheart and point to future directions of structural steel research. The organization of a conference of this magnitude would not have been possible without the support and contributions of many individuals and organizations. The strong support from the Hong Kong Polytechnic University, Professor M. Anson, Dean of the FaTculty of Construction and Land Use, and Professor J.M. Ko, Head of the Department of Civil and Structural Engineering, has been pivotal in the organization of this conference. We also wish to express our gratitude to the Hong Kong Institution of Engineers for sponsoring the conference and the Local Advisory Committee for mobilizing support froTm the construction industry and government departments. Thanks are due to all the contributors for their careful preparation of the manuscripts and all the keynote speakers for their special support. Reviews of papers were carried out by members of the International Scientific Committee and the Local Organizing Committee. To all the reviewers, we are most grateful. We would also like to thank all those involved in the daytoday running of the organization work, including members of the Local Organizing Committee and secretarial staff in the Department of Civil and Structural Engineering. Finally, we gratefully acknowledge our pleasant cooperation with Dr. J. Milne and Mrs R. Davies at Elsevier Science Ltd in the UK. S.L. Chan J.G. Teng
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INTERNATIONAL SCIENTIFIC COMMITTEE The Ohio State University University of Tokyo The Hong Kong Polytechnic University The University of Sydney Delft University of Technology University of Pittsburgh University of New South Wales University of Western Sydney University of Cambridge Xi'an University of Architecture and Technology Purdue University The University of Hong Kong Korea Advanced Institute of Science and Technology M. Chryssanthopoulos, Imperial College Laboratoire de Mechanique et Technologie A. Combescure, Zhejiang University S.L. Dong, University of Surrey P.J. Dowling, EMPA M. Farshad, Fukuyama University Y. Fukumoto, Y. Goto, Nagoya Institute of Technology Washington University, St Louis P.L. Gould, Technisce Universitat, Graz R. Greiner, Monash University P. Grundy, University of Sydney G.J. Hancock, J.E. Harding, University of Surrey K.M. Hsiao, National Chao Tung University J.F. Jullien, INSA Lyon S. Kato, Toyohashi University of Technology S. Kitpornchai, University of Queensland V. Krupka, Institute of Applied Mechanics, Vitkovice T.T. Lan, Chinese Academy of Building Research S.F. Li, Tsinghua University R. Liew, National University of Singapore Xila Liu, Tsinghua University Xiliang Liu, Yianjin University L.W. Lu, Lehigh University Z.T. Lu, SouthEast University E. Lui, Syracuse University P. Marek, Academy of Science of the Czech Republic S. Morino, Mie University D.A. Nethercot, University of Nottingham G.W. Owens, Steel Construction Institute H. Adeli, H. Akiyama, M. Anson, P. Ansourian, J. Arbocz, R. Bjorhovde, M.A. Bradford, R.Q. Bridge, C.R. Calladine, S.F. Chen, W.F. Chen, Y.K. Cheung, C.K. Choi,
vii
USA Japan Hong Kong, China Australia Netherlands USA Australia Australia UK China USA Hong Kong, China Korea UK France China UK Switzerland Japan Japan USA Austria Australia Australia UK Taiwan, China France Japan Australia Czech Republic China China Singapore China China USA China USA
Czech Republic Japan UK UK
J.A. Packer, S. Pellegrino, E.P. Popov, J. Rhodes, J.M. Rotter, H. Schmidt, G. Sedlacek, S.Z. Shen, Z.Y. Shen, T.T. Soong, S.S. Sridharan, N.S. Trahair, T. Usami, A. Wada, F. Wald, E. Walicki, D. White, F. Williams, Y.B. Yang, R. Zandonini, S.T. Zhong,
University of Toromo University of Cambridge University of California, Berkeley University of Strathclyde University of Edinburgh Universitat Essen RWTH Aachen Harbin University of Civil Engineering and Architecture Tongji University Suny, Buffalo, NY Washington University, St Louis University of Sydney Nagoya University Tokyo Institute of Technology Czech Technical University Technical University of Zielona Gora Georgia Institute of Technology University of Wales National Taiwan University University of Trento Harbin University of Civil Engineering and Architecture
viii
Canada UK USA UK UK Germany Germany China China USA USA Australia Japan Japan Czech Republic Poland USA UK Taiwan, China Italy China
LOCAL ADVISORY COMMITTEE Chairman:
J.M. Ko The Hong Kong Polytechnic University
Members: A.S. Beard F.S.Y. Bong A.K.C. Chan H.C. Chan Y.L. Choi W.K. Fung M. Harman I. Kimura M.H.C. Kwong C.K. Lau S.H. Ng W. Tang H. Wu
Mott MacDonald Hong Kong Ltd. Maunsell Consultants Asia Ltd. Ove Amp and Partners (HK) Ltd. The University of Hong Kong Buildings Department, HKSAR Government Architectural Services Department, HKSAR Government British Steel (Asia) Ltd. Nippon Steel Corporation Scott Wilson (Hong Kong) Ltd. Highways Department, HKSAR Government Hong Kong Housing Society The Hong Kong University of Science and Technology Kowloon Canton Railway Corporation
ix
LOCAL ORGANISING C O M M I T T E E Chairman:
S. L. Chan The Hong Kong Polytechnic University
ViceChairman:
J.G. Teng
The Hong Kong Polytechnic University Members: C.M. Chan W.T. Chan T.H.T. Chan K.F. Chung
J.C.L. Ho W.M.G. Ho C.M. Koon M.K.Y. Kwok S.S. Lam J.C.W. Lau Y.W. Mak A.D.E. Pan A.K. Soh K.Y. Wong Y.L. Wong
Y.L. Xu L.H. Yam
The Hong Kong University of Science and Technology Buildings Department, HKSAR Government The Hong Kong Polytechnic University The Hong Kong Polytechnic University Scott Wilson (Hong Kong) Ltd. Ove Amp and Partners (HK) Ltd. Buildings Department, HKSAR Government Ore Amp and Partners (HK) Ltd. The Hong Kong Polytechnic University James Lau & Associates Ltd. Housing Department, HKSAR Government The University of Hong Kong The University of Hong Kong Highways Department, HKSAR Government The Hong Kong Polytechnic University The Hong Kong Polytechnic University The Hong Kong Polytechnic University
CONTENTS
VOLUME I Preface International Scientific Committee
vii
Local Advisory Committee Local Organising Committee
Keynote Papers Unbraced Composite Frames: Application of the Wind Moment Method D.A. Nethercot and J.S. Hensman
A Cumulative Damage Model for the Analysis of Steel Frames under Seismic Actions
13
Z. Y. Shen
Recent Research and Design Developments in ColdFormed Open Section and Tubular Members
25
G.J. Hancock
Behaviour of Highly Redundant MultiStorey Buildings under Compartment Fires
39
J.M. Rotter
Design Formulas for Stability Analysis of Reticulated Shells
51
S.Z. Shen
Ductility Issues in ThinWalled Steel Structures
63
T. Usami, Y. Zheng and H.B. Ge
HighPerformance Steel Structures: Recent Research
75
L.W. Lu, R. Sause and J.M. Ricles
A Unified Principle of Multiples for Lateral Deflection, Buckling and Vibration of MultiStorey, MultiBay, Sway Frames
87
W.P. Howson and F.W. Williams
Beams and Columns ThreeDimensional Hysteretic Modeling of ThinWalled Circular Steel Columns L. Jiang and Y. Goto
xi
101
xii
Contents
Local Buckling of ThinWalled Polygonal Columns Subjected to Axial Compression or Bending J.G. Teng, S.T. Smith and L. Y. Ngok
109
Ultimate Load Capacity of Columns Strengthened under Preload H. Unterweger
117
Chaotic Belt Phenomena in Nonlinear Elastic Beam Z. Nianmei, Y. Guitong and X. Bingye
125
Frames and Trusses
Investigation of Rotational Characteristics of Column Bases of Steel Portal Frames T.C.H. Liu and L.J. Morris
135
Ultimate Strength of SemiRigid Frames under NonProportional Loads B.H.M. Chan, L.X. Fang and S.L. Chan
145
SecondOrder Plastic Analysis of Steel Frames P. P.T. Chui and S.L. Chan
151
Study on the Behaviour of a New LightWeight Steel Roof Truss P. Ma'keldinen and O. Kaitila
159
A Proposal of Generalized Plastic Hinge Model for the Collapse Behavior of Steel Frames Governed by Local Buckling S. Motoyui and T. Ohtsuka
167
Advanced Inelastic Analysis of Spatial Structures J.Y.R. Liew, H. Chen and L.K. Tang
175
Stability Analysis of Multistory Framework under Uniformly Distributed Load C. Haojun and W. Jiqing
183
Space Structures
Studies on the Methods of Stability Function and Finite Element for SecondOrder Analysis of Framed Structures S.L. Chan and J.X. Gu
193
Dynamic Stability of Single Layer Reticulated Dome under Step Load C. Wang and S. Shen
201
Experimental Study on FullSized Models of Arched Corrugated Metal Roof L. Xiliang, Z. Yong and Z. Fuhai
209
QuasiTensegric Systems and Its Applications L. Yuxin and L. Zhitao
217
Connections
The Design of Pins in Steel Structures R.Q. Bridge
229
Contents
xiii
Finite Element Modelling of EightBolt Rectangular Hollow Section Bolted Moment End Plate Connections A.T. Wheeler, M.J. Clarke and G.J. Hancock
237
Finite Element Modelling of Double Bolted Connections Between ColdFormed Steel Strips under Static Shear Loading K.F. Chung and K.H. Ip
245
Analytical Model for EightBolt Rectangular Hollow Section Bolted Moment End Plate Connections A.T. Wheeler, M.J. Clarke and G.J. Hancock
253
Predictions of Rotation Capacity of RHS Beams Using Finite Element Analysis T. Wilkinson and G.J. Hancock
261
Failure Modes of Bolted ColdFormed Steel Connections under Static Shear Loading K.H. Ip and K.F. Chung
269
Design Moment Resistance of End Plate Connections Y. Shi and J. Jing
277
Threaded Bar Compression Stiffening for Moment Connections T.F. Nip and J.O. Surtees
283
Experimental Study of Steel IBeam to CFT Column Connections S.P. Chiew and C.W. Dai
291
Behaviour of TEnd Plate Connections to RHS Part I: Experimental Investigation M. Saidani, M.R. Omair and J.N. Karadelis
305
The Behaviour of TEnd Plate Connections to SHS. Part II: A Numerical Model J.N. Karadelis, M. Saidani and M. Omair
313
Cyclic Behaviour of BeamToColumn Welded Connections E. Mele, L. Calado and A. De Luca
323
Advanced Methods for Modelling Hysteretic Behaviour of SemiRigid Joints Y.Q. Ni, J.Y. Wang and J.M. Ko
331
ColdFormed Steel
Behaviour and Design of ColdFormed Channel Columns B. Young and K.J.R. Rasmussen
341
Section Moment Capacity of ColdFormed Unlipped Channels B. Young and G.J. Hancock
349
Web Crippling Tests of High Strength ColdFormed Channels B. Young and G.J. Hancock
357
Local and Distortional Buckling of Perforated Steel Wall Studs J. Kesti and J.M. Davies
367
xiv
Contents
An Experimental Investigation into ColdFormed Channel Sections in Bending V. Enjily, M.H.R. Godley and R.G. Beale
375
Composite Construction Flexural Strength for Negative Bending and Vertical Shear Strength of Composite Steel SlagConcrete Beams Q.L. Wang, Q.L. Kang and P.Z. Cao
385
ConcreteFilled Steel Tubes as Coupling Beams for RC Shear Walls J.G. Teng, J.F. Chen and Y.C. Lee
391
Experimental Study of High Strength Concrete Filled Circular Steel Columns Y.C. Wang
401
Strength and Ductility of Hollow Circular Steel Columns Filled with Fibre Reinforced Concrete G. Campione, N. Scibilia and G. Zingone
413
Axial Compressive Strength of Steel and Composite Columns Fabricated with High Strength Steel Plate B. Uy
421
Concrete Filled ColdFormed C450 RHS Columns Subjected to Cyclic Axial Loading X.L. Zhao, R.H. Grzebieta, P. Wong and C. Lee
429
Research on the Hysteretic Behavior of High Strength Concrete Filled Steel Tubular Members under Compression and Bending Z. Wang and Y. Zhen
437
Design of Composite Columns of Arbitrary CrossSection Subject to Biaxial Bending S.F. Chen, J.G. Teng and S.L. Chan
443
Effects of Loading Conditions on Behaviour of SemiRigid BeamtoColumn Composite Connections Y.L. Wong, J.Y. Wang and S.L. Chan
451
SteelConcrete Composite Construction with Precast Concrete Hollow Core Floor D. Lam, K.S. Elliott and D.A. Nethercot
459
Testing and Numerical Modelling of BiSteel Plate Subject to PushOut Loading S.K. Clubley and R.Y. Xiao
467
Rectangular TwoWay RC Slabs Bonded with a Steel Plate J.W. Zhang, J.G. Teng and Y.L. Wong
477
Bridges Structural Performance Measurements and Design Parameter Validation for Tsing Ma Suspension Bridge C.K. Lau, W.P. Mak, K.Y. Wong, W.Y. Chan, K.L. Man and K.F. Wong
487
Contents
Wind Characteristics and Response of Tsing Ma Bridge During Typhoon Victor
XV
497
L.D. Zhu, Y.L. Xu, K.Y. Wong and K. W.Y. Chan
Structural Performance Measurement and Design Parameter Validation for Kap Shui Mun CableStayed Bridge
505
C.K. Lau, W.P. Mak, K.Y. Wong, K.L. Man, W.Y. Chan and K.F. Wong
Free and Forced Vibration of LargeDiameter Sagged Cables Taking into Account Bending Stiffness
513
Y.Q. Ni, J.M. Ko and G. Zheng
Stability Analysis of Curved CableStayed Bridges
521
Y.C. Wang, H.S. Shu and J. Ermopoulos
Expert System of Flexible Parametric Study on CableStayed Bridges with Machine Learning
529
B. Zhou and M. Hoshino
Parameter Studies of Moving Force Identification in Laboratory
537
T.H.T. Chan, L. Yu, S.S. Law and T.H. Yung
Seismic Analysis of Isolated Steel Highway Bridge
545
X.S. Li and Y. Goto
Shear Analysis for Asphalt Concrete Deck Pavement of Steel Bridges
553
X. Zha and Q. Xiao
VOLUME II Preface International Scientific Committee
vii
Local Advisory Committee
ix
Local Organising Committee Plates
Strength and Ductility of Plates in Shear
563
T. Usami, H.B. Ge and M. Amano
PostBuckling of Unilaterally Constrained Mild Steel Plates
571
S.T. Smith, M.A. Bradford and D.J. Oehlers
Postbuckling Analysis of Plate with General Initial Imperfection by Finite Strip Method
579
T.H. Lui and S.S.E. Lam
PostBuckling Analysis of Web Plates of Girders by Three Dimensional Degenerated Shell Element Method H. Qinghua, Y. Yue and L. Xiliang
587
xvi
Contents
Shells
Buckling Interaction Strength of Cylindrical Steel Shells under Axial Compression and Torsion H. Schmidt and T. A. Winterstetter
Shell Buckling Design of Austenitic Stainless Steel Cylinders under Elevated Temperatures up to 500~ H. Schmidt and K.T. Hautala
Cylindrical Shells Buckling under External PressureInfluence of Localized Thickness Variation J.F. Jullien, A. Limam and G. Gusic
Stability and Strength of Conical Shells Subject to Axial Load and External Pressure N. Panzeri and C. Poggi
The Nonlinear Stability of SemiThin Spherical Shell Joints under Uniformly Symmetric Circular Line Loads Y.F. Luo, K.S. Huang and Q.Z. Li
The Influence of Circumferential WeldInduced Imperfections on the Buckling of Silos and Tanks M. Pircher and R. Bridge
Experimental Techniques for Steel Silo Transition Junctions J.G. Teng and Y. Zhao
Buckling Strength of TSection Ringbeams in Steel Silos J.G. Teng and F. Chan
Abnormal Behaviour of a Steel Silo Caused by Paddy Rice Storage M.P. Luong
Bifurcation Buckling of Aboveground Oil Storage Tanks under Internal Pressure S. Yoshida
Buckling of Cylindrical Shells Subjected to Edge Vertical Deformation M. Jonaidi and P. Ansourian
On the Nonlinear Analysis of Shells with EigenmodeAffine Imperfections J.G. Teng and C.Y. Song
Postbuckling Analysis of Shells of Revolution Considering Mode Switching and Interaction T. Hong and J.G. Teng
Transition of Plastic Buckling Modes in Cylindrical Shells Y. Goto, C. Zhang and N. Kawanishi
Are the Static Postbuckling Predictions Conservative? A. Combescure
597
605
613 621
631
639 647 655 663 671 679 687
697 705 713
Contents
Plastic Stability of Cylindrical Shells Taking Account of Loading History
xvii 721
V.S. Hudramovych
Design and Construction Prestressing and Loading Tests on FullScale Roof Truss of Shanghai Pudong International Airport Terminal
731
Z. Xiangzhong, C. Yiyi, S. Zuyan, C. Yangji, W. Dasui and Z. Jian
Air Mail Centre at Chek Lap Kok
739
P.H. Lam
Composite Design and Construction of a Tall BuildingCheung Kong Center
747
D. Scott, G.W.M. Ho and H. Nuttall
The Tallest Building in Mexico City: Torre Mayor, Mexico City, Mexico
755
A. Rahimian and E.M. Romero
The Use of Triangular Added Damping and Stiffness (TADAS) Devices in the Design of the Core Pacific City Shopping Centre
775
K.L. Chang, S.J.W. Rees, C. Carroll and K. Clandening
Site Measurement of Vibration Characteristics of Shanghai Jin Mao Tower W. Shi, X. Lu and J. Shen
Design of Steel Scaffolding by Nonlinear Integrated Design and Analysis (NIDA) and the Stability Function
783
791
A. Y.T. Chu and S.L. Chan
Experimental Assessment for Aluminium Alloy Sections in Glass Curtain Walls of Shanghai Jinmao Building
799
L. Tong, Y. Luo, Z. Shen and Y. Wang
Dynamics and Seismic Design Transverse Dynamic Characteristic and Seismic Responses of LargeScale TallPier Aqueduct Y. Li
Dynamic Characteristic and Seismic Response of Semirigid Jointed Frames W.S. Zhang and Y.L. Xu
Nonlinear Seismic Analysis of Flexibly Connected Steel Buildings P. P.T. Chui and S.L. Chan
The Response Analysis of the Transversely Stiffening Single Curvature CableSuspended Roof to the Fluctuating Wind X. Zhao, X. Liu and Y. Dou
Transient Analysis of Stiffened Panel Structure by a Finite StripMode Superposition Method J. Chen
809 815 823
831
839
xviii
Contents
Dynamic Performance of Steel Lightweight Floors M.M. Alikhail, X.L. Zhao and L. Koss
Coupled Truss Walls with Damped Link Elements A. Rahimian
Galloping of Cables with Moving Rivulet L.Y. Wang and Y.L. Xu
Free Vibration Analysis of ThinWalled Members with Shell Type Cross Sections
849 857 873 881
M. Ohga, T. Shigematsu and T. Hara
A Simple Formulation for Free Vibration of FrameShear Wall Tall Building
889
Q. Wang and L. Wang
FlexureTorsion Coupled Vibrations for Tall Building Structures Considering the Effects of Vertical Loads
897
S.H. Bao and S.C. Yi
The Computational Time Efficient Finite Element Method for Large Amplitude Vibrations of Composite Plates
905
Y.Y. Lee and C.F. Ng
Determination of Model Order for Thin Steel Plate Systems Using Vibration Test Data
913
Y.Y. Li and L.H. Yam
Prestressing Study on Tendon Profile on the Analysis and Design of Prestressed Steel Beams
921
G.N. Ronghe and L.M. Gupta
Long Term Analysis of Externally Prestressed Composite Beams
931
A.D. Asta, L. Dezi and G. Leoni
Flexible Connection Influence on Ultimate Capacity of Externally Prestressed Composite Beams
939
A.D. Asta, L. Dezi and G. Leoni
A Fracture Criterion for Prestressing Steel Cracked Wires
947
J. Toribio and M. Toledano
Failure Analysis of Prestressing Steel Wires
955
J. Toribio and A. Valiente
Fatigue and Fracture Experimental Study on Static and Fatigue Behavior of SteelConcrete Preflex Prestressed Composite Beams
965
K. Zhang, S. Li and K. Liu
ObjectOriented Fatigue Reliability Analysis for the Offshore Steel Jacket C. Wang, Y. Shi and S. Li
975
Contents
xix
Fatigue Strength of ThinWalled TubeToPlate TJoints under InPlane Bending F.R. Mashiri, X.L. Zhao and P. Grundy
983
Failure Assessment of BeamtoColumn Steel Joints via LowCycle Fatigue Approaches C. Bernuzzi and R. Zandonini
991
A Method to Estimate PSN Curve for Misaligned Welded Joints G. Deqing
999
Reliability Analysis of Draw Bar of LargeScale Lock Mitre Gate Z.G. Xu, C.Y. Bian and R.L. Wang
1005
Computation of Stress Intensity Factor for Surface Crack in Welded Joint G. Deqing and Y. Yong
1013
Numerical Approach to the Ductile Fracture of Steel Members M. Obata, A. Mizutani and Y. Goto
1021
Fire Performance The First Code on Fire Safety of Steel Structures in China G.Q. Li, S.C. Jiang and J.L. He
1031
Fire Resistance of Concrete Filled Steel Tubes in China L.H. Han
1039
Elevated Temperature Testing of Composite Columns N.L. Patterson, X.L. Zhao, M.B. Wong, J. Ghojel and P. Grundy
1047
Full Scale Fire Test on the New UK Slim Floor System C.G. Bailey, T. Lennon and D.B. Moore
1055
Mechanical Properties of an Austenitic Stainless Steel at Elevated Temperatures J. Outinen and P. Mdkeldinen
1063
Optimization Optical Design of Steel Frames with NonUniform Members A. Mu'ller, F. Werner and P. Osterrieder
1073
Optimal Sizing/Shape Design of Steel Portal Frames Using Genetic Algorithms P. Liu, C.M. Chan and Z.M. Wang
1081
Study on Optimization of Particular and MultiVariable Structures by Wavelet Analysis L. Liu, Y. Zhai and H. Lin
1089
Optimal Analysis of Large Span DoubleLayer Barrel Vaults L. Shan and H. Yan
1099
xx
Contents
Analysis
Determination of Section Properties of Complicated Structural Members
1109
Z.X. Li, J.M. Ko, T.H.T. Chan and Y.Q. Ni
Adaptive Finite Element Buckling Analysis of Folded Plate Structures
1117
C.K. Choi and M.K. Song
Hoop Stress Reduction by Using Reinforced Rivets in Steel Structures
1125
K.T. Chau, S.L. Chan and X.X. Wei
Safety Analysis and Design Consideration for Oil and Gas Pipelines
1133
A.N. Kumar
Prediction of Residual Stresses: Comparison Between Experimental and Numerical Results
1141
Y. Vincent, J.F. Jullien and V. Cano
Soil Structure Interaction
Composite Foundation of Deep Mixing Piles for Large Steel Oil Tanks on Soft Ground
1151
X. Xie, X. Zhu and Q. Pan
An Analytical Study on Seismic Response of Steel Bridge Piers Considering SoilStructure Interaction
1157
A. Kasai and T. Usami
Late Papers
Modelling Hysteresis Loops of Composite Joints Using Neural Networks
1167
J.Y. Wang, Y.L. Wong and S.L. Chan
New Design Methods for Concrete Filled Steel Tubular Columns
1175
Y.C. Wang
Keynote Paper The Implications of the Information Society on the Practice of and Training for Steelwork Construction
1187
G. Owens
Index of Contributors
II
Keyword Index
15
Keynote Papers
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UNBRACED COMPOSITE FRAMES: APPLICATION OF THE WIND M O M E N T METHOD
D A Nethercot 1and J S Hensman 2 ISchool of Civil Engineering, University of Nottingham, University Park, Nottingham NG7 2RD, UK 2Caunton Engineering Limited, Moorgreen Industrial Park, Moorgreen, Nottingham NG16 3QU, UK
ABSTRACT Proposals are given to extend the simplified design technique known as the Wind Moment Method to cover a limited range of composite frames. The range represents that of most interest in practice in the UK. Justification is by comparison with the findings from an extensive numerical study. KEYWORDS
: Composite Construction, Connections, Frames, Joints, Steel Structures, Structural
Design
INTRODUCTION
The Wind Moment Method (WMM) has long been established as a simple, intuitively based, design approach for unbraced frames. More recently, it has been the subject of scientific study, designed to provide a more fundamental understanding of the link between actual frame behaviour and the inherent design simplifications. This work has, until now, been restricted to bare steel construction. In a recent study Hensman, (1998), the authors have examined the case for an extension of the WMM to cover composite steel/concrete frames. Although the approach adopted resembles that used for bare steelwork, a number of particular features have had to be addressed. This paper summarizes the main outcomes from that study. The basis for the extension was numerical modelling, utilising the available body of knowledge on the performance of composite connections, the previous application of the WMM to bare steelwork and the capabilities of the ABAQUS package. It was also found necessary to conduct a detailed examination of the role of column bases  a feature not previously addressed in WMM
D.A. Nethercot and J.S. Hensman
investigations. Several of the findings therefore have relevance to potential improvements in the WMM for bare steel frames. This paper covers: appraisal of the basic source data, outline of the numerical studies, presentation of the key findings and an indication of the resulting design approach. This last item will be presented in a fashion suitable for direct use by designers in a forthcoming Steel Construction Institute Design Guide.
KEY FEATURES OF THE WIND M O M E N T METHOD The approach was originally devised in the precomputer era, when overall structural analysis of unbraced frames represented an extremely challenging and potentially tedious task. It therefore sought an acceptable simplification so that the labour involved in the structural analysis might be minimised. This was achieved by recognising that some simplification in the representation of the actual behaviour would be necessary. Although it is now quite widely accepted that the true behaviour of all practical forms of beam to column connection in steel and concrete construction function in a semirigid and partial strength f a s h i o n  with the ideals of pinned and rigid only occasionally being approached early methods of structural analysis could only cater for one or other of these ideals. Thus the basic WMM uses the principle of superposition to combine the internal moments and forces obtained from a gravity load analysis that assumes all beams to be simply supported and a wind load analysis that assumes beam to column connections to be rigid with points of contraflexure at the midspan of the beams and the midheight of the columns as illustrated in Figure 1. This second assumption permits use of the socalled portal method of frame analysis. Once it became possible to conduct full range analyses of steel frames allowing for material and geometrical nonlinear effects and including realistic models of joint behaviour, studies were undertaken to assess the actual performance of frames designed according to the WMM principles. The findings permitted observations to be made of the two key behavioural measures: 9 That the load factor at ultimate was satisfactory 9 That drift limits at serviceability were achieved. This second point is of importance because, when estimating sway deflections at working load, the WMM normally involves taking the results of an analysis that assumes rigid connections and then applying a suitable scaling factor. Important contributions in the area of bare steel construction are those of Ackroyd and Gerstle, (1982), Ackroyd, (1987), and Anderson and his coworkers at Warwick, Reading, (1989), Kavianpour, (1990), Anderson, Reading and Kavianpour,(1991) NUMERICAL A P P R O A C H All the numerical work was undertaken using the ABAQUS package. Whilst this contained sufficient functionality to cover many of the necessary behavioural features, three items required particular attention: 9 Representation of the composite beams 9 Representation of composite beam to column connections 9 Inclusion of column base effects
Unbraced Composite Frames: Application o f the Wind M o m e n t M e t h o d 4,4'4'4,,1,4,4'4'4'4'4'4'4'4'4'4'4'
~4'
7" r
t
4'4' 4'4' 4'4' 4'4' 4'4' 4'4'4'
(a)
77
t
t
Figure 1 Superposition of gravity and lateral load analyses For the first of these the approach previously utilised by Ye, Nethercot and Li, (1996), that is based on moment curvature relationships developed by Li, Nethercot and Choo, (1993), was employed. Since composite endplates were assumed for the beam to column connections, the work of Ahmed and Nethercot,(1997), in predicting momentrotation response under hogging moment was directly employed. Data on the performance of composite beam to column connections under sagging i.e. opening, moments was, however, almost nonexistent. Previous experience with the Wind Moment Method had, however, suggested that reversal in the sign of the rotation at any connection might be a rather unusual event. An approximate model for composite connection behaviour under sagging moments was therefore devised by examining test data for such connections when subject to cyclic loading. All previous studies of the WMM have assumed rigid i.e. fully fixed column bases. Enquiries among practitioners had, however, already revealed that such an option was not attractive. In addition, there was a widely held belief that all practical forms of "pin" column bases were capable of supplying quite significant amounts of rotational restraint. Accordingly, all relevant information on column base effects  particularly previous experimental studies  was carefully reviewed in an attempt to identify suitable minimum restraint levels likely to be supplied by notionally pinned bases, Hensman and Nethercot, (2000a). The findings were then incorporated in the full parametric study. This point is regarded as particularly important as attempts to justify the WMM approach using truly pinned column bases, Hensman,(1998), had shown that it was almost impossible to satisfy realistic drift limitations due to the greatly enhanced overall frame flexibility resulting from the loss of column base restraint (as compared with the usual WMM assumption of fixed bases). It is believed that the exercise should be r e p e a t e d  since bare steel columns were assumed throughout, it would merely be a case of conducting appropriate analyses on bare steel frames  as a way of similarly relaxing an unattractive restriction in the application of the WMM to bare steel construction.
6
D.A. Nethercot and J.S. Hensman
Because of concem over the adequacy of the modelling of composite beam to column connections under sagging moments, particular attention was paid in an initial study, Hensman, J S (1998), to the occurrence (or not) of reversal in the sign of the connection rotations. Initial studies using the subframe of Figure 2, that was specially configured to represent a typical intermediate floor in a more extensive structure, showed that for realistic arrangements of frame layout, member sizes and levels of gravity and wind loading reversal of rotations, even at the potentially most vulnerable windward connections was extremely unlikely. It was therefore concluded that the full parametric study need not concern itself with further refinement of this feature.
PARAMETRIC STUDY Figure 3 illustrates the basic frame layouts considered and Tables 1 and 2 list the range of variables considered within the numerical study. Although this was based on the equivalent set of restrictions given in Anderson, Reading and Kavianpour (1991) it has been adapted somewhat, both to recognise important differences between bare steel and composite construction e.g. the likely use of longer span beams, and to reflect certain preferences from the industry and recent changes in the UK design environment e.g. issue of a new Code for wind loading. A more detailed explanation of the arrangement of the study, including justification for decisions on joint types, load combinations etc., is available, Hensman and Nethercot (2000b). Full details of the 300 cases investigated covering 45 different frame arrangements, including summary results for each, are available in reference 1. In all cases the approach adopted was to first design the frame using the proposed WMM technique and then to conduct a full range computer analysis to check its condition at the SLS and ULS stages.
MAIN FINDINGS Undoubtedly the most significant overall outcome of the parametric study was the finding that every frame design using the proposed WMM approach was essentially satisfactory in terms of providing an adequate margin of safety against ULS load combinations. This was despite the fact that the actual distributions of intemal forces and moments within the frames often differed significantly from those presumed by the WMM analyses. Only in an extremely small number of cases was any degree of column overstress observed (and then less than 4%)  a comforting feature given that actual end restraint moments obtained from the rigorous analyses were often significantly higher than the assumed 10% of the WMM. The actual values of up to 30% in certain cases might suggest that where gravity loads are high beam sections could be reduced by assuming a largersay 20%  end restraint moment. Before so doing, however, it would be important to check the effect on overall lateral frame stiffness as it might well prove difficult to satisfy drift limitations with this inherently more flexible system. For all cases of frames designed for maximum gravity load and minimum wind load the SLS conditions were met. However, if higher wind loads were introduced, particularly for frames with short bay widths, some difficulty in ensuring adequate serviceability performance might well be experienced.
Unbraced Composite Frames." Application of the Wind Moment Method
7
A general discussion on the findings from the numerical study in terms of possible future modifications to the WMM and links between flame features and observed behaviour is available in Hensman and Nethercot (2000b).
Figure 2: Typical subframe arrangement used for preliminary study (Beam spans vary between 6m and 12m)
8
D.A. Nethercot and J.S. Hensman
Figure 3 9Schematic diagram of alternative flame layouts used in parametric study
Unbraced Composite Frames: Application of the Wind Moment Method TABLE 1 RANGE OF VARIABLES CONSIDERED WITHIN THE PARAMETRIC STUDY
Minimum Maximum Number of storeys 2 4 Number of bays 2 4"1 Bay width (m) 6.0 12.0 Bottom storey height (m) 4.5 6.0 Storey height elsewhere (m) 3.5 5.0 Dead load on floors (kN/m 2) 3.50 5.00 Imposed load on floors (kN/m 2) 4.00 7.50 Dead load on roof (kN/m 2) 3.75 3.75 Imposed load on roof (kN/m 2) 1.50 1.50 Wind loads (kN) 10 .2 40 *2 *' frames can have more than 4 bays, but a core of 4 bays is the maximum that can be considered to resist the applied wind load. ,2 Wind loads = concentrated point load on plane frame at each floor level TABLE 1 RELATIVE DIMENSIONSCONSIDEREDWITHINTHE PARAMETRIC STUDY
Bay width: storey height (bottom storey) Bay width: storey height (above bottom storey) Greatest bay width: Smallest bay width
RECOMMENDED
Minimum 1.33
Maximum 2.67
1.33
3.43
1
1.5
DESIGN APPROACH
The basic design approach is outlined in the chart o f Figure 4. This presents all the relevant steps, including those intended to identify arrangements for which the W M M is not suitable. Some key details for certain of the steps in the actual design procedure are discussed below. Once an initial frame arrangement has been decided upon, global analyses for the three load combinations: 9 9 9
1.4DE + 1.6IL + Notional Horizontal Forces 1.2(DL+IL+WL) 1.4 ( D E + W E )
should be undertaken. Notional horizontal forces should be taken as 0.5% o f the factored dead + imposed load as specified by BS5950: Part 1. Pattern loading should be considered; it may well be
D.A. Nethercot and J.S. Hensman
10 STEP 1 Define frame geometry
NOTE: This flow chart is not a design procedure. It should be used only as a 'first check', to determine if the windmoment method outlined in this document is a suitable design method for the frame in question.
STEP 2 Define load types and magnitude (1) Gravity (2) wind
STEP 3 Design composite beams as simply supported with a capacity of 0.9Mp
The frame design is likely to be controlled by SLS sway. However, a suitable frame design may still be achieved using the windmoment method. Consider increasing the member sizes.
I. . . . . . . . .
. . . . . . . . . I
, I
STEP 4 Estimate required column sections
. . . . . . . . .
l
I
STEP 5 Predict the SLS sway using the method in Section 5
~"
Design the frame as rigid, or include vertical bracing.
~ Is the t~alframe ~ sway oo) 0.234 0.753 1.432
Actual (True EA) 0.233 0.752 1.431
as case 1
as case 1
as case 1
as case 1
0.207 0.705 1.369 0.557 1.503 2.515 0.172 0.563 1.086 0.239 0.779 1.505
0.206 0.698 1.350 0.543 1.479 2.486 0.171 0.562 1.084 0.238 0.779 1.503
0.901 2.363 3.054
0.901 2.362 3.004
0.633 1.893 2.991
As case 5, except that g = 5 is represented exactly as in Figure 2, not by an equivalent diagonal
0.557 1.504 2.519
as case 5
as case 5
As case 8, except that g = 15 is represented exactly as in Figure 2, not by an equivalent diagonal
0.901 2.364 3.054
as case 8
as case 8
Substitute
Datum problem
Datum with the masses of beams of substitute frame lumped at their ends Grinter frame results for datum
Datum and allow for effect of axial forces on column flexure Datum with cladding added (~ = 5)
Datum with both central bay spans doubled Datum with EI of second column from the left doubled Datum with stiff cladding (g = 15) for substitute frame, to represent structural bracing of one bay of actual frame
10
0.234 0.756 1.437 0.234 0.758 1.446 0.234 0.754 1.428 0.208 0.708 1.374 0.557 1.504 2.518 0.172 0.565 1.095 0.240 0.783 1.510
stiffnesses were still calculated using distributed mass. It should be noted that the axial forces were 22.6% of those which would have caused buckling of the substitute frame, i.e. the critical load factor for the substitute frame was 1/0.226 = 4.42. Case 5 gives results when cladding (the mass of which was neglected) represented by bracing equivalent to ~ = 5 at every storey was added to the actual (i.e. multibay) datum problem. The
Unified Principle of Multiples for Lateral Deflection, Buckling and Vibration
95
authors deliberately chose software which does not have coding to represent the spring and rigid cranked beam system of Figure 2 when representing the substitute frame because such a feature is unlikely to be available to a designer seeking to use the substitute frame to undertake a parametric study. Instead the authors used approximately equivalent massless diagonal bracing in each of its bays. By assuming that the beams and columns were inextensible (which is reasonable because the crosssectional area A of the beams and columns far exceeds that of the bracing) the bracing members for storey i were readily shown to have A i = 1.263 yi ~
cm 2
(3)
for the substitute frame and one quarter of this value for each bay of the actual frame, for which all the diagonals were parallel to each other, so that the structure was not symmetric. Cases 6 and 7 were included to show the effects of further deviation from the requirements of the Principle of Multiples. In case 6 the span of the two central bays was doubled, with the substitute beam length being taken as the average of the sum of the actual beam lengths. In case 7 the EI of the second column from the left was doubled at every storey level. Case 8 was solved in order to see to what extent ~ (again modelled by diagonal bracing) could be used in the substitute frame to represent an actual frame which was braced only in the one bay indicated by the dashed lines on Figure 3. These diagonals and those of the substitute frame all have the value of A given by Eqn. 3. Cases 9 and 10 are identical to cases 5 and 8 respectively, except that ~ for the substitute frames was modelled as shown in Figure 2, instead of by the equivalent diagonals of Eqn. 3.
SOME CONCLUSIONS F R O M THE RESULTS OF TABLE 1 All cases of Table 1 (except cases 8 and 10 which are discussed later) demonstrate good agreement between the substitute frame results and those obtained for the actual frame when using extensible member theory, i.e. the true EA' s. This strongly suggests that the first three modes of the actual frame were sway dominated antisymmetric ones, since these are the only modes which the substitute frame can find. The correctness of this conclusion was verified by calculating the natural frequency for the lowest nonsway (i.e. symmetric) mode and, in case 6, eliminating antisymmetric modes between 0.562 Hz and 1.084 Hz for which the mode could be seen upon inspection to be a 'local' mode, i.e. one dominated by flexure of individual members with very little sway occurring. By comparison with case 1, it can be seen from cases 26, respectively, that : the horizontal beam inertias are important but their transverse inertias have negligible effect; the Grinter frame results are very close to the substitute frame ones, so that the use of Grinter frames for structures which obey the Principle of Multiples may only cause very small errors; allowing for the flexural magnification due to axial forces of practical magnitudes causes significant reductions of the fundamental (12% in this case) and higher natural frequencies and these reductions can be calculated very accurately from the substitute frame; allowing for the stiffening effect of cladding can greatly increase the fundamental (by 133% in this case) and higher natural frequencies and again the substitute frame can be used to calculate these increases very accurately.
96
W.P. Howson and F.W. Williams
Note that none of the cases 15 of Table 1 obey the Principle of Multiples because the outer two of the five columns have twice the required properties, but that nevertheless the excellent agreement of the final two columns of results confirms that the inextensible assumption of the Principle of Multiples is extremely accurate. Cases 6 and 7 show that this agreement remains good for frames which depart more radically from obeying the Principle of Multiples. The reason for the substitute frame results for cases 8 and 9 differing so much (by up to 42%) must principally be the extensibility of the beams and columns, because the actual frame with EA>oo gave results almost identical to those of the substitute frame. Physical reasoning suggests that, because only one bay of the actual frame is braced, the extensions and contractions of beams and columns caused by the forces in the diagonal bracing will be largely confined to the beams in the braced bay and the two columns bounding the bay. This further suggests that the beams and columns of the substitute frame should not be treated as inextensible but should instead be given the EA values of an individual beam and column of the braced bay of the actual frame. When this was done the values of 0.901, 2.363 and 3.054 in Table 1 were replaced by 0.607, 1.861 and 2.777, i.e. the maximum difference of +42% from the 'full frame with actual EA' results was reduced to 7% for the third natural frequency and the fundamental was in error by only 4%. (Note that if the bracing of the actual frame is evenly distributed between the four bays, so that each bay has one quarter of the A, the substitute frame is unaltered but the actual frame results of 0.835, 2.248 and 2.992 are much closer to them, as would be expected because the columns of the actual frame will then change length very little.) Hence, the results of cases 8 and 10 lead to the tentative but important new result that the substitute frame method, with an appropriate value of ~ (modelled either via Figure 2 or the diagonals of Eqn. 3) and with appropriate values of EA, gives a useful indication of the natural frequencies for the sway modes of frames which have bracing in a minority of their bays. Finally, comparison of the results of cases 9 and 10 with those of cases 5 and 8 justifies the use of the equivalent diagonal of Eqn. 3 when software incorporating the model of Figure 2 is not available.
FURTHER THOUGHTS
An unbraced frame is usually one of a set of similar frames which are parallel to it and are connected to it by beams perpendicular to it, e.g. to form a building of rectangular planform. The first author, together with a Master's student, have made a very promising preliminary investigation of predicting the sway modes of such structures which sway parallel to the frames, by applying the rules given earlier to obtain a substitute frame, but with the modification that all the frames are used when applying the rules, e.g. the substitute column k is equal to half of the sum of the k's for all actual columns at the same storey level, regardless of which frame the column lies in, etc. This concept is derived from the fact that floors can reasonably be regarded as being rigid in their own planes, so that all frames share the same horizontal displacements. Of course, such substitute frame results will be exact if the frames are identical to one another, are identically loaded and individually obey the Principle of Multiples. This is clearly true, because the frames would then deflect identically to one another even in the absence of floors and the substitute frame would share exactly the behaviour (i.e. lateral displacements, buckling load factor or natural frequencies) of the substitute frame yielded by one frame alone, since all the stiffnesses and loading of the latter substitute frame would be multiplied by the number of frames to give the former one.
Unified Principle of Multiples for Lateral Deflection, Buckl&g and Vibration
97
REFERENCES
Bolton A. (1976). A simple understanding of elastic critical loads. Struct. Engr. 54:6, 213218. See also correspondence 54:11,457462. Bolton A. (1978). Natural frequencies of structures for designers. Struct. Engr. 56A:9, 245253. See also correspondence 59A:3, 109111. Grinter L.E. (1936). Theory of Modern Steel Structures, Vol. 2. Macmillan, New York. Home M.R. and Merchant W. (1965). The Stability of Frames. Pergamon Press, Oxford. Home M.R. (1975). An approximate method for calculating the elastic critical loads of multistorey plane frames. Struct. Engr. 53:6, 242248. Howson W.P., Banerjee J.R. and Williams F.W. (1983).Concise equations and program for exact eigensolutions of plane frames including member shear. Adv. Eng. Software, 5:3, 137141. Lightfoot E. (1956). The analysis for wind loading of rigidjointed multistorey building frames. Civil Engineering and Pubic Works Review. 51:601,757759; 51:602, 887889. Lightfoot E. (1957). Substitute frames in the analysis of rigid jointed structures (Part 1). Civil Engineering and Public Works Review. 52:618, 13811383. Lightfoot E. (1958). Substitute frames in the analysis of rigid jointed structures (Part 2). Civil Engineering and Public Works Review. 53:619, 7072. Lightfoot E. (1961). Moment Distribution. Spon, London. Naylor N. (1950). Sidesway in symmetrical building frames. Struct. Engr. 28:4, 99102. Roberts E.H. and Wood R.H. (1981). A simplified method for evaluating the natural frequencies and corresponding modal shapes of multistorey frames. Struct. Engr. 59B:1, 19. See also correspondence 59B:4, 6465. Williams F.W. (1977a). Simple design procedures for unbraced multistorey frames. Proc. Inst. Civ. Engrs, Part 2 63, 475479. Williams F.W. (1977b). Buckling of multistorey frames with nonuniform columns, using a pocket calculator program. Comput. Struct. 7:5, 631637. Williams F.W. and Howson W.P. (1977). Accuracy of critical loads obtained using substitute frames. Proc. Int. Conf. Stab. Steel Structs., Liege, 511515. Williams F.W. (1979). Consistent, exact, wind and stability calculations for substitute sway frames with cladding. Proc. Inst. Civ. Engrs. 67:2, 355367. Williams F.W., Bond M.D. and Fergusson L. (1983). Accuracy of natural frequencies given by substitute sway frames with cladding. Proc. Inst. Civ. Engrs. 2:75, 129135.
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W.P. Howson and F.W. Williams
Williams F.W. and Butler R. (1988). Simple calculations for wind deflections of multistorey rigid sway frames. Proc. Instn. Cir. Engrs., Part 2 85, 551565. Wood R.H. (1952). Degree of fixity methods for certain sway problems. Struct. Engr. 30:7, 153162. Wood R.H. (1974). Effective lengths of columns in multistorey buildings. Struct. Engr. 52:7, 235244; 52:8, 295302; 52:9, 341346. Wood R.H. and Roberts E.H. (1975). A graphical method of predicting sidesway in the design of multistorey buildings. Proc. lnstn. Civ. Engrs., Part 2 59, 353372.
Beams and Columns
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THREEDIMENSIONAL HYSTERETIC MODELING OF THINWALLED CIRCULAR STEEL COLUMNS Lizhi Jiang and Yoshiaki Goto Department of Civil Engineering, Nagoya Institute of Technology, Gokisocho, Showaku, Nagoya, 4668555, Japan
ABSTRACT An empirical hysteretic model is presented to simulate the threedimensional cyclic behavior of cantilevertype thinwalled circular steel columns subjected to seismic loading. This steel column is modeled into a rigid bar with multiple vertical springs at its base. Nonlinear hysteretic behavior of thinwalled columns is expressed by the springs. As the hysteretic model for the spring, we modify the Dafalias and Popov's boundingline assumption in order to take into account the degradation caused by the local buckling. The material properties for the vertical spring are determined by using curvefitting technique, based on the inplane restoring forcedisplacement hysteretic relation at the top of the column obtained by FEM analysis. By properly increasing the number of springs, the homogeneity of thinwalled circular columns is maintained. Finally this model is used in threedimensional earthquake response analysis. KEYWORDS Hysteretic model, Threedimensional behavior, Local buckling effect, Steel, Thinwalled column, Empirical model, Earthquake response analysis
INTRODUCTION
In the threedimensional earthquake response analysis for thinwalled steel columns used as elevated highway piers shown in Fig. 1, FEM analysis using shell elements is the only direct procedure that can consider both axial force and biaxial bending interaction and local buckling effect. However, it requires a large amount of computing. Herein, we propose a simple threedimensional hysteretic model for thinwalled circular steel columns. To consider the threedimensional interaction, Aktan And Pecknold (1974) developed a filament model. However, their model cannot consider the effect of the local buckling, since they adopt the bilinear relation for the hysteretic model of each filament. The model we propose herein is alike the filament model but uses fewer springs which simulate the threedimensional interaction. As the hysteretic model for each spring, we modify the Dafalias and Popov (1976) boundingline model in order to take into account the degradation caused by the local buckling. The forcedisplacement relationship for each spring is determined by using curvefitting technique, 101
102
L. Jiang and Y. Goto
based on the inplane restoring forcedisplacement hysteretic relation obtained by the FEM shell analysis. Liu et al (1999) also proposed an empirical hysteretic model ,but the application of this model is restricted to inplane case. The validity of our model is examined by comparing with the results of the threedimensional nonlinear dynamic response analysis using shell element.
Fig. l:Thinwalled steel columns of elevated highways in Japan
BOUNDINGLINE M O D E L IN F O R C E SPACE Inplane Hysteretic Behavior of ThinWalled Circular Steel Columns
From the elastic theory, Timoshenko and Gere (1961), the elastic buckling of columns with circular section is governed by two structural parameters R, and 3, .
R, = R. __.ayX/3(1  v 2)
(1)
~ _ 2L ....
(2)
t
r
E
1 ~~ ~
where R and t are the radius and the thickness, respectively, of the thinwalled circular column; cry is the yield stress of steel ; E is Young's modulus;v is Poisson's ratio; L is the height of column and r is the radius of gyration of cross section. In the plastic range, we assume that the hysteretic behavior of thinwalled circular steel columns is influenced by the axial load ratio P/Py (Py Cry,, A and A is the crosssectional area) in addition to the two structural parameters R t and X. As a result of FEM analysis, hysteretic behavior of thinwalled circular steel columns is classified into three types, depending on the value ofR t , as illustrated in Fig. 2 (a), (b) and (c).Herein, the material behavior of steel is assumed to be represented by the threesurface cyclic plasticity model proposed by Goto et al (1998). The material constants used for the threesurface model is shown in Table 1. Considering the sizes as well as the design loads of real columns, three parameters take the values as 0.1 ___P/Py ~ 0  > r'O) cross section parameters, slenderness and buckling reduction factor of the strengthed section (A,W, I  > ~ ) preload ratio o~, strengthening ratio



Figure 3 : Calculation procedure for N R of the strengthed column under preload. To determine the extent of the strengthening plates AA an iteration process is necessary, because W and depends on AA.
Reduction of load capacity due to welded strengthen plates Due to the welding process of the strengthening plates residual stresses are introduced, which lead to a decrease of the buckling load capacity. Their quantities and distribution are in general hardly predictable, due to the high scatter of influence factors. Therefore the influence of the welding process on the load capacity is estimated in an engineering manner. Following the Europian buckling curves the effect of welding on the bearing capacity can be estimated in form of an additional geometric imperfection ev = 0,5. e 0 . Considering this effect in the analytic model (working with Wv* = w v + ev) leads to a maximum decrease of the load capacity of about 12 % for medium slenderness ratios, shown in Figure 4.
122
H. Unterweger
Figure 4 : Reduction fweld of the calculated ultimate load N R due to welding of the strengthening plates for careful execution.
Comparison of suggested and engineering solution To evaluate the suggested load capacity (Equ. 7  9 > NR, ex) with the engineering procedure (Equ. 3 > NR, ing ) a comparison for practical columns is useful. In Figure 5 the increase of the load capacity using the suggested solution referred to the engineering procedure is shown for the two border line cases. On the one hand type 1 buckling about y axis, which is the most effective case for the strengthening plates; and on the other hand type 2  buckling about z axis. In the first case the differencies in load capacity are given for all slenderness ratios, whereas in the latter case with increasing slenderness ratio the results are more and more equal. The differencies increase with growing preload ratio t~ and strengthening ratio ~. For the theoretical case of a preload ratio of t~ = 1,0 and small slenderness we get the highest difference AN = ~ . NR,ing, which is equal to the difference between elastic and plastic section resistance (Eqn. 1, 2). This example shows the economic advantages of the suggested solution.
Figure 5 : Increase of the ultimate load N R using the proposed solution compared to the engineering approach for two border line cases; a.) type 1, y  axis, b.) type 2, z axis.
Ultimate Load Capacity of Columns Strengthened under Preload
123
Overestimation of load capacity due to plastification of the base section The analytical model neglects the effect of plastification of the base section, which grows with increasing preload ratio o~ and also with higher material strength, because of increased plastic zones. To find out the extent of reduction of the load capacity comprehensive finite element calculations with ABAQUS (1996) were made. The web was modelled with shell elements and the flanges and strengthening plates with special beam elements, including progressive plastifications in thickness direction. The highest reduction of load capacity due to plastification are obtained for type 1  buckling about yaxis. Fortunately the decrease of load capacity is very small, e.g. 2  5 % for a preload ratio ct = 0,5. From the results a simplified conservative procedure for practical use in form of a reduction factor fNR, plast (Figure 6) can be given.
Figure 6 : Reduction factor fNR,plast of the calculated ultimate load N R due to plastification of the base section.
Figure 7 : Load bearing capacity N R of strengthen columns using European rolled sections, expressed in form of a modified buckling factor •* a.) type 1  y ; b.) type 2  z.
124
H. Unterweger
Improved solution for practical design For practical design a direct determination of the extent of strengthening plates AA, expressed by the strengthening ratio ~, is desirable. A comprehensive study shows that for every type of strengthened base section the ratios W / W 0 and ~ / 7~0 can be expressed in form of a linear relationship of ~. Introduction of these information in Eqn. 8 and 9 leads to equivalent buckling factors K:* depending on the slenderness of the base section, the preload ratio t~ and the strengthening ratio ~. In Figure 7 for type 1buckling about y axis and type 2  buckling about z axis, using European rolled sections, the load capacities in form of •* are given. The effectiveness of the strengthening plates is easy to survey. In Figure 8 the suggested simple design procedure for direct determination of the extent of the strengthening plates, based on design charts for the global buckling reduction factors K:*, is shown. In Unterweger (1996, 1998) the design charts for rolled European universal columns for type 1 and 2 are presented.
Figure 8 : Starting position and procedure of a practical design of column strengthening (partial factors omitted). References DIN 18800, Teil 1 und 2 (1990). Stahlbauten  Bemessung und Konstruktion. Deutsches Institut f'dr Normung. Eurocode 3 (1993). Design of steel structures; Part 1.1: General rules. CEN. Rebrov and Raboldt (1981). Zur Berechnung von Druckst~iben, die unter Belastung verst~kt werden. Informationen des VEB MLK 20. Unterweger H. (1996). Berechnung von unter Belastung verstarkten stahlernen Druckstaben, unpublished. Unterweger H. (1998). Druckbeanspruchbarkeit von unter Vorbelastung verst~kten Stiitzen. Stahlbau 68: 3, 196 203.
CHAOTIC BELT PHENOMENA IN NONLINEAR ELASTIC BEAM"
Zhang Nianmei 1
Yang Guitong 2
Xu Bingye ~
1Department of Engineering Mechanics, Tsinghua University, Beijing, China 2Institute of Applied Mechanics, Taiyuan University of Technology, Taiyuan, China
ABSTRACT The chaotic motions of axial compressed nonlinear elastic beam subjected to transverse load P = 8P 0( f + coscot)sin rex are studied in this paper. The constitutive equation of the beam is 1 threefold multinomial. The damping force in the system is nonlinear. Considering material and geometric nonlinearity, nonlinear governing equation of the system is derived. By use of nonlinear Galerkin method, differential dynamic system is set up. Melnikov method is used to analyze the characters of the system. The results show that chaos may occur in the system when the load parameters P0 and f satisfy some conditions. The zone of chaotic motion is belted. The route from subharmonic bifurcation to chaos is analyzed in the paper. The critical conditions that chaos occurs are determined.
125
126
Z. Nianmei et al.
KEYWORDS chaos, bifurcation, heteroclinic orbit, periodic orbit, dynamic system, saddle
INTRODUCTION The chaotic phenomena in solid mechanics fields bring more and more interesting. In 1988, F.C.Moon analyzed the chaotic behaviors of beams experimentally first. Then he studied the dynamics response of linear elastic beam subjected transverse periodic load. The chaotic motions of linear damping beams have been studied by many scholars at home and abroad in resent years. The dynamic behaviors of nonlinear damping beams subjected to transverse load P = S P o ( f + coscot)sin m~rx a r e l studied in this paper. The critic conditions that chaos occurs in the system are determined by use of Melnikov method. The results show that the chaotic areas may be limited ribbon zones.
BASIC EQUATIONS The dynamic behavior of a simply supported nonlinearly elastic beam is studied. Two constant compressive loads N are applied at its two ends. The length of the beam is l. The constitutive relation of beam material satisfies:
o =Eo~ +E16 2)
(1)
where, E and E 1 are material constants. We assume that deformation of the beam is still small deformation after buckling. The buckling critical load of the beam is:
~2EI1A1
Here AI = 1 + 3E1602 . 11 stands for inertia moment, 11 = ~ y 2 d A . A is the cross section area of the A
beam. 60 is the strain at neutral surface, it satisfies:
Chaotic Belt Phenomena in Nonlinear Elastic Beam
(
3~2El/a I
E1Eo4 + E18o 3 + 1  ~ AI 2
602 + 60
~'211 AI 2
=0
127 (3)
The beam is subjected to transverse load P =6Po( f +coscot)sin ~r___x_xafter buckling. Then the 1 governing equation of the system is:
c32M
c32w 02w c3w Ow +N ~ +m = 8 P o( f + coscot)sin ~rx _ 6 / . t ~ ~ ax ~ Ox ~ ~ T at at
(4)
where 8/~ is damping coefficient, m is the mass of unit length of the beam. The boundary conditions of the system are:
w(o): wq)o
(5)
w"(0) = w"(l)=O
(6)
Following formula can stand for the strain at the point which distancement from neutral plain is y: O0 c = 60  y c3x
(7)
where 0 is the rotating angle of cross section of the beam at x. It satisfies:
sin 0 =
1
Ow
(8)
1+~" 0 0 x
Submitting geometric relations and physical relations into eq.(4) and omitting the higher order items than three, follow formula can be obtained:
c, 4w_+c212 2 3+6 w
4wl
128
Z . N i a n m e i et al.
V6(t~3W~ 2 {~2W q_,3~2W~ 2 04W 1
,,[
N E11
+
~
m a2w
(9)
~
E11 O t 2
,,sin"" / ~,
=EI & ( f + cosco

Eq.(9) is turned into dimensionless form using dimensionless amounts x = 
coo
,
cOO =
'
ml4
X
t'
m
w=
W
7'
r =COot
'
"
According to boundary conditions (5) and (6), we suppose follow displacement mode:
n
m
w = c,o(v)sin rcx
(10)
Applying Galerkin method to the dimensionless governing equation, differential dynamic system can be obtained:
(11)
=,~q, p~' +,~o (," +,:os,) ~ ~] where
~~(~+~,~), C1 =
A1 , l+eo
Cz =
p,.,.,,(c,_
A1
C4=
,
2(1+ eo) 3
P = ~ ISc~176
~o  P~
Ell
~) 3ELI2
212(1+6o)312
,
N = N 12
Ell
EI1
If system (11) is not perturbed, g = 0. Then eq. (11) is integrable Hamilton system: 'r = v'
(12)
Hamilton function means the total energy of kinetic energy and potential energy. The energy keeps constant on the same orbit:
h =~
2
+
2
+
4
= const
(13)
129
Chaotic Belt Phenomena in Nonlinear Elastic Beam The phase trajectory of undisturbed system may be determined by following formula:
d(p =+~ 2h 
aq92
~(p4 2
(14)
The formula (14) shows that the phase trajectory has closed relation with the value of a, ft. The dynamic response of the system in the case of stable postbuckling path (a > 0) and fl 1, the conditions that Melnikov function has simple zero points is:
2o Poo ;to 0.2
(1)
Institute (1988) has recommended a fullyield surface of hotrolled 1section for compact section bending about the strong axiS, as, in which M and P are moment and axial force acting on the section, Mp is the plastic moment capacity of the section under no axial force and Py is the pure crush load of the section. Initial Yield Surface
The European Convention for Constructional Steelwork (ECCS 1983) has provided a detailed and comprehensive information with regard to appropriate geometric imperfections, stressstrain relationship and residual stress for uses in the plastic zone analysis. The pattern of ECCS residual stress for hotrolled I and Hsections is shown in Fig. 2. The residual stress will result in the early yielding of a section and the initialyield surface can be defined as, Mer = Z e ( Oy  Ores  P / A )
(2)
in which Mer is the reduced moment elastic capacity under axial force P, Ze is the elastic modulus,
(Yy is the yield stress, Crres is the residual stress and A is the crosssection area. In case of no residual stress and axial force, the M~r will become the usual maximum elastic moment (i.e. Mer = Zr Cry). As the normalized force point is within the initial yield surface, the member behaves elastically. The effect of residual stress on the momentcurvature relationship is illustrated in Fig. 3.
P R O P O S E D PLASTICITY METHOD In the traditional plasticzone (PZ) method, beamcolunm members are divided into a large number of elements and sections are further subdivided into many fibres. The solutions by this method are generally considered as the exact solutions. However, the computation time required is much heavier and it is usually for research study, but not for practical design purpose. To simplify the inelastic analysis, a refinedplastichinge method is proposed because of its efficiency.
SecondOrder Plastic Analysis of Steel Frames
153
RefinedPlasticHinge (RPH) Method The proposed refinedplastichinge method is a plastichinge based inelastic analysis approach considering the stiffness degrading process of a crosssection under gradual yielding for the transition from the elastic to plastic states. In the proposed method, material yielding is allowed at nodal section only and can be represented by a pseudospring. The stiffness of the spring is dependent on the current force point on the thrustmoment plane. When the force point does not exceed the initialyield surface, the section remains elastic and the spring stiffness is infinite. If the point reaches on the fullyield surface, the section will form a fully plastic hinge and the value of the spring stiffness will be zero. To avoid computer numerical difficulties, the limiting values of oo and zero will be assigned as 101~ and 10I~ respectively. When the force point lies between the surfaces, section will be in partial yielding and the function of the spring stiffness, t, is proposed to be given by, t  6 E I IMprM I L I M  M rl
when Mer0
(12)
for no axial force, q=0
(13)
for tension, q 2.5), then the design is considered to be reliable. The existing resistance (capacity) factor (q~) of 0.75 for web crippling strength of single unreinforced webs is given by the AS/NZS 4600 and the AISI Specification. This resistance (capacity) factor (q~ = 0.75) is used in the reliability analysis. A load combination of 1.25DL + 1.50LL is also used in the analysis, where DL is the dead load and LL is the live load. Accordingly, the safety index may be given as,
ln/MInFmPm /
F=
0.691~
4V2M + V~ +CpV~ +0.212
(5)
The statistical parameters Mm, F m, V M and V F are mean values and coefficients of variation for material properties and fabrication variables respectively, and these values are obtained from BHP Structural and Pipeline Products (1998), as shown in Table 4. The statistical parameters Pm and Vp are mean value and coefficient of variation for design equations, as shown in Tables 2 and 3 for current design rules and proposed design equations respectively. The correction factor Cp is used to account for the influence due to a small number of tests (Pek6z and Hall 1988, and Tsai 1992), and the factor Cp is given in Eqn. Fl.l3 of the AISI Specification. The safety index in Eqn. (5) is detailed in Rogers and Hancock (1996). The safety indices (13) of the current design rules to predict the web crippling strengths for the four loading conditions are lower than the target safety index, except for the EOF loading condition as shown in Table 2. Safety indices as low as 0.48 were calculated for the ITF loading condition. However, this is not the case for the proposed design equations, the safety indices are higher than the target value for the four loading conditions as shown in Table 3. Therefore, the proposed design equations are much more reliable than the current design rules. The proposed design equations produce good limit state design when calibrated with the existing resistance (capacity) factors (~  0.75).
Web Crippling Tests of High Strength ColdFormed Channels
365
CONCLUSIONS A series of web crippling tests has been conducted to examine the appropriateness of the current design rules stipulated in the Australian/New Zealand Standard (AS/NZS 4600, 1996) and the American Iron and Steel Institute (AISI, 1996) Specification for coldformed steel structures. Tests were performed on high strength coldformed unlipped channels having nominal yield stress of 450 MPa, and the web slenderness values ranged from 15.3 to 45. The specimens were tested using the four loading conditions (EOF, IOF, ETF and ITF) according to the AISI Specification. The test strengths were compared with the current design strengths obtained using AS/NZS 4600 and the AISI Specification. It is demonstrated that the current design strengths predicted by the standard and specification are unconservative for unlipped channels (single unreinforced webs), except that they closely predicted the web crippling strengths for the EOF loading condition in most of the cases. For a certain specimen subjected to ITF loading condition the test strength is only 43% of the current design strength predicted by the standard and specification. Since the design strengths obtained using the current design rules are generally unconservative for unlipped channels, therefore, a set of equations to predict the web crippling strengths have been proposed in this paper. The proposed design equations are derived based on a simple plastic mechanism model, and these equations are calibrated with the test results. It has been shown that the proposed design strengths are generally conservative for unlipped channels with web slenderness values of less than or equal to 45. The reliability of the current design rules and the proposed design equations have been evaluated using reliability analysis. In general, the safety indices of the current design rules are lower than the target safety index of 2.5 as specified in the AISI Specification. Whereas the safety indices of the proposed design equations are higher than the target value. Therefore, it has shown that the proposed design equations are much more reliable than the current design rules for the prediction of web crippling strength of the tested channels. The proposed design equations are capable of producing reliable limit state designs when calibrated with the existing resistance (capacity) factors. ACKNOWLEDGEMENTS The authors are grateful to the Australian Research Council and BHP Structural and Pipeline Products for their support through an ARC Collaborative Research Grant. Test specimens were provided by BHP Steel.
REFERENCES
American Iron and Steel Institute (1996). Structural Members, AISI, Washington, DC.
Specification for the Design of ColdFormed Steel
Australian Standard (1991). Methods for Tensile Testing of Metals, AS 1391, Standards Association of Australia, Sydney, Australia. Australian/New Zealand Standard (1996). Standards Australia, Sydney, Australia.
ColdFormed Steel Structures, AS/NZS 4600:1996,
BHP Structural and Pipeline Products (1998). Pipe, Tube and Structural Products  Mechanical Test Data. Somerton plant, NSW, Australia.
366
B. Young and G.J. Hancock
Hetrakul N. and Yu W.W. (1978). Structural Behavior of Beam Webs Subjected to Web Crippling and a Combination of Web Crippling and Bending. Final Report Civil Engineering Study 784, University of MissouriRolla, Mo, USA Nash D. and Rhodes J. (1998). An Investigation of Web Crushing Behaviour in ThinWall Beams. ThinWalled structures 32, 207230. Pektsz T.B. and Hall W.B. (1988). Probabilistic Evaluation of Test Results. Proceedings of the 9th International Specialty Conference on ColdFormed Steel Structures, St. Louis, University of MissouriRolla, Mo, USA. Rogers C.A. and Hancock G.J. (1996). Ductility of G550 Sheet Steels in TensionElongation Measurements and Perforated Tests. Research Report R735, Department of Civil Engineering, University of Sydney, Australia. Tsai M. (1992). Reliability Models of Load Testing. PhD dissertation, Department of Aeronautical and Astronautical Engineering, University of Illinois at UrbanaChampaign. Winter G. and Pian R.H.J. (1946). Crushing Strength of Thin Steel Webs. Cornell Bulletin 35, Part 1, Comell University, Ithaca, NY, USA. Young B. and Hancock G.J. (1998). Web Crippling Behaviour of ColdFormed Unlipped Channels. Proceedings of the 14th International Specialty Conference on ColdFormed Steel Structures, St. Louis, University of MissouriRolla, Mo, USA, 127150. Zetlin L. (1955). Elastic Instability of Flat Plates Subjected to Partial Edge Loads. Journal of the Structural Division, ASCE 81:795, 124. Zhao X.L. and Hancock G.J. (1992). Square and Rectangular Hollow Sections Subject to Combined Actions. Journal of Structural Engineering, ASCE 118:3, 648668. Zhao X.L. and Hancock G.J. (1995). Square and Rectangular Hollow Sections under Transverse EndBearing Force. Journal of Structural Engineering, ASCE 121:9, 13231329.
LOCAL AND DISTORTIONAL BUCKLING OF PERFORATED STEEL WALL STUDS Jyrki Kesti 1 and J. Michael Davies 2 ~Laboratory of Steel Structures, Helsinki University of Technology, P.O. Box 2100, FIN02015 HUT, Finland 2Manchester School of Engineering, University of Manchester, Manchester, M 13 9PL, UK
ABSTRACT This paper considers the compression capacity of webperforated steel wall studs. The web perforations decrease the local buckling strength of the web and the distortional buckling strength of the section. An analytical prediction of the compression capacity is described. Local and distortional buckling stresses are determined by replacing the perforated part of the web with plain plate of equivalent thickness. The effective area approach is used to consider local and distortional buckling. Comparison between the test results for short columns and the corresponding predictions shows that the method used gives reasonable results for webperforated Csections with or without webstiffeners. KEYWORDS Coldformed steel, wall stud, perforation, compression, local buckling, distortional buckling. INTRODUCTION Webperforated steel wall studs are especially used in the Nordic countries as structural components in steelframed housing. The slotted thermal stud offers a considerable improvement in thermal performance over a solid steel stud. The wall structure consists of webperforated Csection studs with Usection tracks top and bottom and, for example, gypsum wallboards attached to the stud flanges. The sections investigated in this paper are shown in Figure 1. Both types of stud had six rows of slots with dimensions as shown in the Figure. The perforations reduce the elastic local buckling stress of the web and also reduce the bending stiffness of the web which, in turn, results in decreased distortional buckling strength. The aim of this paper is to analyse the local and distortional buckling strength of the perforated steel stud. The local and distortional buckling modes are taken into account in design by using the effective area approach. 367
368
J. Kesti and J.M. Davies
Figure 1: Webperforated Csection and webstiffened Csection (Dimensions in mm)
ELASTIC BUCKLING STRESSES Local Buckling Stress o f the Web o f a Perforated CSection
The depth of the sections considered varied between 150 and 225 mm with a thickness between 1 and 2 mm. The overall depth of the perforations was 58 mm. Local buckling of the perforated region was studied using the elastic buckling analysis available in the NISA finite element software (1996). The analyses were carried out for both the isolated web element, which was assumed to be simply supported, and for the whole section, including the edgestiffened flanges. The width of the flanges was 50 mm and the width of the stiffeners was 15 mm. Individual plate elements and the complete sections of 800 mm in length were modelled, including the perforations. A sufficient length was chosen so that the minimum local buckling stress could be achieved. The elastic local buckling stress, l~rcr.perf, for simply supported perforated plate elements of different widths and thicknesses was determined using the finite element method (FEM). An analytical expression for the local buckling of a perforated plate may be achieved using a buckling factor of k = 4.0 and an equivalent thickness, tr,to~, for the whole plate. The equivalent thickness was determined in a manner similar to Salmi (1998):
/ t r ,loc ",tl
O'cr ,perf.
t
(1)
1 O'cr ,entire
where O'cr,pe~ is the elastic buckling stress of the perforated plate and O'cr,entire is the elastic buckling stress of the entire plate. The elastic buckling stress of the equivalent plate with reduced thickness trtoc is thus the same as that of the perforated plate. The value for tr.toc was found to be in the range 0 . 7 2 t 0.75t for the plates studied. Thus, for design purposes, the equivalent thickness value, tr.toc = 0.72t could be used for the whole range of sections. Local buckling stresses for the whole of the perforated sections, including the flanges, were on average 75% higher than those of the simply supported perforated plates. This indicates that assuming the web part to be simply supported leads to quite conservative results and the contribution of the flanges to the local buckling of the web should generally be considered.
Distortional Buckling Stress Because of the perforation of the web, the, transverse bending stiffness of the section is rather low and the section is sensitive to distortional buckling under compressive load. In the distortional mode of buckling, the edgestiffened flange elements of the section tend to deform by rotation of the flange about the flangeweb junction. The distortional buckling mode occurs at longer wavelengths than local buc.t!ing. Numerical methods, such as the finite strip method (FSM), may be used to determine
Local and Distortional Buckling of Perforated Steel Wall Studs
369
the distortional buckling stress of the section. The Generalized Beam Theory (GBT) provides a particularly good tool with which to analyse distortional buckling in isolation and in combination with other modes. Some approximate manual methods have also been presented, namely the Eurocode 3 (1996) method, which is based on flexural buckling of the stiffener, and a more sophisticated model developed by Lau and Hancock (1987). The most recent method has been presented by Schafer and Pektiz (1999). Schafer's method was used in this study and it was modified to cover the perforated Csections, as shown in Figure 2.
Figure 2: Notations for the perforated Csection and for the flange part alone In the Schafer method, the closedform prediction of the distortional buckling stress is based on the rotational restraint at the web/flange junction. The rotational stiffness may be expanded as a summation of the elastic and stressdependent geometric stiffness terms with contributions from the flange and the web,
+k,)e(k,
(2)
where the subscript f indicates the flange and w the web. Buckling takes place when the elastic stiffness at the web/flange junction is eroded by the geometric effect, i.e., Q =0.
(3)
Using (3) and writing the stressdependent portion of the geometric stiffness explicitly,
ks = kcfe+kc~efcr,d ('kofg atk~c~g) : 0 .
(4)
Therefore, the distortional buckling stress,f~r,a, is
k cge+k c~e k c/g+ k c,,e'
(5)
f cr ,d '~ "~
where the stiffness terms with the notations given in Figure 2 are:
kc/e =
1;4i '[{
2
1112
EIx: (Xo: _hx: )2 + EIw: _Elx~ (xo: _hx: )2 + lyy
k#g = (L ! A: (Xo/h~zy([,I~ I~
!
G1r
(6)
I 1
~ l +) h2x:+YoI _2yo(Xoi_h~ f ~[ IIy: 2 +Ix:+ Iyl
(7)
J. Kesti and J.M. Davies
370
(1'
k~c~g= s th~ ~,Lj 60
(8)
in which L is the critical length for distortional buckling or the distance between restraints which limit rotation of the flange part of the section. The elastic stiffness term for the web is modified to take into account the more flexible perforated part of the web. Thus the perforated part of the web is replaced by a plate of equivalent thickness, tr, which has the same bending stiffness as the perforated web part. For the particular perforation type used in this study, the equivalent thickness was determined by means of the finite element method but, for simpler cases, this may be done by hand calculation. The elastic stiffness term for the web with a different thickness, tr, in the middle part may be expressed as: 1
=
(9)
kc~e 6(l 3o2 b~+ 2b2w w + bwb~ h~ lj+ l2(lEtZrv2 )( 2h~ b2~ bwb~h~lj The critical halfwave length for distortional buckling may now be expressed as:
Lcr = ~r
Ixe (xo  hx )2 +I,o l~y(x ~  hx)2 Iy
11 TM
(10)
The manual calculation method for the distortional buckling of a Csection with reduced thickness in part of the web was verified by comparing its predictions with values given by GBT. Csections with a height in the range 150 to 225 mm and a thickness in the range 1 to 2 mm were used for this comparison. The flange width was 50 mm and stiffener width was 15 mm. The reduced thickness value of 0.39t was used, corresponding to the studied perforation type. The mean ratio between the calculated and GBT value of 0.96 and the standard deviation value of 0.04 demonstrate a good performance for the proposed method. Distortional buckling of the webstiffened Csections is much more complicated and is usually a combination of distortional buckling of the edge and webstiffeners and should be thus determined by using the FSM or GBT.
DESIGN EXPRESSIONS
Current design recommendations do not include the design of perforated sections. In this study, the perforations were considered by using an equivalent thickness for the perforated part when determining the elastic local and distortional buckling stresses. In Eurocode 3, local buckling is taken into account by using effective widths for plane elements and distortional buckling is taken into account by reducing the thickness of the stiffeners. The reduction is based on the Eurocode column design curve ao with ct = 0.13. Schafer and Pek6z (1999) proposed a new design method for considering local and distortional buckling in which local buckling and distortional buckling are seen as competitive buckling modes. Either one of them may be chosen to represent the buckling mode for each plane element. In order to properly integrate distortional buckling into the analysis, reduced postbuckling capacity in the distortional mode and the ability of the distortional mode to control the failure mechanism, even when
Local and Distortional Buckling of Perforated Steel Wall Studs
371
at a higher buckling stress than the local buckling, must be considered. Schafer and Pekoz therefore proposed a method where the critical buckling stress is defined for each plane element as: L r " min Lfcr,f , R d f ~r,a ]
( 11 )
The t e r m fcr, f is local buckling stress based on a buckling coefficient value of k = 4.0 and fcr, d is the distortional buckling stress. The reduction factor for the distortional buckling stress is as follows:
Rd = 1 when ,;ta < 0.673
(12a)
1.17 ~+0.3 Rd Ad+l
(12b)
when ,;ta> 0.673
where 2d = 4 f / f c r , d
"
Finally local and distortional buckling are considered using an effective width approach in which the effective width of each plane element is determined using the wellknown Winter reduction: p=l
when 2 0.673
(13b)
2=4f/fcr
.
Schafer and Pek6z suggested that this reduction could be made for the entire member instead of each element if the buckling stresses were determined numerically. The above approach was examined for the strength capacity of laterally braced flexural members but, in this study, it has also been applied to compression members.
COMPARISON OF TEST RESULTS AND PREDICTED VALUES Results from the compression testing of both section types are available. Salmi (1998) carried out some tests on perforated C and webstiffened Csections. Kesti and Makel~iinen (1999) have also conducted tests on perforated webstiffened Csections. The dimensional notations are shown in Figure 3 and the measured dimensions, yield stresses and failure loads are given in Table 1 for webstiffened Csections and in Table 2 for Csections. The quoted metal thickness value is the core thickness without the zinc layer. The TCJ and TCSSections are from Salmi's test series.
Figure 3: Notations for section dimensions.
J. Kesti and J.M. Davies
372
L CC1.2W1 CC1.2W2 CC1.5W1 CC1.5W2 TCJ1 TCJ2 TCJ3 TCJ4 TCJ5
TABLE 1 TEST DATA FOR WEBSTIFFENEDCSECTIONS bl/b2 el/c2 el/e2 alia2 fl/f2 dl/d2
h
~[rnrn] [nun] [mm]
[nun]
[mm]
[mm]
800 800 800 800
173.7 49.8148.9116.2/16.4 22.8/24.8 173.7 49.7/49.8 16.2/16.1 22.8/24.7 174.1 49.9/50.2 16.2/16.8 23.6/22.9 174.2 49.5/49.8 16.3/16.7 23.0/23.1
800 700 800 700 700
149.5 45.3/46.2 16.7/15.5 22.1/25.3 5.1/5.3 149.1 46.7/48.4 19.9/15.5 12.7/16.3 5.3/5.1 174.6 43.2/44.4 14.9/16.8 33.6/37.9 5.3/5.3 198.9 39.2/40.0 16.9/17.0 33.9/37.8 5.3/5.2 224.3 46.6/46.3 17.0/16.8 35.3/38.01 5.3/5.3
L TCS1 TCS2 TCS3 TCS4 TCS5
h
[mm]
22.4/22.4 22.4/22.4 23.3/22.5 22.6/22.5
[nun]
13.1/10.7 13.1/10.7 10.0/10.9 11.3/10.6
18.0/17.8 4.2/4.2 17.0/17.2 14.0/14.0 17.8/17.8 4.8/4.8 16.8/17.8 17.4/17.4 17.4/18.0 28.8/28.8
TABLE 2 TEST DATA FOR CSECTIONS e t Area A bl/b2 Cl/C2
[mm] [mm] [mm] 800 796 796 798 897
9.3/9.2 9.3/9.3 8.8/8.3 8.9/7.9
149.0 49.7/48.2 173.7 46.2/47.5 173.8 49.1/49.6 199.0 44.0/43.3 223.8 49.0/49.2
[mm]
16.8/15.9 16.4/17.0 16.4/13.4 16.2/16.2 18.9/15.7
[mm] 45.5 57.9 57.9 70.5 82.9
t
[mm] 1.16 1.17 1.95 1.45 1.16
[mm2] 257.1 284.1 476.6 378.1 346.6
Area A
Yield Failure Stress load
[mm] [mm2] [N/mm2] [kN] 1.15 1.15 1.47 1.47
301.3 300.8 377.2 377.1
386 386 380 380
64.4 73.5 96.2 83.1
1.16 1.45 1.17 1.45 1.16
256.1 329.6 282.8 376.9 346.6
387 363 395 366 395
59.9 84.1 63.3 76.6 67.3
Yield Failure Stress load [N/mm2] [kN] 388 52.5 392 55.3 356 108.3 366 74.5 395 57.3
In the analysis of the Csections, the local buckling stress of the perforated section was determined by FEM in order to provide a more exact value for the effective width expressions. As mentioned above, the analytical method for the determination of the local buckling stress is quite conservative if the web is assumed to be simply supported without any contribution from the flanges. The effective width was determined by reducing the area of the whole web, t'hw. Nevertheless, the effective width can be, at most, the width of the entire web section. In the design of the webstiffened section, the perforated web part was ignored and the plate element between the perforated part and plain part was assumed to be an unsupported element for the determination of the effective widths. The elastic distortional buckling stresses for both the section types were determined using the Generalized Beam Theory taking into account the actual column length and the fixedended boundary conditions. The analysis was carded out using the computer program written by Davies and Jiang (1995). Manual calculation methods would require that the column is sufficiently long for the end boundary conditions to be insignificant. The perforated web part was replaced with an equivalent thickness of 0.39t corresponding to the same bending stiffness. The webstiffeners of the TCJsections were relatively small and thus the webstiffeners may buckle at a lower stress than the edgestiffeners. Thus, the distortional buckling stress was determined separately for the modes which included either buckling of the webstiffeners or edgestiffeners. The predicted values were determined using both the EC3 method and Schafer's method. A comparison of the predicted and test values is given in Figure 4. The mean value for the capacity ratio Ntest/Np is 1.02 according to the EC3 method and 1.09 according to Schafer's method. The standard deviations are 0.08 and 0.11 respectively.
Local and Distortional Buckling of Perforated Steel Wall Studs 1,50 1,40 1,30 1,20
//~\
z" 1,1o "~ 1,00 z
373
\ ~"it ,
r
0,90 0,80
"' Schafer 4 EC3 d
0,70 0,60 0,50
r
w''r c..)
c,4
c',4 ,.
~ ,._
c_)
~
r
t.~
r c,.)
Figure 4: Comparison of test results and predicted values
THE DESIGN OF STUDS OF FULL L E N G T H W H I C H ARE RESTRAINED BY SHEETING This study is mainly concerned with the local and distortional buckling of perforated wall studs. These buckling modes are taken into account in the design by using an effective area approach. Flexural buckling modes should also be considered in the design of studs of full length. Minor axis buckling is usually prevented by the sheeting connected to both flanges of the section. In any case, the screw connections have limited shear stiffness and the flexural buckling stress about the minor axis of the stud should be determined as the buckling stress of the compressed strut on an elastic foundation (H6glund, 1998). The web perforations decrease the flexural buckling capacity about the strong axis of the stud due to shear deformations. Allen (1969) presented a buckling load formula for sandwich structures with thick faces when the bending stiffness of the faces is significant compared to that of the whole structure. This sandwich theory was also applied to the perforated steel studs. The buckling load can be expressed as
Ncr ~d = Ne 9
1~ Ne/
N# Nel
S,,
S,, N e
1~
N,
Sv
(14)
N#
Sv
)
where Ne is the Euler load for the column ignoring the effect of shear deformation, Nef is the Euler buckling load of one flange part and Sv is the shear stiffness of the section. Flexural buckling may be taken into account by reducing the yield stress using the column curves given in Eurocode 3. The elastic distortional buckling stress of the perforated stud is quite low when distortional buckling is free to develop. The sheathing screws offer considerable resistance to distortional buckling but the utilisation of this support requires that the sheeting retains its capacity and stiffness for the expected service life of the structure. The resistance of sheathing screws to distortional buckling may be taken into account by using a convenient buckling length in the rotational stiffness equations (6)  (8) or in the numerical analysis. However, the possibility of a failed screw connection in any location should be considered and a minimum buckling length of twice the screw pitch may conservatively be used in the design.
374
J. Kesti and J.M. Davies
CONCLUSION An analytical prediction has been given for the local and distortional buckling capacity of perforated studs. The elastic local or distortional buckling strength may be determined by replacing the perforated web part with a plain plate of equivalent thickness. The effective crosssectional area may be determined according to Eurocode 3 or by using the method suggested by Schafer. A comparison of test results for short perforated columns with the predicted values showed that the method used gives reasonable results for perforated Csections with or without webstiffeners. The analysis also showed that the local buckling stress of the perforated web of the Csection is conservative if it is determined assuming that the web is simply supported without any contribution from the flanges. ACKNOWLEDGMENTS This paper was prepared while the first author was on a oneyear study leave at Manchester University. This leave was supported by The Academy of Finland. The facilities made available by the Manchester School of Engineering are gratefully acknowledged. REFERENCES
Allen, H. (1969), Analysis and Design of Structural Sandwich Panels, Pergamon Press. Davies, J. and Jiang, C. (1995). GBT  Computer program, public domain, University of Manchester. Eurocode 3 (1996), CEN ENV 199313 Design of Steel Structures  Supplementary Rules for Cold Formed Thin Gauge Members and Sheeting, Brussels. Htiglund, T. and Burstrand, H. (1998), Slotted Steel Studs to Reduce Thermal Bridges in Insulated Walls, ThinWalled Structures, 32:13, 81109. Kesti, J. and Makelainen, P. (1999), Compression Behaviour of Perforated Steel Wall Studs, 4 th International Conference on Steel and Aluminium Structures ICSAS'99, Espoo, Finland, 123130. Lau, S. and Hancock, G. (1987), Distortional Buckling Formulas for Channel Columns, Journal of Structural Engineering, 113:5, 10631078. NISA, Version 6.0 (1996), Users Manual, Engineering Mechanics Research Corporation (EMRC), Michigan. Salmi, P. (1998), Uumasta termorei'itettyjen profiilien mitoituksesta, Ter~rakenteiden tutkimus ja kehitysp~iiv~it, 1998, Lappeenranta, Finland. Schafer, B. and PekiSz, T. (1999). Local and Distortional Buckling of ColdFormed Steel Members with Edge Stiffened Flanges, 4 th International Conference on Steel and Aluminium Structures ICSAS'99, Espoo, Finland, 8997.
AN EXPERIMENTAL INVESTIGATION INTO COLDFORMED CHANNEL SECTIONS IN BENDING V Enjily l, M H R Godley 2 and R G Beale2 ~Centre for Timber Technology and Construction, Building Research Establishment, Watford, WD2 7JR, UK 2Centre tbr Civil Engineering, Oxford Brookes Universky, Oxford, OX3 0BP, UK
ABSTRACT The objective of this research was to investigate the postbuckling behaviour of coldformed plain channel sections in bending. 26 coldformed plain sections were tested with their unstiffened flanges in compression with a range of external flange width/thickness ratios (B/t) ranging from 5 to 94. The sections with B/t ratios less than 15 were able to carry the full plastic moment. Sections with higher B/t ratios developed trapezoidal yield lines with ultimate loads accurately predicted by the classical yield line theory of Murray. 24 channel sections were tested with their stiffened webs in compression with web width/thickness ratios (D/t) ranging from 18 to 186. Sections with D/t less than 60 carried the full plastic moment. Initial tests on specimens with D/t ratios between 60 and 100 failed with local web crushing. A modified loading procedure involving applying the load in the middle of the flanges was adopted which produced results with some sections with D/t ratios between 60 and 100 carrying the full plastic moment. All sections with D/t ratios in excess of 100 failed with a 'pitched roof' mechanism in the stiffened web. Murray' s theory was less able to predict collapse behaviour in this case but comparisons with the theory are given. The experimental results are compared with BS5950 and design recommendations drawn up. KEYWORDS Coldformed, steel, channels, bending, design INTRODUCTION Considerable research has been carried out, both experimentally and theoretically, into coldformed channel sections in axial compression, for example, Rhodes and Harvey (1976), Stowell (1951), 375
V. Enjily et al.
376
Murray (1984) and Little (1982) but little research has been reported into bending, Rhodes (1982,1987), Enjily (1985). The objective of this research was to conduct a full experimental investigation of coldformed plain channels in bending and to correlate the results with yield line theory.
UNSTIFFENED COMPONENTS (FLANGES) IN COMPRESSION
Experimental Technique All specimens were tested in four point bending. Load was applied by use of a screw jack and spreader beam so that the load shedding part of the experimental cycle could be followed. The end conditions were simplysupported. Two spans of 1000mm and 500mm were tested with a region of pure bending of 300mm and 200mm in the centre of each specimen. Measurements of central deflection were made by use of a dial gauge on the centre of the web which was in tension and by means of a cathetometer sighted on the webflange junction. Load cells were placed under each support as shown in figure 1.
Figure 1: Experimental Layout and System of Loading To ensure that the sections maintained shape, endplates were welded onto the ends of each specimen. These end plates were made from 4mm thickness steel plate and extended 25mm beyond the outer fibres of the section. They also served to ensure that the end conditions were simplysupported (see figure 2). A screw jack was used in order to apply displacement increments. The load was applied via two rollers nesting on the inside of the web. After each load increment was applied the system was allowed to stabilise. An initial set of experiments was performed on a channel with dimensions 100"50"1.6 and the stabilisation time was varied from between three and eight minutes in order to see if there was any significant creep. The resulting maximum loads only varied by 30N. Hence all experiments reported herein were tested at a stabilisation time of three minutes.
Experimental Investigation &to ColdFormed Channel Sections in Bending
377
i...:. L
r r
PLA___"_.
ilID
ClIOI$IECTI
OI;
lI
PLATE
C U A I i I . [ x. t n r c T I O I t
(
Figure 2" Details of endplates The experimental results are summarised in Table 1. Sample experimental curves are given in figure 3. Note the double 'kink' in the experimental curve of figure 3b. This occurred when one flange buckled before the second. In most cases failure occurred simultaneously in both flanges. TABLE 1 TEST RESULTSFOR THE ULTIMATELOADSOF CHANNELSECTIONSWITHTHEIRFLANGESIN COMPRESSION.
Experiment Reference ] M9
MI0 Mll 'MI2 _ MI3
MI4 Mi5 M16
Q~ Q2 .Q3 Q4 _Q5 9 10 11 12 13 14 ,15 16 Q6 Q7 Q8 Q9 Q10
,
Section Size [Yiek [ Young's (D*B*t) ! Slrer th ] Modulus
30* 8"i.6 45* 16"1.6 60* 24"1.6 75* 32"1.6 90* 40"1.6 105" 48"1.6 120" 56"i.6 135" 64"1.6 160" 80"!.6 210"105"1.6 240"120"1.6 270"135"1.6 3o0"150"1.6 30* 8"1.6 45* 16"i.6 60* 24"1.6 75* 32"1.6 90* 40"1.6 105" 48"1.6 120" 56"1.6 135" 64"1.6 160" 80"1.6 210"105"1.6 240"120"1.6 270"135"1.6 300"150"1.6
](N/Ill 12) ] (N/ram 2)
~ 232.5 198700 jr'232.5 " 198700 jr 232.5 198700 jr'232.5 198700 jr232.5 198700 jr232.5 198700 ~232.5 198700 4232.5 198700 Jr 183.(3 196000 ~_183.(3 196000 J183.G 196000 /183.0 196000 T183.0 196000 4232.5 198700 ~232.5 198700 ]_232.5 198700 ~232.5 198700 [232.5 198700 42325 198700 ~232.5 198700 E232.5 198700 ~183.0 196000 ~183.0 196000 ~183.0 196000 ~'83.0 196000 UI.83.0 196000
ExpeimentalIFuliPlastic Experimental 1BS5950 lExperimental 1;pan
Faih'e Load  Load (kN) (kN) t 0.13 0.58 1.3(3 1.89 2.19 2.46 2.71 3.04 3.80 5.10 5.40 6.40 7.40 0.32 1.34 2.90 4.24 5.02 5.82 6.27 6.80 8.70 11.70 13.10 11.90 12.70
0.122 0.502 Jr I.!54 Jr 2.079 Jr 3.275 Jr 4.743 4 6.684 jr 8.497 Jr 10.495 jr 18.166 Jr 23.773 jr 30.133 Jr36.,724 jr 0.284 Jr 1.172 Jr 2.693 Jr 4.fl50 Jr 7.641 Jr I1.068 ].i5.i29 Jr 19.825 ]24.484 ]_42.388 Jr55.470 ]_70.3"09 185.6"89
 4
Failur~ Load/] Failure I Full Plastic 1 Load i Load
1.065 1.155 . 1.126 0.909 0.669 0.519 0.418 0.358 0.362 0.281 0.227 0.212 0.201 1.123 1.144 1.077 0.874 0.657 0.526 0.414 0.343 0.355 0.276 0.236 0.169 0.148
1.0.122 1.0.475 1.0.660 1.1.153 ~ 1.635 1.1.952 1.2.132 1.2.259 1.2.355 1.2.689 ~ 2.908 13.140 ].3.361 1.0.284 ~1.107 ~1.339 ~2.690 13.816 [_4.556 14.975 15.271 15.494 [_6.273 16.785 ~7.328 17.855
Failure Load/  imm)
BS5950
Failure Load
1.065 1222 1.970 " 1.639 "' 1.339 1.160 1.271" 1.346 1.614 1.897 1.857' 2.038 .,. 2.202 . 1.123 1.211 1.884 1.576 1.316 1.278 1.260 .... 1.290 1.583 1.865 1.931 " 1.624 1.617
l B/t
ratio
[ I ~ i000
t
J ~000 j O00 .[ 000 1. 000 ~ 000 J 000 1. 000 ] 000 1. 000 1. 000 ~ 000 1. 500 I 500 ( 500 1. 500 I 500 1. 500 1. 500 1. 500 ~ 500 ~ 500 I 500 I 500 / 500
1. 14.50(3 1.19.50(3 1.24.500 ~29.500 1.34.500 ~39.500 ~ 49.500 1.65.125 1.74.500 183"875 1.93.250 [. 4.500 ~ 9.500( ~14.500 ~ 19.500 1.24.500 [_29.500 1.34.500 139.500 149.500 165.125 ~74.500 ~83.875 /9~ 250
4.50C
I ~000 1. 9.50c
Theoretical Analysis To enable an understanding of the ultimate post yield behaviour of channels the simple yield line model described by Murray (1984) was used. Channels with flange/thickness ratios less than 15 were able to attain their full plastic moment capability. Longer flanges developed local buckles in the flanges which ultimately produced a series of yield lines as shown in Figure 4 In the analysis the compression elements were divided into a series of strips. Element equilibrium of each strip was used to derive values of the elemental force in each strip and their corresponding moments of resistance.
378
V. Enjily et al.
Figure 3a
Figure 3b
Figure 3c
Figure 3d
Figure 3: Typical experimental curves, flanges in compression
Figure 4: Side Elevation in the region of localised buckling
Experimental Investigation into ColdFormed Channel Sections in Bending
379
These elemental values were then integrated to obtain the total moment of resistance of the compression flange. Full details of the procedure are found in Enjily (1985) and Enjily et al (1998). Figure 5 is a comparison between theory and experiment.
Figure 5: Comparison between theory and experiment (Specimen 14)
Discussion In all cases the channels failed by forming a yield line mechanism in one, or both, of the unstiffened webs. The experimental results are summarised in Table 1. It can be seen that channel sections can carry the full plastic moment for flanges with B/t ratios less than 16. The test results were compared with the maximum loads predicted by BS5950 Part 5(1987). The maximum loads predicted by BS5950 are also given in Table 1. The load factors against collapse calculated by BS5950 are seen to be very conservative giving results that vary from 1.065 to 2.2; the larger discrepancies occurring for the largest flange wide/thickness ratios.
Stiffened component (web) in compression Experimental Technique The initial arrangement of the test rig for specimens tested with the web in compression was identical to that for the case with the flanges in compression, with the exception that the cathetometer was used at midspan. Initially, the load was applied through rollers sitting on the web. However, local crushing occurred under one or both rollers for experiments 18 and P6P9 resulting in premature failure. Owing to the thinness of the webs, and in order to overcome the bearing and tensile stresses exerted on the flanges by this loading, small steel plates were welded to the outer surface of the flanges (close to the free edge) at distances 300mm apart. Holes were drilled through the plates and flanges. Four long stripplates were attached with plate hangers and bolted to the sections. 40mm bars were placed through the plate hangers to provide supports for a loadspreader beam. The screw jack was then placed on top of the spreader and reacted against an independently mounted beam. The force applied by the screw jack was then reacted through this improved loading arrangement into the flanges of the channel section putting it into bending. Figure 6 shows a diagram of the improved loading system. This system was applied to specimens Y1Y11.The results of the experiments with the webs in compression are summarised in Table 2. Typical experimental curves are given in Figure 7. Local crushing was observed in most of the
380
V. Enjily et al.
experiments involving a span of 500mm leading to results significantly below the colTesponding curve for the 1000mm span
TABLE 2 TEST RESULTSFOR THE ULTIMATELOADSOF CHANNELSECTIONSWITHTHE WEB IN COMPRESSION. Experiment Section Size Reference (D*B*t) Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 YIO YI 1 1 2 3 4 5 6 7 8 P6 P7 P8 P9 P! 0
60* 24"1.6 75* 32"!.6 90* 40"1.6 105" 48"1.6 120" 56"1.6 135" 64"1.6 160" 80"1.6 210"105"1.6 240* 120* 1.6 270"135"1.6 300"150"1.6 30* 8"1.6 45* 16"1.6 60* 24"1.6 75* 32"1.6 90* 40"1.6 105" 48"1.6 120" 56"1.6 135" 64"i.6 160" 80"1.6 210"105"1.6 240* 120 * 1.6 270"135"1.6 300* 150* 1.6
(N/mm2)
Yield Young's Experimental F u l l Strength Modulus FailureLoad Plastic (N/ram2)
(kN)
Load (kN)
Full Plastic
Experimental BS5950 FailureLoad/ Failure Load Load
Failure Load
210.0 210.0 210.0 210.0 210.0 210.0 210.0 210.0 210.0 210.0 210.0 232.5 232.5 232.5 232.5 232.5 232.5 232.5 232.5 !83.0 183.0 183.0 183.0 183.0
199300 199300 199300 199300 199300 199300 199300 199300 199300 199300 199300 198700 198700 198700 198700 198700 198700 198700 198700 196000 196000 196000 196000 196000
i.30 2.20 3.30 4.80 6.40 8.20 12.10 19.60 24.20 30.50 33.90 0.26 i.39 3.05 5.48 8.41 11.25 11.68 13.46 18.10 24.00 25.20 27.90 25.90
1.043 1.877 2.958 4.284 5.856 7.674 12.043 20.850 27.281 34.273 41.704 0.285 1.172 2.693 4.850 7.642 11.068 15.129 19.825 24.488 42.388 55.470 70.310 85.689
1.247 !.172 1.116 i.120 !.093 !.068 !.005 0.940 0.890 0.813 0.793 0.913 1.181 1.132 1.130 1.101 1.010 0.772 0.679 0.739 0.566 0.454 0.397 0.302
1.248 1.202 1.118 1.127 1.124 1.138 1.158 1.207 1.202 1.262 1.198 0.913 1.189 1.134 1.132 1.104 1.028 0.803 0.734 0.829 1.151 0.944 0.841 0.651
1.041 1.830 2.952 4.257 5.692 7.209 10.470 16.243 20.130 24.166 28.305 0.285 1.169 2.689 4.842 7.620 10.942 14.537 18.332 21.841 20.855 26.690 33.180 39.801
Experimental Span FailureLoad/ (mm)
D/t ratio
BS5950 1000 I000 1000 1000 1000 1000 1000 1000 1000 1000 1000 500 500 500 500 500 500 500 500 500 500 500 500 500
36.500 45.875 55.250 64.625 74.000 83.375 99.000 130.250 149.000 167.750 186.500 17.750 27.125 36.500 45.875 55.250 64.625 74.000 83.375 99.000 130.250 149.000 167.750 186.500
Theoretical Analysis Discounting the results at 500mm spacing because of local crushing, sections with a web/thickness ratio less than 100 carried the full plastic moment. Sections with larger ratios failed by compression in the web forming a 'pitched roof' yield pattern. Using Murray's. theory and the mechanisms shown in figure 8 a theoretical prediction of the behaviour was made. Full details of the procedure are found in Enjily (1984). A typical comparison between theory and experiment is given in figure 9.
Discussion From the experimental results at 1000mm it can be seen that for web/thickness ratios less than 100 that
Experimental Investigation into ColdFormed Channel Sections in Bending
Figure 7: Typical experimental curves, web in compression
Figure 8" Theoretical model for beams with web in compression
381
382
V. Enjily et al.
Figure 9: Comparison of theory against experiment for specimen Y8 the channels were able to carry their full plastic moment. At 500mm span, a local crushing failure mode occurred before the full plastic moment was reached for web width/thickness ratios between 65 and 100. It is likely that if loading is such as to prevent local crushing that a design approach is to allow full plastic moments to be applied for ratios less than 100. When moments of resistance are calculated by use of BS5950 (see Table 2) it can be seen that again BS5950 is conservative. However as the discrepancy does not exceed 26% the results from BS5950 are a good estimate of failure load. For plain channels with their flanges (i.e. unstiffened elements) in compression the full plastic load can be used for flange/thickness ratios below 16. BS5950 is excessively conservative for flange/thickness ratios above 10. For plain channels with their webs (i.e. stiffened element) in compression full plastic moment can probably be achieved for web/thickness ratios of up to 100. At ratios in excess of this figure BS5950 gives a good conservative prediction of performance. REFERENCES
BS5950 Structural use of steelwork in building Part 5: Code of practice for design of cold formed sections BSI London 1987 Enjily V. (1985). The inelastic postbuckling behaviour of coldformed sections, Ph. D. Thesis, Oxford Brookes University (formerly Oxford Polytechnic) Enjily V., Beale R.G. and Godley M.H.R. (1998) Inelastic Behaviour of ColdFormed Channel Sections in Bending Proc. 2"a Int. Co~f On Thinwalled Structures, Research & Development, Singapore, 1998, 197204 Little G. H. (1982). Complete collapse analysis of steel columns, hTt. J. Mech. Sci. 24, 27998 Murray N. W. (1984). Introduction to the theory of thinwalled structures, Clarendon Press, Oxford Rhodes J. and Harvey J.M. (1976) Plain channel sections in compression and bending beyond the ultimate load. Int. J. Mech. Sci. 18, 511519 Rhodes J. (1982) The postbuckling behaviour of bending elements. Proc. Sixth Int. Speciality Conf. On ColdFormed Steel Structures, St. Louis, 135155 Rhodes J. (1987) Behaviour of ThinWalled Channel Sections in Bending. Proc. Dynamics of Structures Congress '8 7, Orlando, 336351
Composite Construction
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FLEXURAL STRENGTH FOR NEGATIVE BENDING AND VERTICAL SHEAR STRENGTH OF COMPOSITE STEEL S L A G  C O N C R E T E BEAMS Qingli Wang, Qingliang Kang and Pingzhou Cao College of Civil Engineering, Hohai University, Nanjing, 210098, China
ABSTRACT This paper is part of a summary on a series of tests and studies of 6 simply supported and 12 continuous composite steel slagconcrete beams. Using simple plastic theory and conversion of the steel member crosssection shape from " I " to rectangle, calculation formula of flexural strength of continuous composite beams for negative bending is obtained and this formula can provide accurate results no matter the crosssection neutral axis of the composite beam lies in the web or in the top flange of the steel member. Main factors affecting the strength of composite beams for vertical shear such as concrete slab, nominal shear spanratio and force ratio, are discussed in this paper. It is necessary considering the effect of the concrete slab when calculating the strength of composite beams for vertical shear. Bending momentratio should be considered for continuous composite beams.
KEYWORDS Composite steel slagconcrete beams, flexural strength for negative bending, vertical shear strength, conversion of crosssection, nominal shear spanratio, force ratio.
INTRODUCTION This paper is part of a summary on a series of tests and studies of 6 simply supported and 12 continuous composite steel slagconcrete beams. These tests indicate that at flexural failure around the interior prop of the continuous composite beams the concrete slab cracks and the stress in the main part of the steel member exceeds the yielding stress. The simple plastic theory is suitable to calculate the flexural strength of continuous composite beams for negative bending. Main factors affecting the strength of composite beams for vertical shear are approached. It is necessary considering the effect of concrete slab when calculating the strength of composite beams for vertical shear, and the bending momentratio must be considered for continuous composite beams. These are proved by the tests. Comparison of the composite steel slagconcrete beam with the composite steel commonconcrete beam will be presented in another paper. 385
386
Q.L. Wang et al.
F L E X U R A L S T R E N G T H FOR NEGATIVE BENDING
The following method of calculating the flexural strength for negative bending,
M'p, avoids
the
complexities that arise in some other methods when the steel member crosssection is not geometrically symmetric about its centroidal axis as shown in Figure 1 (a). It can provide accurate results no matter the crosssection neutral axis of the composite beam lies in the web or in the top flange of the steel member. The main step of this is a conversion of the steel member crosssection shape from " I " t o rectangle as shown in Figure 1 (a) which is the initial crosssection considered and (b) which is the conversed crosssection and during which following rules must be obeyed: (1) The relative position of the steel member center axis to the composite crosssection keeps unchanged; (2) The steel member crosssection area keeps unchanged and (3) The steel member inertia moment about its center axis keeps unchanged.
Figure 1: Crosssection conversion of the steel member and stress distribution of the composite crosssection New rectangle steel member crosssectional dimensions are given by
ts = x/A.~/O2Is )} ds =~/12Is/As
(1)
where d s and t s = the depth and breadth of the conversed rectangle crosssection respectively; I s = the steel member inertia moment about its centroidal axis; As = the steel member crosssection area. At flexural failure, the whole of the concrete slab may be assumed to be cracked, and simple plastic theory is applicable, with all the steel at its design yield stress of frd for longitudinal reinforcement
and fsd
for steel member respectively. The stresses are as shown in Figure 1 (c), and are separated into two sets: those in Figure 1 (d) which correspond to the plastic moment of resistance of the rectangle steel member alone, M ps, which is given by
M ps = f~a ts d2/4 and those in Figure 1 (e). The longitudinal force, F r , in Figure 1 (e) is
(2)
Flexural Strength for Negative Bending and Vertical Shear Strength
387
Fr =~rfr~
(3)
where A r the crosssection area of longitudinal reinforcement within the effective breath of the concrete slab. =
The flexural strength for negative bending is given by
M'p:Mm + F r ( d  d s / 4 + d t / 2  d r )
(4)
in which d , d r = the depths of the center axis of the steel member and the longitudinal reinforcement below the top of the concrete slab respectively as shown in Figure 1 (a) and
a, =
Asf~a  Fr 2tsf sa
(5)
is the depth of tension zone of the rectangle steel member.
VERTICAL SHEAR STRENGTH
It is very difficult estimating the exact strength of composite beams to vertical shear theoretically for it is influenced by a lot of factors. In reinforced concrete beams, its vertical shear strength is taken into account even concrete cracks, for composite beam the strength of concrete slab for vertical shear should not be neglected too. If the crosssection area of concrete slab and force ratio are relative small and the steel member resists the main vertical shear, then it is feasible and convenient for calculation neglecting the effect of the concrete slab. Whereas a composite beam designed appropriately, the part of concrete slab should not be too small, the result would be too conservative if neglecting the effect of the concrete slab.
Main Influence Factors In this paper vertical shear strength is derived based on test results with theoretical analysis, considering the main influence factors and the calculation model of the reinforced concrete beams. Tests by the author and others show: (1) The vertical shear strength increases as the crosssection area and the axial compressive stress of the concrete slab increase. This is because concrete is not homogeneous material, which leads to the unusual shear stress distribution on cracked section of the concrete slab and very rough interface of crack, and there are friction and occlusive mechanism in the crack which will provide some vertical shear strength; (2) Force ratio, ~ , could embody the contribution of the concrete slab and especially the longitudinal reinforcement inside it to the whole vertical shear strength of the composite beams, which is usually used in the negative moment region of continuous composite beams.
~=Arfry/(Asfy )
(6)
where fry = the yielding stress of the longitudinal reinforcement and fy = the yielding stress of the steel member. The effect of the concrete slab enhances as force ratio increases mainly due to the effect of pin to concrete slab and restriction to crack of the longitudinal reinforcement;
Q.L. Wang et al.
388 (3) Nominal shear spanratio, 2',
,~ '=
M
~
(7)
Vh'
in which M , V = the bending moment and vertical shear on the composite crosssection respectively, h '= the whole depth of the composite beam; There is decrease trend of the vertical shear strength of the composite beams as 2' increases when 2' < 4; (4) The vertical shear strength increases as the transverse reinforcement ratio and the tension yielding stress of the transverse reinforcement increase and (5) Bending moment ratio, m, must be considered for continuous composite beams.
(8)
m
where M  = the negative moment of a point of inflection; M § = the positive moment of a point of inflection. Vertical Shear o f Strength Although there are not effective compositive actions on the prop crosssection of simply supported composite beams and the interior prop crosssection of continuous composite beam, functions of the steel member could be added to that of the concrete slab. The following formulas imitate that of the reinforced concrete beams. For simply supported composite beams subjected to concentrated loads at midspan the vertical shear strength, V,, is provided by the steel member and the concrete slab together V. = Vc + V~
(9)
where Vs = the vertical shear strength of the steel member alone, which is given by (10)
Vs = d w t w f y / V ~
where d , , t, = the depth and breadth of the web of the steel member respectively; Vc = the vertical shear strength of the concrete slab alone, which is given by Vc =
o/a
)
j~+b f ~ + P,~ f r~ bc hc
(11)
wherebc, h c = the effective breadth and depth of the concrete slab; fc = the axial compressive strength of concrete; p,, = the transverse reinforcement ratio; f~v = the yielding stress of the transverse reinforcement; a and b = the coefficients decided by tests, a = 0.2 and b = 1.5. For continuous composite beams subjected to concentrated loads at midspan momentratio must be considered and then
Flexural Strength for Negative Bending and Vertical Shear Strength
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V, = Vc + Vs l+m
(12)
Figure 2 shows the comparisons of V, with V~st and V~est with Vs. The averages of Vu/Vtest and
V,~s,/Vs are 0.935 and 1.401 for simply supported beams (16), 0.978 and 1.3 for continuous beams (712) respectively.
Figure 2: Comparisons of Vu with V,est and V~estwith Vs
CONCLUSIONS (1) Simple plastic theory is suitable for calculating the flexural strength for negative bending of continuous composite steel slagconcrete beams with compact steel member crosssection. Calculation formula presented in this paper can provide accurate solutions even the neutral axis of the composite crosssection lies in the top flange of the steel member. (2) Vertical shear strength of composite beam increases as the crosssection area and the axial compressive stress of the concrete slab and force ratio increase. There is decrease trend of the vertical shear strength of composite beams as nominal shear spanratio increases when 2'< 4. Bending moment ratio must be considered for continuous composite beams. (3) Results of calculation with formula about vertical shear strength of composite beams have good accordance with that of test, concrete slab can provides 28.6%Vu and 23.1%Vu for simply supported and continuous beams respectively.
REFERENCES
1. GBJ1089, Reinforced Concrete Structure Design Code, Construction Industry Publishing Company, Beijing, China, 1989. 2. GBJ1789, Steel Structure Design Code, Construction Industry Publishing Company, Beijing, China, 1990. 3. JBJ12m82, Light Reinforced Concrete Structure Design Rule, Construction Industry Publishing Company, Beijing, China, 1982. 4. Johnson, R. P. (1984). Composite Structures of Steel and Concrete, Volume 1: Beams, Columns, Frames and Applications in Building, Granada, London, England 5. Qingli Wang. Practical Study and Theoretical Analysis on Mechanical Performance and Deformation Behavior of Continuous Composite Beams, Ph.D. Dissertation, Northeastern University, Shenyang, China, July 1998.
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CONCRETEFILLED STEEL TUBES AS COUPLING BEAMS FOR RC SHEAR W A L L S J.G. Teng 1, J.F. Chen 2 and Y.C. Lee 1 1Department of Civil and Structural Engineering Hong Kong Polytechnic University, Hong Kong, China 2 Built Environment Research Unit, School of Engineering and the Built Environment, Wolverhampton University, Wulfruna Street, Wolverhampton WV 1 1SB UK
ABSTRACT Coupling beams in a reinforced concrete coupled shear wall structure are generally designed to provide a ductile energy dissipating mechanism during seismic attacks. This paper explores the use of concretefilled rectangular tubes (CFRTs) as coupling beams and describes an experimental investigation into this form of construction to study their load carrying capacity, ductility and energy absorption characteristics. Results from six tests on simplified CFRT coupling beam models subject to static and cyclic loads are presented. These results demonstrate that CFRT beams have good ductility and a good energy absorption capacity. They are therefore suitable as coupling beams for shear walls particularly if the effect of local buckling is minimised by the use of steel plates of an appropriate thickness. KEYWORDS Coupling beams, concretefilled steel tubes, shear walls, tall buildings, seismic design, ductility. INTRODUCTION Reinforced concrete (RC) coupled shear walls are commonly found in highrise buildings. For buildings subject to seismic attacks, properly designed coupled walls offer excellent ductility through inelastic deformations in the coupling beams, which can dissipate a great amount of seismic energy. It is thus essential that the coupling beams be designed to possess sufficient ductility. The traditional way of constructing a ductile RC coupling beam is to use a large amount of steel reinforcement, particularly diagonal reinforcement (e.g. Pauley and Binney, 1974, Park and Paulay, 1975). However, diagonal reinforcement is only effective for coupling beams with spantodepth ratios less than two. For larger spantodepth ratios, the inclination angle of diagonal bars becomes too small for them to contribute effectively to shear resistance (Shiu et al, 1978). However, deep coupling beams 391
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are often not desirable because their depths may interfere with clear floor height. Furthermore, with the increased use of high strength concrete, it is more difficult to achieve ductility in RC beams as the section size reduces and the brittleness of the concrete increases. Therefore, the exploration of alternative coupling beam forms offering good ductility is worthwhile. As an alternative to RC coupling beams, Harries et al. (1993) and Shahrooz et al. (1993) studied the use of steel Ibeams as coupling beams. As a structural material, steel is much stronger and much more ductile than concrete. However, steel beams may suffer from inelastic lateral buckling and local buckling which limit their ductility. Although local buckling may be prevented by the proper use of lateral stiffeners (Harris et al., 1993), such stiffening is labour intensive and may lead to uneconomic designs. More recently, steel coupling beams encased in normally reinforced concrete have been studied (Liang and Han, 1995; Wang and Sang, 1995; Gong et al., 1997). These studies show that the encasement of concrete leads to increases in stiffness and strength which should be properly considered in design and that the concrete is likely to spall during cyclic deformations. This paper explores the use of concretefilled rectangular tubes (CFRTs) as coupling beams and describes an experimental investigation into this form of construction to study their load carrying capacity, ductility and energy absorption characteristics. Extensive recent research has been carried out on the behaviour of concrete filled steel tubes, particularly as columns (e.g. Ge and Usami, 1992; Shams and Saadeghvaziri, 1997; Uy, 1998). In such tubes, the concrete infill prevents the inward buckling of the tube wall while the steel tube confines the concrete and constrains it from spalling. The combination of steel and concrete in such a manner makes the best use of the properties of both materials and leads to excellent ductility. To the authors' best knowledge, CFRTs have not previously been used as coupling beams, although their use in buildings and other structures, particularly as columns, has been extensive. Apart from ductility considerations, CFRT beams are much simpler to construct than RC beams because both the placement of complicated reinforcement and temporary formwork are eliminated. Compared with steel coupling beams, CFRT beams are more economic due to significant savings in steel.
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Figure 1: Modelling of coupling beams
SPECIMEN DESIGN AND PREPARATION
Modelling of Coupling Beams During an earthquake, the coupling beams provide an important energy dissipation mechanism in a coupled wall structure through inelastic deformations. These beams are subject to large shear forces
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and bending moments, with the effect of axial forces being small. The shear force and bending moment distributions in a coupling beam with the point of contraflexure at the midspan are shown in Figure 1a. These force distributions can be modelled by a cantilever beam under a point load at its free end (Figure l b). This cantilever beam system was thus used in the present study to simulate the behaviour of a coupling beam under seismic loading. The effect of embedment was not considered and the wall was assumed to provide a rigid support to the beam. In practical applications, a sufficient embedment length should be used to prevent premature failures in the embedment zones. An existing approach for designing the concrete embedment for steel coupling beams (Marcakis and Mitchell, 1980; Harries et al., 1993) can be used for designing the concrete embedment for CFRT coupling beams.
Design of Specimens Eight cantilever beams were tested in this study, consisting of two control rectangular hollow section (RHS) tubes and six CFRTs. All tubes had a wall thickness of 2 mm, with a crosssectional height of 200 mm and width of 150 mm. The variable for the CFRTs was the concrete strength, designed to cube strengths of 40, 60 or 90 MPa (referred to as Grade 40, Grade 60 and Grade 90 concrete respectively in the paper). The eight specimens were divided into two series, each consisting of one RHS tube and three CFRTs filled with concrete of different grades. The two series of specimens were tested under static loads and cyclic loads respectively.
Preparation of Specimens The fabrication of the RHS tubes was by coldbending and welding. Two channels were first made from steel sheets using a bending machine. Subsequently, the two channels, with their edges facing each other, were welded together to form a RHS tube with a welding seam at the midheight of each web. Two types of steels with slightly different properties were used (Table 1). These properties were determined by tensile tests using samples from the same plates used for fabricating the RHS tubes. TABLE 1 SPECIMENDETATILS Specimen
RHSs TG40s TG60s TG90s RHSc TG40c TG60c TG90c
Steel properties, MPa Young' s Yield stress Ultimate modulus stress 290 290 290 290 290 290 290 290
441 441 441 441 365 441 441 365
194,000 194,000 194,000 194,000 216,000 194,000 194,000 216,000
Concrete properties, MPa Compressive Compressive Splitting tensile strength, strength, Strength, 28 th days 28th day day of beam test N/A N/A N/A 45.3 45.2 3.03 87.6 84.3 4.57 92.5 94.1 4.68 N/A N/A N/A 41.3 45.1 3.36 87.6 90.62 4.57 112.0 109.5 6.51
Test type Static Static Static Static Cyclic Cyclic Cyclic Cyclic
The fabricated RHS tubes were then filled with fresh concrete. For each of the specimens, six 100x100x100 mm 3 concrete cubes and three concrete cylinders with a diameter of 100mm and a height of 200mm were cast to test their compressive and splitting tensile strengths. Measured concrete properties are shown in Table 1. The actual concrete strength for Grade 60 (Specimens TG60s and TG60c) was as high as that for Grade 90 (Specimens TG90s and TG90c) probably due to mixing problems. While this was undesirable, the specimens were still suitable for the present study and are still referred to using their intended concrete grades (ie TG60 and TG90) in this paper.
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In order to prevent the concrete core from being pushed out when a CFRT specimen was loaded, two 6 mm thick steel plates were welded to the ends of each CFRT beam when the concrete age was 28 days. This simulated the antisymmetric condition at the point of contraflexure in a full coupling beam.
EXPERIMENTAL SETUP The experimental setup for static loading tests is shown in Figure 2. Beam specimens were clamped between two large angle plates, which were in turn fixed on the floor by four high strength bolts. The embedment length of the beams was 440mm. Loads were applied at 460 mm from the fixed end. The span to depth ratio of the beams was 460/200=2.3 which was the smallest value possible because of restrictions of the preinstalled anchor plates on the strong floor. A hydraulic jack was fixed onto the floor to load the beam horizontally for convenience. Displacements at the loading position, the midspan and near the fixed end were measured by electronic displacement transducers. Furthermore, a number of strain gauges were installed near the fixed end (Figure 2). Two displacement transducers were also used to measure the translation and rotation of the fixed end support. The effect of small support movements has been removed in the values of displacements presented in this paper. For cyclic loading tests, two hydraulic jacks were used. Because of this arrangement, the displacement transducer at the loading point was moved to the tip of the beam. The positions of other transducers were the same as in the static tests. The deflection at the loading position was inferred from the measured values at the tip in an approximate manner assuming either the beam deformed elastically or rigidplastically with a plastic hinge at the fixed support. Details are given in Lee (1998). No strain measurement was undertaken in the cyclic tests.
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TEST PROCEDURE
Static Loading Test In static loading tests, the specimens were monotonically loaded until failure. The strains and displacements were recorded at different load levels, from which loaddeflection curves were plotted. These curves were used to determine the values of the 'yield load' Py and the corresponding deflection at the loading position dy (Lee, 1998). Based on observations during the experiments, the 'yield load' was defined as the load when local buckling of the compression flange occurred and corresponds to a strong change in slope of the load deflection curve. This yield load Py and the deflection dy were later used to control the load/displacement levels in the cyclic loading tests.
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Cyclic Loading Test The loading sequence used in the cyclic loading tests is shown in Figure 3. Load control was used before the yield load was reached. Two cycles of reversed cyclic loading were carried out at a load level of P=0.8Py. Three additional cycles were then carried out at P = Py. Thereafter, deflection control at multiples of dy was used. Three complete cycles were carried out at each selected value of deflection until the specimen failed. The loads or displacements were carefully controlled during cyclic tests, nevertheless, some small deviations from the intended values still existed. Displacements were monitored and recorded throughout the test.
Figure 3: Loading history for cyclic tests
Figure 4: Loaddeflection curves under static loading
STATIC TEST RESULTS Figure 4 shows the loaddeflection curves of the loading point for all four static test specimens. The rapidly descending loaddeflection curve after buckling of the RHS tube indicates that its load carrying capacity was reduced quickly, exhibiting very limited ductility. The ultimate strengths of the concrete filled tubes are almost triple that of the corresponding RHS tube. The extended plateaux in the loaddeflection curves alter yielding show that CFRT beams are very ductile. These effects of the concrete infill are well known. The ductile behaviour of the CFRT beams was terminated by tensile rupture of the tension flange which occurred significantly earlier in Specimen TG90s than in the other two CFRT beams. Specimens TG40s and TG60s showed similar ductility, though they were filled with concrete of rather different strengths. The effect of the concrete strength on ductility is thus believed to be small. Although local buckling of the steel tube was observed in all tests, the final failure modes were different for RHS and CFRT specimens (Figure 5). The local buckling of the compression flange near the fixed end occurred at a load of approximately 40 kN, leading to immediate collapse of Specimen RHSs. Shear buckling occurred on both webs at the same load. No crack was found on the tensile flange of Specimen RHSs. For the three CFRT beams, outward local buckling was observed on the compression flanges at a load of approximately 80kN. Shear buckling occurred later on the webs at about 100kN. Clearly, the concrete infill constrained the plate to buckle only away from it, which led to a higher buckling strength, as has been shown by many authors (eg Wright, 1993; Smith et al., 1999). Strain readings showed that the tensile flange had yielded and the compression flange was close to yielding when local buckling occurred. Fracture cracks were found on the tension flanges of CFRT specimens at final failure, indicating the full use of the steel strength. The final failure of CFRT members was by rupture of steel of the tension flange and is referred to as a flexural failure.
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Figure 5: Static loading test: buckling of the compression flange Table 2 shows the experimental ultimate loads for all the static test specimens. The calculated ultimate flexural failure loads according to the approach in BS 8110 (1985) for reinforced concrete beams and using the ultimate stress of steel are also listed for comparison. Clearly, experimental observations are in good agreement with theoretical predictions for CFRT specimens, with discrepancies within 3%. These calculations did not consider local buckling effects, so the calculated ultimate flexural failure load of Specimen RHSs of 93.78kN is more than double the value actually achieved during the test (42.51kN). The chief contribution of the concrete infill is thus to provide constraint to the steel tube. The ultimate strength of CFRT beams increases with the concrete strength. However, this increase is small. Table 2 shows that the concrete strength for TG60s and TG90s is almost twice that for TG40s, but the increase in the experimental ultimate load is only less than 3% while the theoretical increase is less than 6%. TABLE 2 STATICULTIMATELOADS Specimen
fcu, MPa
RHSs TG40s TG60s TG90s
N/A 45.2 84.3 94.1
Test ultimate load, Predicted ultimate kN load, kN 42.51 93.78 117.14 114.2 120.24 119.55 119.57 120.38
Test / Prediction 0.453 1.026 1.006 0.993
CYCLIC TEST RESULTS Test Observations and Failure Modes
Local buckling was observed on both flanges of all the cyclic test specimens. During load reversal, a buckled flange was straightened again under tension. The compressiontension cyclic stresses caused degradation in both steel and concrete, so that the maximum load reached in a cyclic test is considerably lower than that in the corresponding static test. For Specimen RHSc, local buckling was observed in both flanges. No crack developed in the flanges, indicating that the steel tensile strength was not fully utilised. By contrast, cracks developed in both flanges of TG40c and TG60c, and in one of the flanges of TG90c at final failure. Figure 6 shows one
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397
of the flanges for each of the three cyclic test specimens after final failure. All the CFRT specimens failed after 14 to 15 loading cycles.
Figure 6: Failure mode under cyclic loading
Hysteretic Responses The hysteretic loaddeflection responses of the loading position from all four cyclic tests are shown in Figure 7. The load carrying capacity of Specimen RHSc (Figure 7a) was quickly reduced from about 40kN in the first few cycles, to less than 20kN at the 8th cycle and to less than 10kN at the 14th cycle, confirming the lack of ductility as observed in the static loading test. Because the areas surrounded by the hysteresis loops represent the amount of energy absorbed by the test specimen, the energy absorption capacity of RHS tubes is thus very limited and reduces quickly under large cyclic deformations. As observed in the static tests, the ultimate strength of CFRT beams is significantly higher than their hollow counterparts. While the differences in the load carrying capacity in the plastic range are not large between the three CFRT cyclic test specimens, it is worth noting that TG90c, which had the highest concrete strength (Table 1), showed the lowest load carrying capacity. Compared with the results from the static loading tests, the maximum load carrying capacities of CFRT beams under cyclic loading are about 2030% lower, with the difference between the two TG40 specimens being the smallest and that between the two TG90 specimens the largest. This indicates that a CFRT beam with a lower strength concrete behaves better than one filled with concrete of a higher strength. The CFRT beams exhibited strength and stiffness degradations under reversed cyclic loading and pinching is seen for all of them (Figure 7). The main reason is believed to be the degradation of concrete in strength and stiffness when subject to reversed cyclic loading which leads to shear cracks in both directions. Slipping between the steel tube and the concrete may also have been a significant factor. The slipping behaviour may be improved by using shear connectors such as those used by ShakirKhalil et al. (1993). Overall, the hysteretic responses of these beams are good and are better than normal reinforced concrete beams, but are not as good as deep RC beams with proper diagonal reinforcement (Park and
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Paulay, 1975). Significant improvements to the cyclic behaviour of these beams should be achievable by using thicker steel plates so that the effect of local buckling is minimised.
Figure 7: Hysteretic loaddeflection responses at loading position
CONCLUSIONS This paper has explored the use of concrete filled steel tubes as coupling beams for reinforced concrete coupled shear wall structures. Six concrete filled rectangular steel tubes and two rectangular hollow steel tubes have been tested under static and cyclic loadings. The mutual constraints of the steel tube and the concrete infill lead to higher strength and good ductility. The strength and ductility of these beams are insensitive to concrete strength, but cyclic degradation seems to increase with concrete strength. The use of high strength concrete thus seems to be undesirable. The hysteretic responses of these beams under cyclic loads show that they have a good energy absorption capacity. Therefore, these beams are suitable as coupling beams, particularly if local buckling is minimised by using relatively thick steel plates and slipping between the steel and concrete is reduced using some form of shear connectors. Further research is required to better understand this form of coupling beams.
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ACKNOLWEDGEMENTS The authors are grateful to Dr. Y.L. Wong for helpful discussions on the subject.
REFERENCES
BS 8110 (1985). Structural Use of Concrete. British Standards Institution, London. Gong B., Shahrooz B.M. and Gillum A.J. (1997). Seismic Behaviour and Design Of Composite Coupling Beams. Proc. of the Engineering Foundation Conference 1997, ASCE, New York, NY, USA, 258271. Ge, H.B. and Usami, T. (1992). Strength of ConcreteFilled ThinWalled Steel Box Columns: Experiment. Journal of Structural Engineering, ASCE, 118:11, 30363051. Harries K.A., Mitchell D., Cool W.D. and Redwood R.G. (1993). Seismic Response of Steel Beams Coupling Concrete Walls. Journal of Structural Engineering, ASCE, 119:12, 36113629. Lee Y.C. (1998). Concrete Filled Steel Tubes as Coupling Beams for Concrete Shear Walls, BEng Dissertation, Dept of Civil & Structural Engineering, The Hong Kong Polytechnic University, Hong Kong, China. Liang, Q. and Han, X. (1995). The Behaviour of Stiffening Beams and Lintel Beams under Cyclic Loading. Journal of South China University of Technology (Natural Science), 23:1, 2633. Marcakis, K. and Mitchell, D. (1980). Precast Concrete Connections with Embedded Steel Members", PCI Journal, 25:4, 88116. Park, R. and Paulay, T. (1975). Reinforced Concrete Structures, John Wiley and Sons, New York, N.Y. Paulay T. and Binney J.R. (1974). Diagonally Reinforced Coupling Beams of Shear Walls. Shear in Reinforced Concrete: Publication No. 42, ACI, Detroit, Mich., 579598. Shahrooz B.M., Remmetter M.E. and Qin F. (1993). Seismic Design and Performance of Composite Coupled Walls. Journal of Structural Engineering, ASCE, 119:11,32913309. ShakirKhalill, H. and Hassan, N.K.A. (1993) Push Out Resistance of ConcreteFilled Tubes. Tubular structures VI, (ed by Grundy, Holgate & Wong), Balkema, Rotterdam. Shams, M. and Saadeghvaziri, M.A. (1997). State of the Art of ConcreteFilled Steel Tubular Columns. A CI Structural Journal, 94:5, 558571. Shiu K.N., Barney G.B., Fiorato A.E. and Corley W.G. (1978) Reversed Load Tests of Reinforced Concrete Coupling Beams. Proc., Central American Conference on Earthquake Engineering, 239249. Smith, S.T., Bradford, M.A. and Oehlers, D.J. (1999). Elastic Buckling of Unilaterally Constrained Rectangular Plates In Pure Shear. Engineering Structures, 21,443453. Uy, B. (1998). Concrete Filled Fabricated Steel Box Columns for Multistorey Buildings: Behaviour and Design. Progress in Structural Engineering and materials, 1:2, 150158. Wang, Z. and Sang, W. (1995). Beating Behaviour and Calculation Method of Steel Reinforced Concrete Coupling Beams. Journal of South China University of Technology (Natural Science), 23:1, 3543. Wright, H. (1993). Buckling of Plates in Contact with a Rigid Medium. The Structural Engineer, 71:2, 209215.
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Experimental Study of High Strength Concrete Filled Circular Steel Columns Y. C. Wang Manchester School of Engineering, University of Manchester, Manchester M13 9PL UK
ABSTRACT In places where usable floor space is at a premium, it is desirable to use the most structurally efficient load bearing columns. In concrete filled steel tubes, the beneficial interaction between the steel casing and the concrete core gives a load carrying system that is highly efficient. When High Strength Concrete (HSC) is used, the column load bearing performance is further improved. This paper presents the results of a series of parametric experimental study on HSC filled circular steel columns under axial compression. The parameters examined in these tests are: concrete grade, steel grade, column slenderness and steel contribution factor. The objectives of these tests are threefold: 1. To experimentally investigate the performance of HSC filled steel tubular columns; 2. To assess whether the design rules for normal strength concrete (NSC) filled columns can be extrapolated to HSC filled columns, and 3. To examine the structural load bearing efficiency by changing different design parameters. From the results of this experimental study, the following main findings have been obtained: 1. It is conservative to extrapolate the design method for NSC filled steel columns to HSC filled ones; 2. The advantage of HSC in resisting compressive load can be effectively utilised in HSC filled columns, even for slender columns where HSC does not offer much improved rigidity to resist flexural buckling; 3. The improved column strength due to concrete confinement effect is noticeable only for short columns; 4. The confinement effect may be appreciably reduced by a small eccentricity, and 5. The ductility of HSC filled columns is similar to that of NSC filled columns.
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Y. C. Wang
1. Introduction In places where usable floor space is at a premium, it is desirable to use the most structurally efficient load beating columns. Concrete filled hollow steel columns are more structurally efficient in resisting compressive loads than either bare steel columns or reinforced concrete columns. They also have a number of other advantages including rapid construction, enhanced concrete strength and ductility due to the confinement effect and inherent high fire resistance. Normal strength concrete (NSC) filled columns are now being increasingly used in the construction of multistorey and high rise buildings and design recommendations for this type of construction are now firmly established [1,2]. NSC is assumed to have the maximum cube strength of about 60N/mm 2. Using high strength concrete (HSC) can further improve the structural load bearing efficiency of concrete filled columns and improve their durabilit3'. However, before HSC filled columns can be used with confidence and improved economy, their superior load carrying capacity should be confirmed and suitable design guidelines developed. HSC filled steel tubes have been investigated by a number of researchers. For example, O'Shea and Bridge [3] concentrated on local buckling of thin walled tubes filled with HSC. Cai & Gu [4] studied the confinement effect on HSC in short columns. This paper reports the results of a series of tests on HSC (C100) and NSC (C40) filled circular hollow section (CHS) steel columns. The objectives of these tests were threefold: (1) To assess whether the design rules for NSC filled columns can be extrapolated to cater for HSC filled columns; (2) To experimentally study the performance of HSC filled columns, in particular, the confinement effect on the column strength and ductility, and (3) To examine the structural load bearing efficiency by changing different design parameters.
2. Test programme 2.1 Test parameters This series of tests were carried out to examine the influence of a number of design parameters on column performance. In total, 2 pairs of 12 columns were made and tested. Table 1 gives the values of test parameters for each pair of columns. 2.2 Test set up All columns were cast in December 1996 and tests were carried out about six months after casting. For each column, three concrete cubes of 100 mm and two concrete prisms of 90 mm square and 300 mm in height were cast, to be tested on the column loading day. For each concrete mix, three concrete cubes of 100 mm were cast. These cubes were tested after 28 days for quality control. Four strain gauges were attached to the external surface of the steel tube at two opposing sides at each column midheight. Two strain gauges at each side measure the horizontal and longitudinal strains in the steel respectively. For the shortest columns of 500 mm, a vibrating strain gauge was cast in the column centre to measure the concrete axial main.
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Column tests were carried out in the BRE axial test machine with the maximum capacity of 5000 kN. Each column was simply supported about one axis and rotationally restrained in the perpendicular direction using roller supports. The end support increased the column length by about 80 mm at each end. Therefore, the total column length (L) should be the column specimen length (L0) plus 160 mm. All columns were intended to be subjected to compression axial load and were checked to be so by human eyes only. Also each column had some initial imperfections and the end supports had to be adjusted for each test. Inevitably, the column could not be aligned perfectly nor in the central position. This led to a small eccentricity, and bending moment in each column. The amount of this equivalent eccentricity will be evaluated for calculating the column strength. Each column was loaded incrementally until it reached its strength when it could not sustain the applied load. The test was continued to study the column response during unloading at increasing deformation until the column eventually found a stable position.
3. High strength concrete mechanical properties For each column, three cube tests and two prism tests were carried out to determine various properties of concrete. During each prism test, the concrete strain was measured and the complete concrete stressstrain relationship up to the maximum stress was established. Results of the compressive strength, the corresponding strain and the Young's modulus are given in Table 2. The Young's modulus was obtained by using the proposed stressstrain relationship from Cla)~ton [5]. It is observed that the stiffness of HSC is only slightly higher (about 25%) than that of NSC, and also that concrete strain at prism strength is almost independent of the concrete grade.
4. Test observation and results When high strength concrete fails in compression, the failure mode is brittle, this was observed during each prism test when HSC failure was accompanied by a noisy bang. In contrast, HSC filled steel columns failed in a ductile manner, similar to NSC. This was demonstrated by the ability of HSC filled columns to deform under decreasing loads and to find a stable position after reaching the peak strength. Different failure modes were observed for different columns. For short columns (Lo/D=3), the failure mode was clearly local due to extensive concrete crushing and steel yielding. NSC filled columns exhibited very ductile behaviour, with column failure due to splitting of the cold rolled steel tube at the welding edge. HSC filled short columns also behaved in a ductile way. Confinement effect was observed by the fact that the failure strain in HSC was several times higher than the prism crush strain. Nevertheless, the extent of concrete confinement in HSC was lower than in NSC filled columns and no steel tube splitting occurred. Global buckling was the dominant failure mode for the longest columns (Lo/D=25). Due to inevitable eccentricity induced bending effect, global buckling was not very clearly demonstrated in most columns. However, for the two columns that had very little bending moment, column failure was indicated by a sudden large lateral movement.
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All columns w~th the intermediate length (Lo/D= 15) failed in a mixed mode, both axial strain and lateral deflection increased at steady but faster rates until peak applied load. Table 3 presents results for all columns, including the column eccentricity e. To verify the accuracy of the design method and to check the effectiveness of the confinement effect, it was necessary to evaluate the column eccentricity. This value is calculated from the two axial strain readings in the steel tube using the following equation: e
A6.EI N.D
where Ae El N D
[1]
= the difference in longitudinal strain recorded by the two strain gauges =composite section flexural stiffiaess =applied load in the column =column crosssection diameter
Equation (1) is based on elastic analysis, therefore, the value of eccentricity was obtained from the average of the few earlier load increments. In table 3, all design strengths were calculated taking into account the eccentricity and by setting the partial safety factors for steel and concrete to 1.0. Also, the short term concrete modulus of elasticity in Table 3 was used for each column.
5. Analysis of test results The test results have been analysed by a comparison against the predictions of various design methods for concrete filled columns. From this comparison, a number of conclusions may be drawn. This paper presents some of the more important ones. 5.1 Accuracy of current design rules for HSC filled columns The current design rules for concrete filled columns have been derived from test results on NSC. From the comparative results in Table 3, it may be concluded that these design rules give quite accurate predictions for NSC filled columns (TI&T2, T5&T6, T9&TI0). Furthermore, it seems that these design rules may be extended to HSC filled steel columns, as indicated by the overall accuracy in Table 3. Indeed, the current design rules give conservative results for HSC filled columns, thus they are acceptable for safety. Nevertheless, for HSC filled steel cohmms, Table 3 suggests that the accuracy of the NSCbased design rules depends on the column slenderness. While the code predictions are quite accurate for short columns, discrepancy between predicted and test results increases at higher column slenderness. Figure 1 presents the results in Table 3. It is clear that as the slenderness of HSC filled steel columns increase, both BS 5400 Part 5 [2] and Eurocode 4 Part 1.1 [2] predict lower column strength. Whilst this means that both design methods are safe to use for HSC filled steel columns, it also suggests that it is possible to use a higher cohann buckling curve for HSC filled steel columns for improved column efficiency. However, this can only be confirmed atter more extensive experimental studies.
405
Study of High Strength Concrete Filled Circular Steel Columns 5.2 Effect of confinement on concrete
Strength It is now well reax~gnised that when concrete is under triaxial compression, both its load carrying capacity and ductility are increased. Concrete confinement can be obtained through placing hoop reinforcement or using steel casing. For concrete filled columns, although increase in the concrete strength is at the expense of a reduction in the steel strength, the overall effect is a net increase in the column strength. This confinement effect diminishes for slender columns. Although BS 5400 Part 5 [1] gives a limiting length of L/D=25, realistically, the confinement effect is noticeable only for columns of m
L/D not greater than 5. In Eur~xxte 4 Part 1.1 [2], the limiting column slenderness is at 2 =0.5. Nevertheless, in places where the column foot print is large, a L/D ratio of less than 5 is realistic. Thus, it is beneficial to explore the enhancement due to concrete confinement. However, the effect of confinement is greatly reduced by bending in the column. To illustrate the effect of concrete confinement, only Eurocxxle 4 Part .1.1 [2] is used in this paper. Results are given in Table 4 for L0/D=3. Without bending moment, the squash load of a column can be increased by up to 20% due to enhancement. However, with an eccentricity to diameter (e/D) ratio of only 3%, column strength increase due to the confinement effect is reduced by about 30%. For columns in simple construction, BS 5950 Part 1 [6] gives a nominal eccentricit)" of 100mm plus D/2 for beam reactions. For medium rise buildings, this end bending moment can give a significant eccentricity to the overall column axial load, which may completely remove the enhancement due to confinement. For example, for a 10 storey building with 300 mm diameter columns, the column eccentricity (e/D) to the overall axial load of the bottom floor column is about 8%. Therefore, to make use of the enhancement in design, an accurate assessment of the column ~ t r i c i t y should be carried out. Duetili~~ One of the main concerns with using HSC is its lack of ductility and its brittle and explosive failure. However, in the author's tests, no HSC filled steel column suffered from this failure mode and all columns performed in a ductile manner. The ductilit3, of a column is rather difficult to quantify. The unloading slope of the column may give some indication. Figure 2 plots the loadaxial strain relationship for tests T5T8, two of which used HSC and the other two NSC. In this figure, the applied load is norrnalised with regard to the column test strength. The unloading slope seems to be comparable between NSC and HSC filled columns. However, while the two NSC filled column curves are almost identical, there is a great variability in the behaviour of the two HSC filled columns. On the other hand, if the column ductility is measured by the maximum concrete strain reached at the peak column strength, the enhanced strain due to the conflnernent effect may be predicted using the equation obtained by Mander et al [7]:
8cc 
s
1+ 5
I Crcc  1 \ 0"~
[2]
406
Y. C. Wang
Table 5 gives a comparison between test results and predictions using equation (4). Since the confinement effect is negligible for slender columns, the comparison was carried out for short columns (I.o/D=3) only. In addition, the theoretical value of the concrete strength enhancement factor (t~/(rck) has been calculated using recommendations in Eurocode 4 Part 1.1 [2]. Table 5 only indicates a broad agreement between the predictions of equation (4) and test results. Nevertheless, it suggests that the confinement effect can significantly increase the concrete ductility and that equation (4) gives conservative results. Table 5:
Increased concrete strain due to confmement effect
Test ID
2
t~cc/cckaccording to EC4
test
model [7]
T9
0.17
1.774 (1.544)
15.8
4.87 (3.72)
T10
0.17
1.796 (1.56)
13.2
4.98 (3.8)
TII
0.21
1.230 (1.162)
2.03
2.15(1.81)
T12
0.21
1.227 (1.213)
2.36
2.14 (2.07)
T17
0.20
1.60(1.458)
4.0
4.0 (3.29)
T18
0.21
1.571 (1.469)
6.12
3.86 (3.35)
T27
0.18
1.442 (1.18)
5.07
3.21 (1.9)
T28
0.19
1.412 (1.288)
4.5
3.06(2.44)
NB: Values in brackets include the effect of eccentricity. 5.3 Effect of high strength steel One of the original objectives of this series of tests was to examine the effectiveness of using high strength materials, including both high strength steel and high strength concrete. The effect of using HSC has already been discussed in 5.1. It seems that despite only a modest increase in HSC modulus of elasticity., column test strength increases in line with increase in the column squash load regardless of the column slenderness. However, unless the column is short, using high strength steel only gives a small increase in the column strength. Tests T13TI8 are directly comparable to Test T23T28, the only difference being that $355 steel was used in the former and S275 steel was used in the latter. Table 6 gives increases in the column strength due to high strength steel. Clearly, the benefit of using high strength steel diminishes at higher column slenderness. Table 6: Comparison between results for different grades of steel L/D=25 L/D=15 L/D=5 1.075 1.386 1.368 6. Conclusion
This paper has presented the results of a series of compression tests on NSC and HSC filled circular steel columns. From an analysis of the test results, the following conclusions may be drawn:
Study of High Strength Concrete Filled Circular Steel Columns
407
(1) Using HSC can significantly increase the strength of concrete filled columns. This conclusion applies to a wide range of tested column slenderness (2, = 0.21.4 ). (2) Since the modulus of elasticit3' of HSC is only slightly higher than that of NSC, it follows that a higher column buckling curve may be used in design calculations for HSC filled steel columns. However, a large number of tests should be carried out for confirmation. In the meantime, the design rules for NSC filled steel columns may conservatively be used for HSC filled columns. (3) Using high strength steel is far less effective tlwu using HSC in increasing the column strength. (4) The benefits of concrete confinement in increasing the concrete strength and ductility are realised for short columns only. Furthermore, the increase in concrete strength can be reduced by a small ~ t r i c i t y . Therefore, in order to reliably use the beneficial effect of confining concrete, accurate assessment of the column eccentricity should be made in design calculations.
Acknowledgments The tests reported in this paper were carried out by the author at the Building Research Establishment and he acknowledges the technical support of various BRE staff members. He also thanks Mr. Nigel Clayton of BRE for the concrete prism tests.
References 1. Design of composite bridges: use of BS 5400: Part 5:1979 for Department of Transport structures, Department of Transport, London, December 1987 2. Eurocode 4: Design of composite steel and concrete structure, Part 1.1: General rules and rules for buildings, British Standards Institution, London, 1994 3. O'Shea M D and Bridge R Q, "Circular thin walled concrete filled steel tubes", Proceedings of the 4th Pacific Structural Steel Conference, Vol. 3: Steelconcrete composite structures, pp. 5360, 1995 4. Cai, S H and Gu W P, "Behaviour and ultimate strength of steel tube confined high strength concrete columns", Proceedings of 4th International S3~posium on Utilization of high strength/high performance concrete, pp. 827833, Paris 1996 5. BS 5950: Structural use of steelwork in buildings, Part 1: Code of practice for design in simple and continuous construction: hot rolled sections, British Standards Institution, London, 1990 6. Clayton N, "High grade concrete  stressstrain behaviour", BRE Client Report CR44/97, Building Research Establishment, 1997 7. Mander J B, Priestley M J N and Park R, "Theoretical stressstrain model for confined concrete", Journal of Structural Engineering, Vol. 114, No. 8, pp. 18041826, American Society of Civil Engineering, 1988
Y.C. Wang
408
Table 1: Test parameters Test ID TI,T2 T3,T4 T5,T6 T7,T8 T9,TI0 TI1,T12 T13,T14 T15,T16 T17,T18 T23,T24 T25,T26 T27,T28
D(mm) 168.3 168.3 168.3 168.3 168.3 168.3 168.3 168.3 168.3 168.3 168.3 168.3
t(mm) 5.0 5.0 5.0 5.0 5.0 5.0 10.0 10.0 10.0 10.0 10.0 10.0
Lo (mm) 4200 4200 2500 2500 500 500 4200 2500 500 4200 2500 500
Steel 8rade $355 S355 $355 $355 $355 S355 $355 $355 $355 S275 S275 $275
Concrete 8rade C40 C100 C40 C100 C40 CI00 C100 C100 C100 CI00 C100 C100
Cylinder strength
Young's modulus
Table 2: Measured Material Properties Test ID
TI T2 T3 T4 T5 T6 T7 T8 T9 T10 Tll T12 TI3 T14 TI5 T16 TI 7 TI8 T23 T24 T25 T26 T27 T28
Steel yield stress
(Nlmm 2)
438 438 438 438 438 438 438 438 438 438 438 438 i480 480 480 480 480 480 330.5 330.5 330.5 330.5 330.5 330.5
Cube strength
(Nlmrn2) 52 51.8 123.5 121.2 47.5 47.8 116.0 115.3 46 44.7 115.3 113.8 120.7 119.5 113.8 114.2 126.0 120.0 116.8 118.7 113.6 116.3 116.0 114.2
(N/mm2)
40.8 41.3 106.3 91.0 39.0 39.0 97.8 101.0 37.3 36.5 99.5 100.0 92.7 82.3 93.3 90.5 87.8 90.8 , 98.8 98.5 89.3 95.8 91.8 97.0
(N/mm2)
41000 45000 52500 53000 39500 42000 50500 54000 41000 41500 53500 52500 49000 50500 49500 52500 52000 49000 50000 50500 53500 52500 152000 ~i 50500
6max
(mm/m)
2.3 , 2.4 12.83 '2.05 t2.2 2.05 2.45 2.8 2.05 2.15 2.48 2.78 2.4 2.25 2.48 2.20 2.95 2.23 2.78 2.78 2.23 I 2.48 ! 2.38 ' 2.58
Study of High Strength Concrete Filled Circular Steel Columns
409
Table 3: Comparison between design strength and test results Test lD
~
e(mm)
Test
comparison between design calculations and test results BS 5400 Part 5 [ 1] Euroexxte 4 Part 1.1
load
t21 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 Tll T12 T13 T14 T15 T16 T17 T18 T23 T24 !T25 LT26
1.16 1.14 1.'~43 1.35 0.7 0.7 0.85 0.85 0.17 0.17 0.21 0.21 1.36 1.33 0.83
3 5 2.5 5 4 2 5 2.5 5 5 5 1 4.5 1.5 2 0.2 4 3 6 0.5 3 3.5
~ ~ 0.21 1.25 1.24 0.74 0.75
IT,
0.1
lO
, T28
0.19
5
(kN) 900 950 1550 1400 1300 1445 2330 2450 2360 2360 3250 3250 1900 2400 3350 3650 4550 4550 1800 2200 2600 2450
(kN) , 964 ! 963 ~ ~ 1124 i 1382 1465 1858 2004 2002 1988 i 2784 ~ ~ 1661 2855 3032 4326 4386 1429 1605 2333 2314
2,0
2 60
3400
3206
,_..._..__._._
v
~
_pred/test 1.071 0.993 0.754 0.803 1.063 1.014 0.791 0.818 0.848 0.842 0.857 0.950 0.827 0.692 0.852 0.831 0.951 0.964 0.794 0.729 0.897 0.945
(kN~.__pred/test 961 1.068 932 0.981 1153 0.744 ! 1053 0.752 ~ 1448 1.114 1513 1.047 2007 0.854 2197 0.897 1891 0.801 1879 0.796 2944 0.906 32/2 1.007 1481 0.779 1623 0.676 2957 0.883 3076 0.843 3943 0.867 4099 0.901 ~ 0.738 11627 ! 0.739 "2465 ! 0.948 : 1.021
10. 64 0.943
3~
10.952
Table 4: Effect of column squash load increase due to confinement effect Test T9 T10 T1 i TI2 TI7 T18 T27 T28
iD
.
With bending moment 1.145 1.148 1.066 1.087 1.109 1.113 1.056 1.091 .
.
.
.
.
.
Without bending moment 1.207 1.210 1.094 1.092 1.142 1.137 1.137 1.130 .
.
.
,,
;r
1.2
[n 0
1
ea
Grade C40 concrete
t s
0.8

0
00
=:
u)
t! G
Pti .U
0.6

BS5400 joEC4
g!
n.
0.4
1
K
9
0.2
0 0
0.2
0.4
0.6
0.8
1
Nondimensional slenderness
Fig 1: Comparison between predicted and test results
1.2
1.4
1.6
Study of High Strength Concrete Filled Circular Steel Columns
411
A
Z v
"o a 0 .J
C40
C100
I
0
o
I
t
I
t
I
I
2000
4000
6000
8000
 10000
12000
vertical micro strain
Fig
2
9L o a d  s t r a i n
curves, L=2.5m
This Page Intentionally Left Blank
STRENGTH AND DUCTILITY OF HOLLOW CIRCULAR STEEL COLUMNS FILLED WITH FIBRE REINFORCED CONCRETE G. Campione, N. Scibilia, G. Zingone Dipartimento di Ingegneria Strutturale e Geotecnica Universitb, di Palermo, 190128, ITALY
ABSTRACT The focus of the present investigation is the study of the behaviour of hollow circular steel crosssections filled with fibre reinforced concrete (FRC), subjected to monotonic loads. Using the same volume percentages of fibres, the influence of different types of fibres (steel, polyolefin) on the behaviour of the columns was investigated. Results of fibre reinforced composite columns were compared with those of columns filled with plain concrete, showing the advantages of using FRC compared to plain concrete, in terms of both strength and ductility. A simplified analytical model to predict loaddeformation curves for composite members in compression is proposed, and the comparison between experimental and analytical results has shown good agreement. Finally, a comparison is made between the bearing capacity of circular hollow steel columns filled with plain concrete evaluated according to recent European and International codes and that evaluated using the proposed model. KEYWORDS Circular steel columns, fibre reinforced concrete, composite members, strength, ductility, active confinement. INTRODUCTION
In the design of tubular structures, after determining the shell plate thickness satisfying tensile stress, the stability of the shell should be checked for compressive stresses against buckling. A thinwalled cylinder shell subjected to compression may fail either due to the instability of the shell, involving bending of the axis, or due to local instability, as shown in Figure. 1, and also depends on the ratio of the thickness to the radius of the shell wall and on the length of the columns. Failure of this type of structure is due to the formation of characteristic wrinkles or bulges, circular or lobed in shape. To mitigate or prevent this type of failure it is common to encase or fill steel profiles with concrete. The coupling of concrete and steel shapes makes it possible to obtain structural elements which, compared to the single constituent elements, ensure high performance in terms of both resistance and ductility, Cosenza & Pecce (1993). 413
414
G. Campione et al.
Modern codes like the Eurocode relating to composite steelconcrete structures contemplate the use of composite steelconcrete columns made using W shapes or thinwalled tubular profiles with a circular, rectangular or square section, as shown in Figure 2. Concrete, inside or outside the profile, exerts beneficial confinement action against phenomena of local or overall instability and markedly increases the resisting and dissipative capacity of steel columns, Schneider (1998). A further advantage is the increase in fire resistance, Lie (1994), Frassen et al. (1998).
Figure 1: Overall and local instability Dwelling on the case of tubular steel columns, we can observe that, filled with concrete, they present deeply different behaviour from hollow ones and much better performances. There is not only a higher bearing capacity, but also greater safety against flexural instability, which makes it possible to construct particularly slender columns. The concrete is highly confined and hence is more ductile and has a greater bearing capacity, Shakir et al. (1994). The coupling of several tubular sections makes it possible to face very high stresses, greatly increasing the critical load values, both against local and overall buckling of composite columns.
Figure 2: Typical composite members Recent applications of the structural elements concern arch road bridges in which the deck is supported by hollow steel columns filled with concrete. These very slender elements, thanks to the combined use of the two materials, can face the high stresses induced by external loads, with the evident advantage of constituting carpentry during the construction of the structure. Recent theoretical and experimental studies have shown that if traditional reinforcement made up of bars and stirrups is added inside tubes, or these are filled with fibre reinforced concrete (FRC), their
Strength and Ductility of Hollow Circular Steel Columns
415
bearing capacity and ductility increase, Campione et. al (1999). In the last few decades interest in the field of composite materials and especially in fibre reinforced concrete has led to the development of new types of fibres (carbon, polyolefin, kevlar, steel) with different shapes (hooked, crimped, deformed). Due to the bridging capacity of the fibres across the cracks, FRC behaves better than either plain concrete or plain concrete confined with traditional reinforcement in terms of energy absorption capacity, and sometimes strength, especially when a high volume fraction of fibres (12 %) is used, Campione et al. (1999).
EXPERIMENTAL INVESTIGATION In the present section there are briefly mentioned experimental results discussed in detail in a previous investigation by the authors, Campione et al. (1998). The experimental research involved the casting of different types of composite members: steel columns, steel columns filled with normal strength concrete (NSC) and steel columns filled with fibre reinforced concrete (FRC). Different types of fibres (polyolefin straight, hooked and crimped steel fibres) were added to the fresh concrete at a dosage of 2% by volume. The fibres had the characteristics shown in Table 1. TABLE 1 CHARACTERISTICS OF THE FIBRES Type of fibres Poyolefin Hooked steel ~ Crimped steel ~
Shape
""
Diameter equiv. , (ram)
Length. Lf
(mm)
Tensile strength f't (MPa)
Modulus of elasticity Ef (MPa)
Weight density (kg/ m 3)
0.80 0.50 1.00
25 35 50
375 1115 1037
12000 207000 207000
900 7860 7860
The columns had a circular crosssection welded along their length; the yielding stress was 206 MPa and the ultimate stress 324 MPa; the internal diameter was 120 mm and the thickness 3.5 mm, the length of the entire columns was 1000 mm.
Figure 3: Composite columns tested in compression The columns were tested in uniaxial compression utilising a universal testing machine operating in displacement control (Figure 3). A load cell and several LVDT's connected to a data acquisition system were used to record the load P and the vertical deformation 8 of the columns. Monotonic tests were carried out on 100x200 mm cylinders in concrete and FRC both in indirect split tension and in compression to characterise the materials as shown in Figure 4. It is interesting to observe that adding fibres to the matrices the behaviour of the latter changes significantly, especially in
416
G. Campione et al.
the sottening branch, in terms of both energy absorption capacity and residual strength. More details on the strength and strain values and cyclic response of the materials are given in a previous paper, Campione et al. (1998). Table 2 gives the most rapresentative experimental results of the compression and indirect split tension tests, particularly the maximum compressive strength f c, and corresponding strain e0, and maximum tensile strength ft. TABLE 2 EXPERIMENTAL RESULTS FOR COMPRESSIONAND INDIRECT TENSION TESTS Types of fibres Matrix Hooked steel Polyolefin Crimped steel (*) ft=2P/0td h)
E;o
0.0032 0.0061 0.0034 0.0064
f~ (A/IPa) 25.20 27.45 29.34 35.40
(*)f, (MPa) 1.64
3.56 2.42 2.72
Figure 4: Monotonic tests in compression of FRC with 2% fibres Figure 5 gives loaddeformation (P8) curves in compression for steel pipes and composite members in the case of montonic loads. Experimental results have shown that columns filled with FRC exhibit higher strength than those filled with plain concrete. The maximum strength of composite members filled with FRC is 20 % higher than that recorded for steel pipes filled with FRC. After the peak load was reached, failure was due to the crushing of concrete and to local and global buckling of the steel pipes. At this point, the peak load and also the stiffness decreased. By contrast, the addition of fibres ensured better softening behaviour and more available ductility.
Figure 5" Load vs. deformation curves for composite columns with 2% fibres
Strength and Ductility of Hollow Circular Steel Columns
417
STRENGTH OF C O M P O S I T E COLUMNS SUBJECTED TO COMPRESSIVE LOADS Several European and international codes give design rules and simplified formulae to predict the bearing capacity of steel columns filled with plain concrete. These are able to take into account the strength of the materials, and the buckling problems of the composite members. When columns are subjected to axial forces the properties which must be taken into account in the design of members include strength of the constituent materials, local instability, and the capacity to transfer internal stresses between the steel pipe and the concrete core. For composite members having a transverse circular crosssection, EC4 allows one to neglect local buckling problems when the ratio between the diameter d and the thickness t obeys the relationship d/t cable_area(0.7587) > cable_number(O.lO02) AND The prediction of cable_tension IS 84.7976 cable_number(O.lO08) AND The prediction of cable_tension IS 70.9726 1
Figure 1. Moving force on a steel beam bridge An equation of motion in term of the modal displacement q, (t) can be given as 2
~n(t)+Z~,conO(t)+coZqn(t)=~ZPn(t )
(1)
(n = 1,2,...~)
Where n2n2 I ~ co" = 7L
'
C
(" = 2pco, '
nTact
pn (t)= P ( t ) s i n ( ~ )
(2)
are the modal frequency of the nth mode, the damping ratio of the nth mode and modal force respectively. If the timevarying force P(t) is known, the equation (1) can be solved to yield q, (t)
v(x,t)
then the dynamic deflection
can be found from the
q,(t)and
the nth mode shape
function O, (x). This is called the forward problem. The moving force identification is an inverse problem, in which the unknown timevarying force P(t) could be identified based on measuring the displacements, bending moments or accelerations of practical structures. Two methods are developed for the purpose.
Time Domain Method (TDM) As mentioned above, the equation (1) can be solved in time domain by the convolution integral and the dynamic deflection v(x,t) of the beam at point x and time t can be obtained as
v(x't)= ~
2
sinnntzt t L f~e~"~
sinco"(t r)sinnntZr p(r)
(3)
Where co', = co. ~/1  ( 2 , therefore, the bending moment of the beam at point x and time t is
m(x,t) = El
02V(x,t) OX2
=
~2EI1r2n 2 f~ e n=l pL3co' sinnntztL tr176
n~cr L P(r)dr
(4)
Parameter Studies of Moving Force Identification in Laboratory
539
The acceleration at the point x and time t is
ncizzt + ~t h', ( t  r)P(r)sin( n~rcr'dr] a(x, t) = i;,(x,t) = ~ 2_7 ~, (x) [ P(t)sin(~) L ) ,=I pL
(5)
Where
J~,(t) = 1. er176
n)2 _
O.)n'2 ]sin
CO'nt+ [ 2~,(O,(O', ]COSCO'nt}
(6)
(On
Assuming that both the timevarying force P(t) and the bending moment m(x,t)or the acceleration a(x,t) are step functions in a small time interval At, equation (4) or (5) can be rewritten in discrete terms and rearranged into a set of equations as follows
BN•
PNB •
= RN•
(7)
Where, P is the time series vector of timevarying force P(t), R is the time series vector of the measured response of the bridge deck at the point x, such as the bending moment m(x,t) or acceleration a(x,t). The system matrix B is associated with the system of bridge deck and the force. The subscripts N B = L / cAt and N are the numbers of sample points for the force P(t) and measured response R respectively when the force goes through the whole bridge deck.
FrequencyTime Domain Method (FTDM) Equation (1) can also be solved in the frequency domain. Performing the Fourier Transform for Equations (1) and v(v,t) = s ap, (x)q n(t), the Fourier Transform of the dynamic deflection v(x,t) is n=l
2 V ( x, (O) = ~.~ 7jO , ( x ) H , ( (O) P ( (O) n=l
/a,.
(8)
Where H,((O) and P((O)are the Fourier Transform of q,(t)and P(t) respectively. Similarly, the relationships between bending moment or acceleration and dynamic deflection can also be used to execute the corresponding Fourier Transform. Finally, a set of Norder simultaneously equations can be established in the frequency domain. The force P((O)consisted of the real and imaginary parts can be found by solving the Norder linear equations. The time history of the timevarying force P(t) can then be obtained by performing the inverse Fourier Transformation. From the procedures mentioned above, initially, the governing equations are formulated in the frequency domain. However, the solution is obtained in the time domain. Therefore this method is named frequencytime domain method. The above procedure is derived for a single force identification in TDM and FTDM methods. They can be modified for multiforce identification using the linear superposition principle.
EXPERIMENTAL DESIGN The model car and model bridge deck were constructed in the laboratory. An AxleSpacingtoSpanRatio (ASSR) is defined as the ratio of the axle spacing between two consecutive axles of a vehicle to the bridge span length. Here, the ASSR was set to be 0.15. The model car had two axles at a spacing of 0.55 m and it ran on four rubber wheels. The static mass of the whole vehicle was 12.1 kg in which the mass of rear wheel was 3.825 kg. The model bridge deck consisted of a main beam, a leading beam and a trailing beam as shown in Figure 2. The leading beam was used to achieve acquired constant speed of vehicle when the model car approached the bridge. The trailing beam was for the slowing down of the car. The main beam with a span of 3.678 m long and 101 mm x 25 mm uniform crosssection, was simply supported. It was made from a solid rectangular mild steel bar with a
T.H.T. Chan et al.
540 density of 7335 kg / m
3
and a flexural stiffness EI = 29.97kN/m
2 .
The first three theoretical natural
frequencies of the main beam bridge was calculated as f~ = 4.5 Hz, f2 = 18.6 Hz, and f3 = 40.5 Hz.
Figure 2. Experimental setup for moving force identification The Ushape aluminum track was glued to the upper surface of the main beam as a guide for the car. The model car was pulled along the guide by a string wound around the drive wheel of an electric motor. The rotational speed of motor could be adjusted. Seven photoelectric sensors were mounted on the beams to measure and check the uniformity of moving speed of the car. Seven equally spaced strain gauges and three equally spaced accelerometers were mounted at the lower surface of the main beam to measure the response. Bending moment calibration was carried out before actual testing program by adding masses at the middle of the main beam. In addition, a 14channel type recorder was employed to record the response signals. Where Channels 1 to 7 were for logging the bending moment response signals from the strain gauges. Channels 8 to 10 were for the accelerations from the accelerometers. The channel 11 was connected to the entry trigger. In the meantime, the response signals from Channels 1 to 7 and Channel 11 were also recorded in the hard disk of personal computer for easy analysis. The software Global Lab from the Data Translation was used for data acquisition and analysis in the laboratory test. Before exporting the measured data in ASCII format for identification calculation, the Bessel IIR digital filters with lowpass characteristics was implemented as cascaded second order systems. The Nyquist fraction value was chosen to be 0.05.
P A R A M E T E R STUDIES
For practical reason, one parameter was studied at a time. The examination procedure was to examine one parameter in each case to isolate the case with the highest accuracy for the corresponding parameter and then another parameter was examined. The parameters, such as the mode number, the sampling frequency, the speed of vehicle, the computational time, the sensor and sensor locations were considered as variables to examine their effect on the accuracy of force identification. There are two ways to check this kind of effects. One is that the identified results are checked directly by comparing the identified forces with the true forces. However, because the true forces are unknown, it is difficult to proceed. The other way is that the identified results are checked indirectly by comparing the measured responses (bending moments, displacements or accelerations) with the rebuilt responses calculated from the identified forces. The accuracy is quantitatively defined as Equation (9), called a Relative Percentage Error (RPE).
RPE = EJftn, e  ~'dentl x 100%
El/.el
(9)
Equation (9) is also used to calculate the relative percentage errors between the measured and rebuilt responses from identified forces instead of comparing the identified forces with the true forces directly. The measured response (R ....... d) and rebuilt response (Rreb,,izt)are herein substituted for the true force (ftr,,e)and identified force (fide,,)in equation (9) respectively. In the present parameter studies, most of results were from the comparison of the relative percentage errors between the measured and rebuilt response only for the bending moment response. Regarding the results associated with the accelerations, they will be reported separately.
Parameter Studies of Moving Force Identification in Laboratory
541
Effects of Mode Number For comparing the effects of different Mode Number (MN) on identified results in the TDM and FTDM, it was assumed that the sampling frequency (f~) and vehicle speed (c) were constant, and the case of fs = 250Hz, c = 15 Units (1.52322 m/s) was chosen. The data at all the seven measurement stations for bending moments were employed to identify the moving forces. The mode number was varied from MN=3 to MN=10. The identified forces were calculated first, and then the rebuilt responses from the identified forces were then computed accordingly. The Relative Percentage Error (RPE) for both the TDM and FTDM are shown in Figure 3.
Figure 3. Effects of mode number For the TDM, the RPE values at the middle measurement stations are always less than the ones at the two end measurement stations. This is associated with the signal noise ratio of various measurement stations because there are bigger responses at the middle stations than those at the two end stations. It is found that when the mode number is equal to or bigger than four, the relative percentage errors are reduced dramatically. This means the TDM is effective if the required mode number is achieved or exceeded, otherwise, the TDM will be failed. The minimum RPE value case is of MN=5, the maximum RPE value case is of the biggest mode number involved (MN=I 0). This also shows that the case MN=5 is the most optimal case in this kind of comparisons. Similar conclusions are drawn for the FTDM. However, the biggest difference from the TDM is that the RPE is independent of the increment MN after MN=5. By comparing the identification accuracy by the TDM and FTDM in Figure 3, it can be seen that the results are very close to each other when the MN is equal to 5 and 6 respectively, especially at the middle measurement stations. However, it can be seen from the identified forces in Figure 4 that the FTDM is worse than the TDM because it has components with higher frequency noise.
Figure 4. Identified forces (MN=5, fs = 250Hz, c = 15 Units )
Effects of Sampling Frequency The sampling frequency fs should be high enough so that there is sufficient accuracy in the discrete integration in equation (4) and (5) [Law et al 1997]. In the present study, the data was acquired at the sampling frequency 1000 Hz per channel for all the cases. This sampling frequency was higher than the practical demand because only a few of lower frequency modes were usually used in the moving force identification. Therefore, the sequential data acquired at 1000 Hz was sampled again in a few intervals in order to obtain a new sequential data at a lower sampling frequency. Here, a new
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sequential data at the sampling frequency of 333,250, and 200 Hz would be obtained by sampling the data again every third, fourth and fifth point respectively. For the case of the vehicle running at 15 Units, the bending moment data was acquired at the different frequencies of 200, 250 and 333 Hz respectively. The RPE results between the rebuilt and measured bending moment responses are calculated and listed in Table 1 for both the TDM and FTDM. TABLE 1 EFFECTS OF SAMPLING FREQUENCY (c = 15 Units) Sta.
TOM
No.
MN=3 I
1 2 3 4 5 6 7 Case
II
13.8 783. 7.15 609. 6.50 358. 3.61 216. 6.27 359. 7 . 4 1 614. 17.3 780. I, II, and III
FTDM
MN=4 III
I
II
MN=5 III
I
II
MN=3 III
412. 6.44 6.05 6.11 8.86 5.32 3.75 244. 2.81 2.74 2.69 3.34 2.61 2.40 185. 2.74 2.10 1.95 2.87 2.10 1.94 216. 3.15 2.96 2.80 3.72 2.71 2.12 189. 3.16 2.74 2.53 3.58 2.68 2.44 245. 4.74 4.58 4.32 5.42 4.31 3.45 420. 6.61 5.84 5.94 9.36 5.19 4.05 is for 200,250 and 333 Hz respectively.
I
II
188. 167. 163. 165. 164. 169. 187.
1541 1618 1679 1686 1676 1615 1514
MN=4 III
I
1724 6.35 1636 2.68 1647 2.09 1678 2.69 1657 2.61 1654 4.22 1717[ 5.89
MN=5
II
III
I
II
III
16.1 10.3 8.50 8.05 8.62 10.0 16.0
445. 419. 427. 433. 430. 424. 446.
3.38 2.51 2.01 2.09 2.42 2.89 3.92
4.39 2.16 2.14 2.22 2.13 2.70 4.30
2005 1370 1230 1321 1238 1387 2095
For completely comparing the effects of different sampling frequency, the effect of mode number on identification accuracy is also incorporated in the study. It is found that the higher the sampling frequency is, the lower the RPE values are for all the measurement stations in the TDM. This shows that the higher sampling frequency is better than the lower sampling frequency, and the TDM method has higher identification accuracy if the response is acquired at a higher sampling frequency. In Table 1, it is shown that the FTDM method is failed when the sampling frequency fs = 333 Hz and mode number MN=3 because the RPE values are too big to accept for all the measurement stations. However, The FTDM method is still effective for the case in which the mode number is bigger than 3, f~ = 200 Hzandf, = 250 Hz respectively. By comparing the RPE values at a lower sampling frequency f~ = 200 Hz with that at f, = 250 Hz, it is found that the identification accuracy at fs = 200 Hz are higher than one at f, = 250 Hz. It shows that the identified results are acceptable and useful if more mode number and suitable sampling frequency is determined in FTDM method.
Effects o f Various Vehicle Speeds In this section, some limitations on identified methods TDM and FTDM should be considered firstly. In particular, necessary RAM memory and CPU speed of personal computer are required for both the TDM and FTDM. Otherwise, they will take very long execution time due to the bigger system coefficient matrix B in equation (7), or they cannot execute at all due to inefficient memory. As the mode number, the sampling frequency and bridge span length had not been changed for this case, a change of the vehicle speed would mean a change of the sampling point number, namely change of dimensions of matrix B in equation (7). Therefore, in order to make TDM and FTDM effective and to analyze the effects of various vehicle speeds on the identified results, the case of MN=4 and fs = 200 Hz was selected. When the test was carried out, the three vehicle speeds were set manually to 5 Units (0.71224 m/s), 10 Units (1.08686 m/s), and 15 Units (1.52322 m/s) respectively. After acquiring the data, the speed of vehicle was calculated and the uniformity of speed was checked. If the speed was stable, the experiment was repeated five times for each speed case to check whether or not the properties of the structure and the measurement system had changed. If no significant change was found, the recorded data was accepted. The RPE values between the rebuilt and measured bending moment responses are calculated and listed in Table 2. It shows that the TDM is effective for all the three various vehicle speeds. The RPE values tend to reduce for each measurement station as the vehicle speed increases. But, the RPE values are close to each other in the case of 10 Units and 15
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543
Units. It shows that the identification accuracy for the faster vehicle speed is higher than that at slower vehicle speed. However, the FTDM is not effective in the case of lower vehicle speed 5 Units, but the identified results are getting better and better as the vehicle speed increases. Fortunately, the identified result is acceptable in the case of 15 Units in the FTDM. TABLE 2 EFFECTS OF VEHCILE SPEEDS (MN=4, fs = 200Hz ) Station
TDM
FDTM
No.
5 Units
10 Units
15 Units
5 Units
10 Units
15 Units
1 2
6.81 5.54
5.40 2.49
6.45 2.81
1045.69 708.18
101.53 46.57
6.35 2.68
3
5.88
3.03
2.74
621.36
24.83
2.09
4
8.67
3.01
3.15
562.90
45.11
2.69
5 6
4.50 4.66
2.76 3.93
3.16 4.74
586.98 647.38
23.98 44.70
2.61 4.22
7
6.56
7.94
6.61
965.75
94.26
5.89
Effects of Various Measured Station Number This section estimates the effects of measurement station number ( N t ) on the identified accuracy. The N t was set to 2, 3, 4, 5 respectively while the other parameters MN=5, f, = 250 Hz, c = 15 Units were not changed. The RPE values between the rebuilt and measured responses are given in Table 3. The results in Table 3 show that the TDM is required to have at least three measurement stations to get the two correct moving forces for the front and rear wheel axles respectively. But the FTDM should have at least one more measurement station, i.e. 4, to get the same moving forces. However, the errors are increased obviously when the measurement station number is equal to 5 for the FTDM. TABLE 3 EFFECTS OF MEASUREMENT STATIONS No.
FTDM
TDM
Station 2
1 (L/a) * 2 (2L/8) * 3 (3L/8) 2003.03 4 (4L/8) * 5 (5L/8) 2029.36 6 (6L/8) * 7 (7L/8) * Asterisk * indicates
3
4
5
* * 1.91 * 1.50 2.21 2.15 2.08 2.62 2.27 * 2.39 2.48 2.04 2.82 * 2.38 * * the station is not chose.
2
3
4
5
*
*
2.67
27.06
1192.66 *
86.92 115.42
2.35 *
14.82 27.95
1198.49 *
87.52 *
2.49 2.93
14.73 26.62
Comparison of computational time The computational time consists of three periods, i.e., i) forming the system coefficient matrix B in equation (7), ii) identifying forces by solving the equation and iii) reproducing the responses. The above parts are same for the TDM and FTDM. The case described here is of MN=5, f, = 250 Hz,
c = 15 Units, N t = 7 by using a Pentium II 266 MHz CPU, 64M RAM computer. The total sampling points are 700 for bending moment response at each measurement station and the total sampling points are 604 for each wheel axle force in the time domain. Therefore, the dimensions of matrix B are (7 x 700, 2 • 604). The execution time recorded is listed in Table 4 for the comparison on each period of the TDM and FTDM in details. It shows that the FTDM takes much longer than the TDM method in forming the coefficient matrix B. The execution time in other two parts is almost the same for the two methods. The TDM takes shorter time than the FTDM from the point of view of the total execution time.
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T.H.T. Chan et al. TABLE 4 COMPARISONOF COMPUTATIONTIME (in Second) PERIOD TDM FTDM Forming coefficient matrix B 332.69 1059.57 Identifying forces 1837.97 1834.07 Rebuilding responses 55.04 53.99 Total 2225.7 2947.63
CONCLUSIONS Parameter studies on moving force identification in laboratory test have been carried out in this paper. These parameters include the mode numbers, the sampling frequencies, the vehicle speeds, the computational time, the sensor numbers and locations. The study suggests the following conclusions: (1) A minimal necessary mode number is required for both the TDM and FTDM. It should be equal to or bigger than 4. If first five modes are determined to identify the moving forces, the identification accuracy is the highest in the cases studied. (2) The TDM has higher identification accuracy when the higher sampling frequency is employed. However the FTDM is failed if adopting the higher sampling frequency and the lower mode number. (3) The faster car speed is of benefit to both the TDM and FTDM, but FTDM method is not suitable for the slower car speed case. (4) At least three and four measurement stations are required to identify the two wheel axle forces for the TDM and FTDM respectively. (5) The TDM takes shorter time than the FTDM. (6) Both the TDM and FTDM can effectively identify moving forces in time domain and frequency domain respectively, and can be accepted as a practical application method with higher identification accuracy. (7) From the point of view of all the parameter effects on the identification accuracy, the TDM is the best identification method. It should be firstly recommended as a practical method to be incorporated in the future developed Moving Force Identification System (MFIS). ACKNOWLEDGMENT The present project is supported by the Hong Kong Research Grants Council. REFERENCES
1. Briggs J.C. and Tse M.K. (1992). Impact force Identification using Extracted Modal Parameters and Pattern Matching. Int. J. Impact Engineering 12:3, 361372. 2. Chan T.H.T. and O'Connor C. (1990). Wheel Loads from Highway Bridge Strains: Field Studies. Journal of Structural Engineering 116:7, 17511771. 3. Chan T.H.T. Law S.S. Yung T.H. and Yuan X.R. (1999). An Interpretive Method for Moving Force Identification. Journal of Sound and Vibration 219:3, 503524. 4. Fryba L. (1972). Vibration of Solids and Structure under Moving Loads, Noordhoff International Publishing, Prague. 5. Hoshiya M. and Maruyama O. (1987), Identification of Running Load and beam system. Journal of Engineering Mechanics ASCE, 113, 813824. 6. Law S.S. Chan T.H.T. and Zeng Q.H. (1997). Moving Force Identification: A Time Domain Method, Journal of Sound and Vibration, 201:1, 122. 7. Law S.S. Chan T.H.T. and Zeng Q.H. Moving Force IdentificationFrequency and Time Domain Analysis, Journal of Dynamic System, Measurement and Control (accepted for publication) 8. Moses F. (1984). WeighInMotion System using Instrumented Bridge, Journal of Transportation Engineering ASCE, 105(TE3), 233249. 9. O'Connor C. and Chan T.H.T. (1988). Dynamic Wheel Loads from Bridge Strains, Journal of Structural Engineering, 114:8, 17031723. 10.Stevens K. K. (1987). Force Identification ProblemsAn Overview, Proceeding of SEM Spring Conference on Experimental Mechanics, 838844.
SEISMIC ANALYSIS OF ISOLATED STEEL HIGHWAY BRIDGE
XiaoSong LI 1 and Yoshiaki GOTO 2 1 Research Associate, 2 Professor Dept. of Civil Engineering, Nagoya Institute of Technology Gokisocho, Showaku, Nagoya 4668555, Japan
ABSTRACT Seismic isolators with dissipation devices have been widely used for highway bridges in Japan, because they may effectively absorb energy and reduce inertia force induced by earthquake. The main factors that influence the response of the isolated bridge are initial stiffness and yield force of the isolator. These quantities should be appropriately designed. Besides, introduction of the isolators leads to an interaction between the bridge pier and the isolator and increases the computational difficulty due to the nonlinearity that occurs in both the pier and the isolator. The purpose of this paper is to investigate the seismic response of the isolated bridges subjected to ground motions, where we examine how the behaviors of the bridge pier are influenced by the initial stiffness and yield force of the isolator and the elongation of natural period of the bridge. Then, the applicability of the 'Displacement Conservation Principle' for predicting the maximum responses of the piers of the isolated steel bridges is numerically examined. The numerical results show that the application of the 'Displacement Conservation Principle' may be reasonably safe and accurate for the practical design of isolated steel piers.
KEYWORDS
seismic isolation design, nonlinear dynamic analysis, steel highway bridge 545
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INTRODUCTION After the great earthquake happened in Kobe, in 1995, seismic isolators with dissipation devices have been widely used for highway bridges in Japan. Due to the significant increase of the natural period, the isolators may effectively absorb energy and reduce inertia force induced by earthquake. The main factors that influence the response of the isolated bridge are initial stiffness and yield force of the isolator. These quantities should be appropriately designed. Besides, introduction of the isolators leads to an interaction between the bridge pier and the isolator and increases the computational difficulty due to the nonlinearity that occurs in both the pier and the isolator. The purpose of this paper is to investigate the seismic response of the isolated bridges subjected to ground motions, where we examine how the behaviors of the bridge pier are influenced by the initial stiffness and yield force of the isolator and the elongation of natural period of the bridge. Then, the applicability of the 'Displacement Conservation Principle' for predicting the maximum responses of the piers of the isolated steel bridges is numerically examined. The numerical results show that the application of the 'Displacement Conservation Principle' may be reasonably safe and accurate for the practical design of isolated steel piers.
ANALYTICAL MODEL A typical isolated bridge is used as an analytical model, as shown in Fig.1. The total weight of the bridge M=1067ton consists of the weight of the deck Mb=0.95M and the weight of the pier Mp=0.05M. For the height of piers, two values are considered, that is, H=13m for Modell, and H=11m for Model2. The corresponding fundamental natural periods for the two piers are Tl=0.705s and T2=0.549s.
Fig.l: Analytical Model A leadrubber bearing (LRB) is assumed as an isolator with dissipation device that has bilinear yield stiffness as shown in Fig.2. In Fig.2, Qy and Uby are the yield force and yield displacement,
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respectively. K~,~and Kb2 are the elastic stiffness and postyield stiffness with a relation of Kbz=KbJ6.5; Kr~ is the equivalent stiffness and UBe=0.7Ub.... which are suggested by the 'Manual of Menshin (isolation and dissipation) Design of Highway Bridges' (1992).
Fig.2: Hysteresis Behavior of Isolator
ANALYTICAL M E T H O D A numerical method that considers both geometrical and material nonlinearity is used to carry out the dynamic analysis (Li and Goto, 1998). The postyield modulus of material is assumed to be Ep=E/100. The effect of damping is considered by a massproportional damping matrix. The damping coefficient is set to h=0.01 for elastoplastic analysis and h0.05 for elastic analysis. Two standard ground accelerations suggested by Japan Road Association are used for the analysis. One is Type 2 at Soil Group II (hard soil site), the other is Type 2 at Soil Group Ill (soft soil site). Both acceleration waves are illustrated in Fig.3. The time interval adopted in the numerical integration is 0.01s.
Fig.3: Ground Accelerations In order to investigate the effect of the initial stiffness and yield force of the isolator, the calculation is carried out by changing Qy/Py and I~I/K p from 0.2 to 0.9 that is the possible range in practical design, where Py=2( cr yMg/A)/(HB) and Kp=3EI/H 3 are the yield force and elastic stiffness of the pier.
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NUMERICAL RESULTS Typical responses of an isolated bridge are shown in Fig.4. It can be seen from Fig.4 (a) that the displacement of the pier is much smaller than that of the deck due to the isolator. Furthermore, the pier is damaged little and the energy induced by seismic wave is almost dissipated in the isolator, as illustrated in Fig.4 (b). In the following, the effects of the initial stiffness Kb~ and yield force
Qyof the
isolator and the elongation of natural period of the bridge are investigated. Then, the applicability of the 'Displacement Conservation Principle' for predicting the maximum responses of the piers of the isolated steel bridges is numerically examined.
Fig.4: Responses of Model1 with Qy/Py=0.6 and Kbl/I~ =0.5 Subjected to Wave Type 2111 Effect of K~l and Qy on Pier and Isolator With the designated yield force ratios Qy/Py=0.3, 0.5 and 0.7, the maximum response displacements of the pier and the isolator are obtained by changing the initial stiffness ratio Kbl/Kp from 0.2 to 0.9. The relations of ductility factors//p and/1 b of the pier and the isolator vs. the initial stiffness ratio Kb~/Kp of the isolator are shown in Fig.5 for Model1 and Model2 subjected to waves Type 2II and Type 2lit. In this figure, the ductility factors for piers and isolators are defined as /1 p=Upmax/Upy(with solid line) and /1 b=Ubmax/Uby (with dotted line), where Upmax=maximum displacement of pier, Upy=yield displacement of pier, and Ubmax=maximum displacement of isolator and Uby=yield displacement of isolator. It should be noted that the/1 p may be considered as the maximum response or the ductility factor of the pier, while/.t b denotes only the ductility factor of the isolator because the Uby varies with Qy/Py or Kb~/Kp. Similarly, the relations that vary with the yield force ratio Qy/Py are shown in Fig.6 for the designated initial stiffness ratios Kbl/Kp=0.3, 0.5 and 0.7. From Fig.5, it can be seen that the both ductility factors ~t p and /~ b of the pier and the isolator increase with the increase of the initial stiffness ratio Kbl/K.p of the isolator for all cases. However,
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549
some influences caused by the wave types can be found as follows. The maximum responses of the pier subjected to wave Type 211" almost linearly increase as Kbl/Kp increases from 0.2 to 0.9 as shown in Fig.5(a), while those subjected to wave Type 2m" increase a little shapely when Kbl/I~ >0.6. Furthermore, the values of It p for Model2 with a smaller ratio of Qy/Py=0.3 become greater than those with Qy/Py=0.5 and 0.7 when Kbl/Kp~0.6, as shown in Fig.5(c). The relations between the ductility factor It b of the isolator and Kbl/Kp exhibit a different tendency depending on the value of Qy/Py. That is, It b with Qy/Py=0.3 exhibits a large increase, while that with Qy/Py=0.7 shows just a small increase.
Fig.5: Effect of Initial Stiffness Ratio Kbl/Kp of Isolator on Ductility Factors It p and It b
Fig.6: Effect of Yield Force Ratio Qy/Py of Isolator on Ductility Factors It p and It b
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Figure 6 shows that the ductility factor/1 p of the pier increases with the increase of yield force ratio Qy/Py for most values of Kbl/Kp, while the ductility factor/1 b of the isolator decreases. A different tendency, however, is observed for Model2 with Kb~/Kp=0.7 as shown in Fig.6(c). In this case, the maximum value of /.t p is obtained for the yield force ratio of Qy/Py=0.2 that is the smallest one. This phenomenon is identical with that observed in Fig.5(c) and is also identical with those obtained for concrete piers by Kawashima & Shoji (1998). It can be concluded that an isolator with a higer stiffness and a smaller yield force may result in much damage of the pier when subjected to wave Type 2 Ill.
Fig.7: Relations between Energy Ratio Ep/Ef and Period Ratio Tp/Tf
Effect of Elongation of Natural Period The main difference between isolated bridges and conventional bridges is characterized by a significant increase in the natural period of isolated bridges due to the introduction of the isolator that may also effectively absorb energy. As a result, the inertia force induced by earthquake is considerably reduced in isolated bridges. In order to find an appropriate period range for the isolated bridge, an energy ratio Ep/Ef is used to evaluate the damage of the pier, where Ep = 4Ppdup and Ef = ~Pfduf are the energy absorbed in the pier with the isolator and that without the isolator, respectively. And, a period ratio Tp/Tf is used as the abscissa, where Tp = 2nx/m/K e and T f   2 n x / m / K p are the fundamental periods of the isolated bridge and the conventional one, respectively. K e is the equivalent stiffness of the isolated bridge and defined as Ke=KBKp/(KB+Kp), where KB is the equivalent stiffness of the isolator as shown in Fig.2. Figure.7 shows the relations between the energy ratio Ep/Ef and the period ratio Tp/Tf for Model1 and Model2. It can been seen from Fig.7 (a) that a general tendency is that the energy ratio Ep/Ef decreases as the period ratio Tp/Tf increases, except for a few points. The three points over 0.4 are corresponding to the cases: (Qy/Py=0.3, Kbl/Kp=0.8, 0.9) and (Qy/Py=0.2, Kbl/Kp=0.7) for Model2 subjected to wave Type 2Ill. If both Qy/Py and Kbl/Kp are limited to the
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551
range from 0.3 to 0.7 that are favorable in the practical design, the maximum value of Ep/Ef will be less than 0.3 as shown in Fig.7 (b). This result implies that the range from 0.3 to 0.7 is appropriate for both Qy/Vy and Kbl/Kp of the isolator. The corresponding period ratio Tp/Tf is between 2.1 and 3.7. Furthermore, by noting that Ep/Ef approaches zero when Tp/Tf > 3.0, the range from 2.1 to 3.0 of the period ratio Tp/Tf may be more appropriate. This is based on the fact that too much elongation of the natural period may lead to an undesirable large displacement of the deck.
Application of 'Property of Displacement Conversation' As previously described, the interaction between the pier and the isolator results in the computational difficulty due to the nonlinearity that occurs in both the pier and the isolator. To avoid the difficulty, some simple methods have been proposed for predicting the dynamic response of the pier, such as the method based on the 'Energy Conservation Principle'. For the conventional bridge pier that has a shorter fundamental period, the method has been verified applicable and widely used in practical design. For the isolated bridge that has a longer fundamental period, however, it overestimates or underestimates the response of the pier. Here, based on the 'Displacement Conservation Principle' that states 'the maximum elastoplastic deformation of a system with a long fundamental period is approximately equal to the maximum elastic deformation of the same system', the maximum responses of steel piers are predicted by elastic dynamic analysis. In this method, the maximum elastic response displacements of both the pier and the isolator are first dynamically calculated for the isolated bridge with the elastic stiffness Kp of the pier and the equivalent stiffness KB of the isolator. Then, the maximum lateral force of the pier can be obtained by equating the elastic response displacement to the static elastoplastic response displacement.
Fig.8: Comparison of Maximum Responses of Piers Obtained by Simple Method and Analytical Method The comparison of the maximum responses of the piers obtained by the simple method and by the elastoplastic dynamic method are shown in Fig.8 (a), where RF and ~ are the maximum lateral
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552
force and displacement of the pier, respectively. The subscripts e and p denote the elastic dynamic response and the elastoplastic dynamic response. It is observed that the maximum lateral force ratio RFJRFp is closer to 1.0 when Tp/Tf is smaller than 3.0, whilst the maximum lateral displacement ratio d d d p is in the range from 0.77 to 1.86. Similarly, if both
Qy/Pyand Kbl/Kp are limited to the range
from 0.3 to 0.7, RFe/RFp and ~ o/~ p will take the values between 1.0 to 1.81. The approximate results obtained by the 'Displacement Conservation Principle ' may be considered to be reasonably safe and accurate for practical design of the isolated steel piers.
SUMMARY AND CONCLUDING REMARKS
From the above analysis, the following conclusions are obtained. The response of an isolated bridge is greatly influenced by the initial stiffness and yield force of the isolator. The both ductility factors for the pier and the isolator increase with the increase of the initial stiffness ratio Kb~/I