COMPRESSION STABILITY OF HIGH STRENGTH STEEL …

Microsoft Word - Demao Yang PhD Thesis.docLOW STRAIN-HARDENING
Doctor of Philosophy
University of Sydney
Thin-walled steel sections made from high strength thin cold-reduced G550
steel to Australian Standard AS 1397-1993 under compression are investigated
experimentally and theoretically in this thesis.
This thesis describes three series of compression tests performed on box-section
stub columns, box-section long columns and lipped channel section columns
cold-formed from high strength steel plates in 0.42 mm or 0.60 mm thickness
with nominal yield stress of 550 MPa. The tests presented in this thesis formed
part of an Australian Research Council research project entitled "Compression
Stability of High Strength Steel Sections with Low Strain-Hardening".
For the fix-ended stub column tests, a total of 94 lipped-square and hexagonal
section stub columns were tested to study the influence of low strain hardening
of G550 steel on the compressive section capacities of the column members. For
the pin-ended long column tests, a total of 28 box-section columns were tested
to study the stability of members with sections which undergo local instability
at loads significantly less than the ultimate loads. For the fix-ended lipped
channel section columns, a total of 21 stub and long columns were tested to
study the failure resulting from local and distortional buckling with interaction
between the modes.
A numerical simulation on the three series of tests using the commercial finite
element computer program ABAQUS is also presented as part of this thesis.
The post-buckling behaviour of thin-walled compression members is
SYNOPSIS
ii
investigated. The effect of changing variables, such as geometric imperfections
and end boundary conditions is also investigated. The ABAQUS analysis gives
accurate simulations of the tests and is in good agreement with the experimental
results.
Theoretical studies using finite strip methods are presented in this thesis to
investigate the buckling behaviour of cold-formed members in compression.
The theoretical studies provide valuable information on the local and
distortional buckling stresses for use in the interaction buckling studies. The
finite strip models used are the semi-analytical and spline models.
As expected from the stub columns tests, the greatest effect of low strain
hardening was for the stockier sections where material properties play an
important role. For the more slender sections where elastic local buckling and
post-local buckling are more important, the effect of low strain hardening does
not appear to be as significant. The pin-ended and fix-ended long column tests
show that interaction, which is between local and overall buckling in the box
sections, and between local and distortional buckling in the open channel
sections, has a significant effect on their member capacities.
The results of the successful column tests and ABAQUS simulation have been
compared with the design procedures in the Australian/New Zealand Standard
for Cold-Formed Steel Structures (AS/NZS 4600) and the North American
Specification for Cold-Formed Steel Structural Members prepared by the
American Iron and Steel Institute. The stub column tests show that the current
design rules give too conservative predictions on the compressive section
capacities of the column members; whereas the long column tests show that the
SYNOPSIS
iii
current column design rules are unconservative if used in their current form for
G550 steel.
Three design proposals are presented in this thesis to account for the effects of
high strength thin steels on the section and member capacities.
PREFACE
iv
PREFACE
This thesis is submitted to the University of Sydney, Australia, for the degree of
Doctor of Philosophy. The work described in this thesis was carried out by the
candidate during the years 1999 to 2003 in the Department of Civil Engineering
at the University of Sydney under the supervision of Professor G.J. Hancock,
BHP Steel Professor of Steel Structures.
In accordance with The Bye-Laws of the University of Sydney governing the
requirements for the degree of Doctor of Philosophy, the candidate submits that
the work presented in this thesis is original unless otherwise referenced within
the text. The complete experimental programs performed including the design
of the specimens and imperfection measurement rig are claimed as original. The
simulations of the complete experimental programs, in particular the analysis of
stress distribution on the lipped channel cross-section, are also claimed as
original. The design proposals for local buckling, interaction of local and overall
buckling, and interaction of local and distortional buckling are claimed as
original.
Twelve supporting papers and three research reports which are based on the
work presented in this thesis have been written. They are:
1. Compression Tests of Box-shaped Cold-Reduced High Strength Steel
Sections, Proceedings, 6th Pacific Structural Steel Conference, Beijing,
October, 2001, Seismological Press (Yang and Hancock)
2. Stability and Ductility of Thin-High Strength G550 Steel Members and
Connections, Keynote paper, Proceedings of the 3rd International Conference
PREFACE
v
Structures-Advances and Developments-Elsevier 2001) (also published in
Thin-Walled Structures Vol. 41, 2003.) (Rogers, Yang, Hancock)
3. Compression Tests of Cold-Reduced High Strength Steel Stub Columns,
Proceedings, 16th International Specialty Conference on Cold-Formed Steel
Structures in Orlando, October, 2002 (Yang and Hancock)
4. Compression Tests of Cold-Reduced High Strength Steel Long Columns,.
Proceedings, 16th International Specialty Conference on Cold-Formed Steel
Structures in Orlando, October, 2002 (Yang, Hancock and Rasmussen)
5. The Behaviour of High Strength G550 Steel Sections as Used in Residential
Construction, Keynote paper, Proceedings, 2nd International Symposium on
Steel Structures, Seoul Korea, November, 2002 (Hancock, Rogers and Yang)
6. Stability of High Strength G550 Steel Compression Members, Keynote
paper, Proceedings of the 3rd International Conference on Advances in Steel
Structures, Hong Kong, December, 2002 (Yang and Hancock)
7. Compression Tests of Cold-Reduced High Strength Steel Channel Columns
Failing in the Distortional Mode, Proceedings of the 5th International
Conference on Steel and Aluminium Structures (ICSAS) & 7th International
Conference on Steel Concrete Composite Structures (ASSCCS03), Sydney,
June, 2003 (Yang and Hancock)
University of Sydney, Department of Civil Engineering, Research Report
R815, 2002 (Yang and Hancock)
PREFACE
vi
9. Compression Tests of Cold-Reduced High Strength Steel Long Columns,
University of Sydney, Department of Civil Engineering, Research Report
R816, 2002 (Yang, Hancock and Rasmussen)
failing in the Distortional Mode, University of Sydney, Department of Civil
Engineering, Research Report R825, 2003 (Yang and Hancock)
(accepted for publication in ASCE) (Yang and Hancock)
12. Compression Tests of Cold-Reduced High Strength Steel Long Columns,
(accepted for publication in ASCE) (Yang, Hancock and Rasmussen)
failing in the Interaction between Local and Distortional Modes, (accepted
for publication in ASCE) (Yang and Hancock)
14. Developments in Design for Distortional Buckling of Thin-Walled
Members, Keynote paper, The Fourth International Conference on Thin-
Walled Structures, ICTWS 4, Loughborough, England, UK, June, 2004
(Yang and Hancock)
Columns, Proceedings, 17th International Specialty Conference on Cold-
Formed Steel Structures in Orlando, November, 2004 (will be submitted)
(Yang and Hancock)
ACKNOWLEDGMENTS
vii
ACKNOWLEDGEMENTS
I wish to express my sincerest thanks to my supervisor Professor Gregory J.
Hancock for his supervision, enthusiastic guidance and continual
encouragement throughout the course of my candidature. I am indebted to his
understanding, tolerance and precious support.
The scholarship provided by a joint Department of Civil Engineering and Centre
for Advanced Structural Engineering Scholarship during the course of this work
is gratefully acknowledged.
This thesis forms part of an ARC research project entitled “Compression
Stability of High Strength Steel Sections with Low Strain-Hardening” being
carried out in the Department of Civil Engineering at the University of Sydney.
I would like to thank the Australian Research Council and BHP Coated Steel
Australia for their financial support for these projects performed at the
University of Sydney.
The tensile specimens were milled in the William and Agnes Bennett
Supersonics Laboratory in the Department of Aeronautical Engineering. The
compression specimens were fabricated in the J.W. Roderick Laboratory for
Materials and Structures in the Department of Civil Engineering. I would like to
thank Mr. Todd Budrodeen for fabricating the specimens and designing the rig
for imperfection measurement. The finite element analyses were carried out on
UNIX terminal using ABAQUS in the Department of Civil Engineering at the
University of Sydney. I wish to thank Dr Tim Wilkinson for his advice and
assistance on using ABAQUS.
ACKNOWLEDGMENTS
viii
I wish to thank Associate Professor Kim J.R. Rasmussen for his advice and
suggestion on this work. My thanks go to Dr. Young Kwon, Dr. John
Papangelis, Dr. Lip Teh and Dr. Ben Young for their advice and suggestions.
My thanks go to Ms. Gwenda McJannet for her help on various aspects of the
presentation of this work.
My thanks go to my friends and colleagues in the Department of Civil
Engineering at the University of Sydney for their friendship and valuable
discussions, in particular Michael Bambach, Michael Dong, Elisha Harris,
Kelvin Ye and Wilson Yuan.
I wish to thank my parents and my brothers for their support, encouragement
and patience throughout the years of my postgraduate study.
I wish to thank my wife, Xiangyun Zhao, and my daughter, Zhaoyu Yang, for
their love, emotional support and encouragement as well as their appreciation of
my study. I am indebted to my wife so much for looking after our daughter for 4
years in China without my presence.
This thesis is dedicated to my parents Mr. Nianshen Yang and Mrs. Mingxiang
Han.
1.2 OBJECTIVES OF THIS THESIS..............................................................................4
2.2 LOCAL BUCKLING, POST-BUCKLING OF PLATES AND INTERACTION OF PLATE ASSEMBLIES...................................................................................19
2.2.1 General............................................................................................................................................. 19 2.2.2 Local buckling of plate ..................................................................................................................... 19 2.2.3 Post-buckling and interaction of plate assemblies ........................................................................... 21
2.3 BUCKLING OF COMPRESSION MEMBERS.....................................................25 2.3.1 General............................................................................................................................................. 25 2.3.2 Local buckling .................................................................................................................................. 25 2.3.3 Overall buckling ............................................................................................................................... 27 2.3.4 Distortional buckling........................................................................................................................ 30
CONTENTS
x
2.4.2 Interaction of local and overall buckling ......................................................................................... 35 2.4.3 Interaction of local and distortional buckling .................................................................................. 38
2.4.3.1. Linear interaction buckling.................................................................................................................39 2.4.3.2. Non-linear interaction buckling ..........................................................................................................39
3.1 INTRODUCTION......................................................................................................58
3.5 ANALYSES ................................................................................................................67 3.5.1 Elastic local buckling analyses......................................................................................................... 67 3.5.2 Test results and comparisons with design standards........................................................................ 68
CONTENTS
xi
4.1 INTRODUCTION....................................................................................................107
5.1 INTRODUCTION....................................................................................................142
CONTENTS
xii
6.5 SUMMARY ..............................................................................................................213
7.2 REDUCTION FACTOR MODIFIED TO 0.9FY ..................................................252 7.2.1 Limitations on the use of high strength steels................................................................................. 252 7.2.2 Brief description of the comparison of the test and ABAQUS results with Design Standards ....... 254 7.2.3 Modified reduction factor............................................................................................................... 256 7.2.4 Reliability study .............................................................................................................................. 257
7.3 INTERACTION OF LOCAL AND OVERALL BUCKLING............................261 7.3.1 The strength prediction of members subjected to flexural buckling ............................................... 261 7.3.2 Brief description of the comparison of the test and ABAQUS results with Design Standards ....... 263 7.3.3 Modified design curve .................................................................................................................... 265
7.4 INTERACTION OF LOCAL AND DISTORTIONAL BUCKLING.................266 7.4.1 The strength prediction of singly-symmetric sections subjected to distortional buckling .............. 267 7.4.2 Brief description of the comparison of the test and ABAQUS results with Design Standards ....... 270 7.4.3 Proposed design methods ............................................................................................................... 273
7.4.3.1 Method 1: based on Kwon & Hancock Equation .................................................................................... 274 7.4.3.2 Method 2: based on AS/NZS 4600 Clause 3.4.6(b) ................................................................................ 275 7.4.3.3 Comparison with the experimental results .............................................................................................. 275
7.5 SUMMARY ..............................................................................................................276
8.4 DESIGN RECOMMENDATIONS ........................................................................298
Ae effective of area (mm2)
Aeq equivalent effective of area (mm2)
b web width (mm)
CP correction factor
fn design stress (MPa)
fy yield stress (MPa)
F stress function defining the median fiber stress of the plate
Fm fabrication factor mean value
h flange width (mm)
k plate buckling coefficient
NOTATION
xv
Lm mean live load
Ln nominal live load
Mm material factor mean value
n number of tests.
Ncred reduced member capacity (kN)
Nl , Nd theoretical local and distortional buckling loads (kN)
Nol elastic buckling load (kN)
Ns nominal section compression capacity (kN)
Ns0.75 nominal section compression capacity based on 0.75 fy (kN)
NsRb nominal section compression capacity based on Rbfy (kN)
Pc , Pkh distortional buckling strengths (kN)
Pcr elastic buckling load of test (kN)
Pcrl, Pcrd, elastic local and distortional buckling load respectively (kN)
Pe Euler load (kN)
Pne inelastic long column buckling load (kN)
Pnl , Pnd resulting limiting strengths (kN)
Pnld nominal interaction axial strength (kN) accounting for
interaction of local and distortional buckling
Pt ultimate test load (kN)
Py squash load (kN)
Q total load
NOTATION
xvi
r radius of corner (90o) (mm) or radius of gyration of
cross-section
R radius of corner (135o) (mm)
Rn nominal resistance.
t thickness (mm)
GREEK LETTERS
γ0, γ reduction factor of radius of gyration
γD, γL load factors for dead and live load respectively
φ resistance (capacity) factor
0φ current resistance factor
cφ calculated resistance factor
1.2 OBJECTIVES OF THIS THESIS..............................................................................4
1.1 Statement of the problem
The use of high strength steels with yield stress values up to 550 MPa is
increasing rapidly, particularly for steel framed houses with sections as thin as
0.4 mm. Steels with high yield stress usually have little or no strain hardening in
the stress-strain curve, and low ductility unlike conventional structural steel that
is highly ductile and strain hardens as shown in Fig.1.1. Strain hardening is
important in the stability of thin-walled sections and so the high strength steels
are likely to have their stability significantly affected by the lack of strain
hardening.
For high strength steel sections made from thin zinc-coated or aluminium/zinc-
coated cold-reduced steel to Australian Standard AS 1397-1993, no specific
investigation has been performed. Mainly due to lack of knowledge on their
structural behaviour, the 1996 Australia/New Zealand Standard AS/NZS 4600
for Cold-Formed Steel Structures and the 2001 North American Specification
(NAS) for Cold-Formed Structural Members have generally limited the design
stress for high strength low ductility steels to 75 percent of their yield stress or
tensile strength as applicable. The NAS further restricts the use of this steel to
multiple web configurations such as sheeting and decking.
A research project on these steels in tension, which was carried out by Rogers
and Hancock (1996), has shown that they have substantially reduced ductility
but this may not affect the net section strength of perforated sections. Steels of
Chapter 1: INTRODUCTION
3
this type are similar to Structural Grade 80 steels in the USA according to the
ASTM A653 (1997) and ASTM A792 (1994) standards. A research project led
by Professor W-W Yu at the University of Missouri-Rolla to investigate the
strength of ASTM A653 steel when formed into decking sections and subjected
to bending has demonstrated that their local and post-local buckling capacities
may be significantly influenced by the lack of strain hardening. In particular, the
ultimate moments of panels with slender sections (b/t>100) were lower than the
design moments calculated based on a conventional effective section model.
However, no significant definitive testing has been performed for sections
composed of AS 1397 steel in compression. The AS 1397 steel may be zinc-
coated or aluminum-zinc coated. Those studied in this thesis were aluminum-
zinc coated similar to ASTM A792.
Normally, one of three basic types of buckling, local, overall, and distortional,
can occur in thin-walled steel sections as shown in Fig. 1.2. However, the basic
types may interact with each other. For doubly symmetric box sections, local
and overall buckling or interaction between them rather than distortional
buckling may occur. It is customary to consider that a column may buckle in
either one of two ways: (a) by plate buckling of its component webs and flanges
in shorter half-waves (local or plate buckling) or (b) by deflection of the entire
column in a half-wave of length equal to the effective column length (overall
buckling). For a given column, buckling is supposed to occur at the lower of the
two critical stresses, local or overall. In reality, however, there is an interaction
between these two modes of buckling, so that the failure stress will be smaller
than the overall buckling stresses even if the column has no imperfections.
Imperfections play a significant role in interaction buckling. Column strength is
characterized by the maximum axial force that can be supported without
excessive lateral deformations. In view of the fact that post-buckling strength of
a flat plate is available for structural members to carry additional load, cold-
4
formed steel sections are normally designed on the basis of the post-buckling
strength of the plate elements rather than based on the local buckling stress.
In practice, singly symmetric open sections, such as channel-sections, are
commonly used in cold-formed steel design. For sections of this type, especially
for sections fabricated from such thin (less than 1 mm) sheet steel, distortional
buckling can occur and becomes a significant failure mode. The wavelength of
distortional buckling is generally intermediate between that of local buckling
and overall buckling. Research into the distortional mode of buckling has
attracted considerable attention in recent years. However, as discussed in Kwon
and Hancock (1992), the design method in the AISI Specification is
unconservative for distortional buckling of channel sections composed of high
strength steel of yield stress 500MPa, and so alternative design methods are
necessary to design against distortional buckling in this case. Design rules for
distortional buckling of compression and flexural members were included in the
1996 Edition of AS/NZS 4600.
1.2 Objectives of this thesis
The main objective of this thesis is to investigate the stability of high strength
steel sections and to determine the influence of the lack of strain hardening on
their capacity to resist buckling, particularly in the inelastic range. Several
research projects have been conducted which clearly indicate that the strength of
high strength sections is reduced by the lack of strain hardening. For high
strength quench and tempered steels to ASTM A514 fabricated by welding to
form conventional box, cruciform and I-sections of moderate thickness, a
research program on local instability by Rasmussen and Hancock (1992) has
shown that the lack of strain hardening influences the strength of the sections,
5
particularly for stockier sections where the lack of strain hardening eliminates
the usual rise in strength above the squash load which is the product of the
section area and the yield stress. A related objective of the thesis is to determine
whether the 75 percent limit on yield stress as expressed in the NAS (2001) and
AS/NZS 4600 is valid for these thin AS 1397 steels, and if not, whether it can
be increased, and to what extent.
The G550 sheet steels are very thin 0.4 mm~1.0 mm and so the effect of
slenderness is important. A related objective is therefore to investigate the
influence of the slenderness of the G550 sheet steels on section stability.
1.3 Outline of this thesis
1.3.1 General
This thesis consists of four main parts: experimental investigations, numerical
investigations, theoretical investigations and the design recommendations.
1.3.2 Experimen tal investigation
A series of compression tests was performed on thin-walled box and channel
section columns. The tests were performed on pin-ended box shaped long
columns and on fixed-ended stub and channel section columns. The main
purpose of the tests was to investigate local, overall and distortional stability.
The main objective of the local stability investigation using stub columns was to
determine the adequacy of the design rules in Section 2 (Elements) of AS/NZS
4600 and Section B of the NAS (2001). The investigation of the overall stability
6
and the interaction of local and overall buckling was carried out using pin-ended
columns to improve the design rules in Section 3.4 of AS/NZS 4600 and
Section C4 of the NAS (2001). The tests performed on the lipped channel
sections were performed on fixed-ended sections mainly to investigate local
buckling, distortional buckling and the interaction between them to determine
the adequacy of Clause 3.4.6 (Distortional Buckling) of AS/NZS 4600 for high
strength steels.
An accurate instrument was designed and used to measure the geometric
imperfections of the specimens.
1.3.3 Numerical investigation
The finite element non-linear analysis program “ABAQUS” is used to simulate
the geometric and material nonlinear behaviour of the columns. Two different
models are established, one for simulating the pin-ended box section and one for
the fixed-ended channel sections. The experimental data is very important in
calibrating and implementing the finite element non-linear analysis. Once this
has been performed, the finite element non-linear analysis can be used to extend
the range of test data, and to investigate the effect of changing variables, such as
stress-strain characteristics, residual stresses, geometric imperfections and
section geometry.
The 4-node doubly curved thin or thick shell, reduced integration, hour-glass
control, finite membrane strain element, type S4R, is used. The ratio of the
length to width of each element is kept approximately 2:1. For the long
columns, different mesh densities are adopted. In the longitudinal direction of
the column, the nodes were concentrated towards the middle of the column so
that a finer mesh is obtained around the centre. In the transverse direction, the
7
finer mesh is used at the corners based on the concept of effective area. The
material behaviour data used in the ABAQUS model was obtained from the
stress-strain curves of coupon tests in tension. For the ends, two different types
of boundary conditions are used to simulate the test situation in the stub column
tests. The ends are divided into an immovable end and a movable end, which
was the loaded end. Two different ways were used to introduce geometrical
imperfections into the ABAQUS analyses. The first is to use the initial out-of-
plane deflection at mid-length of the column based on the imperfection
measurement and Walker’s (1975) suggested expression. The second is to
introduce the imperfection based on an eigenvalue buckling analysis again with
Walker's suggested expression for the amplitude.
1.3.4 Theoretica l studies using finite strip methods
The semi-analytical finite strip method of buckling analysis of thin-walled
sections is a very efficient tool for investigating the buckling behaviour of cold-
formed members in compression and bending. It assumes that thin-walled
sections buckle with simply supported ends free to warp but with section
distortion prevented at the ends. The program THIN-WALL (TW) (1998) was
developed to perform a semi-analytical finite strip buckling analysis of thin-
walled sections under axial compression and bending. It can be used to
understand the general local, distortional and overall buckling behaviour of thin-
walled sections. It is applicable to both open and closed sections and quantifies
the different buckling stresses.
In order to account for the fixed-ended boundary conditions in the tests, the
Spline Finite Strip Method (SFSM) was developed for buckling analysis by Lau
and Hancock (1986). The method uses spline functions in the longitudinal
direction and can account for a range of end conditions including fixed and free.
8
It is used in the thesis to accurately simulate the test boundary conditions.
The TW analysis gives the elastic local and elastic distortional buckling stresses
at given half-wavelengths, whereas the SFSM analysis gives the actual buckling
stress of a given section length between fixed ends.
The theoretical studies also provide valuable information on the local and
distortional buckling stresses for use in the interaction buckling studies.
1.3.5 Design recommendations using the effective width method and direct
strength method.
The effective width method is an elemental method since it looks at the
elements forming a cross section in isolation. It was first proposed by von
Karman (1932) and calibrated for cold-formed members by Winter (1947). It
accounts for post-buckling by using a reduced (effective) plate width at the
design stress. This method is used in the thesis for computing the predicted load
for all test sections.
For high strength steels, Clause 1.5.1.5 (b) of AS/NZS 4600 and Section A2.3.2
of the NAS Specification (2001), have a reduction to 75% of the yield stress. It
appears from the series of stub column tests of box shaped sections described in
the thesis that a modified reduction factor should be used to better utilize the
strength of material. So a modified reduction factor of 0.90 can be used in place
of the reduction factor 0.75, which is specified for G550 steel with the thickness
being less than 0.9 mm in AS/NZS 4600.
In order to take account of interaction of local and overall buckling of long
columns a modified design method based on the column design curve of
9
AS/NZS 4600 is proposed. A reduced radius of gyration is used in Clause 3.4.2
of AS/NZS 4600 to replace the normal radius of gyration (Section C4.1, Eq.
C4.1-1 of the NAS Specification) when the design stress exceeds the local
buckling stress.
The Direct Strength Method (DSM) was proposed by Schafer and Pekoz (1998)
and summarized by Hancock, Murray and Ellifritt (2001) who also
demonstrated its applicability. The DSM determines the strength for local and
overall (L+E) interaction and distortional and overall (D+E) interaction and
takes the lesser of the two as the strength. This method is used for computing
the test strength of the lipped channel sections. However, it does not account for
the interaction of local and distortional buckling and underestimates some of the
test strengths.
Based on AS/NZS 4600 Clause 3.4.6(b) and the Kwon & Hancock equation
(1992), two simple design methods are proposed to account for the interaction
of local and distortional buckling in thin sections of high strength G550 steel.
To determine the nominal axial strength (Pn) of the lipped channel section at
intermediate lengths, it is proposed that the interaction of local and distortional
buckling is taken into account with the distortional mode treated as an overall
mode in the DSM.
300
600
Strain
2.2 LOCAL BUCKLING, POST-BUCKLING OF PLATES AND INTERACTION OF PLATE ASSEMBLIES...................................................................................19
2.2.1 General............................................................................................................................................. 19 2.2.2 Local buckling of plate ..................................................................................................................... 19 2.2.3 Post-buckling and interaction of plate assemblies ........................................................................... 21
2.3 BUCKLING OF COMPRESSION MEMBERS.....................................................25 2.3.1 General............................................................................................................................................. 25 2.3.2 Local buckling .................................................................................................................................. 25 2.3.3 Overall buckling ............................................................................................................................... 27 2.3.4 Distortional buckling........................................................................................................................ 30
2.4 INTERACTION OF MODES...................................................................................35 2.4.1 General............................................................................................................................................. 35 2.4.2 Interaction of local and overall buckling ......................................................................................... 35 2.4.3 Interaction of local and distortional buckling .................................................................................. 38
2.1 RESEARCH ON HIGH STRENGTH STEEL
The use of high strength steels is increasing rapidly. The combination of greater
strength and low cost leads to advantages in many fields of industry including
house construction. Cold-reduced steels with high yield stress usually have little
or no strain hardening in the stress-strain curve, and low ductility unlike
conventional structural steel that is highly ductile and strain hardens as shown in
Fig. 1.1. Strain hardening is important in the stability of thin-walled sections
and so the high strength cold-reduced steels are likely to have their stability
significantly affected by the lack of strain hardening.
In past decades, many papers & research reports have been published on high
strength steels. Although, in recent years high strength steels have become
available, some standards & specifications have limited the use of some high
strength steels due to their lack of ductility. For example, the 1996
Australia/New Zealand Standard AS/NZS 4600 for Cold-Formed Steel
Structures and the 1996 American Iron and Steel Institute (AISI) Specification
for Cold-Formed Structural Members have limited the design stress for high
strength cold-reduced steels to 75 percent of their yield stress or tensile strength
as applicable. The AISI Specification has recently been revised in Supplement
No.1 (1999) to allow values higher than 75 percent for multiple web
configurations, the value depending mainly on plate slenderness.
Priest and Gilligan (1954) re-examined the design specifications for structural
Chapter 2: LITERATURE REVIEW
14
carbon steel, particularly in regard to buckling and elastic stability. The essential
principles of structural design were discussed and some formulas, charts and
tables were developed to assist engineers in designing for high strength steels.
Dhalla et al. (1971) investigated the influence of low-ductility on the tension
and connection behaviour of cold-formed members under static loading. A
series of coupon and connection tests were carried out. The concepts of local
ductility and uniform ductility were introduced. Most of the low ductility steels
investigated showed significant local ductility, but very limited uniform
ductility. Dhalla and Winter (1974) suggested alternate approaches for
measuring local and uniform ductility and further presented ductility criteria to
ensure satisfactory structural performance of thin steel members under
essentially static load.
εun ≥ 3.0%
For local elongation in a ½ in. (12.7 mm) gage length across the fracture
ε1/2 ≥ 20%
σu / σy ≥ 1.05
Maricic (1979) presented briefly the analysis and examples of galvanized high
strength steels. The analysis showed that the thinner the steel, the lower the
elongation will become. Currie (1989) examined the developments in research
and design of light gauge cold-formed steelwork used in the construction
industry, and reviewed the applications of these new developments in design
standards around the world.
Hancock et al. (1987) presented in detail the tests of thin-walled high tensile
15
steel columns, fabricated from nominal 5 mm thick Grade 350 hot rolled steel
plate. Strength tests of thin-walled high tensile steel columns consisted of
welded I-sections, welded channel sections and cold-formed square hollow
section. The use of these test strengths for selection of column curves in the
Australian Steel Standard AS 4100 (1998) for columns where local and Euler
buckling interact was explained.
Macadam et al.(1988) conducted a series of coupon, beam and column tests to
characterise the material such as the determination of local and uniform
elongation and to determine the effect of the low strain hardening on the
behaviour of members. All the specimens were fabricated from the high yield
strength carbon steel with low strain hardening. The coupon tests showed that
O-steel (original) had the ratio (σu /σy) of about 1.4, an uniform elongation of 4
to 6 percent and a total elongation of 15 percent, for R-steel (further reduced)
had the ratio (σu /σy) of about 1.01 to 1.02, an uniform elongation of 0.8 to 2.7
percent and the total elongation of 9 or 10 percent. The column tests showed
that the results for the intermediate length columns were predicted
unconservatively. As a result of their work, it was concluded that the ductility
requirements in the specification should be amended to permit the use of the
low-strain-hardening ductile steel, but that its application be limited to use as
flexural members.
Pan and Yu (1988) presented tests of sections with unstiffened elements
fabricated from materials with yield strengths from 581 to 1057 MPa. Modified
effective width formulae were proposed to predict the post-buckling strengths of
unstiffened elements with high yield stresses. The test results showed that the
yield strengths in tension and compression were not significantly different.
Rasmussen and Hancock (1992) studied the local instability of high strength
16
quench and tempered steels to ASTM A514 fabricated by welding to form
conventional box, cruciform and I-sections of moderate thickness. The coupon
tests showed that the ratios of tension and compression 0.2% proof stress are
about 0.89 and 0.98 for 5 and 6 mm thickness steels respectively. The analysis
showed that the lack of strain hardening influences the strength of the sections,
particularly for stockier sections where the lack of strain hardening eliminates
the usual rise in strength above the squash load which is the product of the
section area and the yield stress.
Bernard et al. (1992) performed a series of deck tests with intermediate
stiffeners. The decks were fabricated from steel sheets conforming to AS 1397
Grade 550 steel sheets with 0.60 mm nominal thickness. These steels are similar
to ASTM A653 Structural Grade 80 steel. The effects of stiffener size on the
ultimate moment capacity and buckling modes on the loss of section stiffness
were studied. The local and distortional buckling modes were investigated and
design rules proposed.
Kwon and Hancock (1992) conducted a series of compression tests on lipped
channel sections. All specimens were formed from a cold-reduced zinc-coated
steel conforming to AS 1397 grade G500 with a total coated thickness of 1.2
mm. The tests were carried out between fixed ends and investigated post-
buckling in the distortional and mixed local-distortional modes. The tests by
Bernard et al, and Kwon and Hancock were performed on sections for which the
yield stress was significantly higher than the distortional buckling stress and so
a substantial post-buckling reserve of strength occurred.
Rasmussen and Hancock (1995) presented a test program on long columns
fabricated from high strength steel plates with nominal yield stress of 690 MPa.
13 box and I-section specimens were tested. The analysis showed that the effect
17
of residual stresses was less detrimental to the strength of high strength steel
columns than to the strength of ordinary steel columns.
A research project led by Professor W-W Yu at the University of Missouri-
Rolla (1995, 1996) investigated the strength of ASTM A653 Structural Grade
80 steels when used as decking. The tension coupon tests indicated that the
0.2% offset yield and tensile strengths of the steel both in the rolling direction
and perpendicular to the rolling direction increases with a decrease in the
thickness of steel sheets, while the ductility of the steel decreases with the
decrease in the thickness of steel sheets. When the steel sheets were formed into
decking sections and subject to bending, they demonstrated that their local and
post-local buckling capacities may be significantly influenced by the lack of
strain hardening. In particular, the ultimate moments of panels with slender
sections (b/t>100) were lower than the design moments calculated based on a
conventional effective section model.
Rogers and Hancock (1996, 1997a) conducted a series of tensile coupon tests on
perforated & unperforated specimens. High strength G550 sheet steels which
ranged in base metal thickness from 0.40 to 0.60 mm were tested as tensile
coupons with various size and shape of perforations. The test results indicated
that the G550 sheet steels do not meet the Dhalla and Winter (1971, 1974)
material requirements regardless of direction, except for uniform elongation in
the longitudinal test specimens but this may not affect the net section strength of
perforated sections and the net cross-section ultimate tensile strength can still be
adequately predicted using current design provisions. Rogers and Hancock
(1997b) also investigated bolted connections of thin G550 and G300 sheet steels
in 0.42 mm and 0.60 mm thicknesses. The test results indicated that the
connection provisions in the AISI Specification (1996), Eurcode and AS/AZS
4600 cannot be used to accurately predict the failure mode of bolted
18
connections constructed using G550 and G300 steels. The design rules cannot
be used to accurately determine the bearing resistance of bolted test specimens
based on a failure criterion for predicted loads.
Earls and Galambos (1998) presented numerical studies on HSLA80 (High
Strength Low Alloy) wide flange beams. Two distinct inelastic flexural modes
emerged as the dominant modes occurring at failure in HSLA80 beams under
moment gradient: little out-of-plane deformation accompanying localized
buckling which occurs close to the mid-span stiffener and a great deal of out-of-
plane deflection associated with the regions near localized buckling in the
compression flange. The study showed that the structural ductility of the beams,
as quantified by plastic hinge rotation capacity, is very much dependent upon
which of the two mode shapes governs at failure and concluded that the design
specifications which are based on the behaviour of mild carbon steel will not be
applicable to the design of HSLA80 flexural members.
Hancock (1997, 2003) reviewed the changes of the main Standards and
Specifications in the world. Major research developments around cold-formed
steel structures were summarized. Gresnigt and Steenhuit (1997) gave an
overview of the developments towards higher strength structural steels, which
focused mainly on the research done in Europe and the consequent standards
and codes of practice. Hancock and Rogers (1998) summarized the research
performed at the University of Sydney on ductility of cold rolled high strength
steel sections and distortional buckling and reviewed the material standards
specified in the AISI Specification 1997, Eurocode 3 Part 1.3 (1996) and
AS/NZS 4600:1996.
INTERACTION OF PLATE ASSEMBLIES
2.2.1 General
A plate element subjected to compression, bending, shear or a combination of
these stresses in its plane may buckle locally or distort at a low stress level
(local buckling stress). Although the local buckling load may not be the design
basis since the post-buckling load can be much greater, local buckling may have
a very significant effect on the member strength, especially for members which
suffer interaction of local and overall buckling or local and distortional
buckling. Therefore, the local buckling behaviour, post-buckling behaviour and
interaction between plate elements have to be considered in structural design.
2.2.2 Local buck ling of plate
Bryan (1891) first discussed the problem of the buckling of a simply supported
thin-rectangular plate uniformly compressed in the longitudinal direction and
obtained the solution from the fundamental differential equation for the
deflection of the plate. Timoshenko (1936) discussed the stability of plates
under various conditions of support at the two edges parallel to the longitudinal
compressive forces and showed the application of the theory to the investigation
of the plate elements of the steel columns. The general soution based on Bryan’s
equation for the elastic critical local buckling stress (fol) is given by:
2.1
where k is a plate buckling coefficient which depends on the support conditions.
Lundquist and Stowell (1942) assumed that restraint among structural members
is provided by a specially defined elastic restraining medium and derived a
formula for the critical compressive stress at which buckling may be expected to
occur in outstanding flanges and presented a general chart for determining the
value of k.
This theory was applied to the member cross-sections which are composed of
various connected elements by many researchers (Stowell et al. 1952, Bleich,
1952). Stowell et al. (1952) presented the calculation of the buckling stresses of
flat unstiffened plates and integral flat-plate sections and gave the buckling
stresses in the form of theoretical charts. Bleich (1952) presented the
relationship between k and the plate aspect ratio in graphical form and gave a
generalized form of the theory for critical local buckling stress. The equation for
the elastic critical buckling stress was modified by the plasticity reduction factor
and varied with the type of loading and support condition.
A generalized approach to the local instability based on the small deflection
theory of plate bending was developed by Chilver (1953). Application of this
method to any particular plate resulted in the derivation of the exact elastic
instability stresses. This method was used to deal with the flexural and torsional
effects of reinforcing flanges of struts with open section.
Timoshenko and Gere (1961) presented the values of k for rectangular plates of
high aspect ratio with different types of boundary conditions and stresses. More
cases of buckling were considered and more tables for calculating critical
stresses were added.
Bulson (1970), in his well-known book, summarized the available work on plate
buckling and extended it further. The values of k and the critical stresses of
unstiffened and stiffened rectangular plates with various loading and boundary
conditions were given.
2.2.3 Post-buckl ing and interaction of plate assemblies
Local buckling causes a loss of stiffness and a redistribution of stresses.
Uniform edge compression in the longitudinal direction results in a nonuniform
stress distribution after local buckling. The buckled (called post-buckled) plate
has the maximum stress occurring at the longitudinal edge supports and the
minimum at the centre. The redistribution of stress normally continues until the
stress at the edge reaches the yield point of the steel and then the plate begins to
fail. The post-buckled plate can carry further load. In some cases, yielding may
occur at the centre of the plate as a result of combined bending & compression.
The post-buckling behaviour of a plate can be analyzed by using large
deflection theory. Schuman and Back (1930) presented the experimental proof
to show that the ultimate load was higher than the buckling load, which
previously assumed that the buckling load was the highest load a plate could
carry.
The differential equations for large deflection buckling of a plate were
introduced by von Karman in 1910.
)2(2 2
∂ ∂ ωωω 2.2b
where F is a stress function defining the median fiber stress of the plate
22
The solution of the differential equation is too complicated to be used for
practical design. von Karman (1932) presented a theoretical study of a simply
supported rectangular plate. A method for determining the post-buckling
strength was developed based on the concept of ‘effective width’ which is
illustrated in Fig. 2.1.
= 2.3
where be is the effective width, fcr is the buckling stress, fy is the yield stress; in this method, it
is assumed that the total load is carried by a fictitious effective width b.
Based on many tests and studies of post-buckling strength of the long plates that
are stiffened along both longitudinal edges, such as webs of channels and I-
beams, Winter (1948) presented a modified formula for calculating the effective
width be.
b t
f Etbe −= 2.4
Coan (1951) studied the post-buckling of a simply supported rectangular plate
with a small initially curvature. Boundary conditions were stress free supported
edges and uniformed displaced loaded edges. Levy’s series solution of von
Karman’s compatibility equation taken in conjunction with the energy method
replacing the von Karman’s equilibrium equation was used to calculate the post-
buckling response assuming the nonuniform edge displacements that are
characteristic of stress free edges. Yamaki (1959) extended Levy’s and Coan’s
work to the nonlinear problem which was solved extensively under various
boundary conditions combining two kinds of loading and four kinds of
boundary conditions. Numerical solutions were obtained for the deflection, edge
shortening and effective width of a square plate in compression. Formulae for
the ultimate load of a square plate were derived by using the maximum shear
theory for the beginning of yielding.
Walker (1968) used the von Karman large deflection equations to carry out the
analysis of buckling and post-buckling of single plates and assumed that a short
channel column may be treated as a collection of individual plates connected
appropriately along their common edges. The analysis provided an engineering
estimate of the maximum load capacity of a channel. Walker (1969) analyzed
the post-buckling behaviour of flat square plates loaded along two opposite
straight edges using the von Karman equations, trigonometric series and
Galerkin’s method. The effects of initial geometric imperfections were studied.
A perturbation method was used to solve approximately the non-linear algebraic
equilibrium equations occurring in the analysis of uniformly compressed square
plates.
Abdel-Sayed (1969) presented an approximate theoretical approach for the two
cases of plates where the longitudinal edges are restrained to remain straight or
are free to move in the plane of the plate. The effective width of a wide thin
plate, under compression in its plane, was examined by solving the von Karman
governing differential equations. Formulas showed the effective width of the
plate to be reduced when there is a small deviation from flatness or the edges
parallel to loading are free to move in the plane of the plate.
Dawson and Walker (1972) studied the post-buckled plate behaviour and the
effects of different generalized geometric imperfection parameters on the
expression for the ultimate load. The explicit expressions for the collapse, end
shortening and stiffness of simply supported plates with stress free edges were
24
derived.
Walker and Murray (1975) described the manner plates buckle and studied the
behaviour of analogous mechanisms consisting of rigid links and springs. Based
on these studies, a plate mechanism was derived. The analysis showed that
membrane elastic energy plays a significant role in determining the post-
buckling behaviour of a thin plate.
Kalyanaraman et al. (1977, 1978, 1979) presented the experimental and
analytical investigations on the local buckling of unstiffened compression
elements in the elastic range. The effects of initial imperfection and rotational
edge restraint on the local buckling of compression elements were considered.
Rhodes (1981) presented an approach to the analysis of elastic-plastic behaviour
in plates subjected to uniaxial compressive loading. The general approach was
to base the analysis on equations derived for perfect linear elastic plates and to
use simple routine modification procedures to take into account material
yielding.
Based on the studies of Pekoz (1986a), the 1986 Edition of the AISI
Specification uses the following equation in the calculations of uniformly
compressed stiffened elements.
where ρ=(1-0.22/λ)/λ 2.7
)/)(/)(/052.1( EftwK=λ 2.8
An unstiffened element has one longitudinal edge free and one supported.
Despite behavioural differences, the same method for stiffened elements can be
adopted for unstiffened elements. The elastic local buckling and post-buckling
behaviour of unstiffened elements was studied by many investigators. Pekoz
(1986b) proposed that Winter’s effective width equation could be used for
unstiffened elements providing the appropriate value (k) for the buckling
coefficient was adopted.
2.3.1 General
Steel sections may be subject to one of three basic types of buckling mode:
local, overall (Euler or flexural-torsional) or distortional buckling. Local
buckling is particularly prevalent in cold-formed sections and is characterized
by relatively short wavelength buckling of individual plate elements. Overall
buckling is a long wavelength mode in which the plate elements forming cross
sections undergo significant translations without cross-seectional distortion.
Distortional buckling is buckling which takes place as a consequence of
distortion of the cross section. The wavelength of distortional buckling is
generally intermediate between that of local buckling and overall buckling as
shown in Fig. 1.2.
2.3.2 Local buck ling
26
Local buckling of plate elements of columns represents a particular case of plate
instability where the adjacent plates forming the section buckle at the same
stress. The column design has to take into account the stability of the plate
elements. Bleich (1952) presented a theory to demonstrate the fundamental laws
which control the behaviour of compressed plates under various conditions of
restraint to which the plate element of a column are subject. The theory provides
the basis for reliable rules to compute the required thickness of plates for
practical purposes rather than requiring a tedious investigation of the condition
for the occurrence of local buckling in each individual case. A coefficient of
restrain (ζ) was introduced to study the rectangular plate. Design formulae for
the required thickness of plate elements of columns were given in order to
prevent premature failure of compression members by local buckling.
Walker (1966) studied plates and channel sections subjected to eccentric
compression by means of an approximate solution using the Ritz-Galerkin
method. In order to determine the channel load-bearing characteristics, the
corresponding plate properties were determined and combined to obtain
compatibility conditions along the adjoining edges. An approximate series
method was used to calculate the buckling loads of flat plates which are
subjected to linearly-varying compressive-load actions in the plane of the plate
along two opposite edges.
Reiss and Chilver (1968) studied the local buckling of columns using some
researchers’ test results of different shape sections. The study showed that the
assumption that strength of a section is the sum of the separate strengths of the
component plates, assuming these are simply-supported or free on the
longitudinal edges, may be in error on the unsafe side by as much as 20% for
some sections.
Venkataramaiah and Roorda (1978) presented a generalized method of
theoretical analysis for the problem of local buckling of thin walled channel
sections under eccentric compression. The theoretical analysis finds critical
stresses of the web and flange elements at their common edge by using the
criterion of vanishing combined stiffness at this edge.
Parks et al. (1988) investigated the local buckling of both stiffened and
unstiffened curved elements through the use of short stub column tests. Three
different curvatures of stiffened and unstiffened curved elements were formed
from high strength steels. The empirical expressions were developed for
predicting the local buckling stress.
Landolfo et al. (1999) conducted a series of stub column tests to study the
ultimate strength of aluminum alloy channels. The local buckling of internal and
outstand plate elements constituting the member cross section was analyzed by
considering the restraining action derived from their interaction. The test results
were used to investigate the degree of accuracy of the effective thickness
approach.
2.3.3 Overall bu ckling
A slender axially loaded column will tend to fail by overall buckling with
different type of buckling for different cross-section: flexural buckling, torsional
buckling and torsional-flexural buckling. Doubly symmetric shapes and closed
shapes may fail by overall flexural buckling. Singly symmetric shape columns
may fail by either flexural or a combination of flexural and torsional buckling.
Point symmetric shape columns may fail by flexural or torsional buckling. In
the elastic range, the elastic critical buckling stress (foc) for a long column can
be determined by the Euler formula. Two methods have been used for
28
determining the inelastic critical buckling load: the tangent modulus method and
the reduced modulus method.
where r is the radius of gyration of cross-section
Bleich (1952) gave a good prediction of the inelastic critical flexural buckling
stress (fcr) using the tangent-modulus theory together with the Euler formula for
critical flexural buckling stress. Three simultaneous differential equations of
buckling by torsion and flexure in their most general form were given. These
equations may be applied to columns with pinned, fixed, or free ends as in the
usual theory of buckling.
Chilver (1961) discussed the basic material properties of cold-formed steel
sections and of the geometry of formed sections. The flexural and torsional
properties of open thin-walled sections were reviewed. The strengths of tension
members, columns and beams were studied. A torsional-flexural buckling
parameter, which defines the mode of buckling and the buckling load, was
suggested. Vlasov (1961) and Timoshenko (1961) developed a general theory of
flexural-torsional buckling. Trahair (1993) provided an up-to-date treatment of
modern methods of analysis of flexural-torsional buckling and a detailed
summary of knowledge on flexural-torsional buckling.
A general formula proposed by Chajes and Winter (1965) investigated the
torsional-flexural buckling of centrally loaded thin-walled members and
presented a simple method of accounting for torsional-flexural buckling. Based
on the use of an interaction type of equation, a relatively uncomplicated and
straightforward procedure for determining the torsional-flexural buckling load
was described. The basic theory was limited to hinged and fixed end columns
but can be extended to include other than fixed or hinged boundary conditions
by using an effective length. Hone (1967) carried out a detailed analysis of the
differential equations of torsional-flexural buckling and established the critical
solutions for three types of end conditions. The solutions were converted into a
suitable non-dimensional form and various parameters were used in a series of
design charts. Barta (1967) reviewed the development of fundamental research
on the elastic flexural-torsional buckling of thin-walled bars and discussed the
fundamental differential equations governing the problem. The Chwalla (1950)-
Witte (1957/59) formula was improved and simplified. Two methods, which are
the energy method and the simplified finite-difference approach, were used to
solve the simplified equation.
Pekoz and Winter (1969) studied the general behaviour of thin-walled singly
symmetric open sections under eccentric axial loading on the plane of
symmetry. The analysis showed that for a given shape, depending on
eccentricity and slenderness, a variety of behaviour modes is possible. The
effect of pre-critical deflections and other modifications were introduced to the
basic theory to analyze these modes.
Wang (1986) presented a rational method for determining the torsional-flexural
buckling load for thin-walled columns of open cross sections to deal with the
battened and unbattened members under concentric or eccentric load. The
influence of pre-buckling deflection was taken into consideration.
Yang (1986) derived the differential equations of equilibrium for the stability of
thin-walled beams of an arbitrary open cross section. Various nonlinear effects
due to the geometric change were considered. These equations can be used to
study various torsional-flexural buckling problems of a thin-walled bar.
Seah and Rhodes (1990), studied the behaviour and the strength of edge-
stiffened beam sections which were subjected to pure bending and bent in such
a way that the stiffeners were in compression. Relatively long edge stiffened
equal-flanged channel beams were tested under four point bending. The
behaviour for lateral torsional buckling was investigated in the elastic range.
Two series of tests on edge stiffened beam sections of various geometries were
described. Design procedures were also proposed based on the effective width
concept and theoretical findings for prediction of the ultimate strength of the
edge stiffener beam with a limiting plate width to thickness ratio of 60. Seah
and Khong (1990) further compared their results with those obtained by semi-
analytical, semi-numerical approach adopting the Rayleigh-Ritz method based
on the energy principle and found close agreement between them.
2.3.4 Distortiona l buckling
Thin-walled channel sections and other mono-symmetric sections may undergo
a distortional buckling mode which is the lip-stiffened flange of the section
rotating about the flange-web junction or the lip of the section rotating about the
lip-flange junction.
Sharp (1966) reported some experimental data on buckling of hat sections
formed from aluminum sheet and summarized available information on the
strength and behaviour of plates with stiffeners oriented parallel to the direction
of applied loads. Three buckling modes were identified. As stated in the paper,
when the rigidity of restraining elements to a flanges becomes large, a torsion
buckling analysis that does not include the effects of flange flexibility becomes
highly unconservative. Therefore, an approximate modification to the restraint
31
term, which is rotational restraint offered to flange by other elements of the
section, was given.
Williams and Wittrick (1972) presented the numerical results for the buckling,
under uniform longitudinal compression, of a series of panels with unflanged or
flanged integral stiffeners, or with Z-section stiffeners. The results showed that
torsional buckling modes contain a substantial amount of deformation of the
stiffener cross-section and the assumption, which the cross-section does not
distort, can lead to considerable over-estimation of the torsional buckling stress.
Thomasson (1978) conducted a series of tests on the lipped channel sections
with/without stiffeners in the web. The interaction between different modes was
studied. The torsional mode of the stiffener, which is characterized by the fact
that the flange of the primary stiffener is displaced sideways at the same time as
the wide flange is deformed, was investigated. The associated critical load was
designated a torsional buckling load.
Hancock (1978) gave a definition of distortional buckling mode, which was
considered as a mode occurring at a half-wavelength between local and lateral
buckling. The finite strip method was used to study the local, distortional and
flexural-torsional buckling of I-beams bent about their major axis. The estimate
of the distortional buckling load for the beam subjected to continuous lateral
and torsional restraint on the tension flange was compared with the values
produced using Bleich’s “pony truss chord model”.
Desmond et al. (1981a) tested a variety of edge stiffened channel sections with
edge stiffening lips turned inward or outward at an angle to the flange. Two
interrelated yet fundamentally different buckling modes characterized the
behaviour of edge-stiffened elements; namely the stiffener buckling mode (more
32
recently called distortional buckling) and the local plate buckling mode (local
buckling). The local buckling interaction between web and flange elements was
not fully investigated, both the analytical and experimental work were limited to
those assemblies in which local instability is initiated in either the flange or the
stiffener.
Sridharan (1982) developed the finite strip method to study post-buckling in the
local-torsional (now called distortional) mode. The examples illustrated that one
likely consequence of buckling of an edge stiffener in its own plane would be of
plastic yielding. The yielding of a member which has been the main source of
stiffness cannot but hasten the collapse of the structure.
Mulligan and Pekoz (1984) tested a series of channel sections in either
concentric or eccentric loading in order to study the local buckling on the
overall buckling behaviour. General failure modes for the test specimens were
gradual lateral deflection without twisting of the cross section. Several
specimens exhibited significant local-torsional (now called distortional)
buckling rather than the interaction mode between local and overall buckling. A
simple limiting stress approach was proposed. To prevent the local-torsional
buckling mode, Mulligan and Pekoz proposed an effective section method with
the stress at the flange-stiffener junction being limited to the local buckling
stress with k=4.0. The proposed method was shown to be conservative when
compared with their limited test data on the local-torsional mode.
Hancock (1985) carried out a series of tests to study the effect of the distortional
mode on cold-formed lipped channel sections with rear flanges using the
material with thicknesses 1.6 & 2.0 mm and the yield stresses 523 & 487 MPa.
The distortional stresses were from 375 to 180 MPa. The experimental tests
showed that there was very little post buckling strength available for distortional
buckling mode probably because the sections yielded.
Lau and Hancock (1987) derived explicit analytical expressions to predict the
distortional buckling stress of thin-walled channel sections columns with a
range of section geometries. The rigorous analytical expressions were first
developed for an approximate model using the flexural-torsional buckling
theory of undistorted thin-walled columns.
Lau and Hancock (1988, 1989) tested a series of channel columns of different
section geometries, thicknesses and steel strength grades under uniform
compression in a fixed-ended boundary conditions. The section geometries
consisted of lipped channels, hat sections and storage rack sections. The lengths
of the test columns ranged from short stub columns which failed mainly in
inelastic local buckling to long columns which failed in the elastic or inelastic
flexural-torsional buckling mode. Based on the test results, two different design
methods using the effective width equation were proposed for the distortional
buckling and yielding.
Kwon and Hancock (1992) performed a series of compression tests on lipped
channel sections with/without intermediate stiffeners in the web and
investigated post-buckling in the distortional and mixed local-distortional
modes. The section geometry and yield strength were chosen to ensure to ensure
that a substantial postbuckling strength reserve occurred in the distortional
mode. Material with 1.0 mm thickness and 590 MPa yield stresses was used.
The distortional buckling stress (from 88 to 45 MPa) was much less than the
yield stress. The experimental tests showed a significant post-buckling strength
reserve in the distortional mode, even when local and distortional buckling
occurred simultaneously.
Davies and Leach (1994) introduced the second-order terms associated with
geometric nonlinearity into the basic equation of Generalized Beam Theory.
Simple explicit equations for the load to cause buckling in individual modes
under either axial load or uniform bending moment were derived. The explicit
procedure showed how to consider the linear interaction between local,
distortional and global buckling modes.
Schafer (1997) used finite strip and finite element analysis to demonstrate that
the distortional mode has greater imperfection sensitivity than local modes. The
results showed that distortional failure has a lower post-buckling strength than
local failures. Schafer and Pekoz (1999) developed new hand methods to predict
the critical buckling stress in both the local and the distortional mode.
A new design procedure for cold-formed steel members called the ‘Direct
Strength Method’ (DSM) was proposed by Schafer and Pekoz (1998) and
summarized by Hancock, Murray and Ellifritt (2001) who also demonstrated its
applicability. The method employs elastic buckling solutions for the cross-
section, instead of the element-by-element plate buckling solutions used in
traditional design. The method uses the entire cross-section in elastic buckling
determination and incorporates local, distortional and global buckling into the
design process.
Prola and Camotim (2002a, 2002b) presented investigations on the elastic
distortional post-buckling behaviour cold-formed lipped channel steel columns
and beams using spline finite strip analyses. The analysis showed a post-
buckling asymmetry phenomenon with respect to the cross-section initial
distortions.
2.4.1 General
To determine the capacity of a cold-formed member, consideration must be
given to local, overall and distortional behaviour. However, interaction between
buckling modes may result from the occurrence of simultaneous or nearly
simultaneous buckling loads and generally produces adverse effects on the
strengths of members with slender plate elements. The interaction between
buckling modes has been studied extensively over decades.
2.4.2 Interaction of local and overall buckling
Research into the interaction of local and overall buckling has been conducted
by many researchers. Bijlaard and Fisher (1952, 1953) analysed thin plated
columns of square or H-type cross section and produced some closed form
solutions using their semi-intuitive method of ‘split rigidities’ in conjunction
with the principle of virtual work. The tests showed that these columns buckled
elastically in a flexural mode at a load higher than the local buckling load but
less than the Euler buckling load. Considerable interaction buckling effects were
demonstrated.
Graves-Smith (1967) investigated the behaviour of locally buckled box section
columns and examined their load carrying capacity using an elastic-plastic
analysis. The column was assumed initially straight and a short length was
analyzed using a Rayleigh-Ritz energy method. The interaction between local
plate deformations and the spread of plasticity was treated on an approximate
basis because of the lack of a general relationship between stress and strain in
the plastic plates. The reduction in strength of longer columns caused by overall
36
buckling was assessed by obtaining the apparent internal bending stiffness of
the locally buckled section.
van der Neut (1969) studied initially straight columns containing only local
imperfection and columns having both local and overall imperfections. The
study showed in the case of an idealized column severe imperfection-sensitivity
when the ratio of Euler load to local buckling load is about one but the
imperfection of the column axis appears to have a minor effect upon the load
carrying capacity. van der Neut (1973) gave a correction and re-evaluated in the
case of an idealized column the reduction of the failure load due to the
imperfection of the column axis. The analysis showed that whereas initial
waviness of the composing thin walls reduces the buckling load mainly when
the ratio (R) of Euler load to local buckling load is about one, the imperfection
of the column axis has its strength reducing effect over a much wider range of R,
more in particular in the range R>1.
Skaloud and Zornerova (1970) carried out a series of column test and concluded
that the buckling of the plate elements diminished the effective section of the
column and consequently accelerated the column’s flexure. Interaction between
local and overall buckling was observed.
DeWolf et al. (1974) studied the behaviour of cold-formed columns subject to
local buckling and presented an analytical approach to account for the combined
effects of local buckling, column buckling, and nonuniform material properties
in compression members which was based on the tangent modulus concept and
the effective width expression.
Gibert and Calladine (1974) used a graphical method to extend Calladine’s
work and to make an exhaustive study of the combined effects of both local and
37
overall imperfections. The effects of imperfections were well described by the
celebrated Perry formula in conjunction with a single imperfection parameter
compounding simply the local and overall imperfections. The reduction of the
peak load due to local and overall imperfections together was much less than the
sum of the separate effects.
Little (1979) discussed the problem of predicting theoretically the strength of
square thin-walled steel box columns which are pin-ended and loaded centrally
at the ends. A general method for generating design curves for the local-flexural
interaction situation were described. The analysis showed that in a local-flexural
interaction situation, the stocky column strength is not usually the same as that
stress-value which, if reached in the critical element, is sufficient to cause rapid
failure of more slender columns. This is in contrast to the behaviour of simple
columns, for which both these values are simply the yield stress.
Hancock (1981a, 1981b) proposed a simple design method for I-columns which
are subjected to both local and flexural buckling phenomena. The finite strip
method was extended to include the nonlinear response of imperfect plate strips
under longitudinal compression and proposed a method to calculate the
reduction in the flexural buckling load of imperfect box and I-section columns
in the region of the local buckling load. For the box columns rather than for I-
section columns, the reduction calculated using the effective width finite strip
analysis agreed closely with the calculated using the effective width formula.
Bradford and Hancock (1984) presented a nonlinear finite strip method for the
post-local buckling of geometrically imperfect plate assemblies. The method
was used to provide an accurate alternative to the Winter effective width
formula for obtaining the effective section of a simply supported I-beam in the
post-buckling range.
38
Mulligan and Pekoz (1984) presented an effective section method for analysing
the effects of local buckling on the overall buckling modes of behaviour of
singly symmetric thin-walled columns and beam-column. The method
recognized the post-local bucking of the component plate elements and the
associated shift of the centroid.
Toneff et al. (1987) studied the interaction of local and flexural or flexural-
torsional buckling modes by adopting a finite element for a thin-walled beam-
column of arbitrary cross sections to include local degrees of freedom. The
method allowed distortion of the cross section.
Sridharan and Zeggane (2000) studied the interaction of local and overall
buckling in plate structures and stiffened shells using a specially formulated
shell element. Amplitude modulation, a key feature of the interactive buckling,
was incorporated in the element formulation. A procedure was outlined for
incorporating the key secondary local mode in the interactive buckling model.
Young and Rasmussen (2000) described a technique for determining the overall
flexural and flexural-torsional bifurcation loads of locally buckled cold-formed
channel columns. The inelastic nonlinear finite strip buckling analysis was used
to determine tangent rigidities of a locally buckled section which were used in
the overall flexural and flexural-torsional equations.
2.4.3 Interaction of local and distortional buckling
The interaction between local and distortional bucking modes, and distortional
and flexural-torsional modes has not been widely investigated. Hancock (2002)
summarised the research on the mode interaction and categorized the mode
interaction into: Linear interaction and Non-linear interaction. Linear
39
interaction is the interaction of the two modes which occurs at the same half-
wavelength and Non-linear interaction is that where the short half-wavelength
mode (usually local) occurs over multiple half-wavelengths and interacts with a
long half-wavelength mode by reducing its effective stiffness against buckling.
Therefore, the following review of the researches on the mode interaction can
be classified according to this definition.
2.4.3.1. Linear interaction buckling
Williams and Wittrick (1972) showed the coupling between the different modes.
The coupling between different modes was weak if these predominantly
consisted of the local or overall mode. When the torsional mode was dominant,
the effect due to coupling between the modes can be considerable.
Davies and Jiang (1996) presented an analysis on thin-walled columns with the
properties typically used in practice using GBT and showed that the GBT can
provide a particularly appropriate tool to analyze distortional buckling in
isolation and in combination with other buckling modes. For a column of
symmetrical cross-section, there is no modal interaction between symmetric and
asymmetric buckling and the degree of interaction with other bucking modes
depends primarily on the buckling half-wavelength. For distortional buckling in
columns, it is generally sufficient to consider a single buckling mode involving
symmetric rotation of the flanges about the flange/web junctions.
2.4.3.2. Non-linear interaction buckling
Pekoz (1991) presented an overview of research on thin-walled steel and
aluminum structures conducted at Cornell University. The first part of the paper
was aimed at developing sufficiently accurate and simple design formulations
for laterally unsupported cold-formed steel compression flanges. Possible
failure modes were illustrated in the paper. The interaction of local and
distortional buckling was discussed.
Kwon and Hancock (1992) conducted tests on sections of high strength steel for
which the buckling stresses were significantly less than the yield stress of the
material. Although a significant post-buckling strength reserve was observed, no
adverse interaction between local and distortional buckling occurred for these
sections with thickness down to 1.0 mm.
Serrette and Pekoz (1995a) discussed an analytical method that was formulated
for estimating the elastic distortional buckling stress of flexural members with
laterally unsupported compression flanges. The elastic buckling finite strip
program was used to study the interaction of local and distortional buckling.
The analysis showed that for some specimens, the interaction resulted in a
significant decrease in buckling stress. However, the relationship between the
degree of the interaction of buckling modes and the specimen of geometric
properties could not be identified from this study. Serrette and Pekoz (1995b)
presented the results from two experimental studies and proposed two design
recommendations for the interaction between local and distortional buckling in
standing-seam panels. The methods differ only in the procedure by which the
elastic distortional buckling stress is computed.
2.4.4 General st udies on mode interaction
Two different approaches have been adopted to predict post-buckling
phenomena. One approach is to make use of the concept of effective width to
account for the post-buckling strength of the locally buckled component plates.
The representative papers are such as Loughlan and Rhodes(1979), Hancock
41
(1981a, b), and Bradford and Hancock (1984) which are based on the empirical
formula suggested by Winter and the nonlinear finite strip analysis. Another
approach is on the basis of the general Koiter theory (1945). Hereafter, a brief
review on the second approach is given.
The general post-buckling theory of elastic structures was described by Koiter
(1945). The concept of initial imperfections was introduced to explain the
significant reductions of the critical load and the discrepancy between linear
theories and experiments on shells. Graves-Smith (1967) and van der Neut
(1968) first studied the interaction between local and Euler buckling in thin-
walled compression members by using mathematical models which included the
nonlinear membrane stiffness of the plates forming the columns. After those
pioneering works, the phenomenon of post-buckling was extensively studied by
many researchers who include Tvergaard, Byskov and Hutchinson, Sridharan
and his co-workers, Pignataro and his co-workers, and Kolakowski and his co-
workers.
Tvergaard (1973) studied the initial post-buckling behaviour of a wide,
integrally stiffened panel subjected to compression. Two buckling modes
occurred simultaneously and were investigated. Byskov and Hutchinson (1977)
presented a preliminary investigation of mode interaction in axially stiffened
cylindrical shells under axial compression. An initial post-buckling analysis
method was proposed for simultaneous and nearly simultaneous buckling
modes. The influence of a given level of imperfection on the optimum was
explored.
Sridharan (1983) developed a semi-analytical approach for the study of doubly
symmetric interactive buckling of plate structures subject to axial compression
using the theory of mode interaction and the finite strip concept. Sriharan and
42
Ali (1985) developed a new analytical model for a study of the response of
thin-walled beam-columns having doubly symmetric cross sections with some
novel features which are: incorporating the interaction of overall buckling with
two companion local modes; accounting for the phenomenon of amplitude
modulation and modeling any set of realistic end conditions. The singularity
problem in interactive buckling, which was bypassed in the previous study, was
considered. Sriharan and Peng (1989) described a new analytical model for
stiffened panels which was developed for the study of nonlinear interaction of
local and overall instabilities of axially compressed stiffened plates. The
phenomenon of amplitude modulation and the triggering of the secondary mode
w

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