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)
8. Compression Tests of Cold-Reduced High Strength Steel Stub
Columns,
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)
10. Compression Tests of Cold-Reduced High Strength Steel Channel
Columns
failing in the Distortional Mode, University of Sydney, Department
of Civil
Engineering, Research Report R825, 2003 (Yang and Hancock)
11. Compression Tests of Cold-Reduced High Strength Steel Stub
Columns,
(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)
13. Compression Tests of Cold-Reduced High Strength Steel Channel
Columns
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-
Chapter 1: INTRODUCTION
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,
Chapter 1: INTRODUCTION
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
Chapter 1: INTRODUCTION
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
Chapter 1: INTRODUCTION
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.
Chapter 1: INTRODUCTION
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
Chapter 1: INTRODUCTION
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
Chapter 2: LITERATURE REVIEW
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
Chapter 2: LITERATURE REVIEW
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
Chapter 2: LITERATURE REVIEW
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
Chapter 2: LITERATURE REVIEW
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.
Chapter 2: LITERATURE REVIEW
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
Chapter 2: LITERATURE REVIEW
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
Chapter 2: LITERATURE REVIEW
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
Chapter 2: LITERATURE REVIEW
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
Chapter 2: LITERATURE REVIEW
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
Chapter 2: LITERATURE REVIEW
)/)(/)(/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
Chapter 2: LITERATURE REVIEW
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
Chapter 2: LITERATURE REVIEW
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
Chapter 2: LITERATURE REVIEW
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
Chapter 2: LITERATURE REVIEW
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
Chapter 2: LITERATURE REVIEW
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
Chapter 2: LITERATURE REVIEW
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
Chapter 2: LITERATURE REVIEW
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
Chapter 2: LITERATURE REVIEW
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
Chapter 2: LITERATURE REVIEW
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
Chapter 2: LITERATURE REVIEW
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
Chapter 2: LITERATURE REVIEW
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
Chapter 2: LITERATURE REVIEW
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
Chapter 2: LITERATURE REVIEW
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