Flexural-Torsional-Buckling-of-Structures-New-Directions-in-Civil-Engineering-.pdf

Flexural-Torsional-Buckling-of-Structures-New-Directions-in-Civil-Engineering-.pdf

1. Contents

Public by E EN Sp

= Cies Road, Bhopbriges. Glasgow Preface ix

Chapman & Hall Japan Thomson Publishing Japa, Hiraawacho Nemo Units and conversion factors xiii
Se ae a Glossary of terms w
Melbourne, Vitoria 3205, Au Principal notation wil
man & Hal nda aon sal 1 Introduction 1
a 11 General 1

12. Historical development 3

Paba in the USA and Canad by CRC Pres Ine, 200 Corporate Bouleva 13. Structural behaviou 5
NW, Boca Raton, Pri y 14 Bucklin u
Fir ion 1993 1.5. Design against buckling 14
1993 NS. Tai 16 References 15
Tricio n North Amer 2 Equilibrium, buckling, and total potential 7
Typeset in 1O/2pt Times by Thomson Pres (Ind) Lid, N oh oa Ir
Pinto in Grat Brain by St Edmundsbury Press Ld, Dry S Edmunds, Sol 32 Equi 17
ison 105 23 Total potential 23
A ii bial via 24 Virtual work 24
im or review ss x 25 Nature of equilibrium 28
DD ls publication may nor beprduce, sore or or 26. Bifurcation buckl 31
: In-plane behaviour of beam-columns 3

; 28 Flexu nal bucklin 37

» 29 References 48

3 Buckling analysis of simple structures 4

7 General 4

2 The energy method 49

33. Choosing the buckled shape 53

Libary of Congres Cataloging in Publicacion Daa avait Problems 66
37 References 68

4 Finite clement buckling analysis 6

41. General 6

2 The finite element method of buckling analysis 6

#3 Flexural-torsional buckling of columns n

44 Flexural-torsional buckling ofmonosymmetrie beam-columns 78

2. 46 Computational considerations
i E
48 References

5 Simply supported columns
51 General
52 Elastic buckling analysis
53 Doubly symmetric sections
54 Monosymmetric sections
55 Asymmetrie sections
56. Problems
57 References

6 Restrained columns
61 Gener

aint stifinesses

63. Bucklin

64. Continuous restraints

65. Discrete restraints
66 Problems

67 References

7 Simply supported beams
7.1 General

Uniform bending

3 Moment gradient

7.4 Transverse loads

5 Transverse loads and end moments

{6 Effects of load height

77 Problems

18 References

8 Restrained beams
81 General

82. Restraint stifinesses

83 Buckling analysis

84 Continuous restraints
85 Discrete restraints
86 Problems

87 References

9 Cantilevers
91
92
93
94 :
95 Continuous restraints

165
166

169
169
169

9.6 Discrete restraints
9.7. Problems
98 References

10

Braced and continuous beams
10.1 General

102. Interaction buckling

10.3 Approximate methods

104 Elastic buckling of braced and continuous beams
105. Problems

106. References

11 Beam-columns
111 General
112. Uniform bending
m
114 Transverse loads
11.5 Restraints
116 Problems
11.7 References

Moment gradient

12 Plane frames
121 General
122 Joints
123 Frame analysis

124. Portal frames
125 Other frames
126. Problem

127. References

13 Arches and

13.1. General
132. In-plane behaviour
133. Energy equation for lexural-torsional buckling

134. Differential equilibrium eq
13,5. Arch bucklin

136 Ring
137. Effect of load height on arch buckling
138 Finite element analysis

139 Proble

13.10 References

under uniform compression and bending
wuckling under uniform compression

14 Inelastic buckli
14.1. General
142. Tangent modulus theory of inelastic
143. Pre-buckling analysis of in-plane
144 Inelastic buckling analysis

buckling

3. viii Contents
145 Tnelastic buckling predictions
146. Problems
147 References

15 Strength and design of steel
15.1 General
152 Beam strength
153 Working stress design of steel beams
154. Limit st
155. Limit states strength desi
156 Limit states strength design of steel beam-columns

es strength design of steel beams

of steel columns

15.7. Problems
158 References

16 Special topies
16.1. Stepped and tapered members
162 Optimum beams
163 Secondary warpi
164 Interaction of local and flexural-torsional buckling
16.5 Web distortion
166. Pre-buckling deflections
16.7. Post-buckling behaviour

168. Viscoclastic beams
169 Flexural-torsional vibrations
16.10 Erection buckling

16.11 Directed loadin
16.12 Follower loading
16.13 Problems

16.14 References

17 Appendices
1 In-plane bending

Uniform torsion
Warping torsion

17.4 Energy equations for lexural-torsional buckling

175. Differential equilibrium equations for the buckled position

17,6. References

Index

Preface

iuctural failure in which one or more
members of a frame suddenly deflect and twist out of the plane of loading.

Flexuraltorsional buckling is a mode of

Because Mexural-torsional buckling reduces the load-carrying capacity of
ing additional bracing, or by

structure, designers must prevent it either by prov
using larger members.

ith

in S.P. Timoshenko's widely used textbook Theory of Elastic Stability, last

published (with JM. Gere) in 1961, and later in F. Blelch's Buckling Strength of

Metal Structures, published in 1952. These cover a wide range of structural

stability topics, a

few chapters.
These books were both written before the advent of the electronic di

From the 1930s, the subject of flexural-torsional buckling has been d

do their coverage of flexural-torsional buckling is limited toa

tal
computer in the 1950s, and the subsequent explosion in published research on the

subject. Consequently, more recent treatments of stability theory have been more
limited either in their scope or in their depth. Strangely, there have been few, if
any, recent books published which provide thorough treatments of Mexural-
torsional buckling

“This book is intended to provide both an up-to-date treatment of modern
methods of analysing flexural-lorsional buckling
detailed summar

‘and also to provide sufficiently
s of knowledge on flexural-torsional buckling that it can be
used as a source book by both designers and researchers.
It may

advanced level undergraduate courses, the teacher will want to simplify and edit
the material given. Such a course may introduce the theory of structural stability
given in Chapter 2, and the hand and computer methods of analysis dealt with in
Chapters 3 and 4. Subsequent mate

lso be used for teaching purposes as a text or reference book. In

al on column, beam, and beam-column
buckling may be selected from Chapters $, 7 and 11, and on design against
flexural-torsional buckling from Chapter 15.

For normal level post

aduate courses, these topics may be presented in

ater detail, and expanded with material on restrained buckling from Chap-

ters 6,8,9, 10and 12, and possibly with a treatment of inelastic buckling base
Chapter 14, For advanced course

¡ese topics would be studied more thorough
1y, while additional topics may be introduced from Chapters 13 and 16.

Itis not often these days that a researcher is allowed to develop a fascination for
a subject and spend the amount of time on it that I have on flexural-torsional
buckling, and I count myself as being privileged in this regard. My first introduc-

tion to the subject was through a Structures Honours course given to me as an

4. x Preface

undergraduate by the lateJ.W. Roderick in 1955. Asa masters research student, a
topic on the flexural-torsional buckling of beam-columns was suggested to me by
one of my co-students,P.G. Lowe, and my work on this, which was supervised by
Roderick, confirmed my interest in the subject. In early 1961 my curiosity on the
lateral buckling of beams was stimulated by the British Standard BS 449:1955.
Th ls Association of Australia for the
development of an Australian code for the design of steel structures, and into my
doctoral study of the fexural-torsional buckling of frame structures,

led me into work for the Stand

supervised by Roderick
In 1968 1 had the good fortune to spend a study

feat Washington University

with T.V. Galambos, who inspired me with his own fascination forthe subject. In
theearly 1970s, Heart much with JM. Anderson, S. Kitipornchai,P. Vacharajt
tiphan, S.T. Woolcock, T. Poowannachaikul and B.R. Mutton, an exceptional
group of graduate students at the University of Sydney. My good fortune
continued in 1974-75, when I spend another study leave, this time at the
University of Sheffield where I collaborated with D.A. Nethercot. In the 1980s, I
‘again had a number of outstanding students, M.A. Bradford, P.E. Cuk and J.P.

Papangelis, while two of my colleagues at the University of Sydney, G.J. Hancock
and NL. Ings, collaborated with me on flexural-torsional buckling research, as
did MA. Bradford and S. Bild as post-doctoral fellows, and later YıL. Pi.

Tn the early 1970s I began teaching the same Structures Honours course that
Roderick taught me in 1955,
Stability. | gave a related stabil
Jus of preparing and de

and also a post-graduate course on Structural
‘course atthe University of Alberta in 1985, The

sti Joping these courses and of challenging a

feacting to my students has done much to extend my own understanding 0
subject, as well as to give me a broad outline for this book

My studies of flexural-torsional buckling have not been limited to research,
‘and my work for nearly 30 years with the Standards Association of Australia on
the preparation of codes for the design of steel structures has given me a keen
appreciation of the need to translate research findings on flexural-torsional
buckling into forms that are easily understood and used by designer
reatly influenced in the preparation of this book by my teaching
‘and research experiences in Australia at the University of Sydney, in the USA at
Washington University, in the UK at the University of Sheffield, and in Canada
at the University of Alberta. While a significant proportion of the material in this

Thave been

book has been developed by me and my colleagues and students, much ofitis not
original, but has been gathered from many sources. Unfortunately, it is very

Sificult or even impossible to acknowledge all individual sources, and so the

references given in this book are restricted to those which the general reader may
wish to consult for further information.

Y would like to thank the School of Civil and Mining Engineering of the
University of Sydney for the facilities that it has made available to assist in the
preparation of this book. The manuscript was expertly typed by Jean Whittle

and
ms prepared by Ron Brew and Kim Pham,

Cynthia Bautista, and the

Preface xt

Many valuable comments a

students at the Univ

sa ity ol Sydney, especial by KAR, Rasmussen,
Final, wish acknowledge th fing help and support of my wie ely
without whom the writing of this book w a aa

not have been possible

Nicholas Trahair

5. Units and conversion factors

Units

While most expressions and equations used inthis book are arranged so that they
are non-dimensional, there are a number of exceptions. In almost al ofthese, SI

units are used which are derived from the basic units of kilogram (kg) fo

‘metre (m) for length, and second (s) for time,

The SI unit of force is the newton (N), which is the force which causes a mass of
1 kg to have an acceleration of 1 m/s?. The acceleration due to gravity is 9.807
is 9.807 N
ascal (Pa), which is the average stress exerted by a

m/s? approximately, and so the weight of a mass of 1 k

The SI unit of tres is th

force of IN on an area of Im?. The pascal is too small to be convenient in

structural engineering, and itis common practice to use
(1 MPa = 10° Pa) or the identical newton per square millimetre (1 N/mm
10° Pa). The megapascal (MPa) is used gene

Table of conversion factors

To Imperial (British) Units To SI units
Ike 14.59 kg
im 03048m
0.025 4m
Imm 304.8 mm
(0.039 37 in 254 mm
IN 02248 b = 4448 N
LEN 0224 8 kip 4448 kN
0.100 36 ton 964 kN
1 MPa 0.1450kip/in{ks) 1 kipyin 6895 MPa
0.064 75 tonyin. Lon/in. 1544 MPa
IKNm 0737 6kipft I kip ft 1356 kNm

03293 ton ft ton ft 3037 kNm
ile Ge Mu rip mt Yung msi acy vx

6. Glossary of terms

A member curved in the plane of loading,

A member which supports transverse loads or moments only
column A member which supports transverse loads or moments which
ending and axial loads which cause compression.

le. A member which supports transverse loads or moments which cause

bending and axial loads which cause tension
Brace A secondary member which prevents or restrains deflection or twist
rotation of a main member.
Braced beam A beam with a number

cross-sections braced against lateral

deflection and twist rotation.
18 A mode of failure in which the
n or plane normal to that of the loads or moments acting

is a sudden deformation in a

Cantilever A member with an end which is unrestrained against lateral
deflection and twist rotatio

Capacity factor A factor used to multiply the nominal capacity to obtain the

design capacity
Column A member which supports axial compression loads
Conservation of energy A principle describin

structure and its loads may deform without any change in thet

he conditions under which a

al energy of the

Con

inuous beam A beam which iso

Design capacity The capacity of the structure or element to resist the design
loads. Obtained as the product of the nominal capacity and the capacity factor.

Design oad The combination of factored nominal loads which the structure is

required to resis.
Distortion A mode of deformation in which the cross-section of a member
ch

es shape
Eflective length The length of an equivalent simply supported member which
has the same elastic buckling load as the actual member.

Elastic behaviour Deformations without yielding

Energy method A method of buckling analysis based on the principle of
conservation of en

F
complete structure is divided into a number of elements of finite size.

First-order analysis Elastic linear analysis in which equilibrium is formulated

ite element analysis A

yputer method of numerical analysis in which a

7. xvi Glossary of terms

for the undeformed position of the structure, so that the moments caused by
red.

products ofthe loads and deflections are ig
First-ield moment The value of the bending moment which nominally causes
he first yield ofa cross-section
Flexural-torsional buckling A mode of buckling in which a member deflects

and twists

Frame A skeletal structure consisting of a number of members connected
together at joints

Frame buckling A mode of buckli
participate.

Geometrical

Inelastic behaviour Deformations accompanied by yielding

In-plane behaviour The behaviour of a member which deforms only in the
plane of the applied loads.

Lateral buckling Flexural-torsional buckling of beams.

Limit states design A method of in which the performance of the
structure is assessed by comparison with a number of limiting conditions of
Usefulness. The most common conditions are the strength limit state and the

in which all the members of a frame

nperfections Initial crookedness or twist

Load and resistance factor design The limit states method of design in which
the factored (reduced) resistance is compared with the factored (increased) loads.
Load factor A factor used to multiply a nominal load to obtain part of the
design load.
Local buckling A mode of buckling which occurs locally (rather than gen-

erally in a thin plate element of a member,

Member One-dimensional structural element which supports transverse or
tudinal loads or moments.

Member buckling A mode of buckling involving the complete length of a
member.

‘Nominal capacity Capacity of a member or structure computed using the
formulations of a design code or specification,

‘Nominal load Load magnitude determined from a loading code or specifica

Non-uniform torsion The general state of torsion in which the twist of the

member varies non-uniformly

‘Out-of-plane buckling The buckling of a member out of the plane of loading.

Plastic analysis A method of analysis in which the ultimate strength of a
structure is computed by considering the conditions for which there are sufficient
plastic hinges to transform the structure into a mechanism,

Plastic hinge A fully yielded cross-section of a member which allows the
member portions on either side to rotate under constant moment (the plastic
moment).

Plastic moment The value ofth

iclded
behaviour Behaviour after buckling

«e bending moment which will cause a section to

become fully
Post-buckl

Glossary of terms vit

Pot
datum.

Pre-buckling behaviour Behaviour before buckling

Parlin A horizontal member between main beams which supports roof sheet
ing

Reduced modulus The modulus of elasticity used to predict the buckling of

ial energy Energy associated with height ofa gravitational load above a

inelastic members under constant applied load, so called because itis reduced
below the elastic modulus.

Residual stresses The stresses in an unloaded member caused by uneven
cooling after rolling, flame cutting, or welding

Resistance Capacity

Restraint An element which restrains the deflection or twisting of a member.

‘Second-order analysis Non-linear analysis in which equilibrium is formulated
for the deformed position of the structure, so that the moments caused by
products of the loads and deflections are included.

Shear centre The point in the cross-section of a beam through which the
resultant transverse force mu

act ifthe beam is not to twist.
Shear modulus The intial modulus of elasticity for shear stresses.
Squash load The value ofthe compressive axial load which will cause yielding
throughout a short member

Strain energy Energy associated with the straining ofa structure.
Strain-hardening A stress-strain state which occurs at stresses which are
greater tha
Strength limit state The state of collapse or loss of structural int

the yield stress.

ngent modulus The slope ofthe inelastic stress-strain curve which is used to
predict the buckling of inelastic memt ing load.

Total potential The sum of the strain energy of a structure and the potential
energy ofthe gravitational loads acting on it

Uniform torque That part ofthe total torque which is associated with the rate

‘of the angle of twist rotation of the member.

m torsion The special state of torsion in which the twist ofthe member
varies linearly

under increa

Virtual work A principle used to assess whether a structure isin an equilibrium
position.

Warping A mode of deformation in which plane cross-sections do not remain
plane

Warping torque The other part ofthe total torque (than the uniform torque),
which only occurs during non-uniform torsion, and which is associated with
changes in the warping ofthe cross-sections.

Work Energy transferred during the movement ofa force.

Yield stress The average stress during yielding when significant straining takes
place, Usually, the minimum yield stress in tension specified for the particular
sted

us The initial modulus of elasticity for normal stresses.

8. Principal notation |

‘The following is the principa
meaningis assigned to
‘one are possible, then the correct one will be

used.

A
AA]
tro) [Aca]
B

ral
CANTA]

CANTA]
Cu

c

notation used in this book. Usually, only one
bol, but in those cases where more meanings than

vident from the context in which itis,

Matrices for linear and quadratic potential energy contribu

tions of {a}

Matrices for linear and quadratic potent
tions of (0)

Flange width, or
Bimoment |
Matrix for in-plane generalized strains

Matrices for linear and quadratic g

Matrices for out-of-plane generalized strains
adient factor for beam-columns (equation 11.31)
uation 14,55)

Matrix for out-of-plane nodal deformations
Matrix for in-plane nodal deformations
Overall depth of cross-section

Vector of restraint point shear centre deformations
Generalized elasticity matrix |
In-plane generalized elasticity matrix

Out-of-plane generalized elasticity matrix

Generalized init

‘Young's modulus of elasticity
Reduced modulus of elasticity
Strain-hardening modulus of elasticity

Tangent modulus of elasticity
Translational restraint force
Ultimate tensile strength

Yield stress

Shear modulus of elasticity
Relative stifinesses of beam restraints at ends A, B
Strain-hardening shear modulus of el

Tangent shear modulus of elasticity

M:
My My
M,

M.

My

M,

M,

M,

M,

M,
Myo May M
M

Mo
M.
MM,
My

M!
[M]

obal stability matrix

Second moments of

ansformed element stability matrix

ea of bottom and top flanges

Second moment of arca of elastic core
Polar second (+A

noment of area = (I,
Section prope

y = Janta? + 92)da
Warping section constant

Second moments of area about the x,y axes
Second moment of area of compression lang

Torsion section constant
Torsion parameter = J(n?El.,/GJL?)

Beam parameter = (n° El J3/4GJL)

Global out-of-plane stiffness matrix
Transformed element out-of-plane stiffness matrix
Global in-plane stiffness matrix

Global in-plane tangent stiffness matrix

Length of member

Effective length

Length of restraining segment

Moment, or

Concentrated mass

Design bending moment

Bottom and top flange minor axis end moments
Nominal member moment capacity

Disto

Elastic buckling moment
Flange moment

Limiting moment at fist yield
Maximum value of M,

Full plastic moment

Bending and torsional restraint mon
Nomini

Uniform torque, or
Ultimate moment capacity, or

Unbraced buckling moment

Ultimate moment capacity for uniform bending
Warping torque
Bending mom

Moment at nominal first yield

Uniform bending buckling moment = J{P,GJ(t
Torque about the longitudinal axis
Out-of-plane matrix of powers of 2/L, or

Mass matrix

K

9. M;] In-plane matrix of powers of 2/L 1 w Work done, or

N Axial tension force Wagner stress resultan

[m Matrix relating (u, } x Global axes

ENGEN] Matries relating (0)? 5) 2.2, Elastic section moduli about x,y ax

IN? Matrix relating (0) to u co ous power of aL

1 Axial compression force | A Vector of two powers of z/L

po Design axial compression force A Load distance along beam

Po Elastic buckling load E Distance from shear centre

P, Failure load ia Vector of coefcients of powers of

» Inelastic buckling load h Width of thin rectangular element, or

P, Reduced modulus buckling load | Distributed bimoment per uni length

» Strain-hardening buckling a Depth of narrow rectangular section

» Tangent modulus buckling load, or | Vector of shear centre deformations
Nominal tension capacity h Translational restraint force per unit length

», Ultimate axial foree capacity we Element stability matrix

Por, Column flexural buckling lo LP ELL h Distance between flange centroids

Py Squash load | Effective length factor

», Column torsional buckling load = (GJ +x? El, /L?) Load height effective length factor

0 Concentrated loa Minor axis bending effective length factor

East buckling load | Warping elective length factor

© In:plane global nodal forces Element out-of-plane stifiness

o Value of @ for elastic buckling with rigid torsional support Element in-plane tangent stiffness matrix
restraints | Moment factor

R Radius of curvature ma Bending and torsional restraining moments per unit length

R Vector of discrete restraint ations m Uniform torque per unit length

is} Matrix oferos-sction coordinates of P I m Warping torque per unit length

T Flange thickness n Integer

13 Outof plane transformation matrix A Load per uni

tr In-plane transformation matrix a Value of for an unrestrained beam

ñ Sain energy {a Element in-plane distributed loads

uKw Defletions in global X, Y, Z directions rar Radio gyration about x,y axes

u, Element out-of-plane strain energy É i

Un Flange bending strain en E

U Ineplane strain energy

u Discrete restraint strain energy 2 y

u Total potential e Vector of continuous restraint actions

u Uniform torsion strain energ s Distance along section mid-thickness line, or

v Potential ener, or Distance between discrete restraints, or
Shear fore, or Distance along curved shear centre axis
Volume ‘ Thickness of thin-walled section, or

7 Element outofplane potential energy Thickness of web, or

y In-plane potential energy Time

v Shear force in y direction le Distance from mid-thickness surface

u Shear centre deflection in X direction

10. Buckling component of u
Bottom and top flange deflections in X direction

Initial erookedness
Deflections of Pin X, Y, Z directions
Vector of deformations

Shear centre di
Buckli

plane nodal deflections of cross section

lection in Y direction
-omponent of

Value of

al centroid
Shear centre deflection in Z direction
Principal centroidal axes

Coordinates of centre of buckling rotation

Coordinates of shear centre
Distances of diserete rotational restraints from centroid
Distances of continuous rotational restraints from centroid
Dist

Distances of continuous

es of diserete translational restraints from centroid

Distance to centroid
Distance of concentrated load from centroid
Distance of distributed load from centroid
Longitudinal axis through centroid
Angle, or

Beam torsional stiffness
Stiffness of critical segment
Limiting value of stfiness
Buckling factor for beams
Stiesses of ange minor axis rotational end restraint, or
Stiines of restraining segment

Stiesses of discrete bending and torsional

Stiinesses of continuous bending and torsior
Slenderness reduction factor

Stffnesses of discrete translational restraints
Stffnesses of continuous translational restraints
Stiffness of discrete warpin

Stitiness of continuous warping restraints

Discrete restraint stiffness matrix
Continuous restraint stifiness matrix

Ratio of end moments

Monosymmetry section constant = Ip,/l, — 2}

meters

Major and minor axis end restraint pa
Shear strain, or

End moment coefficient

‘Stiffness factor for moment distribution

Stiffness factor for restraints at far end
Shear strain at P

Load height factor for continuously restrained beams

(equation 8.24)

Central or end deflections

Initial central crookedness

Global out-of-plane nodal deformations
Global in-plane nodal deformations
Element out-of-plane nodal deformations
Element in-plane nodal deformations
Normal strain, or

Dimensionless distance of load from centroid
Normal strain at P

Residual strain = o,/E

Strain-hardening strain

Yield strain

Generalized strain vector

Generalized in-plane strain vector
lized stiffness strain vector

lized stability strain vector
Crookedness parameter, or
Cocliient of viscosity

Rotation

Initial central twist rotation

Distortional twist rotation of fa
Warping rotation of fa
Rotations about global X, X,Z axes

Curvature
Buckling load factor

Zero interaction buckling load factor of critical
Zero interaction buckling load factor of nth segment

Zero interaction buckling load factor of restraining segment
(PEL)

Poissons ratio

Density, or

=1,/1

Perpendicular distance from shear centre

Normal stress

Allowable workin

Maximum norm:

Residual stress
Normal stress at

11. Warping normal stress
Generalized out-of-plane stress vector
Generalized in-plane stress vector

Shear stress at P

Rotation about global Z axis
Twist rotatio

Capacity factor
Buckling component of $
Initial twist

Nature frequeney of vibrations
Section warping function

1 Introduction

1.1 General

Thin-walled structural members may fail in a flexural-torsional buckling mode,
in which the member suddenly deflects laterally and twists out of the plane of

ber which has low lateral
plane of

loading. This form of buckling may occur in a mem
bending and torsional stifnesses compared with lts stiffness in the

loading

The most common form of flexural-torsional buckling is for I-section beams
which are loaded in the planes of their webs, but which buckle by deflecting
Jaterally and twisting, as shown in Figures 1.1 and 1.2a. Flexural-torsional

buckling may also occur in concentrically loaded columns. This can be regarded
as a general case, of which flexural buckling without twisting is one limiting
‘example (Figure 1.2b). Some columns may buckle torsionally without bending

(Figure 12c) which isthe other limiting example ofthe flexural-torsional buck
ling of columns. Beam-columns bent in a plane of symmetry may also buckle in a
flexural-torsional mode,

Flexural-torsional buckling is not confined to individual members, but also
occurs in igid-jointed structures, where continuity of rotations between adjacent
‘members causes them to interact during buckling

Flexural-tor
structures as it may reduce the load-carrying capacity. Unless it is prevented by
using either sufficient bracing or members which have adequate flexural and

¡onal buckling is a primary consideration in the design of steel

torsional siffnesses, then larger members must be used to avoid premature

failure, The determination of these larger members will be dominated by
considerations of flexural-torsional buckling

tion to Mexural-torsional buckling. An

This chapter provides an introdi

historical survey is made in section 1.2, which is followed by general reviews of

structural behaviour in section 1.3, of buck]

against buckling in
‘Chapter 2 provide

flex

methods of predicting elastic lexural-torsional buckling
The buckling of individual columns, be

in section 14, and of design
tion 1.5.
eral treatment of buckling with particular reference to

torsional buckling, while Chapters 3 and 4 present hand and computer

ms, and beam-columns is described in

Chapters 5-9 and 11, while the buckling of continuous beams, frames, and arches
(Figure 1.3) and rings is discussed in Chapters 10, 12, and 13.

Inelastic buckling is dealt with in Chapter 14, while the use of Mexural-torsional

buckling predictions in the determination of design strength is described in
Chapter 15. A number of special topics are briefly discussed in Chapter 16

12. Introduction

Figure 1: Flexuraltorsional buckling of a cant

Historical development 3

12 Historical development

121 ELASTIC BUCKLING THEORY

‘The initial theoretical research into elastic Nexural-torsional buckling was pre-
ceded by Euler's 1759 treatise [1] on column flexural buckling (Figure 1.4),
Which gave the first analytical method of predicting the reduced strengths of
slender columns, and by Saint-Venant's 1855 memoir [2] on uniform torsion
(Figure 1.40), which gave the first reliable description ofthe twisting response of
members to torsion.

However, it was not until 1899 that the first treatments were publishes
flexural-torsional bückling by Michell [3] and Prandıl [4], who considered the
lateral buckling of beams of narrow rectangular cross-section. Their work was
extended in 1905 by Timoshenko [5,6] to include the effects of warping torsion in
I-setion beams.

‘Subsequent work in 1929 by Wagner [7] and later work by others led to the
developm I-orsional buckling, as stated by

nt of a general theory of flexur

a
i
AY

gael (a Tost 1 Euler Buckling UL SS Venant Torsion

mer buckling

Figure 1.4 Euler buckling and St Venant torsion

13. 4 Introduction

Timoshenko [8] and Vlasov [9],
Timoshenko [10] and Bleich [11]

‘Specific studies of exural-torsional buckling were made by many researchers,
but prior to the 1960s, these were limited by the necessity to make extensive
calculations by hand. Some ofthese are included in the 1960 survey by Lee [12]

This situation changed dramatically with the advent of the modem digital
‘computer, and the 1960s saw an explosion in the amount of published research.
Ava result, the focus of research moved from the flexural-torsional buckling of
isolated members under various loading conditions to the effects of end restraints
vid-jointed frame as a result ofits continuity with
‘adjacent members. Many of he s are summarized in the 1971 survey of
the Column Research Committee of Japan [13]

The extension of the general finite element method of structural analysis [14]
to lexural-torsional buckling problems by Barsoum and Gallagher in 1970 [15]
‘ava further change, in that it was no longer necessary to publish comprehensive

most any particular

and incorporated in the textbooks of

exerted on a member of a ri

results of elastic lexural-torsional buckling studies, since a
-neral purpose computer pro;

situation could now be analysed using a
development is similar to that which occurred in the in-plane analysis of plane
Fieid-jointed frames, in which the tabulations of solutions used in the 1930s were

replaced by general purpose plane frame computer analysis pro
Many of the developments of the theory of flexural-torsional buckling have
been made by extensions of the previously accepted theories, as expressed either

by the differential equations of elastic bending and torsion or by the energy
al accept

‘equation for buckling. Not all of these extensions have received gene
ance, and so a number of attempts have been made through the 1980s to produce

a generally acceptable theory of flexural-torsional buckling. This book includes
Such a general theory which i based on the use ofthe second-order relationships
between the deformations and strains that take place during bending and torsion,
the concept of the total potential, and the principles of virtual work and

equilibrium, and of conservation of energy during buckling. This approach has
been used, for example, to re-examine the flexural-torsional buckling of arches,
carly studies of which were reported by Vlasov [9] and Timoshenko [10]

jonal buckling

While the historical development of knowledge of flexural-
undoubtedly was initiated by the need to prevent premature failure of steel
Structures in this mode, this isnot well documented. It seems likely, however, that
carly design procedures for preventing the lateral buckling of steel beams
followed and were closely related to those used for preventing the flexural failure

‘of columns.

inst lexural-torsional buckling was the
buckling which

The need to be able to design aj
catalyst for the development of a theory for flexural-torsiona
Would allow the successful prediction of failure. Early theoretical research was

into the elastic buckling of perfectly straight members, some of which was verified
experimentally. However, the very straight and slender members used for these
Experiments were unrepresentative of the ral steel beams used in pr
which showed that their strengths were reduced below those
<dastic buckling theory

actce, tests of
dicted solely by

Theoretical research therefore extended from the elastic buckling of straight
members to study the influences oferookedness, yielding, and residual stresses on
the strengths of real steel beams, and to determine how to incorporate these into
the procedures used in design. These developments tended to follow behind the
corresponding developments from the elastic flexural buckling theory to the
Strengths of real steel columns. Early research on the inelastic lateral buckling of
Steel beams was carried out by Neal [16] and Galambos [17]. Flint [18] was one
ofthe early researchers studying the
fon the lateral buckling of beams and b
Some of the early well-documented experiments on the lateral buck
steel beams were ca

ects of intial erookedness and twist [19]
-am-columos.

ng ofreal
-d out by Hechtman, Hattrap, Styer, and Tiedmann [20]
Fukumoto and Kubo [21-23] reviewed and produced a data base of the

experimental studies prior to 1977 on the lateral buckling of real steel beams.
Early rules for designing steel beams a rally
ning columns against flexural buckling, with
perhaps the first proposal based on flexural-torsional buckling being made in
1924 by Timoshenko [24]. The first modern treatment was probably given
Kerensky, Flint and Brown [25] as the basis for the British Standard
BS153-1958 [26]. More recently, most countries have or are transforming their
design standards into the limit states format [27]. Current design criteria are
reviewed in [28-30]

ainst lateral buckling were ge

transpositions of rules for des

13 Structural behaviour
134 PLASTIC DEMAVIOUR

1341 L

ar behaviour

The simplest and most widely used model of the behaviour of a structure under
Static loads assumes that all of the deformations are proportional to the magni:
tude of the load set acting on the structure, so that the relation between load and
response is linear, as shown by Curve 1 in Figure 1.5

For this linear model to be valid, the material itself must have a linear
ship between stress and strain, Such a material is usually described as
elastic. (Strictly, elastic means perfect recovery on unloading, so that an elastic

material may be non-linear. However, most elastic mater

a are linear.) Most
Structural stecls are linear, at least for stresses less than the yield stress Fy

as shown in Figure 1.6, while many other structural materials are regarded as

being linear over most of the range of working load,

14. 6

(O Material and geometric
non-linearities

Deformation

15 Structural behaviou
Figure

Stress

SS Strain-hardening Es

N Plastic

~ Elastic E

/

Tensile fracture

[Not to scale]

Structural behaviour 7

analysis [31]. These allow the deflections of the structure under load to be
ign under the working loads, and the member end
Actions to be approximated for strength design,

assessed for serviceability de

131.2 Now-linear behaviour

linear elastic model of itself does not allow the strength ofthe structure to be
assessed. For this purpose it is necessary to know of any material and structural

s before the real behaviour of the structure and its maximum
Joad-carrying capacity can be approximated.

Structural non-linearities cause the deformation response ofthe structure to
load to become non-linear as shown by Curve 2 in Figure 1.5, even when the
material remains lin
ciated with additional moments caused by the products of the loads and the

deflections of the structure or member as shown in Figure 1.7, Such

re allowed for when equilibrium is formulated for the deformed
geometry ofthe structure under load [32,33], instead of the unloaded position,
and so these non-linearities are usually described as being geometric, or second:
orde
Geome
indefii

eects

ie non-linearities may cause the load-deformation behaviour of an
¡own by Curve 2in

Oey Strain

Figure 16 ldealized stress-strain relation for structural tel

The structure itself must also behave linearly for the linear model to be valid,

No structure is truly linear, but many are approximately so, provided the

deflections are small
The modern popularity of the linear elastic model of structural behaviour

arises from the widespread availability of computer programs for linear

irst-orde

order behaviour

15. 8 Introduction

Figure 1.5. This limit (Curve
structure, Real structural b
material becomes non-linear as shown by Curve 4 in Figure 1.5, and a

in Figure 1.5) isthe elastic buckling load of the
jour will depart from this asymptotic behaviour

maximum load capacity will usually be reached, after which the load capacity will
p

buckling behaviour ofa structure can be regarded as the limit ofthe
near behaviour, In elastic buckling, the primary or pre-buckling

last non-
response of the structure is in a diferen direction to the buckling response. For

example, the pre-buckling response of the compression member shown in
Figure Aais due to longitudinal shortening w, while the buckling response is one
of transverse bending deflections e. Thus the buckling response v remains zero
until the buckling load P, is reached, when the buckling response may initiate
and continue indefinitely. For the buckling response to remain zero until the

‘buckling load is reached, there must be no real or equivalent loads which would
cause a primary response in the buckling direction.

A structure such as a concentrically loaded column or a beam loaded in the
plane ofthe web may exhibit real load-deformation response which differs only
Slightly from the idealized buckling respons, in that the response in the buckling,
direction remains small until the buckling load is approached. Examples include
Straight concentrically loaded columns with small transverse loads, and columns

‘with initial crookednesses which cause small transverse bending eflects. For such
load may provide a quite accurate assessment of

members, the elastic bucklin
the strength, especially for slender members, for which small transverse loads or
cerookednesses are less important.

After the buckling load is reached, the post-bucklin
may remain constant, or may rise or fall. This is caused by chant

ad-deformation curve
er
during buckling, which may lead to redistributions of the

actions through the structure, Large deformations, for which there are gross

changes in the chord lengths of ome members and their rotations,may also affect

the post-buckling behaviour.

1.32 INELASTIC BEHAVIOUR

1.321 Inelastic materials

Al structural steels have a limited range over which the stress-strain behaviour is
uctural steel exhibit a horizontal yield plateau once the yield
re 1.6, followed by a slowly rising

linear, Normal

stress Fy has been reached, as shown in Figus
sin-hardening region. Cold-formed and stainless steels and aluminium all

tahibit stress-strain curves which are rounded after a limit of proportionality is

reached, as shown in Figure 1.8

1.322 Inelastic stress distribution

The actual stress distribution at a member cross-section depends on the geometry
rial stress-strain curve, and the

of the section, its structural actions, the m

Structural bohavi

Ultimate tensile

strength

0.2% Proof stress

| Fracture

Stress (MPa)

Le
cal elongation
not to scale

200

100

0 a om 5
ZI POT 002 003 00% 005 00 007 00
Strain on SOmm gauge length

Figure 18 Stress-steain curve of cold reduced steel shoot.

residual stresses present before loading, such as those es
manufacture of the member.

When the stresses are low so that the member remains elastic, the stress
distributions caused by axial force and bending actions are linear, as shown in
Figure 1.9. Under bending actions, the maximum stresses occur at the extreme
fibres, and when these reach the yield stress, a redistribution of the stresses
commences. Useful structural limits (in the absence of local buckling effects) for
structural steel members are provided by the momer
Which the cross-section becomes fully plastic

The presence of residual stresses such as those caused by uneven cooling after
hotrolling or welding causes early initiation of yield, and generally affects the
inelastic stress distribution. Because such residual stress distributions must be
sell-equilibrating so that they have zero axial force and bending actions in the
unloaded member, they have no effect on the section full plastic capacities Mp
and Py.

used by the method of

Mp or axial force Py at

1323 Inelastic members and structures

Inclastic effects on the behaviour of mé
are best described sepa

nbers and structures subject to buckling
ately in terms of their eflets on the pre-buckling
behaviour, on the buckling behaviour, and on the non-linear behaviour of
members with geometrical imperfections such as initial crookedness or twist

16. 10 Introduction

H 5

Tris ee

ergo O ee O Fest mi Date O rar, O santos

Figure 19 Moment-curvaturo relationships for steel beams.

Inelasticeffets change the distributions ofthe bending moment, and to lesser
extent of Ihe axial force, in an indeterminate structure before buckling. Since
buckling depends on the pre-buckling distributions of these actions, these
nelasti effects may be important.

Inelastic effects also increase the deflections. When there are significant

‘geometric non-linear effects in the pre-buckling regime resulting from the pre-
buckling deflections ofthe structure, these non-linear effects may be increased by
additional deflections caused by inelastic behaviour, The advanced analysis of
Structures which accounts for geometric and material non-linearities including
the effects of geometrical imperfections and residual stresses is described in [34].

While inelastic behaviour affects the buckling actions as described above, it
also reduces the buckling resistance below the corresponding elastic resistance.
Yielding causes local reductions in the cross-section stifiness which when ag-
grogated over the complete member or structure may substantially reduce its
buckling resistance. For example, one simple model of the inelastic flexural
buckling of columns ignores the stiffness of any yielded regions ofthe column, so
that its buckling resistance is based only on the regions of the column which
remain elastic.

Thelastic behaviour also affects the strength of a member with small geometri-
cal imperfections such as initial crookedness and twist. While the member
remains elastic, its load-deformation behaviour asymptotes towards the elastic
buckling behaviour of a perfectly straight member, as shown in Figure 1.5. The
actual behaviour departs from this when the member first yields, and a maximum
Toad is reached which is less than the elastic buckling load. This maximum load,
whieh depends on the inelastic material properties, is sometimes approximated
by the load at first yield

Buckling 11
14 Buckling

141 GENERAL

Buckling has already been described as the behaviour in which a structure or a
structural element suddenly deforms in a (buckling) plane diferent to the origin

(pre-buckling) plane of loading and response. Member buckling (Figure 1.1)
fhvolves a single member, and may occur in flexural, torsional, or flexural

torsional modes. The half wave length of the buckle is of the same order as the
member length

‘Buckling may involve all the members of a frame, with interactions between
the individual members. The buckle half wave length may be ofthe same order as
‘a member length, or may be ofthe order ofthe frame size.

‘On the other hand, local buckling (Figure 1.10) usually takes place over a short
length of a member of the same order as the cross-section width or depth.
Distortional buckling (Figure 1.11) lies between member and local buckling, and
is usually of a half wave length intermediate between the member and the
cross-section dimensions.

“These various forms of buckling are described in more detail in the following
sub-sections.

focal buckling of an section column.

17. 12 Introduction

igure 1.11 Dior

142 FLEXURAL BUCKLING

Feral bucling ofa member (Figures 1.26 and La) may involve trans
Elsplactmens or va the mmbereoseseton, andi rested by the Nexura
Hell Bl or EI, ofthe member. I occurs when the second-order moments
ad bythe product of the axial compresion force P with the place,
D ae equal everywhere to thenternal bending rstanes Eu
SEN ER Flexural bucking san be regarded as Tinting ase of
Bexural-trsiona buckling

Maur buckling may vive a single member, a group of members, or a
comple ame In braced rames, buckling i usual concentrated near one
Ser wie ls dry restrained by interactions with the adjacent
members and indirectly by the more emote members. In unbraced multi-storey
Muere buckling oxcurs a one tre, and involves al the columns of that

Buckling 18

storey, which are restrained by the beams and columns of the adjacent storeys
‘ther unbraced frames may buckle in modes which directly involve many or all
ofthe members.

143 TORSIONAL BUCKLING

‘Torsional buckling (Figure 1.2c) of a member involves twist rotations $ of the
member cross-sections, and is resisted by the torsional
warping rigidity EL. It occurs when second-order torques Pr
the axial compression force P and the twist d/dz are equal everywhere to the
sum of the internal torsion resistances GJ dg/dz and — EI„@’#/dz?. Torsional
buckling can be regarded as a limiting case of flexural-torsional buckling.
‘Torsional buckling may also occur in complete frames. Often the buckling
resistance of these is dominated by the flexure ofthe individual members, as for
example in tower frames whose horizontal cross-sections rotate.

144 FLEXURAL-TORSIONAL BUCKLING

Flexural-torsional buckling, which is the subject of this book, involves both
displacements u, vand twist rotations $, and is therefore resisted by combinations
ofthe bending resistances El, d*u/dz and — El,4*o/dz* and the torsional resis-
tances GJ d¢/dz? and — El 4%

While doubly symmetric columns whose centroidal and shear centre axes
coincide buckle in either a flexural or a torsional mode, monosymmetrie and
asymmetric section columns may buckle in flexural-iorsional modes. In these

cases, the separation of the centroidal and shear centre axes causes these axes to
become skew during buckling, so that the axial compression force acting along
the centroidal axis has transverse components which create torques acting about
the shear centre a

The flexural-torsional buckling of beams (Figures 1.1, 12a) involves lateral
Aisplacements u out of the plane of bending and twist rotations ¢. In this
case the twist rotations ¢ cause the applied moments to have components actin
out ofthe original plane of bending, while the lateral rotations du/dz cause the
applied moments to have torque components about the axis of twist through the
shear centre,

Beam-columns bent in a plane of symmetry may also buckle in flexural-
torsional modes which combine those of columns and beams

Flexural-torsional buckling may occur in frames (Figure 1.30), where there are
interactions between the adjacent members during buckling. In continuous or
braced beams (Figure 1.3a) one span or segment is usually the most critical, and
is restrained by the adjacent spans or segments. In Ihree-dimensional frames, the
members in each primary load-carrying plane interact during out-of-plane
flexural-torsional buckling, and may be restrained by transverse members be-

tween adjacent primary frames.

18. 14 Introduction

Arches loaded in their plane (Figure 1.3c) may also buckle in a flexural
torsional mode by deflecting out ofthe plane and twisting

145 LOCAL BUCKLING

Local buckling of a thin plate element (of thickness 1) of a structural member
involves deflections of the plate out ofits original plane, as shown in Figure 1.10.
Local buckling is resisted by the late flexural rigidity £r°/12(I — v), and occurs
when the second-order actions caused by the in-plane compressions and the
out-of-plane deflections are equal everywhere to the internal resistances of the
plate element to bending and twisting.

‘Local buckling is usually concentrated near one particular cross-section of a
member where the in-plane compressions of the plate clements are greatest,

although multiple local buckles may occur in members whose stresses are
constant along the member length. The half wave length of the buckle is of the
order of the plate width, Local bucking effects may reduce the resistance of a
member to flexural-torsional buck!

Local buckling may occur in plate and shell structures, as well as in the
structural members used in frame structures. Examples of plate structures
5 and rectangular and trapezoidal tanks, while

include stifened plate gird
shell structures include cylindrical and spherical containment structures. These
may buckle locally in the more highly stressed regions, as well as in a more

global fashion, involving larger regions ofthe structure.

146 DISTORTIONAL BUCKLING

Distortional buckling (Figure 1.11) describes a buckling mode intermediate

between those of local and member buckling. In member buckling, the cross-
section is assumed not to distort and buckling involves the whole member length
‘while local buckling involves relative displacements of the component plates over

2 short length of the member.

Distortional buckling often involves web flexure and corresponding rotations
of the langes which vary slowly along the member length, as shown in Figure
1.11. Distortional effects may reduce the flexural-torsional buckling resistances of
thin-web beams.

1.5 Design against buckling

Methods of designing against flexural-torsional buckling are essentially of two
types. For the first type, buckling is avoided, and the member in-plane capacity
is fully utilized. One way of achieving this is to use sections which are not
susceptible to buckling. For example, closed sections have very much higher
torsional rigidities GJ and higher flexural rigidities El, than corresponding open
I-section members, and rarely buckle in a flexural-torsional mode. Les effective

References 18

js the use of I-sections with comparatively wide flanges, which have higher
flexural El, and warping El, rigidities than I-sections with narrow fang

À second way of avoiding buckling sto increase the bracing, ether by reducing
its spacing, or else by increasing its effectiveness, as for example when a lateral
deflection brace is made more effective by using it also to restrain the twist
rotation $.

For the second method of designing against buckling, a reduced capacity is
determined which accounts forthe effects of flexural-torsional buckling, in which

fase the members in-plane capacity is not fully utilized. Capacity reductions
“depend on the slenderness of the member, on the initial crookedness and twist, on
the residual stresses, and on any loads which act above the shear centre, and are

usually determined in accordance with a design code [29]. I is advantageous to
{ake account of the effects of moment distribution and restraints which generally
increase the buckling resistance, and many design codes allow this to be done.

16 References

1. Euer, L. (1759) Sur a force des colonnes. Memoirs Academic Royale des Sciences et
Belle Lettres, Berlin, 13, partial translation by Van den Brock, JA. (1947) Ame
Journal of Physie, 15, 309-18,

Seint-Venant, B (1855) Memoire sur la torsion des prismes. Memoires des Savas
Etrangers, XIV, 233-560.

3. Michel, AG. (1899) Elastic stability of long beams under transverse forces.
Philosophical Magazine, 48, 298-309.

4, Prandtl L. (1899) Kipperscheinungen, Thesis, Munich

$5, Timoshenko, SP. (1953) Einige Stabiltactsprobleme der Elatzitactsthcoro, in
Collected Papers of Stephen P. Timoshenko, MeGraw-Hil, New York, pp. 1-50.

6 Timoshenko, SP (1983) Sula stabilite des systemes elastiques, in C
Stephen P. Timoshenko, McGraw-Hill, New York, pp. 92-228

7, Wagner, M. (1936) Verdrehung und Kaickung von offenen Profilen (Torsion and
Buckling of open sections), NACA Technical Memorandum No. 807.

8. Timoshenko, SP. (1953) Theory of bending, torsion, and buckling of thin-walled

à Papers of Stephen P. Timoshenko,

ected Papers]

members of open cross-section, in Colle
MeGraw-Hill, New York, pp. 559-609,

9, Viasow, VZ (1961) Thin-Walled Elastic Beams, 2nd ed, Irae Program for Scientific
Translation, Jerusalem.

10. Timoshenko, SP. and Gere, LM. T
New York

1, Bleich, F (1952) Buckling Strength of Metal Structures, MeGraw-Hil, New York

12. Les, GC. (1960) A survey of literature on the lateral instability of beams, Welding
Research Council Bulletin, No. 63, August.

13. Column Research CG (1971) Handbook of Structur
Corona, Tokyo

14. Zenkicwicz, O.C. and Taylor, RL. (1989, 1991) The Finite Element Method, Volume
1 Basic Formulation and Linear Prob Solid and Fluid Mechanics
Dynamics and Non-Linearity, 4th edn, McGraw-Hill, London.

lasie Stability, 2nd edn, MeGra-Hil

mittee of Jay

u. Volume

19. 16 Introduction

15. Barsoum, RS. and Gallagher, RH, (1970) Finite element analysis of torsional and
torsional exural stability problems. International Journal of Numerical Methods in

Engineering, 2,

Neat BG (1950) The lateral instability of yielded mild steel beams of rectangular

ren section, Philosophical Transactions, Royal Society of London, A, 242 January)

197-242,

17. Galambos, TL. (1959) Inelastic lateal-torsional buckling of eccentrcally loaded
Wide-Tange columns, PhD Thesis, Lehigh University, Bethlehem, PA.

18. Flin AR. (1950) The stability and strength of slender beams. Engineering, 170
(December) 545-9.

19. Trahair, NS. (1969) Deformations of geometrically imperfect beams. Journal ofthe
Structural Division, ASCE, 95 (STD) 1475-96.

20. Hechiman, RA, Hattap 15, Styer, EF and Tiedmann, TL. (1955) Lateral buckling
‘Of role steel Lbeams. Proceedings, ASCE, 81 (Separate N. 797, September

21. Fukumoto, Y and Kubo, M. (1977) A survey of ets on lateral buckling strength
‘of beams in Preliminary Report, 2nd International Colloquium on Stability of Stel
Siructures, ECCS-IABSE, Liege, pp. 233-40.

22. Fukumoto, Y. and Kubo, M (1977) A supplemer
buckling strength of teams, in Final Report, nd In
of Stel Structures, ECCS-IABSE, Liege, pp. 15-7

23. Fukumoto, Y. and Kubo, M. (1977) An experimental review of lateral buckling of

beams and girders, in Proceedings, International Colloquium on Stability of Structures

Under Sttie and Dynamic Loads, SSRC-ASCE, Washington, pp.S61-62.

‘Timoshenko, SP. (1924) Beams without lateral support. Transactions, ASCE, 87,

2-7

25. Kerensky, O.A, Flint, AR. and Brown, W.C. (1956) The basis
plate girders in the Revised British Standard 153. Proceed

Engineers, Part UI, S (August) 396-521

Bath Standards institution (1958) BS 153 Steel Girder Bridges, Part 3B: Stresses,

Part 4: Design and Construction, BSI, London.

37. Ravindra, MK and Galambos, .V.(1978) Load and resistance factor design for see.
Journal of the Structural Division, ASCE, 104 (ST) 1337-54,

28, Galambos TV. ed) (1988) Guide to Stability Design Criteria or Metal Structures, ath
can, John Wiley and Sons, New York.

29. Beedle, LS, (ed) (1991) Stability of Metal Stru

Structral Stability Research Council, Bethlehem, PA.

Thai, NS. and Bradford MA (1991) The Behaviour and Design o Steel Structure,

revised 2nd eda, Chapman and Hall, London.

Harrison, HB. (1990) Structural Analysis and Design Some Mlerocomputer Applica

tions, Parts 1 and 2, 2nd edn, Pergamon Press, Oxford

32. Harrison, HB, (1973) Computer Methods in Structural Analysis, Prentice-Hall

to a survey of tests on lateral
mational Collqulum on Stablity

15-4 World View, 2nd edn,

Englewood Ci, NI

3. Hancock, GJ. (1991) Elastic method of an
second order eflcs, in Structural Analysis to AS4100, School of Civil and Mining
Engineering, The University of Sydney, pp.2.1-232

34, Clarke, MU, (1991) Advanced analysis, in Structural Analysis to AS4100, Schoo! of

The University of Sydney, pp.61-655.

Civil and Mining Engineer

2 Equilibrium, buckling, and total
potential

24 General

In this chapter, the conditions for the equilibrium of a structure are discussed

generally, and then related to the conditions at bifurcation buckling.
Equilibrium is considered in section 22 for a structure of a linear elastic

material, The deflection of the structure under inet

{elation to the changes which occur in the energ

asing load is then discussed in

ofthe system used to apply the
foads, in the strain energy stored in the structure, and in the potential energy of
the loads.
‘The principle of virtual work for a structure in equilibrium is stated in
section 24. This is then related tothe fist variation ofthe total potential defined
and used to d

in section 2. ive the principle of stationary total potential

‘The conditions of stable, neutral, and unstable equilibrium are discussed in
section 25. The relationship between the condition of neutral equilibrium and
the principle of conservation of energy during buckling is established. Both of
these are expressed in terms of the second variation of the total potential, and
restated by the theorem of minimum total potential

The nature of bifurcation buckling, in which there is a sudden change at the
bifureation buckling load from the pre-buckling load-deflection path to a new
“and diffrent buckling path, is considered in section 2.6. The neutral equilibrium
conditions of bifurcation buckling under constant load are discussed, and the
relationships between the virtual work principle for equilibrium of the buckled
position and the conservation of energy condition for neutral equilibrium at the
buckling load are established, These relationships are summarized in Figure 2.1
for the bending and flexural buckling of beams and columns.

The relationships between pre-buckling equilibrium, buckling and non-
behaviour are exemplified in section 2.7 by considering the in-plane behaviour of
beam-columns. The corresponding relationships for flexural-torsional buckling
ae stated in section 2.8

22 Equi

brium

221 MATERIAL BEHAVIOUR

The structure shown in Figure 22a is composed of a material which is homo:
Beneous,isotropic, elastic, and has the linear relationship.

20. 18 Equilibrium, buckling, and total potential

ee] [5]

[En |
Fe | Fe
een 171.5) | | AT

Figure 21 Theory of buckling

E

ecto 1.9) a Er Go)

son 1719)

Equilibrium 10

\
Figure 22 Structure, loads and material behaviour.
o=Ee en

between the stress ¢ and the strain e, as shown in Figure 22, in which E is the
Young's modulus of elasticity

Tn the unloaded state, the material is unstressed (0 = 0) and unstrained
In the loaded state, small strains e develop, with corresponding stresses a

(e=0),

‘When loads Q are added slowly (so that there are no dynamic effects) to the
structure shown in Figure 22a, it deflects v as shown in Figure 23. In the

deflected position, the structure is said to be in static equilibrium, in that the
Toads Q are exactly balanced by reaction forces and the deflections » remain
constant with time.

The variations shown in Figure 2.3 of the deflections » with the loads Q define
the static loud-deflection behaviour. This may be linear, when the deflections v
are proportional tothe loads Q, or non-linear. The deflections may vary smoothly
with the loads; or the rate of change of the deflection may suddenly change, in
which case the load-deflection behaviour bifurcates; or the deflection itself ma
50 that the structure ‘snaps’ to a new position at which

suddenly change
equilibrium is re-established.
223 STRAIN ENERGY

As the structure defects, the strains and stresses in the structure change, and the
internal strain energy of the structure changes. The increase SU in the strain

21. 20 Fquilibrium, buckling, and total potential

tte

energy U is defined by
fi.) e
su= | [["oac}ar es

ds
which Viste volume of the structure, and Land I represent the initial and
fia strain sates of the Structure, as shown in Figure 24a. For a linear elastic

Equilibrium 21

the total strain energy is

in which a and £ now represent the stress and strain in the final state II, as
indicated in Figure 24b.

Both the stress-strain and the strain-deflection relationships may be non.
linear, in which case Ihe strain energy U contains deflection terms of cubic order
‘of higher, so that

es

Wien both the stress-strain and the strain-deflection relationships are linear,
then a, =4 = --=0 and the strain energy U is a quadratic function of the
deflection 0

224 work

As the loads Q on the structure and its deflections y increase, work must be done
by an external power source in applying additional loads to the structure and in
increasing the deflections. The work done by the external power source

a transfer of energy from the power source to the structure and its

The work W done on loads Q which deflect vin the directions of action of the
loads is given by

w=-E ("040 5

inwhich the minus sign indicates that negative work is done ifthe load Qis moved
in its own direction, and positive work is done if the load Q is moved in the
‘opposite direction. Negative work done on a load when it is moved in its own
direction may also be thought of as positive work done by the load, in the sense
that energy equal to this positive work is transferred from the structure and its
loads to the external power source.

Ifthe load Q remains constant while it displaces vin its own direction, then the
external work done on itis obtained from equation 25 as

W=-0 es

in which Q and v represent the values for the final state I, For example, the lifting
ofa gravity load Q up through a distance h from the ground to the structure
shown in Figure 2

requires positive work Oh to be done on the load. I this load
isthen added to the structure, causing it to deflect downwards by», then a further
amount of (negative) work — Qvis done on the load. Thus the total work done on

22. 2 Equilibrium, bucking and total potential
a
io fa
I „|
T Td
€ i
— a
i
22 ee Ben
; n A

Figure 25 Establishment of an equilibrium position,

the load Q during these operations is
W = Qh-0ù en

Ifa load Q increases linearly with its deflection v, so that Q = ko, then the external
work done on it is obtained from equation 25 as

W=-100 es

«das the work which must be done on

‘The potential energy Y ofa load Qis defi

the constant load to move its point of application from an initial reference

position to its final position. Thus

v=W=-Q0 es
when the (downwards) gravity load Q is moved (downwards) though a distance u
Note that the sense of isthe same as the direction of Q, and so the potential

energy decreases as the deflection increases. Equation 2.9 defining the potential
takes the convenient but arbitrary reference position which corresponds

to the zero displacement position (= 0) of the load.

An equilibrium position ofa structure is established by steadily adding load to it
gure 23. It is assumed that this

until the position is reached, as indicated in F

ic; thus there are no dynamic effet

the kinetic ene
s supports are frictionless and elastic, th

process is qua
Femains zero, the structure and it

ho energy losses due o fiction or plasticity, and each load continues to act in its
d is conservative. In this case energy is conser

total energy of the system remains constant

Care is needed in the interpretation of this statement, since the system must
include the power source used to move the loads. For example, in Figure 25, the
power source wil first do work } Or, in taking the weight ofthe load, where is

the upwards extension of the power hoist, and will then do work Qh in lifting the
Joad into position. It will finally do negative work — Q(e, +») as the force Q in
the hoist decreases to zero during the transfer ofthe load to the structure, Thus

the total work done by the power source is

AGRADE ein
4 W=Q(h— 30). Qu)
rites

r-0m-9 esa

Wala Pt es)
a

Ber pa
E u=10 e

and so the structure stores tr o the negati

in energy equal

‘of the work ~ 40»

done on a force which increases from 0 to Q as its deflection increases from 0 tov

23 Total potential

The total potential U ofa structure and its loads [1,2] is defined as the sum of
the strain energy U of the structur
that

“and the potential energy V of the loads, so

Ur=U+V 216

Usually, the potential energy datum istaken as the unloaded, unstrained position
ofthe structure, in which case U = 0 for this position

In general, the total potential depends on the load Q acting on the structure
and the deflected position v assumed, so that

Ur= (age? + aye! + au +) Ov em

23. 24 Equilibrium, buckling, and total

Figure 28 Total potential.

after using equations 2.4 and 29. The particular case where U = aze? is demon-
Strated in Figure 2 6a by the variations of U, — V = Qo, and U with Q and v. The
total potential U has a stationary value when dU,/de=0, in which case

0-
Stationary values of U are shown in Figure 268 for loads 0,04, and Qs.
The significance of the stationary condition of Uy will be discussed in sec
tion 243 following
‘Usually, only one value of o will correspond to the actual equilibrium position
of the structure under the action of Q, so that all other values of» are fictitious.
Thus, in general, the total potential represents a function whose value depends on
ssumed for the structure. The total potential therefore
ation of the actual

0)

the fictitious position v
does not of itself represent a useful concept in the conside
equilibrium condition of the structure. However, when itis combined with the
and conservation of energy discussed in the following

principles of virtual work
Sections, it becomes a powerful tool which can be used to determine the
‘equilibrium position ofthe structure, and the nature of the equilibrium condition,

24 Virtual work

24,1 RIGID noDIES

The principle of virtual work is most simply applied to a rigid body which i in
equilibrium under the action ofa set of constant forces Q,, Qs, ... and moments
M Mas shown in Figure 27a. Its assumed that the body and its actions

Virtual work 25

a rn E >

Figure 27 Virtual work and equilibrium

undergo virtual deformations 5,,50..., 60,, 602, These assumed deforma
tions are generally fictitious, are infinitesimally small, and are consistent, which in
this case means that they correspond to a single set of rigid body displacements
and rotations of the body and the points of application of its actions.

The principle of virtual work then states thatthe original position of the body
and its actions is one of equilibrium ifthe virtual work —E(Q do + M 40) done o
all the actions Q, M is zero for every consistent set of virtual deformations do, 6
Thus

SW = - Y (030 + M0) =0. (10)
For example, for the virtual deformations shown in Figure 27a,

50(Q2 sina + Q5) —6u(Q, + 0,0052) —60(—_M + bQ cos a + a0,) =0
(2.20)

and since this must hold for all sets 4v,5w,68, then each term must vanish
independently, whence

Q,sina +0, ea)
0,+Q,cosa=0, en)
M+bQ,cosa+aQ,=0, 2)

Which are the three equilibrium equations for the rigid body

242 DEFORMABLE BODIES
‘The extension of the principle of virtual work to a deformable body requires
the inclusion of the increase ÖU in the strain energy stored in the body as a result
ofits straining resulting from the virtual deformations. The principle of virtual

24. 26 Equilibrium, buckling, and total potential

work for a deformable body states that the original position of the body and its

ions sone of equlibrium ifthe sum of the increase 6U inthe train energy stored
in the body and the virtual work 8H done on its actions is zero for every consistent
set of virtual deformations. Thus

SU +5W=0,

656,

Ihe tension member shown in

For example, if the strain energy stored in
Figure 2.7 is written as

U = jean

for real displacements w= ,2/L, then for virtual displacements 5w = 31, 2/L,

the strain energy increases by
SU =(EA/Lyw 6, 226

and so the virtual work condition (equation 2.24) be

(EA/L) 5m, — O50, =0
hence
Q= EAw JL (28)

Which describes the linear relationship between the load Q and the equilibrium
displacement w, of the load Q acting on the tension member.

243 PRINCIPLE OF STATIONARY TOTAL POTENTIAL

The principle of virtual work for deformable bodies may be transformed to the

principle of stationary total potential by noting that in a conservative potential

energy system, the work dW done on the actions Q during virtual deformations
50 is equal to the change SV in the potential energy of these actions, so that
equation 224 becomes.
au +5v=0 (29
8U;=0 230)
after using equation 2.16.
Thus the principle of virtual work is identical with the principle of stationary
total potential, which states that an equilibrium position is one of stationary total
potential

Some writers describe equation 229 as an illustration of the principle of

conservation of energy in a conservative system, in that it requires a
SU in the strain energy to be matched by a corresponding decrease — 6V in the

potential energy. However, this interpretation isan artificial one, since it

changes which occur when virtual deformations 80 take place from the real

Virtual work 27

qulibriam postion oto an assumed poston (o +0) which is generally not one
equilibrio

When the total potential Uy varies with v and Q as in equation 2.17, then
equation 230 leads tothe equilibrium condition

= 20,0 + 3ay0? + dao 231)
which describes the load-deflction behaviour of the structure.
Conditions of this type are illustrated in Figure 28. When the total potential
Uy is quadratic (a, =a,=-~=0), the load-deflection behaviour of equa-
tion 231 is lin
However, when the total potential isc

ar, as shown in Figure 2.6b.

(a = =0) the load-deflection
behaviour is quadratic. When a, is positive so that d?Q/de? is positive, the
structurestifens under load, as for example is the case for tension members with
bending moments. When a, is negative so that d#0/dv* isnegativ

stilfness decreases with load, as isthe case for compression membe

with bending

moments. In this case, the load Q reaches a maximum value
When the total potential is quartic, the load-deflection behaviour is cubic.
When a tive and a, is positive, then the early decrease in stiffness is
followed by a subsequent increase in stiffness. A maximum load followed by a
snap will occur if a, <3a3/8a,, so that there isa load for which dQ/de = 0.

25. 28 Equilibrium, buckling, and total potential

2.5 Nature of equilibrium

25.1 STABLE, NEUTRAL, AND UNSTABLE EQUILIBRIUM

tral or

An equilibrium position of a structure under load may be stable,
‘unstable, One method of determining the type of equilibrium is to consider the
behaviour of the structure and itsloads when an infinitesimally small disturbance
is ist applied to displace the structure while its loads remain constant, and then
removed.

Ti the structure returns to its original position for every disturbance, then the
original equilibrium position is said to be stable. For example, consider the
Small sphere shown in Figure 29a atthe low point at Position 1 ofthe spherical
hollow, which is an equilibrium position for the sphere under the action ofits
‘own weight and the support reaction Ifthe sphere is displaced to Position 2 and
then released, it will roll down the hollow towards Position 1, and so this
‘equilibrium position is stable.

If, however, there is any disturbance for which the structure remains in the
displaced position after the disturbance is removed, then the original equilib
‘ium position is said to be neutral. Thus the sphere shown on a horizontal plane
in Figure 29b will remain in the displaced Position 2. Because the structure
remains in the displaced position after the disturbance is removed, this too is one
of equilibrium.

Finally i there is any disturbance for which the structure displaces further
from the original equilibrium position when the disturbance is removed, then the
equilibrium position is unstable. This is illustrated in Figure 2.9¢ for a sphere
‘which is initially in equilibrium at the high point of Position 1 of a spherical
mound, When this is displaced to Position 2 and then released, it continues to
move away from the equilibrium position

The nature of an equilibrium position may also be determined by considering the
energy input to or output from a structure and its loads during an infinitesimal

PR Tt A

Figure 28 Types of equilibrium.

Nature of eq

disturbance, in which the structure displaces dv to an adjacent equilibrium
position. Ifa force ÖF in the same direction as the displacement d is required to
Fhaintain equilibrium in the adjacent position, as shown in Figure 2.10a, then the

power source providing tis force does positive work

BW =48F60 232)

‘on the structure and its loads. This work done causes an incre
ff the strain energy and the potential energy of the system, so th

{ow = QU +8) e)
(note that SU + 8V =0 because the original position is one of equilibrium). In
this case the original position is stable
Ts, however, a force öF in the opposite sense to de is required to maintain
‘equilibrium (BF is negative), as shown in Figure 2.10%, then work — 192 W
Fv is done by the structure

and its loads, and energy is transfered to the
“+ 6°V) in the sum of the strain
the original position is

power source equal to the decrease —1(6
energy and potential energy of the system. In this ca
unstable.

The original position is one of neutral equilibrium when the work done
JóFóvis zero. In this case 6F =0, and the structure and it loads can be deflect du

Under a zero disturbance from the original position, as shown in Figure 2.106
While it does so, 1924 = 0, and so

HEU +EV)=0 234)

so that there is no change in the energy of the structure and its loads. Thus the
structure and its loads obey the law of conservation of energy while they deflect

F a

N |

| —> oF DE N
If toro GE 1

Figure 240 Equilibrium and conser

26. 30 Equilibrium, buckling, and total potential

from the original position of neutral equilibrium

algo ©

of equilibrium,

The conservation of energy condition at neutral equilibrium expressed by
equation 2.34 can be related directly to the total potential U (see equation 2.16)
by

15%0,=0. 635

When the equilibrium position is unstable, 16? U < 0, while 36*U > 0 when the

equilibrium position is stable. This is restated in the theorem of minimum total

potential, according to which the stationary value of Ur (for which SU = 0) of an
‘equilibrium position is a minimum when the position is stable.

This is illustrated by Curve 3 in Figure 28 for the case where
Up aye? + a — Qe

The neutral equilibrium conditions of SU, = Oand $57U

and a is negati
require that

o es

and
aa, 238)

so that
Qu= -aißa,. 239)

The equilibrium variation of @ with given by equation 2.37 is shown by Curve 3
in Figure 28, with the neutral equilibrium values of Q. and 2, given by
‘equation 2.39 and 2.38. It can be seen that the neutral equilibrium condition
corresponds to the maximum value Q, of Q.

The second variation of the total potential may be expressed as

all of) (240)

when o <q then $3?U;, > 0 50 that the equilibrium is stable,

FU, <0 so that the equilibrium is unstable. This

It can be seen tha
and that when
unstable condition “can be interpreted physically by noting that in this
condition ( >), the structure and its load require an external supporting force
(negative ÖF) to Keep them in equilibrium when they are disturbed do

‘Similarly if the total potential Uy is quartic as indicated for Curve 4 of
Figure 28, then the stiffness dQ/dv may steadily de

s the load increases

towards Q. and become zero at the maximum load Q,, at which the structure is
in neutral equilibrium. Alter this the load-deflection Behaviour shows negative
süffness and is unstable until a minimum load is reached, when the

load-dellection curve again starts rising. Any increase in load above O,, is

Bifurcation buckling 31

accompanied by a snap from the neutral equilibrium position », to the rising
foad-deflection curve where the structure is again stable

tion buckling

2.6 Bifures

261 LOAD-DEFLECTION PATHS

Under idealized circumstances, the load-deflection path of a structure may
suddenly branch or bifureate, as shown in Figure 2.11b and c. Thus the deflection
Suddenly becomes multi-valued atthe bifurcation load. After the bifurcation load
js passed, the original path, which isthe upper path, is unstable, and any small
disturbance will cause a snap to the lower path. Thus there is a change in the
Joad-deflection behaviour from the original pa
behaviour is called bifurcation buckling.
Bifurcation occurs when the lower branch or buckling path , is independent of
{or orthogonal to) the pre-buckling path x, so that there is no component in v
Which is of the same shape as y, Strictly, this idealized situation never oceurs in

10 the new path. This branching

fealty, because of the presence of small imperfections in real structures. These
il generally contain a small component of the same shape as ty, which is
Amplified as the bifurcation load is approached. The resulting load deflection
path changes smoothly, as indicated by the dashed line in Figure 2.1 lc. Neverthe
less the bifurcation buckling behaviour shown by the solid line of Figure 2.11c
can be regarded as the limiting behaviour of certain types of structure with
vanishingly small imperfections.

The load at which bifurcation buckling takes place is called the buckling load,
“and sometimes the critical load, to distinguish it from the maximum load of a
smoothly changing load -deflction behaviour.

The mode of bifurcation buckling considered in this book is one for which the

load initially remains constant along the buckling path, as shown in Figure 2.1 1e

buckling.

27. brium, buckling, and total potential

It occurs when the buckling behaviour is indey
negative) ofthe buckling path 0

dent of the sense (positive or

262 PRE-BUCKLING POSITION

At the buckling load, but before the change of path takes place (0, =0), the
‘deflected position v on the original load-deflecion path is one of equilibrium.
Thus, the principle of virtual work requires ¿Up =0 (Equation 2.30) for every
set of virtual displacements 6, as indicated in Figure 2.12a, which allows this
position to be found

2.63 BUCKLED POSITION

When the structure buckles under constant load from a pre-buckled position 10
‘buckled position (0 -+ y) then this buckled position is also one of equilibrium,
and so the principle of virtual work requires SU yy = 0 for virtual displacements
60, from this buckled position (0+ 0), as shown in Figure 2.12b.

Because the position » at buckling is one of neutral equilibrium, then the
principle of conservation of energy requires ¿6% x = 0 for displacements from
the pre-buckled position v. This second variation from the pre-buckled position v

may be considered as being equivalent to a first variation du, from the buckled
position (o + 1), which itself corresponds to a first variation x, from the unbuck
led position . Thus for bifurcation buckling, the principle of virtual work for the
equilibrium of the buckled position and the
buckling are equivalent. This equivale
Figure 2.12b

The principle of virtual work SUxy =0 can be used to obtain the differential
‘equations for equilibrium of the buckled position 1, These can also be obtained
by first using the law of conservation of energy 45°Up=0, and then using the

law of conservation of energy during
is illustrated diagrammatically in

$, TS

Figure 232 Pre-buckling equilibrium and bifurcation buckling,

In-plane behaviour of beam-columns 33

calculus of variations. These methods are applied in section 27.2 to demonstrate

the equivalence ofthe principle of virtual work and the law of conservation of
y for the example of the flexural buckling of a column,

In-plane behaviour of beam-colum

a

271 BENDING OF BEAMS

The elastic bending of doubly symmetric beams is analysed in section 17.1. For
the beam shown in Figure 2.13a which has equal and opposite end moments M,
the differential equation of equilibrium obtained by using the principle of virtual
work is

Mi=0 ean

in which

M,=—El,v e)
and which is subject to the boundary conditions
(MQ. = (Mo = M. (sa

Equation 241 expresses the equilibrium between the transverse distributed
load effect M° of the moments M and those of (zero) axial load N and (zero)
distributed load 4, (see also equation 17.19)

If equation 241 is integrated twice and the boundary conditions of
equation 2.43 substituted, then it is found that

M,=M (4

Figure 213 Bending ofa beam.

28. 34 Equilibrium, buckling, and total potential

and so
Elo =M 045)

after substitution into equa
between the internal moment of resistance — Elu
ect of the applied moments M.

ion 242. Equation 245 expresses the equality
generated by the bending of

the beam and the
The deflected shape is obta

d by integrating equation 2.45, and when the
beam is simply supported (vo = ë, = 0), then this is given by

Elo= (ML #/L) 246)
so that the central deflection v, is
1, =ML?/SEL, em

This linear relationship between M and y, is shown in Figure 2.136.

1 Pre-buckling beh

y be obtained

The elastic behaviour of the column shown in Figure 2.14a

from the analysis given in section 17.1. The pre-buckling deflections are given by
.=0 es)
and
w= (Q/E. (249)
T
Hb
Le er

Figure 214 Buckling ofa column.

In-plane behaviour of beam-colur 3
Equation 249 is derived from the approximate equilibrium equation
EAw'=—9 250)

which expresses the equality between the internal axial force resistance EAw
erated by extension of the column and the tensile axial force resultant
N=—Q of the applied load Q. The linear relationship between Q and —w,

‘obtained from equation 2.49 is shown in Figure 2.14b

2722 Buckling behaviour

‘The bifurcation buckling of the column from the pre-buckled position {0,w} to
(o vw + 1) is analysed in section 17.1.7, where iti shown that the condition of
neutral equilibrium leads to

es)

This is the energy equation for column buckling. It expresses the principle of
conservation of energy during buckling which requires the increase in the strain
energy 1£ELOgPdz to balance the decrease in the potential energy

FENG? de caused by the applied load Q(=—N) moving downwards
through a distance

This distance is equal to the apparent axial shortening of the projection of the
deflected column on the original Z axis resulting from the slope e, of the column
(note that X, KZ are used to represent the original axes of the undeformed
member, and, ),2 to represent the axes after deformation)

Fora simply supported column (9, = 0, =0) a possible buckled shape is given
by

L

Which leads to an estimate of the buckling load as

O=P,= ELLE ess
for any magnitude n, ofthe buckled shape. The corresponding axial shorteningis
Wyn = AL ess

The buckling behaviour is shown in Figure 2.14b. For loads Q less than P,, the
P,, this becomes indeterminate,

transverse displacement u, is zero, but at Q
Similarly, at loads less than P,, the axial displacement —1 inc
with Q, but becomes indeterminate at P,. Thus the behaviour bifurcates at
P, from the prebuckling displacements (0,1) 10 {mew +), as shown in
Figure 2.14b.

cases linearly

29. 30 Eauilibrium, buckling, and total potential

tis shown in section 17.1.8 that the differential equation for equilibrium of the
buckled position can be obtained either by using the virtual work equation or by
culus of variations [3] on the energy equation (equation 2.51), This

(ELLE (Né =0 es,
which is subject 10
LEE + Nido = 0.) de
Lits =0.$ e
Equation 2.56 expresses the equality between the resistance (El aj)” o transverse

distributed load generated by the bending ofthe column during buckling and the
transverse distributed load component (No) of the tensile axial force resultant
N. The second boundary conditions of equation 2.57 express the zero momer
conditions atthe simply supported ends, and the fist conditions express the zero.
shear conditions at the ends required by the symmetry of the column and its
loading.

It is readily verified that for a simply supported column, these equations are
satisfied by the buckled shape of equation 2.53 when the applied load Q(= — N)
qual to the buckling load given by equation 2.54

27.3 NON-LINEAR BENDING OF BEAM-COLUMNS

wiour of a beam-column is a

The elastic be lysed in section 17.1. If the
beam-column has equal and opposite end loads and end moments, and is simply

supported so that vy =u, =0 as shown in Figure 2.1Sa, then the differential

T ad

Non-linear bending ofa beam-column

Flexural-torsional buckling 37

equilibrium equations are

Mi (Ney =0
N'=0

in which
M,=~El,v (2.60)
N=EAw +03/2) 261)

and which are subject to certain boundary conditions. Equation 2.58 expresses

the equilibrium between the transverse distributed load effect — M: of the
bending moment M, and the corresponding effect (Nv) ofthe tensil
resultant N (See also equation 17.19), Equation 2.59 expresses the

between the longitudinal distributed load effect N” ofthe axial force resultant N
“and the (zero) distributed longitudinal load g, (se also equation 17.20
Equations 2.58-2.61 can be solved approximately by assuming that

N= Edw as)
Which leads to
N=-0 es
so that — w/= (Q/BA) zas in equation 249, and
EE)
van (3, (5) Jon J es)
EEE \ 265)

‘at = L/2. The dimensionless non-linear relationships between M and v, given by
Sb. At low loads Q, the behaviour approxi:
‘mates the linear beam behaviour of equation 2.47, but as the loads inerea

this equation are shown in Figur
the

behaviour becomes increasingly non-linear, and asymptotes towards the column
buckling behaviour as the load Q approaches the buckling load P, of equa:
4

2.8 Flexural-torsional buckling

28.1 GENERAL
The clastic bending behaviour of members with longitudinal and transverse
forces and bending moments is analysed in section 17.1, while the elastic uniform

30. 58 Equilibrium, buckling, and total potential

analysed in sections 17.2 and 17.3. The approximate energy and differential
al rium equations are derived fr Nexuralorsonal buckling in sections 17.4
sa Ts These approximate equations are discused in this section for the
Dear ence of concentrically loaded columns, of beams bent in principal

282 PRE-BUCKLING BEHAVIOUR

The pre-buckling displacements consist of longitudinal displacements w in the
0), in-plane transverse dis
+ 0)

case of concentrically loaded columns (u
placements in the case of beams bent in the.
and both v and win the case of beam-colum
W=6=0.

The pre-buckling displacements induce longitudinal normal strains at a point

principal plane (u
bent in the yz plane of symmetry

P in the cross-section, which may be approximated by
gow yo 2.66)

and corresponding stresses
p= Eby en

These have stress resultants

2.68)
2.69)
70)
so that
N= Edw
M,=—El,v

W= —Elyw'— Ely,

in which Ay ly Ip are

em

Flexural-torsional buckling 30

|
2+y) da, |

relationships allow the stress y to be expressed by
My! 19
The stress resultants of equations 2.68-2.70 consist of an axial tension force N an
in-plane bending moment M, acting about the x axis, and a quantity W which
may be referred to as a "Wagner. While this attr stress resultant plays no part in
either the linear approximation for the pre-buckling stresses ap (equation 2.73),

mination ofthe approximate pre-buckling deflections v and w from

œ=N

orin the d
equations 271 and 2.72, yet it does play an important role in the flexural
torsional buckling behaviour, as will be seen in section 283,

283.1 Energy equation

When the axial forces inducing compressive stress resultants P= — N in an
axially loaded column are large enough, the column may bifurcate from its

pre-buckled position (0,0,1,0) by deflect and twisting dy 10
{us Bow + Ms). The energy equation for the buckled position is approximated
by (ee section 17.46)

MARS

x +868 2x0

Wig} dz=0. (276)

In this equation, P is positive for columns in compression.

The first term ofthis equation represents the additional strain ene
the column during buckling, and includes components associated with bending
curvatures u and of about the y and x axes, warping “wistatures' dj, due 10

non-uniform’ twisting, and twists dí. The second term in equation 2.76

corresponds to the decrease in the potential energy of the axial forces during
buckling as the column shortens due to rotations uk and about the y and x axes
and twists dí, and includes contributions which allow for the difference between
the shortening of the shear centre axis (xo, o) and that ofthe centroidal axis (0,0)
along which the average axial forces act.

The components ofthe potential e
ly, by considering the displacements

y decrease may be interpreted physical

open +) $

31. 40 — Equilibrium, buckling, and total potential Flexural-torsional buckling a1

vu xx lO

N
wor °F

si hs wal Figure 247 Components of axial force P.
Substituting op = — P/A and integrating leads to
shown in Figure 2.16 of a point P(x,3) in the cross-section to P, caused by the
shear centre displacements 14,04, and the twist rotation d, about the shear Que = Pi, + yb) 1 ak
centre. A longitudinal element de x A through P rotates Qn = Plo, x061) J à
= (y pod
oan + (= x em) Moo =P

for small displacements uv, diles
transverse displacements u0z and v.52 (Figure 2.18), and so do work equivalent

Lo potential energy decreases of

and so the sorce ayó acting through P has components 4 5A up, 07648
acting normal to the rotated element, as shown in f

ure 2.17, These components,

give rise to transverse forces

ae air] Quide= sf Pla? rodas, |
| em ee | es)
0, f oda | )

32. 42 — Equilibrium, buckling, and total potential

e 247 N bz
|
| “ns Dire scene

It can be seen that equations 2.83 and 2.84 provide work components wh

‘equivalent to the potential energy decrease of equation 2.76.

‘Alternative physical interpretations ofthe potential energy decrease of equa-
tion 2.76 may be made by considering the rotations uy 0} (equation 2.78) of an
element through a point P(x, )) in the cross-section, These rotations cause the
projection of the element on to the Z axis to shorten by

chare

Bde = Su? + of) 52

approximately, as demonstrated in F 9. Thus the work done by a force

0,54 acting on an element of area 5A t

5 Wp = Ku? + 0f)0p5ASz 286)

The total work done is ther!

wo [| sont? +1P)aae um

Substituting equation 2.78 and 9, = — P/A also leads to work terms which are
equivalent to the potential energy decrease of equation 2.76,
In the special case where the shear centre and centroid coincide, the work done

Bz cosup
211-272) 7

Up

62

Ku

reduces to

the average shortenings

a

288)

interpreted as the work done by P due to

feeds and ifs de resulting from the rotations

and vj, The third term corresponds to the work done by P due to the average

shortening $5 (1/4) $4
More generally, the shortening due to the twist dh ca

sulting from the twist dí

considering the rotation 2,0%, of a longitudinal element
point Pl, y) ata dista

CENTRE)

be determined by
fea 5A through a

289)

from the shear centre as shown in Figure 220, Thus the shortening of the

a
5p = 5 (00605:

and the total average shortening is
LTT ua
34) osea fora

s+ y3} 92:

Thus the work done by the load P due to the twist dí is given by

weil Pag 4x3 +38

2.90)

es)

33. 44 — Equilibrium, buckling, and total potential

Figure 220 Rotation due to twist

It may be noted that the component [5 PUly/A)gde in equations 276,

284,288 and 292 may be re-expressed as — 4/5 Wide in which W is the

"Wagner stress resultant obtained from equations 271,273 and 2.75 as
W=—Ply/A 293)

when M, =0. Thus the work done during buckling includes a contribution from
the‘ Wagner’ stress resultant. This contribution was first reported by Wagner [4],
and is responsible for the torsional buckling (u, = v, = 0) of doubly sym

28.3.2 Differential equilibrium equations

The differential equilibrium equations for the flexural-torsional buckling of an
axially loaded column are approximated by Gee section 17.5.1)

vod} =0, 94)

Flexural-torsional buckling 45
Eloi) + (Plo, x =0, 095)

(EL. $0) — (C6) + (Pau — (Prat) + (PUJA +

+ydé} =0.
2.96)

In these equations, P is positive for columns in compression.
The first of these equations expresses the equality between the internal
istributed load in the x direction generated by

resistance (E1,u)" to transverse

‘bending about the y axis of the column during buckling and the transverse

load component — {P(u, + od) of the compressive axial force resultant P
(Gee equation 281) caused by th

rotation (1, + yogi). The second equation
{equation 295) expresses a similar equality for bending about the x axis.

The third equilibrium equation (equation 296) expresses the equality between
the sum (El. $4) —(GIG,) of the internal warping and uniform torsion
resistances 10 distributed torque by the warping and twisting of

the column during buckling and the distributed torque components of the
compressive axial force stress resultant P (see equation 282) caused by the
rotations ul, 2% and the twist dí

284 BIFURCATION BUCKLING OF MONOSYMMETRIC BEAMS
284.1 Energy equation

When the transverse loads and moments inducing stress resultants M, in a
monosymmetric beam bent in the principal YZ plane are large enough,
the beam may bifurcate from its pre-buckled position (0,2, ,0) by deflecting
laterally u, and twisting 6, to (4, + Pa + Woy) The energy equation for the
buckled position is approximated by (ee section 17.4.7)

EI u? + ELLE + GG

Ne
AS

elle 290)6 de =0 en

for shear centre loading.
The first term of this equation

presents the increase in the strain energy

stored in the beam during buckling due to bending curvatures u about the y axis,

he second term in
equation 2.97 is equal to the increase in the potential energy of the loading
system, This second term is negative, and so corresponds to a decrease in the
potential energy, equal to the work done by the bending moment M, as the beam
deflects 1, due to the combined effects of the lateral deflection u, and twist

A physical interpretation of the first part —¿52M,ópugdz of the decrease

in the potential energy of the loading system given in equation 297 can be

obtained by considering the curvature cy = 614 in the YZ plane of bending

34. so um, buckling and total potential
x z
===
Curvature vy"= uy
(a) Section (e) Elevation
led shear

2 Curvature u

(b) Plan

Figure 221 Curvature of buckled beam,

resulting from the minor axis curvature uf and the twist rotation dy shown in
Figure 221. This curvature causes the moment M, to do work
2M bp uk de 2.98)

AfS2M,@yuidz of the
stem given in equation 297 can
be obtained by separating it nto three components

1
IT Maids

MG $y) de,
299)

The frst component }[$.M, ul dz arises from the transverse forces
Qu=M, 100)
obtained by substituting 07 =M,y/1, into equation 279, The transverse forces

undergo differential transverse displacements 1,52 as shown in Figure 2.18, and
so do work equivalent to a potential energy decrease of

Qurisds = À | Madrid 101)

The second component $f M,uidi de of equation 299 arises from the first

raltorsional buckling 47

portions of the torques
Mu = Mau M (pull

Dé 2.102)

obtained by substituting a = M, y/L, into equation 2.79. The torques undergo

differential twist rotations 6,62 as shown in Figure 2.18, and so do work

equivalent to a potential energy decrease of

Mod | (Mando Multi 2908

ar (2103)

For a doubly symmetrie section, Ip, = yo = 0, and so

nei"

109

Maud de

EM bar of
rotations — d(— yujdu)/dy= uk, which occur during buckling as a result of
longitudinal buckling displacements — jugé. Differential rotations (uy) 5
cause the moments M, to do work equivalent toa potential energy de

SEM dy) de

Fora monosymmetri section, the additional decrease {EM (l./L,
4 in the potential energy of he loading system given in equation 297 arises
from the second portions ofthe torques of equation 2.102. Thus the potential
energy decrease resulting from the diflerential twist rotations 4402 of these
torques is given by equation 2.103,

"t may benoted that the component [5 M,(/,/I,)6 dz in equation 297 may
be re-expressed as À fé Wi de, in which WV is the Wagner stress resultant
obtained from equation 2.73 as W = M,[p,/1, when P = 0. Thus the work done

The third component equation 299 arises from small

during buckling includes a contribution from the ‘Wagner’ stress resultant.

284.2 Differential equilibrium equations

‘The differential equilibrium equations for the flexural-torsional buckling of a
‘monosymmetric beam bent in the yz principal plane are approximated by (see
section 17.52)

(Ely UY + (Mid =0, 2105)

(EI (GI) + Maa — (Mallo ll, —29065)'=0. (2109

The first of these equations expresses the equality between the internal resistanc
(EI ui)" to transverse distributed loadin the x direction generated by the bending
about the y axis ofthe beam during buckling and the transverse load component
—(M,¢4)" of the bending moment M, caused by the twist rotation dj.

The second equilibrium equation (equation 2.106) expresses the equality be-
tween the sum (EL. 6)" — (6/6) of the internal warping and uniform torsion

resistances to distributed torque generated by the warping and twisting of the
beam during buckling, and the distributed torque components of the ben
moment stress resultant M, and the ‘Wagner’ stress resultant W = M,Ip,/l,

35. 40 — Equilibrium, buckling, and total potential

285 BIFURCATION BUCKLING OF MONOSYMMETRIC

285.1 Energy equation

When the axial loads and the transverse loads and moments bend a
monosymmetric beam-column in the yz plane of symmetry (xp =0), the beam-
column may bifureate from its prebuckled position (0,2,,0) by deflecting
laterally u, and twisting d 10 (uy, 0+ 0 1+ WP). The energy equation for the
buckled position is approximated by (see section 17.4.7)

e

+ Eli? + GIG) de

10 tu + (p/A+

&=0

246 2.107)

+ (Ura

for shear centre loading. In this equation, P is positive for beam-columns in
compression. Equation 2.107 is a combination of equation 2.76 for column
buckling (with v= 0 and x, =0) and equation 2.97 for beam buckling.

285.2 Differential equilibrium equations

The differential equilibrium equations for the flexural-torsional buckling of
a monosymmetric beam-column bent in the yz plane of symmetry are
‘approximated by (see section 17.52)

(Eu) + {PU + yo}! + Mad)” =0,

2.108)
(E1,66)" (CIE + (Pro) + {PUy/A + OY
0

+ Mau + (Moll. + 2906 2.109)

These equations are combinations of equation 2.94 for column buckling with
equation 2.105 for beam buckling, and of equation 2.96 with equation 2.106,

2.9 References

1. Langhaar, HLL (1962) Energy Methods in Applied Mechanic, John Wiley, New York

2. Rubinstein, ME. (1970) Structural Systems ~ Statistics, Dynamics, and Stability, Prem
tice Hal, Englewood Clif, NJ

3. Courant, R (1948) Diferental and Integral Calculus, Volume I, Blackie, London.

4. Wagner, H, (1936) Verdrehung und Knickung von offenen Proflen (Torsion and
Buckling of open sections) NACA Technical Memorandum, No. 807

3 Buckling analysis of simple
tructures

31 General

Flexuraltorsional buckling depends on many parameters, including those
defining the structural geometry, the support and restraint conditions, the
material properties, and the load arrangement. Because of these many par-
ameter, it is mot possible even in the most detailed survey to include al of the
solutions that may be needed. I is therefore necessary to present methods of
analysing a wide range of situations. A computer method of analysing very wide
range of lexural-torsional buckling problems which is capable of high accuracy is
presented in Chapter 4

In this chapter, the energy method [1-3] for the analysis of flexural-torsional
buckling is presented in a form which is suitable for hand use. Such a method is
limited to problems whose solutions do not require excessive computational
effort from the analyst. For this to be so, the problems must be comparatively
simple, and solutions of moderate accuracy must be acceptable,

While these conditions may seem to be restrictive, they nevertheless allow a

wide range of engineering problems to be solved with an order of accuracy which
is consistent with the accuracy with which many of the controlling parameters
can be determined. Even when higher accuracy is required, approximate sol
tions usually allow the more important parameters to be identified, and may
suggest methods of closely approximating more accurate solutions obtained in
other ways.

The energy method is presented and demonstrated and its accuracy discussed
in section 32. The choice of suitable buckled shapes is considered in section 3.3,
five examples are worked in section 3.4, and methods of increasing the accuracy

are given in section 3.5.

3.2 The energy method

32.1 BUCKLING AND CONSERVATION OF ENERGY

The condition of neutral equilibrium at bifurcation buckling follows from the
principle of conservation of energy (section 2:5) As the structure under a fixed set
of loads buckles from an unbuckled position in a quasi-static manner to an
adjacent buckled position which is one of equilibrium, the increase in the stain
energy 40°U stored in the structure is matched by an equal decrease in the

36. 50 Buckling analysis of simple structures

potential energy — 182 of the loads. Thus the energy equation at bucklin

be expressed as

(

AU + FV) =0. EN]

For the flexural-torsional buckling of a column (section 28.3) these changes are
given by

(section 24.5) in which u, , d are the buckling deflections and twist rotation, and

P is positive for members in compression

‘The strain energy changes $5°U in these equations include the strain energy
changes due to bending about both or one of the x,y principal axes, and to
warping and uniform torsion caused by the twist rotations $. The potential

energy changes 19%V are expressed in terms of the stress resultants P and M,

induced by the applied loads

322 DEMONSTRATION OF THE ENERGY METHOD.

method will be used to find the
approximate flexural compressive buckling load Q of the simply supported
column (M, =0) shown in Figure 3.1. For this mode of buckling, »= 6 = 0, and
50 equations 3.1 and 32 simplify to

For demonstration purposes, the ene

64

Aer replacing the compressive stress resultant P by Q-

The first step in solving this equation approximately sto guess a buckled shape
which satisfies at least the kinematic boundary conditions (see section 3.3.1),
which relate to the deflection and rota

jon constraints. In this problem, these

The energy method 51

À Xu

(a) Unicaded DI Unbueteg Buckled

Figure 22 Flex

boundary conditions are those of displacement prevented at the column ends, so
that

uy =u,=0. 0s

A very simple

ui

66)

which satisfies the boundary co

ditions of equation 3.5. In this equation, dis the
gnitude of the central deflection w,3. The value of à does not need to be
appear in both terms of equation 3.4 and cancel out.

known, since it
The Second step is to substitute the guessed buckled shape into the energy
equation and integr

e. In this problem, this leads to

5 [eras%- 212? 2149)

whence
EL 2 L— $Q(49)4L/L? UL + 4/318) =0, 68)
which simplifies to
Q= ELL? 69

which is an approximate solution for the buckling load
This approximate solution is 22% higher than the correct solution of

O= MEN 610)
which corresponds to a buckled shape

u=ösin em

37. 52 Buckling analysis of simple structures

This buckled shape also satisfies the boundary conditions of equation 3.5, and
when substituted into equation 34 leads to

BELA RL (L/2)—$O54R/L)(L2)=0 613
method will
led shape is
assumed. However, thisis notin itself useful conclusion, since itis almost always
the case that the correct buckled shape is unknown. It does, however, allow it to
be inferred that the more accurate the assumed buckled shape, then the more

which simpliie to equation 3.10. This demonstrates that the
give the correct solution for the buckling load when th

accurate is the buckling load calculated

323 ACCURACY OF THE ENERGY METHOD

The energy method gives a buckling load solution which is more accurate than
the buckled shape guessed, provided this is reasonably close to the true buckled
shape. This is because the energy method always provides an upper bound to the
true buckling load when the guessed buckled shape satisfies atleast the kinematic
boundary conditions (see section 33.1) as is shown in section 17.110.

“The upper bound nature of the calculated buckling load is demonstrated in
Figure 32 which indicates its variation with the buckling shape guessed. Ifthe
true shape is guessed, then the true buckling load is calculated, which is equal to
the minimum value ofall the calculated loads,

El Lera
cg os

5 | e

3 al gi

Guessed buckled shape

Figure 92 Variation of calculated load with guessed shape

Choosing the buckled shape 53

The upper bound nature ofthe calculated buckling load is responsible for the
comparatively high accuracy of the buckling load predicted. Provided the
guessed shape is reasonably close to the true shape, then the calculated load,
Which varies slowly with the guessed shape inthis region, is quite close tothe true
buckling load,

3.3 Choosing the buckled shape

331 BOUNDARY CONDITIONS

The accuracy ofthe buckling load calculated by the energy method is related to
the accuracy of the buckled shape assumed. Ideally, the buckled shape should
satisfy all the boundary conditions imposed by the method of support and
constraint of the structure, These boundary conditions are usually described as
being either kinematic or static.

Kinematic boundary conditions relate to the geometrical constraints at the
supports of the structure. These geometrical constraints prevent one or more

deflections or rotations at the support points. Thus deflection prevented, rotation
prevented (Figure 33), and warping (deflections) prevented at a support are
examples of kinematic boundary conditions.

Static boundary conditions relate to th
moments, shears, bimoments and warpin

values of stress resultants such as
torques at the supports. Thus the
static conditions ata frictionless hinge may be expressed by zero moment, and
those ata free end by zero moment and zero shear (Figure 33),

The buckled shape guessed should satisfy the kinematic boundary conditions,
as shapes which ignore geometrical constraints are likely tole
predictions which are much lower than the true buckling load,

to buckling load

On the other hand, the guessed buckle shape need not satisfy the static
boundary conditions, which are local expressions ofthe equilibrium equations.
Thisis consistent with the fact thatthe guessed shape will not generally satisfy the

ed parabolic
buckled shape of equation 3.6 does satisfy the kinematic boundary conditions

differential equilibrium equations. For example, while the y

Kinematic

dary conditions fora cantilever

38. 54 Buckling analysis of simple structure

|

eye ee eee

p=, = 0, lts curvature u’ = ~86/L? does not satisfy the static zero moment
boundary conditions El,us = Eli =0.
332 POWER SERIES

Perhaps the simplest buckled shapes can be obtained by using a limited power

afl

u=o+taz/L+a,

in which the values of the coefficients da... a, are selected to satisfy the
kinematic boundary conditions. The coc

les of limited

power series are shown in Fi

0 the ease of ints

The simplicity ofa limited power series is du
products which aris in

Buckled shapes may also be obtained by using a limited sine series

L+

u= a, sin n2/L + a,$in 2

a, sin nL 0)

Each term of such a ser

les automatically saisies the boundary conditions
allows a single

p= u, = 0, which ol

The terms of such a series are orthogonal (section 17.1.9), which sometimes

leads to simplifications when the square:

the series is to be integrated as part of
the solution ofthe energy equation. For example, it is assumed that

+a, sin 3nz/L e

for the column buckling problem of section 322, then

[rar = mn + 8103 6.10

Choosing the buckled shape 55

with the coefficient of the eross-produet a, a, equal to zero
More generally, however, the use of trigonometric series may lead to integra
tion difficulties as in beam lateral buckling problems where the beni

M, varies asa limited power series. For example, when

M,=(al?/2

em

as it does for a simply supported beam with uniformly distributed loading 4,
then the work done term $
type

DM ‚Su de of equation 33 will have terms of the

AM sn
These dicts cn be overcome by malin we ld vals of
Table Em BR and mu 18

Y sin“
7 ar

L

Table 3. Definite integrals

= æ + I 2 3 :
102500 Om) osé os 0563
12 O2 009 OS 020! 0204
1 3 0006 006 0) 0286 -03726
2 2 0200 O6) oma ie ums
3 5 om où? aie au -00810
01 EY) (EE ee
A mo 2 3 :
1 0000 0754 1399 55561
3 000 02829889 Soc
3 0000 0000 DIS ake 20080
E JMS E
3 0000 03056 03000 6968
3 0m ommse 10m 92013

39. 56 Buckding analysis of simple structures

1 02500 ONE 00806 00608 00626
1 061-0002 007 —0068S -008
00000 006 0038 OMIS QI
2 0250001968 02 OS) 04382
2 0010 dia O7 026
3 02500 0017 O0 020s
aaa
momo 2 3 3
1 0500007854884 13926
12 0mo 03537 Ir 37542
103 0m ao 03125 53716
2 2 Osmo OS 17074 1092
2 3 Gao 03810 100 $5095
3 000 OS 1627 10280

3.4 Worked examples

34.1 SIMPLE BEAM IN UNIFORM BENDING

The energy equation for Mexural-orsional buckling of an elastie beam (sec
section 17.4.7) is given by

Lu egur+p,6) dz

Ito 01

For doubly symmetric beams, f, = yo =0. For a simply supported beam in
uniform bending (Figure 35a), M; = M and q, = Q, = 0.
Kit is assumed that

w/o = 4/0 = 2/L 620

>)

==

{al Simple Bean in Ur

ral Load at Shear

which satisy the kinematic boundary conditions

Won = bou =0 621)

(but not the static boundary conditions Myo, =Elyuj.=0 and Boy
El, 65 = 0) then

Lau 2} EPA + EIA + SILLA) de
Mmes a.
1 asp + EL +000
Los
Ly meee ston
—
11 [eV Tir, m 1föl_,
facto} | wre Hof

‘This is satisfied when
iene -M

=0, 524
-M (GI+12EL, Lai

40. 58 Buckling analysis of simple structures

whence
M 625
which is reasonably lose tothe exact solution (ee Section 7.2.1)
(( EE) 3.26)
mu, (E (6 26
Alternatively, it may be assumed hat
a = 6/0 = sin 627)

which satisfies the kinematic boundary conditions of equation 321. In this case

P+ ELA Ley sin? xz

eu =} | or

os? nz/L} de

2MB0(— x3/L) sin? nz/Ldz

Using the appropriate integrals from Table 3.1 and substituting leads to

OUEST M fl
22110 NO

which is satisfied by the exact solution of equation 326

342 SIMPLE BEAM WITH A CENTRAL SHEAR CENTRE LOAD

concentrated load

Fora simply supported doubly symmetric beam with acen
Q (Figure 3,5b), M, = 0z/2 while 0<z < L/2. Ifthe parabolic buckled shapes of
‘equation 3.20 are assumed, and half of the symmetric beam is analysed, then

sum ESO + non + 610

as in section 34.1, while

(Q2/2)(280)(@/L— 5050/96,

1 {VTT men
arte} | sous os

sou 7/8)

0 629)
TOS

Worked examples 50
so that

ot, ISR (o, REN
16 (E) (a+ BE)
OL _ 135 (PE (Gy PEL)
er

ü + RE JL?)

AL) (92 EL/L2) + O4GS

asin section 34.1, and

1

(0:/2)(250(

2)sin® xz/L dz = — Q60(n x 02759)

after using the appropriate integral from Table 3.1. Thus the energy equation

% „1m (SE) (os PB) en

Ifthe central concentrated load of the simple beam analysed in section 342 acts

low the centroidal axis as shown in Figure 3.50, then the

at a distance

potential energy change during buckling must be increased to

de [phone]

to account for the work done by the concentrated load Q moving upwards
through the small distance yo¢ 2/2.
IF the parabolic sha

es of equation 3.20 are assumed, then this leads to
5050 1, pf1_(1Y1

F 56 +30

11 f6)"[ 12E1,/12
235105 L- ser

son a)

(Gd +12E1,/12 + 30Ly9/16) | 05

o 639

41. 60 Buckling analysis of simple structures
OL {En (os. REL, (The) |, 228), q

sl 7 + 630
EN Ya) 7
If the half sine wave bucked shapes of equation 327 are assumed, then this
leads to

La = — 0808 x 02159) + 10g,

12 (3) (8x 02759) 01/4
33010) | -«x02759/m9144 (014 PEL /L? +20L ele)

The solution of this can be expressed as

oL

Bt 1:00 |[1+ (05075 jrs) 0
in which
Myo L2)(G1-+ PELL) 63
and
py EJ 640)
solution
le ou

obtained from section 7.6.1

344 SIMPLE BEAM UNDER MOMENT GRADIENT

Forthe simply supported doubly symmetric beam with unequal end moments M

and AM shown in Figure 364,
M,= MU (1 + By/L 642

For uniform bending, B= — 1, and the beam buckles with the half sine waves of
equation 3.27, while for double curvature bending (f = 1), the deflected sha
antisymmetrica, while the twist shape is symmetrical, as shown in Figure 3.60

This suggests that the buckled shapes n

be approximated by

= Osin xz/L | ae
u= 6, sinz/L + 5,sin2nz/L5

Worked examples 61

>> oc u

Wit rotation ®

{9 Sucios 5

ps
Figure 86 Beam under moment gradient
so that 6, = 0 when f = —1,and 3, =O when f= + 1. Equations 3.43 satisfy the

kinematic boundary conditions of equation 321
Usi

leads to

g the buckled shapes of equation 343

*p1,/L*)| (6, sinaz/L + 46, sin

1
ET,
2 + GI6*)dz EPL,
if 1 gl
[Pao gut dz = — 3200/12)
x (6, sin? na/L+ 43, sin de

MO(OSó,—(1 + M[O78545, —4 x 028296, Ur}

so that the energy equation becomes

le BIL mi -(1+ m2) (6,
mes 6 OTOSMA +1) o
o La +) -orosma+ 6 (+ Ren) Jo
64
This is satisfied when
1-4 AR? p
Ê OR , ss
PELL 16 ET JE

42. ut = = (3.46)
M78: 1.058 + 0.36? <25 (347)
i
Pa ne pe
Live! [ESO + En PO + GPL)
1 gp? 48101 4400080 | say
yal (Yorn sitet 602»ée= — 0066. |
ree 2 flo ay
of [note wars mue)

y accurate approxi:

This solution is substantially hi
mation of

OL//(EL,61)=395+352,/(*El,JGJL2)

obtained from section 9.3.1, This error arises because ofa conflict in the boundary
conditions at the support, where the kinematic condition of end warping

prevented requires d = 0 (equation 349a) for a cantile

Increasing the accuracy 88

ing rigidity El, Howev
boundary condition has no significance, and di, #0 (a more suitable twist shape
than equation 348 is used in section 3.5.2 to obtain a more accurate solution for

1, = 0) For cantilevers for which both GJ and El, are significant, the value of dí

fora cantilever with zero warping rigidity El, such a

usually increases very quickly from zero at the support, but then varies only
slowly. In this case itis difficult to select suitable twist buckled shapes for hand
and it is better to use a computer method to obtain solutions of

analysis,
reasonable accuracy,

In the case of a cantilever with El, =0, a more accurate solution can be
obtained by using the buckled shapes

umór/L 6:53)
4 6530)
for which
¡SU (SEL PL? + GPL)
and
82V = — 006/3.
Thus
1 féVTaEIL —0L/3](6)
{oy {3} 3.54
aL of | -oLg os Jos" oom
whence
OL?/(E1,G)) =6 659)

which is much closer than equation 3,51 (with EI, = 0) to equation 3.52

3.5 Increasing the accuracy

3.5.1 USING THE MINOR AXIS BENDING EQUATION

The a
buckling problems can often be significantly improved with comparatively litle
extra effort by making use of the minor axis bending differential equilibrium
‘equation, which often takes the form (see also equation 2.105)

racy of the hand energy method of solving flexural-torsion:

El = M6 659

except when minor axis end restraints induce restraining end moments and
shears. Equation 3.56 can be used to obtain

w= —M,@/El, 657)

so that a buckled shape for u is not required. In this case the accuracy of u is
directly related to that ofthe guessed twist shape 4, and is often more accurate
than the result of differentiating twice a guessed deflected shape u, This method is
sometimes referred to as Timoshenko's Energy Method [1]

When equation 3.57 is substituted into the appropriate terms of the energy

43. 64 Buckling analysis of simple structures

‘equation (equation 3.19) it leads to

and
EL:
À lernsez

65)

Equation 359 can be evaluated directly from equation 358, so that one less
integration needs to be carried out
For example for the simple beam in uniform bending analysed in section 3.4.1
LE leads to

the assumption of $/0

Thus the energy equation becomes

0, 6.60)

Lc OEI, 48h 1 + 0780} Tanya)

181) (or 28 )} ast

A

vil

Which is much closer than equation 3.25 to the e

solution of equation 326.
In the case of the simple beam with a central shear centre load analysed in

section 3:42, the assumption of 4/0 = 2/L—

[3 leads to

el 2/1

2M26%/E1 ds

(Q?6*L?/EI,)(29/26880)
‘Thus the energy equation becomes

((QPL?/E1, (29/53 760) + 4E1,/L? +

Lao (PEN (as REL) a)
N Can)

which is much closer than equation 3.30 to the accurate solution of equation
331

The accuracy ofthe hand energy method can also be improved by extending the
buckled shapes so that they satisfy the static boundary conditions as well as the

Increasing the accuracy 65

kinematic boundary conditions (see section 33.1). This method generally in
creases the number of terms in the buckled shape
amount of effort required to reach a solution,

ons for th

and therefore increases the

For example, the static boundary condi

€ simple beam in uniform

bending analysed in section 34.1 are
usu, =0 (64a)
for zero end moments M,, and
d=di=0 ab)

for zero warping restraint. These and the kinematic boundary conditions of
‘equation 32.1 are satisfied by the buckled shapes

Ws=9/0 TA 669

which have three terms instead ofthe two of equation 3.20 used in section 34.1 to
satisfy the kinematic boundary conditions only, Substituting equation 3.65 in the
energy equation leads to

1 LOVE 2401/51? VMpsL 758)
f , bo so
2 of | —17M/35L (176J/35L+ 24E1./51>)J10/ L
which s satisfied when
(oser 988261.) er
mm {CB (04282) os

Which is very lose to the exact solution of equation 3.26
In the case of the cantilever with 1,=0 and an end load analysed in see
tion 3.45, the static boundary condition of

6-0 (6:58)
for zero end torque M, is satisfied by

$/0=22/L —23/L:

G69)

which has two terms instead of the single term of equation 3.53b. Subst

equations 3.69 and 3.53a in the energy equation leads 10

1 féVTaEr Le — 91/2] (3)

a of L-one «css Jos” em

which is satisfied when
012//(E1,6N=361

Which is much closer than equation 3.55 to equation 3.52 (with El, =0).

353 MINIMIZING THE BUCKLING LOAD

The accuracy of the hand energy method can be improved by extending the
buckled shapes so that they contain an arbitrary parameter. The calculated

44. 66 Buckling analysis of simple struct

buckling load then depends on th a be minimized to

obtain the most accurate solution possible for all the buckled shapes defined in

arbitrary parameter, and

this way. This method requires an additional effort in the minimization process,
which is generally not justified for a hand calculation.

In the case ofthe simple beam in uniform bending analysed in section 3.4.1, the
parabolic buckled shapes of equation 3.20 can be extended to

u6= 6/0=(@/L- 2/12) + ae/L 24

in which « is an arbitrary parameter. Substituting these in the energy equation
leads to

Lapp ao Ma 24525097 Aa
alo | aras censo (090120520090 arr + ese, Jef
en
so that
MPL? _12(1-+a4/9){GI(t + 22/5 4202/35) + IDE 402/52)
El ++ 22797
e

Although this equation for M?L?/B1, can be minimized formally with respect to
a, much tedious manipulation can be avoided by programming equation 3.73
and finding a numerical minimum by rial and error. Such a procedure forthe two
limiting cases of GJ=0 and EI, =0 discloses minima for #= 1.0759, and
substituting this into equation 3.73 leads to

HE) 0m

which is very close to the exact solution of equation 3.26.

M

3.6 Problems

A uniform simply supported column has a compression load Q acting at one end,
and a second compression load Q acting at mid-height, which is unbraced, as
shown in Figure 37a. Determine approximate values of PL?/xEI at elastic
flexural buckling,

(a) by assuming u = 45(2/L — 23/12)

(0) by assuming u =ósinz1=/L

(©) by assuming an asymmetrical buckled shape which reflects the asymmetri
cal loading:

(@ by assuming buckled shape which satisfies both the static and the
kinematic boundary conditions; and

>>
— fi
LE

—— +
ST

wre 27 Problems 91-96.

(e) by assuming a variable buckled shape with an undetermined parameter,
and minimizing the calculated buckling load,

A non-uniform column is builtin at its base and free at its top, and has
compression loads per unit length q uniformly distributed down its length, as
shown in Figure 3.76. The flexural rigidity varies linearly from El at the bottom
to El/2 at the top. Determine an approximate value of dgL/n° EI at elastic
flexural buckling.

PROBLEM 33

A uniform simply supported column has a compression load Q at one end and
an elastic translational restraint at mid-height which exerts restraining force
At as shown in Figure 3/7, Determine the variation of Q1%/x"E] at flexural
buckling with the dimensionless brace stiffness aL>/El,

A simply supported column has a compression load Q and clastic rotational
restraints at both ends which exert restraining moments au, as shown in
Figure 3/4, Determine the variation of QL/x"EI at flexural buckling with the
dimensionless brace stiffness al.

45. 68 Buckling analysis of simple structures

A simply supported monosymmetric beam has a central concentrated load Q
acting a distance yg below the centroid at mid-span, which is unbraced, as shown,

in Figure 3.7e, Determine the variation of QL/4M,, at elastie Nesural-torsional
buckling with yoP,/M,. and B,P,/M,

(a) by assuming parabolic shapes for u and 6;

(b) by assuming half sine waves for u and q
(e) by assuming a parabolic shape for $ and using the minor axis bending
‘equation; and

(& by assuming shapes which satisfy both the static and the kinematic

boundary conditions.

A simply supported doubly symmetric beam (fi, = 0) has a uniformly distribute
load per unit length q acting at a distance y, below the centroid, as shown in
Figure 3,7. Determine the variation of gL*/8M,, at elastic Nexural-orsional

buckling with y,P,/M,

(a) by assuming parabolic shapes for u and y;

(0) by assuming half sine waves for u and 6;

(© by assuming a parabolic shape for ¢ and using the minor axis
equation: and

(d) by assuming shapes which satisfy both the static and the kinematic
boundary conditions.

ding

3.7 References

e, IM. (1961) Theory of Elastic Su

1. Timoshenko, SP. and G dition,

McGraw-Hil, New York,

2 Temple, G. and Bickley, W.G. (1956) Rayligh’s Principle and Its Applications to
Engineering, Dover Publications, New York

3. Bleich, F. (1952) Buckling Strength of Metal Structures, McGraw-Hill, New York.

4 Finite element buckling analysis

41 General
In Chapter 3, the energy method of analysing flexural-orsional buckling was
presented in a form which is suitable for hand use. The need to avoid excessive
Computational effort by the analyst limits the use of this hand method to
problems which are comparatively simple and for which solutions of moderate
“accuracy are acceptable

This chapter presents a computer n

1od of analysing flexural-torsional
buckling which is capable of very high accuracy, and which can be applied to a
very wide range of structures. The method is the same energy method as that

discussed in section 2.5 and used in Chapter 3, but is presented in the form of the
finite element method [1-5]

very widely u

The application ofthe finite element method to elastic buckling problems [6]

is discussed generally in section 42, and then partiularized to the flexural-
torsional buckling of columns in section 43, and to the Nexural-torsional bucı

ling of monosymmetric beam-columns in section 44

The results of many finite

ement analyses ofthe lexural-torsional buckling of

columns, beams, cantilevers, continuous beams, and beam-columns are included

in Chapters 5-11. Extensions of the finite element method of elastic buckling
Chapter 12 on frame buckling, in Chapter 13 on the
of arches and rings, and in Chapter 14 on inelastic buckling

analysis are discussed
bucklin

4.2 The fin

element method for buckling analysis

‘The application of the finite element method to buckling problems in general
[1-6], and to flexural-torsional buckling problems in particular [7-10] involves
replacement of the quantities in the energy equation for the complete structure

EU +82) =0 (ay

by the sums ofthe approximations $4°U 4
rk done on each of a finite number of elements into which the struct

V, forthe strain energies stored in
and wo

divided, so that

28V)=0 (42)

in which $5°7, represents the work that would be done on an initial load set, and
is the buckling load factor by which the initial load set must be multiplied to
‘obtain the buckling load set.

46. 70 Finite element buckling analysis
Ir

i

*

Figure 41 Division of a structure into elements

The fist sep is therefore to divide the complete structure into a number of
finite elements connected at nodes, as for example in Figure 4.1. Approximations
are then made for the variations of the buckling deformations u,v, over the
length of atypical element, which are then related to the element values (5,} of
the buckling nodal deformations. This allows the element strain energy stored
to be written as

19%, =4(8,)Tk.1(0,} 43)
and the element work done as
av =) 44)

in which [K,] is the element stiffness matrix, and [g,] is the element stability
matrix associated with the initial load set.

To facilitate the summations of equation 42, the element nodal buckling
deformations {5,}, which are defined in relation to the local element axis system,
are transformed to the global nodal buckling deformations (A) at all the nodes
defined in relation to the structure axis system. These transformations take the

form of
BJ = (TI) us
in which [7] is a transformation matrix.
Thisallows the element local stiffness and stability matrices to be transformed
to the element global matrices
(KI (TMT,
[= TTS

46

so that
you,
av.

HA)TEK.IC
safay"EG,3{a}.J

an

Flexural-torsional buckling of columns 71

These element global matrices can then be summed to form the structure stiffness

and stability matrices
tk]

(48)
16]

so that the ene

ay equation (equation 42) becomes
AJULK] +A1G])(8) =0. 49)
This summation may be carried out in stages after each element matrix [£.] or
La] is transformed,

The boundary (support) conditions of the structure will require that some of
the global nodal buckling deformations {A} are zero. In this ease, the global
stifness and stability matrices [X], [G] are reduced by eliminating the rows and
columns corresponding to the zero buckling deformations, These reductions

may be done in stages at each of the nodal deformation transformations of
equation 45.

Equation 49 is described as a ‘generalized linear cigen-problem [4]. When the
matrices are of order n, then it has eigenvalues 2, and n non-zero eigenvectors
{A}, which satisly it. The lowest eigenvalue 2, defines the load set at which the

structure first buckles, and the corresponding eigenvector (A), defines the

corresponding buckled shape or buckling mode of the structure, Numerical
procedures for determining the lowest eigenvalue 2, and the corresponding
eigenvector {A}, are discussed in [2,4, 5.

Alternatively, the
determinantal equation

values A, can be found by finding the roots of the

K+2G]=0. (4.10)

Methods of finding the lowest root are discussed in [11, 12}

4,3 Flexural-torsional buckling of columns

43.1 BUCKLING DEFORMATION FIELDS

A wide range of approximate deformation fields may be assumed for the shear
centre buckling deformations u, v, $ which occur during the flexural-torsional
buckling of column. It is most common to use cubic deformation fields of the
form

uma, +a,(e/L) +4 Hal) ay

in which Lis the length ofthe element (Figure 4.1b) and 2

clement. Equation 4.11 can be written as

thedistance along the

u=(Z}"{a} (412)

47. Finite element buckling analysis

(413)

as

ZI)
This simple field has four undetermined constants ay — as. These can be replaced

by the four nodal deformations

€ ends 1,2 ofthe element by modifying the field formulation. This sd
ing appropriate diffrentiations and substitutions so that

(8,) =1C,J(a)

in which

Equation 4.16 can be inver

Thus the deformation field can be rewritten as
u=(Z} TCI,

Similar cubies can be used to represent the other buckling deformations u
(u) = MIE (6.

in which

n=| {97 zit {or
{or {oT

(ag)

(4:16)

“m

CCR

19)

(420)

(421)

ua

423)

(424)

Flexural-torsional buckling of columns 73
and
AO
[cy 1 (cr m 425)
(II rea

432 ELEMENT STIFFNESS MATRIX
“The increase in the strain ene
flexural-torsional buckling (se

y stored in an element length Lduring column
ion 28.3.1)

180,4 | (ele? + Ble? + £107 + God (426)
can be expressed inthe form of Equation 43 by fist transform
ei 427)
in which (,} is a generalized
(428)
and [D,]is a generalized elasticity matrix
m, 0 0 s]
tele Beer ob (429)

0 0.60
o 0 0 EL]

The str

can be obtained by appropriate differentiation of the

deformation fields of Equation 4.21 as
6.) = [B,JLCT*(ó, (430)
in which
or for
(ey .
| or 431)
and
(ZY"= (0,1/L,2:/12,32/1) (432)
{ZY = (0,0,2/22,6/L (433)

Substituting equation 4.30 into equation 427 leads to

sur (larme) (430

48. 74 Finite element buckling analysis

and this is identical with equation 43 ifthe element stiffness matrix is written as

CARE | (8,J"0D,J08,Jdz TT] (435)

433 ELEMENT STABILITY MATRIX

The work done on an axial force acting on an element of length L during
83.1) can be expressed as

lexural-torsional buckling (section
1

90° - 20000 + 20114) dz

(436)
in which P isthe inital axial compression force and 2 is the load factor by which
the initial force must be multiplied to cause column buckling,

Equation 4.36 can be transformed to

re} TED Mo Jdz (437)

Ls
28,

in which {c,) is another generalized strain vector

= vey? (438)
and [D,] isa generalized initial stress matrix
Poo Pro
wj=-|0 P Px (439)
Pro —Pxo Plly/A+x3+99)

deformation fields of equation 4.21 as

The stra can be obtained by appropriate diflerentiation of the

fe) =[8,J1CJ*(5) (440)
in which
Er or or
1.3=| OF (27 OF aa)
or OF zy

Substituting equation 4.40 into equation 4.37 leads to

peru yucr( [EI ITB Mde)eCr* (3.1 442)

and this is identical with equation 44 if the element stability matrix is written

tod=ter( | to, 10,108.34 )1c1"" (43)

Flexural-torsional buckling of column:

434 DEFORMATION TRANSFORMATIONS

Before summing the strain energy changes in and work done on all the elements

of a structure, the element nodal buckling deformations {6,) are transformed
from the local element axis system to a global system which is common for all the
structure

Usually, the elements of a structure subject to column flexural-torsional
buckling have collinear z axes (structures whose elements are not collinear
usually have bending moments) although their local x, axes may be inclined at

angles «to the global X, Y axes, as shown in Figure 4.2. In this ease, the element
nodal buckling deformations

{en} = {ty (444)
at node n are related to the global buckling nodal deformations
{a} =( AJ} (445)
in which
(84) = {Une Vs par Oz (446)
by
[hal (4, am
Sn

49. 7% Finite element buckling analysis

cosa 0 sin 0 -» 0
0 cz 0 sima 0 -)0
sing 0 cosa © x 0 J
il 9 sn 0 cosa 0% ws)
0 0 0 0 1 0
9 0 0 0 o :

Most of the deformation transformations of equations 444-448 are obvious,
except for the transformation

6-02 (449)

Physically, this can only be interpreted in terms of compatibility of the end
warping displacements w = 06’ of two elements which have a common node.
Equation 4.49 can only represent this warping compatibility if theres no conflict
between the warping functions o of the two elements in those regions of the XY

ross-sections of the two elements coincide, This condition is

plane where t
most commonly satisfied when the two elements have identical and coinciding
cross-sections. When this is not the case, then continuity between the ni
coinciding cross-sections would normally only be met by the provision of a rigid
node element that prevents the warping (5° = 0) ofeach element.

435 BOUNDARY CONDITIONS

The boundary conditions for th
require one or more ofthe global nodal deformations {A} (see equation 4.46) to
bbe zero, corresponding Lo the prevention of the deformation. There should always

eflexural-torsional buckling ofa column usually

be a sufficient number of boundary conditions to prevent rigid body deflections
(U,V) or rotations (0,0x,22) of the column, which would occur under zero.
loads. For example, at one end of a cantilever, the boundary conditions

{Uo:0v0: Vor 0x0: 820 (430)
are suficient to prevent rigid body buckling motions,
The effects ofthe boundary conditions are allowed for by eliminating the rows

and columns of the global stiffness and stability matrices corresponding to the

zero nodal buckling deformations, usually as part of the nodal defor

transformations described in section 4.34.
Sometimes, boundary conditions are specified for an off-axis point Op(X Ya)
obal XYZ axis system,
} of

and directions 24 which do not coincide with those oft
Jobal nodal deformations |
deformations

as shown in Figure 43, In such a case, the
‘equation 4:46 can be transformed to the corresponding

ee (si)

Flexural-torsional buckling of columns 77

x Un xn

Up ay

Exa
Figure 43 Offaxis boundary con
by
à} = CT] ( (452)
in which
ia may 8 Tae ony 8 eas

436 INTERNAL HINGES

Some structures may have internal hinges which allow independent element
sata common node, For example, Nexural hinges ensure that
are zero, so that the rotations vu’ of

nodal deformation
ement moments M, or M, atthe hi
ent at the hinge are unrestrained, Other examples of hinges include

and

torsional hinges, which ensure zero values of the torques M, at the hin

50. 78 Finite clement buckling analysis

allow the twist rotations $ of each element to be unrestrained, and warping
hinges, which ensure zero values of the bimoments B at the hinge and allow
warping displacements w proportional to 6’ to be unrestrained.
al conditions in a practic

allow a particular type of hinge often imply other types as well. For example a
torsional hinge (M, = 0) often implies flexural hinges (M,, M, =0) while flexural
hinges (M,, M, = 0 often imply a warping hinge (B= 0), and vice versa

In static problems without stability effects ([G] = [0]), internal hinges are
usually allowed for by condensation techniques [2,4], in which the free nodal
deformation associated with the hinge is eliminated from the element stiffness
matrix [k,]. The use of such a technique in a stability problem will convert the

problem of equation 49 into a non-linear eigen-problem, which will

invalidate the numerical procedures commonly used to extract the ei

I should be noted that the phys structure which

and the eigenvector [2,4, 5]

Internal hinges in stability problems are therefore best handled by increasing
the global nodal deformation vector {A} by the element nodal deformations 5,
allowed by the hinges, and by modifying the deformation transformation matrix
CT, ion 448) appropriately. However, care must be taken not Lo create

(see equa
unconstrained global nodal deformation, as this will lead to a corresponding

rigid body deformation under zero load.

44 Flexural-torsional buckli metri

beam-columns

1g of monosy

44,1 BUCKLING DEFORMATION FIELDS

Cubie buckling deformation fields (see section 43.1) may also be assumed for the
buckling deformations u, 6 of a monosymmetric beam-column (x = 0). These

are given by equation 4.

[aa [er (6,
in which
i (459)
CE dde (455)
2" (or
Mm as
vn el 456)
and
O]
ta [ ] «sn
TS]
in which {Z)* is given by equation 4.13, and [C,]~* by equation 419.

Flexural-torsional

442 ELEMENT STIFFNESS MATRIX

The increase in the strain et

rgy stored in an element of length L during
beam-column flexural-torsional buckling (section 2.8.5.1)

+ Ed + G16} de (458)

ea} = (40,6, 67" (459)
El 0 0
(=| 0 Go (4.60)
0 0 El,
so that
LA [CC
in which
ro {or
a=] OF ZF OF (461)
or or zT

which leads to equation 435,

ma=rcr "(| CB" (2,1 08.142 )tc]"*

in the same way as for the flexural-torsional buckling of columns.

The work done on the actions acting on an element of length L during flexural
torsional buckling (sections 28.5.1 and 174.7) can be expressed as

AA

= Yo

(452)
in which P, M,, and q are the initial axial compression force, bending moment,
and distributed load, and 2 is the load factor by which the initial force, moment,
and distributed load must be multiplied to cause beam-column buckling.

51. 80 Finite element buckling analysis

Equation 4.62 can be expressed in the form of equation 44 by using

(as)
Po 0 Pr
0 0 My, o
x , 464)
a=) 0 My an o
Pre 00 PUNA RIE) EMAIL 20)
.) (BIC,
ex or
ea (465)

BI=| or or {27 (7

op or {or {7}

which leads to equation 443,

toa=ter( [ tar 10,100,30)1c1

in the same way as for the exural-torsional buckling

444 DEFORMATION TRANSFORMATIONS

In many cases, the elements of a structure subjected to beam-column flexural
torsional buckling have collinear z axes and parallel y axes (structures with
elements whose y axes are not parallel have biaxial bending actions) In these
cases the element nodal buckling deformations

Ben) = {arts bar Ga (4.6)
are identical with the element global buckling deformations
Ag} = {Un es Ozun 0a)? (467,

and the deformation transformations of equation 447
Bu) = Tan]
tothe global nodal deformations of equation 445
(a) (RTS SL

become very simple
“The transformation of equation 449
PET

again can only be interpreted in terms of the compatibility of the warping

Pre-buckling analysis @1

displacements w= og at a node common to two elements, and can only
represent this warping compatibility i there is no conflict between the warping
functions w» for the two elements

445 OFF SHEAR CENTRE NODAL LOADS

When off shear centre loads AQ act at distances yg below the centroidal axis,
additional work

(4:68)

NERV ME 0 0»

is done which must be included in the energy equation (equation 42). Ifthere are
nodes at all ofthe load points, then the element twist rotation at each load
point corresponds directly to the global nodal deformation 0, atthe load point.

Thus the effects of off shear centre loads AQ can be included by adding the terms
QU vols to the diagonal elements of the global stability matrix [G] of
equation 49 which correspond to 4.

446 BOUNDARY CONDITIONS AND INTERNAL HINGES

‘The discussion of section 43.5 of the boundary conditions for the flexural:
torsional buckling of columns, including off-axis boundary conditions, also
applies to the flexural-torsional buckling of monosymmetric beam-columns, with
some simplifications that result from the elimination of 1,0, from the global

buckling deformations.
Similarly, the discussion in section 43.6 of internal hinges also applies to
monosymmetric beam-columns

4.5 Pre-buckling analysis

45.1 GENERAL

For the flexural-torsional buckling analysis ofa structure, the distributions of the
axial compressive forces P and in-plane bending moments M, must be known,

For statically determinate structures, these can be determined by statics.

for statically indeterminate structures, such as propped cantilevers,
continuous beams, portal frames and the like, the in-plane pre-buckling deflec-
tions of the structure under load must be analysed. It is appropriate to use a
computer method for this in-plane analysis when a computer method is being
used for the out-of-plane buckling analysis. Usually it will be sufcient to carry
out a first-order lines

effects

clastic in-plane analysis [13), for which in-plane stability

are ignored. When these need to be considered, then a second-order

52. 82 Finite element buckling analysis

non-linear elastic in-plane analysis [14] should be made. If inelastic buckling is
being considered, then an inelastic in-plane analysis should be made [15].
When a finite element method is used to analyse lexural-torsional bucklin
is logical to use a finite element method for the in-plane pre-buckling analysis. A
first-order linear finite element method of in-plane analysis is described below.

For the first-order in-plane analysis of an elastic structure, the conditions of
equilibrium can be expressed using the principle of stationary total potential
derived from the principle of virtual work (see section 24) as

(6U,+5%y=0 (4.69)

in which

(Uy =U (4270)

isthe first variation ofthe sum of the in-plane strain energies in the elements ofthe

structure, and

(Vy = DIV. + LAV am

isthe first variation ofthe sum of the potential energies ofthe distributed element
ated nodal loads (0,
nent strain energy can be expressed as

Toads (9) and the concent

The first variation ofthe ele

su

lement nodal deformations, (35) are the virtual

in which (6,.) are the in-pa
nodal deformations, and [k, isthe element in-plane stiffness m
global sytstem by using the

be transformed from the local element system to ü

in-plane transformation matrix [T, in

Eu) = (Tel {4} 473)

in which {A,), are the in-plane global nodal deformations. The transformed fist
lement strain energies can be summed to obtain

variations of the:

Vie = (6A) “m
in which
(k= SUF Chall (475)
variation ofthe element potential energy can be expressed as
IV,g= ~ (85,)" CAC] a (476
Which can transformed and summed to obtain
tag am

Pre-buckling analysis a8
in which
(0) =EITITA (a (478)

are the nodal loads equivalent to the distributed loads {a}.
Thus the virtual work condition of equation 4.69 can be c

A} [K,} {Ai} = {84} (9 (479)
in which

CR (4380)
Since equation 4.79 must hold for all admissible sets of virtual in-plane displace

ments (54,), then
[KI 18) =(0, (sn,

which are the in-plane equilibrium equations.

452 IN-PLANE DEFORMATION FIELDS

The in-plane deformations », w may be approximated by cubic and linear
displacement fields (see section 4.3.1) so that

(0) = EMAC

(482)
in whieh
(483)

(484)

(455)

(486)

an

and

and [C,]"* are given by equations 4.13 and 4.19.

453 ELEMENT STIFENESS MATRIX

The fa vacation of the strain energy stored in a element of length J is
given by ú
¿Un |" | copos: us)

53. 84 Finite element buckling analysis

and the stress 07 at a point P(x,y) are approximated

Won (490)
Gr= Es, «on
Equation 4.89 can be written as
sus a)" fo) de (492)
in which
6) =(0,w)" (493)
) =1D te (494)

are the generalized in-plane strains and stresses, and

BI, 0 id
wa-[% 2] 49

p= | 0,4 an

Man | apra 498)

are the in-plane stress resultant

The strain vector (e
deformation fields of equation 4.82 as

can be obtained by appropriate differentiation of the

DAT" (6% (499)
in which
zur ior
mf 4.100)
(a | fi | (a

Substituting equations 494 and 499 492 leads to

su

este [rsrroareae)rea-s 100)

and this is identical with equation 4.72 if the element stiffness matrix is written

Pre-buckling analysis 85

«J=1CJ*( | 18,7'D,108,J4= )1c,J7* (4.102)

454 EQUIVALENT NODAL LOADS

The first variation of the potential energy of constant distributed loads
iting on an element of length L

e |, (60) a, (4.103)
can be writen as
aM yea ( ("regresa (4104
which so the form of equation 476 With |
getan [gres (4.105)
Transformation and summation led o
rt zea mareas. (4109
This equivalent to
(Noy =—(64)* (0x) (ion
wien
04) =Sena"tca*( [*taara) (a, (4.108)

Which are the nodal loads equivalent to the distributed loads (4,}.

455 SOLUTION

Alter substitution of the bound

y conditions, the linear in-plane equilibrium

jon 481) can be solved for the global nodal displacements (A,

caused by the load set {4}, (Q,,). The element nodal displacements (¿,) can be
obtained from equation 473, and these can be used to determine the strains ¢p
from equations 499 and 490, the stresses a, from equation 4.91, and the stress
resultants M,, — P from equations 494-496.

54. 86 Finite element buckling analyst
4.6 Computational considerations

46.1 INTEGRATION
When cubic field are used for the buckling displacements u,v, the integrations
that must be performed to determine the element stiffness and stability matrices
[lc]. Lac] (ee equations 4.35 and 443) are simple enough to be carried out
formally [10]. However, itis often considered to be more reliable o use a general
‘numerical integration method which does not need to be modified for different

displacement fields. Gaussian numerical integration methods [1] can be of very
aken along the element,

high accuracy if sufficient sampling (Gauss) points are
For example, only n sampling points are required for the exact integration of a
polynomial of degree (2n- 1).

462 NODE SPACING AND ACCURACY

While the finite element method always predicts an approximate buckling factor
which is greater than the true value, the accuracy can be improved by increasing

he numberof elements into which the structure is divided. Generally atleast two
elements should be used to represent each member of the structure when cubic
‘deformation fields are used. This is because a cubic field can have only one
and often the most critical member ofthe structure will buckle as

inflexion point
¡felastically restrained at both ends so that it has two inflextion points. Studies in

[10] suggest that using two elements per member will often lead to errors of less
than I

However, exp
‘gence, and suggests that at least four elements per cantilever should be used.

fence with cantilevers [16] indicates a lower ra

463 HIGHER ORDER ELEMENTS
An alternative method of increasing the accuracy 10 that of increasing the number
of elements is to use higher order elements, based on quintic or even higher order
polynomials, than the cubies discussed in section 431. The use of higher order

elements is discussed in [17,18].

4.7 Problems

In Problems 4.1-46, a finite element computer program is to be developed for
ous columns. The program is

analysing the elastic flexural buckling of con
to use a data file which defines the member nodes and constraints, the element

properties and connections, and the initial axial forces. The solutions for
Toad factor and the buckled shape are to be written into a

lowest buckl
solution file.

Problems 87

Write a computer routine for calculating the element stiffness matrix from the
clement data

Write a computer routine for calculating the le
clement data and the initial axial forces.

nt stability matrix from the

Write a computer routine for transforming from the element to the structure
coordinate system.

Write a computer routine
stability

summing the transformed element stiffness and

Write a computer routine for using the boundary constraints to reduce the
structure stiffness and stability matrices.

Write a computer routine for finding the lowest buckling load factor and the
corresponding buckled shape

Extend the solutions of Problems 4.1-46to develop a program for analysing the
clastic in-plane flexural buckling of plane rigid-jointed frames

Write a computer program for the
bending of plane rigid-jointed frames.

Jastie first-order analysis of the in-plane

Integrate the computer programs developed for Problems 47 and 48 so that the
in-plane bending program determines the initial axial forces to be used in the
buckling program.

55. 88 Finite element buckling analysis

Extend the solution of Problem 49 to allow for continuous clasti restraints (see

section 63)

Extend the solution of Problem 4.10 to allow for concentrated elastic restraints

(Gee section 63)

Adapt the solution of Problem 49 so as to produce a computer program for

¡tic lexural-torsional buckling of continuous monosymmetric

‘analysing the el
beam-columns.

Adapt the solution of Problem 49 so as to produce a computer program for
sie flexural-torsional buckling of continuous asymmetric

analysing the el
columns

4.8 References

1. Zienkiewice, O.C. and Taylor, RL. (1989) The Finite Element Method, Volume
1. Basic Formulation and Linear Problems, (1991) Volume 2~ Solid and Fluid
Mechanics, Dynamics and Non-Linearity, MeGraw-Hil, New York.

2. Cook, RD, Malkus, DS. and Plesha, ME, (1989) Concepts and Applications of Finite
Element Analysis, 3 edn, John Wiley, New York

3, Gallagher, REL (1975) Finite Element Analysis Fundamentals, Prentice Hal, Engle
wood Cl, NT

4, Bathe, K-J. and Wilson, EL. (I
Prentice-Hall, Englewood CH NT.

5. Irons, BM. and Ahmad, S. (1980) Techniques of Finite Elements, Elis Horwood,
Chichester

6. Galambos, TV. (ed (1988) Guide to Stability Design Criteria for Metal Structures, At

eda, John Wiley, New York.

Barsoum, RS. and Gallagher, RH. (1970) Finite clement analysis of torsional and
stra stably problems. International Journal of Numerical Methods in Engineering

213), 335-82

Powell, Gand Klingner, R. (1970) Elastic lateral buckling of steel beams. Journ

the Structural Division, ASCE, 9ÓST3) 1919-32

Nethereot, DA. and Rockey, K.C (1971) Finite element solutions f

Columns and beams International Journal of Mechanical Sciences, 13, 985-9.

1) Numerical Methods in Finite Element Analysis,

the buckling of

References 89

10. Hancock, Gand Trahait, NS. (1978) Finite element
of continuously restrained beam-columns Civil Engineering Transactions, Institution
of Engineers, Australia, CE20(2) 120-7.

masi ofthe lateral buckling

11. Gallagher, R.H.(1973) The init element method in shell stability analysis. Computers
and Structures, 3, 593-1 ‘

12. Cedolin, Land Gallagher, RH, (1978) Frontal-based solve for frequency analysis
International Journal of Numerical Methods in Engineering, 12, 1659-66.

13. Harrison, HLB. (1990) Structural Analysis and Design - Some Micro Computer
Applications, Parts 1 and 2, 2nd edn, Pergamon Press, Oxford

14. Harrison, HB, (1973) Computer Med
wood Clif NJ

15, Clarke, MJ. (1991) Advanced analysis, Lecture 6 of Structural Analysis to AS4100,
School of Civil and Mining Engineering, University of Sydney, November.

16, Trahair, NS, (1983) Lateral buckling of overhanging beams, in Instability and Plastic
Collapse of Stee! Structures, (ed. LJ, Morris), Granada, London, pp. 503-18,

17. Papangelis,J.P and Trahair, NS. (1986) In-plane finite element analysis of arches, in
Proce, 1 Pace Stuer Stel Conte, Avia, August, Vol 4

18 Papangeli, LP. and Trahair, NS. (1987) Finite element analysis of arch lateral

buckling. Civil Engineering Transactions, Institution of Engineers, Australia, CE29(1)

xs in Structural Analysis, Prentice-Hall, Engl

56. 5 Simply supported columns

5.1 General

A concentrically loaded column may buckle Nexurally by deflecting 2
buckle torsionally by twistin wn in Fig
independent for columns of doubly symmetric
buckle atthe lowest of the
However, for
dependent, and th
with the indep

the column will
with each of these three modes.
lumos of asymmetric cross

the modes are inter

buckling

57. 92 Simply supported

very thin-walled open sections which have very low torsional and warping
rigidities GJ and EI, are likely to buckle in a predominantly torsional mode,
while sections with very low minor axis flexural rigidity El, are likely to buckle

ina predominantly flexural mode.

The treatment given in this chapter of the elastic lexural-torsional buckling of
simply supported columns discusses the effects of cross-section type on the mode
ff buckling. It is assumed that the columns are perfectly straight, and that the
axial forces are applied concentrically

The beneficial effects of end and intermediate restraints are discussed in
Chapter 6, and the effects of end moments caused by eccentric axial forces in
Chapter 11

5.2 Elastic buckling analysis

The ends z = 0, L ofthe axially loaded columns shown in Figure 5.1 are simply

supported, so that

ou. 61

and

Om t= 9h = bol 62)
If the column buckles laterally u,v and twists @ as shown in Figure 52a into an
equilibrium buckled position, then the differential equilibrium equations
(section 28.3.2) are

(ELLE = — (Pa) + Px 63)

(ELY = — (Pu) +(Py09) 64

(Pray +:

(ELOY —(GI4 (Pyou) 65

torsional buckling behavi

Figure 52 Flex

Elastic buckling analysis 98

in whieh

EU ALA RA 66

and P is taken as positive when it acts in compression. Equation 5.3 represents
the equality between the bending actions — (Po and (Px acting about the x
axis and the corresponding flexural resistance (E1,0")’, while equation 54 pro-
vides a corresponding representation for bending about the y axis. Equation 5.5
represents the equality between the torsion actions —(Pr3g’J, (Pxor‘), and

(Pyqit) and the warping and torsional resistances (El, $1)" and ~ (GJ6')
Tt can be verified by substitution that equations 5.1-5 are satisfied by the
buckled shapes

u/5,= 0/8, = 9/0 =sin 2 62
where 6, 8, and O are the values ou, and ¢ at mid-height, provided that the

axial force P satisfies

(P.-P)
o o 68)
Px,
in which
P, 69)
r, 610
P,= (GI + REIN sn

are three reference buckling loads associated with flexural buckling about the x
axis (P,) flexural buckling about the y axis (P,) and torsional buckling (P,)
Equation 58 can also be obtained by substituting the buckled shapes of
on (section 28.3.1),

equation 5.7 into the energy equal

[ter + Ente + EE + 6167) dz

1
LT

+ RG — 2x06!

which represents the equality at buckling between the flexural, warping, and
torsional strain energy stored and the work done by the axial force
Equation 58 can be written as the cubic equation

JAP) = Pr y) — PAP P, PAP)

Pri{P,P, + P,P, + P,P.) (PAP PA =0 (513)

which has three solutions P,, P,,P,. It can be shown that these are real, and that

the lowest solution is always less than or equal to the lowest ofthe three reference

58. Figure 5 Solution of cubic equation for buckling loa
buckling loads P,, P, P,. For example, consider the case where P, <P, < Py as
3 for a 150mm x 100mm x 10mm angle section column. It

0.Thusa

shown in F

can be shown that /(0) =— P,P,P,r3 < 0 and f(P,)=(P, — P,)P2y
solution (at which f(P) = 0) must occur between O and P, so that 0 < P, <P,
The relative magnitudes of the buckled shapes u, 1, @ can be obtained by
substituting the solution of equation 5.13 into equations 5.3-S.5, whence
50 =PyallP,— P) 619
50 = — Pxo(P,—P) 619

The relative magnitudes of the buckled shapes determine an apparent axis of

rotation which can be found by noting that during buckling, a general point P(x, y)

ofthe cross-section moves to a final position Pfx+ u — (y — Joly) + » + x — xo)0)

as shown in Figure 52a. However, the apparent axis of rotation through R(X, 3.)

does not move, and so its coordinates x, y, must satisfy u—(y, — yo) = and
+ (exo) =0, so that

X= PPP, (516)

P,y9hP,—P. 617

The apparent centre of rotation of a 150mm x 100mm x 10mm steel angl
section column of length L = 2000mm is shown in Figure 544

At the axial force P defined by the solution P, of equation 5.13,
rmations are defined in shape but are indeterminate in their absolute
as indicated in Figure 52b. When the
‘equation 5.13, then the only solution of equations 5.1-5. is

buckling

ial force does not satisly

u=v=6=0 (618)

bly symmetric sections 95

{
/
LT
Vie © CN
Y PE ga mo
J

indicating that the column remains straight and untwisted until the lowest buck!
ing load P, is reached, as shown in Figure $.2b. Thus the state of equilibrium
bifurcates at the lowest buckling load P, from the stable position given by
equation 5.18 to neutral equilibrium positions given by equation 5,

5.3 Doubly symmetric sections

For doubly symmetric sections, xy = yp=0, and the off-diagonal terms of
equation 58 vanish. In this case equation 5.13 simplifies to

(P,—PNP,—PNP,—P)=0 (519)
and so the three solutions are

Pp, (520)

The column therefore buckles at the lowest of these three reference loads and in a
corresponding mode. Thus buckling occurs either by flexure about the x axis at
P,, or by flexure about the y axis at P, or by torsion about the z axis at P,
The variations with length of the reference buckling loads P,, P,, P, of an
Lsection are shown in Figure 5.5. It can be seen that in this case the lowest
reference load is always P, so that this section always buckles in flexure about the

59. 96 Simply supported columns

Figure 55 Buckling of doubly symmatric section columns.

axis. This is generally the case for simply supported 1-section columns, for which
P, > P, while h> 0.58B, and [,/r2 > 1, while hc > B*T. The exceptions are likely
10 have low values of h/B (which decrease the ratios of P,/P, and P,/P,) low
values of /T (which decrease the ratio of P,/P, and low values of /h and T/B
(which decrease the contribution made by the torsional rigidity GJ to P.)

‘The variations ofthe reference buckling loads ofa cruciform section are shown
in Figure 5.5b. It can be seen that P,is often the lowest reference load, so that the
column frequently buckles in a torsional mode. A cruciform section has zero
‘warping rigidity El, and so depends entirely on its torsional rigidity Gu for ts
resistance to torsional buckling, which is then independent of the column length
L. Because the torsion section constant J(=Ebr'/3) varies as the cube of the
thickness 1, the resistance of a cruciform column to torsional buckling reduces
rapidly with the thickness , Thus torsional buckling is common in thin-walled
cruciform section columns, and itis only for long cruciform columns that the
flexural buckling loads P,, P, may decrease below the torsional buckling load P,
and the mode may change to flexural buckling.

‘On the other hand, the torsional rigidity GJ is always higher than the flexural
rigidity EI for a circular hollow section, and the two are comparable for most
rectangular hollow sections. Because of this, the torsional reference buckling
loads P, of practical hollow section columns are always higher than the corres
ponding flexural loads P,, P, and so hollow sections buckle in a flexural mode
at the lower of P, and P,

Monosymmetric sections 97
5.4 Monosymmetric sections

For monosymmetric section columns which are symmetric about the x axis so
that yo =0, equation 5.13 simplifies to

(P= PIB + BP PAP. P)—(P xo?) = 0 (621)
which has the solutions
P,=P, 622
Œ+P, x}
pts x

in which r3 = (J, + 1,/A. Such a column therefore buckles at the lowest of the
three loads P,, P;, or P,,cither in a flexural mode about the y axis at P, or in a
flexural-torsional mode by flexure about the x axis and torsion at the lower of
P,P

The solutions of equation 5.13 for monosymmetrie section columns which are
symmetric about the y axis so that x= 0 can be obtained by interchanging x
and y in equations 521-523,

The lower flexural-torsional buckling load P, of equation 523 can be expressed
rnon-dimensionally in terms of a modified harmonic mean buckling load
V{P_PAl +x2/r)}. The dimensionless buckling load Py//{P,P,(1 +xUr})
then varies as shown in Figure 5.6a with the parameter §(P, + P(1 + x8/13)
V/{PsP.+x3/r2)} which may be thought of as the ratio of a modified arith-
etic mean buckling load }(P, + P,)(1 + x3/r3) to Ihe modified harmonic mean.

Figure 56 Flexural-torsional buckling of monosym

60. Monosymmetric sections 99

than P,
P,, For very
predominantly

61. 100 Simply su

flexural-torsional mode is predominantly torsional, The mode changes at
L=4565mm.
On the other hand, torsional effects may dominate in thin-walled mono-

symmetric lipped channel and hat section columns, as is demonstrated in
Figure 58. The geometries of these sections are such that in both cases P, is
much less than P, and P, so that the columns buckle in lexural-torsional modes
which are predominantly torsional at loads P., which are a litle less than the

torsional reference loads P,. Similar behaviour is shown by the thin-walled

angle section columns shown in Figure 590.

monosymmetri lipped

5.5 Asymmetric sections

For asymmetric sections, the cubic equation 5.13 cannot be simplified, and so the
lowest buckling load must be obtained numerically. This can easily be done
ating linearly using the values of f(P)for P = Oand for

approximately by inte
the lowest value P, of P,, Py P, Thus
Ps)
», 629
Een)

The accuracy ofthis approximation can then be improved by again interpolating
linearly, this time using f(P.) calculated for the approximate value of Pj and
f(P), and this can be repeated until a sufficiently accurate estimate of Pr, is
“obtained. This Pa is always less than the lowest of P,P, P, and corresponds to

th the x and y axes and

a flexural-torsional mode involving flexure about b

torsion about the z axis.

Torsional effects are often important in thin-walled angle section columns,

‘and J decreases rapidly with the w

Il thickness. The variations

because I
with lo

and the three reference buckling

59%. For this

ith of the lowest buckling load P,
loads P., PP, of an asymmetric angle section are shown in Figur
eater than P,, and so the buckling mode chan,

from

section P, is much g
nantly flexural (about the y axis) for long
For intermediate length columns, the buckling load

predon columns to predominantly
torsional for short column

is significantly less than both P, and P,

5.6 Problems

A simply supported steel (E = 200.000 MPa, G = 80000 MPa) cruciform section

column is fabricated from four plates 120mm x $ mm x L (Figure 5.100),

(4) Determine the variations of the elastic flexural a
loads P,, P, P, with the length L

d torsional buckling

Problems 101
pay y
j
E | 4
1 ;
4 Las |

(0) Find the length Lat which P, =P,
(e) Compare the torsional buckling load P, with a value obtained from the
plate elastic local buckling stress

Ek
DU 67

in which k = 0425 for a plate outstand b of thickness 1, and v= 025,

A simply supported tee-seetion column (Figure 5.10b) is formed from half of the
[section shown in Figure 7.23 by cutting it along the web centre line, The depth
of the teesection is 305 mm, and its properties are yp = 65.3 mm, A = 7970 mm
1, = 68.5 x 10°mm*, 1, = 19.7 x 10° mm, J = 771 x 10° mm,

(a) Determine the variations of P,, P,, P, with the length L
(b) Determine the variation of the minimum elastic buckling load with the
length L

A simply supported steel (E = 200000 MPa, G= 80000 MPa) angle section
column has overall leg widths of 125mm and 75mm and a thickness of mm
242mm, À = 1500 mm’
31.7 x 10° mm

(Figure 5.100) Its properties are yy = 333 mm,
1, =268 x 10°mm*, 1, 0.399 x 10°mm*, J

(a) Determine the variations of P,, P,, P, with the length L.
mine the variation of the minimum elastic buckling load with the
hL

62. 102 Simply supported columı

5.7 References

1. Timoshenko, SP. and Gere, IM. (1961) Theory of Elastic Stability,
Hil, New York

2. Vlasov, VZ. (1961) Thin Walled Elastic Beams, 2nd edn, Israel Program for Scientific
Translations, Jerusalem.

3. Bleich, F (1952) Buckling Strength of Metal Structures, MeGraw-Hil, New York

4 Chajes, A. and Winter, G (1965) Torsional-exural buckling of thin-walled members.
Journal of the Structural Division, ASCE, 91 (STA) 103-24

dedn, McGraw

6 Restrained columns

6.1 General

connected to other elements which participate in the buckling
d significantly influence the buckling resistance. Braces are provided

the purpose of increasing the buckling resistance (Figure 6.1), but
many other elements, such as sheeting, which are intended primarily for other

purposes, may also have important restraining actions.

These elements induce restraining actions (Figure 62) which restrict the
buckled shapes of the column, and increase its buckling resistance. Diserete
restraints act at points where braces or other restraining elements are connected
tothe column and induce actions which resist the buckling deflections, rotations,

and warping displacements, These restraints are usually assumed to be elastic in
which case they may be characterized by their elastic stifinsses

In some cases the discrete restraints may be assumed to be rigid, so that they
prevent one or more of the buckling deformations. In the special case where rigid
restraints prevent the lateral deflections and twist rotations of the cross-sections
at which they act, then the column may be described as a braced column which
consists of column elements between the restraint points

Continuous restraints are usually considered to be uniform along the length of
a column, and are often used to approximate Ihe actions of restraining elements.
which are connected to the column at closely spaced intervals, as in the case of
wall sheeting (Figure 63). Continuous restraints are similar to discrete restraints,
in that they induce actions which restrain the buckling deformations (Figure 62).
They also are usually assumed to be elastic, and so may be characterized by their
clastic stiffinesses. When continuous restraints acting in both principal planes can

Figure 61 Column restraint

63. Restrained columns

Figure

Figure 03 Restraint actions of contin

be assumed to be rigid, then they enforce a longitudinal axis about which the
column cross-sections rotate during buckling.

This chapter is concerned with the influence of restraints on the elastic
buckling of columns. The buckling of restrained beams is treated in Chapter 8, of
restrained cantilevers in Chapter 9, of continuous beams in Chapter 10, and of
restrained beam-columns in section 11.5.

Restraint stiffnesses 105
6.2 Restraint stiffnesses

621 CONTINOUS RESTRAINTS

A column may be continuously rest

ined as shown in Figure 62b by transla
tional restraints of stiffness %..2,, which act at distances , x, from the centroid,
by rotational restraints of stiffness #,,,2,, which act at distances y,,x, from the
centroid, by torsional restraints of stiffness 2,., and by warping restraints of
stiffness à

The actions exerted by these restraints can be replaced by the shear centre

(9) = ety fan Mes Mes)" (1)

shown in Figure 62c, in which b is Ihe bimoment per unit length. For these
actions to be statically equivalent to those ofthe restraints they must be related
to the shear centre deformations (Figure 6.28)

(= tn, éd (62)
by

(= alla) (63)

& 0 0 0 4 0
0 a, 0 0 0 a
0 0 w% 0 æ 0

64)
0 0 0 a 0 a |
ta 0 a 0 à o!
o 0 aw 0
in which
au al)
at |
El BA } 65

A column may have discrete restraints which have translational stiffnesses

Ann, (at distances )y, xy from the centroid), rotational stiffnesses ay ay (at

64. 106 Restr lor

distances yyy xy from the centroid, torsional stiffness ag, and warping stiffness
The actions exerted by these restraints can be replaced by the shear centre

= {Fre Mayo Frye = Maso Maes BY? 69

in which Bis the bimoment. For these actions to be statically equivalent to those

int, they must be related to the shear centre deformations

(D) = (uu00, 6.07 (67)
by
LR} = Cas] {D (68)
in which the discrete restraint stiffness matrix is
te 0 0 0 4 0
0 ty 0 00 a
mal 9% 0 % 0 a
0 0 0 ame 0 ty
ea 0 tye 0 tee 0
[BE ae Fer av
in which
ape = nr

ayy = yO
yg = On 3 \

22 = A TEEN { QU
an = Ale = Xo) |

qu aa (Ya Jo)

6.3 Buckling analysis

In the general case of restrained columns, exact solutions for the buckling loads
cannot be ob

ined, and a numerical method must be used to obtain approximate
solutions. Ifthe approximate energy method is used (Chapter 3) then the strain
energy stored in an element during buckling should be increased to

El,u? + Elo"? + El, $" + G16)

(DJL) {0} + *[a]{d) de “un

to account for the strain energy stored in the discrete and continuous restraints

acting on the element.
Ifthe finite element method of computer analysisis used (Chapter 4) then each
matrix should be augmented by transforming the term

which accounts for continuous restraints, and the column
stiffness matrix should be augmented by including a transformation of the term
}{D}" [ay] {D} for each discrete restraint

64 Continuous restraints

64.1 GENERAL SOLUTION

A simply supported column (up, = 0,1 = do. =0,4%,, = 6,2 = Do, = 0) with

uniform continuous restraints buckles into half sine waves so that

6/0 sin

(612)

The load which causes elastic buckling can be obtained [1,2] by substituting

these buckled shapes into the energy equation

Tone
Lou +óv)=0 (613)

in which the strain energy 482 U stored during buckling is given by equation
6.11 and the work 4 *V done on the applied compression force P is given by

Lay 2x0 d +2ypu'd'} de. (614)
This leads to an equation of the form
a, 0 ay
0 ay: 0 |=0 (615)
in which
a Enp,—P, |
+ dy?) +n P,P,

a Pr + GI + PR, \ (616)

eL Jia — Pr
de OP )
a he a aig aati
lads tote lowed valu ofthe buckling lad Poa be determined

65. 108 Restrained columns

642 DOUBLY SYMMETRIC COLUMNS WITH SHEAR CENTRE

For doubly symmetric columns (x 0) with shear centre restraints

Der a, =0), equations 6.15 and 6.16 reduce to
Eu PNP yy PAP = PE =0 (617)

Ppp SIP, + Oy + Lt, (618)

Po = Py Og + 8 LAN 619)

Pu = (GI + nm EL 4 a + 0, La (620)

Thus there are three possible buckling modes; flexure about the y axis at P,,
flexure abou

the x axis at P,,, and torsion about the z axis at P,,. The actual

bbuckling load will be the lowest of Pye, Pars Par

These buckling loads depend on the number n of half waves in the buckled
shapes. Each of equations 6.18-620 is of the form

Pump + Py Py (621)
and the lowest value of P, can be closely approximated by

Pain = 2(P,P.)+ Py (622)

Figure 84 Buckling of doubly symmotric columns with continuous shear centre

except when P,/P, <1, in which cas

this approximation is conservative, as

Shown in Figure 6.4. An accurate solution may then be obtained by using n = | in

equations 6.18-620 or 621

643 OTHER COLUMNS

For other columns with more general restraints, equations 6.15 and 6.16 cannot

be so simplified, and the actual buckling load must be determined by finding
the valu
will correspond to a flexural-torsional buckling mode in which the column

& of n which gives the lowest solution of P. In general, this solution

ineously deflects 4,0, and twists 4.

644 BUCKLING WITH AN ENFORCED CENTRE OF ROTATION

When rigid translational or rotational restraints act along a line (x,y) then the
column buckles with this line as an enforced centre of rotation, so that

=. 700, (623)
5,= (ne x). (624)
In this case the only restraint strain energy arises from the torsional and
warping restraints 2, 2,, and so the energy equation (equations 6.11,6.13,6.14)

P,P) 0 Py
o WPF) Ps,
Per at o
Pr Pr (Gr 2

(625)

When equations 623, 6.24 are substituted, then equation 6.25 can be rearranged

Parte + P+ Pn? (626)
in which

PP Oo) Pa Xo)? + ELLE]

P,=(63 } (62m
Pal

and
ner (628)

Equation 626is ofthe same form as equation 6.21, and can therefore be solved in

66. 110 Restrained co

the same way. When P, < P,, then th

»

minimum solution is given by
nia = P+ Py t Pe (629)

When P,> P, then the minimum solution may be closely approximated by

Prin = 2 (PP) + Py (630)

Discrete restraints

65.1 END RESTRAINTS

Restraints acting at the supported ends (u = 0 = = 0) ofa column may restrain
end rotations u,v' [3] and warping displacements proportional to 4 [4]. The
particular case of a doubly symmetric column (xo
equal shear centre rota

0) whose ends have
nal and warping restraintstifesses of 4, ju and y

is shown in Figure 6.54.

In this case, the buckled shapes are defined by three equations of the type
u _ u _ 6 _cos(nz/kL ~x/2k) —cos(n/2h)
LG 0 (= cos(r/20) 7
here satisics equations of the type
E EZ 32)
Bn Hes (632)

In these equations, the values ky, kw ky, of k are associated with uv, and

fo ss >
En 638

Es

fs Due

Figure 65 Buckling of columns with equal end restrain

Discrete restraints 111
asyL/EL ,24,L/EL,,2yL/El,, respectively. These define three column effective
lengths K, L,k,L,K, L, equal to the distances between the inflexion points of the
buckled shapes for 6.

‘Substitution of equations 6.31, 6.32 into the energy equation

(633)

leads to
Pa PP Pa PIB = 0 (639

in which
Py =A EL GI
Pam PEL J?

Poy = Gd +R EL EL

which can also be obtained from equations 59-5.11 for an unrestrained column
by substituting the appropriate effective lengths kL for the actual length L.

The relationship obtained from equation 632 between the effective length
factor k and the dimensionless end restraint stiffness 2L/El is shown in

Figure 6.5b. It can be seen that k varies almost linearly from 1.0 to 0.5 as
(@L/ENNU +aL/EI) increases from zero (for an unrestrained column) to
unity (for a rigidly restrained column). This suggests that k m
approximated by

y be closely

+aLjEI

(638)
2L/El J

as shown in Figure 6.
‘The effect of unequal minor axis flexural restraints 2yyn at the ends A,B
of a column is to increase the elastic minor axis flexural buckling load P, to the

value of Py given by equation 6.35 with the effective length factor k, being the
solution [3] of

=)

IO TA 1 39)

in which
Ga =2EL,Jagyalo

(640
2El,/tgyuL. J J

Ge

These solutions are shown in Figure 66b.

67. 112 — Restrained columns.

Figure 68 Buckling of columns with unequal end restraints

‚These solutions may also be used forthe effective length factors k, of columns
with unequal major axis end flexural restraints dex BY Substituting x for y
Similarly, solutions may also be obtained for the eflective length factors k, of
columns with unequal end warping restraints wa, yx by substituting ay for
ax In for Ly and k, for k,

652 INTERMEDIATE RESTRAINTS

Intermediate restraints acting between the ends of a column may restrain the
lateral deflections 1,» [3] and twist rotations & [4], The particular case of a
doubly symmetric column with centr

úcentroidal restraints is shown in Fig
idsoth = Oat the restraint point,
the column buckles an saving an
effective length of L/2, as shown in Figure 6.7b. Thus the elastic buckli
resistances are given bY P as Pas Pax (equations 635-637) with k=05
When the restraints are elastic, the column may buckle in a symmetrical
mode as shown in Figure 6.7a, The buckled shapes are given by the three

ure 67a. When the restraints are ri

symmetrically into two half waves, each

equations [3]

®_6_2__sines/kL) ean
5, "3, OL (ajk)cos a/2h)

while 0 <2 <L/2, where k satisfies equations of the type
al? __(x/2k)*cot(n/2k) nn

T6ET (5/20 cov(a/2k) —1

Disorete restraints 139,

Figure 67 Column with central restraints

In these equations, the three values k,kysk, are associated with 10,4 and
Atay ns and 1,1, 1, respectively

‘The relationship obtained from equation 642 between the reciprocal of the
effective length factor k a

14 the dimensionless central restraint stiffness
21? (16EI) is shown in Figure 676. It can be seen that 1/k increases from 1.0
towards 2.86 approximately

as the restraint stiffness increases from zero towards
infinity

Also shown in Figure 67e is the solution 1/£=2 for the antisymmetric
buckling mode ofa column with a rigid restraint, which intersects the symmetric
mode solution ata limiting restraint stifines given by

a LINGEN = (643)

Since buckling always occursin the mode which has the lowest buckling load, a
column with a >, always buckles antisymmetrically in two half waves, and at
the lowest load given by P,a, Pas Pox with k= 0,5. On the other hand, a column
with a < a buckles symmetrically ina single half wave, and ata load obtained by
using the Solution of equation 6.42. This solution may be closely approximated
by

1

1+ Thala, = OMe,

(644)

in the range 0 < a <a

68. 114 Restrained columns

The influence ofn, equal stifness centroidal restraints spaced at equal intervals.
5=L/(n, + 1)alonga doubly symmetric column has been studied [5,6]. For high
restraint values, the column may buckle between restraints into (n,+1) half
waves ata load given by the lowest of Pj Pay, Pox With

k=s/L (645)

For smaller restraint values, the column will buckle in a complex mode into a
lesser numberof half waves at reduced buckling load. Solutions [7] for columns
with translational restraints a are shown in Figure 68, and compared with the

corresponding solutions for columns with continuous restraints. Approximate

solutions may also be obtained by ‘smearing’ the discret

into an equivalent uniform continuous translational restraint of stiffness

a= naq/L (646)
and using this value in equations 6.17-6.22.

The limiting restraint stiffnesses required to ensure that a column buckles
between discrete restraints in (n, + 1) half waves is approximated by

03800, 41)" (647)

: u a
: thy Fe

: Elo

4 4

Figure 0: Buckl

which is somewhat higher than the correspondi

proximation

025n (648)

for a column with continuous restraints which buckles in n half waves. These
approximations are also shown in Figure 68

6.6 Problex

Adapt the solution of Problem 4.13 so as to produce a computer program for
clastic lexural-torsional buckling of continuous columns with

A simply supported I-section column (Figure 69a) whose properties are
Figure 7.23 is continuously restrained along its centre line so that u = v= 0.

Determine the variations of its modified dimensionless torsional buckling

toad
pa = Per 2)
%z 4
(a) Problem 6.2 (b) Problem 6.3

69. 116 Restrainod eolum

ind the number n of buckle half waves with the dimensionless torsional restraint
stiffness af = 2, L*/n ‘El

The tee-section column of Problem 5.2 has an enforced centre of rotation about
the tip ofits stem (Figure 6 9b). Compare its elastic buckling load with half ofthe
torsional buckling load ofa simply supported I-seetion column whose pro}

are given in Figure 723,

6.7 References

1. Tra
The Profession

ir, NS. (1979) Elastic lateral buckling of continuously restrained bearm-columns
à Civil Engineer (eds Campbell-Allen, D. and Davis, EH) Sydney

University Press Sydney, pp 61-73

2, Trahair, NS. and Nethercot, D.A. (1984) Bracing requirements in thin-walled struc
tures, in Development in Thin-Walled Structures-2, eds Rhodes, J. and Walker, AC),

Elsevier Applied Science Publishers, pp 93-130,

Trahair, NS. and Bradford, M.A. (1991) The Beh

revised 2nd edition, Chapman and Hall, London.

‘Svensson, .E.and Plum, CM. (1983) Steer eet on torsional buckling of columns,

Journal of Structural Engineering, ASCE, 109 (3), 758-72.

5. Home, MAR. and Ajmani, JL. (1969) Stability of columns supported laterally by
side-als, International Journal of Mechanica Science, 1, 159-74

Medland, LC. (1979) Flexural-torsional buckling of interbraced columns. Engineering

Structures, À (April, 131-38,

7. Winter, G. (1958) Lateral bracing of columns and beams, Jo
Division, ASCE, 84 (ST2) 156L.1-22,

Design of Stel Structures,

al ofthe Structural

7 Simply supported beams

7 General

A beam which is bent in ts stiffer principal plane may buckle out of that plane by
deflecting laterally u and twisting $, as shown in Figures 7.1 and 7.2b. These
deformations are interdependent. For example, a twist rotation ¢ of the beam

cross-section will cause the in-plane bending moment M, to have an out-of-plane
component M, as shown in Figure 7.3a, which will cause lateral deflections u
Conversely, lateral deflections u will cause the moment M, to have a torque

component M ul as shown in Figure 7.3b, which will cause twist rotations 6.

The resistance to out-of;plane buckling depends on the resistance to lateral
bending and torsion. Thus slender beams with low values of E1,/L?, GJ, and
El. 12 arelikely to buckle in the elastic range under quite low loads, as indicated
in Figure 74, Steel beams with moderate resistances to lateral bending and
torsion are likely to yield and then buckle inelastically, while stocky steel beams
with high resistances will il in some other mode, such as in-plane collapse when

fully plastic.
The elastic buckling of simply supported beams is treated in this chapter, while
Chapter 8 deals with restrained beams, Chapter 9 with cantilevers and C

ter 10 with braced and continuous beams, Some special topics on elastic buckling
are treated in Chapter 16. The inelastic buckling of steel beams is discussed in
Chapter 14, and the use of 1 design of steel beams

against Nexural-torsional buckling is treated in Chapter 15,

The beams considered in this chapter are simply supported in-plane and
out-ofplane. Beams which are simply supported in-plane are single span beams
inst in-plane transverse deflections (09 = u, = 0) but are

whose ends are fixed

unrestrained against in-plane rotations 5, v,. The ends of beams which are
simply supported out-of-plane a

twist rotations (uo =u, = do = dí, = 0), but are unrestrained a
rotations u, (50 that ws =u; = 0) and against warping displacements propor.
tional to des 9, (60 that 65 = 4% = 0) as shown in Figure 7.5.

fixed against out-of-plane deflections and
inst minor axis

“The beams are assumed to be perfectly straight and untwisted before loading
(crooked and twisted beams are treated in Chapter 15) and to be loaded by
ons only in the plane of loading. It

moments or loads which initially cause def

is also assumed that the directions of the loads or of the planes of the applied
moments remain unchanged during buckling (the effects of loads whose dire

tions change are considered in Chapter 16).

70. 118 Simply supported beams

Figure 741 Floxuraltorsional buckling of a beam,

7.2 Uniform bending

72.1 DOUBLY SYMMETRIC SECTIONS

Uniform bending is induced in a simply supported beam of doubly symmetric
cross-section by equal and opposite end moments M as shown in Figure 7.58, o
that M, = M. If the beam buckles laterally u and twists $ into an adjacent
position as shown in Figure 75c and d, then for this position to be one of
‘equilibrium, the differential equilibrium equations (section 28:42)

(EL HH = 0 a
and
(ELOY (616) + Ma =0, (2)

3 IR, bos BY were equilibrium
LL |
.. 2 [of equilibrium >
9 in-plane deflection v e

Out-of-plane ue

(a) In-Plane Bending lb) Out-of Plane Buckling

Figure 72 Elastic bending and buckling,

(Fag oes

N e [past rotations o,
onen He Twist ane of buckled beam
detections u

mp

B

(a) our.

ane Moment Bi Torque

Figure 73 Interdependence of u and 6

and the boundary con

Yon == do = 4, = 0 03

must be satisfied. Equation 7.1 represents the equality at equilibrium between the
out-of-plane bending action —(M,¢)" and the flexural resistance (El,u’Y', and
equation 72 represents the equality between the torsion action — Mu’, and the
warping and torsional resistances (EL, 4°) and — (GJ)

It can be verified by substitution that these equations are satisfied by the
buckled shape

1 0

71. Buckling resistance

Simply supported beams

Strain
hardening

Inelastic

\

buckling

\ Fully plastic

—

SElastic buckling

Beam slenderness

Figure 74 Efe

derness on buckling resistance

(a) Elevation

(b) Part

rts prevent end rotation (0-0)
and allow end warping (0"=0)

Plan_on Longitudi

Uniform bending 121

where 5 and 0 are the values of u and ¢ at mid-span, provided the value of the
applied moments M is given by

Mya = V/{(REI,/L2) (GI + EL IL?) as)
M, ro (PP 0s
in which
LyA, en
L/L 718)
and

Pam (GI + MEILE) 9

are the minor axis flexural and torsional buckling loads of a simply supported
column (see section 52) can also be verified that the magnitudes of the buckled
shapes are related by

3/0=M,JP, 0.10)

which defines an axis of buckling rotation ofthe eross-section (se
distance

Figure 7.6)ata

Wo =M,,/P, em

below the z axis.
These results may also be obtained by substituting the buckled shapes of

ani

sl

Position of centre of rotation {ajay,

72. 122 Simply supported beams

‘equation 7: into the energy equation (section 284.1)

Lu"? + ELO” + G16) di

which represents the equality at buckling between the flexural, warping and
torsional strain energy stored and the work done by the bending moment
M,=M.

At the buckling moment M,, defined by equation 7.5 the buckling deforma-

tions are defined in shape by equations 7.4 and 7.10, but are indeterminate in
magnitude, as indicated in Figure 7.2b. When the applied moments M are less
than M,. then the only solution of equations 7.1 and 72 is

indicating that the beam remains unbuckled until M,, is reached, as shown in
2b, Thus the state of equilibrium bifurcates at M = M,, from the stable

Figure
position given by equation 7.13 to neutral equilibrium positions defined by
equations 7.4 and 7.10.

The relationship given by equation 7.6 between the buckling resistance M,,
and the harmonic mean of the column buckling loads P,, P, demonstrates the
interdependence of the flexural and torsional resistances in providing the beam
buckling resistance

The expression for the buckling resistance giv
thatthe resistance decreases as the beam length L increas

by equation 7.5 demonstrates
and at arate which

depends on the value of the torsion param

K=/(r*El,JGIL2) a

F
is inversely proportional to L. In this case 1, =dP/12 and J
E/G = 25 for steel, so that equation 7.5 becomes

arrow rectangular section beams, À = 0,and so the buckling resistance M,
0/3, while

M„z04P,L

For -section beams, I, = 1/44, where his he distance between flange centroids.

For very thin-walled beams, J->0, and equation 7.5 approaches

M,,%P,W/2 0.19)

so that M,, is nearly inversely proportional to L?. This is similar to the flexural
buckling behaviour of columns, and has led to the modelling of beam buckling i
terms of the column buckling resistance P,/2 of the compression flange. Such a
model is valid in this limiting case, since the axis of buckling rotation (at
ye=M,,/P, = h/2) is at the tension flange, which therefore does not make any
‘contribution to the flexural and warping resistances,

However, this model is inaccurate for the usual range of hot-rolled steel
I-section beams, for which both /, and J are significant. In this case the buckling
resistance varies in a less simple way with the beam length L, as indicated by

Uniform bending

equation 75, while the distance to the centre of rotation y. varies between OL

approximately for narrow rectangular beams and 1/2 for very thin-walled
[section beams, The variation of 2y./h with K (K = K for doubly symmetric

beams) is shown by the dashed line in Figure 7.6

The solution given by existance of a doubly

uation 7.5 for the buckling

symmetric beam in uniform

bending ignores the effect of the pre-buckling in
plane deflections, which transform the beam into a ‘negative arch’, and increase
its buckling resistance. The resistance M is more accurately

section 166) y

en by (see

M 1
Mm,” JU ER JEL TOI

TAPTREL))

For many practical I-section beams, EI, is much greater than El,, GJ, EL./L
and the more at given by equa
ore this effect. However, this is not the case for

hollow rectangular sections, where El, and GJ are often comparable with El,
and the buckling resistance may be greatly increased. Equation 7.17 indicates
that the resistance ofa circular hollow section (EI, = El,) is infinitely large, and
that beams bent about their minor axes (EL, <El,) do not buckle. These
conclusions for doubly symmetric beams in uniform bending are summarized in
Figure 7.7

Sections which are point symmetric, such as zed-sections, have coinciding
centroidal and she

entre axes (xo = Yo = 0). When such a beam is bent in its,

1
0 | o rem
o =, | T
a ur
4 A
NET mal —

73. 124 Simply supported beams

Bat
| ae ad

First v in Y2 plar Simultaneous Constraints enforce
Then u.@ out of plane ue cocina =
and © = 0

(a1 Buckling (by Biaviat Bending cl Constrained

174 Behaviour of ed-beams

stifer YZ principal plane, it may buckle laterally u, $ out of this plane of bending

asindicated in Figure 7.8. The differential equilibrium equations (and the energy
equation) for point symmetric beams are the same as those for doubly symmetric

‘beams, and so their elastic buckling resistances under uniform bending are also
given by equation 7.5, provided appropriate values ofthe section constants 1 J,
and I, are substituted.

However, zed-beams bent in principal planes are rare in practice. More
‘commonly, a beam is loaded intially in the plane of the web as shown in
Figure 7.8b,in which case it bends biaxiall, by simultaneously deflecting up and

Alternatively, a zed-beam may be constrained to deflect in a particular plane
Se. In this case the constraints

such as that of the web, as shown in Figure
prevent any buckling effects.

3.1 Sections bent about an axis of symmetry

Sections which may be bent about an axis of symmetry (yp = 0) include channels,
equal leg angles, and monosymmetric sections, as shown in Figure 7.7c. The
differential equilibrium equations and the energy equation for these ae the same
as those for doubly symmetric beams and so the elastic buckling resistances

under uniform bending are also given by Equation 7.5. For cases where El, is not
large compared with EL, GJ, EL./L?, then equation 7.17 may be used for a more
accurate estimate of the buckling resistance

Uniform bending 125
723.2 Sections bent in a plane of symmetry

Sections which may be bent in a plane of symmetry include monosymmetric
Trsections, tro dons, and equal I
these sections, additional terms enter into the equilibrium equations

s, as shown in Figure 7.7d. For

Elu) +(M,4) = 0, as)
(EL (GI) + Mat (MB Y =0, 019)

Maga + MsB.9)d2. (120)

‘These additional terms are associated with a monosymmetry property of the

Be= (t/t) | y +y?)d4—2)

aa)

The buckled shapes which sty equation 718,719, and the boundary

Mat dane ‘buckling, whch can be obtained by substituting
cation 74 into equation 718 and 719 onto equation 720,1 given by

M PLAN

ut VU +) +

(722)

in which M,, and P, are given by equations 7.5 and 78, while the centre of

rotation during buckling lies at

y= M/P, +» (a)

M given by
are shown in Figure 79. These solutions correspond to positive

The positive solutions for the elastic buckling resistance

moments M which cause compression in the top fibres of the section. The
ative solution, which corresponds to. negative moment which causes tension

in the top fibres, i identical with the postive solution when the beam is inverted
50 that f changes sign. For positive M, the buckling resistance increases with the
dimensionless monosymmetry parameter f,P,2M,.. For negative values of
BaP,/2M y for which the shear centre S lies below the centroid C of the
cross-section, the buckling resistance is low, while th

values of fP,/2M,, for which the shear centre is above the centroid

reverse is true for positive

A better understanding ofthis may be gained by noting that tensile longitu
nal stresses reduce twisting, as indicated in Figure 7.102 by the action of the

childs spinning toy, while compressive stresses increase twisting as demonstrated

in Figure 7.10b by the torsional buckling of a water tower. This later effect is the
same as that which causes torsional buckling in cruciform-section columns
(section 53) For doubly symmetric beams, the tensile and compressive bending

74. 126 — Simply supported beams

puis acl

Figure 7.8 Monosymmetrc beams in uniform bending,

}

tral

fs Water Tower

Uniform bending 127

stresses are equal, and so the increased buckling resistance caused by the tens

stresses is balanced by the incre

ed buckling action caused by the compressive
stresses. However, this balance is upset in monosymmetric beams, such as the
I-section shown in Figure 7.106, Here the compresiv foros Cin th larger top
flange is approximately equal to the tensile force Tin the bottom flange, but
because this top flanges closer tothe axis of twist through the shear centre Sits
warping rotations during twisting are less than those of the smaller bottom
ange, and its disturbing transverse compon

ess than the restoring compo-
nent of the tensile bottom lange force. The disturbing torque exerted by the top
flange component about the close shear centre axis is even less than the restoring
torque ofthe more distant bottom flange component, and so the effet of the force
inthe smaller flange dominates. In this and so the buckling
resistance is increased, Conversely, when the larger flange is in tension, the
resistance to buckling is decreased, as shown in Figure 73.

The solution of equation 7.22 for the elastic buckling moment requires the use
ofthe section constants JJ, fe and fi. While these can be determined from the
section dimensions, simple approximations which can be evaluated by hand are

ase it isa tensile fore

often sought. For monosymmetri L-sections, these can be expressed in terms of
the value of

ph ¢

7]

the ratio of the minor axis second moments of area of the compression flange and
the complete cross-section, Thus the monosymmetry property can be approxi

mated by [1]

9H2p — DEI) 025

while
Tem pl pi 029
and

where band rare the width and the thickness ofeach narrow rectangular element
of the cross-section

Approximate values of ML/,/(E1,GJ) determined from these equations are
plotted in Figure 7.11 against the torsion p

R= REICH) (728)

for selected val

of p. For a tee-beam with the flange in compression (p = 1.0),
the dimensionless buckling resistance is significantly higher than that for an
equal flanged I-beam (p = 0.5) with the same value of K, but the resistance is
greatly reduced for a tee-beam with the flange in tension (p = 00).

The positions of the centre of rotation during buckling may also be approxi
‘mated by using equation 7.25, and these are shown in Figure 7.6. It can be
seen that these are close to the bottom flange for high values of K, but move
lower as K decreases

75. Simply supported beams

N L T T
05 10 1520 25 30

Dimensionless bucking moment ML/VIELG)

Torsion parameter R = Winde) nR/ucu

Figure 711. Monosymmetric I-beams in uniform bending.

The differential equilibrium equations and the energy equation for asymmetric
sections (xo, yo #0) bent in their stier principal planes are the same as
equations 7.18-7.20 for monosymmetric beams (xp=0) bent in the plane of
symmetry, and so the buckling resistances under uniform bending are also given
by equation 7.22. The most common example of an asymmetric section is an
unequal angle (Figure 7.7).

However, angle beams bent in principal planes are rare in practice. More
‘commonly, an angle beam is loaded in the plane of a leg, in which case it bends
biaxiall in the same way as does the zed-beam shown in Figure 78b. Alternative-
ly, it may be constrained to deflect in the plane of a leg, in much the same way as
the zed-beam of Figure 7.0 is constrained to the plane of the web, in which case
the constraints prevent any buckling effects

7.8 Moment gradient

A simply supported beam is under constant moment gradient when its bending.
moment distribution varies linearly, as in the case of the beam shown in
Figure 7.12a, which is bent by end moments M and PM, so that

M,=M—M( + Bil 029

Moment gradient 120

SEE TM
b) Bending Moment nal Fate g

Figure 712 Doubly symmetric beams under moment gradient

In this case, some ofthe terms in the differential equations of bending and torsion
(equations 7.18 and 7.19) have variable coefficients, and it is much more difficult
to obtain solutions than previously. Approximate solutions may be obtained by

using. the hand energy method discussed in Chapter 3, while more accurate
solutions may be found by using finite clement computer programs, such as that
described in Chapter 4

‘Some numerical solutions for doubly symmetric sections obtained by this latter
method are shown in Figure 7.126, in which the variations of a moment factor

a= M/My,

0)

are plotted against the end moment ratio P for selected values of the torsion
parameter K (equation 7.14). It can beseen that while qs almost independent of
K, there are substantial variations with the end moment ratio f, and that the
buckling resistance of a beam in double curvature bending (3 = + 1) is approxi

mately 25 times the resistance for uniform bending (£ = — 1), Approximate lower
bound values for 4, are given by
5 + LOSB + 0.362 <25 731)
1406-049) <25, (732)
while approximate mean values are given by

151//(1— 1.408 + 0.89%) <26. 733)

The incre

sin buckling resistance that occur with inex
ratio f are associated principally with chat

inthe end moment
ges that occur in the buckled deflected
shapes, which change from a symmetric half sine wave for uniform bending

76. , baron fare
(>) ===>

PES op ame
=>
15

(al Beam

[c) Buckled Shapes

Figure 719 Buckled shapes for double curvature bending (f= 1

(B= — 1), Lo the anti-symmetric double half wave for double curvature bending
(B= +1) shown in Figure 7.13e, In this later case, the anti-symmetry ofthe My
distribution requires the de uto be anti-symmetricif the twist rotation 6 is
symmetric (see equations 7.1 and 7.2). Because of this anti-symmetricu the lange
deflections u + hó/2 are biased towards the compression regions of the beam
(top left and bottom right), as shown in Figure 7.130

‘The buckling of monosymmetric I-beams under moment gradient has been
studied in [2], and approximate solutions for double curvature bending (f= + 1)
are shown in Figure 7.14. In this case it does not matter which flange is the larger
(e. the curves for p and (1— p) are identical), since each flange is similarly
nt from the behaviour shown in Figure 7.11 for

— 1), where one flange acts only in compression,

stressed. This is quite dife
beams in uniform bending ($
and the other only in tension. Approximate solutions for other moment gradients
are given in [2], which suggests that these may be approximated by

ML
Ver,ch

where

anl I +, 03%
W

fy = HB + PP 035
fa = auf ~ B/2-+ BH + BID 039

‘Transverse loads 181

Figure 714 Monosymmetric -beams in double curvature bending.
and

= 91 + pl +R) 037)
7.4 Transverse loads

741 CONCENTRATED LOADS

The bending moment M, in a beam with transverse load varie

ng the beam,
and so the differential equilibrium equations again have some variable coef
cients, and are difficult to solve. Numerical solutions obtained by the finite
element method for doubly symmetric beams with central concentrated loads Q
are shown in Figure 7.15. For the case where the load acts at the shear centre;
these may be closely approximated by using

Mn = 2M, (738)
with

ig 135 039)
and

M.=0L/4, (740)

When the shear centre load Q acts at a distance a away from mid-span, then
equation 7.38 can again be used, but with

135 + 04(2a/L)? a

77. Figure 728 Buckling of beams with cen ntrated load

and A
Mau = (QL/4)(1 — 40214),

Equation 741 is plotted in Figure 7.16, and it can be seen that the resistance is
lowest when the load is at mid-span, so that the maximum bending moment is
also at midespan.

When ther

centre,

are two shear centre loads Q, ea a distance a from the

du = 1.0-+0.35(1 - 2a/L)? a)
and

Ma =(QL2\(1

1. a

Equation 7.43 is also plotted in Figure 7.16, and it can be seen that the moment
at the b

factor a is lowest when a= L n is in uniform bending, and

highest when a=0 so that the length of the uniform

ding region is zero.
However, even though a, decreases as 24/L increases, the value of Q at buckling
increases. The approximate solutions of equations 7.38-7.44 are collected to-
gether in Figure 7.17

Solutions for the buckling of monosymm
are reported in [3-5], These solutions indicate that the interaction between the

Transverse loads 198

20

itt

L L 1 1 J
0 0.2 04 06 08 10

Dimensionless load distance 2a/L

moment distribution and monosymmetry is complex, especially for T-beams,and
that it is no

y to develop simple approximations for the buckling resistance
Nevertheless, a wide rang:

u + ete) can

except for very m

nosymmetric sections for which p approaches 0 or

78. 194 Simply supported beams

Seon Segoe! [roman Diarios [on [Rese

rrserospea3p 25 | pa

iosoasino-2e/i? | 02e

cuyo [rt
a 02a
La ë

DS [ns san]
Of ma ste 35010 | o<peoes
pe) Ny at |

Fi a 1230 base
113-0108 T O<fi<0.7

42 UNIFORMLY DISTRIBUTED LOADS

‘The buckling resistances of doubly symmetric beams with uniformly distributed
loads q are shown in Figure 7.18. For shear centre loading, these may be
approximated by using equation 7.38 with
BL am
and

Ma = al? 748)
‘This result is lower than the buckling resistance given by equation 7.39 for beams
with central concentrated loads, because the bending moment distribution for
distributed loading is more nearly uniform, This reinforces the conclusion
reached in section 7.41 that the moment factor aq decreases as the moment
distribution approaches that of uniform bending
equations 7.47 and 7.48 is included in Figure 7.17.

he approximate solution of

no

80

60

Torsion parameter K=Vin"El,/GJL2)

Figure 748 Buckling of beams with uniformly distributed loads,

Solutions for monosymmetric I-beams are reported in [3-5]. For beams with
uniformly distributed shear centre loading, these can be approximated [4] by
using equation 7.45 with Ma = 4L4/8, zu = 1.13, and

f=}

14 (749)
Once again, these are reported to be of good accuracy

xcept for very monosym-

7.5 Transverse loads and end moments
751 GENERAL

While isolated simply supported beams rarely have in-plane end moments, they
may be used to represent the individual segments or spans of braced or continu-
ous beams, as shown in Figure 7.19. The appro»

supports (u=0= 9, u° =

mations for out-of-plane simple
=") ignore any continuity actions which may

79. 136 — Simply supported beams

imations for braced and continuous beam

restrain buckling of the individual segment or span. The effects of buckling
restraints are treated in Chapter 8, while approximate methods of estimating
their effects in braced and continuous beams are given in Chapter 10. When these
buckling restraint effects are ignored, then approximate estimates ofthe clastic
buckling resistances can be obtained from section 7.5.2 or 7.5.3 following.

‘The magnitudes ofthe in-plane end moments acting on a beam may be conveni

ently expressed in terms of the corresponding end moments of a builtin beam.
Thus for a beam with equal moments at both ends, the corresponding fixed end
moments for central concentrated load Q are QL/S, so that the actual end
moment may be expressed as JOL/8, where 0 < < 1. The maximum moment
Ms always occurs at mid-span, and is given by

Ma=(1—P/D0L/4. (730)

For t where the load Q acts at the shear centre of a doubly symmetric
Tsection beam, the value of M at elastic buckling may be approximated by using
ith

equation 7.38

an = 1.35 +0366. LAN

sam with zero moment at one end, the fixed end moment is3QL/16,and so
the actual end moment may be expressed as 3POL/I6, where 0 < f'< 1. The
maximum moment occurs at mid-span while 0 < $ <8/9, and is given by

My.=(1—38/8)QL/4 32)

and occurs at the end while 8/9 < f-<1, and is given by

M = 3QL/16. as

Transverse loads and end moments 187

2357

Moment factor anatta/Nyz

0 02 04 06 08 10
End moment factor ß

Figure 7:

car centre of a doubly symmetric

For the case where the load Q acts atthe sh
I-section beam, the value of M, atelastic buckling may be approximated by using

equation 7.38 with

diy = 135-4 0.158 (754)
while 0 < p< 089, and by
12+30p 755)

while 0.89 <
The approximations of equations 7.50-7.55 are collected together in Figure
17, and plotted in Figure 7.20.

53 UNIFORMLY DISTRIBUTED LOADS

For a beam with equal end moments and uniformly distributed load y, the fixed
end moments are L*/12, and so the actual end moment may be expressed as

80. 138 Simply supported beams

Bat2/12, where O<< 1. The maximum moment occurs at mid-span while
0<P<075 and is given by

Ma = (1 — 29/34 756)
and occurs atthe end while 0.75 < P<1 and is given by
Ma = Bal? 12 sn

For the case where the load q acts at the shear centre of a doubly symmetric
I-section beam, the value of M a elastic buckling may be approximated by using
equation 7.38 with

113-0128 058)

while 0</<075, and by
au 238 +488 05)
while 075 << 1
Similar approximations ha
one end, and these are collected together with those of equations 7.56-7.59 in
Figure 7.17, and plotted in Figure 7.20

been developed for beams with zero moment at

7.6 Effects of load height

761 CONCENTRATED LOADS

The buckling resistance ofa simply supported beam may be significantly affected
by the distances of transverse loads from the shear centre axis. Most examples of
offshear centre loading involve freely swinging gravity loads, as for example in
the case where a crane runway gitder supports loads acting at its top flange, or
when a monorail supports loads acting at its bottom flange

When a transverse concentrated load Q acts at a distance (yg — yo) below the
shear centre and moves with the beam during buckling, as shown in Figure 721a,
it exerts an additional torque — Qlvg— Ye about the shear centre axis. This
additional torque opposes the twist rotations 4 of the beam, and increases the
resistance to buckling, as indicated in Figure 7.15. Conversely, when the load acts
above the shear centre, then the additional torque amplifies the twist rotations,
and reduces the buckling resistance ofthe beam.

This latter effect may be very dangerous, as for example in the case of a beam
with 1, > 1, which might not be expected to buckle. Ifthe torsional sifiness of
such a beam is low, then buckling may occur in a predominantly torsional mode.
In the limiting case in which there is no lateral deflection and the work done by
the bending moment M, is negligible, hen the buckling load may be evaluated by
setting the work —Qlve—yo)$2/2 done by the load equal to the strain
energy 16? stored in torsion, where a is the torsional stifines of the system, so

Effects of load height

0/2

Flexural

Buckling (0) Torsional Buckling

Figure 7.21 Effects of load height

that

= 4/09 Yo) 0.0)

‘The negative signin this equation indicates that downwards load causes buckling
when it acts above the shear centre so that (yg Jo) is negative, For simply
supported doubly symmetric beams with central concentrated load, the torsional
stiffness is

= (4GJ/LY{1 — [tan h(a/2K)]/(a/2K)) (761)

This stiffness may be reduced in monosymmetric beams with negative values of
Ba such as the trough girder shown in Figure 721b, by the effect of the
monosymmetry term —¿(£M,P,6dz in the energy equation (equation 720) in
which case the buckling resistance will be correspondingly reduced.

The torsional buckling resistance given by equation 7.60 is reduced when the
minor axis flexural rigidity El, is low, in which case significant lateral deflections
u occur during buckling. Finite element solutions for doubly symmetric I-beams
with central concentrated loads Q are shown in Figure 7.15, in which the
dimensionless buckling loads OL?/,/(EI,GJ) ate plotted against the torsion
parameter K. Two sets of curves are shown, one suitable for I
which the load height is represented by the parameter

tion beams in

&=2yo/h as

so that = + 1 for bottom or top flange loading, and the other of more general
application which uses the parameter

ve |(#) as

Ela

81. 140 — Simply support

For doubly symmetric Esection b

K (764)

Figure 7.15 indicates thatthe effect of the load height parameter 2yq/h increases
with the torsion parameter K, demonstrating that load height effets are more
important in short beams with high EI, and low GJ.

The solutions shown in Figure 7.15 for doubly symmetric beams may be
closely approximat

+ 28tavo} ass

Mtz)

Numerical solutions for doubly symmetric beams with off-centre and two:
point loads are given in [4], These indicate that equation 7.65 can again be used
to obtain approximate solutions, provided the appropriate expressions for M,
and a, given in Section 7.4. are used, Additional numerical solutions for beams
ven in [6-9],

with end moments and central concentrated loads are

The elfects of load height on the buckling resistances of monosymmetric
T-beams have been tabul raphed in [4]. This latter study
suggests that good approximations can be obtained for central concentrated

ed in [3,5] and

Figure 722, Approximation for load height effects on monosymmetrc Lbeams.

loads on monosymmetrical beams (except for extreme monosymmetry) by using

0.0)

where
an

ion 7.66 is plotted in Figure 7.2.

sistances of

and fs is given by equation 7.46. E
Approximations for the effects of load height on the buckling
monosymmetre beams with of-centre or two point loading are also given in [4]

762 UNIFORMLY DISTRIBUTED LOADS

The effects of load height on the buckling resistances of simply supported beams
with uniformly distributed loads q are similar to those of concentrated loads.
Numerical solutions for beams of doubly symmetric cross-section are shown
in Figure 7.18, and these may be closely approximated by using equation 7.65
with

an = 113 0.58)
and

My = al 05)
Numerical solutions for monosymmetric I-beams have been tabulated in [3,5]
and graphed in [4]. This ki be approximated
(except for very monosymmetric beams) by using equations 7.66 and 7.67 with
equations 7.68, 7,69 and 7.49. Additional numerical solutions for beams with end

re given in [6-9]

er study suggests that these m

moments and uniformly distributed loads a

7.7 Problems

Determine the elastic Aexural-torsional buckling moment of a de

properties are given in Figure 7.23, and which isin uniform bending and simply

supported over a span of 120m, as shown in Figure 7.24a
T tm 7 a

82. 142 — Simply supported beams

17.24 Problems 71-718.

Evaluate the effect of the pre-buckling in-plane deflections on the elastic flexural
torsional buckling moment of Problem 7.1

Determine the elastic Nexural-torsional buckling moment of a narrow rectangu-
lar (50 mm x 10 mm) steel beam (E = 200000 MPa, G = 80000 MPa) which isin
uniform bending and simply supported over a span of 20m, as shown in
Figure 7.24.

A monosymmetric beam consists of an I-section member whose properties are
given in Figure 7.23 with a 300mm x 20mm plate welded to one flange. Deter
mine the elastic Nexural-torsional buckling moments for the beam which is in
uniform bending and simply supported over a span of 120m, as shown in
Figure 7.24,

(a) when the plated flange is in compression;
(0) when the plated flange is in tension

A tee-section beam 305mm deep is formed by cutting the [section shown in
Figure 7.23 along its centre line. Determine the elastic flexural-torsional buckling

Problems 148

moment forthe beam which is in uniform bending and simply supported over a
span of 3.0m, as shown in Figure 7.246,

(a) when the flange is in compression;
(0) when the flange is in tension.

Determine the elastic flexural-torsional buckling moment M of a beam whose
properties are given in Figure 7.23, and which is simply supported over a span of
120m with unequal end moments M and 0.4 M which cause double curvature
bending, as shown in Figure 7.244.

Determine the elastic Nexural-torsional buckling load ofa beam whose properties

are given in Figure 7.23, and which i simply supported over a span of 120m. The
beam has a concentrated load at mid-span, where bracing prevents lateral

deflection and twist, as shown in Figure 7.24

Determine the elastic exural-torsional buckling load ofa beam whose properties
are given in Figure 7.23, and which i simply supported over a span of 120m. The
‘beam has a concentrated load at the shear centre at mid-span, which is unre

strained, as shown in Figure 7.24

Determine the elastic flexural-torsional buckling load of a beam whose
properties are given in Figure 7.23, and which is simply supported over a span of
as shown in

120m. The beam has uniformly distributed shear centre loading

Determine the clastic Nexural-torsional buckling load Q of a beam whose
properties are given in Figure 7.23, nd which is continuous over two equal spans
of 12.0m, Each span has a central concentrated load Q acting at the shear cent
which is unbraced, as shown in Figure 7.24.

Determine the elastic flexural-torsional buck!

are given in Figure 7.23, and which is simply supported over a span of 12.0 m. The

load ofa beam whose properties

83. 144 Simply supported beams

beam has a concentrated load acting on the top flange at mid-span which is
unbraced, as shown in Figure 7.241,

Determine the elastic Mexural-torsional buckling of a steel beam (E =
200000 MPa, G = 80000 MPa) of circular hollow cross-section (outside diameter
610mm, wall thickness 127mm, 1, = 1060 x 10mm‘, J = 2130 x 10° mm‘),
and which is simply supported over a span of 60m. The beam has a concentrated
load acting at a height of 1500mm above the beam axis through a rigid stub
column welded to the beam at mid-span, as shown in Figure 724).

‘The monosymmetric beam of Problem 7.4 is simply supported over a span of
120m, as shown in Figure 7.24k. The beam has a concentrated load acting

() at the top flange; or
(i) at the bottom flan

at mid-span, which is unbraced. Determine the elastic exural-torsional buckling
loads

(a) when the plated flange is in compression;
(0) when the plated flange is in tension

7.8 References

1. Kitipornchai, $, and Trahair, NS. (1980) Buckling properties of monosymmetic I
beams. Journal ofthe Structural Division, ASCE, 10ST), 941

Kitipornehai, . Wang, CM. and Trahair, NS. (1986) Buckling of monosymmerrio

I-bcams under moment gradient. Journal of Structural Engineering, ASCE, MAS,

3. Anderson, JM. and Trahair, NS. (1972) Stability of monosymmettc bea
cantilevers, Journal of the Structural Division, ASCE, SHST), 269-86

4. Wang, CM. and Kitiporacha, S. (1986) Buckling capacities of monosymmeti 1
beams, Journal of Structural Engineering, ASCE, LC), 2373-91

5. Roberts, TM. and Burt, CA. (1985) Instability of monosymmetric I-beams and
cantilevers, International Journal of Mechanical Sciences, 215), 313-24

6. Austin, WJ. Yegian, S. and Tung, T.P. (1955) Lateral buckling of clastcally end
restrained beams. Proceedings, ASCE, 81 (Separate No. 673, 1-25,

7. Trahair, NS. (1965) Stability of 1-bcams with elastic end restraints. Journ
Institution of Engineers, Australia, 316), 157-88

8. Trahair, NS. (1966) Elastic stability of beam elements in rigid-jointed structures
Journal ofthe Institution of Engineers, Australia, 7-3), 171-80

9. Trahair, NS, (1968) Elastic stability o propped cantilevers. ii Engineering Transac
tion, Institution of Engineers, Australia, CET), 94-100

ms and

the

8 Restrained beams

8.1 General

A beam is often connected to other elements which participate in the buckling
action, and significantly influence its buckling resistance. Braces are provided
specifically for the purpose of increasing the buckling resistance (Figure 8.1a), but
many other elements, such as sheeting, which are primarily intended for other
purposes also have important restraining actions.

‘These elements may induce restraining end moments which act in the plane of
loading (Figure 8.1b, as for example in built-in beams and propped cantilevers,
and also in continuous beams. These restraining end moments change the
in-plane bending moment distribution and modify the buckling resistance.
The influence of in-plane end restraining moments on the elastic buckling of
simply supported beams is discussed in section 7.5, and of continuous beams in
Chapter 10.

‘Out-of-plane restraining actions (Figure 82) restrict the buckled shape of the
beam, and increase its buckling resistance. Discrete restraints act at points where

braces or other restraining elements are connected to the beam, and induce
actions which resist the buckling deflections, rotations, and warping displace
ments. These restraints are usually assumed to be elastic, in which case they may

be characterized by their elastic stiffnesses.

In some cases the discrete restraints may be assumed to be rigid, so that they
prevent one or more of the buckling deformations. In the special case where rigid
restraints prevent lateral deflections and twist rotations of the restrained cross
sections, then the beam may be described as a braced beam which consists of
beam elements between the restraint points and the supports. The elastic
buckling of braced beams is treated in Chapter 10.

Continuous restraints are usually considered to be uniform along

the length of
a beam, and are often used to approximate the actions of restraining elements
which are connected to the beam at closely spaced intervals, asin the case of roof
sheeting (Figure 83). Continuous restraints are similar to discre

that they induce actions which restrain the buckling deformations (Figure 8.2)
‘They also are usually assumed to be elastic, and so may be characterized by their

clastic stiffnesses. When continuous restraints can be assumed to be rigid, then

they enforce a longitudinal axis about which the beam cross-sections rotate
during buckling

This chapter is concerned with the influence of out-of-plane restraints on the
clastic buckling of simply supported beams, The buckling of restrained columns

84. 140 Restrained beams

He pl

Figure &1 Beam restraint

is treated in Chapter 6, of restrained cantilevers in Chapter 9, of continuous

beams in Chapter 10, and of n d beam-columns in section 11.5.

8.2 Restraint stiffnesses

821 CONTINUOUS RESTRAINTS

A beam may be continuously restrained as shown in Figure 8.2b by a

tional restraint of si h acts at a di

ness a w 2 y, below the centroid,
minor axis rotational restraint of stifines y which acts at a distance yb

int of stiffness 2, and by a warping res
stes

Restraint stiffness 1
ES nf EA

The actions exerted by these restraints can be replaced by the shear centre

ent per unit length. For this set of
actions to be statically equivalent to 1 they must be related

to the shear centre deformations (F

82)
A= tal td rc)
wl » : > es

which have translational stiff

xis rotational stiffness ay, (at a distance

85. 148 — Restrained beams

The actions exerted by these restraints can be replaced by the shear centre

iR) May Mas, BJ" es

in which Bis the bimoment. For this set of ations to be statically equivalent to
those of the restraints, they must be related to the shear centre deformations

{D} = {uw 0)" 69
by

R} = [aa] {D} en
in which the discreto restraints’ sifiness matrix is

nalysis

In the general case of restrained beams, exact solutions for the buckling loads
cannot be obtained, and a numerical method must be used to obtain approximate
then the strain

solutions. Ifthe approximate energy method is used (Chapter

energy stored i ‘buckling should be increased 10

an element during

Lau el | erw? + Ely"? 46162 )d

15 {D}" (za) {D} dd (89)

{y

stored in the discrete and continuous restraints

to account for the strain ener
the element

acting
the finite element method of computer a

element stiffness matrix should be au

[aq] {d}dz which accounts for continuous restraints [1] and the beam

ysis is used (Chapter 4) then each

nented by transforming the term

Stiiness matrix should be augmented by including a transformation of the
HDY"[as]{D} term for each discrete restraint.

inuous restraints

84.1 UNIFORM BENDING

A simply supported beam (u,
uniform continuous restraints bus

dou = 0. = 0) in uniform bending with

los into n half sine waves [2] so that

L (8.10)

The moment which

uses elastic buckling can be obtained by substitutin
buckled shapes into the energy equation,

EU+EN=0 en

in which the strain energy $5°U stored during buckling is given by equation 89

and the work done 13° Von the applied loads is given by

(Mou + MB,0)d: (60)

This leads to

m

613)

In general, a number of trials must be made before the int

y value of which
leads tothe lowest value of the buckling moment M can be determined.

Doubly symmetric beams (A, = 0 = yo) with continuous warping and torsional
restraints only (= 0 = 4) buckle ina single half wave (n = 1) at a moment

LR) (8:14)

M= {PGI + ELLE + a

It can be seen that these restraints contribute directly to the effective torsional
stiffness of the beam.

Doubly symmetric beams with continuous minor axis rotational restraints
also buckle in a single half wave (n = 1) at a moment given by

feet Bie} EH RGE} an

Solutions obtained from equation 8.15 for the dimensionless buckling resistance
M/M, are plotted in Figure 84 against the dimensionless restraint position
,P,/Mys for specified values of the dimensionless restraint stifiness 2, /P,

86. When the restraint acts above the sh
buckling

centre (y, P,/M,.is negative) the elastic
ment M increases indefinitely with the restraint st
for beams with restraints which act below the shear centre (y,P,/M,. is positive),

However,

the buckling moment increases with increasing restraint stiffness towards a
limiting value M. which is given by
Mo_l
My 2|

M
| 18)

My. Py

This limiting value, which corresponds to Ihe case where the beam buckles with

an enforced centre of rotation (at the restraint position y), is also shown
aphically in Figure 84, It is of inter

1.0) when y, P,/M,_ = 1. In this ase, the restraint has no effect because it acts

at he axis of cross-section rotation (see Figure 7.6) of an unrestrained beam. For

beams with restraints which act below the unrestrained axis of rotation, the

restraints are comparatively ineffective unless they act at some considerable

st 10 note that M../My, is a minimum

distance below the unrestrained axis of rotation
Doubly symmetric beams with continuous translational restraints buckle in n
half waves at a moment given by
[M al? nf
Vpn RFP, My $Y

619)

Ma, =n? P (GI + PRET ILE) (820)

Continuous restraints 181

Equation8.19 has the same form as equation 8.15 for beams with minor axis
rotational restraints, and so Figure 8.4 can also be used for beams with transla
tional restraints, provided M/M,, 2,/P, and y, P,/M,, are replaced by M/M,

a LMP, and 1 y,P,/M,

preted for beams with tran

rotational restraints,

respectively. Figure 84 can therefore be inter.
jonal restraints in the same way as for beams with

However, equations 8.19 and 8.20 indicate that there are a number of different
elastic buckling moments whose values vary with the number n of half waves into
which the beam buckles. The lowest value of M or given values of %L?/x2P, and
¥4Py/Myz may be determined by calculating successive values of M/M,, for n

1,2,3,.... In general, these will decrease at first until the minimum is reached
and then increase. Thus the successive calculations may be terminated as soon as
there is an increase in M/M,,. Some solutions of equation 8.19 are plotted in
Figure 8.5. It can be seen that in general, the number n of half waves at buckling
increases with the dimensionless restraint stifiness %L*/xP,, but decreases as the
dimensionless restraint position y,P,/M,x and the torsion parameter K increase
Thus the number of half waves at buckling will be high in beams with low warping

rigidity EL, which are highly restrained at points high above the shear centre,

3)
o © 2 30 70 sT 30

Dimensionless restraint stiffness ayl2/n?P,

87. Restrained beam:
842 UNIFORMLY DISTRIBUTED LOADS

The elastic buckling of beams with uniformly distributed loads q acting through
sheeting which also provides continuous restraints has been studied in [3,41
do. =0,4%,,=6%,, = 0) Which
areloaded and restrained against minor axis rotation at the Same distance y, from
the shear centre are shown in Figure 8,6. The two diferent sets of curves shown
for K = 001, indicate that the elects ofthe torsion parameter K on these curves
are very small, and may be neglected.
The similarity between the results shown in F

Solutions for simply supported beams (uo

gure 8.6 and those in Figure 84
for uniform bending has led to the approximation

(a2 toi fal? 20) (2 te (ity

lam, * P, m.) em." PS 18M,” pP, \M,

in which
[one E
+ Gr, pi a
e
{op - (00182 Y) _f

fie À {oss , 0624
11109 (Goer Re) 10" a,

Figure 86 Uniformly loaded beams with continuous minor ax

Continuous restraints 159,

055—005K*/(0.08 + K3) for y, <0,

r (825)
005K?

(008 + &

which is reasonably accurate provided that ~2 < y,P,/M,, < 2and a,/P, < 100.

Solutions for beams with uniformly distributed loads and major axis restrain
ing moments acting at one or both ends are also given in [3]. These indicate that
the high efficiency of minor axis rotational restraints acting above the shear
centre falls off as the amount of negative bending caused by the end moments
increases. It is also demonstrated that minor axis rotational restraints are most
effective near the supports where (u — ,6 is usually greatest, and least effective
near mid-span.

‘The elastic buckling of very thin-walled beams (K -+ co) with rigid continuous
restraints which enforce an axis of buckling rotation at the loaded top flange was
studied in [4]. The predicted maximum moments M, at the elastic buckling of
beams which are prevented from deflecting and twisting at their ends
{oz =0= 40,1) are shown non-dimensionally in Figure 8.7 against the load
parameter qL/8M and the end moment ratio f. The relationships between M,
and the maximum end moment M or the distributed load q are shown il
Figure 88. The dimensionless buckling resistance M„L2/[r? (El, El) shown
in Figure 8.7 s usually higher for gravity loading (when the restrained top flange
is generally in compression) than it is for uplift loading (when the unrestrainec
bottom flange is often in compression).

PRRZAN? N
= 085+ (2-23) [015- )horn>0,

Figure 87 Buckling of thin-walled beams with restrained Mangos
y .

88. 154 Rostrained beams

8.5 Discrete restraints
85.1 UNIFORM BENDING

85.1.1 End restraints

bou = 0) of a beam may restrain

Restraints acting atthe supported ends (uo,,

minor axis rotations u’ and warping displacements proportional to $. Thi
particular case of a beam in uniform bending whose four flange ends have equal
minor axis rotational end restraints of stiffness a is shown in Figure 89a. These

exert restraining moments Mz,y=2,U/r,» where the lange rotations at one end
are ty = w+ hg/2, and store strain energy during buckling

CAT 629

1
Up = dal +d!

this is compared with

Us = (D) "Lan }{D}

obtained using equation 88, then the restraint siffnesses at one end ca

expressed as

ar = | ss
ty = 4 J 629
with = 0.
In this case, the buckled shape is defined by [5]
xf steep) ca Pen
where k satisfies
ob A a
sar ot (630)

These equations are the same as the corresponding ones for braced columns with
equal rotational end restraints (section 6.5), and define a beam effective length
KL equal to the distance between the inflexion points of the buckled sh:
shown in Figure 8:96. Note that this effective length (ofthe buckled shape) is not
related to the length between any inflexion points o
Substitution of equations 827-8.30 into the

1 deflected shape.

wde=0 (831

leads to

ñ = =

Figure 89 Buckling of be

89. 156 — Restrained beams
and
52M aay
07 PERE el
which can also be obtained from equations 75 and 7.10 for an unrestrained beam

by substituting the effective length ÆL for the actual length L

The relationship obtained from equation 8.30 between the effective length
factor k and the dimensionless end restraint stiffness ay L/El, is shown in
Figure 89e. It can be seen that varies almost linearly from 1.0 to 0.5 as
(2aL/EL,/Q + ax L/EL) increases from zero (for an unrestrained beam) to unit
(fora rigidly restrained beam). This suggests that k may be closely approximated
by

au L/El,

* 2420 L/El LE

as shown in Figure 896,
The correspondence of equations 8.29 and 830 with equations 6:31 and 6.3

for columns with equal end restraints has led to the suggestion that the effects of
unequal flange end restraints aa am, at the ends A,B of a beam should be
approximated by using the solutions shown in Figure 8.10b obtained from

Figure 6.6b for braced columns with unequal end restraints. This suggestion has
proved to be satisfactory for a wide range of braced and continuous beams (se
Chapter 10).

Sometimes constraints may act at the ends of a beam [
nt end warping (#5,,=0) The increased bucklin

7] which effectively

pres resistance may be

Figure 810 Buckling 0

Disc

te restraints 187

o 0 10 5 20 30

ameter K=V (nel, /GI

pproximated by

in which the warping restraint effective length factor is given by ky =0.5, as
shown in Figure 8.11

If elstic end restraints
neously, the buckling resistance may be estimated from

inst minor axis rotation and warping act simulta-

where the two ends are identicall restrain

effective length factors ky ky may be approximated by
44m,
Kay 1 (837)
LEI
and
44 Ca + nk /4)L/ Ble =

kw =
4+ Ya + ay 0/4) JE

90. Restrained beams

Itisusually assumed that the twist rotations at the ends ofa beam are prevented

by rigid torsional end restraints. When these restraints are not rigid, then the

buckling resistance is decreased. Numerical solutions for the reduced resistance

may be approximated by us

Manier

MY (OS FASK TG e

These approximate solutions are shown in Figure 8.12. It can be seen that the

resistance falls off rapidly for beams with high values of K, and that
mparatively hig

significant reductions below M,, for such be

co h values of torsional resraint stifines are required to prevent

sn the points of support may restrain lateral
deflections u and twist rotations ó. The particular case of a beam in uniform

Approximate

straint parameter [ol /6J1/(1+0g2L/05)

Discrete restrain

a
A
an +

ding with central restraints is shown in Figure $.13a. When the restraints are

rigid so that u = & = Oat the restraint point, the beam buckles antisymmetrically
into two half waves, each having an effective length of L/2, as shown in

Figure 8.13b, Thus the elastic buckling resistance is given by

(6:40)

with k=0S.
When the

the beam may buckle in a symmetrical mode, as

shown in Figure 8.132, An analytical solution can be obtained for the special case
when the restrain stifinesses ae related by
ar ETA _ anal?
1 1 sa
Al N
in which
are? __(n/2K) couts 24) m
16EL, ~ A eot(s 20 -1 es
Je = MIEL CL) 645)

and M is given by equation 8.40,

91. 160 Restrained beams

Equations 840-8.43 correspond to the symmetric buckled shape defined by

10]

u_ó_ 2 sintr/kL)
370" L Mesh

649

L/2, and to an effective length KL. This buckled shape is the
same as that for a column with a central elastic translational restraint
Equations 840-843 with y, = 6/0 satisly the energy equation for the restrained
beam.

The relationship obtained from equation 8.42 between the reciprocal of the
effective length factor k and the dimensionless central restraint stiffness
due L'AGEL, is shown in Figure 8.13c. It can be seen that 1/k increases from 1.0
towards 2.86 approximately as he restraint stifiness increases from zero towards
infinity

Also shown in Figure 8.13c is the solution 1/k=2 for the antisymmetric
buckling mode of a beam with rigid restraints (Figure 8.130), which intersects the
symmetric mode solution ata limiting restraint stiffness given by

ana L/16E,

Since buckling always occurs in the mode which has the lowest buckling load, a
beam with ag. > A, always buckles antisymmetrically in two half waves, and at
a moment obtained from equation 8.40 by using k= 0. On the other hand, a
beam with ap, < gu buckles symmetrically in a single half wave, and at a

tion 8.40 by using the solution of equation 8.42. This
solution may be closely approximated by

649

moment obtained from eq

Tin =O ina 69
Equations 8 and 84 indicate thatthe values of an, and ay. required 10
moves higher. The valo ofa, decreases lo zero when

ES h ae
HT DAR? fh

for which the value of a is given by

144K)
(+8K

16E

(848)

Thus it can be concluded that a top (compression) flange translational restraint
(r= —h/2) alone of stifness ax =16x* EJ,/L will be sufficient to cause anti-
symmetric buckling with k = 0.

When a central translational restraint acts alone, then the limiting restraint
stillness a, required to cause antisymmetric buckling increases with the restraint

Diserete rest

(5 161
position yy. The solutions of [9] for beams with shear centre restraint (py =0)
in this case the limiting restraint stifiness may be approximated by

aE? _ 1531 + SK?)

EI, © (1+ 074K)

suggest t

(849)

For a > &yı, then k=05, while approximate solutions for M for a <a, may

be obtained from
M _ [2400+ SterL/EI,)

«ff (650)
MN | 24000513761, J

The effect of a central torsional restraint acting alone has been studied in
[9,10}. The buckling mode was found to change from a symmetric half wave
Lo two antisymmetric half waves at limiting restraint sffnesses a, which may
be approximated by
Bank
GJ

100x (851)

For ax, > na, then k =0.5, while approximate solutions for aq <a may be
obtained from

y tre (Ma _ 5 (8.52)
mullite) 4
in which M isthe value of M for k= 05,

The tales of m equal sem matris spaced at equal intervals
s= Ef +1) has been studied in [1-13] For high restraint values, he eam
may bucle between traits ito (y+ 1) half waves a a moment Bien by
equation 40 with

k=s/L. 653)

The limiting stiffness ay ofeach restraint required to cause this mode of buckling
may be determined approximately by treating the beam as continuously re-
strained (section 8.4) and determining the limiting stiffness 41 of the continuous
restraints required to cause buckling at the moment determined from equa-
tions 8.40 and 8.53. The limiting sifiness zu. ofeach discrete restraint may then
be approximated by ‘concentrating’ the continuous stiffness ay, according to

a = La | (859)
For & < au, the beam buckles in a complex mode which is characterized by
twisting at some or all of the restraint points. The buckling resistance may be
approximated by ‘smearing’ the discrete restraints into equivalent
continuous restraints of stifiness

a= mal (655

and using this in the continuous restraint solution of equation 8.13.

92. 102 Restrained beams
This method is generally conservative for low values of n, the number of
restraint points. For example, the limiting stiffness calculated by {is method fora
al(n,~ I) translational restraint at the shear centre is given by
dL? HU + 5K?)

EL, ~ +R)

beam with a ce

(850)

which varies between 1.41 and 191 times the value obtained from equation
849, which the buckling resistance for less than the limiting stiffness is given

by
(a

1

+(e)

‘Similarly, the limiting stiffness calculated for a beam with a central rotational
ana L/GI = 312 (1 + SK) (858)

which varies between 099 and 1.48 times the value obtained from equation 8.51
ice for less than the limiting stifness is given by

while the buckling resista
equation 8.52 It can be expected that the accuracy of the method will increase

with the number of restraint points n,

852 NON-UNIFORM BENDING

8521 End restraints

There have been a number of studies [14-19] ofthe effects ofelasticend restraints
against minor axis rotation (ax), warping (aw), and twist rotation (24) on the
elastic buckling of beams in non-uniform bending. The scope ofthese studies is
summarized in Figure 8.14

Generally, the studies indicate that these effets are qualitatively similar to
those on beams in uniform bending (section 8.5.1.1). This observation leads to the
suggestion that the elects of elastic lange end restraints might be approximated
by using

(859)

for the maximum moment at elastic buckling, in which the moment factor a, is
approximated as in sections 7.3,7.4 or 7.5 for simply supported beams with
centroidal loading y, yg =0),and the effective length factor kisapproximated as
in Figure 8.10.

A similar suggestion may be made for estimating the effect of load height (see
section 7.6) on the buckling of end-restrained beams from equations ofthe form
of equation 7.65, but with M,,,P, calculated using the approximate effective

Discrete restraints 168

e — iy : Fr, i ler 1
are eS
spec} pee

Figure 814 Studies of beams with end restraints

length kL determined from Figure 8.10 in place ofthe actual length. This leads to
predictions of somewhat variable accuracy, but these may be tested by comparing
them with the numerical solutions reported in the studies summarized in

Figure 8.14,

85.2.2 Intermediate restraints

‘There have been a number of studies [8-11, 13,20] of the effects of central
translational or torsional restraints on the buckling of simply supported beams in
‘non-uniform bending, including both the limiting stfinesses required to cause
antisymmetrical buckling, and also the buckling resistances when the rest
stiffesses are less than the limi

g values. The scope of these studies is sum-
marized in Figure 8.15. While quantitative solutions for many specific cases can
be obtained from these references, some more general but qualitative conclusions
can be drawn,

First of all, when the central restraint exceeds the limiting stiffness, then the
buckling resistance is determined for a half beam which generally has a less
uniform moment distribution than the complete beam. Because of this, the
benefits of central limiting restraints are usually greater for beams in non-uniform
bending than they are for uniform bending. As a result of this, the limiting
restraint stfinesses for beams in non-uniform bending are generally higher than
for uniform bending,

93. 164 Restrained beams

G— | es | eel 4

u, ET

The limiting translational restraint stiffnesses y, required to cause antisym-

‘metrical buckling increase as the restraint point moves lower, or as the load point

‘moves higher. Antisymmetrical buckling often cannot be achieved when the
restraint point is below the load point, but top flange restraints with
ay L>/El, > 1612 will cause antisymmetrical buckling when the loads act at or
below the top flange. Approximate solutions for translational restraints with

21 <4, may be obtained from

My ar [(Ma

‘ ea)
a) tl

in which M, is the maximum moment in an unrestrained beam at elastic
buckling, and M, is the maximum moment for antisymmetrical buckling.
The limiting torsional restraint stiffinesses aya required to cause anlisymmetri
cal buckling appear to increase as the load point moves higher. Approximate
nal restraints with 29, < ap may be obtained from

solutions for tors

M aes {(Mu\*_ 4} a
(ie) rate) Y eet

where My, My are defined similarly to the values used in equation 8.60 above.

sints spaced at equal intervals have been

The effects of equal stillness re
studied in [13]. For centroidal loading, these may be approximated by assuming
that the beam isin uniform bending and using the approximate method proposed
in section 85.1.2. The accuracy ofthis wll increase with the numberof restraints,

since the bending moment distributions between the restraints will become more
uniform.

8.6 Problems

A beam whose properties are given in Figure 7.23issimply supported over a span

of 120m, as shown in Figure §.16a, It has a uniformly distributed uplift load

acting at the top flange, which is continuously restrained by elastic minor axis

rotational restraints of stifiness &y/P, = 10. Determine the elastic flexural
buckling load.

Determine the effects of rigid end warping restraints on the elastic Nexural-
torsional buckling moment of a beam whose properties are given in Figure 723,
“and whieh isin unifor

shown in Figure 8.16b.

ending and simply supported over a span of 120m, as

A beam whose properties are

ven in Figure 723 is in uniform bending over a
span of 120m, as shown in Figure 8.16b. One end of the beam is simply

supported out-of-plane, and the other has elastic minor axis rotational and
warping end restraints whose stifesses are defined by aq, = 42 /h? = EI, /L.

Determine the elastic flexural-torsional b

A beam whose proper

is are given in Figure 7.23 is in uniform bending over a
span of 120m, as shown in Figure 8.16b. Both ends are prevented from deflecting
laterally, fre to rotate laterally and to warp, but are elastically restrained a

are defined by axe L/GJ = 08.

twist rotation by restraints whose stifines

94. 166 Restrained beams

Determine the reduction in the elastic flexural-torsional buckling moment from
the value for full restraint against twist rotation,

A beam whose properties are given in Figure 7.23 is in uniform bending over a
span of 120m, as shown in Figure 8.16b. Both ends are simply supported, and the
mid-span is elastically restrained against lateral deflection and twist rotation by

restraints whose stiffnesses are defined by

tee

Determine the increase in the elastic Nexural-torsional moment caused by the

aL? (,_ 2x
A

restraints,

7 References

1. Hancock, GJ-and Trahair, NS. (1978) Finite element analysis ofthe lateral bucking
of continuously restrained beam-columns, Cll Engineering Transactions, Institution

of Engineers, Australia, CE 20,2), 120-2

Trahai, NS (1979) Elastic lateral buckling of continuously restrained beam-columns

The Profession ofa Civil Engineer (ds. D. Campbell-Allen and E.H. Davis), Sydney

University Pres, Sydney, pp. 61-73.

Hancock, GJ. and Trahair, NS. (1979) Lateral buckling of roof purins with dia

phragm restraints Ciel Engineering Transactions, Institution of Engineers, Australia,

CE 10-15,

Ings, NL and Trahair, NS. (1984) Lateral buckling of restrained roof purlins

Thin-Walled Structures, 2, 285-306.

5. Trahait, NS. and Bradford, MA. (1991) The Behaviour and Design of tee Structures,
revised 2nd edition, Chapman and Hal, London.

6 Vacharajittphan, P. and Trahai, NS. (1974) Warping and distortion at section
joints, Journal ofthe Structural Division ASCE, 100 (ST), 47-64.

Ojalvo, M. and Chambers, RS, (1977) Elle of warping resrains on I-beam buckling

Journal ofthe Structural Division, ASCE, 103 (STI?) 2351-60,

8. Mutton, BR. and Trahair, NS. (1973) Stflness requirements for lateral bracing
Journal ofthe Structural Division, ASCE, 99 (ST10), October, 2167-82

9. Netheroot,D.A. (1973) Buckling oflateraly or torsionally restrained beams. Jounal of
the Engineering Mechanics Division, ASCE, 99 (EMS), 713-91

10. Taylor, A.C. and Ojalvo, M. (1966) Torsional restraint of lateral buckling. Jo
the Structural Division, ASCE, 92 (ST2), 115-29.

11. Home, MR. and Ajmani, I. (1969) Stability of columns supported laterally by
side-rals. International Journal of Mechanical Sciences, 1, 159-74

12. Milner, HR, (1975) The buckling of equal fanged beams under uniform moment
restrained torsionally by stiff braces in Proceedings, Sth Australasian Conference on
the Mechanics of Structures and Materials, Melbourne, pp. 405-20.

B

References 167

Medland, LC. (1980) Bucking of interbraced beam systems. E
2,2), 90-6

Austin, WA. Yegian, S. and Tung, T.P. (1985) Lateral buckling of elastic
restrained I-beams. Proceedings, ASCE, BI (Separate No. 673, 1-25.
Trahair, NS. (1965) Stability of I-beams with elastic end restraints, Journal ofthe
Institution of Engineers, Australi, 37 (6, 157-68

Trahair, NS, (1966) Elastic stability of I-beam elements in rigd-jointed structures.
Journal ofthe Institution of Engineers, Australie, 38 7-8), 171-80

Trahair, NS. (1968) Elastic stability of propped cantilevers, Ciil Engineering
Transactions, Institution of Engineers, Australia, CEIO (1, 94-100.

Nethereot, DA. and Trahair, NS (1976) Lateral buckling approximations fr elastic
beams. The Structural Enginer, 546) 197-204,

Trahar, NS, (1983) Lateral buckling of overhanging beams. Instability and Plastic
Collapse of Steel Structures (ed, LJ. Morris, Granada, London, pp. 03-18.
Trabai, NS. and Netherco, DA. (1984) Bracing requirements in thin-walled struc
tures, in Developments in Thin-WalledStructures-2ed J. Rhodes and A.C. Walken),
Elsevier Applied Science Publishers, pp. 93-130,

end

mal

95. 9 Cantilevers

9.1 General

re usually considered to be flexural members which in the plane of
builtin

I the support (ey = d = 0) and free at the other end (t,t
109.13, However, in this book, the word canti

lever is used to describe the out-of-plane end conditions of a Rexural member
at the support
ined), as

shown in Figure 9.10. Such

atthe other end (up

restrained end (0,0, unrestrained), since any in-plane support usually provides

On the other hand, if there are o

formations uy, dr
vent end

t-of-plane restraints which pre

deflection and twist (u, = 6. = 0) as shown in Figure 9.lc, then the behaviour is

nilar to that of a simply supported beam which has restraints a

axis rotation and warping at one end (4 = di =0). This case is treated in
Chapter 8
The absence of out-of-plane restraints at the free end of a cantilever

substantially changes the buckling mode and buckling resistance, as shown in
Figure LL. In an end lo

further, rather than
(Figure 7.)

od cantilever it is the tension flange which buckles the

he compression flange of a simply supported beam

This chapter is concerned with the elastic buckling of cantilevers of doubly

symmetric or monosymmetric cross-section under end moments, end loads, or
uniformly distributed loads, and the effects of load height from the shear centre

axis. The effects of concentrated and continuous restraint are also discussed.

The buckli

18 ofa cantilever which is continuous with theres of the structure at
its support (4 # 0,6% # Dis treated in Chapter 10, while the inelastic buckling of

9.2 Uniform bending

Cantilevers with end moments which cause uniform bending are rare in practice
since most loading actions are by loads rather than moments. One possibl
source of an end moment is an axial load which acts cocentrically so that i

96. it is closely related to those of canti
transverse loads only, which have no end moments to do work

IS Tis much more likely that an end moment would rotate dy with the end of a

cantilever, as shown in Figure 92b. In this case the work done during buckling

swith,

wert u must be increased by the work (M 9,1) done by the end moment to
b) Cantilever rained Bea 2M ul dz + Mibu, ony
Qut-ot-Plane of-Plane
Figure 84 Cantilever boundary condition In the case of the cantilever of Figure 9.2, My = — M.

The addi
evaluated by r

onal work — Mouw, done by the end moment M can also be
presenting the end moment M by stat
M/h acting at the top and bottom flange end centroids, as shown in Figure
with such a loading is a beam-column, and so the cantilever represents the 9.2b. These fange end centroids displace relatively in

limiting case where the load tends to zero and its eccentricity to infinity. Despite bu

statically equivalent to a concentric axial load and an end moment. A member

the transverse direction by
udinally F (1/2), 0 that the work done by the end forces

its rare practical occurrence, end moment loading sat least of theoretical interest, is Ak M/M(F hou), or — Móra as in equation 9.1. (It may be noted that
since the closed form solution for simply supported beams with equal and Figure 92 ls dram for positivo $e, alihough ln blo caso the actual 4, À
‘opposite end moments (section 7.2.1) is so widely used as a basis for design. negative, as shown in Figure 9.3a)

Some care must be taken to define the plane of action ofa

moment acting at the When the end moment rotates with the cantilever, then the relative
free end of a cantilever (Figure 92a). The general expression ~3f42M Hu d the buckled cantilever and the end moment
ig |

ositions of
the same as those of half of a
in section 174 for the work — 16*V done during buckling is consistent MA, simply supported beam in uniform bending (see Figure 7.5) (the actual positions
with an end moment M which does not rotate ¢, with the end ofthe cantilever, as spond to a rigid body deflection u, and rotation ¢, of the half beam and
shown in Figure 926, but which remains in a plane parallel to the original moment). Thus the buckled shapes of the cantilever can be derived from those
undeformed YZ plane. While itis dificult to imagine a practical loading situation |

e bx with the end of tl

given by equation 7.4 for a simply supported beam as

where the end moment does not ro santilever, this case

1/5 = 6/0 = 1 — cos 02
Substitution of these into the energy equation (section 284.1 or 17.47)
E 1 Delft
+ e Ll (enue? + 21.02 + 616%) 0241 ['amqur de o 3)
5 Ar ,
+ | \ N h leads to
A ML i
A ML Fern 30) er
is KL IY where
F 4 NE Es K JH), 65
/ 7 40 = amp, es
PRES) ‘
| where
e P,= RE en
! Solutions of equation 94 are compared in Figure 935 with those of
ious a Le ie ia | Mall [Er en +K es

97. Figure 83 Buckling of cantilever with end mom
vation 7.5 for simply supported beams. It can be seen that the
antilever buckling resistance M ishalfof My. or K = 0,and that it increases only

slowly with the torsion parameter K
‘Numerical solutions for the elastic buckling resistances and buckled shapes of

1d moments which do not rotate ds are also shown in
ies almost linearly with the torsion

3. The buckling resistance va

ter K, and may be approximated by

ML/JEL,GD=1.6+08K 09)
are somewhat greater than those of equation 9.4 for cantilevers
(61, but still ess than those of equation 9.8 for simply

These resistance:

whose end moments rot

supported beams.
The buckled shapes of cantilevers with K = 1 are compared in Figure 93a,
These shapes have been normalized so that the value of uy = is unity. It ean be

seen that for end moments which do not rotate dy, the lateral deflections u are

somewhat less than for moments which rotate 4, but the twists ¢
greater and of opposite sign. For equal lange I-beams, the buckled shapes may be

Shown as the top and bottom flange deflections uy » = u + h6/2. It can be se
that while both flanges of the cantilever with rotating end moment deflect in the

same sense, the reverse is the case for the cantilever whose end moment does not
rotate. When the moment rotates, itis the bottom (compression) lange which
buckles the further, but when the moment does not rotate, then itis the top

(tension) flange which buckles the further

922 MONOSYMMETRIC SECTIONS

‘Numerical solutions forthe dimensionless buckling resistances ML//(E1,G) of
monosymmetric cantilevers with end moments M which do not ro

ate are
reported in [1], and these are plotted in Figure 9.4 ag

inst the beam parameter
R= REICH) 0.10)

for selected values ofthe monosym

netry parameter

p= bl, oan

These curves are somewhat similar to those shown in Figure 7.11 for simply
supported beams in uniform bending, in that the
increases with the monosymmetry parameter p (that i, asthe relative stifines of
the compression flange increases) However, there is a reversal near p = 09, and

ıckling resistance generally

the buckling resistances of tee-sections with cir flanges in compression (p = 1.0)
are less than those of some slightly ess monosymmetrie sections This is because
the tension flange is more important in a cantilever than it is in a simply
supported beam (in an equal flanged cantilever with an end moment that does not
rotate itis the tension fan

whereas itis the co

hich buckles the further, as shown in Figure 9.3,

ession flange which buckles the further in a beam, a
shown in Figure 7.1), Thus the monosymmetry benefit which results from
having a very large compression fat

‘of having no tension flange in a

mewhat reduced by the disadvantage

98. 174 Cantilevers
9.3 End loads

93.1 DOUBLY SYMMETRIC SECTIONS

Numerical solutions for the buckling resistances of doubly symmetric cantilevers
with end loads Q acting at distances yy below their centroids are reported in [2],

and are shown in Figure 9.5. These solutions may be approximated by

oL f 12) f 126-0) |
= on L+ax-2l1+ 5 \
Yeon “Vta EA Ts eof
(0.12)

in which

vo |(EL) _ 20K

e 1 013)

la)

roidal loading (¢ = 0), the variation of the buckling resistance with the

torsion parameter K is approximately linear, as its for end moments which do

not rotate dy. A comparison of Figures 9.36 and 9.5 indicates that the buckli
nent, Thisis
the

resistance under end load is substantially greater
so because the bending moment M, caused by end load is high only ne

restrained support
The effect of load height yo is demonstrated in Figure 9.5, and it can be seen
that while bottom flange loading significantly increases the buckling resistance,

top flange loading may reduce it substantially, especially for cantilevers with high
es of K. The non-linear effect of load height is suggested by the approximate

formulation of equation 9.12. This effect is alittle diferent from that shown in

Figure 722 for simply supported beams, where the buckling resistance increases

indefinitely with yo. For cantilevers, the approximation of equation 9.12 ug
that the resistance may approach an upper bound.

932 MONOSYMMETRIC SECTIONS

Numerical solutions for the buckling resistances of monosymmetric cantilevers
with end loads Q acting at distances (yg — Jo) from the shear centre are reported
in (1, 3,4], and approximate equations in [1]

For shear centre loading, the buckling resistances vary with the beam par

ameter K and the monosymmetry parameter p in a somewhat similar manner to
that shown in Figure 94 for uniform bending, although the resistances are

higher for end load because the moment M, is high only near the

generally
support

For bottom fa
that for shear cent

nerall increased over

ge loading, the buckling resista
loading, with much the same pat

of variation with K

and p. However, this pattern changes considerably for top flange loading, with
very signifi

nt reductions in stifiness for high values of p (
small tension fa

forcantilevers with
5). It appears that the maximum resis

for sections with nearly equal flanges (p 70.5) and thi

dons are almost independent of their a
sion, or p =O, lange in tension).

de (p= 1, flange in compres

9.4 Uniformly distributed loads

94.1 DOUBLY SYMMETRIC SECTIONS

Numerical solutions for the buckling resistances of do

bly symmetric cantilevers
with uniformly distributed loads acting at distances y, fr

‚m their centroids
are reported in [2], and are shown in Figure 9,6. These solutions may be

99. wor

mensionless bi

Figure 88 Buckling of cantilevers with distrib

approximated by

aw 14-09] af 13-09 |
af Vacio 21

Teen Jm 1 136-005)

0:14)

in which e=(2y/MK/x is a dimensionless load height parameter, For centroidal
loading (¢=0), the buckling resistance again varies almost linearly with the
torsion parameter K, and is higher than the resstane for end load (Figure 9.5,

serally lower. For load

because the bending moment i
centroid, the resistance changes non-linearly with the dimensionless load height

as suggested by the approximate formulation of equation 9.14

9.42 MONOSYMMETRIC SECTIONS
‘Numerical solutions for the buckling resistances of monosymmetrie cantilevers
with uniformly distributed loads q are reported in [1,3], and approximate
equations in [1].

Discrete restraints 177

Again, the bucklin

resistance for shear centre loading varies with the beam
parameter K and the monosymmetry parameter p in a somewhat similar manner
to that shown in Figure 94 for uniform bending. For bottom flange loading, the

the
same pattern of variation with K and p. For top flange loading, the pattern

changes considerably, and it again appears that the resistance of equal flanged
sections is close to optimal, and that the resistances of tec-sections are almost
independent oftheir attitude (lange in compression or tension),

9.5 Continuous restraints

The actions of continuous restraints on cantilevers may be modelled as in
section 82.1, and ther effects on elastic buckling may be analysed numerically by
using the adaptation of the finite element method described in section 83,
The effects of continuous elastic restraints on the elastic buckling of doubly
metric cantilevers were studied in [5]. The results reported for end louds
acting at the centroid indicate that in many cases a cantilever acts as if rigidly
restrained if the translational restraint stiffness per unit length exceeds 2,
25001, /L* approximately, provided the torsion parameter K = (*El,/GJ}
is not small (ie. K > 05).
Rigid continuous translational restraints enforce an axis of buckling rotation

at the height y, at which the restraints act. The results of [5] indicate that for
cantilevers with high values of K (ie. K> 1.5), tension flange restraints are

ge restraints, and that the buck
ose for unrestrained cantilevers,
especially when the end load acts at or below the centroid,

markedly more effective than compression

ling resistances are significantly higher than €

9.6 Discrete restraints

96.1 SUPPORT RESTRAINTS.

9.6.1.1 Minor axis rotation rest

The support restraints for cantilevers are generally assumed to be rigid, so that
minor axis end rotations ug and end warping displacements proportional to dí
are prevented, in addition tothe usual support conditions that lateral deflections

up and twist rotations do are prevented.

However, a rigid restraint is not required to prevent the minor axis end
rotation u when end twistin

do is prevented, since under these circumstances
the minor axis end moment — (My is automatically zero. Because ofthis, any

non-zero elastic restraint stiffness will be sufficient to ensure tp = 0. Indeed,
the limiting case of zero rotation restraint, for which the cantilever theoretically

100. 178 — Cantilevers

Lk———Bearings on vertical axis

permit rotation u' and ensure
u"=0

Beam is free
to rotate in
horizontal plane

Figure 87 Rigid body buckling mode (0 = 0)

buckles at zero applied loadin a rigid body mode by rotating around the support
79.7, does not lead to structural failure, since the cantilever

is shown in

remains unstrained. Instea sible for the cantilever to support non-zero,

loads without structural failure, even though it may rotate do as a ri

about the support. Thusit may be assumed tha
isprevented, even where there is no minor axis restraint, provided end

End warping is usually assumed to be prevented atthe supports of cantilevers by
ke very significant contri

rigid restraints to the flanges. Warping restraints m

butions to the buckling resistance of cant nd when these are removed

O, then the buckling resistances are

realy reduced, as is

ures 98, 9.9

loading, the buckling resistance à

K, indicating that end warping restraint is required to ensure that
idity El, contributes to the buck

warp at the support (or ‘overhangi

demonstrated in F 19.10, These figures show that for centroidal

warping

resistance. Cantilevers which are fee to

segments) buckle with an almost linea

0 05 10 15
orsion parameter K:

101. 100 Cantilevers Discrete restraints 181

orsion parameter KeV(n?l,/GIL?)

Figure 930 Buckling of overhanging segments with uniformly dist

08

Torsional restraint parameteı

variation in the twist rotation das shown in Figure 98a, so that the warping 02
ain energy stored ${5Ef4(9") dz is very small

The elastic buckling resistances of overhanging segments with 4% =0 shown in
gures 98, 9.9 and 9.10 have been approximated by [2] Figure 811 Ovechanging segments with torsional restraints

ML/VEL, GI) = 164 005K, 019) reduction in the stiffness of the torsional restraint at the support will cause a

ov _f. 1-00 | af 1 corresponding reduction in the buckling resistance
Tann” LUC amps AR Te The reduced buckling end loads Q of overhanging segments with elastic
(016 torsional support restraints are shown by the non-dimensional ratios 0/0, in
Figure 9.11, in which Qi the buckling load for an overhanging segment with
a se-09 | { 2-09 | rigid torsional support restraint (see equation 9.16) These may be approximated
4-2) > ii PP a pproximated

Seren Turise- Vt TE] by

Qu)

O 1 Mag nel | |f snl /G.
PEA y L/GI |

in which ¢ = (2yy/h)K/x isa dimensionless load he
On UL SV (SFA +R) Fa LION)

‘es are significantly lower than those predicted by the corresponding equa
tions 99, 9.12 and 9.14 for built-in cantilevers for which 4, =0.

parameter. These resistan

(9.18)

which is accurate for bottom flange and centroidal loading while K <
conservative for top flange loading while K <2 and a,,L/GJ > 25.

9.6.13 Torsional restrain

Supports which are capable of providing full minor axis rotation and warping

restraints will usually be capable of providing full torsional restraint also, so that

o=0. However, when a support is not capable of providing any warping The effects of rigid restraints acting at the free ends of
aint, then it is possible that fll torsional restraint may not be available. Any

9.62 END RESTRAINTS

antilevers have been

studied in [6,7]. Cantilevers with free end restraints which prevent lateral

102. displacement u, and twist rotation dy are similar to simply supported beams with

end restraints ion up and warping deflections propor.
tional to dí. They may be treated as in section 8.5.2.1, as may cantilevers with full

The elastic buckling of doubly symmetric and monosymmetrie cantilevers with

ints was sudid in [68]. The cantilevers had either
concentrated end loads Q, or uniformly distributed loads q, acting at the top
flange, centroid, or bottom flange (2y,/h=—1,0,1). For doubly symmetric

cantilevers, the optimum distance from the support of. restraint which prevented
both lateral deflection and twist rotation (u =$ = 0) varied between 0.35L and
085L approximately, depending on the loading, load height, and torsion par
ameter K, and the benefits from optimum bracing were substantial,

of K and top fange loading.

cally for

milevers with high va

The effectiveness of rigid translational or torsional intermediate restraints was

lso studied in [6]. In general, translational restraints (w= 0) were found

comparatively ineffective, except when acting at the end of the top fang
Torsional restraints ( = 0) were generally more effective, especially when they
acted at a distance from the support between 0.451. and 0.75L approximately

inst both

although not as effective as optimum restraints ag

9.7 Problems

he elastic lexural-torsional buckling moment ofa cantilever whose

in Figure 723, and which is in

Determine

cantilevered length of 6.0m, as shown in Figure 9.124,

(a) when the moment rotates with the twist rotation of the free end of the
cantilever

(6) when the moment does not rotate

E
the monosymmetrie section of Problem 7.4, and which has a cantilevered length
of 60m as shown in Figure 9.12b, and which has end mo
rotate with the twist rotation of the free end,

mine the elastic Aexural-orsional buckling moments of a cantilever with

ca À 5 I
2)" 2 3)" ==

Determine the elastic flexural-torsior

al buckling load of a narrow rectangul
0 mm) steel cantilever (E = 200000 MPa, G = 80000 M Pa) which has

a cantilevered length of 1.0m and a concentrated end load acting on the top
surface ofthe cantilever, as shown in Figure 9.12c.

Determine

he elastic Mexural-torsional buckling load of a cantilever whose
properties are given in Figure 7.23, which has a cantilever length of 60m as

shown in Figure 9.1
along the top flange,

Determine the elastic flexural-torsional buckli
ment having the same properties, len

h, and loading as the cantilever of
Problem 9.4, but whichis free to warp at the support, asindicated in Figure 9.12e

Determine the stiffness ofthe torsional restraint required at the support of the
overhanging segment of Problem 9.5 if the reduction in the elastic flexural

torsional buckling load below that for fll orsional restraint isnot toexceed 10

103. 104 — Cantilever
9.8 References

1. Wang, CM. and Kitiporncha, S. (1986) On stability of monosymmetric cantilevers.
Engineering Structures, 83), 169-180.

Trahar, NS. (1983) Lateral buckling of overhanging beams, in Instability and Plastic

apse of Steel Structures, (ed. LJ. Mortis), Granada, London, pp.503-18.
nderson, JM. and Trahait, NS. (1972) Stability of monosymmeiric beams and
levers, Journal ofthe Structural Division, ASCE, 9NST1), 269-86,

TM. and Burt, CA. (1985) Instability of monosymmerie beams and
als 3-2.

5, Assadi, M. and Roeder, CW. (1985) Stability of continuously restrained cantilevers

ing Mechanics, ASCE, 11(12), 1440-56.

6 Kitipornchai 8, Dux, PF-and Richter, NJ. 1984) Buckling

mal of Structural Engineering, ASCE, LOST), 2250-62.

Nethercot, DA. (1973) The effective lengths of cantilevers a
buckling The Structural Engineer, SI), 161-8

8. Wang, CM, Kiipornchi, 5. and Thevendran, V. (1987) Buckling of braced

symmetric cantilevers, International Journal of Mechanical Sciences, 249) 321

cantilevers, Internat

10 Braced and continuous
beams

10.1 General

The restraining actions between adjacent segments in continuous beams are
not easily predicted, because there are interactions between the segments at buck
ling, as indicated in Figure 10.1. These interactions allect the sliffnesses of the
restraining segments, which depend not only on their geometric and material
properties, but also on the relative importance of thet loading. Because of this,
the information given in Chapters $ and 9 for elastically restrained beams and
cantilevers cannot be used directly to predict the elastic buckling loads of
continuous beams.

This chapter is concerned with these buckling restrainin

actions, and presents
approximate methods for predicting their effects. The methods may also be
applied to elastic beams which are prevented from deflecting laterally and
‘isting (w= 6 = 0) at supports and brace points as shown in Figure 102b. An

extension ofthese methods to beams with overhanging segments (Figure 1020) is

A | B É
E
|

lug

{b) Segment AB (e) Segment BC

Figure 184 Interaction buckling

104. lle dde do 4

b) Braced Beam (€) Overtangino,

also presented. The extension of the methods to inelastic beams is referred to in
sections 14.54 and 14.55

10.

Interaction buckling

The continuity between the adjacent segments of a continuous beam during
buckling induces interactions between them, Several diferent buckling modes:

possible for the two-span beam shown in Figure 103a, depending on the loading

(dl Mode 3, Zero Interacio

arrangement. For the mode shown in Figure 10,3, the left-hand segment is

avily loaded, and dominates in the buckling action. During buckling, this

segment is elastically restrained by the right-hand segment, and there isa point of

inflection in the deflected shape of the left-hand segment. For the mode shown in

Fig

deflected shape of the heavily loaded right-hand segment.
When th

buckling, then the inflection point is at the interior support, as shown in

10.3, the reverse is the case, and 1

re is a point of inflection in the

Figure 10.34. In this special case there is no buckling int

More generally, however, one segment is elastically restrained by the other, as

shown in Figure 10.1, In this ease, the left-hand segment AB has the inflection

point, and is positively restrained, in the sense thatthe restraining minor axis end
moment My provided by the right-hand s
ofthe left-hand segr

ment BC opposes the end ro
ent. Thus the buckling resistance ofthe left-hand restrained
d'abov ed in Figure 10.4
ions of the dimensionless buckling load

gmentisinen he value for zero restraint, as indi

This figure [1,2] shows the v

1*//(EI,GJ) of one segment of a narrow rectangular section beam with the

Ba = (ane L/ELNG + ana L/ EL) (10.1)
B= (ty L/EL3-+ 2ayL/EL), (102)

in which
A (103)
Any = — Mya (104)

105. 108 Braced and continuous beams

Note that the values of ay equal to 0 and co corresponding to zero and rigid
to corresponding values of f and f equal to 0 and 1. Thus the

es from the zero minor axis restraint value (or = 0)

restraints le
buckling resistance incre
to the rigidly restrained value (or B= 1) as the restraint stiffness 07, increases

from 0 to «e
When the left-hand segment of Figure 10.1a is positively restrained (so that t

end moment M,

right-hand segment is negatively restrained, in that the minor axis end moment

M acts in the same sense as the end rotation u, as shown in Figure 10.1c. This

‘opposes the minor axis end rotation 44), then the restraining

negative restraining action decreases the buckling resistance of the restraining

right-hand segment below the value for zero restraint ($, = 0),as indicated by the
lower curves in Figure 10.4 for negative values off

The particular case where Q, = 20; and L, = L, may be examined in detail
Analysis ofthe in-plane bending of the beam leads to

),

and
140118
+ Lil

Br =1.50,

while minor axis continuity and equilibrium at the interior support require that

(uy), = — (ays Which leads to
ANA AN
= + LE) TA,
18 (Pa), =0 for zero interaction leads to (f,),=0 and 0/0, =
37.012542 1.46 (see Figure 104) instead of 05. A better guess is ($), = — 20 50

20.498, which is very close to 0.5. Thus
ase the dimension-

that ($), 20:49 and Q3/0, > 13/2
the effects of interaction between the two segments are to
less buckling load of the left-hand segment from 254 for zero interaction to 27.3
approximately and to decrease that of the right-hand segment from 37.0 to 13.6
approximately

This example demonstrates the general characteristic of interaction bucklin
that sufi action takes place to increase the buckling resistance of the
restrained segment and decrease that ofthe restraining segment until their ratios
equal to the ratio ofthe applied loads. Thus all segments participate in the buck!
ing action, and buckle simultaneously, even though one segment is restrained
by

The buckling load combinations for the two-span beam of Figure 10.1 with

Ly = Ly are shown as an interaction diagram in Figure 10.5. It can be seen that
the buckling load Q, increases slowly with ©, until a maximum value is reached
near the zero interaction point Q, = Q,. Soon after this both Q, and Q, reduce,
ases from zero is

Q, rapidly and Q, slowly. The slow increase in Q, as

Interaction buckling 189

—-— y=0 approximation
5 approximation
aight line approximations

Y load OLE

buckli
EX

Dimensiontess

ling Load Qt}

Kling of a tn

primary due tothe change in the major axis end restraint parameter ($), which
increases from 0. to 1.0 when Q,/0, increases from 0 to 1.0. Figure 10.4 indicates

that the increase in the buckling resistance due to this cause is greater than the

decrease due to the decreas
100,

in the minor axis restraint parameter ($) from 0.5

Also shown in Figure 10.5 are
tions obtained by assuming

h-dot curves for the buckling load predic-

ero interaction (B; =0) and using the lower of the

buckling load factors for each span. These predictions are generally conservative,
except at Qs = Q,, where they are exact. In this case the beam buckles with zero
interaction between the segments,

Another set of predictions is shown by the dashed curves in Figure 10.5 for
Which it is assumed that the

straint provided by the restraining segment is
unaffected by its loading. Thus aq, =3EL,/L is used so that f¿=05. These
predictions are generally unconservative, although they are close when either Q,
or 3 is zero.

It can be seen from Figure 10.5 that in this case both of these assumptions
produce approximations which are reasonably close to the accurate solutions,
This is because in this case of a narrow rectangular beam, the minor axis

restraints have their smallest effects. More gen

ally, the minor axis restraint
effects willbe greater, and these approximations may not be sufficiently accurate.

approximate methods are discussed in section 102.

106. 190 Braced and continuous beams
10.3 Approximate methods

103.1 STRAIGHT LINE METHOD

ion diagram of Figure 10.5 suggests that close approx

mations can be obtained by drawing the straight dotted lines between the zero
interaction load combination and the buckling loads obtained for the cases
0, =0 and Q, =0 by assuming that the restraint provided by the restraining

segment is unaffected by its loading. Unfortunately, this method has proved to be

00 complex for use in routine design, possibly because the available tabulations

ofelastic buckling loads are not only insufficient to cover al the required loading

and restraint conditions, but also because they are too detailed to enable them to

be easily used

10321 Gener

10d [3], it is assumed that there are no minor

In the zero interaction met
actions between adjacent segments at the brace and
‘buckles as if simply supported and indepen

bending or warping int

support points, Thus each segm
ts. This is equivalent to assuming that each se
th factor of k= 1

its adjacent segme

between adjacent brace or support points has an efletiv

The buckling load of e

h segment may then be estimated by using the

available data for simply supported beams (Chapter

Ttis usually convenient to

ad factor given by

express this as the buckl

aot (105)
M
moment in the segment at elastic buckling, and

in which M is the maximun
Mais the corresponding m

diferent estimates 2, for the buckling load

This procedure leads toa numb

bound estimate ofthe buckling load factor 2
ned from the lowest of

factor, one for each segment. A lowe

it which buckling actually takes place may then be obta
these estimates, so that

A=), (106)

The other values of 2, generally overestimate the true buckling load factor

Thus for the two-span beam of Figure 10.1 with Q, = 20s, Ly = Las (
015, (a =1.5, and an intial load set corresponding to Q,L3/,/(E1,G)) = 1.0,
0213), (1,6) = 0.5, the zero interaction assumption of (4), = (Ps), =0 leads
to buckling oad factors of 4, = 25.4/1 and 4, = 37/0. (see Figure 104). Thus the

lower bound estimate is À = 254, which is alittle lower than the actual buckling

load factor of 27.4. The value 2, = 74 is much higher

The zero interaction method may be summarized as follows:

(a) Determine the properties EI, GJ, Ely, L of each of the n segments,

(b) Analyse the in-plane bending moment distribution throughout the beam

for an initial load set, and determine the initial maximum moment M for

(6) Assume that the effective le

(4) Estimate the maximum moment M, at elastic buckling for each segment,
and the corresponding buckling load factor 2, = (M./M,

(e) Determine a lower bound estimate of the be:

th factor k of each segment is un

The properties and load
calcu

of a braced beam are shown in Figure 10.6. The
jon steps (all in N and mm units) in the zero interaction method of
approximatin

(a) El, =20 x 101%, GJ =49.33 x 10%, El, = 2500 x 10°, L
(b) For an initial load set of M = 1.0, Mu; = 1.0, Maia = 05.
© ky = 1.0, ky = 10.

(@) Using

1 = L = 4000,

let Gs) (y, PEN
m ram
2 (3489 x 109, /(1-+ 1.768%)

M,

(b) Bending Moment

107. 192 Braced and continuous beam
1.30, My, =921 x 106 2, = 921 x 10°
Mua = 1240 x 105,2,

= -05,

9-04, 180 x 10°

(0 4=921 x 10°
This result is 2074 less than the accurate solution of 2 = 1156 x 10° determined by

2 finite element computer program of the type described in section 4.4,

1033. General

A more accu

ate estimate of the buckling load factor may be obtained by

the interactions which take place between adjacent
nts buckle simultaneously

estimating the effects of
segments during buckling [4-6]. Because all segn
these effects need only be estimated for one segment, and itis usually most
if ths isthe segment which has the lowest zero interaction buckling

load factor. This segment is identified as the critical segment, and its adjacent
ments generally provide

nents. The restraining se
ment, and increase its bucklin

positive restraints to the critical
The increased buckling resistance of a critical
Ih factor. It is assumed that this is related to the

determining its effective le
relative end stiffness ratios G,, Ga as shown in Figure 10.7 in the same way as for

Figure 107 Buckling ofrestrinted segments

Approximate methods — 193

compression members with elastic end restraints [5]. Thus

4% =QEL/L), os)

is the minor axis flexural stfness of the critical

segment, and aga, gy are the
effective stifnesses of the restraining segments,

The effective stifiness ag ofa restraining s 4 by

nent may be approxi

29 = (EL Dir (109)

in which yy depends on the bending moment magnitude and yy depends on th
conditions at the far end of the restraining segment. The factor y is equal
when the segment has to provide an equal restraining moment at its
equal to 3 when there is zero moment at the far end, and is equ
end is built in, as shown in Figure 108.

The factor 7 varies from 1 to 0 as the bending moment magnitude for the
restraining segment varies from zero to a level suficient to cause it to buckle with

zero interaction. For restraining compression members [5], the corresponding
factor is closely approximated by (1 — P/P,)in which Pis the actual compression
force in the restraining segment, and P, is its zero interaction) flexural buckling
load, The accuracy ofthis approximation reflects the linear relationship between
the buckling resistance and E/,, and indicates that the stiflness available for

108. 194 Braced and continuous beams

restraining the critical segment decreases almost linearly as the axial force

increases towards P,

bers, the relationship

MEIC SEL
N 0)

However, in flexural me

Ca Mio (10.10)

forthe buckling resistance of an I-section member varies between proportional to
(EL) and proportional to (El Jas x?EI,}"/4GJL? varies between 0 and vo. It
might therefore be expected that the factor 7, for the stifiness available for

nent should vary between (1—M2/22 M2, and

restraining the critical se
(= Moa Mide
Such expressions are not easy to use, since (Ma) is unknown, and an iterative

procedure must be used, starting with an assur

104 value of (Man. A simpler
computational procedure is to assume

w= (1 Al) (10.11)
in which 2,2 are the zero interaction buckling load factors for the critical and
restraining segments. This approximation is accurate when J, i equal to 0 oF y,
and provides estimates which often lie between those of (1 ~ M2J/2M2,)_ and
(1—Mg/aqM, x Hallows a direct calculation for 7, 10 be made and avoids the
iterative procedure

improved method may be summarized as follows:

{a-e) As for the zero interaction method (section 103.21)
(0) Identif the segment with the lowest buckling load factor (A) as the
critical segment AB, and its adjacent segments as the restraining seg

(@) Use Ay Zens Mag and the stiffness approximations of Figure 108 in

equation 109 10 calculate #4, agp or the restraining segments
(0) Use equation 10. to calculate a for the critical sc

he stifness ratios Gy, Gy

mine the effective length factor k for the critical

(6) Use equation 10.7 to calculate
G) Use Figure 10e to dete

(&) Use k to estimate the maximum moment My. in the critical segment a
clastic buckling, and the corresponding buckling load factor 4 = M
M,

103.32 Example

The calculation steps (all in N and mm units) in the improved method of
approximating the elastic buckling moment of the braced beam shown in
Figure 10.6 are as follows.

(a-e) As in section 10.322.

(0 ABis

he critical segment, with 2, =921 x 10%, and (Zq)yc =2480 x 10

Blastic bucklin

244 = 0 by inspection
sun = GEIL) — fan) = 1886 EL
(i) = 2BI/L

© Gx= CEIL)O= co

y= QEI/L(1.886 E1/L) = 1.06
(i) B=0875 from Figure 107
& zen) |/, „El
R x A
Me mV!" Greit)

= (1.30 x 3489 x 10"J0.875),/(1 + 1.768%/08754)
1169 x 10°
A= 1169 x 10

which is 27% higher than
the accurate solution of 2 = 1156 x 10°

action approximation, and within 1.2% of

10.34 FURTHER IMPROVEMENTS AND EXTENSIONS

Further improvements in the approximate methods for calculating the elastic

buckling loads of braced and continuous beams

e given in [7]. A nun

effective length charts are provided for segments under constant momen
ent, which are generally more accurate than the si
the bending moment distribution factor yy is approximated more accurately

(10.12)

Very accurate solutions are obtained after two or three iteration cycles.
The method described in section 10.3.3 has been extended [8,9] to allow the

approximate analysis of the inelastic buckling of braced and continuous beams
(see sections 14.54, 14.55)

10.4

lastic buckling of braced and continuous beams
1041 BRACED BEAMS

104..1 General

Ther
beams, and the scope of these is summa

have been a number of studies [4,7, 10,11] of the elastic buckling of braced

ally, these
dies demonstrate the conservative nature of the zero interaction method of
approximation (section 10.3.2), the increased accuracy of the improved method

ized in Figure 109. Genet

(section 10.33) and the high accuracy of the method given in [

109. 196 Braced and continuous beams

ous jenzz | u
ose forgo | mm
Optimum [010343 | 101

(a [ram widspan [10
Eu hu 4 mmm] — Midspan [10 '
q fea = foes faz ía
aa 0-05. [ouo3o| m

0122 | 1

— | 0 [os Jouoso| m

Figure 109 Studies of braced beam:

104.1.2 Optimum brac

A study [10] has been made of the elects ofthe brace position on the buckling ofa
braced beam under moment gradient, and these are shown in Figure 10.10a as

nee M/M,, in which M, is the

values of the dimensionless buckling resis
buckling moment
brace ge:

he unbraced beam. It can be seen that the provision of a

ally leads toa substantial increase in the buckling resistance,
values of K,

Optimum brace positions which maximize the buckling resistance are shown

in Figure 10.10. These lie in the half of the beam which has the higher end

moment M, and appear to be close to the brace positions for zero buckling

braced and continuous beams 107

interaction between the segments. A simple method of approximating the
‘optimum brace position is suggested in [10] in which the brace isso placed that it
divides the bending moment diagram into two equal areas (without regard for the
sign of the moment).

104.13 Concentrated m

The elastic buckling of braced beams with concentrated moments at the brace
point (Figure 10.11) has been studied [11]. These beams develop internal re

straints at the brace against either minor axis rotation or warping. These

restraints are associated with the antisymmetry about the brace point of the
bending moment diagram, and with the jump in the bending moment caused by
the concentrated moment

110. 190 1s beams

This behaviour is quite different from that of bea

th symmetrical bending
ams, whose buckled shapes u, tend to be antisymmetrical about
ent segments of these beams are identical and
have identical symmetrical bendi ams, the segments buckle with
zero interaction, and both segments act as if unrestrained, as shown in Fig.
ure 10.12.

moment di
the brace point. When the

DATE)

Figure 10.12 Buckled shapes for symmetrical moment distributions

Bending Moment Distribution

Figure 1013 Buckled shapes for antisymn

Figure 104 Buckling of braced beams with concentrated moments.

sof beams with antisymmetrical bending mi

The buckled shap
are shown in Figure 10.13, For these, either the lateral deictio
about the brace point, so that each segment acts asif fully restrained

u = 0) or warping (9 =0)

against either minor axis rotati
A method of approximating the buckling resistances of braced beams with

concentrated moments has been developed in [11], which has been shown to give
accurate solutions for the beams shown in Figure 10.14, and conserva

042 CONTINUOUS BEAMS

10421 General

he elastic buckling of

There have been a number of studies [1,2,4, 12-14] of
continuous beams, and the scope of these is summarized in Figure 10.1.

these studies demonstrate the conservative nature ofthe zero interaction methoc

of approximation, and the increased accuracy of the improved method, as was

‚cam of Figure 105

104.22 Continuous roof purl

The elastic buckling of continuous thin-walled roof purlins which are continu:
ously restrained against minor axis rotation at the loaded top flange has been

111. Braced and continuous beams

Elastic buckling of braced and continuous beams

Figure 10.18 Buckling loads for thin-walled roof purins

top flange, so that the load height effect (section 7.6) is to resist twisting of the

ing. Conversely, the load height effect of gravity loading at the
top flange isto increase the twisting during buckling. The load height effect is
evidently more important in these continuous purlins than that of the continuous

Puta Eed los | vas | um fo [im E
ay ay varies | 0 ast =
te jas | ufo

la a

E h E E = beam during buck

studied [13]. The buckling resistances under gravity and uplift loading ofsingle-,
two-,and three-span purlin with K = 5 are compared in Figure 10.16 with those
of single-span beams in uniform bending.

The resistant 1 of the restraint stiffness when uplift

is almost independ.

loading acts near the top Mange, since this is close to the centre of buckling
rotation of an unrestrained beam (see Figure 7.6). However, the resistances of
purlins under gravity loading increase significantly with the restraint stiffness,
especially for single span purlins. The effect is less pronounced for two- and

three-span purlins, whose unrestrained bottom flanges are in compression n
the interior supports,

It may be noted that the resistances of these continuous

purlins are always
higher for uplif than for gravity loading, despite the fact that uplift loadin
induces compression in the unrestrained bottom fa

he uplift loading bei

near mid-span. These

Higher resistances result from applied at the restrained

1043 OVERHANGING BEAMS

The elastic buckli

‘overhanging beam consists ofa supported segment and an overhangi

of overhanging beams has been studied in [15,16]. An

egment
(section 9.6.1.2) which is continuous over a support with the adjacent supported

segment, as indicated in Figure 10.2c. The continuity between the segments
generally leads to a restraining action against warping at the common support
The limiting cases are those where the supported segment is so stiff that it
effectively prevents end warpi
common support

("= 0) of the overhanging segment at the

and where the overhanging segment isso tif hat it effective
prevents end warping (= 0) of th two

supported segment. Between the
extreme cases lies the special zero interaction case for which there is no buckling
restraint action at the support. In this ease, both segments act asi free to warp

(6° = 0) at the support. The buckling r gments free to

stances of overhangin

warp are discussed in section 9.6.12, and of supported segments free to warp in
Chapter 7

112. ction between segments during buckling causes the

ment. In
ained by

nt to restrain the more critically loaded se

les critically loaded se

es, a lower bound estimate ofthe buckling resistance can be ob

he lower load factor calculated by assumi
y support. This method is sin

warp (6° =0) at
esented in section 10.32 for braced and continuous beams

The results of the application of this lower b
nging beams with equal end loads Q acting atthe shear centre are shown
Is with K = 2and short overhangi

overh
by the dashed lines in Figure 10.17. For beam
rain warping of the supported segments, the lower bound

segments which r
solutions are reasonably close to the accurate solutions shown by the solid line.

(For beams with K = 0, the lo

beams have no

x bounds are exact because thes
if free to warp.) However, the lower bounds

warping stifness, and always act
are increasingly conservative for beams with short supported segments which

ay be obtained by assuming that both segments are

An upper bound solutio
prevented from warping (= 0) These solutions are shown by the dash-dot lines
in Figure 10.17, and overestimate the buckling resistance, except in the limiting

rigid

‘ase where the supported segment isso short that it provides an effective

An

jecurate approximate method is suggested in [16], in which a warping
nt is approximated by

05 +05/(QoL/M,.) (10.13)

Figure 1047 Interaction buckling of ove

Problems

and the buckling load Q is approximated by

= Qu? ~1/ky) + Qu = 1) (10.14)
in which Qo is the buckling load of an overhanging segment which is fe
(section 9.6.1.2) Q . is the buckling load of a cant
the support (section 9.3.1), and M,, is the buckling moment of a supported

jo warp

+ prevented from warpingat

segment which sree to warp (section 7.2.1) Thus Qo and M,, are as calculated in
the lower bound method by assuming #" = 0,and Q., is as calculated inthe upper

bound method by assuming d =0. The predictions of this approximate method

are quite lose to the accurate solutions, as is demons

d'in Figure 10.17,

105 Problems

Determine the elastic flexuraltorsional buckling load of a continuous be

whose properties are given in Figun
120m, and uniformly distributed shear o

The beam has two equal spa

loading, as shown in F

Determine the value of @ at el.
m whose properties are given in Figure 7.23. The supported span has a shear
centre oad of 4Q at mid-span, which is unbraced, and a load of Q at the end ofthe

1 shown in Figure 10.18b.

ic lexural-torsional buckling for an overhanging

cantilever, which is braced,

Determine the value of @ at elastic exural-torsional buckling for a braced beam

whose properties are given in Figure 7.23. The beam span is 18.0m, and is braced

each load point, as shown in Figure 10.18.

Figure 1048 Problems 104-108.

113. 204 Braced and continuous beam
10.6 References

1. Trahair, NS. (1968) Elastic stability of propped cantilevers, Ciel Engineering
Transactions, Institution of Engineers, Australia, CE1O(1), 94-100
Taha, NS. (1968) Interaction bucking of narrow rectangular continuous beams
Civil Engineering Tr 5. Institution of Engineers, Australia, CE1O(2), 167-172.
3. Salvadori MG. (1951) Lateral buckling of beams of rectangular cross-section under
bending and shear, in Proceedings, First US National Congress of Applied Mechanics,

pp. 403-5.
4. Nethercot, DA. and Trahair, NS (1976) Lateral buckling approximations fr clastic
beams. The Structural Engineer, S46), 197-208

5, Trahair, NS. and Bradford, M.A. (1991) The Behaclour and Design of Stee Sutures,

revised 2nd edition, Chapman and Hall, London.
6, Nethercot, DA. and Trahair, NS. (1977) Lateral buckling calculations for braced
beams. Cio Engineering Transactions, Institution of Engineer, Australia, CE1%2),
2114

Dux,P.F and Ki
Journal

jornchañ.S. (1982) Elastic bucking inuous beam,
he Structural Dicision, ASCE, 10{ST9), 20

8. Nethereot, DA. and Trahair, NS. (1976) Inclati lateral buckl

beams. Journal of the Structural Division, ASCE, 102(STS), 701-1

9. Dux, PF and Kitiporncha,S. (1984) Buckling approximations for inelastic beams
of Structural Engineering, ASCE, 1100), 559-74

jonc, $. and Richter, NJ. (1978) Elastic lateral buckling of -beams with

actions, Institution of Engi

intermediate restraints. Cil Engineering Tra

ers, Australia, CE202), 105-11

11. Cuk, PEE. and Trahair, NS. (1983) Buck

Journal Structural Engineering, ASCE, 1056), 1387-401

abate. NS. (1969) Elastic stability of continuous beams. Journal ofthe Structural

Division, ASCE, 9SST6) 1295-3

13. Hancock, GJ. and Trabair, NS, (1979) Lateral buckling of roof purins wit dia

Australi,

‘hag restraints Ciel Engineering Transactions Institution of Engine
C2, 10-5,

4. Trahair, NS. (1966) Elastic stability of L'beams elements in rigd-jointed

urna! of the Institution of Engineers, Australia, 380-8), 171-80.

15. Nethercot, DA. (1973) The effective lengths of cantilevers as governed by lateral
buckling. Phe Structural Engineer, SU) 161-8

16. Trahait, NS. (1983) Lateral buckling of beams, in Instability and Plastic Coll

LA. Morris Granada, London, pp. 03-18,

11 Beam-columns

11.1 General

Beam-columns are m

bers with compressive axial forces and transverse loads
or moments. Beam-columns which are bent in a plane of symmetry may fail by
excessive bending in that plane, or may buckle out of the plane by deflect

ally u and twisting 6. This flexural-torsional buckling behaviour is an
interaction between the buckling behaviour of columns discussed in Chapters 5
and 6 and that of beams discussed in Chapters 7 and 8 and extended to cantilevers
in Chapter 9 and braced and continuous beams in Chapter 10

The beam-columns considered in his chapter are initially straight and untwis
ted, the axial forces act concentrically through the centroid, and the transverse
loads and moments actin the plane of section symmetry

11.2 Uniform bending

The beam-column of doubly symmetrie cross-section shown in Figure 11.1 is
simply supported and has compressive end forces P and equal and opposite end
moments M which induce approximately uniform bending in the YZ plane. For
in-plane equilibrium, the in-plane deflection v of the beam-column must satisfy
the differential equilibrium equation (section 27.3)

(ELOY + (Puy =0 ain
and the boundary conditions
Der (12)
The deflected shape which satisfies these equations is given by
b= (M/P){cos az + sin pztanuL/2) (13)
in which
se =P/EL, ne)
and the bending moment M, = —EI,v" is given by
M,/M = cos + sin tan (UL (15)

In many cases, beam-columns have compressive loads P w
less than the in-pla

column buckling loads

P=" EL/L Mm}

114. 206 Beam-columns

Im

in which case the bending moment M, is reasonably well approximated by using
the constant value
mn Y (1
Pp,

4 moments M are high enough
For the

When the beam-column compressive load P a
then it may buckle out-of-plane by deflecting laterally w and twistn
en the differential equilibrium

buckled position to be one of equilibrium,

(ELY + (Pa + MS =0 us)
and
LEROY (GI + {PLAY + Mau =0 us)
and the boundary conditions
Om ty. = = Gon = 6 (11.19)

must be satisfied. Equation 11.8 represents the equality at equilibrium between

nal resistance (E/,10) to distributed transverse load and the sum ofthe
o M,, wh

transverse load components of the axial force P and the mom

ation 11.9 represents the equality between the sum of the intern

(EG) and —(GIG' to distributed torque and the sum of the distributed
torque components ofthe axial force P and the moment M,.
It can be verified by substitution that when M, in these equations is taken as

as in the approximation of equation 11.7, then they are satisfied by

buckled shapes given by

and
#0= MP, P} aa

in which 5 and O are the values of w and ¢ at mid-height, provided the load P and
oment M, satisfy

M.\*_(,_?)
M) 5)

In this equation, P, and P, are the column flexural and torsional buckling

resistances (section 5.2) given by

PR EI (0.19)
and

PA (GI + EL ILE) (ay)
in whieh

FLA (110)
in general (but with yo =0 in this case of a doubly symmetric section), and M,, is

the beam uniform bending buckling resistance section 72.1) given by

Mur (PP) (ua

tions 11.11 and

À [era + EL 6"? + Gud

nts the principle of conservation of energy during buckling through

moment M

nd the increase in the work done by the compressive load P and the

solutions of equation 11.13 for the moment M, at buckling increase from

2210 as the axial compressive force P decreases from the lower of the column
buckling loads P, and P, to the beam buckling moment M,, when P= 0, as
shown in Figure 11.1. Further increases take place after the axial load P becomes.

115. 208 Beam-columns

Although the derivation of equation 11.13 can include the approximation of
tation 11.7 for the amplification by the axial force P of the in-plane bending

‘om M to M, it neglects the eflect of the pre-buckling deflections v on
mm-column into a “negative arch’, and
ion 13.53). It has been

the curvature, which transform the b
increase its out-of-plane buckling resistance (see se
shown [1] that the conditions at buckling are more accurately approxi

Gt) (1 ral ale =) (119)

in which
M,

IG ATEN (64 + PEL EVEL)

M, 20)

n buckling resistance (sections 7.2.1 and

«curate expression for the be

16.6) which includes the efect ofthe pre-
Equation 11.19 is often simplified by replacin
= P/P,)by (1 — P/P,)(1 — P/P,) which isconservative when P, > P, > Pasi
often Ihe case. When this is done, equation 11.19 can be replaced by
PAM
PPP JM,

uckling curvature
Myx BY My, and by replacing

simpler

=1 aa)

nsile axial forces

This approximation may also be used for beam-ties with i
ve P) is given by

(negative P}, although it may become increasing!

more negative. In this cas, a safer approximation (for nega
PM ot (ua
PM,

112.21 Sections bent about an axis of symmetr

Monosymmetric beam-columns which are bent about the x axis of symmetry
as channels and equal leg angles, do not buckle suddenly out

(=0) su
cof plane, but bend biaxially and twist as soon as loading is commenced. This
biaxial bending and twisting occurs because of the presence of additional terms
(xd) in the in-plane bending equilibrium equation, which cause coupling
between the in-plane deflection v and the out-of-plane twist rotation 6 at any
non-zero value of P. These additional terms also appear in the corresponding
equations for monosymmetrie columns (yy =0) where they lead to coupling

between the buckling deformations v and ¢ (see sections 52 and 54).

11.222 Sections bent in a plane of symmetry

For a monosymmetrie beam-column bent in a plane of symmetry (x) =0) the

additional terms described above in section 11.221 disappear, and there is no

Uniform be

coupling between v and 4. However, other additional terms arise and lead to
additional coupling between the buckling deformations u and ¢, so that the

differential equilibrium equations (section 28.5.2) become

Eu + (Pe + yoo} +(M,9)"=0

(EIS (GIB) + {Pyou + (PUp/A + 300

+ Mya" — (MU

ob =0 (11.24

equation (section 28.5.1) becomes

Llera + Bg"? + 614) de

pu + (Ira

(1125)

These additional terms also appear in the corresponding equations for mono:

symmetric columns (sections 5.2 and 54) and in those for monosymmetric beams

The buckled shapes which satisfy equations 11.23 and 11.24 and the boundary
conditions of equation 11.10 are given by equation 11.11 and

5/0 =(M, + Pya)/(P,—P) (126)

M, is taken as
and 11

jonstant, and the conditions at buckling obtained from
4 or from equation 11.25 become

Are

PUP,

pls

¡CANO! Jj, ua

and r] =r3 + y]. This reduces to equation 11.13 for doubly symmetrie sections
for which yy =0 and f,=0.

The solutions of equation 11.27 for the combinations of (M,/M,)
(+ Pyo/M,) and P/P, at buckling are shown in Figure 11.2 by curves for given
values of P,/P,)(1 — MBa/Pr? nilar to those of Figure 11.1
for doubly symmetric beam-columns (yy = 0,8, = 0), except for the additional

ccurves shown for negative values of (P,/P,) (1 — M,ß,/Pr?) which occur when

Mf, > Pr}. In these cases, the dimensionless moment (M,/M,,)(I + Pyo/M,)is

not unbounded, but increases as P decreases from P, untl a maximum value is
hed, and then decreases towards zero as P/P, d

M,S./Pr)

The bi ally

because the value of (P,/P,) (1 — M,ß,/Pr}) does not remain constant for a

eases towards 1/{(P,/P.)

‘Kling curves shown in Figure 11.2 are not easy to interpret physi

117. 212 Beam-columns

The flexural-torsional buckling of beam-columns of monosymmetric hat
section has been studied in [2], and of monosymmetric I-section in [3-5]

11.3 Moment gradient

The simply supported doubly symmetric beam-column shown in Figure 11.54
has unequal end moments M and PM, so that the variation of the bending

moment M, is approximately linear, and is given by

M,= M = MI + B)e/L (11.29)

when the components Po due to the axial force P and the in-plane deflections v are
(ed In this case some ofthe terms in the differential eq

and torsion (equations 11.8 and 11.9) have variable coefficients, and itis much.

more difficult to obtain solutions than previously. Approximate solutions may be

obtained by using the hand energy method discussed in Chapter 3, and more

tions of bending

accurate solutions by using finite element comput
described in Chapter 4,
Numerical solutions for the buckling of beam-columns under moment gradi-

programs such as that

ent are given in [3-6]. These may be approximated by using the interaction
equation
M P P
( )

(ax) (11.30)

Figure 11.8 Buckling of doubly symmetric beam-columns under moment gradient

Transverse loads 213

with values ofthe beam-column moment factor C4. given by

(1=092P/P,) fo 55 (1+
o (>

ion 11.31 is plotted in Figure 11.5b, which shows that Cy, generally
€ very hi

nd moment
load r
that the value of C also increases si

o B (except h moment gradients)
P/P,, The more accurate solutions of [4,5] indicate
ly with the value of K =/(r*El,/GJL2)
but that the approximations of equations 11.30 and 11.31 are generally quite
curate, except for some conservatism at high values of X, and P/P,

and with the ax

114 Transverse loads

The elastic Mexural-torsional buckling of simply supported beam-columns of
doubly symmetric cross-section under central concentrated load Q or uniformly
distributed load q has been studied [4], When the transverse load acts at the shear

centre, the elastic buckling conditions may be approximated with reasonable
racy by using

Forcentral concentrated loads Q, the approxi

naximum bending moment is

OLA (1133)

and the moment factor Cy, which varies slightly with P/P, is approximated by

c (1134)
For uniformly distributed load q,

M2 4L28 (135)
and

CE (136)

When the transverse

oad acts away from the shear centre
1 elastic buckling solutions of [4] a

distance yy below
well

the centroid, then
approximated by

Ma [Ma 08C..¥9/,_ P\ Pe a
Tutt Cor À 2 el ne 2 a m

This equation reduces to those given in section 7.6 for
(-=0)

118. 214 Beam-columns

1

5 Restraints

A beam-column may have continuous or discrete restraints which restrict its
buckled shape and increase its resistance to flexural-torsional buckling. These
restraints may be translational (z), minor axis rotational (x,), torsional (4), or
warping (2), as discussed in section 8.2)

resistance of a beam-colu

The effects of restraints on the buckling may
be determined after including additional terms 12{D}"[291{D} and
4 JE (a) [a J(d) dz in the energy equation of equation 11.18, in which [as], (25)

are the stiffness matrices for discrete and continuous restraints and (D}, (d} are

the vectors of buckling deformations given in section 8.2

à beam-column in uniform bending with uniform continuous

A simply supp
ints buckles into n half sine waves [7], so that

1/5=4/0=sinnnz/L (11.38)

The axial force P and moment M, which cause elastic buckling can be obtained
equation obtained by

by substituting these buckled shapes into an

a Ts] (a) de discussed in

section 11.5.1. This leads to

¡enting equation 11.25 by restraint terms

M+ Py +4, ao yo En

(rr, -P nx) (Gd +n* El JL Mu

ae) In] (11.39)

nd P, and can be solved directly for the elastic

which is a quadratic in M,
buckling values. In general, a number of trials must be made before the inte

value of n which leads to the lowest values can be determined.
Doubly symmetric beam-columns (yy = f, = 0) with continuous warping (2,)
0) buckle in a single half wave (n= 1)

and torsional restraints (a, ) only (x,
when

M2=(P,—P)(GJ-+ m2 Ely/L? — Pro +e a Le) (140)

It can be seen that these restraints contribute directly to the effective torsional

stifness of the beam-column, by increasing its torsional column buckling load P,

PEP, + (040, JRR au)
and its beam buckling moment M, to

My=ro (1.42)

Discreto restraints 218

Thus the solutions shown in Figure 11.1 for unrestrained beam-columns can also
be applied to restrained beam-columns by substituting P for P, and M3 for
M,

Boubly symmetric

restraints (2) only also buckle in a single half wave (n = 1) when

form as equations 8.15-8.17 for beams (P= 0), but with P,

substituted for P,,and M,,, for M. Thus the solutions shown in Figure 8.4 for

beams may also be applied to beam-columns by making these substitutions.

buckles with an enforced cen

of rotation at the restraint position y, the
moment M,, at buckling can be obtained from equation 8.18 as
Ma _1 Pe Mun)

Mrs 2 UMyap DES AT
which has a minimum value of M,/M,,, = 1 when y,P,,/M,

he restraints have no

= 1. In this case

1 axis Of cross-section rotation

ect because they act at

11531 End restrai

Restraints at the supported ends (u=@=0) of a beam-column may restrain

tions u and warping displacements proportional to 4, For a
doubly symmetric beam-column in uniform bendin

hose four lange ends hav

equal minor axis rotational end restraints of sffness 2, the buckled shape is

defined by (see also sections 6.5.1 and 8.5.1.1)

u_@ _cos(az/kL~x/2k) — cos(n/2k) ha
570 1 cos(r/24) (RES)
where k sais
mL
z (1149)

119. The axial force P and moment M, which cause elastic buckling can be obtained

Aion 1131) which ads o
M ry,
Y 10
SR ,
in mich
Pq PELE «sn
Pan (GI + PE JOLI us
a
Maar Para) (usa

which can also be obt or an unrestrained

ined from equations 11.13-11.17

beam-column by substituting th th KL for the actual length L
Solutions of equation 11.49 are shown in Figure 6.5b and Figure 896, and are

closely approximated by
sgl EI,

Dag L/EL

(ss)

The clastic buckling of beam-columns with unequal rotational end restraints ma
on 85.1.1 for b
ied for

the same method discussed in sect

be approximated usin sams,

Thus end restraint flexibility ratios G = 2E1,/2,,Lare calcul

length factor k

beam-column, and used in Figure 8.106 to determine the effect
to be used in equations 11.50-11.53

11.532 Intermediate restraints

Intermediate restraints acting on a beam-column may restrain its lateral dec
tion u and twist rotation 4. When n, equally spaced restraints are rigid so that
1 = d = Oat the restraint points, the beam-column buckles between the restraint
points at an axial load P and moment M, that satisfy equations 11.13-11.17 with

the member length L replaced by the distance between restraints L/(n, + D.

The buckling resistances of beam-col

mns with n, equally spaced elastic

restraints of equal stiffness may be taken as the lower ofthe value determined for

buckling between the restraint points, and that approximated by ‘smear

restraints into equivalent uniform continuous restraint of stifiness

anal (1155)

and using these in equation 11.39. The effects of intermediate restraints away

from the shear centre which prevent lateral deflection and restrain twist rotation

have been studied in [8]

11.6 Problems

A simply supported [section beam-column whose properties are given in
Figure 7.23 has a span of 60m, axial compressions Q KN, and equal and opposite
end moments 1.5 kN m, as shown in Figure 11.6a. Determine the value of Q at
clastic Nexural-torsional buckling,

(a) using equation 11.13;
(0) using equation 11.19;
(©) using equation 11.21

A simply supported I-section beam-tie whose properties are given in Figure 7.23
hasa span of 6 0m, axial tensions Q kN, and equal and opposite end moments 1.5
QkNm, as shown in Figure 11.6b. Determine the values of Q at elastic flexural:
torsional buckling,

(a) using equation 11.13
(b) using equation 11.19,
(6) usin

(d) using equation 11.22

A simply supported beam-column has the lipped channel section given in
Figure 58a (xo = — 78.7 mm, A = 560 mm?, 1, = 6720 x 10° mm“, 1, = 5364 x

120. 218 Beam-column
10° mm, J = 0.7467 x 10° mm“, 1, = 1.0619 x 10 mm, f, = 171.6 mm). The
beam-column has a span of 2.0m, axial compressions Q KN mm, and equal and
‘opposite end moments Qe KNm causing bending a

F the variations of Q at elastic flexural-torsional buckling
with e

¡bout the y axis, as shown in

ure 11.66. Determit

(a) when Qe causes compression in the lips
(0) when Qe causes tension in the lips

A simply supported beam-column whose properties are given in Figure 7.23 has a
span of 12.0m, axial compressions QKN, and a concentrated load 0.5 QKN at
the beam-column is braced, as shown in Figure 11.6d. Deter

mid-span, wh
mine the value of Q at elastic flexural-torsional buckling

A simply supported beam-column whose properties are given in Figure 7.2:

Span of 120m, axial compressions QKN, and a concentrated shear centre load
OSQKN at mid-span, where the beam-column is unbraced, as shown in Fig
ure 11.66. Determine the value of Q at elastic flexural-torsional buckling

References

ck, ST, and Trahai, NS. (1974) Elect of it
mation on lateral buckling Journal of Structural Mechanics, 31), 29-60.

2. Pekoz, TB. and Winter, G.(1969) Torsional-Rexurl bucking of thin-walled sections
under eccentric load. Journal of the Structural Division, ASCE, 9S(STS) 941-63.

3, Kitipornchai S. and Wang, CM. (1988) Out-of plane buckling formulae for beam
volum tc Beams. Journal of Structural Engineering, ASCE, 114(12) 2773-89

4. Kitiporncha,S. and Wang, CM. 1988) Flexural-torsional buckling of monosymmet-
rie beam-columns/tie beams. The Structural Engineer, 66(23) 393-9.

5, Wang, CM. and Kitipornchai, S. (1989) New se of buckling parameters for mon
symmetric beam-columns/te beams. Journal of Structural Engineering ASCE, 115(0)

6. Cuk, PE and Trahair, NS. (1981) Elastic buckling of beam-columas with unequal
nd moments. Cul Engineering Transactions, Institution of Engineers, Australi
CER) 166-71

7. Trahar, NS. (I

1. Vacharaitiphan, P, Woo
dt

East lateral buckling of continuous} restrained beam-columns,
in The Profession ofa Ciil Engineer (eds D. Campbel Allen and EH. Davis) Sydne
University Press, Sydney, pp. 61-73,

$. Home, MR. and Ajmani LL. (1969) Stability of columns supported laterally by

ational Journal of Mechanical Sciences, 1, 159-74

12 Plane frames

12.1 General

Plane frames are planar structures consisting ofa number ofindividual members
connected together at joints. The members are usually subjected to both axial

force and bending actions, and are often described as beam-columns, with beams

(no axial forces), and columns (no bendin
flexur

actions) being the limit

torsional buckling behaviour of individual isolated members was dis

cussed in Chapters 5 (columns), 7 and 9 (beams and cantilevers) and (beam:
columns)

The flexural-torsional buckling of a plane frame loaded in its plane will
generally involve all the members of the frame, with interactions between the
members resulting from continuity at the joints connecting them, Some members

will act as if positively restrained by the adjacent members, in which case their
behaviour may be similar to that described in Chapters 6 and $ for restrained

columns and beams. However, the restraining actions between frame members at
buckling are no less difficult to predict than are those between the adjacent

segments of braced and continuous beams discussed in Chapter 10, and it may

not be sufficient to consider only the behaviour of individual members

The prediction ofthe Nexural-torsional buckling behaviour of plane frames is
further complicated be

use the interactions between adjacent mem
much on the details ofthe joints connectin
le joint, warping deforma

em and their rest

ns in one member are compatible with

cross-section distortions in the perpendicular member, as shown in Figure 1

121. while bimoments in one member induce distortional moment pairs in the other

effects may need to be accounted for
+orsional buckling under in-plane

Thus distortion
per is concerned with the flex
ine frames whose members sie principal pla
frame. The nature of the joints between the members and their
joint compatibility and equilibrium are first considered, and then

of plane

s coincide with the

methods for the accurate and approximate analysis of the buck
frames are discussed. Finally, elected results determined for portals and other

12.2 Joints

1 IN-PLANE COMPATIBILITY

general the member in-plane axes y, ata joint will be inclined a to the fram

In
axes Y,Z, as shown in Figure 12.2. Ifthe member is rigidly connected to the joint,

then compatibility requires that its end in-plane pre-buckling deformations

Joints 221
be related to the global joint in-plane deformations
A= (VV, WyF (122)
(8) } (123)
in which
cosa 0 sin
Imj-| 0 1 0 (24)

These in-plane compatibil
analysis of the fr

y conditions are used in the pre-buckling in-plane

Compatibility of a member

rigidly connected to the joint j shown in
Figure 122 requires that its out-of-plane buckling deformations

be related to the joint out-of-plane deformations
A} =(U,U, 0)" (26
by
5)=[T1(A
in which
1 0 0
(T}=|0 cose —sina (128)

O sina cosa

These out-of-plane compatibility conditions are used in the out-of-plane buck

ling analysis ofthe frame.

12231 General

The out-of-plane buckling analysis of
warping conditions at the ends of In general, there may be diferent
conditions for each member connected to a particular joint, and it is not always

also requires consideration of the

122. 222 Plane frames

possible to associate each set of member warping displacements proportional to
the vist d with a single deformation of the joint. For this reason, the member

warping degrees of freedom ¢ should remain as independent local degrees of

freedom, instead of being associated with the global joint degrees of freedom. The
following sub-sections deal with particular examples of member warping condi

12232 Warping free

In some cases, a member is connected to a joint in such a way that there is no

restraint of warping, and the end warping stresses (= Eo

y =0 (129)

Such a condition occurs for examp
through ts web only, leaving is fa

when an I-section member is connected

S free to warp. Studies of warping free joints
between channel and zed-section m

discussed in [1,2]

33 Warping prevented

It a member is rigidly connected to a joint which itself is suficienty ri

effectively to prevent member end warping deformations w(= ax), then member

end warping may be assumed to be prevented, so that
6-0. (12.10)

Such a condition may occur for example, when the langes of an I-section membe

12234 Warping restrained

If à member is rigidly connected to a flexible joint, then end warping displace

ments w(= cp’) of the member will be elastically restrained by bimom
B(= E1,"), This elastic restraint may be characterized by the relationsh

Elgg” = au (24)

in which ay is the warping restraint stiffness and the negative sign is used to
indicate that for positive warping restraint, the bimoment B opposes the warping
deformations w. A zero value for 2 corresponds to warping free with 4" = 0,

while very large values of ay correspond to warping

Restrained warping at section joints such as those shown in Figure 12.3 has
been studied in [3,4]. For the unstifened joints considered (Type A), the dimen
sionless warping restraint stiffness ay/(Ely paratively low, varying
from 05 to 0.15

J) is cor

back to 04 as the included angle a between the members
ares from 60° to 120° to 150°, and may conservatively be taken as zero, so that
the member can be considered to be f

Two Pairs of Stiffeners e Pairs of Stiffeners

The value of ay/,(EI„GD) is

(Type B), and varies between 1 and 10 for joints with

proximately 1 for joints with one stiffen

stiffeners (Type C)
Joints with three stiffeners (Type D) effectively prevent end warping,

w
(ogetherat a joint, the warping end displacements w(= cg are the same foreach
member, and a

membe

two collinear members of the same cross-section are rigidly connected

continuous through the joint. In this case each of the two
‘warping degrees of freedom 6° ca
freedom by using

be transformed to a global d

123 Fran

nalysis

123.1 IN-PLANE PRE-BUCKLING ANALYSIS

Since an out-of-plane

clastic buckling analysis req
distributions ofthe axial forces and in-pl

a knowledge of the
moments, a pre-buckling analysis of
the frame is required. This can often be done by using a first-order linear elastic
analysis (section 45 and [S) for a set of initial loads. The out-of-plane buckling
analysis then determines the lowest factor by which these initial loads must be
multiplied to cause out-of-plane buckling

123. Sometimes the frame may have significant elastic non-linear behaviour in-
plane before out-of-plane buckling occurs In this case a second-order non-linear
elastic analysis [6] may be made, which allows for the effects of the second-order

moments caused by the products ofthe applied loads and member forces with the

structure and member displacements. Such an analysis must be carried out
iteratively for a particular set of loads. The out-of-plane buckling analysis then
determines the lowest load factor by which this load set and all axial forces and

moments must be multiplied to cause out-of-plane buckling. Because there is a
‘non-linear relationship between the loads and the axial forces and moments, the
in-plane analysis must be repeated for the new set of loads predicted by the
buckling analysis, and a new out-of-plane buckling analysis must then be made.
This process should be repeated until a satisfactory conv

12.

There are a number of methods used when carrying out elastic buckling analyses
to model flexible joints be
In these c

seen members which induce elastic warping restraints.
cs, warping of one member at he joint is elastically restrained by the
resistance to distortion of a perpendicular member atthe joint (section 122.34).

The most rigorous method of elastic buckling analysis requires the inclusion of

distortional d

of freedom corresponding to the warping degrees of freedom
in the perpendicular members, and includes the effects of distortional moments
(Ma in Figure 12.1) which are in equilibrium with the lange moments (My)
equivalent to the bimoments in the perpendicular members. While the effects of
web distortion on the flexural-torsional buckling of beams and beam-columns

have been studied (sce section 16.5), they do not appear to have been considered
in frame buckling,

An alternative approximate

nethod is toincorporate clastic warping restraints

‘atthe member ends whose stifnesses y are based on studies such as those of
whose results are approximated in section 12.2.34. Computer methods have been
developed for this [7-9] which are based on the finite integral method [10]. A
finite element method may be developed by extending the method of [11]
discussed in Chapter 4, to include the effects of the concentrated elastic end
restraints iy, by using section 8.22.

d less precise method is to idealize the warping conditions at member
ends as being either free to warp (6 = 0) or continuous ($ =), or prevented

12.4 Portal frames

There have been a number of studies ofthe flexural-torsional buckling of portal
frames (7,8, 12-16

1 the scopes of these are summarized in Figure 124, They
include studies of the combinations of loads which interact to cause frame

er Y ||

Figure 124 St

Portal frames

ling of portal frame

124. tions, Parts and
6. Harrison, H.B (1973) Compu

2nd edn, Pergamon Press, Oxford
Methods in Structural Analysis, Prenice-Hal E

oo wood Chi NA
“AE 7. Vaharajtiphan, Pad Tah, NS. (1973) Estate buckling portal frames
à. Vacarsphan, and Take NS (973) Mal ss of lateral buklng in plane
= {ames Jounal ofthe Sacra icon, ASCH 01 STP, 197-516
mw | 9. Vacharaftiphan, P and Tra, NS 197) Die Snes Analy of Later

t Buckling, Journal Structural Mechanics, 32), 107-3
Brown, PT. and Trahait, NS. (1968) Finite integral solution of diferetil equations

Civil Engineering Transactions, Institution of Engineers, Australia, CE10(2), 193-6
Figure 126 Studies of other frame 11. Hancock, G.and Trahar, NS. (1978) Finite element analysis of the lateral buckling

of continuously restrained beam-columns. Ci Enginering Transaction, Institution
of Engineers, Australia, CE20(2), 120-7

buckling. The load combinations shown in Figure 12.5 for the buckling of a 12. Hartmann, AJ.and Munse, W.H (1966) Flexurl-torsional buckling of planar frames.

pitched roof portal frame and of a two-bay portal frame under horizontal and Journal of the Engineering Mechanics Division, ASCE, Vol. 92(EM2), 37-59.

vertical loads are typical, and indicate that as one load increases, the other load at 13. Cuk, PEE Rogers, D.F. and Trabair, NS. (1986 Inelastic buckling of continuous ste!

buckling decreases. beamn-columns, Journal of Constrctional Stee! Research, 6, 21-52.

14. Bradford, MA, and Trahair, NS. (1986) Inelastic buckling tests on beam-columns.
Journal o Structural Engineering, ASCE, NA), 538-49.
mes 15. Hancock, Gi. (1976) Tests of lightly restrained portal ames. Steel Construction,
Australian Institue of Steel Construction, 10(2 10-6
The scopes of some studios of other plane frames [8,9, 17,18), including mul 16. Kren, S. (1990) Constrained lateral bucking of -beam gabl frames Jour
nd trlangudsted fan Structural Division, ASCE, 116(12), 3268-84
17. Argytis, LH, Balmer, H., Doltsinis, 1St, Dunne, PC, Haase, M, Kleber, M
Malgjannakis, GA, Miejnck, H.-P, Muller, M. and Scharpf, D:W. (1979) Finite
Element method - The natural approach. Computer Methods in Applied Mechanics
and Engineering, 17/18, 1-106
8. Kouhia, R. (1990) Nonlinear finit element analysis of space frames. Report 109,
PROBLEM 12 Department of Structural Engineering, Helsinki University of Technolog

12.5 Other fr

12.6 Problem

Adapt the solution of Problem 4.12 so as to produce a computer program for
analysing the elastic Me

al-torsional buckling of plane frames whose member
sections are symmetric about the plane ofthe frame.

12.7 References

Baigent, AH, and Hancock, GJ. (1982) Structural analysis of assemblage
walled members. Engineering Structures, 4, 207-16
2. Hancock, 6). (1985) Portal frames composed of cold-formed channel and zed

125. 13 Arches and rings

13.1 General

This chapter is concerned with the out-of-plane flexural-torsional buckling of
hare loaded in-plane. Each of theses,

planar arches, curved beams, and ri

re 13.1, Constant curvature is common, but
ported

‘curved in its plane, as shown in
other profiles such as parabolic may be used. Arches are st both ends,
ly in such a way that deflections of the ends away from each other are

nted, either by the nature of the supports, or by tie members.

Arches whose ends are free to move apart are better described as curved beams.
Rings are closed arches with 360° included angles, so that there are no ends.

Arches whose ends are prevented from moving apart are often used st

turally because of their high in-plane stiffness and strength, which result from
their ability to transmit most ofthe applied loading by axial force actions, so that

the bending actions are reduced, Curved beams whose

nds are free to move apart
have much more significant bending actions, and are es tiffand strong in-plane
Rings are sel-reacting, and have high in-plane stiflnes and strength with respect
to uniform radial loading.

The resistances of arches and rin onal buckling
depend on the rigidities El, for lateral bending, GJ for uniform torsion, and EI,

to out-of-plane flexura

arch may be reduced by is

for warping torsion, The buckling resistance of an
in-plane curvature, and so it may require

ay method to the Mexural-torsional buckling of

ificant lateral brac

This chapter extends the en

arches and rings. Clos ns are developed for circular arches and

rings in uniform compression or bending. The finite element method of buckling

)

+

Figure 184 Arch, curved beam, and rin

126. 230° Arches and rings

analysis developed in Chapter 4 for straight members is extended to arches, and

used to obtain solutions for arches with concentrated loads.

13.2 In-plane behaviour

Under in-plane loading, the shear centre ofa cross-section of a
R may deflect v radially and
case the centroidal longitudina

o

3] by

in arch of radius
gentially as shown in Figure 13.4, in which
i strain e and the curvature x are approximated

WEHR + pale +W/R) aan

KUH WR (132)
in which ' indicates differentiation with respect to the shear centre distance s
around the arch. The longitudinal strain e at a point P ata distance y from the

centroid is approximated by

The corresponding longitudinal stresses
09= Ely

N= odd
and E

My= | opyda,
whence y

N= EAU, —0/R) + pale" + W/R)]
and

M,=— Ello" + WR)

(133)

(54)

(135)

(136)

(13)

(138)

Figure 132 Displacements and uniform compression and bending.

Uniform compression P= — N occurs in an arch when uniformly distrib
radial loads q= P/R cause cor

tant radial displacements » only (w,=0), as

shown in Figure 13.2b, so that M, =0. It can be seen that the ends of the arch

tly towards cach other. Arches whose ends are prevented from moving

er have moments induced by horizontal end reactions,
Uniform bending occurs in a curved bea

n when only equal and opposite end
atthe ends of the arch

re 1326. Itcan be seen 1
x. Arches whose ends are prevented from deflecting

moments M act as shown in Fi

The in-plane behaviour of arches under other loadings than those of u

compression and uniform bending is best analysed by the finite element method
discussed in section 13.8.1

133

ergy equation for flexural-torsional buckling

The energy equation for lexural-torsional buckling u, ¢ for the monosymmetrie

section arch of developed length L and constant radius of curvature R shown in

A| El, (qu + 6/R)? + Et (6 WIR” + CHE —w/R)?} ds

$ OUR) +936")

À kann + 678+ 8°) + a10,—r0e

in which EI, is the section minor axis flexural rigidity, EI, is
rigidity, Gs the section torsional rigidity, and r, are the mi
radi of gyration, y, is he shear centre coordinate and , is the monosymmetry
section constant. The arch has radial concentrated loads Q, = and distributed

loads q, = q acting yy and y, from the centroid and axial loads Q, and in-plane

moments M. These loads and moments have in-plane axial compression force

resultants P and bending moments M. For zero values of 4,0, and 1/R,

2.107 for the buckling of straight monosym

13.4 Differential equilibrium equations

nes

equation by using the calculus of variations, accordin

The differential equilibrium equations for flexural-torsional buckling can b

obtained from the en

127. 232 Arches and rings

to which the functions u, which make

stationary satisfy the condit

Substi

equation 13.11 leads 10.

(EL,

+9/R)

Pu +

a

aie)

ELA

w'/RYRY

WIRYR)

OT
IR)

o,

w/RYRY

g the energy equation (equation 139) into equation 13.10 and using

aan)

Arch buckling under un

WIR — {GI WIR + El, fu + d/R)/R

Pyou + {Pr +r? +788) — {Pr2w/R}

Pyod/R

Mu" — {M Bd) + MGR +40, 06 =0, (13:13)

which are the differential eq

librium equations for minor axis bending and
torsion for the buckled position u, 6

ly diffe
ained in [4-7],

differential equilibrium equations for arches have been

Arch buckling under uniform compression and bending

Arches which are simply supported laterally so that up = u, = 0,1 = =0,
do = $.= 0,05 = 6 = 0 under constant P,M,=M,g have possible buckled
shapes defined by

4/5 = 9/0 = sin nasjL (13.14)
which correspond to n buckled half waves around the arch length 4. Substituting
these into equation 139 leads to

TE 30S Les kas JU

and
ka = Pye PPP JR,

yz = yy = — (Pro + M) (El, +2 PAR,

az =P. —P)+MB,+ 40, ll

(1319

(Py +M) Ln

El,L

after making the appro

i (13:17)

Pp
P, (13.18)

Uniform compression P is produced in an arch by uniform radial loading
a =P/R,asshown in Figure 132b. In this case, equation 13.16 can be written for

128. 224 Arches and rin
doubly symmetre sections (y= 0) as
kPa = 1 — PIP )
KualMyn = (lb. rad | (3.19)
alt 3P..)=1—P/P y+ 03/63, |

in which

a= Lina,
b= nr al Pal |
MP
Equation 13.15 is satisfied when
which leads to
GP + ad -PIP + 03/63) =, 0b) (18

‚don the number n of half waves, and so the

The solutions P ofthis equation depe

lowest solution must be determined by a trial and error process, by increa
until the solution increases. The lowest solution usually occurs for n= 1

Solutions of equation 13.22 for the particular case for which

PP, a

are shown in Figure 134 by the variations of P/P,, with b, and a, For a, =0, the
ht line, and buckles at the lowest compression

arch degenerates into a sta
member flexural buckling load P,,. For a,=1.0 and n= 1, the arch is semi
circular, and buckles under ze:

ro load in a rigid body mode, because the torsional

end restrain which ensure $, = 0 do not prevent rotation about the diameter

joining the arch ends.

1353 DOUBLY SYMMETRIC CURVED BEAMS IN UNIFORM BENDING

Uniform bending is produced in a curved beam by equal and opposite end
moments M when the ends of the arch are free to move together or apart, as

shown in Figure 1326, In this case, equation 13.16 can be written for doubly

symmetric sections (8, = 0) as

ku/Pa=1+ 0303
RM qu = — MM un Quad | (1324

Mr Pa) = 1 + (MM, deb, + a

Once again, equation 13.15 issatisied by equation 13.21, which in this case leads

142/02 +(0,/0,) M/M oa) (1 + 0262) =(0,/0, + yy + MM) (1325)

Arch buckling under uniform compression and bending 235

0 920% 0508

Value of à:

Lan

Figure 134 Doubly symmetric arches in uniform compression

The solutions M of this equation depend on the number n of half waves, and so
the lowest solution must be determined by

‘and error. The lowest solution
usually occurs for n= 1

Solutions of equation 1325 are shown in Figure 13.5 by the variations of

M/M; with band a, For a, = land = 1,thecurved beam is semi-circular, and
body mode, because the torsional end

restraints do not prevent rotation about the diameter joining the ends. For a, = 0,

buckles under zero moment in a ri

the curved beam degenerates into a straight beam, and buckles at the lowest beam
buckling moment M. For negative values of a, the arch is inverted. The
resistance to buckling generally increases as a, decreases, and is higher for

‘negative’ curved beams whose outer (convex) surface is in tension than its for
positive’ curved beams whose outer surface is in compression.

54 MONOSYMMETRIC ARCHES

The flexural-torsional buckling of monosymmetri arches in uniform compres
sion or uniform bending has been studied in [3] and results for monosymmetric
sections formed by reducing one flange of an I-section member are shown in
Figures 136 and 137.

129. 236 Arches and rif

Figure 13.6 Monosymmetrc arches in uniform co

For the arches in uniform compression, the variation of the dimensionless
buckling force PP, 9 with the arch parameter a, = L/nzR and the monosym-
metry ratio 2yo/h is shown in Figure 136. In
nr EL/L? calculated for the doubly symmet
distance between the flange centroids. Positive values of

the smaller. It can be seen in Fig

ls figure, Po is the value of

ie cross-section and h is the
ya/h correspond to the
136 that while

outer (convex) flange bei

Ring buckling under uniform compression 297

there are significant reductions in the buckling resistance as the section is made

more monosymmetric by reducing one flange, the resistance when the outer
(convex) flange isthe larger (negative 2yq/4) is only slightly greater than when the
inner flange is the larger (positive 2yu/).

For curved beams in uniform bending, the variation of the dime
buckling moment ML/J/(EI,GJ}, with the parameter a,=L/nxR and the
monosymmetry parameter his shown in Figure 137. In this figure, (El, Gl) is

«tion. It can be seen that

the value calculated for the doubly symmetric cross
en the compression flange isthe larger (positive B,/h) is greater

the resistance w
than when the tension

ange is the larger (negative f,/).

13.6 Ring buckling under uniform compression

136.1 UNRESTRAINED RINGS

R), solutions for doubly symmetric sections
ditions require

In the case of complete rings (L

can be obtained from equation 13.22 by noting that continuity

130. 298 Arches and rings

there to be an even number » of half waves. The value of n = 2 corresponds to
a, = L/nr R = 1.0 and the rigid body mode already noted in section 13.52. I this

Figid body mode is prevented, then the lowest solution corresponds to n= 4, so

that a, =0.5, Solutions for a, = 0.5 are shown in Figure 13.4

1362 RESTRAINED RINGS

Rings are often connected continuously to cylindrical, conical, or spherical shells,
which may effectively prevent transverse displacement of the rings at the point of
attachment, as shown in Figure 138 [8,9]. The ring then buckles with an
enforced centre of rotation atthe point of attachment y,, so thatthe shear centre

5=(,—y0)0 (320)

in which case equation 13.15 becomes

2y,— vodka + ka} 0=0.

€ x

Centroid ——= |} ———
5 Yo
Shear centre —— Ye

Centre of rotation ——

y

ling with an enforced centre of rotation

Effect

on arch buckli

In the special case of doubly symmetric sections (o = 0) with centroidal loading

Pan PH HER, (329
in which
EEE ERA)
CRETE (1329
and
Part PtP, (1330

When P, > P,,the minimum sol
m, and can be approximated by

on depends on the numberof even) half waves

P=2(P,P)+, (1332)

If the connected shell which enforces a centre of rotation during buckling also

stilfess a, then r3P, increases to

PRP = EI, +2, (1333)

13.7 Effect of load height on arch buckling

The effects of load height
supported curved beams with central concentrated load Q
139 [10]

the load bel

9 on the flexural-torsional buckling of simply

e shown in F

can be seen that the buckling resistance increases as the distance of

ow the curved beam increases, until limiting value is approached at
curved beam buckling mode changes from a single half wave to two half
waves. Semi-circular curved beams (L/xR = 1.0) buckle in a rigid body mode

j/k = 1/0)

The effects of load height y, on the buckling of arches in uniform compression

which th

under zero load when the load acts on the ine joining the supports (

re 13.10 [10]. Again, the buckling resistance increases as the

The results of experiments [11,12] on the buckling of doubly symmetric

curved beams and monosymmetrie arches under central concentrated loads are

show very good

theoretical predictions obtained by finite element analysis, and provide general
support for the theoretical findings summarized in this chapter

132. 242 Arches and ring
13.8 Finite element analysis

138.1 IN-PLANE PRE-BUCKLING ANALYSIS

mate, and the distributions of the in-plane
best
€ element

Many arches are statically ide
stress resultants P, M, depend on the deflections v,w, In this case, the
determined by using à computer method of analysis such as
method [1, 10, 12, 13]

The finite element method for the in-plan
deseribed in section 4.5 uses linear displacement fields forthe in-plane longitudi
I has been found [14] that linea fields may cause significant
errors in the analysis of arches, but that improved accuracy can be achieved by

ments. One study [1] has suggested that where 10 quintie

analysis of straight elements

elements may give an accuracy of within 1%, 60 cubic elements may be required
sts is simplified by the use of

curacy. The use of higher order elemé
ation [13] instead of formal integration,
order elements leads to higher order nodal deformations

for the same:

Gaussian numerical inte

generally associated with stress resultants, These are additional to
those deformations used to ensure geometric continuity between adjacent el.

cements, The enforcement during assembly of the elements of fictitious ‘compati

bility” conditions in which the higher order nodal stress resultant deformations
forone element areequated to the corresponding ones fr the other element at the

ay introduce errors, especially when there are slope or area changes at a

node between elements.
One method of

deform

elements at the node, so that the erro

woiding these errors is to condense out the higher order

jons before the assembly. Another meth

[1] is to use short length
are reduced,

138.2 OUT-OF-PLANE BUCKLING ANALYSIS
The finite element method described in sections 42 and 44 for the clastic
flexural-torsional buckling analysis of structures composed of straight elements

has been extended tot the energy
uation 13,9 was substituted forthe straight element equations 4.58

e curved elements of arches [10,12]. For

equation of
and 46

deformations
sed out, but
procedure is similar to that recommended in section 4.3.6 for internal hing

{uintic deformation fields were used for the out-of-plane buckling
3. The higher order nodal deformations u”, 4” were not conden:
so that the eigenvalue problem remained linear. This

Studies [10,12] have suggested that convergence is much slower for arches
with in-plane bending than for uniform compression, and that up to 16 elements

may be required to obtain an accuracy within

References 248
13.9 Problem

Adapt the solution of Problem 4.12 so as to produce a computer program for
analysing the elastic flexural-torsional b

ickling of an arch whose cross-sections
are symmetric about the plane of the arch

13.10 References

1. Papangelis, .P. and Trahair, NS, (1986) In-plane finite element analysis of arches.
Proceedings, Pacific Structural Steel Conference, Auckland, August, Volume 4
pp. 333-80

Papangelis, LP. and Trahair,
Journal of Structural Eng

NS. (1987) Flexural-torsional buckling of arches.
ering, ASCE, 113(8), 889-906

Trahair, NS. and Papangeli, JP. (1987) Flexural-torsional buckling of monosym

metric arches. Journal of Suctural Engineering, ASCE, 13(10), 2271-88

4, Yang, Y-B.and Kuo,S-R. (1986 Static stability c
of Engineering Mechanics, ASCE, 12 (8), 821-41

5, Rajasekaran, S.and Padmanabhan, S. (1989) Equations of curved beams. Journal of
Engineering Mechanics, ASCE, MS (5) 1094-111

Yang, Y-B, Kuo,S-R.and Yan, J-D. (1991) Use of steaight-beam approach to study

bucking of curved beams, Journal of Structural Engineering, ASCE, 117(7) 1963-78.

Kuo,S-R.and Yang, Y-B. (1991) New theory on buckling ofcurved beams Journal of

Engineering Mechanics, ASCE, 117 (8, 1698-717

3, Teng, JG. and Rotter, JM. (1988) Bucking of restrained monosymmetrie rings
Journal of Engineering Mechanics, ASCE, 114 (EM10) 1651-71

9. Teng, LG. and Rotter, LM. (1989) Buckling of rings in column-supported bins and
tanks. Thin alld Structures, 7 3-4) 257-80,

10. Papangeli, LP. and Trahair, NS. (1987) Finite element analysis of arch la
bucking. Civil Engineering Transactions Institution o Engineer, Australia, CE29 1)

11. Papangelis, 1. and Trahai, NS, (1987) Flexurl-orsional buckling tests on arches,

ing, ASCE, 113 (7) 1433-43,

pangelis, JP. and Trahar, NS. (1988) Buckling of monosymmet

point loads. Engineering Structure, 10 (8), 257-64

13. Zienkiewiez, O.C. and Taylor, RL. (1989) The Finite Element Method, Volum
Basic Formulation and Linear Problems; (1991) Volume 2 Solid and Fluid Me
cs, Dynamics, and Non-Linear, th edn, McGraw-Hill, London

14, Dawe, D4.(1974) Numerical studies using circular arch finite elements. Computers and
Structures, 4 (4), 729-40,

urved thin-walled beams Journal

Journal of Structural Engin

ches under

133. 14 Inelastic buckling

14.1 General
A slender beam or beam-column which has low resistances to lateral bending and
torsion may buckle while still elastic by deflecting and twisting out ofthe plane of
loading. This elastic flexural-torsional buckling was discussed in Chapters 7-11

The resistance of a member to elastic buckling increases as its slenderness

decreases, and a steel member of moderate stiffness may yield before its elastic
buckling
induced by the applied loads with any residual st
manufacturing process is completed. Yielding reduces the effective out-of-plane

ofthe stresses

load is reached. Yielding is caused by a combinati

ses which remain after the

rigidities, and decreases the buckling resistance below the elastic value, as shown,
in Figure 74,

A theoretical model of inelastic lexural-torsional buckling is developed in his
chapter, and a computer method of analysis is presented. Comparisons are made

lastic buckling experiments

of the theoretical predictions with the results of ine
Following this, the effects of the support conditions and the loading

ments on the inelastic buckling of beams and beam-columns are discussed.
The members considered in this chapter are steel I-seetion members which are

perfectly straight and untwisted before loading. Real members

ivatures and wists, which further reduce their strengths. Design

assumed to
have initial
methods which account for the

-ductions caused by these geometrical imper

tions are discussed in Chapter 15,

st

14.2 Tangent modulus theory of à c buckling

142.1 FLEXURAL BUCKLING OF COLUMNS

Most inelastic buckling predictions are based on the tangent modulus theory,
to predict the flexural buckling of inelastic

which was originally develope
columns [1,2]. Strictly speaking, u
non-linear stress-strain curves are elastic so that the loading and unloading
paths are identical, as shown in Figure 14.1a. In this case, the tangent

stress level in the column may be

only to members whose

is theory appli

modulus E, appropriate to the avera
stituted for the initial modulus £ in the expression x*ET,/L? for the elastic
load P, of a simply supported column to obtain the tangent

flexural buckling
modulus bucklin

I, as

134. 246 Inclastic buckling

als is appai

The application ofthis theory to inelastic mate ntly invalid, since
these have unloading paths which are parallel to the initial slopes E of the
Siress-strain curves, as shown in Figure 14.1b. This observation led to the
development ofthe reduced modulus theory of inelastic buckling, for which E, is
used only in the loading region of the column (where compressive bending
tresses increase the total strain during buckling), and the initial modulus E is

{ised for the unloading region (where tensile bending stresses decrease the total
Strain) Under these circumstances, a reduced modulus E, can be determined from
E, E, and the geometry of the cross-section, and used to calculate a reduced

modulus inelastic buckling load
P= ELL? (142)

The reduced modulus E, always lies between E and E, so that

> PP, (143)

as shown in Figure 14.16
Although the tangent modulus theory appears to be invalid for inelastic
materials, careful experiments have shown that it leads to more accurate predic

tions than the apparently rigorous reduced modulus theory. This paradox was
Fesolved by Shanley [3], who reasoned that the tangent modulus theory is valid
When buckling deflections are accompanied by simultaneous increases in the
applied load of sufficient magnitude to prevent strain reversal, as shown in
Figure 14e, When this happens, all the stress and strain increments are related

by the tangent modulus E, and so the buckling load is equal to the tangent
modulus load P,. Thus lateral deflection initiates at P, and increases with
increasing load (and decreasing E,) until a maximum load is reached, as shown in

nt modulus theory of inelastic buckling 247

The tangent modulus load P, is the lowest load at which lateral deflection
straight column can commence. Itis theoretically possible for buck

at higher ods he ting ae bea lr modal oad
buckling initiates without any change in the appli ‘cin Fle

od load, as shown in Fig

1422 APPLICATION TO STEEL COLUMNS

The application of the tangent modulus theory to th
columns with stress-strain curves

inclastic buckling of steel
the type shown in Figure 14.2 led at first to
nonstrated in F

predicted buckling

jure 14.30. This figure shows that the
ads for slender columns increase as the slenderness de:

eases until the squash load Py = AF, is reached, when the buckling load drops
Suddenly to ero, corresponding tothe sudden change in E from E to zero when
general yielding occurs. If, however, the general stress level reaches the train
hardening range, there is a sudden increase from zero correspondi
sudden change in E from zero 10 £,

That these a
hat the and general yielding which led to P, = Odoes not occur. There are
wo reasons for this, the first being associated with the residua
those shown in Figure 144 which

jomalous predictions are not realized in practice is due to the fact

tresses o, such as
ire induced in most structural steel members

ring their manufacture, The va
section cause yielding to ini

ions of these residual stresses across the

ate locally, at points where the resultant compressive

135. 240
\
\

3 8. |

3 si
=
a
Jure gen and
A]

and EE,

DI Yield Et,
(a) Yield G0 LL Yield Gf,

Figure 142 Tangent modulus predictions for ste! columns

Range shown is mean
+ standard deviation
4 for 34 beams

Range shown is mean
21 standard devia

for 23
vane À)
Î ¿o
I art
MA
7
Walz ‘Measured 12) eine Measured SI
1a t-te 1) Wet Beans

stresses are greatest. As the load increases further, there are corresponding
th ases in the remaining elastic core where

increases in the yielded areas, and decr
E, =E. The decreasing elastic core leads to decreases in the tangent modulus

rigidity

(En, asa)

Tangent modulus theory of inelastic buckling 249

where, is computed for the elastic core, and consequent decreases inthe tangent
‘modulus buckling load. Thus the residual stresses cause early yielding as shown
in Figure 14.3b, and the tangent modulus load diverges inereasingly from the
elastic buckling load as yielding progresses through the column.

The second reason why yielding does not occur suddenly and gene

lly is
associated with the actual yielding process of steel. First it should be noted that
the curve shown in Figure 14.2 does not represent the local variations between
stress and strain, but rather the mean variations averaged over the instrumented

volume of the test piece for which it was determined. Locally, yielding progresses
asa series of sudden discontinuous lips in the matrix between the grains [4] from
the elastic condition to the strain-hardened condition. Thus when the average
stress-strain curve indicates that yielding is taking place, some of the material is
clastic (E, =), and the rest is strain-hardened (E, = E,). Thus the average E,
representative of the whole yielded region decreases steadily from E to E, as the
strain increases from yielding at ey to strain-hardening at ¢,. This has led to the

‘conservative proposal that E, should be used for E, inthe yielded regions.
The approximation of E, for E, in the yielded regions explains why short length
moved by annealing do not

columns which have had their residual stresse
‘buckle suddenly when they first become fully yielded. The approximation is
also in agreement with observations of other buckling phenomena. For example

‘many beams are capable of reaching the fll plastic moment Mp and maintaining
it over a considerable range of deformations before local buckling occurs. Since
the flanges are fully yielded before Mp is reached, it can be concluded that the
local buckling strength is not reduced to zero by yielding

It can be seen that the tangent modulus buckling theory for steel columns
should allow for the presence of residual stresses when determining the elastic,
yielded, and strain-hardened regions of the column, and should use E for the
clastic regions and E, for the yielded and strain-hardened regions. Thus the

tangent modulus flexural rigidity (EL), will be equal to the elastic rigidity of an
ion obtained by transforming the yielded and strain-hardened
cording to the modular ratio E/E

equivalent sect
regions

1423 FLEXURAL-TORSIONAL BUCKLING

The application of the tangent modulus theory to the inelastic lexural-torsi
buckling of a steel member requires only one further extension of the theory for
the flexural buckling of columns. This extension is needed to determine appropri
ingent shear modulus G, to be used when evaluating the

d strain-hardened regions to the effective torsional

ate values of the
‘contributions of th
rigidity (GI)
Although the incremental theory of perfectly plastic solids suggests that the
shear modulus of any yielded material is initially equal to the elastic modulus 6,

ielded a

experimentson yielded steel members [5,6] indicate that it decreases rapidly with
even quite small strains. An examination [7] ofthe theoretical and experimental

136. 250 Inclastie buckling

evidence led to the conclusion that the shear modulus of yielded struct
could be

be approximated by

6 2

G THEMEN (45)

where vis the elastic Poissons ratio.

14.3 Pre-buckling analysis of in-plane bending

Before an inelastic out-of-plane buckling prediction can be made, the in-plane
bending must be analysed so that the distributions of the elastic, yielded, and
strain-hardened regions throughout the member can be determined. The effective

the inelastic buckling resistance can

out-of-plane rigidities which contribute t
then be evaluated using these distributions.

When the member is statically determinate
in two separate stages. First, the variation of the axial force and bending mom
along the member can be determined from states. Following this, the locations of
ided, and strain-hardened regions within selected

the in-plane analysis can be made

the boundaries of the elastic,
cross-sections can be determin:
properties, residual stresses, and the axial force and bending moment

‘When an inelastic member is statically indeterminate, the two stages cannot be
separated, because the material non-linearity eloses the chain of dependence of

sing the cross-section geometry, material

Yielding on stress resultants, on redundant actions, on deflections, on stiffness,
and on yielding, In addition, it may be necessary to consider the effects of
arity, as for example when there are significant in-plane

instability effects of some sway frames.

the bendi
\. computer method of analysing the in-plane behaviour of steel

A finite elem
cribed in [8]. This method allows forthe effects of residual su
Actions. The following sub-sections

frames is de
yielding, strain-hardening, and finite d
for this method, and show how the cross-section and

discuss the data requir
member analyses are performed, and how the yielded and strain-hardened

1432 SECTION GEOMETRY AND MATERIAL PROPERTIES

The
Figure 14.54. For simplicity, all radii and fillets are ignore

taperin
An idealized tensile stress-strain relationship for structural steel is shown in
‘and é,- The existence of an upper

sometry of an idealized doubly symmetric section member is defined in

Figure 142, with nominal values for E, E,

Pre-buckling analysis of in-plane bending 251

yield stress is ignored, This stress.

in curves usually assumed to apply to all of
the material in the cross-section, even though the web yield stress is usually
significantly greater than the lange yield stress.

1433 RESIDUAL STRESSES

1433. Hotrolled beams

Longitudinal residual stresses are induced in a hot-rolled steel member during
1g after rolling, and as a result of any mechanical processes used to
straighten it [9]. During coolin

ly exposed regions of the
ction at the flange tips and web centre cool more rapidly. These early
cooling regions shrink, inducing matchin

1 plastic lows in the high temperature
web junctions, which have correspondingly low
during final cooling ofthe high temperature

‚web junctions are resisted by the regions already cooled

late-cooling regions atthe fa

yield stresses. Subsequeı
regions at the flang
which have developed high yield stresses, and induce residual compressions in
them, as shown in F 1

e 1442. Equilbrating residual tensions are induced in tl
late-cooling flange-web junctions. Cold-workin
causes local yielding, and fu

by mechanical strai
x modifies the residual stress pattern.
distributions of resid

I stress vary considerably with the

cross-section geometry and with the cool
Idealized and measured distributions of th

and straightening processes

sidual stresses in hot-rolled beams

are shown in Figure 1444. The distributions across the flanges and web are often

approximately parabolic, being compressive att

arly-cooling flange tips and
web centre, and tensile at the late-cooling flange-web junctions. It has been
suggested [10] that the maximum residual stresses in hot-olled beams may be

137. 252 Inelastie buckling
approximated by

04 137502 ARBT)Nmm

100(—03-+ 4/28T)N mm

833 (08 + 4/287)Nm

(148)

While the flange and web distributions are approximately parabole, it has been

ee [UE dt quer dits fhe orm
oul Fy = as + 03/8) + 0,2%/8)
e 4 (147)
aaa? oa
shoul be weds tha the conditions
feaa=o dus
PTE tu»

ses havea

are satisfied. The first ofthese conditions ensures that the residual
Is about the x and y axes are

zero axial force resultant (zero moment result
automatically ensured by symmetry). The second condition was su

gested to

ensure thatthe residual stresses have a zero axial torque effect when the mem
is twisted. If this condition is not satisfied, then the effective elastic torsional
dit of the cross-section changes to (GJ — J o,(x? + yA).
In practice, some residual stress distributions may differ sig
idealized pattern discussed above, as can be seen from the measured distributions
shown in Figure 144a [12]
It has been suggested [13] that comparative numerical analyses should be

carried out using ay = 0 and
oy =—035F) E
ann =050Fy, | (14.10)
so that
s 0 ov 1
ne 0 0 | ma
10 000 (om ii
| 3 45 105 ji en
| EA |
35105 525 [Al 300 /]
where
Ay/Ay= WARBT= 1) can

Pre-buckling analysis of in-plane bending 258

143,32 Welded beams

Residual stresses are induced in welded beams by uneven heating and cooling asa
result offlame-cutting and welding the flanges and webs. The process is similar to
that described above for hot-rolled beams, with large residual tensions b

induced inthe late-cooling lange-web junctions, and equilibrating compressions
in the flanges and web, as shown in Figure 14.4,
The magnitudes and distributions of residual stress again vary with the

cometry and with the cutting, weldi

cooling and straightening
esidual stress distribution shown in Figure 144, the
compressive stresses are uniform across the flanges and web, except near the
flange-web junctions. It has been suggested [14] that the maximum tensile stress

ssumed to be equal to the parent metal yield stress Fy, and that the

processes. Inthe idealized

compressive stresses at the flange tip can be determined from

aye = 0.1B,S(4y/A) (14.13)

in which A, isthe area of weld metal added in the biggest single pass at the weld

site, by using their sug y supplied per unit volume of
electrode wire

Again, practical residual stress distributions may differ significantly from this
idealized pattern, as can be seen from the measured distributions shown in

Figure 144b [15],

1434 CROSS-SECTION ANALYSIS

The in-plane displacements, wy of any point P(x,y)in the doubly symmetric

cros-setion shown ln Figur can be pres interns the displacement
po —Á
is | 2

138. where

(14.15)

(14.19

The effects of finite rotations are included in these equations by the use of sind

and cos 0 inst

second-order stra
Figure 147, and can be ob

The inclusion ofthe second-order terms (e? + w/2)/2allows the effets ofin-pl

1d of the usual small rotation approxi

instability to be incorporated into the analysis.

The correspondin

a= | "(Ga

clement of length L can be approximated by

jons of @ and 1
aE a

nd the

lp My a shown in

oe (a ae
"TS cm ER CAL
ee La f 4

(14.17)

(14.18)

(14.19)

at the displacements v, w of the elastic centroidal axis of an

(1420)

Pre-buckling analysis of in-plane bending 255

zy (0
» 421)
A [ip 2
Z) = (1,2/L,22/L2, 24/19), and

contains eight arbitrary coefficients. The

li is os Mp Wo YT (1422)

can be obtained from equation 14.20 in the form of

[ea (a) (14.23)

which enables the coefficients {a} to be eliminated from equation 1420, whence

(0) = MILCH" (6, (14.24
The second-order strains gy given by equation 14.18 are non-linear functions of
the derivatives vv, w, which can be obtained from equation 14.24 in the form of

de) = LEB IEC (, (1425)

which enables the stresses a,
the nodal displacements {à}

ven by equation 14.19 to be expressed in terms of

ment, the

For equilibrium of the fetal wich priori
[ETAIENT (1426

where the symbol indicates a virtual displacement, and the generalized in-plane

nodal frees {0,,} corresponding to {6,} are the static equilibrants ofthe stresse
a This can ber

55,3" CCI -" LBA {b} Ldz — {66

o am

in which {b} are non-linear functions of (3,) and of the generalized stress

rare

(1428)

[ opxtaa

Performing the variation in equation 1427 and n +) is arbitrary

139. 256 Inelastic buckling

leads to set of eight simultaneous non-linear equations of the form of
Qu = Pir Duar,

(1429)

The terms in these equations are formed by integrations over the volume of the
nt, as indicated by equations 14.27 and 1428. If Gaussian int
[16,17] is used with respect to z then the integrations over the:
be carried out at the Gauss points.
The tangent stifness equations for the element can be expressed as

Ua] {A5} = (AQU (1430)

area A need only

in which the symbol A indicates an increment, and {Aÿ,), {AO} are the
increments from the equilibrium values (54). {Qe}, and [Kj] is the elemer

tangent stiffness matrix, An approximation for [k,,] may be obtained from
‘equation 14.29 by assuming that there are no unloading and no changes in the

Yransformed clement to be established for the stress levels corresponding 0 (,}
Aou tangent stifles can be obtained from equation 1429 as
fe Ao. 0M as, (uan

A0.=

Further details of his process are given in [8]
The tangent stiffness equations for each clement can be transformed and

combined to form the global tangent tines equations
[K,3184) = (80) (1432)

(40) are the increments in the structure’ nodal displacements
and forces and [K;,] isthe global stiffness mat

in which (AA,

The load-deflecion behaviour ofthe structure is non-linear in
solution for a particular set of applied nodal forces {Q,} can be obtained
iteratively by using the Newton-Raphson method [16,17], starting from an
initial set of global displacements (Ay). This initial set may be arbitrary, but
faster convergence can be obtained iit is calculated from a conventional linear
clastic analysis, or extrapolated from a converged analysis corresponding to a

lower set of nodal forces, as indicated diagrammatically in Figure 14 8.
A typical cycle of the solution process is illustrated diagrammatically in
Figure 148b. The cycle starts with the global displacements {A}, which are
transformed to the ‘displacements 5). These are used in equation 1429
to obtain the element forces (Qu), which are transformed and combined into
the global forces (Q,). If these are sufiiently close to the applied forces (0)
then the process has converged, and the solution has been obtained.
‘When {Q,,} is not sufficiently close to {Q,), the incremental global forces

AQ.) = (0,

an be substituted into equation 1432, which can be solved for the incremental

y (1433)

Pre-buckling analysis of in-plane bending 287

JO tinesr aproxi:
IQ soirs
© extrapolation

geo _”
Oy he By Dance

global displacements (AA). The new and more accurate set of global displacements
aes} = {Bu} + (AA) (1434)

can then be used to begin a new cycle of calculations. The process
convergence is obtained.

peated until

Geometrical shear centre axis

host sen

Flange-W

Figure 149 Yielded and strain-hardened boundaries.

140. 258 Inclastic buckling
1436 YIELDED AND STRAIN-HARDENED BOUNDARIES,

The converged solution for the element nodal displacements (3,) can be used in
equations 14.25 and 14.18 to find the cross-sectional variations ofthe strain a at
the Gauss points. The positions of the yielded and strain-hardened boundaries

where & = 6, Can then be determined.

Typical variations of these boundaries with the major axis moment M, for

beams (P = 0) are shown in Figure 149 [18]. The top fla

atthe flange tips at 0.5 My (My is the nominal first yield moment FyZ,) since the
residual compressive stress is 0.5 Fy in this case, and is fully plastic at 1.09 My
The bottom flange commences yielding at 0.7 My at its centre where the residual

tensile stress was 0.3 Fy), and is fully plastic at 108 My. Yielding of the web
‘commences at the bottom at 0.76 My, and progresses slowly until the flanges
are fully yielded at 1.09 My. The process then accelerates, and the web is virtually
fully yielded at 1.14 My, when strain-hardening commences in both flanges.

14.4 Inelastic buckling analysis

144.1 METHODS OF ANALYSIS
A number of methods have been used to analyse the inclastic Nexural-torsion:
buckling of steel section members, including the finite difference, finite integral
transfer matrix and init element methods, These are summarized in [19]. In this
chapter, only the inelastic finite element method is described. This method is an
adaptation of the method presented in Chapter 4 for elastic buckling.

1442 ELEMENT DEFORMATIONS

The elastic finite element method of Chapter 4 models the deformatior
clement in terms of the deformations of a one-dimensional line. For an elastic
e axis for this ine, since this allows

lement, it is convenient to use the shear ce
the firstorder relationships between the stress resultants and the curvatures and
twist to be decoupled, and simplifies the buckling equations.

centre axes. The first is the
centre of the

For inelastic elements, there are two she
geometrical shear centre axis, which passes through the shea
‘geometrical cross-section, as shown in Figure 14.5a, The second is the inelastic

shear entre axis, which is the locus ofthe shear centres of sections transformed
from the geometrical section by using the tangent modulus ratio E/E = E/E for
ions, as shown in Figure 14,5b. The position

the yielded and strain-hardened re
re axis varies with the yielding of the section, as shown in

‘ofthe inelasticshear o
Figure 149. For this example, the she
flange as M, increases from 0.5 My to 1.08 My. This happens because the be
flexural rigidity of the top flan

‘entre axis moves towards the bottom

is the more effective asa result of the significant reductions in the effective
‘caused by early yielding near the flange tips

Inelastic buckling analysis

1]
(yielding at the centre of the bottom flange causes only minor reductions in its
rigidity). For M, > 108 My, both flanges are completely yielded or strain
hhardened, and so the effective section is doubly symmetric again, and the in
shear centre moves back to the geometrical axis,

While the inelastic shear centre axis seems the natural one to use when
describing the resistances of an inelastic element to lateral bending
there are significant disadvantages. First of all, the axis is often inclined to the

geometrical axis, to which the stress resultants are usually referred, which
requires the variable position of the axis to be included in the formulation
Secondly, the first-order decoupling between the stress resultants and the curva:
tures and twist for elastic elements is largely destroyed for inelastic elements, even
when the second-ord

flexural-torsional coupling is unimportant, and the
significance of the elastic shear centre axis as the axis of pure torsion
appears.

‘On the other hand, while the use of the geomé

ical axis requires som
when formulating the lateral bending and torsional resistances, its position does.
ot change with the amount of yielding, but remains aligned with he stress resultant
axes. For these reasons, the inelastic fin

e element method is most simply
developed from the displacements u, & of the geometrical shear centre axis

1443 STRAIN ENERGY
The flexural strain energy stored in the flanges during inelastic buckling ean be

ax E
Un=z| KEN + hg PP + (EAU — hd")

le (1435)

where (Ely), (Ely) are the tangent flexural rigidities of the top and bottom
flanges, and h is the distance between the flan

entroids. The tangent rigidities
may be determined by transforming the yielded and strain-hardened regions of
the cross-section using the tangent modulus ratio E/E, as shown in Figure 14.5,
For this purpose, the positions of the boundary planes between the elastic and
yielded zones are obtained from the in-plane pre-buckling analysis, as described

in section 14.3.6, For sections with residual streses, these boundary planes are
almost perpendicular to the flange mid-plane, and so for convenience they are
assumed to be perpendicul

The tangent torsional rigidity (GJ), can be obtained by using G, forthe yielded
and strain-hardened regions so that

CENT

C (1436
where b and are the actual width and thickness of each zone (elastic, yielded, or
strain-hardened) of the cross-section, and C is a correction to allow for end,
junction, and finite thickness effects. For simplicity, C is usually taken as the
elastic value given in [20]. The uniform torsional strain energy store

141. 260 astio buckling

inelastic buckling can then be written as

(Ged (1437)

The total strain ene
obtained from equa

stored in the element during buckling can therefore be
jons 14.35 and 1437 as

1
Lu, [eS Date} de (1438)

where
ur, $.) (1439)

[ (Ely + Eh © ]
iD, 0 Gd, (14.40)

(Er ELM 0 (El + Eis |

The off-diagonal terms —(El — Ely h/2_ allow for monosymmetry eff
caused by unequal yielding of the anges. The use of equation 14.38 in which
tangent rigidities vary along the element automatically allows for the non-
uniformity of the inelastic element.

Typical variations of (Ely + Ela)» and (GJ), with moment M, are shown
non-dimensionally in Figure 14.10 or the case of a beam (P = 0), These remain at
their elastic values until yielding commences at M, = 0.5 My, and then decrease

towards their strain-hardened values as yielding extends through the cross
tion. The intial rate of deerease of the flange flexural rigidities is fastest forthe

se early yielding at the Range tips, and

top flange, where the residual stresses ca
consequent large reductions in (El), On the other hand, early yielding at the

centre of the bottom flange has very litle early effect on (Ely),

1444 work DONE

The work done on the applied forces acting on a monosymmetric element (xy = 0)

can be determined from the second-order normal strain (equation 17.163)
1 1
er 2) + youl’ + = yoo? + du) + 208 +37) (1441)
1
Lay a nid + ya 6 + DE + = 70° 167) dA de
ab 306°) 2 (1442)

where ay is the longitudinal normal stress (tension positive). This equation is

independent of the elastic moduli, and is valid for both elastic and is

Inelastie buckling analysis 261

04 08 16

Ratios of rigidities

Figure 1410 Variations of the inelastic rigidities with moment

on the forces and their displacements. 5
Ir the inc

ations across the section are carried out, then equation 1441

+ yo} de (1443)

Where Gi, Ga is and oy are the stress resultants defined by equati

n 1428.

For doubly symmetric sections (yp = 0), equation 14.42 can be e

[DI {ede (1444)

1,08) (1443)

142. 262 — Inelastic buckling

and
a 0 0 ao
wa-| © % o
x (1446)
o 0) o 2
ou © 0 ue au)

1445 SOLUTION METHOD

The element stifiness and stability matrices (kJ, [9] which correspond to the
element strain energy 19*U, and work J6*V, may be obtained for a particular

results ofthe in-plane analysis in equations 14.38 and 14.44

load set by using th

These equations involve integrations along the length of the element, and ifthese
are done by Gaussian integration [16, 17) then the quantities in equations 14.40,
“and 14.46 need only be evaluated at the Gauss points. It should be noted that the
stress resultants 04, 0,0; and gj, are automatically obtained in the in-plane
analysis.

The element matrices [kJ C9.] may be transformed and assembled to form
global stifiness equations.

((K1 + CD) {A} = (0 (1447)

in which G isan initial stability matrix corresponding to an initial load set, and À
isa load factor. In elastic buckling problems, [K] and [G] are independent ofthe
load level 2, and need only be evaluated for the initial load set (2 = 1). In this case,
equation 1447 is a generalized linear eigenvalue equation, for which efficient
methods have been developed for the extraction ofthe buckling load factors and
thecigenvectors {A} which define the buckled shapes. These methods are referred
to in Chapter 4

However, in inelastic buckling problems, the stiffness and stability matrices
[K] and A[G] must be recalculated for each load level because ofthe changes in
the inelastic section properties and the stress distributions, and so 1
value extraction methods lose their efficiency. In general, a more reasonable

computation procedure is to iterate through a series of load levels towards a
solution. At each load level the in-plane analysis is performed, and the results are
then used to establish the matrix [X + AG] and the value ofits determinant is

calculated. This is repeated for different load levels until a zero value for the
determinant is found, which defines a buckling load set, and enables a corre-
sponding buckling mode (A) to be determined. Some care must be taken with this
method to ensure thatthe lowest buckling load is not missed, This can usually be
done by steadily increasing the load level from first yield by reasonably small

Inelastic bue

145 Inelastic buckling predictions

145.1 EFFECTS OF RESIDUAL STRESSES

The effects of residual stresses on the inelastic lateral buckling of steel beams
= 0) may be conveniently studied by considerin

simply supported doubly
symmetric (xp = yo =0) beams in uniform in-plane bending (q =0), firstly be
cause in this case the modifying effects of moment gradient and non-uniform
yielding are excluded, and secondly because the conditions at inelastic buckling
can be expressed by the comparatively simple equation

ia = EL, — Ely) h/2L
PEL, + Ely) f ¡CAY |

+ Eh Gay + Hos +6, (1448)
; | n GH tdi (aa

à ro 44 by assuming that the buckled
shapes are half sine waves, as they are for elastic buckling. This equation is most

whieh is bai n equations 1438 and I

easily solved for the span length L for which inelastic buckling occurs at a
specified moment M, = 0, since the tanj

ent modulus rigidities and the stress
resultants 0, 4 can be determined directly for this moment, The solutions are
conveniently displayed in plots of the type shown in Figure 14.11, in which the
dimensionless inelastic buckling resistance My/My is plotted against a modified

slenderness /(My/M,) in which My = M,, is the elastic buckling resistance, and

Eo %
u Basic theoretical model —
> ph
3 06 =
ES PA
8 ul CD tot-rated section
ë ad Uc te
Mead rel res, Gt
0 02 04 06 08 40 12 14 16

Modified slenderness ¥(My/Me)

Figure 1411 Effects ofa made for inelastic buckling mod

143. 264 Inelastic buckling
My =FyZ, on 1
plotting largely removes the influences of cross-section and yield stress.
The effects of the residual stresses, cross-section, yield stress, and tangent
moduli assumptions have been studied by many investigators [19], and some of
of the assump.

is the nominal frst yield moment of the section. This method of

these are demonstrated in Figures 14.11 and 14.12. The influence
tions made concerning the inelastic rigidities are illustrated in Figure 14.11 fora
ized residual stresses [20], The curve labelled

hotrolledI-section beam with ide
“asic theoretical model’ was calculated using the assumptions discussed earlier
in section 142. It shows an almost linear increase in the inelastic buckling
moment with decreasing slenderness, commencing at M/My =05 when the
Compression flange tips first yield to M/M y = 1.08 when both flanges are nearly
fully yielded. For My/My> 1.11, the beam acts as if completely strain-hardened.
The other curves of Figure 14.11 show that the assumption of G,=G in the

e increases in the buckling moment, and that the

yielded regions lead to mode
Assumption that E,= 0 in the yielded regions leads to slight decreases. More
important are the overestimates of buckling moment caused by neglecting the
“ects of monosymmetry (associated with (El, — Ely)) in the yielded cross

section.
The effects ofthe magnitude and the shape ofthe residual stress distribution are
demonstrated in Figure 14.12. The most important factor is the magnitude ofthe
ses lead to early

seal stew atthe fang tp High compres esiua st
Sing the top Compression Range pe, wi sipieant reductions in (El)
à moderate decreases in

te Full plasticity Mp
S10 iii
Eu

= Flange) - Constant

ME an
tal fo} a= és
= 203X203 UC 46

A a à u

Modified slenderness YiMy/Me)

Figure 14.12 Effects of residual stress on inelastic buckling

Inelastic buckling predictions 208

(Eh will cause substantial decreases in Mp since most terms are affected
‘adversely. On the other hand, the small decreases that occur in (£/,), caused by
tensile yielding in the bottom flange have very little efect on M, because there are
‘compensating monosymmetry terms.

Related explanations for the adverse effects of yielding at the tips of the top
flange can be obtained by considering either the position ofthe centre of buckling
rotation, or the effects of monosymmetry. The centre of buckling rotation is often
close to the bottom flange, as shown in Figure 7.6, so that it makes only a small
contribution to the buckling resistance. Thus the top flange is the more import
ant, and so reductions in its rigidity will cause substantial reductions in the
buckling resistance, Alternatively, it can be reasoned that the larger reductions in
the top lange rig

dity move the shear centre downwards and create an unfavour
able monosymmetry effect, so that the reductions in the buckling resistance are
accentuated, as shown in Figures 79 and 7.11

Residual stresses may increase the resistance to inelastic buckling, asin the case
of welded beams where flame-cutting ofthe flanges induces high residual tensile
stresses at the flange tips. This is because the tensile residual stresses delay
yielding of the top (compression) flange tips, thereby maintaining its flexural
rigidity at higher moments. On the other hand, early yielding at the tips of the
bottom (tension) Range has a smaller effect because the tension flange deflects less
during buckling and makes a smaller contribution to the buckling resistance. The
effect of the tensile residual stresses at the flange tips is therefore to change the

sense of monosymmetry, causing the shear centre to move upwards, and increas
ing the buckling resistance,

Also important in the elects of residual stresses is their distribution across the
flanges, as the more nearly constant isthe residual stress, then the more dramatic
is the decrease in the top flange flexural rigidity (El), once yielding commences.
Thus, while the peak residual compressive stresses are often higher in hot-rolled
beams than in welded beams (Figure 144), causing earlier reductions in the
buckling strengths of intermediate slenderness beams, yet the more constant
residual compressive stresses in welded beams cause greater reductions for low
slenderness beams, as can be seen in Figure 14.12

1452 SIMPLY SUPPORTED BEAMS

145.21 General

Many theoretical studies [19] have shown that the effec of variations in the
major axis moment distribution along a beam on its inelastic buckling resistance
is very important, This is because yielding takes place only in the high moment
regions, and so the consequent reductions in the section rigidities are localized.
The inelastic buckling resistance depends markedly on the location and extent of
these regions of reduced rigidity. T

these theoretical findings

e following sub-sections summarize some of

144. 8 Inelastic buckling

145.22 Moment gradients

buckling predictions for hot-rolled beams with ui

shown in Figure 14.13 [21]. The effect of the end moment ratio Bis.
stic buckling moments of beams in uniform bendi

„while those for beams

Tela
M, pM a
very important, as the ine
(B= — Nate reduced substantially below the e

val
tly reduced, at least until he full

with high moment gradient (f+ 1)are only sig
plastic moment Mp = FyS, is reached. This is because a high moment gradient
Ensures that yielding is confined to short regions atthe ends of the beam. In this

for buckling resistance

1, which is mostly responsib

case the central re
remains elastic, and there is only a small reduction in buckling strength

Also shown in Figure 14.13 are simple approximations given by
M, 0301-070 My/My)

x S = (1449)
(0610308 + 0.076%)

070

in which My = 2 M, is the elastic buckling moment for the same value of $
+ 7) These approximations, which are valid in the range 0.7 < My

ancies which

M, < 1.0, are generally conservative, except for the small dise
‘occur for uniform bending ($ = — 1). They can be used to determine approximate
limiting modified slendernesses at which the inelastic buckling moment M, is

equal tothe fll plastic moment Mp. These approximate limiting values are

(My/MJe= {039 + 0308 —0.0762)/0.70} (14.50)

jes from —1 to +

nd vary from 0.17 10094 as
Experimental results for beams with central concentrated braced loads (which
re 14.13

are equivalent to half-length beams with f= 0) are also shown in F
Ihese are reasonably close to the theoretical predictions.

[22] It can be seen tha

É 00 "m

A Yan

Y y ii ei results (2

3° Errar se shee \
0 02 0. 06 08 10 2

Modified slendernes

Inelastic buckling predictions 207
145.23 Transverse loads

Inelastic buckling predictions for beams with unbraced central concentrated
loads Q acting at the geometrical axis are shown by the curve 7 =0 in Fig:
ure 14.14 [21]. These predictions are only alittle higher than those for uniform
bending (8 = — 1), because while yielding i confined to the mid-span region, it is.
here that the reductions in the rigidities have their greatest effect. A reasonably

accurate approximation for the inelastic buckling moment
M,=OL/4 ass)

is obtained by using = — 0.70 in equation 1449.
Predictions have also been made for beams with uniformly distributed load, or
with two loads, each placed a distance a off-centre. In

ach case, the maximum
moment at inelastic buckling may be approximated by using equation 14.49 with
f= — 090 for uniformly distributed load, or

B=-07-0.6a/L (14.5)

for ofcentre loading.
It has been suggested [21] that the effects of transverse loads acting away from
the beam axis could also be approximated by using equation 14.49, provided that
the moment M, used in equation 14.49 includes the effect load height on elastic
buckling (section 7.6). Some limited experimental evidence for this is shown in
Figure 14.14 [23]

145.24 End moments and transverse loads

The effects of equal in-plane end moments ¿QL/8 on the inelastic buckling of
hot-rolled beams with central concentrated foads Q are shown in Figure 14.14

A rer

buckling Meg

EOF m \
‘i S

145. 268 Inelastic buckling

2 chaviour is similar to that of beams
[21]. For high end moments (2.0), the behaviour is

th high moment gradient (0.5), since yielding occurs only atthe ends wher
it ig relatively unimportant. These loading arrangements may be re
Équivalent unequal end moments, so that equation 14.49 may be used with

P=-07+03 (1453)

The effects of unequal end moments 7,Q1/8 and 740L/8 on the inelastic
buckling ofhotrolled beams with central concentrated loads Q were investigated
in [24], The results ofthis study are summarized by the equivalent end moment

re 14,15

ratios f shown in F

1453 CANTILEVERS
The inelastic buckling of hot-rolled cantilevers with concentrated end loads or
uniformly distributed loads has been investigated in [25]. This study includes the
tect ofthe height ofthe loading relative Lo the geometrical axis ofthe cantilever
In all eases, yielding was confined to the support region, and ct

reductions from the elastic buckling resistance which were similar to those for

used substantial

simply supported beams with central concentrated loads.

14.5.4 DETERMINATE BRACED BEAMS

mnalysing the inelastic buckling

Approximate methods have been developed f
of determinate beams which are prevented from deflecting and twisting at their

a at centro)

ut (2 y

Dimensionless end moment Ya

Inelastic buckling pre

ictions 200

supports and brace points. The simplest method is to ignore the buckling

interactions which take place between adjacent segments during buckling, which
are similar to those which occur in clasic braced beams (Chapter 10). A lower
bound may then be obtained from the lowest of the inelastic buckling load |

calculated for the segments of the beam. The information given in section 14.52
may be used to assess these load factors. The accuracy of this method is
demonstrated in Figure 14.13 by the braced beam experimental results of [22] for
f= —10, —07.

When a more accurate predict

on is required, then the eflects of interactions
between adjacent segments may be approximated by using an extension [21,24]

‘of the elastic method described in Chapter 10. The accura
demonstrated in Figure 14.16. A more elaborate exten

n is developed

14.55 CONTINUOUS BEAMS

A continuous be

mis statically indeterminate, and as yielding progresses in the
beam, its in-plane bending moments are redistributed from the elastic distribu-
tion towards that ofthe plasticcollapse mechanism, (However, this redistribution
may not be as rapid as is suggested by simple plastic theory because of the effects

of strain-hardening) The redistribution allows the loads to increase to levels
which are often significantly higher than the nominal loads which cause fist yield
in beams without residual tresses, Because ofthis, there are usually several à

ofthe beam with appreciable yielding. The changes in both the moment and yield
region distributions may be favourable or unfavourable with respect 10 the
inelastic buckling resistance.

Theoretical predictions ofthe inelastic buckling loads of two-span continuous

N JL
7 ET
Slenderness ratio of central segment 13/1,
Figure 1446 Inelastic buckling of braced beams

146. 270 Inclastic buckling

Legend — 01/07
vo
08
08
10
56
20
25

Elastic buckling Og

9105 007 08 03 12
Modified slenderness VAy/0g
Figure 1447 Inelastic buckling prediction for continuous beams

hot-rolled beams are shown in Figure 14.17 with the correspond:
ing predictions for simply supported beams with central concentrated loads or
uniform bending. All ofthe plotted points (with the exception of those for the load
ratio 0,/0, = 1.36) are very closely grouped, and for modified slendemesses
7/0/1049 = 06 are quite close to the curve for simply supported beams with
Sentral concentrated loads. This is because all of these continuous beams yield

fist at one of the load points at mid-span. For decreasing slenderness the plotted
an the curve for simply supported beams, principally

points rise more rapidly t
because yielding spreads moreslowly from the load points as a result ofthe higher
moment gradients, particularly towards the interior support

For continuous beams with load ratios of 0,/0 = 1.56, yielding occurs first at
small reductions from the

elastic buckling resistance, in much the same way as do the end moments of
Simple beams with high moment gradient (Figure 14.13. Thus the inelastic
ams are noticeably higher than those ofthe other

continuous beams shown in Figure 14.17.
‘Some of the theoretical predictions for two-span continuous beams (for
L, = 2:44 m) are also shown in Figure 14.18, together with the experimental
results of [28]. The experimental results are surprisingly low, even though the
ire 14.14 for simple beams of the same

buckling resistances ofthese

corresponding results shown in Fi

ross-section [23] are in reasonable agreement with theory. No satisfactory

explanation has yet been advanced for these discrepancies

Approximate theoretical predictions for the inelastic buckli
beams with central conce

loads may be made by using the methods for

raced beams described in section 14.54 provided that the in-plane

moment distribution can be determined, It is suggested that two approxim
should be obtained, one for the elastic moment distribution, and one for the
distribution at plastic collapse. If necessary, the final approximation may be
determined by interpolation,

1456 BEAN-COLUNNS

1456. Isolated braced beam-columns

The inelastic buckling of isola

ed hot-rolled beam-columns with unequal end
moments M, BM was studied theoretically in [13]. The predi

pared with approximations obtained from the linear interaction equation
EM
PQ PF) Mu Ga
with
Ca = 06-048 > 04 (1435)

In these equations, Py is the inelastic buckling load of a simply supported column,
and Myo isthe uniform inel

tie buckling moment 0
Approximations for these were develope
‘of hot-rolled I-sections as

simply supported beam.
from theoretical predictions for a wide

P/Py = 1.035 —0.181 /Py/P)— 0.128 Py/P, < 1.0, (1456)

147. 272 Inelastie buckling

Mia/ My = 1.008 —0245 Mp/M,, < 10 (1457)

in which P,, Py are the elastic flexural buckling and squash loads of the column,
and M, My are the uniform elastic buckling and full plastic moments of the
beam, Equation 14,57 yields slightly different results from those obtained from
equation 14.49 for = — 10.

Tt was found that these approximations were generally conservative, and
especially so for high moment gradients ($ > 0.5). The reasons for this conserva.

tism were attributed to the use of a linear interaction equation instead of a
parabolic one (see equation 11.13), and to the use of a C,, factor which was
independent ofthe axial load P (see equation 11.31)

Because of this it was decided to use a modification of the clastic parabolic

interaction equation (equation 11.30) to

(5) (ls) (1458)
where
E) (E

These equations proved to be of high accuracy, as is demonstrated in Figure

1419.

Figure 1419 Improved interaction equation for inelastic buckling

Problems 273

Continuous peam-column
i

>

Strong beans

Figure 1420 Arrangement of a threespan beam-column

145.62 Continuous bean-columns

The inelastic buckling of beam-columns continuous over three spans has been
studied experimentally [29] and theoretically [30,31]. The beam-columns were
of hot-rolled I-section and were loaded by the end forces P,, P,, the brave force
P,,and concentrated in-plane moments developed by the forces P,, P.

shown in
Figure 14.20. These actions caused significant in-plane yielding of the continuous
beam-column, reducing is resistance to out-of-plane buckling. Because of this,
restraining actions developed by the weak out-of-plane beams played more
important roles in determining the beam-column strength.

The purpose ofthe experiments was to obtain data which could be used to test
inelastic buckling theories. The values of the ratios P,/Py of the theoretical
predictions obtained from [30] with the experimental failure results are shown in
Figure 1421, which indicates extremely close age

14.5 Problems

A simply supported I-section beam whose properties are given in Figure 7.23 has
a span of 0m and equal and opposite end moments, as shown in Figure 14.22
and a yield stress of 250 MPa, Determine the inelastic buckling moment.

A simply supported I-section beam whose properties are given in Figure 7.23 has

a span of 10.0m and unequal end moments M and 0.4 M which causes double

148. Specime

k sv, (ee
se |" Number h

P 2 0082 | 0.96

2 om | 096

= las 00

09

4 p,| a 0 106

A o 10

Pr = FA
Tm;
=: o

Determine the value of M at inelastic buckling

A simply supported [section beam whose properties are
‘concentrated load at th
ion and twist

de

Determine the maximum m

€ centre ofits

and a yield stress of 250 MPa.

ivenin Figure 7.23 has

in Figure 14,22, and a yield stress of 250 MPa

inelastic buckli

t—) (Y
|
ha u m Er

A simply supported I-section beam whose properties are given in Figure 7.

‘concentrated load atthe shear centre fits 60 m span which is unrestrained, as

shown in Figure 14.224, and a yield stress of 250 MPa. De
inelastic buckling

has

ermine the maximum

A simply supported [section beam whose proper!

uniformly distributed shear centre loading, as shown in Figure 14.226, and a yield
stress of 250 MPa. Determine the maximum moment at inelastic buckli

A two-span continuous section beam whose properties a
has equal concentrated loads at the shear centre
which are unrestrained, as shown in Figure 1422
Determine the maximum moment at ine

given in Figure
the centre of both 60m spans

nd a yield stress of 250 MPa.
¡ing

(a) by assuming an elastie distribution of the in-plane bending moment;
(6) by assuming a plastic collapse mechanism distribution of the in-plane

149. 276 Inclastic buckling

A simply supported I-section beam whose properties are given in Figure 7.23 has
a top flange concentrated load at the centre of its 60m span which is un-
restrained, as shown in Figure 14.22g, and a yield stress of 250 MPa. Determine
the maximum moment at inelastic buckling

A simply supported Isection beam-column whose properties are given in
Figure 7.23 has a span of 60m, axial compressions Q KN, and equal opposite end
moments LSQKN m, as shown in Figure 14.22h, and a yield stress of 250 MPa.

Determine the value of Q at inelastic buckling.

A simply supported Isection beam-column whose properties are given in
Figure 7.23 has a span ol 100m, axial compressions Q KN, and end moments 1.5
QkNm and OKN m, as shown in Figure 14.22 and a yield stress of 250 MPa,
Determine the value of Q at inelastic buckling.

1. Galambos, TV. ed) (1988) Guide to Stability Design Criteria for Metal Structures, th
fed, John Wiley and Sons, New York

2. Trahair, NS. and Bradford, M.A.(1991) The Beha

revised 2nd eda, Chapman and Hall, London

Shanley, ER. (1947) Inelastic column theory Journal of Aeronautical Sciences, 145,

4, Lay, MG.(1965) Yielding o uniformly loaded stein
Division, ASCE, 9USTO) 49-6.

Haaijer, G. (1957) Plate buckling in the strai-hardening range

Engineering Mechanics Division, ASCE, BEM), 1212147.

Massey, PLC (1963) The torsional rigidity of steel Lbeams. Civil Engineering and

Publi Works Review, 58 680}, 367-71; S8(681), 488-92.

Lay M.G. (1965) Flange local buckling in wide-lange shapes. Journal

Division, ASCE, 9MSTO) 95-116.

8. El-Zanaty, M.H. and Murray, DW. (1983) Nonlinear finite element analysis of tel

frames. Journal of Structural Engineering, ASCE, 109(2), 353-68

Lay, MG. and Ward, R. (196

our and Design of Stee Structures,

una of the

6

ye Structural

Residual tresses steel structures. Steel Construction

310,221

10. Young, BW. (1975) Residual stresses in ho-rolled sections, in Proceedings, Interna
ional Colloquium on Column Strength, JABSE, Vol. 23, pp. 25-36.

11. Lee, GC, Fine, ES. and Hastretes, W.R. (1967 Inelastic torsional buckling of

Hol the Structural Division, ASCE, 93(ST5) 295-307.

29,

31

References — 277

Fukumoto, Y. Itoh, Y. and Kubo, M. (1980) Strength variation of laterally unsup-
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Bradford, MA. and Trahai, NS, (1985) Inelastic buckling of beam-columns with
unequal end moments. Journal of Consructional Stel Research, 50), 195-212
Dwight, JB. and White, J.D. (1977) Prediction of weld shrinkage stresses in plated
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Fukumoto, Y. and lob, Y. (1981) Statistical study of experiments on welded beams

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Bathe, KJ. (1982) Finite Element Procedures in Engineering Analysis, Prentice-Hall

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Kitipornchai, 5, and Trahair, NSS. (1975) Buckling of Inelastic 1-Beams Under

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Trahair, NS. (1983) Inelastic lateral bucking of beams, in Beams and Beam Columns

(ed. R. Narayanan), Applied Science Publishers, Barking, pp 35-69.

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moment. Journal of the Structural Dion, ASCE, 9(STI1), 2551-66.

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Dax, PF. and Kitiporncha, S. (1983) Inelastic beam buckling experiments Journal of

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150. 15 Strength and design of
steel members

151 General

Steel members which are loaded in a principal plane may fail in that plane as a
result ofexcessive yielding or local buckling, or by excessive bending and twisting

out of the plane of loading caused by flexural-torsional buck!

The out-of-plane strength M, is significantly influenced by el

buckling, but is reduced below the buckling stren

h My or M, as a result of
such as initial crookedness or twist, as shown in
ly or twist until the

geometrical imperfecto

Figure 15.1. Perfectly straight members do not deflect latera

clastic or inelastic buckling load is reached, and failure occurs then or soon after

On the

defects

lis

th

stresses,and out-of-plane deflection and twist, Thereafter, a lower load -deforn
M, is reached which i eq

her hand, an elastic member with realistic initial crookedness or twist

on-linearly, with the deflection asymptoting towards the elastic buck

gload. This elatic behaviour continues until the irs yield occurs, asa result of
& combination of stresses due to in-plane bending and compression, residual

to the out-of-plane strength.

Designers may avoid flexural-torsional buckling completely, either by using
sections which are not susceptible to this form of buckling, or by providing

sulicient bracing to prevent buckling. In these cases, the member's
capacity willbe governed by in-plane considerations, such as excessive plasticity
or local buckling.

Designers who do not avoid flex

torsional buckling altogether must design

by estimating the members capacity to resist this mode of failure, and by ensuring
that this exceeds the design action.
The out-of-plane strengths of beams which an
ling failure are considered in section 15,

usceptible to flexural-torsional
while code rules for designi

against the flexural-torsional buckling of beams are discussed in sections 15.3

and 15.4, The strengths and design of columns which fil in a lexural-torsional
mode are considered in section 15.5, and of beam-columns in section 15.6.

15.2 Beam strength

1521 ELASTIC BUEKLINC
The clastic flexural-torsional buckling behaviour of straight beams is discussed in
Chapter 8 (restrained beams), Chapter 9

Chapter 7 (simply supported beam

151. 200 Strength and design of steel members
Moment
4

Elastic buckling

Me Tinelastic
buckling Elastic bending
and twisting
Mn <— Initial crookedness
Me and twist

\
x

Yo Residual stresses

Lateral deflection or twist

Figure 154 Behaviour of real beam

(cantilevers), and Chapter 10 (braced and continuous beams). The resistance to

clastic buckling depends principally on

(a) the beam rigidities El, GJ, and EI,
(0) the beam span or segment length between braces L;
(© the bending moment distribution M,;

(@) the load height yo;
(e) the restraints against lateral deflection u, lateral rotation du/dz, twist
rotation 9, and warping proportional to dé/dz.

The maximum moment M,, in a beam segment at elastic lexural-torsional

buckling provides an upper bound to the be

im strength,

1522 INELASTIC BUCKLING

A straight steel beanrof intermediate slenderness fails before its elastic buckling
load is reached, asa result of premature yielding caused by the in-plane actions.

Beam strength 281

“and the residual stresses induced in it by the method of manufacture. The inlu-
ence of residual stresses on inelastic buckling is discussed in section 14.5.1. For
simply supported beams in uniform bending which are of hot-rlled section, the
inelastic buckling moment M is approximated by (see also equation 14.49)

Ma/My = 0.70 + 0.30(1 —0.70Mp/M,,)/098 as»
according to which M, decreases from Mp to 0.7 Mp as the modified slenderness
Y (M1,/M,,) increases from 0.169 to 1.195, as shown in Figure 14.13 by the curve
for B= 1. At /(My/M,,)= 1.195, the approximate inelastic buckling moment
M is equal to the elastic buckling moment M,,, and so elastic buckling at M,
controls for higher slendernesses.

Tt was shown in section 14,52 that the effect ofthe bending moment distribu:
tion on inelastic bending is very important. Beams in uniform bending have their

resistances reduced substantially below their elastic buckling resistances because
yielding takes place uniformly along the beam. On the other hand, yielding is
Confined to the end regions of beams in double curvature bending, and the

reductions from their elastic buckling resistances are much smaller, as shown in
Figure 14.13 by the curve for f= 1

For simply supported beams with unequal end moments M and PM, the
inelastic buckling moment M, was approximated by

M, 0.3001 —0.70Mp/M
oa 000 fo! My)

152
Mp (0:61 0304 +0078% “a
in which
M¿=2,M, (153)
is the elastic buckling moment and
iq = 1.75 + 1058-4 0.38? < 2.5 usa

is the moment modification factor of section 7.3. These approximations, which
are valid in the range 0.7 < M,/Mp-< 1.0, are shown in Figure 14.13.

1523 BENDING AND TWISTING OF CROOKED BEAMS

Real beams are not perfectly straight, but have small initial erookednesses and
twists which cause them to deflect laterally and twist at the beginning of loadin
Ian elastic simply supported beam with equal and opposite end moments M has

initial erookedness and twist given by

0 2 90 = sin ass)
de 8, 1

in which 4, is the central crookedness and 0, is the central twist, then its
deformation can be analysed by considering the minor axis bending and torsion
equations [1,2}

M + do) (1586)

152. 202 Strength

5010 = M, JP ass

in which

M,

VASE, LANG +? ELLE) (159)

is the elastic beam buckling moment, and
P= L/L (15:10)

is the elastic column flexural buckling load, then the soluti
and 15.7 which satisis the simply supported boundary conditions of

.=0, sed
(9/4) = (89/8), = 0)
is given by
in pm sin (15.12)
in which
5_0_ MM LS
50 MM,

The variations of the dimensionless central deflection 5/5, and twist 0/0,
shown in Figure 152, and it can be seen that deformations begin at the
‘commencement of loading
M, is approached.
The bending and tw

18 of beams with central concentrated loads which have

cither intial crookedness and twist or are loaded eccentrially to the plane ofthe

web have been analysed in [1], and similar results obtained to those above for

beams in uniform bending. The results shown in Figure 15.2 can therefore be
expected to be indica
As the defor

The maximum longitudinal stress in the beam in the absence of residual st

ve of a wide range of situations

nations increase with the applied moments M, so do the stresses.

the sum of the stresses due to in-plane plane bending, and

warping, and is equal to

Beam strength 203

The limiting m

M=oZ, (1513)

at which the maximum stress reaches first yield so that

F, (15:16)
is piven by
Pol, Pyhi2
mm AZ MA (517)
MC 2 0, M5
in which
Marz, (15:18)
When the central crookedness and twist are given by
Pd lA ay s
M, "TT HP AM, aM, nee)

7 can be solved for the dimensionless lim
125+ My/My\? My]
mS

My _ (125+ My/My

My "| 2 y

520)

153. 284 Strength and design of steel members

Figure 153 First yield of crooked and twisted

The variation of the dimensionless limiting moment M,/My is shown in
Fi ntity /(My/M,) plotted on the horizontal axisis a
modified slenderness of the beam. It can be seen that the limiting moments of
short beams approach the nominal yield moment My, and underestimate the

re 15,3, in which the que

‘capacities of real beams which can be expected to reach the full plastic moment
Mp, provided local buckling does not cause premature failure. While the limit
moments M, of long beams shown in Figure 15.3 approach the elastic buckling
moment M,,, the values for beams of intermediate slenderness may overestin

the strengths of real beams, whose residual stresses cause premature yielding

1524 EXPERIMENTAL STUDIES

More than 450 flexural-torsional buckli
‘welded steel I-beams have been reported in six studies reviewed in [3], The results
‘ofsome ofthese tests (on beams in uniform or near uniform bending) are shown in

tests of commercial hot-rolled and

Figures 154 and 15.5, where variations in the elastic ran
‘conditions, and restraints have bec

My

in each figure, demonstrating not

due to beam geometry
eliminated by plotting
gainst the modified slender

support and loadin
the non-dimensional maximum moment M,
mess /(My/My). There is considerable scatt
only the effects of variable initial crookedness and twist and load eccentricity on
strength, but also those of variable residual stress, moment distribution and

lateral continuity between adjacent segments on inelastic buckling,

Modified sl

Points representing the mean minus two standard deviations of the hot-rolled
test results of Figure 15.4 are shown in Figure 15.6. Also shown in Figure 15.6
is the uniform bending inelastic buckling

approximation of equation 15.1, and
a modification of equation 15.20 for the frst yield of crooked and twisted beams,
for which the first yield moment My has been repla
Me

It can be seen that neither the inelastic buckling approximation nor the
modified first yield curve is completely successful in representing the lower
bounds of the est results. The modified first yield curve isa litle low for hig!

by the full plastic moment

and

154. 286 Strength and design of steel members

N Elastie buckling Hye
E

Figure 185 Comparison of strength approximations for ho-olled beams in bending.

low modified slendernesses /(Mp/M,,) and the inelastic buckling curve is too

e unt dol inion tae
Moog! NY 1 Me)
E

as shown in Figure 156,

Working stress design of steel beams

In the working (or allowable) stress method of designing steel beams, the
inequality

oa, (15:22)

must be satisfied, in which a is the maximum stress determined by analysing the
naximum allowable stress given in

usually obtained by

structure for its working loads, and , is the

the code. The values given for the allowable stress 0, a
dividing the nominal maximum stress at failure 0, = M,/Z, by a factor of safety
which is often taken as 1/06.

Working stress design methods are somewh
failure is carried out at loads approximately equal to 60% of those which cause
placed by the limit states design methods discussed in

logica, in that design against

failure, and are being

ath design of steel beams 287

sections 15.4-15.6following. The working stress design of steel structures accord-

ing 10 the American [4], British [5], and Australian [6] design codes is discussed
in detail in [7]

15.4 Limit states strength design of steel beams

154.1 GENERAL

The limit states (or load and resistance factor) streng
[2,8,9] requires the inequality

design of steel beams

Mt <M, (1523)

failure states and under all loading sets. In this

to be satisfied for all mom

equation, M* is the design moment, M, is the nominal city, and is

the capacity factor
The design moment M* is determined from the analysis ofthe beam under the

ant design load combination obtained from the nominal g

and other loads multiplied by appropriate load factors which take into account
the possible overload situations that may occur, The nominal moment capacities
M, are specified in the design codes for the various failure states which may occur,
including failure by in-plane plasticity, local buckling or flexural-torsional
buckling, and represent the compromises made by the code writers between the
conflicting requirements of accuracy and simplicity. The capacity factor (which

is often close to 09) is chosen by the code writers so that it will lead to a

satisfactory evel of safety when used with M* and M, in equation 1523.1t can be

t of as taki

understrengih.

‘beams include yielding and local buckling ofthe cross-section, and
onal buckling ofthe be

thou account of the possibility that the structure may be

es that must be considered for the limit states strength

plastic collapse or flexural-tors as a whole.

For the eross-section limit states, the design moment M* is the value for the

cross-section under consideration, while the nominal moment capacity M,
depends on both yielding and local buckling considerations. Beams whose
compression flanges have low width~ thickness ratios do not buckle locally, and

bed as compact if they can reach the full plastic moment M,
taken as the section capacity. Beams whose compression flanges

are often des
which is the
h width-thickness ratios buckl

have hig locally, and are described as slender.
Their moment capacities are reduced below the nominal first yield moment My
Beams whose flanges have intermediate width-thickness ratios may be described
as non-compact, and their capacities are often interpolated between Mp and
My

The member limit state of plastic collapse is restricted to beams which do not
fail prematurely by local or Rexural-torsiomal buckling, and which can reach and
sustain the full plastic moment at the arly-forming hinge locations. The member

155. 208 Strength and design of steel members

limit states for other beams are taken as those of section failure due o yielding or
Local buckling, provided there is sufficient bracing to prevent premature failure

by flexural-torsional buckling.

For the flexural-torsional buckling limit state, the design moment M* is taken

‚ment under consideration,

as the maximum moment in the beam or beam s
while the nominal member capacity M, depends on the beam slenderness, the
bending moment distribution, the load height above the shear centre, and the
restraints, The formulations given by a number of design codes [10-14] for the
effects of these on the nominal member capacity are discussed in the following

1542 SLENDERNESS

The effects of slenderness on the nominal moment capacities M, of compact
beams in uniform bending according to diflerent design codes [10-14] are
ith the lower bound approximation of equation 15.21

compared in Figure 15.7 a

to the test results shown in Figures 154 and 156.
The Canadian nominal moment capacity [10] is given by

0<J(Mp/M,)<0683, Myy/Mp=10, |

0.683 < /(Mp/M,,) < 1224, Myy/My=1.15(1~0.28My/M,,))

1224 < /(Mp/My Myy/Mp=Mya/Me

Figure 15,7 Effect of slenderness

design of steel beams 200

The second of equation 15.24 provides a parabolic transition from the full plastic
moment M, to the elastic buckling moment M,
[11] i given by

The British nominal moment capas
0< J(Mp/M,,) 504, Myy/Mp= 10,

Mie My,/My \

ES eng (529
in which
Gn/Mp= {1+(1 +) M,,/Mp)/2 1526)
and
n=0007 J(e? E/Fy)(/(My/M,,) — 04). (1527)

The second of equation 1525 and equations 1526 and 1527 correspond to a
modification of the first yield condition of equation 15.20 with My replaced by
My and 1.25 by (1 + nM,,/M). Equations 15.25-15.27 are shown in Figure 157
for the case where E = 200000 MPa and Fy = 250 MPa

The American nominal moment capacity [12] is given b

0<L
Lp SLS Ly Mu/Mp= 10

M/Mp= 10,

MM) LL, — Lo), ) (1528)

DEL MEN, |
in which |
Me = tt —10F 9M usa

moment Mp to the elastic buckling moment M,.. Equations 15.28-15.30 are
shown in Figure 15.7 or the ease of an 18WF50 beam with E = 29 000 ips in
and Fy = 36 kipsfin

The Australian nominal moment capacity [13] is given by the lower bound
approximation of equation 15.21 to the test results shown in Figure 15.4 and 15.6
while 0.26 < /(Mp/M,.)

The European nominal moment capacity [14] is given by

OS /(Mp/M,)<02, Mya/Mp= 10,
M,

02< J(My/M,2, (1531)

in which a/My is given by equat

156. 200 Strength and design of steel member

and

n= 0.021(/(My/M,,) 02) (ss:
Equation 1531 is the same modification as that of the British equation 1525
jon 15.20 with My replaced by Mp and 1.25

ofthe first yield condition of equa
by (+ 9M, Mn)

The comparison shown in Figure 15.7 of the diferent nominal moment
capacities shows that the highest capacities are predicted by the Canadian and
‘American codes, which appear to ignore any effects of initial crookedness and
twist, and to be based entirely on the elastic and inelastic buckling strengths. The
Canadian capacities are close to upper bounds ofthe test results of Figure 154,
while the American capacities are closer to the means of the

The British capacities shown in Figure 15.7 appear to be the lowest, but this is

st results

deceptive because ofthe use of a high capacity factor of $ = 1.0 compared with
the values of $ =09 and 1/1.1 used by the other codes. The Australian capacities
‘lose to the lower bounds ofthe test results of Figure 15.4, while the European
‘capacities lie between the lower bounds and means ofthe test results

1543 MOMENT DISTRIBUTION

bution on the flexural-torsional buckling of beams

The effects of moment diste
with unequal end moments M, PM are accounted for in the Canadian, American
and Australian codes (10,12, 13] by using the moment modification factor

iq = 1.75-+ 1.059 + 0.3082 (1533)

for elastic buckling presented in section 7.3, with maximum values of 2.5, 23 and
25 respectively. In the American and Australian codes, is applied directly to

the unform bending nominal capacity My,, whereas ts application in the
Canadian code isto the elastic buckling resistance M,,. This later method is
Conservative, and does not take account of the important eect of localized
Jiciding on inelastic buckling, which was discussed in section 14.522

“The effects of unequal end moments are accounted for in the British code [11]

by using a moment modification factor

1/m= 10.570.338 + 0.08%) < 1/043 (1534)

tor is applied

whose values are close to the corresponding values of dy. Th
‘ly to the uniform bending nominal capacity My, of qual Manged be
uniform section

The
[14] by using

dir

ual end moments are accounted for in the European code

dig = 1.88 + LAON +0,52? <270 (1539)

er than equation 1533, However, equation 15.35 is
nee My

which is somewhat h
conservatively applied to the uniform bending elastic buckling resist

instead of the uniform moment capacity My

Limit states stren

design of steel beams 201

The effects of non-uniform moment distributions caused by transverse loads
are allowed for approximately in the British, Australia, and European cod:
principally by using values for the factor a, oF its equivalent which depend on the
bending moment distribution. For the Australian code, a simple approximation
can also be obtained by using

M,
VO + ME

(1536)

in which M, is the maximum moment,

ind M,, My, and M, are the moments at
the quarter, mid-, and three-quarter points of the segment length, The British and
European codes apply these factors conservatively to the uniform elastic buck
ling resistance M,,, but the Australian code is more realistic in applying them to
M

1544 LOAD HEIGHT

The destabilizing effects of gravity loads which act at the top fange (see
section 7.6) are allowed for approximately in the British and Australian codes
[11,13] by calculating increased lengths

LekL (537)

for use in calculating the elastic buckling resistance M,,, in which the load height
factors k, are approximately 1.2 or 1. for beams, and higher for cantilevers

A more accurate method is permitted in both the Australian and European
codes [13,14], in which the effect of load height on the elastic buckling resistance
is calculated more accurately by using approximations such as those of
equation 7.5,

No account of load he
110,12).

in the Canadian and American codes

1545 RESTRAINTS

The effects of end restraints against lateral rotation du/dz out of the plane of
bending are allowed for in the British and Australian codes [11,13] by
calculating decreased effective lengths

Ly= kyl (1538)

in which the lateral rotation restraint factor ka, is 0.7 or 0.85 depending on
whether the restraint is ull or partial or whether there are restraints at both ends
or only one end. The European code [14] decreases the 07 factor to 0.5, the
theoretical value for full restraint at both ends (section 85.1.1) The Canadian and
American codes [10,12] do not account for the effects of lateral rotation

The effects of partial torsional restraints which allow some end twisting are

accounted for approximately in the British and Australian codes by increasing

157. 292 Strength and design of steel member

the effective length L,. In the British code the increase is twice the beam depth for
partial restraints at both ends, while the increase given in the Australian code also
depends on the ratio 1/Tof the web and flange thicknesses. The Australian code
also allows more accurate approximations to be made for partial torsional
end restraints which are based on those of section 8.5.1.1. The Canadian and
American codes do not account for the effects of partial torsional end

The
accounted for, except in the European code, which allows

n85.1.1) are not generally

fects of end warping restraint (se

e use of reduced
effective length
L=kuL (1539)

without specifying values of ky

1546 DESIGN BY BUCKLING ANALYSIS

A method of designing against flexural-torsional buckling by using the results of
clastic buckling analysis is allowed implicitly in the British and Europ

‘codes [11, 14]. The implication is made through the use of expressions for the

clastic buckling resistance for a range of loading, support and restraint condi-

tions. Thus the maximum moment at elastic buckling determined by the

is used in place of the uniform bending buckling resistance M, in the design

alysis

process,
Design by buckling
Because this code multiples the elasti uniform bending buckling resistance M,

analysis is allowed explicitly in the Australian code [13].

by the moment modification factor 2, the maximum moment at elastic buckling

Mas determined by buckling analysis is reduced to

Mas Malt,

so that this value of M, can be used instead of M,, in calculating the slenderness
factor 0, (= Myq/M,) in equation 15.21. The value ofa, is then reintroduced in

the final calculation of the nomin:

‘moment capacity

My =a Mp < Mp (15.40)

Limit states strength design of steel columns

The American code [12] provides for the design of columns against torsional or
exural-torsional buckling. This is carried out by substituting the load Py, at

elastic flexural-torsional buckling (sections 5.25.5) for the value of P, at elastic
flexural buckling in the formulations for the column strength P,, This substitu
tion is an example of design by buckling analysis, and is based on the method
ven in [15] of designing cold-formed columns against flexural-torsional buck
Ting, Thus the dimer pacity P,/Py is given by the

onless nominal design

imit states strength design of steel beam-columns 299

Figure 15 Nominal column design capacity for fexural-torsional buckling

American code as

= 0.6587"/F« (1541)

while J (Py/P.) <1.5,in which Py = AFy is the squash load, and by

P./Py =0877/(Py/Ps) (1542)
while J(Py/P4) > 1.5, as shown in Figure 158.

The Canadian code [10] requires a rational analysis to be made, which may be
taken to be the elastic buckling analysis method used for the American code. The
European code [14] refers to flexural-torsional buckling of columns, but the
British and Australian codes [11,13] do not. These latter codes protect indirectly

against the torsional buckling of angle and cruciform section columns through
their rules against the local buckling of thin outstands,

15.6 Limit states strength design of steel beam-columns

Steel beam-columns must be designed against each of three modes of failure:

(a) local failure at a cross-section due to yielding and/or local buckling;

(0) in-plane failure due to excessive bending caused by the in-plane moments,
including the second-order effects due to moment amplification by the
axial compression;

(6) out-of-plane failure due to fexural-orsional buckling under the in-plane

158. 204 Strength and design of steel members

For each of these failure modes, it is common to use interaction equations to
represent the design inequality, and at the simplest level these usually take the

form of the linear interaction equation (see Figure 159)

pr Me

+ (1543)
oP, OM, J

in which P* and M* are the design axial force and maximum amplified moment,
and P, and M, are the nominal capacities in axial compression and bending

In some cases the same interaction equation is used for two or more of the three

ind out-of-plane failure, but in each case

they reflect the capaci

failure
the nominal capacities are interpreted differently so tha
tiesto resist the particular failure mode being considered. Thus when designing
P, is interpreted as the out

en as the capacity P,

nodes of cross-section, in-plane a

against out-of-plane flexural-torsional buckling
of plane column buckling capacity, which is usually ta

—-— Linear
Canadian
America ES

(P*70P,20)

0 02 04 08 08 10
Dimensionless moment M/OM,

Limit states strength design of steel beam-columns 208

associated with the load P, for elastic flexural buckling out of the plane of
bending, while M, is interpreted as the beam nominal capacity M,, associa

with the elastic Aexural-torsional buckling moment M,
The de

ign moments M* used in equation 15.43 generally account for second:
order effects on the in-plane bending moment distribution. They may be
determined by amplifying the moments calculated from a first-order analysis, or

in many codes [10, 12-14] by using a second-order in-plane analysis,
Non-linear interaction equations are used as a higher tier in some codes,

The Canadian code [10] uses

usually for compact section

Pe ossMs

10 (1549)
PTA

provided M? < 4M,., while the American code [12] uses

ps sm

o 1545)
dro 3am, \
while P9/6P,,>02, and
ps Me
PET ML (15.46)
while P*/6P., <02. The Australian code [13] uses
M: <6M, (547)

in which

Ms = a. Moro {tl = P*/6P. 1 (1548)
1p) (LAB, nl A

ón (>) (50) (01-02 2) (1549)

and Mo is the value of M, for uniform bending. These formulations are based

on experimental and theoretical studies of the Nexural-torsional buckling of
bbeam-columns with unequal end moments M, PM [16,17]. The linear and
non-linear inter ‚tions 1543-1549) for the design of beam

tion equations (eq

Most design codes ignore the strengthening effects of axial tension on the

resistance to out-of-plane flexural-torsional buckling (see sections 11.2.1 and

11.22) and only require the moment condition of equation 1523 to be satisfied
The Australian code [13] does allow for this strengthening effect through the
interaction equation

pr me

10 15:50)
OP, OM. (5%)

in which P, isthe tension capacity, provided the section and in-plane capacities
are not exceeded,

159. 206 Strength and design of steel member

15.7 Problems

Problems for designing against flexural-torsional buckling may be based on
many of the elastic and inelastic buckling problems given in other chapters. These
design problems will require the determination of the design capacity according
to an appropriate design code [10-14]. The design data will need to include an
appropriate value of the yield stress F (often equal to 250 MPa). The following
subsection lst suitable problems.

15.72 SIMPLY SUPPORTED BEAMS

Problems 7.1,3,4,5,6,7,8,9,10, 11,13.

1573 RESTRAINED BEAMS

Problems 8.1,2,3,4,5.

1574 CANTILEVERS

Problems 9.1,2,3,4,5.

Problems 10.1,2,3.

157.6 peaM-coLUMNs

157.7 INELASTIC BUCKLING
Problems 14.1,2,3,4,5,6,7,8,9.

15.78 STEPPED AND TAPERED MEMBERS

Problems 16.1,2

15.8 References

ral ofthe

1. Trahair, NS (1969) Deformations of geometrically imperfect beams. J
Structural Division, ASCE, 9S(ST7), 1475-96
Trahair, NS. and Bradford, M.A. (1991) The Behaviour and Design

revised nd ed, Chapman and Hal, London

References 207

Nethereot, DA, and Trahair, NS. (1983) Design of laterally unsupported beams, in

Developments in the Stability of Structures, Volume 2, Beams and Beam-Columns ed. R

Narayanan} Applied Science Publishers, Barking, pp. 70

American Institute of tel Construction, (1969) Specification forthe Desi

om, and Erection of Structural Stel fr Building, AISC, New York

British Standards Institution (1969) BS449: Part 2: 1969, Specfcai

Structural Steel in Building, BSI, London.

Standards Assocation of Australia (1975) AS125

SAA, Sydney

Trahai, NS. (1977) The Behaviour and Design of Steel

and Hall, London.

Ravindra, MK. and Galambos,T.V. (1978) Load and resistance factor design fr ste

Journal ofthe Structural Division, ASCE, 104(ST9) 1337-54,

‘Yura, J, Galambos, TV. and Ravindra, MK. (1978) The bending resistance of steel

‘beams, Journal ofthe Structural Division, ASCE, 1O4(STS), 1355-70.

Canadian Standards Assocation (1989) CAN /CSA-SI6.1-M&9 Limit States Design of

Steel Structures, CSA, Toronto.

British Standards Institution (1990) BS5950: Part 1: 1990, Struct

in Building, Part 1, Code of Practice for Design in Simple and Continuous Construct

Hot Rolled Sections, BSI, London.

American Institute of Steel Construction (1986) Load and Resi

Specification for Structural Stee! Buildings, AISC, Chicago.

Standards Australia (1990) AS4100, Stel Structures, SA, Sydney.

Commission of the European Communities (1990) Euracode No. 3: Design of Stee!
Building, CEC

American Iron and Stee! institut (1991) Load and Resistance Factor Design Specific

ion for Cold-Formed Stel Structural Members, AIS, Washington, DC

Cuk, PE, Rogers, DIF, and Trahair, NS. (1986) Inelastic buckling of cont

steel beam-columns, Journal of Constructional Steel Research, (1) 21-52

Bradford, M.A. and Trahair, NS. (1985) Inelastic bucking of beam-columns wi

‘unequal end moments. Journal of Consructional Stel Research, 53), 195-212

ws Fabrica

5, SAA Stel Structures Code

uctres st edo, Chapman,

160. 16 Special topics

16.1 Stepped and tapered members

16.1.1 GENERAL

Non-uniform stepped and tapered members are often used because of the
nomies produced by reducing the cross-section in the low moment re
ons are usually determined by it

While the cross-section red lane bending

considerations, these reductions may significantly reduce the member’ resistance
onal out-of-plane rigid
orsional buckling of stepped and tapered

to flexural-torsional buckling, by reducing the cross-sec
ities El,,GJ, and El. The fexur

members has been reviewed in [1

‘Stepped members (Figure 16.1a) have sudden changes in the cross-section,
usually produced by increasing the flange thickness of a welded beam or by
welding additional flange plates to a hot-rolled Lsection. Sometimes, stepped

have sudden changes in flange width. Web depths are never changed
although short tapered tra
nt depth,

sitions may be used to join sections of

Tapered members (Figure 16.1b) have gradual changes in the cross-section

dimensions. Tapered web depths are very common, but less common are tapered
Range widths. Flange or web thicknesses are never tape

steel structures, but sometimes are in aerospace struct

«in civil engineer

s, when savings in
weight can produce significant economies

cross-section of a member depend on the cross:
and the location of the reduced cross-section along
proximate effects of dimension changes are

8 the flange width B or thickness T ca

The effects of reduci

section dimension reduce
member, The
Table 16.1 It can be seen that chat

moderate to large changes in the out-of-plane rigidities, that changing the web
depth h has no effect on EI, small effect on GJ, and a large effect on EL, and that
changing the web thickness £ causes no changes, except for moderate changes to
Gu.

The effect

of reductions in the Rexural rigidity El, are greatest in the region
where the buckled beam has the greatest buckling curvature u, which is usually
at mid-span. The effects of reductions in the warping rigidity EL. are usually
greatest also at mid-span, However, the effects of reductions in the torsional
rigidity GJ are greatest where the twist dé/dz is greatest, which is usually at

161. 300 Special topics

section Heavier flanges Additional

(a) Stepped Beam Sections

Figure 162 Stepped and tapered bea

‘Table 16.1. Approximate effects of dimension changes on buckling rigidities

Approximate Change in Rigidity

Dimension changed El, [7 El,
» erate(B) Large (B°)

Flange width (8) Large (B*)_ Moderate(B) Large

Flangethickness() Moderate(T) Large(T?) Moderate(T)

Wed depth (1) 0 Small)
Web thickness (0 0 Moderate (£)

1612 sreerao nus
Inder onal Concerts fads was Sai a The Mans were stepped a
oti ran a meson bing ds QL E10) wth e

Stepped and tapered members 301

= nes steppe

o 02 OF 06 08 10
Section parameter 10 620 4h Hay
Figure 162 Reduction factors for stepped and tapered beams.

loads acting at the top flange, centroid, or bottom flange for a range of beam
parameters K = (EI /GJ1?)

It wasfound that the buckling loads varied almost linearly with he length ratio
and that reasonably close approximations could be determined by linear
interpolation between the buckling loads of uniform members. Thus

CCR TONER] (161)
in which Q,,, is the buckling load for a uniform member having the full
cross-section (= 7 = 1) and Q,, is the buckling load for a uniform member

having a reduced cross-section corresponding to the values of the width and
thickness reduction ratios f and }

For the elastic buckling moments M, of stepped beams of constant depth h, [5]
uses the approximation

Mae = M, (162)
in which

a. =1.0—24a(1— fy) (163)
and M,, is the elastic buckling moment of a uniform member having the full

eross-section. Equation 16.2 is compared in Figure 16.2 with the mean values for
the width and thickness stepped beams of [1]

162. 202 Special topics
16.13 TAPERED MEMBERS

The elastic flexural-torsional buckling of tapered simply supported equal flange
rated load was studied in [2]. The

I-section members under central cono
section lange width B, thickness T,or web depth h tapered nearly from maximum

ure 16.16,

values at mid-span to minimum values at the supports, as shown in F
following the corresponding reductions in the bending moments, It was predicted
¡ly and confirmed experimentally that the elastic buckling resistance

theoreti
reduces almost linearly with the lange width and thickness reduction ratios and

but is virtually independent of any reduction in the web depth h. Reference [5]
{Adapts the approximation of equation 162 by redefining 2, as

ay = 1.0 = 0.6{1 — PO + 04 gil hn): (164)

The elastic buckling of cantilev
from maximum values at the support to minimum values at the free end was
studied in [6,7], and tabulations were given for concentrated end and uniformly
distributed loads acting atthe top flange, centroid, or bottom flange. The el
buckling of simply supported beams with equal and opposite end moments or
uniformly distributed load was also studied
The elastic buckling of simply supported beams whose flange width Band web
support to minimum values
aphs was given for beams with

s whose cross-section dimensions taper linearly

depth h taper linearly from maximum values at 0
at the other was studied in [8], and a series of
‘unequal end moments M at the larger end and PM at the smaller end. Also

studied were simply supported beams tapered from a maximum section at mid-
span to minimum sections at the supports under uniformly distributed load.
The elastic buckling of simply supported beams which tapered linearly from
‘one end to the other was also studied in [9], which provided the basis for the
‘design rules of [10]. The effects of restraints on continuous beams were investi-
gated in [11].
The elastic buckling of monosymmetricI-beams was studied in [12] for simply

supported beams with central concentrated load and which tapered from mid:
span to the supports. The buckling of tapered and haunched beams was discussed
in [13], and test results reported in [14]

A finite element method of analysing the elastic buckling of tapered monosym-
continuous restraints in [15], and to inelastic buckling in [16].

beam-columns was presented in [4], and extended to beam-columns with

16.2 Optimum beams

Itis always desirable to distribute the material in a beam so as to minimize its cost
€ represents the results of

mize its efficiency

a cost optimization process in which designs are successively adjusted so as to
achieve the most economica solution. This usually depends on a very wide range

nany cases, current prac

Optimum beams 308

‘of parameters which affect the design, manufacture, erection, and use ofthe beam,
“and not just on the weight of material used,

Optimization may be carried out by minimizing the weight of material used
when this dominates in the total cost structure. In this ease, the optimization
should consider the distributions of material both within the cross-section and
along the ler a
its size is var

ath ofthe member, Very often Ihe cross-sectional shape is fixed, and
along the length

The optimization must be carried out while ensuring that the members able to

resist all failure modes, including in-plane yielding and local flange or web

‘buckling, as well as fexural-torsional buckling. Often the proportions of the

ross-section will be fixed so that local buckling will not occur, in which case the
optimization can be made with respect to in

ane yielding, and out-of-plane
buckling. In-plane yielding is usually considered to be governed by the in-plane
bending moment resistance, and the bea

is very simply optimized so that its
section moment capacity follows the bending moment distribution. In the
case ofa beam subjected to uniform bending, this wll result in a constant section
beam,

Optimization with respect to member buckling will distribute the mate

ial
along the length ofthe member o as to maximize ts buckling resistance. This will

result in material being more concentrated in the regions where the local
|
Figure 163 Column optimized against flexural buckling and yielding

163. 304 Special topics

contribution to the buckling resistance is greatest. In the case of the flexural
buckling of columns [17], the internal resistance to flexure EI, d?u/dz? is greatest
mn ofa simply support
€ optimum column has minimum cross-sections in the end regions (to
prevent premature yielding), and elsewhere has cross-sections which increase
towards the mid-length, as shown in Figure 163. The volume ratio ofthe optimal
column to the uniform column having the same strength varies from 0.866 for
very long columns which are dominated by buckling to 1.0 for short columns

in the central r d column of constant cross-section, and

which are dominated by yielding

For a simply supported narrow rectangular section beam in uniform bending,
the flexural-torsional buckling resistance depends on the internal resistances
EI, d2u/d=? to flexure and GJd¢/dz to uniform torsion. For a uniform section
beam, El,d*u/dz* is greatest at the beam centre and zero at Ihe supports, but the
reverse is true for GJdg/dz, In this case the optimal resistance to flexural
torsional buckling is produced by a uniform section beam

For a simply supported I-section beam in uniform bending, the internal
atest al the centre ofa uniform

resistance to warping torsion — El, d°4/d2° is
section beam, and least at the supports. For the case where GJdg/de is negligible
in comparison with — El, d*9/d*, the constant depth optimal beam will have ts
flange material distributed as in Figure 16.3, with its Range volume ranging from

(0866 to 1.0 times the flange volume of a uniform beam of the same strength.
The optimization of narrow rectangular and I-section beams against flexural
studied in [18, 19}, which consider cantilevers and

torsional buckling has b
simply supported beams under concentrated or uniformly distributed loads, or
‘unequal end moments.

16.3 Secondary warping

Thin-walled open section members of rectangular, angle, tee, or cruciform section
have zero warping rigidity ET,, when calculated u
‘open sections (section 17.3. Each ofthese sections consists ofa series of rectangu-

Jar elements which radiate from a common point, as shown in Figure 164. Thus

y the theory of thin-walled

any warping shears in their mid-surface planes are concurrent, and so the
p/dz°, which is the torque resultant of these shears, is

warping torque — El, d

The thin-walled theory predicts zero warping displacements during the torsion
‘of members ofthis type of section, and so boundary conditions associated with
warping (such as free to warp, or prevented from warping) cannot be invoked
when 1, =0. Because ofthis, theoretical solutions forthe twisting ofthe member
show ‘Kinks’ at points of concentrated torque or at built-in supports, as shown in
Figure 165, which do not occur in practice.

While this does not cause any theoretical difficulties, a smoothing out of these
kinks can be predicted by including the effects of secondary or thickness warping

Secondary warping 305

184 Section with radiating elements

Figure 165 Smoothing effect of secondary warping

[20,21] which are neglected in thin-walled theory. The non-zero section warping
constant /,, can be obtained by considering the displacement

usé (165)

of an clement 1,65 of the typical rectangular eleme
Figure 164, This element will develop a sh

ar component

5V= E(6st3/12)u" (166)

Which has a torque effects about the axis of twist through the common point
S.Theinteg he i
torque

ed effect ofall such torque effects is equal tothe secondary warping

My, =~ El,,d*/4 (167

164. 306 Special topics

Thus
(168)

so that
Taree. (169)
Normally, the secondary warping section constant I, is so small that the torsion

parameter K = (x? Elq/GJL?)is very small, Mathematically, ths leads to rapid

Changes in the twist d@/dz in the regions of concentrated torques, as shown in

Figure 16.50, which in the limit of K = 0 become the kinks shown in Figure 16.5.
Doubly symmetric circular and hollow rectangular sections behave similarly
to the sections with radiating elements discussed above, but for different reasons,

Such sections do not warp during twisting, because the warping deflections due to
twisting are exactly balanced by the warping deflections due to shear straining,
The shear stresses produced by the shear strains are equivalent to a circulating

orsion. Thus doubly

shear flow which is very effective in resisting uniform
Symmetric hollow sections have very large torsion section constants J, and zero
farping sections constants L. The kinks which appear at points of concentrated
b are consistent with the sudden changes in the

sudden cha

orgue shown in Figure 16.
circulating shear strains which occur when there
uniform torque, as at concentrated load points. Such kinks do not appear in
practice because of rapid changes in the stress distribution, which can be
predicted by using the theory of elasticity in place of thin-walled theory

ction of local and flexural-torsional buckling

164 Int

Elastic local buckling of a very thin compression flange will significantly reduce
the resistance of a beam in uniform bending to flexural-torsional buckling [22].
In this case of uniform bending, local buckles appear along the whole length of
h local failure is postponed by the lange

the compression flange, and even thou
postlocal buckling resistance, the flexural and torsional stfnesses ofthe fla
are reduced, so thatthe eff ne rigidities El,, GJ, and El, of the
beam are also reduced along the whole length of the beam [23]. The reduced

an be predicted by trans-

resistance of the beam to flexural-torsional buckling,
forming it into a monosymmetrie beam which has these reduced rigidities.

The interaction between local and flexural-torsional buckling for other than
uniform bending has been studied [24]. However, lange local buckling is then
ons, and it effect on flexural-torsional buckling

confined to the

is reduced

When the elastic local buckling load of the compression flange is close to the
clastic flexural-torsional buckling load, the actual strength may be reduced by
imperfection sensitivity ects. Some small reductions were reported in [25] for
thin-lange beams in uniform bending. In practice, very thin flanges are rarely

Web distortion 307
used in beams in uniform bending, and itis unlikely that strength reductions wil
slenderness, so that the local buckling streı rá

¡this ofthe same order as the inelastic
Rexural-torsional buckling capacity. An experimental study [26] of simply
supported steel beams with © ir

tral concentrated loads showed that their
strengths could be closely predicted by substituting a reduced cross-section
moment capacity for the full plastic moment capacity when determining the
Nexural-torsional buckling capacity (see section 154.2), the red
moment capacity allowing for 4

based on test results

d section

effects of yielding and local buck

ing and being

16.5 Web distortion

Itis usually assumed inthe analysis of lexural-torsional buckling that there is no
change in the shape ofthe cross-section during buckling. Web distortion (Figure
1660) may significantly reduce the flexuraltorsional buckling resistance.
Flexural-torsional buckling with web distortion is often deseribed as distortional
buckling. It should be noted that distortional buckling need not be associated
with flexural-torsional buckling, as for example in the case of the buckling of
‘columns with thin webs (see Figures 1.11 and 16.7), where the buckling mode
is of intermediate wavelength between those of local and member buckling
(27,28)

7

Figure 166 Effect of distortion of flexural

165. 208 Special topic

ltz Lx

1 100 100 10000
Buckle half wavelength
Figure 167 Column buckling modes
Web distortion allows larger deflections of he critical lange than usual. The

‘sistance to buckling is reduced through a corresponding reduction inthe strain
‘wists, Web distortion,

ergy stored during buckling caused by reduced fa
may occur in beams with flexible webs, due either to their thinness or to holes. It
also occurs in beams whose compression flanges are unrestrained atthe supports
[29] as shown in Figure 16.6d, near the supports of continuous composite
beams [30] as shown in Fig girders [31], as shown in
Figure 16.6.

The distortional buckling of beams in uniform bending has be

re 16.60, and in trou

by using the finite strip method [32]. A simple approximate solution has been
obtained for I-section beam-columns in uniform bending by assuming that the
web distorts ina cubic shape [33]. This assumption has also been used to analyse
beam-columns of general, thin-walled open cross-section under general loading
conditions [29, 34]. This method has been applied to a wide range of situations
and extended to inelastic buckling problems. A recent study [35] has used a super
distortional, and lateral buckling. An

finite element to include the elects of loca

extensive review is given in [36]

16.6 Pre-buck

ng deflections

The usual method of analysing lexural-torsional buckling neglects the effects of
€ of bending, which increase the buckling

the pre-buckling deflections in the plan

Post-buckling behaviour — 309

Fora simply supported beam in uniform bending the increased elastic
buckling resistance is given by

M,
M a 5 5 6.10)
JOE ENANA an 69

in which
My. = (EL, )LNGI + Ely, (16.11)

is the classical resistance (section 7.2.1),
The pre-buckling deflections transform th
coneave curvature ofthe deflected beam increases its buckling resistance, just as

the convex curvature of an arch section 13.5.3) decreases its buckling resistance

cam into a ‘negative’ arch, The

The resistance of the beam in uniform bending increases with the ratio J,/1, and

theoretically becomes infinite when 1, = 1,. This is consistent with the common
assumption that a beam with 1, > 1, does not buckle, because it finds it easier to
remain in the more flexible plane of bending than to buckle out-of-plane by
deflecting in the stiffer plane, (Note that this common assumption does not
always hold true, as for example when loads are applied well above the shear
centre axis (see section 7.6}, which may lead to buckling in a dominantly torsional
mode)

Early work on the effets of pre-buckling deflections was reviewed in [37]
which considered simply supported beams in uniform bending. This was
extended toa wide range of load, support, and restraint conditions in [38]. Finite
lement methods of analysis have been developed in [39-41]

16.7 Post-buckling behaviour

Slender statically determinate beams which remain elastic have slowly rising
load-deformation paths after flexural-torsional buckling [42], as shown in
Figure 168. This behaviour is similar to that of the behaviour of slender columns
after flexural buckling [43]. The increased load carrying capacity over that
Predicted by the small deflection theory of elastic buckling is associated with
disturbing effects at large deflections which are not as large a

s those predicted by

the small deflection theory
However, slender redundant beams [44-46] may have significant increases in
load capacity above those predicted by the small deformation theory of flexural:

torsional buckling. The twist rotations that occur aftr initial buckling reduce
locally the effective bending rigidity in the plane of loading, which causes a
redistribution ofthe bending moments. In continuous beams, the moments near
greater part in resisting

the untwisted regions at Ihe supports increase, and play
the applied load.
The buckling resistance i

erally affected by the bending moment distribu-
3-7.5 and Figure 7.17), and in this case the redistribution is

favourable, and increases the buckling resistance, The buckling resistance may

166. 310 Special topics

Redundant

Large deflection theory

cod

Figure 168 Post-buckling behaviour of beams

mptote towards a limiting value (Figure 168), but then may continue to

increase, but only slowly, as in the case of statically determinate beams
‘Significant increases in strength after post-buckling are only realized in slender

rectangular beams or in very slender [section beams. Yielding effects usually

cause practical
achieved [46]

ams to fail before any significant post-buckling reserve can be

16.8 Visco

lastic beams

Materials which creep under sustained loading may be modelled as being

iscoclastic [47]. Perfectly straight viscoelastic members may buckle at
some time after the application of load, while the deformations of members with
initial crookedness or twist will increase with time, and may become excessive, or

may lead to material non-linearity, and even higher deformations.
ime () relationship for the three-element

model shown in Figure 169a is given by

rote neon

= Ec+m 1612
Es)" Esa "a un

The material strains & = 0/E, immediately a stress o is applied, and then creeps

viscously towards a final strain e, = o(E, + E5)/E, Ey, as shown in Figure 169b.

When the material is under constant stress, it can be regarded as having an
effective modulus of elasticity which decreases with the time from the initial
application of load from an initial value Eg = E, towards a long term value

Viscoelastic beams 311

E. = E, ENE, + Es), Thus the possiblity of ereep buckli
stress can be investigated by first finding the

under constant

ticity for which

elastic buckling would o
determining th

and then
ime at which the effective modulus would have reduced to this

167. 312 Special topics

tial crookedness

"The deformations of viscoelastic members with i nd twist
have been investigated in [48] by using the correspondence principle, which
allows the viscoelastic problem to be transformed into a corresponding elastic
problem. Some of the results of this study are shown in Figure 16.10 fora simply
supported beam in uniform bending. When the applied moment M is equal to the
Jong term buckling moment M.,, the deformations increase linearly with time,

and can be expected to cause the material to become non-linear at some later
time, Higher values of M than M,, cause deformations that accelerate with time,
but lower values cause the deformations to asymptote to those that would be
predicted (se section 15.2.3) by using an elastic model with the long term moduli
E.G

16.9 Flexural-torsional vibrations

169.1 GENERAL

The vibrations of structural systems [49] are generally introduced by considerin
ee of freedom system (Figure 16.113) for which the
werms the variation of the displacement v with

the behaviour of a single d
equation of motion which

Mé +amv= 00 (16.13)
d

in which M is the mass, ap, isthe spring constant, Q(0) is the driving force a
0 = (fd). For free vibrations, Q(0 1 the solution of equation 16.13 is

=A sin at) (16.14)

= /(2,/M) (1615)

a

{al Single Degree of
Freedom Syst

{BL Transverse Vibrations
sf à Celung

Figure 18.11 Free vibrations

Flexuraltorsional vibrations 318
isthe natural frequency, and the intial conditions determine the amplitude A and
the phase angle 0
For continuous systems such as the simply supported column shown
in Figure 16.11, the equation of motion for free transverse flexural vibra
tions is
mi + Ela + Po" =0 (16.16

ibrations

in which mis the mass per unit length. Free

b= A sin (mz/L) sin (Qt)

have natural frequencies

$” 1

a VA 0 7 )} (16.18)

Continuous systems may be analysed by using the energy equation
LOU REV), (16.19)
in which 48% is the change in the strain energy, 182V the change in the

potential energy of the applied loads, and }52KE is the change in the kinetic

energy. This equation isa statement of the principle of conservation of eneray

during the free vibrations, according to which the total energy o does
not vary with time, The total energ

and potential energies whe

equal to the sum of the maximum strain
the kinetic energy is zero (zero velocity), and also
equal to the maximum kinetic energy when the strain and potential ene
zero (zero displacement).

The energy equation can be used to develop a finite element method of
Analysing free vibrations, which can be expressed in

HAYKD+ALGI- RIM) (a) =0 (6
in which [X] is the stiffness
structure, [G] is the stability matrix associated with the potential energy of an

inital set of loads, [M] is the mas
the masses,

trix associated with the strain energy of the

matrix associated with the kinetic ene

A) isthe vector of nodal deformations of the structure, and À is the

load factor. The natural frequencies ofthe structure satisfy

K+3G-0%M|=0. (162

1692 STRUCTURAL VIBRATION MODES

Structural members may vibrate in axial, flexural, or torsional modes, or in
combinations of these. Axial vibrations involve axial displacements w. For
uniform members fixed at one end and free at the other (Figure 16.11c), the
displacements are given by

A sin (Qn —1)n2/2L.~0) 162

168. 4 Special topics
when the mass per unit length m is constant, and the natural frequencies Q by
9? = En IP? EA/AmL (1623)

cause of the

in which mis an integer. These frequencies are usually quite h
high axial stiffness EA/L
flexural vibrations v about the x axis of simply supported members
‚tions 16.16-16.18, Corresponding equations
govern the free Nexural vibrations u about the y axis.
Torsional vibrations ¢ of members of doubly symmetric cross-section which

The fr
ith axial loads are governed by ©

are prevented from twisting but free to warp at both ends take the form of

$= Asin sine) (1629
L
in which
nef, El
o ii (1625)
A)
ti da (1626)

is the rotary inertia, which depends on the distribution of material of density p
through the

mass arca A attached to the member, and is uniform along the

‘Structural vibration modes may be coupled flexural-torsional modes. The
ich case its

coupling may be caused by the axial force or bendin;
fexural-torsiomal buckling of co

e coupling leading to th
members (Chapter 5, beams (Chapter 7), and beam-columns (Chapter 11)
ample when the

Coupling may also be caused by the mass distribution, as for e

centre of mass does not coincide with the shear

Axial forces and bending moments change the natural frequencies of vibration of
columns, beams, and beam-columns, This ís caused by the change in the effect
stifness of the member from the este stifness associated with the strain energy

U (see equation 16.19) caused by the change in the potential energy V of the

loading system. Thus in equation 1620, the stability matrix 2[G] indicates the

change caused by the loading from the elastic stiffness [K ].

The n

column decrease as the axial compr

cachos the column flexural buckling load (see equation 16.18) These decreases
by th th

the destablizing effects ofthe axial compression P.

ural frequencies © for transverse vibrations of a simply supported
sion P increases, and become zero when P

are caus reduction in the effective flexural stifness EI,/L? caused by

Flexuraltorsional vibrations 315

‘The natural frequencies of flexura
monosymmetrie (xy = 0) be

torsional vibrations of a simply supported
m-columa in uniform bending

can be obtaines

ing the vibration mode as

1581
fu) (8) „m

lof los
in which case the terms of the energy equation (equation 1620) become
A} = {6,0}, (1628)

Gr ELL) 0
[KJ=1 $
|

0 (GIL entr El, (623)

sey o Po WEM + na? Pll
MB?)
(1630)
em (631)
[var
Toy= | pw oda } (1632)

da |
o= | Pi +0?) da

are mass integral of the density (p distribution over the mass area A connected
to the member.
The frequency equation (equation 16.21) can then be expressed as

aye o (1633)
in which
a: = SELS nt PILE — OL )
a3 EM né Pya/ LÀ + ly /L |
(MG/L +n EL JL) p 06

Per}

net. |

This leads to a quadratic equation in 0%, which can be solved for the natural
frequencies 0,

169. 316 Special topics

The natural frequencies of cantilevered beam-columns have been investigated
in [50],

16.10 Erection buckling

Spreader beams used for lifting during erection as shown in Figure 16.12a may
buckle in a flexural-torsional mode, The inclined upper cable causes compression
in the central portion of the beam, while the loads Q being lifted cause bending in
the vertical plane

no supports which prevent lateral deflection or twisting, but these

There are
buckling deformations are resisted by the loading system itself. Rigid body
rotation about the vertical axis through the cable suspension point A is un
restrained, but this rigid body mode is of no structural significance. Rigid body

horizontal deflections are restrained because the Mexibilitis of the upper and
lower cables require each pair of attachment points B, B’and C,C to the beam to
the suspension point A, as shown in

isting twist rotations are provided by

remain in a vertical plane contain
Figure 16.12b, Because ofthis, torques
the upper cable forces acting above the beam, and by the loads Q being lifted
whieh act below the beam

The elastic flexural-torsional buckling of spreader beams has been analysed in
[51], Ie was found thatthe optimum arrangement of the cable attachment points
B,B is close to that for which the cable line intersects the lifting beam
load attachment points C,C, in which case there are no bendin

xis at the

actions, The

a
A
a |

column simply supported between the load attachment points C,C. The buck
Jing capacity reduces rapidly as the cable points B,B' move away from the
optimum positions

The elastic flexural-torsional buckling of cambered I-section beams during
‘rection has also been analysed in [52],

16.11 Directed loading

Not all structura

loads have the characteristic of gravity loads of remaining
parallel to their original lines of action as the structure deflects. Directed loads
‘exerted through tension cables or by articulated jacking systems act through a
fixed point, and their lines of action become inclined as the structure deflects
la

rally. The resulting lateral components of the directed loads may exert
restoring or disturbing actions on the structure, and may i
buckling resistance

e or decrease the

The effect of directed loading on the flexural buckling ofa cantilever column is
shown in Figure 16.13 [53]. The buckling load is given by

P/P, =(uL/xy (1635)
in whieh
P= EL (1636)

isthe flexural buckling load of simply supported column unk

er gravity loadin

/ 3
|
Figure 1643 Flexural buckling of cantlovered columns with directed loading.

170. 318 Special topics
and y is the solution of
(an wl yu) =( 0/1) (1637)

in which b defines the point through which the load is directed. For L/b=0, the
directed load is a gravity load, and P/P, = 0.25, As L/b increases, the buckling
Toad increases to P= P, for L/b= 1.0, and then approaches a limiting value of
2045 P, For negative values of L/b, the directed load exerts a disturbing
component, and the buckling load decreases from 0.25 P, towards zero.

The effects of directed loading on the elastic flexural-torsional buckling of
simply supported beams and cantilevers with concentrated lo

ds have been

investigated in [54]. For downwards loading, the buckling resistances are

increased when the load direction points are below the load application points,
and decreased when above. Beams and cantilevers with top flange loads directed
from above may have very low buckling resistances, while ther

are often very
significant increases in the buckling resistances when bottom flange loads are
directed to lower points.

16.12 Follower loading
‘Some static loading methods cause structures to buckle suddenly in a dynamic

manner, which cannot be predicted by using the concepts of equilibrium or
conservation of energy based on strain and potential energy alone. These include

follower loading, as shown in Figure 16.14, in which the applied load P acts
tangentially at the end of the member [55,56]. A simple demonstration ofthis
made by holding a garden hose some distance from is

low until instability occurs. In this case the follower
loading is supplied by the reaction of the water as it leaves the hose. The same
problem may be caused for aerospace vehicles by their rocket thrusts. Related
ns may be caused by the thrusts of jt engines mounted on the cantilevered

dynamic buckling can b
end, and increasing the wat

Follower loads are examples of non-conservative loads, for which the work
done during deformation is deformation path-dependent. They difler signifi
cantly from conservative loads, such as gravity loads for which the work done
depends only on the vertical distance between the initial and final positions, and
is independent of the deformation path. The principles of static equilibrium and
of conservation of energy based on strain and potential energy alone can be
applied to conservative load sy:

The path dependence of the work done by a follower load can be demonstrated
by considering the two alternative deformation paths shown in Figure 16.14. The
first path shown in Figure 16.14a consists of an initial end rotation (during which
a small amount of work is done), followed by a final horizontal deflection in
which substantial negative work is done by the horizontal component of the load

For the second path shown in Figure 16.14, a small amount of work is done by

Follower loading 319,

Rotation and Translation

Translation and Rotatio:

Figure 16.14 Deformation paths fr fllower loading

the initial horizontal defection, followed by another small amount of work
during the end rotation.

The behaviour of non-conservative load system can be analysed dynamically

by considering the natural frequency of vibrations, Ifthe follower load point of

Figure 16.14 has a concentrated mass M, then it can be shown [55] that the
natural frequency of vibration © is given by
gt = _(#LP EN/ML es
Sin pL — uLeos pl
in which
(LP = PL/EL (1639)
The natural frequency © increases with P from ,/(EI/3ML‘) at P=0 until it
becomes infinite when
sin pL pl cos ul =0 (16.40)
20.19 EL? (1641)

Beyond this load, N° becomes negative, so that © becomes imaginary, and the
vibration proportional to sin 2: changes from one of steady vibrations to one of
exponential div

snce proportional to €", This divergence is fed by kinetic
By extracted from the follower loading.
The load at divergence is approxima

ly eight times the value of x

with gravity loads. I can be seen that the restoring component ofthe

171. 320 Special topics

al Probes (b) Problem 162

follower load has a substantial effect on the buckling resistance of the cantilever.
However, the application ofa static equilibrium buckling analysis to this problem
155] predicts an infinite buckling resistance, Thus, while idealizing the follower
load as a gravity load is overly conservative, using a conventional stability
analysis is quite dangerous.

A survey of early work on follower loading is given in [55], which includes
studies of the flexural-orsional behaviour of beams and cantilevers under
concentrated loads. More recent work is reported in [50,57]

16.13 Problems

A simply supported beam whose properties are given in Figure 7.23 is
hened by welding a plate 300 mm x 20 mm to each flange over the central
50 m portion ofthe 12.0 m span. The beam has a concentrated load atthe shear

centre at mid-span, which is unrestrained, as shown in Figure 16.15a. Determine
the increased elastic Nexural-torsional buckling load

A welded steel beam (E = 200000 MPa, G = 80000 MPa) has two 300 mm x
20 mm flanges welded to a 10 mm web which tapers so that the overall depth
reduces from 800 mm at mid-span to 500 mm atthe supports. The beam i simply
supported over a span of 12.0 m, and has a central concentrated load at mid-span,
which is unrestrained, as shown in Figure 16.156. Determine the elastic lexural-
torsional buckling load,

16.14 References

1. Trahair, NS, and Kitiporachai, . (1971) Elastic lateral buckling of stepped I-beams
mal ofthe Structural Division, ASCE, 97(STI0) 2535-48

2. Kitipornehai, Sand Trahair, NS. (1972) Basie st

References 321

bility of tapered I-beams, Journal

of the Structural Division, ASCE, 98(ST3), 713-28

Galambos, TV. (e) 1988) Guide to Stability Design Criteria for Metal Structures, 4th

cain, John Wiley, New York

Bradford, MA. and Cuk, PE. (1988) Elastic buckling of tapered monosymmetric

Lbcams Journal of Structural Engineering, ASCE, 114 (9), 977-96

Standards Australia (1990) 44100-1990 Steel Structure, Standards Australia, Sydney

Netheret, D.A (197) Lateral buckling of tapered beams. Publications, IABSE, 334

173-92,

Brown, T.G. (1981) Laterabtorsional buckling of tapered I-beams. Journal of the

Structural Division, ASCE, 107 (ST) 689-97

Bradford, MA. (1988) Stability of tapered I-beams, Journal of Constructona

Research, 9 (3), 195-216.

Lee, G.C, Morell, ML. and Ketter, RL. (1972) Design of tapered members. WRC

Bulletin No. 173.

American Institute of Steel Construction, (1986) Load and Resistance Factor Design

Specification for Stractural Steel Buildings, AISC, Chicago.

Morell, M.L.and Lee, G.C (1974 Allowable tress for web-apered beams with lateral
straints. WRC Bulletin No. 192. February

Kitipornchai,S. and Trahair, NS. (1975) Elastic behaviour of tapered monosymmet

rie Lbeams Journal of the Structural Division, ASCE, 10 (STS), 1661-78.

Horne, MR, Shakir-Khali, H. and Akhtar, $. (1979) The stability of tapered and

haunched beams. Proceedings, Insttwion of Ciel Engineers, 67 (2) 677-94

Horne, MR, Shakir-Khali,H. and Akhtar, S, (1979) Tests on tapered and haunched

beams, Proceedings, Institution of Ciel Engineers, 67 (2) 845-50.

Bradford, M.A. (1988) Lateral stability of tapered beam-columns with elastic re

straints, The Structural Engineer, 66 22), 376-82

Bradford, M.A. (1989) Inelastic buckling of tapered monosymmetri -beams.

neering Structures, 1 (2) 119-26.

Trahair, NS. and Booker, LR. (1970) Optimum elastic columns. International Journal

of Mechanical Sciences, 12 (11), 973-83.

Wang, C-M. Thevendran, V, Teo, KL. and Kitipornchai, S.(1986) Optimal design of

tapered beams for maximum buckling strength. Engineering Structures 8 (4), 276-84

Wang, C-M. Kitipornchai, Sand Thevendran, V. (1990) Optimal designs o l-beams

agains lateral buckling. Journal of Engineering Mechanics, ASCE, 116(9), 1902-23.

Bleich F (1952) Buckling Strength of Metal Structures, McGraw-Hill, New York.
Attard, MM. and Lawther, R. (1988) Effect of secondary warping on lateral buckling,

in Proceedings, 1th Australasian Conference on the Mechanics of Structures and

Materials, University of Auckland, pp. 219-25.

Cherry, 5. (1960) The stability of beams with buckled compresion Manges. The

Structural Engineer, 38 9), 277-85.

Bradford, MA. and Hancock, G4. (1984) Elastic interaction of local and lateral

buckling in beams. Thin-Walled Strctues, 2, 1-25,

Wang, ST, Yost, M. and Tien, Y.L.(1977) Lateral buckling of locally buckled beams

using finite element techniques. International Journal of Computers and Structures,

469-75

Menken, CM, Groot, WJ. and Stallenberg, G.AJ. (1991) Interactive buckling of

beams in bending. Thin-Walled Structures, 12, 415-34

172. 222 Special topics References 328

2, Kubo, M.and Fukumoto, Y.(1988) Lateral torsional buckling ofthin-waled1-beams 49. Clough, R.W. and Penzien, (1975) Dynamics of Structures, McGraw-Hill, New Y

Journal of Structural En ASCE, 1148) 841-35 $0: Attard, MM. and Somervall, LI (1987) Stability of thin-walled open beams under
fr, Hancock, G.. (1978) Local, distrtional, and lateral buckling of -beams. Journal of ‘non-conservative loads. Mechanics of Structures and Machines, 18(3), 395-412.

the Structural Division, ASCE, 104 (STI), 1787-38. St. Dux, PF. and Kiipornchai, $. (1990) Buckling of suspended I-beams. Journal of
2%. Lau, SCW and Hancock, GJ. (1990) Inelastic buckling of channel columns in th Structural Engineering. ASCE, 16 (7) 1877-9

distortional mode. Thin-Walled Structures, 10, 59-84, 52. Peart, W.L., Rhomberg, EG, and James, RW. (1992) Buckling of suspended cam:
9. Bradford, M.A. and Trahai, NS. (1981) Distortional buckling of -beams. Journal of bere girders. Journal of Structural Engincering, ASCE, 118 (2 505-528

ral Engineering, ASCE, 107 2), 355-70. 53. Simitss, G (1976) An Introduction to the Elastic Stability of Structures, Prentice

Yo Johnson, R.P. and Bradford, M.A, (1983) Distortiona lateral buckling of unstiffened Hall, Englewood Cif, N)

composite bridge girders, in Proceedings, Michael R. Horne Conference on the 4. Ings, NL and Trahair, NS. (1987) Ream and column buckling under directed loading,

Instability and Plastic Collapse of Steel Structures, Granada, London, pp 569-80 Journal of Structural Engineering, ASCE, 113 (6, 1251-63,

31 Seah, LK. and Khong, PM. (1990) Lateral-torsional buckling of channel beams 55. Bolotin, V.V. (1963) Nonconsevatie Problems of the Theory of Elastic Stabi,
Journal of Constrctional Steel Research, 17() 265-82. Pergamon Press, New York

32. Cheung, .X. (1976) Finite Strip Method in Structural Analysis, Pergamon Pres, New 56. Ziegler, H. (1968) Principles of Structural Stability, Blaisdell Publishing Co, Waltham,
York MA
Hancock, G., Bradford, M.A. and Trahair, NS. (1980) Web distortion and flexural 57, Dabrowski, R. (1991) Two examples of instability under follower load, Journal of

torsional buckling. Journal of Structural Engineering, ASCE, 106 (7), 1551-71 CConstrutional Steel Research, 192), 153-61,
34. Bradford, MA. and Trahair, NS. (1982) Distorional buckling of thin-web beam.
columns. Enginering Structures, (I) 2-10,
453) Chin, CK., Al-Bermani, FGA. and Kitipornchai, 8 (1992) Stability of thin-walled
ange shape and flexible web. Engineering Structures, 14

36) Bradford, M.A. (1992) Lateral-dstrtional buckling of steel section members.
Journal of Constutional Steel Research, 23 1-3) 97-116
Trahair, NS. and Wooloock, ST, (1973) Ele of major axis curvature on beam
stability. Journal ofthe Engineering Mechanics Division, ASCE, 99 (EM) 85-98
38. Vacharajtiphan, P, Wooleock, ST. and Trahair, NS. (1974) Effect of in-plane
deformation on lateral buckling. Journal of Structural Mechanic, 3 (1), 29-60.
39. Roberts, EM. and Azizian, 2. (1983) Influence ofpre-buckling displacements on the
clastic critical loads of thin-walled bars of open crosssetion In
Pi, YL. and Trahair, NS. (1992) Prebuckling defections and lateral buckling-theory
Journal of Structural Engineering, ASCE, 118 (1), 2949-66,
41. Pi, Y.L and Trahair, NS, (1992) Prebuckling deflections and lateral buckling-appli
cations. Journal of Structural Engineering, ASCE, 118 (11), 2967-85
42. Wooleock, ST. and Trahair, NS. (1974) Post-buckling behaviour of determinate
af the Engineering Mechanics Division, ASCE, 100 (EM2) 151-71
Zndedn, MeGraw

beams, Jou
43, Timoshenko, P. and Gere, LM. (1961) Theory Elsie Stability
Hil, New York.
44. Masur, EF. and Milbradt, K.P. (1957) Collapse strength of redundant beams after
lateral buckling. Journal of Applied Mechanics, ASME, 24 (2), 283-8.
45. Woolcock, ST. and Trahair, NS. (1975) Post-buckl
beams. Journal ofthe Engineering Mechanies Division, ASCE, 101 (EM4) 301-16
46. Woolcock, ST. and Trahair, NS (1976) Post-buckling of redundant I-beams Journal
Engineering Mechanics Division, ASCE, 102 (EM2) 2
'W. (1967) Viscoelsiciy,Blisdel Publishing Co, Waltham, MA.
48. Booker, JR. Frankham, BS. and Trahair, NS, (1974) Stability of visco-clasti
structural members. Ciil Engineering Transactions, Insitation of Engineers,
Australia, CEI6 (1), 45-5

of redundant rectangular

173. 17 Appendices

1711 In-plane bending

17.1. MEMBER AND LOADING

‘The member of length L shown in Figure 17.1 is of doubly symmetric eross-
section. The x, y principal centroidal axes are defined by

| zaa= | yaa= | »a4=0. ann

The following cross-section properties are defined

(172)

The member is acted on by end loads Q,, Q, >, 0,1. 2 and distributed loads per
nit length q,.q, which act at the centroid, and moments M,,, Ma. These actions
move with the cross-section ofthe member, but remain parallel to their original
directions and planes defined by the original Y, Z axes of the straight membe

The centroid of a cross-section undergoes small deflections o, w parallel to the
Y, Z axes under the applied actions, as shown in Figure 17.2. The cross-section

rotates — approximately (‘= d/d:), and the z axis has an approximate
curvature — 0
The displacements rp, ws in the Y, Z directions ofa point Pin the cross-section
which has coordinates x, y are given by
wp (73)
i (7a)

‘The longitudinal normal strain sat P parallel to the z axis may be defined in
terms of the rates of change ofthe deflections, 1 along the element, as shown in

174. an % so that
RER = à: we
E am
Y The longitudinal normal stress at the point P in the cross-section is given by

UsowydaS "LS

in which Nis the internal tension and M,istheinternal bending moment, whence

ty E A | N = EA(W +272] an

M,=-El, S

7
| JE eu and 17.8, Thus
fe >
% p= N/A + Map, (741)

Figure 172 In-plane displacements, a
17.1.5 TOTAL POTENTIAL

= Z The total potential U (see section 23) ofthe trained length L of the member and

Ur
in which U is the strain energy stored in the member, and V isthe potential energy

pP = wpóz ‘of the loading system measured from the straight position of the member

A |
E ye 2 EIRE way

and when equations 17.7, 178, 17.10, and 17.11 are substituted, this becomes

1 i a
u LEAG + 02/2)? + EL) di 744)
Y |

nal normal strain at P. after using

jons 17.1 and 17,

175. 328 Appendices

The potential e
expressed as

gy V of the loading system (see section 22.5) may be

| Got amde— 500+ Ow— Ma) (1715)

17.16 VIRTUAL DISPLACEMENTS AND EQUILIBRIUM

Suppose the member undergoes a set of virtual displacements de, öw from an
equilibrium position 1, w while under the action of constant loads and moments.
For equilibrium of the positon e, w, the principle of stationary total potential

‘equivalent to the principle of virtual work (see section 2.4) requires that

8U;=0 (1716)

for all sets of virtual displacements do, dw
Substituting equations 17.12, 17.14, and 17.15

[toe (egw + 02/2) + Den + an {BAW +

09, — öng,)de

— F600, + 800, — M) =0. MA)
Integrating by parts leads to

[coter — tz ~4,) +00 N ale

+ [60(Nv) — dí M, + 5uM;, + GwN IE — 3800, + dn, —5vM,) =0

0718)
after substituting equation 17.10,

Since this must hold for all admissible sets of virtual displ
then

ements dr, dv,

Mi (No) =4,, (17.19)
N'=3, (1720)

Which are the differential equilibrium equations, and

UM + Nu,

(1720)

For the stability of an equilibrium position defined by v, w to be neutral (see
sU;=0

‚ments (dv, dv) which takes pl

for any set of small buckling displa
equilibrium position under constant loads and moments, In this case, the
adjacent buckled position [ + 52, w + 44) is also one of equilibrium.
Using equation 17.15 and noting that the loads 4, 4, 0, Q, and moments M,
remain constant and that the second variations of, w, and «vanish so that
18V =0 (172:

then equation 17.22 for neutral equilibrium leads to

[zac + 2640 vo + EL

in which {vw} = (60, 0w) are the buckling displacements

In the special case of inextensional buckling for which the stress resultants N

=0 (1726)

and if the pre-buckling transverse disp 0 (Le. 9,4, M, are

zero), then equation 17.25 becomes

NC} de =0 a

for infinitesimal buckling deflections ty. This is the energy equation for
inextensional buckling. The sig

convention for N in this eq
positive, and so N is nega

e for a member in compression.

LS EQUILIBRIUM OF THE BUCKLED POSITION
Because the buckled position {0 + 019 + w,) is one of equilibrium, this position
can be found by considering a set of virtual displacements doy, ów from it. Thus
using equation 17.17

[ Cou, (EA bw + ++ 21 +00) + GG EL (o +05)

+ bw, [EA + mk + (0 + 093/2)) — Bq, —óm9,Jdz

Y (60,0, +50, ~ 54M.) =0. (1728)

176. 330 Appendicos

Equation 17.17 can also be applied to the prebuckled equilibrium position {o, w

LEAG + 03/2} + 6 EL a" + Sw EA(w + 72/2) — doy,

E (60,0, +50, — 604M.) =0.

Subtracting equation 17.29 from equation 17.28 leads to

Lx EA(w

) + dc El de =0

assuming that the buckling disp fe infinitesimal

Integrating by parts leads to

N + öm{ (EA

[ont + Lag" (8

iN) + Bo EALw, + og) + GEL ax) — do VEL Y

Su (BAW, + 00) 5 =0 ara)
after substituting equation 17.10. Since this must hold for all admissible sets of
virtual displacements {öv, Öw,), then

(ELY (Ne) EA + 0050) =0 ur.

EA + ve} = 0 az.

which are the buckling differential equilibrium equations, and
LE) + Nu, + EA + 0) ou = 0]

[Elio =0 | (1734)
TEA + vou =0)

which are the boundary conditions for these equations.
These buckling equilibrium equations can also be obtained from the

g to which the functions x, w, which make

Inplane bending 331

Thus using the energy equation of equation 1725in place of equation 17.35 leads
also to equations 1732 and 17.33
In the special case for which the pre-buckling transverse displacements »

remain zero, equation 17.32 simplifies to
(ELY — =0 (1738)

which isthe differential equilibrium equation
two of the boundary conditions of equation Y

or the buckled position, The first

34 reduce to
(ELY + Noos = 04)
[Ellon =0.5

which are the boundary conditions for equation 17.3.

19 ORTHOGONALITY RELATIONSHIPS

The differential equilibrium equation for the buckled position (equation 17.38)
an be written as

(en,

ang =0 (1740)

in which N now represents an initial distribution of axial stress resultants and 2 is
à load factor. This equation has an infinite number of distinet solutions (eigen-
functions) tp each of whichis associated with a load factor (eigenvalue) , These

eigenvalues may be arranged in ascending order so that
<<< <A

The equality of equation 17.40 can be used to show that

y)Jdz=0, (1741)

A No MERE) N

[tte

whence

BLEI aj, AgNO} — fo EL o, + 0, (1742)

after integration by parts and substi
equation 17.3.

Thus

tion of the boundary conditions of

Nuit? =0 (1744)

177. 332 Appendices

A similar argument starting from

an be used to show that

ral (EL [2p NOS) = tel (EL) Py — (No

El vi, de =0, (17.46)

when J, # À. Equations 17.44 and 1746 are the orthogonality relationships

with equation 17.8.

17.110 RAYLEIGH'S QUOTIENT
The en ation for inextensional buckling (equation

1727) can be

Ufer + ANG) de =0, (an

in which N is again an initial distribution of axial stress resultants and J, is one of

the load factors at buckling. Thus

- (17.48)
By writing

In (1749)
then

fi (17.50)

An approximation

oo = La) ms)
forthe buckled shape o,, associated with the lowest eigenvalue 2, may be formed

from the eigenfunctions », and a set of arbitrary multipliers a, The correspond:

(17:32)

Insplane bending 393

which simplifies to

[eE vds

ET

after using the orthogonality relationships of equations 17.44 and 17.46, Thus

a half
a= eit (1754)

substituting equations 17.49 and 17.50, and so
452 (1755)
This result shows that Rayleigh's quotient 2, is never less than the lowest

The in-plane bending behaviour discussed in sections 17.1.1 to 17.18 may also be
presented in a matrix formulation, The normal strain 8, of equation 17.7 can be

expressed in terms ofthe generalized strains (c) by

a= (5) (756
in which
MA! ae
“Af lof
and
SF = {1,9} (17.58)
Equation 17.57 can also be expressed as
(B.+ Bol) ans)
in wich
0) = rw? (e
o 01]
13 761)
ame 0 al (1762)

Tecan be shown t

[5Bq1{®} =[8,J(50) (17.63)

178. 384 Appendices
Generalized stress resultants (0) corresponding to the generalized strains
given by equation 17.57 may be defined by

{8e}"(0}=[ önarda (1764)

(ei = tt) u209
101= | (syre(sjaa (1766)
so that y
EA 0
e] 2 re

The generalized stresses (0) defined in this way are identical with those of

9 and 17.10,
energy component U of 4

The str

total potential U

y given by

after using equations 17.56, 17.65, and 17.66,

The potential energy component V of the total potential Uy given by
equation 17.15 can be expressed as.

v ef" {a} de — E fo)" (0} (1769)
in whieh ! +

0) = {9.4.07 (17.70)

(0)=(0,0.M,)"

when q Oy M,
can be expressed in the form of

roid. When 4, 9, act away from theo

[AU = Ag] 0 (17.73)

[546] (0) = [Aa] {50}

The virtual work equilibrium requirement of equation 17.16 can be expressed

[có to) — (69) (a))42— E (5e}" {0} =0. (1776)

6e) =[B, +28.) (50)
and

(50) = [Ay + 24] (50) (17278)

which make use of equations 17.59, 3, and 17.75

I is now assumed that the displacements v, w can be expressed in the form of
wy = EN] (3) (17.9)
in which

5 wa)? (1780)

and the elements of [N] are functions of =. In this case, (0), (0), and

(0) = LN] (3 (st)

= Ng] (1782)

a)" CAT (1783)

Substituting these relationships and equations 17.77 and 1778 allows
a | (UNITS, +2861" fo} EN ITA + 24a" fa} e

No Tia + 24001 (0102) )=0. (1784

this must hold for all sets of virtual deformations defined by (55), the

[corto + 2807" (@) INTA, + 2407 (a)

NJ Tte (OT, 08)" = (0 (1785)
which are the non-linear equilibrium equations.
The neutral equilibrium condition of equation 17.22 is equivalent to
+ (5%}"o}) — (Fe) {ad o «789

in whieh
{6%} = [2584] (60) (787)

179. 336 Appendices
and

50} = (0 (1788)
Using equation 17.81 with equations 17.63 and 17.65 leads to

$8)"[K,](88) =0 (1789)

LK) =0 (1790)

in which

Ue] = [EN J'TE, +28,I'TOJ(B, +284]+[MDINJdz (1791)

h makes use ofthe identity

L6BoT (o}

N00
ma=|o 00 (1793)
000

The neutral equilibrium condition of

in whieh

Kıl=o (1794)

“and the load sets which satisfy this equation are the buckl
Because the adjacent buckled position is one of equilibrium, the condition
brium may also be obtained by using virtual work to analy

for neutral equi
equilibrium ofthe bucklı
the buckled position e +

position. Thus if 6, 5 ae virtual displacements from

er + my. then,

Bo + Boy] LN,
ICh)

(00) =[B,

(1795)

and

= (0) + (DUB, +28,J00.J(5,) (1796)

Substituting into the virtual work equilibrium condition of equation 17.76 leads

55)"( |, (UNTER: +2B4J" fo} —ENJUTA, + 2407" (4))dz

[No Lie + 24901" (10,)%,(0,)"7"

| NET" (0) + CB + 2807 LIEB, +2B0]0N

l'agent

Uniform to

which uses the identity [24qa]"() = [Mg] EN] 3g}, which is similar to equa
tion 1792, If equation 17.84 for the equilibrium of the pre-buckled position is

substituted, then this simplifies for infinitesimal buckling displacements to

DESTE

o (1798)

d since this must hold for all sets of virtual displacements (35), then
[Kel{5,} = (0} (1799)

which are the buckling equilibrium equations.

17.2 Uniform torsion

1
The n
thickness, 1, and length L. The member is acted on by uniform end torques
Mu M and distributed uniform torques per unit length m,. Any warping
torques M, M, and warping torques per unit length m, (see section 17.3) are
assumed 10 be

row rectangular section member shown in Figure 174 is of width 6,

ligible

Figure 174 Uniform torsion

180. 338 Appendices

E

A member cross-section rotates through a small angle $ about the Z axis, as
shown in Figure 17.5. When the distribute
member, so that the twist = do

he X, Ydirections of

a point Pin the cross-section which has coordinates x, . These displacements are
p= = 49 (17.100)

The point P also displaces wp in the Z direction, where w is a function of the
coordinates x, y, so tha

w= VD) (17.102)

The first-order shear strains in the member are related to the displacements tp,

vp through

el + Gul (17.103)
so that
0, )
WELT — ») | (17.104)
ero)
4 SHEAR STRESSES AND UNIFORM TORQUE
The shear stresses are related to the shear strains through
t= Gi
=} (17.105)
B= Gyn $
in which G is the shear modulus of elasticity, so that
COTE)
») (17.106)

Gele.

While the shear stress distributions may be obtained by solving the equilibrium

o (17.107)

for the warping function W(x,y) these are more commonly found in terms of a

stress function Ox, ) [1] defined by

= co, |
(17.108)

=~ 20/ax.§
Contours of
Figure 17.6.
‘maximum shear stresses, while the spacing between the contours is inversely

m shear stress for a rectangular section are shown in

ts to the contours indicate the directions of the
oportional to the maximum shear stress magnitude
gular section, the stress function is approximated

0x Gel /4—x2) (17.109)
The corresponding shear stresses are given by
o |

(17.110)
t= 690%)

so thatthe shear stress contours become st

ght lines as shown in Figure 17.6.

181. 10 Appendices

el

ile his approximation forthe rs function quit acurate over mas of
here the

the narrow rectangular section, itisin error near the ends ofthe section,
shear stresses become parallel to the x axis, as shown in Figure 17.6a. The
se shear stresses leads toa serious underestimation ofthe uniform.

omission oft
torsion stress resultant M,,

A more accurate approximati
shown in Figure 17.6¢ [3], which leads to shea

ined from the ‘mitred’ contours

ogr(?=2-1) azım

in the end regions. The torque resultant ofthese and the shear stresses, is given

by
M,=6)6 (TE)

in which J is the torsion section constant approximated by

(vt az
Us
The maximum shear stress
an Got arm
M. (17.115)
J

More accurate approximations may be obtained from the numerical solutions

Uniform torsion 941

reported in [1] as

2 (10674! +006! 17.116)
ania FO (
and
M,
+ 7 an
GPA — 0.66675 + 028827)
1725 OTHER THIN-WALLED OPEN SECTIONS
For very narrow rectangular section, ¢/b+0, and so
Jeep. az)

Other very thin-walled open sections can be considered as be
series of very narrow rectangular sections, so that

composed of a

Zorn, (7.419)

The use of this relationship in equations 17.112 and 17.115 will lead to quite

accurate solutions, except for the shear stresses at junctions, where re-entrant

Corrections for finite thickness, end effects and junction effects for other

thin-walled open sections can be obtained from [4].

1726 TOTAL POTENTIAL, VIRTUAL ROTATIONS, AND EQUILIBRIUM

The total potential U; (see section 2.3) ofa twisted member and its loading system
is given by

U;=U+H (17.120)

in which the strain en

(72

anız

Substitu

leads to

ing equations 17.110 and 17.111 into equation 17.

va! (capt ant

When the member under otations 66 from an equilibrium
position $ while under the action of constant torques, the principle of virtual

182. M42 Appendicos
work requires that

6U,=0 (17.124)

(G10) - 56m,}d2—T 56M, =0 (17.125)

and integrating by parts leads to

| 564-6169 mas + 8610114 56M, =0. (11.126)

Since this must hold for all admissible sets of

(G26) =m anız
and
(= Mas]
i (17.128)
(= Mu
Equation 17.127 is the first uilibrium equation for uniform

totsion, and equations 17.128 are the boundary conditions.

17.3 Warping torsion

The member of length L shown in Figure 17.7a is of thin-walled open cross
section. The principal centroidal axes x,y of the cross-section are shown in
Figure 17.7b, The positions ofthese axes are defined by equation 17.1

The position of the shear centre (xo, yg) of the cross-section shown in

re 17.7b is defined by the conditions

o, (17.129)

(17.130)

in whic
centre to the mid-thickness tangent, a

thickness line. Equation 17.130 satisfies

@dA=0, (7431)

IN N
SR =
K Ne > |

The member is acted on by end warping

orques My, M, and end bimoments
B,, By, and distributed warping torques per unit len
Figure 178, It is assumed that

‘Mey as shown in

torques (section 17.2) are negligible

183. 344 Appendices

A member cross-section rotates about the shear centre axis through S(xo, Yo)
small, so that

shown in Figure 178, It is assumed that any shear strains

warping due to shear can be neglected. In this ease, the longitudinal warping
displacements are due to twisting alone.

As a result of the rotation ¢, a line through a point P(x, y) ata distance

45 = (lx (17133)

n the shear centre rotates through an angle ayd¢/de as shown in Figure 173,

an element of the wall

=p Los an.
de

as shown in Figure 179, Integration leads to

Edi (17135)

Ifthe arbitrary constant of integration wo is chosen so thatthe average warping
displacement is zero, then equation 17.135 becomes
a
m=o (17.136)
de

ion 17.130 which satisies

in which « is the warping function given by eq
‘equation 17.131

Figure 179 Warping of an element da dx duo to tw

ñ

The first-order longitudinal normal strains at the point P resulting from the

warping displacements wp are

4-04 (17.137)

All shear strains are neglected

17.34 STRESSES AND STRESS RESULTANTS

ep
r= Etr (17.138)
Because of equations 17.129 and 17.131
{ oda =0,|

[ arxaa=0,) aras)

{ opyaa=o]

and so the axial force and bending moment stress resultants of are all zero. The
non-zero stress resultant of 0 is the bimoment

B= | oyoda=E1,4' (17.140)

Although the shear strains have been assumed to be so small that they can be

neglected, they are generally not zero, and correspond to warping shear stresses

tp which are induced by variations of the longitudinal normal stresses 0 along
the length ofthe member, inthe same way that bending shear stresses are induced

by variations of the bending normal stresses along the member. Thus

Eg" [tds azıa)

The warping shear stresses tp have a warping torque stress resultant

Mu= | potetds (17.142)

which can be expressed as [
Mu= (E16) (17.143)
Substituting equation 17.140 leads to

M,=—B (17.144)

184. 346 Appendices

The total potential U (se section 2.3) ofthe twisted length of then
loading system is given by

U¡=U+V (17.145)

(17.146)

and the potential
position ls given by

Substituting equations 17.132, 17.137, and 17.138 into equation 17.146 leads to

(17.148)

Whi
position $ whi
pi

n the member undergoes a set of virtual n

56 from an equilibrium
der the action of constant torques and bimoments, the

nciple of virtual work requires that

SU¿=0 (17.149)

| (0¢"E1.0°—d4m,}d2—¥ (66M, +54 B)=0 (17.150)

and integrating by parts leads to

SOLEIL. OY -m,)dz + [56'E1,0" —SH(EL, 67

Y (66M, +54'B) =0. azıs)

Since this must hold for all admissible sets of virtual rotations ö

(EISY=m, (17.152
and

(EL Ó Y = Mya

Ed =M,

(EL), = Bs, (17133)

(EL,4") = B, )

jonal buckling 347

Energy equations for flexural-o

Equation 17.152 is the first-order differential equilibrium equation for warping

torsion, and equations 17.153 are the boundary conditions,

‘gy equations for flexural-torsional buckling

1741 MEMBER AND LOADING

The member of length L shown in Figure 17.7a is of thin-walled open cross:

section. The positions of the principal centroidal axes x, y of the cross-section

Figure 1710 End actions