STABILITY OF BEAMS WITH DISCRETE LATERAL RESTRAINTS

STABILITY OF BEAMS WITH

DISCRETE LATERAL RESTRAINTS

A thesis submitted to Imperial College London for the degree of

Doctor of Philosophy

by

Finian McCann

B.E.(Civil) M.Sc. D.I.C.

Department of Civil and Environmental Engineering,

Imperial College of Science, Technology and Medicine

London SW7 2AZ, United Kingdom

Declaration

I confirm that this thesis is my own work and that any material from published

or unpublished work from others is appropriately referenced.

Signature:.........................................................................

1

Abstract

The current work analyses the lateral stability of imperfect discretely-braced steel

beams using variational methods. To facilitate the analysis, Rayleigh–Ritz ap-

proximations are used to model the lateral deflection and the angle of twist. The

applicability of the methods is initially demonstrated for the cases of unrestrained

and continuously restrained beams by comparison with both analytical and nu-

merical solutions of the governing differential equations of the respective systems.

The method is then applied in full to the case of a discretely-braced beam. Ini-

tially, it is assumed that the degrees of freedom (DOFs) can be represented by

single harmonics; this is then compared to the more accurate representation of

the DOFs as full Fourier series. After carrying out a linear eigenvalue analysis

of the system, it is found that the beam can buckle into two separate classes of

modes: a finite number of modes, equal to the number of restraints provided,

which involve displacement of the restraint nodes and interaction between dis-

tinct sets of harmonics, and an infinite number of single harmonic internodal

buckling modes where the nodes remain undeflected. Expressions are derived for

the elastic critical moment of the beam, the forces induced in the restraints and

the threshold stiffness, i.e. the minimum stiffness required to enforce the first

internodal buckling mode, whereupon the beam attains its maximum achievable

critical moment. The analytical results for the critical moment of the beam are

validated by the finite element program LTBeam, while the results for the de-

flected shape of the beam are validated by the numerical continuation software

Auto-07p, with very close agreement between the analytical and numerical re-

2

sults. Design formulae, from which practical design rules can be developed, are

given for the critical moment, restraint force and threshold stiffness. The design

rules return values close to those predicted from theory. When compared against

equivalent design rules developed based on analogies with column behaviour, it

is found that the column rules are generally overly conservative for restraints

attached close to the compression flange and considerably unsafe for restraints

attached close to the shear centre.

3

Acknowledgements

First and foremost, my most heartfelt gratitude is extended to my project su-

pervisors, Dr. Leroy Gardner and Dr. M. Ahmer Wadee, who throughout my

four-and-a-half years at Imperial College as both an M.Sc. and Ph.D. student

have always made themselves available to respond to my concerns, have always

been there to guide my progress and, most importantly, have always been friendly,

patient and genuinely interested in my work. I honestly could not have asked for

better supervisors and I am deeply indebted to both of them for their guidance.

I extend my gratitude also to my undergraduate lecturer at NUI, Galway, Dr.

Piaras O hEachteirn, who first awakened my interest in structural steel design.

I would also like to thank the Engineering and Physical Sciences Research Council

and the Department of Civil and Environmental Engineering at Imperial College

for funding my research (the former through project grant EP/F022182/1). I

extend a huge thank you also to Ms. Fionnuala Nı Dhonnabhain who from my

first day in the department has always been a friendly face and a great help.

I thank the members of Imperial College Gaelic Athletics Club for making my

time at Imperial completely unforgettable; the fact that the club has developed

so much from such humble beginnings is an immense source of pride for me.

I thank my parents Frank and Imelda and my brothers Fergal and Ciaran for

their unwavering support throughout all my academic endeavours.

4

Finally, for all the support and affection that remained constant throughout all

my research at Imperial College, my special thanks must go to my girlfriend Anna

Symms.

5

Contents

1 Introduction 26

1.1 Outline of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.1.1 Overview of stability analysis . . . . . . . . . . . . . . . . 29

1.1.2 Demonstration of analytical methods . . . . . . . . . . . . 29

1.1.3 Analysis of beams with discrete restraints . . . . . . . . . 30

1.1.4 Validation of discretely-braced beam model . . . . . . . . . 30

1.1.5 Design formulae . . . . . . . . . . . . . . . . . . . . . . . . 31

1.1.6 Worked examples . . . . . . . . . . . . . . . . . . . . . . . 31

1.1.7 Conclusions, findings and further work . . . . . . . . . . . 32

2 Overview of stability analysis 33

2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.1.1 Definition of stability . . . . . . . . . . . . . . . . . . . . . 35

2.1.2 Equilibrium paths . . . . . . . . . . . . . . . . . . . . . . . 37

6

2.2 Treatment of stability in design codes . . . . . . . . . . . . . . . . 39

2.2.1 Buckling resistance . . . . . . . . . . . . . . . . . . . . . . 41

2.2.2 End restraints . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.2.3 Lateral restraints . . . . . . . . . . . . . . . . . . . . . . . 43

3 Analytical methods 46

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.1.1 Calculus of variations . . . . . . . . . . . . . . . . . . . . . 47

3.1.2 Harmonic analysis . . . . . . . . . . . . . . . . . . . . . . 47

3.2 Unrestrained beams . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2.1 Previous results . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2.2 Perfect case . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.2.3 Imperfect case . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.2.4 Postbuckling . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.3 Continuously restrained beams . . . . . . . . . . . . . . . . . . . . 65

3.3.1 Previous work . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.3.2 Potential energy formulation . . . . . . . . . . . . . . . . . 66

3.3.3 Harmonic analysis . . . . . . . . . . . . . . . . . . . . . . 69

3.3.4 Moment–stiffness curves . . . . . . . . . . . . . . . . . . . 70

7

3.3.5 Forces induced in restraint . . . . . . . . . . . . . . . . . . 72

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4 Discrete restraints 76

4.1 Previous work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.2 Single harmonic representation . . . . . . . . . . . . . . . . . . . . 84


4.2.2 Rayleigh–Ritz analysis . . . . . . . . . . . . . . . . . . . . 86

4.2.3 Critical moment . . . . . . . . . . . . . . . . . . . . . . . . 87

4.2.4 Threshold stiffness . . . . . . . . . . . . . . . . . . . . . . 88

4.2.5 Deflected shape and restraint forces . . . . . . . . . . . . . 94

4.2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.3 Full harmonic analysis . . . . . . . . . . . . . . . . . . . . . . . . 96


4.3.2 Equilibrium conditions and mode separation . . . . . . . . 98

4.3.3 Deflected shape for node-displacing modes . . . . . . . . . 100

4.3.4 Critical equilibrium conditions . . . . . . . . . . . . . . . . 103

4.3.5 Restraint forces . . . . . . . . . . . . . . . . . . . . . . . . 119

4.3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

8

5 Validation 123

5.1 Validation of critical moment results . . . . . . . . . . . . . . . . 124

5.1.1 LTBeam software . . . . . . . . . . . . . . . . . . . . . . . 124

5.1.2 Comparison with LTBeam results . . . . . . . . . . . . . . 126

5.2 Validation of results for deflected shape . . . . . . . . . . . . . . . 136

5.2.1 Description of analysis method . . . . . . . . . . . . . . . 136

5.2.2 Results for lateral deflection . . . . . . . . . . . . . . . . . 139

5.2.3 Results for angle of twist . . . . . . . . . . . . . . . . . . . 146

5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6 Design formulae 148

6.1 Auxiliary calculations . . . . . . . . . . . . . . . . . . . . . . . . . 149

6.1.1 Approximations for κ . . . . . . . . . . . . . . . . . . . . . 149

6.1.2 Conditions for “fully restrained” design . . . . . . . . . . . 151

6.1.3 Modifications for non-uniform moment . . . . . . . . . . . 152

6.2 Design formulae for threshold stiffness . . . . . . . . . . . . . . . 153

6.3 Approximations for critical moment . . . . . . . . . . . . . . . . . 156

6.3.1 Single restraint . . . . . . . . . . . . . . . . . . . . . . . . 157

6.3.2 Two restraints . . . . . . . . . . . . . . . . . . . . . . . . . 159

9

6.3.3 Three or more restraints . . . . . . . . . . . . . . . . . . . 160

6.4 Restraint forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

6.4.1 Calculation of K∞ . . . . . . . . . . . . . . . . . . . . . . 167

6.4.2 Calculation of (F/P )∞ . . . . . . . . . . . . . . . . . . . . 168

6.4.3 Modifications for µ < 1 . . . . . . . . . . . . . . . . . . . . 173

6.5 Optimisation of stiffness and strength requirements . . . . . . . . 175

6.6 Comparison with equivalent column rules . . . . . . . . . . . . . . 176

6.6.1 Critical moment for beams with a single restraint . . . . . 177

6.6.2 Threshold stiffness . . . . . . . . . . . . . . . . . . . . . . 178

6.6.3 Restraint forces . . . . . . . . . . . . . . . . . . . . . . . . 178

6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

7 Worked examples 183

7.1 Example 1: Design of lateral restraints . . . . . . . . . . . . . . . 183

7.2 Example 2: “Fully restrained” beam . . . . . . . . . . . . . . . . 186

7.3 Example 3: Partially-braced beam . . . . . . . . . . . . . . . . . . 190

8 Conclusions, findings and further work 193

8.1 Conclusions and findings . . . . . . . . . . . . . . . . . . . . . . . 193

8.2 Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

10

A Proofs of identities 198

A.1 Summation of sin2 n terms . . . . . . . . . . . . . . . . . . . . . . 198

A.2 Summation of sinn sinm terms . . . . . . . . . . . . . . . . . . . 200

A.3 Finite-termed representation of S2 . . . . . . . . . . . . . . . . . . 201

11

List of Figures

1.1 Distribution of stresses in a doubly-symmetric beam bending about

its major axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.2 Demonstration of lateral-torsional buckling, involving the lateral

displacement of the shear centre u and the angle of twist of the

cross-section φ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.1 Model of a simply-supported axially-loaded strut. The dashed line

denotes the configuration of the strut at buckling. . . . . . . . . . 34

2.2 Typical plot of the variation of the total potential energy V with

a generalised coordinate Q. At local maxima and minima where

dV/dQ = 0, the system is in equilibrium. . . . . . . . . . . . . . . 36

2.3 The “rolling–ball” analogy of stable and unstable equilibrium. . . 37

2.4 Equilibrium paths for different types of structural systems: a)

Stable-symmetric, b) Unstable-symmetric, c) Asymmetric. The

thick lines correspond to the idealised perfect case. The finer lines

are imperfect equilibrium paths. The circles denote points where

the stability of the system changes. . . . . . . . . . . . . . . . . . 40

12

2.5 Lateral-torsional buckling curves as defined by BS 5950, AS 4100

and EN 1993-1-1 for rolled sections, where χLT is the moment

reduction factor and λLT is the generalised slenderness. . . . . . . 42

3.1 Cross-sectional geometry, degrees-of-freedom, loading configura-

tion and system axes of models studied in the current work . . . . 49

3.2 Values of κ for UB sections across a range of span-to-depth ratios 54

3.3 Where the curvature χ of the beam is equal to initial curvature

χo, the beam is said to be in a strain-relieved state (Thompson &

Hunt, 1984). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.4 Cross-sectional geometry of beam with an initial lateral imperfec-

tion uo, the dashed lines indicating the position of the perfect case. 58

3.5 Normalised load-deflection curve corresponding to the nth harmonic 60

3.6 Postbuckling paths for an unrestrained beam, as determined by

Rayleigh–Ritz analysis. . . . . . . . . . . . . . . . . . . . . . . . . 64

3.7 Postbuckling paths for an unrestrained beam, as determined by

the numerical continuation software Auto. . . . . . . . . . . . . . 64

3.8 Typical cross-sectional deformation of a continuously restrained

beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.9 Strain energy stored in a linearly-elastic restraint . . . . . . . . . 69

3.10 Moment–stiffness curves for a continuously restrained beam (κ = 5) 71

3.11 Plots of restraint force ratio against restraint stiffness for different

restraint heights and torsion-warping parameters (M = Mcr,2) . . 74

13

4.1 Typical normalised moment–stiffness curve for beams with dis-

crete restraints, indicating the threshold stiffness and associated

threshold moment. . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.2 (top) Cross-section at a restraint node; (bottom) Model of beam

with nb discrete elastic restraints. . . . . . . . . . . . . . . . . . . 85

4.3 Typical critical mode progression for beams with discrete restraints

when assuming single harmonic functions for the DOFs. . . . . . . 88

4.4 Typical variation of γs,T with η when assuming single harmonic

functions for the DOFs for a > alim. . . . . . . . . . . . . . . . . . 90

4.5 ∂γA/∂η > 0 at η = η∗ implies that η∗ < 1/√

2. . . . . . . . . . . . 91

4.6 Sequential critical mode progression for nb = 3 since γs,T increases

for each value of n. . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.7 Critical mode skipping for nb = 6 since γs,T is at a maximum at

n = 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.8 Restraint force ratios for a singly-restrained beam. . . . . . . . . . 95

4.9 Comparison of mode shapes of u as calculated using single har-

monic and Fourier representations of the DOFs for a beam with

three restraints (a = 0.5, κs = 0.31, γ = 17.5, µ = 0.75). . . . . . . 102

4.10 Typical load deflection curve for a beam with discrete restraints

when representing the displacement components as Fourier series

(m = 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.11 Comparison of mode progression behaviour as predicted by Fourier

series and single harmonic representation of the DOFs. . . . . . . 106

4.12 Sequential critical mode progression (nb = 3, a = 0.5, κs = 0.5). . 107

14

4.13 Loss of sequential critical mode progression for a < alim (nb = 3,

a = −0.225, κs = 0.5). The threshold stiffness of the beam is that

associated with m = 1 rather than m = nb. . . . . . . . . . . . . . 108

4.14 Loss of full bracing capability a < aNT (nb = 3, a = −0.25, κs =

0.5). While them = 2 andm = 3 curves intersect µ = 1, them = 1

curve is lower than them (apart from a brief range of stiffnesses)

and lower than µ = 1 and is thus the critical mode. . . . . . . . . 110

4.15 As aNT is only slightly below alim (with a maximum difference of

0.095 at κs = 0 and nb = 1), aNT can be taken conservatively

as equal to alim. The physical consequence of this is that provid-

ing restraint at a level below alim implies that full bracing is not

achievable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.16 As the restraint is moved further from the compression flange, the

stiffness required to brace the beam fully is considerably greater. . 112

4.17 Moment–stiffness curves for a single restraint (a = 0.5, κs = 0.5). . 114

4.18 Moment–stiffness curves for two restraints (a = 0.5, κs = 0.5). . . 115

4.19 Moment–stiffness curves for three restraints (a = 0.5, κs = 0.5). . 116

4.20 Moment–stiffness curves for four restraints (a = 0.5, κs = 0.5). . . 117

4.21 Moment–stiffness curves for five restraints (a = 0.5, κs = 0.5). . . 118

4.22 Moment–stiffness curves for six restraints (a = 0.5, κs = 0.5). . . . 118

4.23 Boxplot of ratios between threshold stiffnesses as predicted by the

smearing technique and those predicted by full harmonic analysis

for a range of restraint numbers. . . . . . . . . . . . . . . . . . . . 119

15

4.24 Restraint force ratio curves assuming single and multiple harmonic

imperfections when loaded at M = MT (a = 0.5, κ = 5, nb = 3). . 121

5.1 Normalised moment–stiffness curves for varying restraint heights,

nb = 1, L = 10.5 m. . . . . . . . . . . . . . . . . . . . . . . . . . . 128


nb = 3, L = 10.5 m. . . . . . . . . . . . . . . . . . . . . . . . . . . 129


nb = 5, L = 10.5 m. . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5.4 Normalised moment–stiffness curves for varying beam spans, nb =

3, a = alim. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.5 Normalised moment–stiffness curves for varying beam spans, nb =

3, a = 1.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.6 Normalised moment–stiffness curves for varying numbers of re-

straints, L = 10.5 m, a = alim. . . . . . . . . . . . . . . . . . . . . 134

5.7 Normalised moment–stiffness curves for varying numbers of re-

straints, L = 10.5 m, a = 1.0. . . . . . . . . . . . . . . . . . . . . 135

5.8 The piecewise stiffness distribution function kf , for a beam with

three restraints, and a restraint width of L/50. . . . . . . . . . . . 138

5.9 Typical graph of u/L against x/L for R2 > 0.999 (L = 7 m, a = 0,

nb = 5, M/MT = 0.676 and K/KT = 0.5). . . . . . . . . . . . . . 140

5.10 Typical graph of u/L against x/L for 0.99 6 R2 < 0.999 (L =

10.5 m, a = 1, nb = 3, M/MT = 0.484 and K/KT = 0.125). . . . . 141

16


8.75 m, a = 0, nb = 2, M/MT = 0.949 and K/KT = 0.875). . . . . 142


12.25 m, a = 0, nb = 3, M/MT = 0.706 and K/KT = 0.25). . . . . 143

5.13 Typical graph of u/L against x/L for 0.90 6 R2 < 0.96 (L = 14 m,

a = 0, nb = 4, M/MT = 0.575 and K/KT = 0.125). . . . . . . . . 144

5.14 Example of a highly divergent graph where R2 < 0.90. A close

match between the curves is not possible due to the highly asym-

metric nature of the curve produced by Auto. . . . . . . . . . . . 145

6.1 A comparison of actual smax values for UB sections with the cor-

responding approximate values as calculated by Equation (6.10). . 152

6.2 A comparison of actual and approximate values of γs,T for nb = 1. 155



6.5 A comparison of actual and approximated values of µcr for nb = 1

and a = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158


and a = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

6.7 Comparison of actual µtr values for various restraint heights a with

the approximate value obtained from Equation (6.22). . . . . . . . 160


and a = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

17


and a = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161


and a = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162


and a = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163


and a = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164


and a = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164


and a = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165


and a = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

6.16 Demonstration of the calculation of minimum stiffness requirement

to achieve a desired restraint force; in this example, the value of

F/P is to be limited to 1.5(F/P )∞, hence a minimum stiffness of

K = 3K∞ is required. . . . . . . . . . . . . . . . . . . . . . . . . . 167

6.17 A comparison of actual and approximated values of γs,∞ for nb = 2

and µ = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

6.18 A comparison of actual and approximated values of γs,∞ for nb = 3

and µ = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

6.19 A comparison of actual and approximated values of (F/P )∞ for

nb = 1 and µ = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

18


nb = 2 and µ = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 172


nb = 3 and µ = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

6.22 Values of µ, assuming the beam is fully braced and there is no

overdesign i.e. MEd = Mb,Rd. The curves depend on the value of

αLT from §6.3.2.2 of the Eurocode, which defines different buckling

curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

6.23 An example of a scenario where K∞ < 0. For positive, i.e. real,

restraint stiffnesses, the value of F/P approaches (F/P )∞ asymp-

totically from beneath. Hence, a reasonably accurate conservative

estimate can be obtained by taking F/P = (F/P )∞. . . . . . . . . 175

6.24 A diagram showing the concept of the initial imperfection and the

lateral deflection of a member being represented as an equivalent

horizontal UDL, qF , in the restraining system. . . . . . . . . . . . 181

19

List of Tables

5.1 Values assumed for the parameters in the validation using LTBeam.127

5.2 Values assumed by the parameters in the validation using Auto. 138

5.3 Distribution of R2 values between analytical and Auto results for

the lateral deflection. . . . . . . . . . . . . . . . . . . . . . . . . . 140

5.4 Distribution of R2 values between analytical and Auto results for

the angle of twist, φ. . . . . . . . . . . . . . . . . . . . . . . . . . 147

6.1 Values of νb,T used for calculating KT . . . . . . . . . . . . . . . . 154

6.2 Maximum, minimum and average percentage errors between actual

and approximate values of γs,T as shown in Figures 6.2 to 6.4.

Negative errors correspond to ranges of restraint height where the

approximate formula underestimates the threshold stiffness. . . . 154

6.3 Maximum and average percentage errors between actual and ap-

proximate values of µcr shown in Figures 6.5 and 6.6. . . . . . . . 157


and approximate values of µcr for two restraints as shown in Figures

6.8 and 6.9. Negative errors correspond to ranges of stiffness where

the approximate formula overestimates the critical load. . . . . . . 162

20

6.5 Maximum, minimum and average percentage errors between ac-

tual and approximate values of µcr for three restraints as shown

in Figures 6.10 and 6.11. Negative errors correspond to ranges of

stiffness where the approximate formula overestimates the critical

load. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

6.6 Maximum, minimum and average percentage errors between ac-

tual and approximate values of µcr for four restraints as shown

in Figures 6.12 and 6.13. Negative errors correspond to ranges of

stiffness where the approximate formula overestimates the critical

load. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163


and approximate values of µcr for five restraints as shown in Figures

6.14 and 6.15. Negative errors correspond to ranges of stiffness

where the approximate formula overestimates the critical load. . . 166

6.8 Values of νb,∞ used for calculating K∞. . . . . . . . . . . . . . . . 168


and approximate values of γs,∞ as shown in Figures 6.17 and 6.18.

Negative errors correspond to ranges of restraint heights where the

approximate formula underestimates the actual value. . . . . . . . 170


and approximate values of (F/P )∞ for as shown in Figures 6.19

to 6.21. Negative errors correspond to ranges of restraint heights

where the approximate formula underestimates the actual value. . 171

6.11 Values of the limiting slenderness, below which µmax < 1, corre-

sponding to the imperfection factors of EC3. . . . . . . . . . . . . 173

6.12 Values of κlim. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

21

Nomenclature

Roman symbols

A Cross-sectional area

a Height of restraint relative to shear centre of cross-section

a Nondimensional restraint height

b Cross-sectional breadth

E Young’s modulus of steel

e Vector of imperfection amplitudes

en Amplitude of nth imperfection harmonic function

F Force induced in a discrete restraint

f Distributed force induced in a continuous restraint

fy Yield stress

G Elastic shear modulus of steel

H Hessian matrix

H Set of interacting harmonics

h Height of cross-section

hs Distance between shear centres of flanges

I Moment of inertia of cross-section

It St. Venant’s torsion constant

Iw Cross-sectional warping constant

Iy, Iz Major axis and minor axis moments of inertia, respectively

iz Minor axis radius of gyration

22

K Elastic stiffness of a discrete restraint

KT Overall threshold stiffness of a restrained beam

KT,n Threshold stiffness corresponding to the nth mode

k Elastic stiffness per metre of a continuous restraint

L Span of member

M Applied moment

Mb,Rd Design buckling resistance moment

Mcr Critical moment of a beam with discrete restraints

(Fourier series representation)

Mcr,n Critical moment of an unrestrained beam for the nth

buckling mode

MK,n Critical moment of beam with discrete restraints for the nth

buckling mode (Single harmonic representation)

Mk,n Critical moment of beam with a continuous restraint for the

nth buckling mode

Mob Critical moment of unrestrained beam

MT Threshold moment

M Nondimensional moment

m Buckling mode identifier

nb Number of discrete restraints

P Compressive load

PE Euler buckling load

p Distributed compressive force in beam flanges

Q Generalised coordinate

r Radius of rotation

S1, S2 Infinite sums

s Restraint spacing

tf , tw Flange and web thicknesses, respectively

U Strain energy

Ub Strain energy due to bending

UR Strain energy stored in the restraints

23

Ut Strain energy due to torsion

Uw Strain energy due to warping

u Vector of displacement amplitudes

u Lateral deflection of shear centre of beam cross-section

uo Initial lateral imperfection

un Amplitude of lateral deflection harmonic

un Value of un normalised by hs/2

V Total potential energy

v Horizontal deflection of shear centre of beam cross-section

Wy Major axis section modulus

X Compression or extension of a restraint

x Ordinate in direction parallel to longitudinal axis of the beam

xt Torsion parameter

y Ordinate in direction parallel to major axis of beam cross-section

z Ordinate in direction parallel to minor axis of beam cross-section

Greek symbols

α Ratio of F/P to (F/P )∞

αtr Value of K/KT for transition from first mode to

second mode buckling

β Ratio of K/K∞

γ Nondimensional discrete restraint stiffness

γc Nondimensional continuous restraint stiffness

γs Value of γ normalised for a restraint spacing s

γs,T Threshold value of γs

∆ Displacement of applied load

δn,m Sign operator function

24

δq Deflection of restraining system

η Normalised mode identifier

Φ Work done by applied load or moment

φ Angle of twist of cross-section

φn Amplitude of nth twist angle harmonic

κ nondimensional beam parameter

κs Value of κ for a restraint spacing s

λ Beam slenderness

µ Nondimension moment, normalised by the threshold moment, MT

ν Poisson’s ratio of steel

σ Correction factor

θ Rotation of applied moment

χ Curvature of a beam

χo Initial curvature of a beam

χLT Lateral-torsional buckling moment reduction factor

Script symbols

L Lagrangian

` Cross-sectional parameter related to torsion and warping rigidities

U Beam parameter

V Buckling parameter

25

Chapter 1

Introduction

As the use of steel in construction continues to propagate around the world, the

need for increased safety, accuracy and efficiency in designing steel structures is

paramount. Modern design codes such as the British BS 5950 (British Standards

Institute, 2000), the Australian AS 4100 (Standards Australia, 1998) and the

Eurocode EN 1993-1-1 (Comite European de Normalisation, 2005) employ a limit

state design philosophy. A limit state defines the situation whereby a structure

fails to perform to its requirements. Typically there are two types of limit state:

ultimate limit states, related to the safety of the structure (and its inhabitants),

and serviceability limit states, related to the functionality and usability of the

structure. When considering an ultimate limit state, the general philosophy is

that the structure should be able to withstand all the loads and combinations

of loads it is expected to be subjected to within its design life. If the structure

or, more specifically, a structural element, is no longer able to fulfil this purpose

then it is deemed to have failed (Comite European de Normalisation, 2002).

When designing structural members, there are a number of ways by which they

can fail that must be considered. Failures can be due to a loss of equilibrium

within the structure, or when the structure transforms into a mechanism upon

creation of a plastic hinge, or when the material itself is fractured. A loss of

26

MM

Tension flange

Compression flange

Deflected shape of beam

C

T

Stress profile

Figure 1.1: Distribution of stresses in a doubly-symmetric beam bending about

its major axis.

ϕ

u

Figure 1.2: Demonstration of lateral-torsional buckling, involving the lateral dis-

placement of the shear centre u and the angle of twist of the cross-section φ.

stability is also another route by which failure can occur. A beam under load

must be able to resist the bending moment produced within the beam and the

shear force created internally. The individual elements that make up the beam

must also be able to resist localised buckling or else their stiffness and hence load-

carrying capacity is compromised. Another phenomenon that must also be taken

into account is lateral-torsional buckling. If a beam is loaded such that it bends

about its major axis, the bottom portion of the beam is in tension and the top

portion is in compression. Figure 1.1 shows this effect for a doubly- symmetric

I-shaped section. This compression can lead to a change in stability that causes

the beam to deflect laterally and also causes it to twist about its longitudinal axis

(see Figure 1.2). The lateral-torsional buckling capacity of a beam is improved

by restricting its ability to deflect laterally and/or to twist. This is achieved

27

through the provision of restraints attached to the beam. The restraints can

be either continuous, e.g. metal sheeting, or discrete, e.g. purlins attached to a

rafter. Additionally, distinction can be made between types of restraint based on

the deformations they restrict. Lateral restraints – which are the subject of the

current work – inhibit a cross-section of the beam from deflecting relative to its

shear centre. Torsional restraints prevent twisting of the cross-section about the

longitudinal axis. Rotational restraints prevent in-plane rotations and warping

restraints prevent rotation of a cross-section about its vertical axis.

As is detailed in subsequent chapters, in the current work, the restraints are

modelled as elastic springs, rather than being rigid, implying that they have a

finite stiffness. The magnitude of this stiffness obviously determines the require-

ments for the design of the restraints but also that of the main member, since

stiffer restraints allow the beam to carry greater loads. Conversely, as will also

be seen, the magnitude of the imperfection of the main member has an effect on

the strength requirements of the restraints.

This concept of restraints possessing both stiffness and strength is fundamental

but it does not seem to be treated satisfactorily in design codes. Therefore, the

aim of the current work is to develop convenient design rules based on analytical

studies, which are themselves validated by numerical methods. The following

section provides a brief summary of the contents of the current work.

1.1 Outline of thesis

A brief summary of the chapters in the thesis is given below.

28

1.1.1 Overview of stability analysis

A brief overview of stability theory is provided in Chapter 2, showing initially the

work of Euler (1744) in finding the buckling load of an axially-loaded strut. A

definition of equilibrium is provided for a single degree-of-freedom system, based

on the total potential energy of the system, which is then extended to multiple

degree-of-freedom systems. The concept of systems being in either stable, un-

stable or neutral equilibrium is presented, and the concept of a bifurcation point

is explained, leading to the definition of critical loads. The distinction between

linear and nonlinear systems is defined and then the various types of postbuckling

behaviour for nonlinear systems is described. The effect of imperfections is then

discussed.

An overview of the treatment of the design of members against instabilities is

covered in the second section, comparing the approaches of the British, Australian

and European codes. Initially, the general approach to designing against lateral-

torsional buckling is detailed then the inclusion of the effects of lateral restraints

is examined.

1.1.2 Demonstration of analytical methods

The two analytical methods used in the current work are presented in Chapter 3,

namely the calculus of variations and the Rayleigh–Ritz method. After detailing

the development of lateral-torsional buckling theory, the potential energy formu-

lation for an unrestrained beam is given. It is then demonstrated using both

of the methods mentioned how to determine the critical moment for a simply-

supported, unrestrained beam. Next, imperfections are included, allowing the

deflected shape of the beam to be determined. Finally, a nonlinearity is included

so that the postbuckling behaviour of the beam can be portrayed. A similar study

is conducted for the case of the beam with a continuous lateral restraint. Rela-

tionships between the critical moment and the restraint stiffness are established

29

and expressions are obtained for the forces induced in the restraining system.

1.1.3 Analysis of beams with discrete restraints

The model studied in Chapter 3 of a beam with a continuous restraint is modified

in Chapter 4 to model a beam with an arbitrary number of discrete restraints

located at regular spacings along its length at an arbitrary height relative to

the shear centre of the beam. After a synopsis of previous work relating to the

subject of discretely-braced beams, an analysis of the stability of such a beam

is carried out. Initially, the degrees-of-freedom of the model, namely the lateral

deflection and the angle of cross-sectional twist, are assumed to be in the form of

a single harmonic function, as has been assumed in previous studies. Expressions

for the critical moment and the restraint force are derived, on the basis of a single

half-sine wave initial lateral imperfection. The progression of the critical buckling

mode with restraint stiffness is examined.

Next, the degrees-of-freedom are modelled using Fourier series so that an exact

solution for the deflected shape may be obtained. The results for the critical mo-

ment and the restraint force are compared with those found for a single harmonic

representation and discrepancies are found and discussed.

1.1.4 Validation of discretely-braced beam model

The results found in the previous chapter are validated using two separate com-

puter programs in Chapter 5. The results for the critical moment are validated

using LTBeam (Galea, 2003), a program that uses the finite element method to

determine the elastic critical moment of a beam with restraints. A comparison

with the results of the current work shows that there is close agreement between

the analytical and numerical methods.

30

Next, the results for the deflected shape are validated using the numerical con-

tinuation software Auto-07p (Doedel & Oldeman, 2009). Using the calculus of

variations, the governing ordinary differential equations of the model are found

and are solved by Auto using numerical techniques. It is again found that there

is close agreement between the analytical and numerical results.

1.1.5 Design formulae

Practical formulae are provided in Chapter 6 to be used for designing a structural

system comprising beams and lateral restraints, based on the results of Chapter

4. A method for determining the optimum number of restraints such that lateral-

torsional buckling can be ignored is outlined. Expressions for the critical moment

of the beam, as well as expressions for the minimum stiffness required to ensure

the restraint nodes do not deflect, are given. Finally, methods to determine the

force induced in the restraints are provided.

These formulae are then compared against equivalent design rules intended for

use with columns, since the current best practice is to base values of restraint

force and required restraint stiffness conservatively on column rules. However, it

is shown that while the rules are conservative for restraints positioned close to

the compression flange, they are in fact unsafe for lower restraint heights.

1.1.6 Worked examples

Three worked examples are presented in Chapter 7 which demonstrate the prac-

tical application of the design rules presented in Chapter 6.

31

1.1.7 Conclusions, findings and further work

In the final chapter, a summary of the principal findings and general conclu-

sions based on the work shown throughout the thesis are presented, along with

suggestions and guidelines for future work.

32

Chapter 2

Overview of stability analysis

The intention of the current chapter is to introduce the analysis of the stability of

structural systems. §2.1 provides a brief overview of the development and main

concepts of stability analysis and the derivation of properties of interest to a

design engineer. The application of these properties by design codes is examined

in §2.2.

2.1 Background

The first investigations into the stability of structural elements were conducted

by Euler (1744), who examined a simply-supported strut under an axial load P

(see Figure 2.1). The investigation was part of a series that Euler performed

that concerned the calculus of variations (Fox, 1987), which is described in more

detail in §3.1.1 and demonstrated in §3.2.2. In his analysis, Euler formulated

the governing ordinary differential equation (ODE) of a simply-supported strut

of length L, which in the notation of the current work can be written thus:

EIu′′ + Pu = 0, (2.1)

33

P

L

P u(x)x

Figure 2.1: Model of a simply-supported axially-loaded strut. The dashed line

denotes the configuration of the strut at buckling.

where EI is the flexural rigidity of the strut and primes denote differentiation

with respect to x, the longitudinal ordinate. The equation makes use of the

approximate expression M(x) = −EIu′′ for the internal bending moment M .

The general solution for the lateral deflection u(x) is given by:

u(x) = A1 sin

(x

√P

EI

)+ A2 cos

(x

√P

EI

). (2.2)

By applying the conditions of zero deflection at the supports i.e. u(0) = u(L) = 0,

a solution is arrived at:

A1 sin

(L

√P

EI

)= 0. (2.3)

Setting A1 = 0 would produce the trivial result representing the pre-buckling fun-

damental equilibrium solution; therefore an expression for the minimum buckling

load is found by noting sin(nπ) = 0. The lowest, or most critical non-trivial value

is obtained at n = 1, resulting in:

PE =π2EI

L2. (2.4)

For loads less than this amount the only other solution to the equations above is

the trivial case of u = 0, so what Euler had found was the minimum compressive

load that causes the once-straight strut to deflect laterally, i.e. the load at which

it loses stability and buckles. This minimum load is known as the Euler buckling

load of the strut and underpins the analysis and design of axially-loaded members.

Further expressions taking different boundary conditions into account can be

obtained in a similar manner.

34

2.1.1 Definition of stability

Structural engineers, when designing a structure or an element thereof, apply the

principle that the body to be designed is in equilibrium under conservative static

loading, meaning that the vector properties of the loads remain constant. A state

of equilibrium can be said to prevail based on a number of different definitions, but

the one that is probably most familiar is that all the various forces and moments

acting upon the body – whether they are applied externally or are reactions –

are resolved such that their algebraic sum is equal to zero, i.e. that there are no

resultant forces or moments acting upon the body. This definition was used by

Euler to solve for PE, as shown in the example in the previous section. Another

more fundamental definition, from which the previous one can in fact be derived,

is that the total potential energy V in equilibrium of the system does not change

with respect to the generalised coordinates of the system. The total potential

energy of a system under load is given by:

V = U + Φ (2.5)

where U is the strain energy stored within the structure and is obtained by

integrating the product of the stress and strain tensors of the structure across its

volume, and Φ is the work done by the applied loads, which is more commonly

denoted as −P∆ or −Mθ. Figure 2.2 demonstrates points of equilibrium in a plot

of V against a generalised coordinate Q of a single degree-of-freedom (SDOF)

system. A generalised coordinate can be perceived as a measure of a related

displacement component of the system under analysis. In mathematical notation

the condition necessary for a state of equilibrium for a SDOF system is:

dV

dQ= 0. (2.6)

In multiple degrees-of-freedom (MDOF) systems, this condition is extended for

all the generalised coordinates, so that if ∂V/∂Qi = 0 ∀ i then the system is in

equilibrium.

Further to this definition comes the classification of states of equilibrium as being

35

Q

V dVdQ= 0

dVdQ= 0

Figure 2.2: Typical plot of the variation of the total potential energy V with a

generalised coordinate Q. At local maxima and minima where dV/dQ = 0, the

system is in equilibrium.

either stable, unstable or neutral. If a system is in stable equilibrium it returns

to this equilibrium state if perturbed slightly. In contrast, a system in unstable

equilibrium deviates significantly from its initial state if perturbed. A system

in neutral equilibrium can be perceived to be stable but on the cusp of insta-

bility. The classic analogy of a rolling ball demonstrating the different types of

equilibrium is shown in Figure 2.3. In the left hand diagram, the ball is at the

bottom of the bowl in a stable position since the ball returns to this position if

it is moved away. In the right hand diagram, the ball, at the top of the bowl

in unstable equilibrium, undergoes a large displacement away from this position

if perturbed slightly. Mathematically, the stability of a system (or lack thereof)

can be established by examining the second derivatives of V with respect to the

generalised coordinates. In a SDOF system:

d2V

dQ2< 0 ⇒ Stable equilibrium.

d2V

dQ2> 0 ⇒ Unstable equilibrium.

d2V

dQ2= 0 ⇒ Higher derivatives must be

examined to classify the equilibrium state.

When a structure loses stability, i.e. when it becomes unstable, the dynamic

process of buckling occurs where the structure deflects noticeably. In the previ-

36

Figure 2.3: The “rolling–ball” analogy of stable and unstable equilibrium.

ous example of a flexible strut under compression, it was seen how initially the

unloaded strut is in a stable state. As the load is increased, the strut remains

stable until the load reaches PE, whereupon the strut loses stability and buckles

laterally. The point at which this transition from stable to unstable equilibrium

occurs is defined by the third condition above, i.e. when the second derivative

vanishes. For MDOF systems, the derivative in the conditions above is replaced

by the determinant of the Hessian matrix of the system, H, a square matrix com-

posed of the partial second derivatives ∂2V/∂Qi∂Qj, so that the condition for

critical equilibrium is det(H) = 0. This is what is known as a bifurcation point

and the lowest load at which such a point exists is the critical load of the system.

In a structural system, this load level defines the maximum load the system can

support before buckling occurs and is thus important to design engineers.

2.1.2 Equilibrium paths

From the definition of the conditions necessary for an equilibrium state it is pos-

sible to determine the deflected state of a structure for a given load or conversely,

given a deflected state, the load applied to the structure, if so desired. These

load–deflection relationships are termed equilibrium paths. In a geometrically

perfect model, it is assumed that there are no deflections in the system when it

is unloaded, i.e. the generalised coordinates of the system Qi = 0 and the strain

energy U = 0 when the applied load P = 0. When analysing structural mem-

bers such as columns and beams in their perfect state, it is found that the total

potential energy formulation does not contain linear terms in Qi due to strain

energies being of the form 12kijd

2ij, where kij and dij represent general components

37

of stiffness and displacement fields, respectively. Generally kij and dij are nonlin-

ear functions of the DOFs of the model but they are often linearised in order to

simplify calculations, such as assuming small deflections or linear elasticity. This

underpins the difference between geometrically linear and nonlinear analyses. In

a nonlinear analysis, upon evaluation of ∂V/∂Qi = 0, the solutions Qi = 0 exist

alongside a function P ≡ P (Qi). However, in a linear analysis, the largest power

of Qi is two and so the solution for P is a constant. In either case, the solutions

Qi = 0 describe a trivial fundamental path.

Upon investigation of the conditions for critical equilibrium, a bifurcation point

of the system is found at the point P = PC on the fundamental path, which

can be obtained directly from a linear analysis of the system. If the loading

history of the system is examined, it is seen that, starting from the unloaded

state P = 0, the load–deflection curve follows the fundamental path with no

deflections present in the system until the critical load P = PC is reached. The

structure subsequently loses stability at this point, begins to deflect and exhibits

its postbuckling behaviour. It is important to note that only through a nonlinear

analysis can information be obtained about the postbuckling behaviour of the

system; a linear analysis can only predict the location of the bifurcation points

on the fundamental path. The postbuckling behaviour can be categorised based

on the stability of the equilibrium path.

• In stable-symmetric postbuckling, the structure is able to carry additional

loads beyond its critical point. An axially-loaded plate is an example of

such a structural system.

• In unstable symmetric postbuckling, after the bifurcation as the deflec-

tions increase the capacity of the structure decreases. A cylindrical shell

in compression is the classic example of a system with unstable-symmetric

postbuckling behaviour.

• In asymmetric postbuckling the stability of the postbuckling path is depen-

dent on the sense of the initial postbuckling deflection. Asymmetric frames

38

are a practical example where this type of response can be seen.

The models discussed previously describe geometrically perfect, idealised struc-

tures. Real structures however contain imperfections which affect their capacity

and load–deflection relationships. It can be shown that these imperfect equi-

librium paths are asymptotic to their perfect counterparts, as the imperfections

lead to reduced capacity. Figure 2.4 shows equilibrium paths for the three post-

buckling scenarios mentioned above, along with the corresponding asymptotic

imperfect equilibrium paths. When an imperfection is included the system does

not bifurcate in the same way. In stable and unstable symmetric systems, the

bifurcation point for the perfect case is known as a pitchfork bifurcation while

in asymmetric systems it is a transcritical bifurcation. With the imperfection

however there is a continuous equilibrium path that contains a limit point where

the transition from stable to unstable equilibrium occurs smoothly; this type of

bifurcation is known as a saddle-node bifurcation.

For further details on nonlinear structural stability, the reader is directed to

Thompson & Hunt (1973), Thompson & Hunt (1984), Bazant & Cedolin (1991)

and Wadee (2007). A summary of the history and development of the subject is

given by Bazant (2000). For an introductory text on bifurcation theory the reader

is referred to Glendinning (1994). A comprehensive treatment of practical issues

regarding bifurcation theory in engineering and associated numerical methods is

given by Seydel (1994).

2.2 Treatment of stability in design codes

When designing a system comprising of a primary beam laterally restrained by

discrete secondary members, e.g. roof rafters restrained by purlins, there are three

issues that must be addressed: (i) the strength of the primary member (i.e. the

beam) to withstand design loads, (ii) the stability of the primary member and

39

Q

P

PC

Q

P

Q

P

Stable symmetric Unstable symmetric

Asymmetric

increasing imperfections



PC PC

a) b)

c)

Figure 2.4: Equilibrium paths for different types of structural systems: a) Stable-

symmetric, b) Unstable-symmetric, c) Asymmetric. The thick lines correspond

to the idealised perfect case. The finer lines are imperfect equilibrium paths. The

circles denote points where the stability of the system changes.

40

(iii) the strength of the secondary members to withstand loads transferred from

the primary member. The first issue is one that is covered comprehensively by

all steel codes. Individually, the second and third issues are generally quite well

understood but when examining a combined system it is clear that design codes

do not treat the issue comprehensively.

2.2.1 Buckling resistance

The theoretical critical load at which a structure buckles is the basis on which

the design of members against instability is based. However, as mentioned in

the previous section, in reality its value does not represent the actual resistance

of the member due to imperfections and nonlinear effects. In modern design

codes, such as the British BS 5950-1:2000 (British Standards Institute, 2000), the

Australian AS 4100-1998 (Standards Australia, 1998) and the Eurocode EN 1993-

1-1 (Comite European de Normalisation, 2005), the influence of imperfections on

the actual buckling load is taken into account through the use of Perry–Robertson

curves defined by a related imperfection factor. These factors are intended to

provide a safe lower bound to experimental results. The work of Beer & Schultz

(1970) led to the formulation of the imperfection factors as well as the form of

the curves themselves. Taras & Greiner (2010) have suggested improvements that

can be made to these curves in terms of reliability and accuracy by including an

extra modification factor related to the plastic and elastic section moduli of the

beam, based on a geometrically nonlinear analysis with imperfections.

In BS 5950, there is a single lateral-torsional buckling Perry–Robertson imper-

fection factor specified for beams, while more distinction is made for compressive

buckling curves for struts with different material thicknesses and cross-section

profiles; however, the form of the curves varies depending on the the yield stress

fy. The formula itself is not provided explicitly in the main body of the code,

but is included in an appendix; pre-formulated tables were provided in the design

code that directly gave the buckling strength of the section.

41

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3

BS 5950 – S275

BS 5950 – S355

AS 4100

EC3 – Curve a

EC3 – Curve b

EC3 – Curve c

EC3 – Curve d

TheoreticalLTχ

λLT

Figure 2.5: Lateral-torsional buckling curves as defined by BS 5950, AS 4100 and

EN 1993-1-1 for rolled sections, where χLT is the moment reduction factor and

λLT is the generalised slenderness.

In AS 4100, there is also a single lateral-torsional buckling formula for members

in bending, from which a slenderness reduction factor is calculated. This factor

is then applied to the in-plane bending resistance. For column buckling, five

member section constants are specified, based on the cross-section profile and

flange thicknesses.

In the Eurocode there are four different curves specified for lateral-torsional buck-

ling and five for strut buckling, the choice of which depends on the height to

breadth ratio and buckling axis of the members. In the case of the lateral-

torsional buckling curves, the differentiation between the curves is only specified

for I-shaped members; for all other sections the most conservative curve is recom-

mended. From equations provided by EC3 a reduction factor χLT is applied to

the in-plane bending resistance of the beam to arrive at the buckling resistance

moment Mb,Rd. Figure 2.5 shows the various curves for lateral-torsional buckling

defined by the codes.

42

2.2.2 End restraints

The effect of end restraint conditions is accounted for in BS 5950 by specifying

effective length factors with which to modify the span of the members; a similar

practice is employed by AS 4100. In the Eurocode, there is no specific method

outlined to account for the end restraint; in fact there is no explicit method

specified for determining the critical moment of the beam. Rather, the code

advises that the value for critical moment should take the loading conditions, the

moment distribution and the lateral restraints provided into account.

2.2.3 Lateral restraints

In treating the influence of lateral restraints, there are some differences between

the codes. The British and Australian codes both state that a cross-section

can be assumed to be restrained laterally if the intermediate restraint at that

section is sufficiently stiff to inhibit any lateral deflection of the compression flange

relative to the supports (§4.3.2.1 in BS 5950; §5.4.2.1 in AS 4100). Examining the

treatment of restraint strength in BS 5950, §4.3.2.2.1 states that the restraints

should be able to withstand a total force of 2.5% of the maximum compressive

force in the beam, divided amongst the restraints in proportion to their spacing.

This is only a provision for one or two intermediate restraints as §4.3.2.2.2 states

that for three or more restraints provided along the span of the beam, each

restraint should be able to withstand 1% of the maximum compressive force (or

the 2.5% force from §4.3.2.2.1 again distributed in proportion to the restraint

spacing). AS 4100 is more onerous, stating that all bracing elements should be

able to withstand a 2.5% force, but allows for distribution of a 2.5% force if

the restraints are spaced closely enough so that the capacity of the beam (or

beam segment) is greater. In both codes, meeting these requirements allows the

designer to divide the beam into segments with pin-joints assumed to exist at the

bracing points. These segments are then checked individually to ensure stability

43

throughout the beam.

In its treatment of restraint against lateral-torsional buckling, EC3 is more vague

than AS 4100 and BS 5950, merely stating that beams with sufficient restraint

to the compression flange are not susceptible to LTB. It does not specifically

state that the beam can then be analysed as separate segments like the British

and Australian codes, but rather that the value used for the elastic critical mo-

ment should include (amongst other things) the influence of lateral restraints,

thus leaving the designers to decide whether to divide the beam into laterally-

unrestrained segments, or to use a method of determining the elastic critical

moment of a laterally-restrained beam. In its treatment of the design of bracing

members, it does not specify one particular restraint force ratio that the brac-

ing members must be able to withstand. Instead, an equivalent stabilising force

based on the size of the initial bow imperfection of the primary beam is deter-

mined, and the bracing system as a whole should be able to withstand this. For

one member restrained, assuming the initial imperfection to be L/500, this rule

equates to a distributed restraint force ratio of 1.6%. This value is in fact a lower

bound, since it is based upon the assumption that the restraining system does

not deflect.

Hence, on balance it can be said that while there are differences in phrasing and

some specific provisions, the three codes more or less share the same approach in

designing a laterally restrained beam: ensure the restraint is of sufficient stiffness

so that the bracing points can be considered to remain in their initial positions

and ensure the restraint is strong enough to transfer a specified percentage of the

maximum compressive force in the beam, varying from 1% to 2.5% in the British

and Australian codes. However, they all fail to provide a method of determining

the stiffness required of the restraint to ensure that buckling occurs between the

restraints. Examining the analogous situation of a column restrained at mid-

height, it is noted that if the stiffness of the restraint is below a certain threshold

stiffness (often approximated as 16π2EIz/L3), then the buckling mode will involve

displacement of the bracing point. If the stiffness exceeds this threshold value

44

then the buckling mode will switch to that assumed by the codes, i.e. the column

buckles in between the bracing points. If the situation arose in the case of a

braced beam that the restraints were not of sufficient stiffness to enforce buckling

between the bracing points, then assuming the opposite (as the codes do) would

lead to an overestimation of the actual capacity of the beam.

45

Chapter 3

Demonstration of analytical

methods

3.1 Introduction

In the current chapter, the analytical methods that will be used in Chapter 4

to study the stability of beams with discrete restraints are introduced for an

unrestrained beam and for a beam with continuous restraint. As opposed to

determining the governing equations of the system from direct equilibrium, the

two methods detailed here are based on variational principles. Initially the total

potential energy of the system is established and then one of two alternative ap-

proaches for evaluating equilibrium and critical stability states are used: applying

the calculus of variations to obtain the system of governing differential equations,

and performing a harmonic analysis whereby the degrees-of-freedom (DOFs) are

assumed to be the superpositions of harmonic functions.

46

3.1.1 Calculus of variations

Although having being used in some form by the prominent mathematicians of

the 17th and 18th centuries - such as Johann Bernoulli, Fermat, Newton and

Liebniz - to solve variational geometric problems, the calculus of variations was

first refined as a theory and given this name by Euler in the 18th century after

correspondence with Lagrange (Ferguson, 2004). In its application to the prob-

lems discussed in the current chapter, the calculus of variations is used to find the

first variation of the total potential energy, δV , of a system in order to establish

equilibrium conditions. As mentioned in §2.1.1 in order for the system to be in

equilibrium, δV = 0 and so the equilibrium differential equations are obtained

once the appropriate boundary conditions are applied.

3.1.2 Harmonic analysis

In cases where closed-form analytical solutions of the governing differential equa-

tions of a system are unobtainable or where obtaining the equations themselves is

not achievable, the Rayleigh–Ritz method is an alternative route to determining

an approximate solution to the system. Mode shapes are assumed for the ap-

propriate displacement components and the total potential energy is minimised

with respect to the amplitudes of these mode shapes which are generalised co-

ordinates, as opposed to functions of some spatial ordinate as performed in the

calculus of variations. Although it is an approximate method, the accuracy of the

solutions can be much improved if the modal approximations satisfy the system

boundary conditions. Throughout the current work, Fourier series are used as

Rayleigh–Ritz approximations and due to the displacement functions comprising

a superposition of sinusoidal functions, this method is termed a harmonic anal-

ysis. Also, as will be seen in the following section, the differential equations for

load-carrying members often have periodic solutions and so applying a harmonic

analysis can in fact lead to exact results.

47

3.2 Unrestrained beams

In this section, the basic model of a simply-supported I-beam under constant

major-axis bending moment (Figure 3.1) is analysed in terms of its stability. Af-

ter a synopsis of the historical treatment of the lateral buckling of beams, the two

analytical methods outlined in the previous section are applied to the model to

highlight the convergence between them in terms of results. An imperfection is

then included into the analysis to demonstrate the asymptotic nature of imper-

fect load–deflection curves compared to the perfect system. Finally, a geometric

nonlinearity is included to demonstrate the weakly stable postbuckling response

of beams.

3.2.1 Previous results

It is noted that the results found in the current chapter for the critical moment

and to some extent the restraint forces are previously known; the intention is to

demonstrate how the same results may be obtained using the methods described

previously such that they can be implemented in the analysis detailed in Chapter

4.

The first investigation into the lateral stability of structural members in bending

was performed by Prandtl (1899), who studied the lateral buckling of a beam

with a rectangular cross-section and verified his findings experimentally. Michell

(1899) also provided a solution for a special case of lateral buckling. Reissner

(1904) developed methods to formulate the flexural and torsional equilibrium

equations inherent to the problem of the lateral stability of beams.

Timoshenko (1910) determined the buckling loads of columns, beams and shells,

derived from applying variational methods to the total potential energy of each

system as opposed to determining the equilibrium equations directly. Expanding

upon this work, Timoshenko (1911) studied further applications of this type of

48

u

r

-v

ǿ

M M

y

L

z

x

Figure 3.1: Cross-sectional geometry, degrees-of-freedom, loading configuration

and system axes of models studied in the current work

buckling analysis using energy methods to problems involving prismatic beams.

The seminal reference book for linear problems of stability of structural members

is that of Timoshenko & Gere (1961). In that monograph, the stability of bars,

beams, shells and plates was examined, including the case most pertinent to

the current work, a simply-supported I-beam under pure bending. The system

was defined as having three displacement component functions – horizontal and

vertical displacement of the shear centre (u(x) and v(x)) and the cross-sectional

angle of twist φ(x), where x is the longitudinal spatial ordinate (Figure 3.1) –

and thus three associated differential equations. This definition of the functions

also assumes that the cross-section of the beam remains rigid and that there are

no shear deformations. These assumptions comprise Vlasov beam theory that

underpins most classical analyses of beams. If small deflections are assumed,

resolving the applied moment M into components My and Mz along the deflected

cross-sectional axes leads to My ≈ M and Mz ≈ φM . Hence, the major-axis

equilibrium equation is independent of u and φ, which in turn leads to the critical

moment of the beam being independent of the major-axis flexural rigidity. This

is valid for beams where the major-axis flexural rigidity is much larger than

that of the minor-axis, an assumption the current work also makes. Reissner

49

(1904) discussed the effects major-axis deflections have when the relative flexural

rigidities are of a similar magnitude, with Timoshenko (1910) providing numerical

solutions for the particular case of a cantilever beam.

Timoshenko & Gere (1961) document how such an I-section exhibits different be-

haviour to the beam of rectangular cross-section studied by Prandtl (1899) due to

warping of the flanges. Different loading conditions and end restraint conditions,

as well as plastic buckling, were also examined but these are outside the scope

of the current work. The well-known solution for the critical lateral-torsional

buckling moment, Mob, of a simply-supported beam under uniform moment was

also provided, given here using the notation of the current work as:

Mob =π2EIzL2

√IwIz

+L2GItπ2EIz

(3.1)

where E, G are the Young’s and shear moduli, respectively; Iz is the minor-

axis second moment of area; It and Iw are the torsional and warping constants,

respectively, and L is the span of the beam. In the following section it will be

seen how this result can also be found using the variational methods previously

described.

3.2.2 Perfect case

As in the model detailed by Timoshenko & Gere (1961), the system is assumed

to have two DOFs: the lateral displacement of the centroid of the cross-section,

u(x) and the angle of twist of the cross-section φ(x), as shown in Figure 3.1. In

keeping with classical investigations of the stability of beams, the Vlasov beam

conditions mentioned in the previous section are assumed to prevail and hence

web distortion is not accounted for. White & Jung (2007) state that for standard

doubly-symmetric I-sections, e.g. Universal Beam (UB) sections, the effects of

web distortion on the lateral buckling load may be neglected. Owing to flanges

being stockier than webs for standard doubly-symmetric I-sections, this implies

that flange distortion can also be neglected and so the two DOF formulation used

50

in all the models in the current work is deemed appropriate. Assuming that the

flanges remain rigid, the effects of web distortion can be included by adapting

the approximate Rayleigh–Ritz analysis of Hancock et al. (1980).

Potential energy formulation

The general form of the potential energy formulation for an open-section thin-

walled beam is given by:

V = Ub + Uw + Ut −Mθ . (3.2)

where V is the total potential energy, Ub is the strain energy in bending, Uw is

the strain energy in warping, Ut is the strain energy in torsion and the Mθ term

represents the work done by the applied moment. Adapting the energy equation

presented by Pi et al. (1992) – linearised by assuming small deflections – to model

a beam with equal and opposite end moments, Equation (3.3) is obtained:

V =

∫ L

0

1

2

(EIzu

′′2 + EIwφ′′2 +GItφ

′2 + 2Mφu′′)

dx , (3.3)

where primes denote differentiation with respect to x. This is the basic po-

tential energy formulation to be used in the current work. One point to note

in using this particular energy equation is the form of the work done term,

Mθ = −M∫φu′′dx. Pi et al. (1992) have stated that some authors have re-

placed this term with +M∫φ′u′dx in their formulations and that this “alter-

native” formulation leads to errors in determining buckling loads and moments

when considering non-uniform bending moment distributions. While such distri-

butions are not the focus presently, it is nevertheless important to be aware of

this should the current work be adapted for different loading conditions.

51

Calculus of variations

Equation (3.3) is now stated in the form of a Lagrangian, i.e. a function of u and

φ and their derivatives:

V =

∫ L

0

L(u, u′, u′′, φ, φ′, φ′′)dx. (3.4)

The first variation of V is given by:

δV =

∫ L

0

(∂V

∂uδu+

∂V

∂u′′δu′′ +

∂V

∂φδφ+

∂V

∂φ′δφ′ +

∂V

∂φ′′δφ′′)

dx. (3.5)

Substituting the physical values of the derivatives leads to:

δV =

∫ L

0

[(EIzu′′ +Mφ)δu′′ +Mu′′δφ+GItφ

′δφ′ + EIwφ′′δφ′′] dx . (3.6)

Noting that δφ′ = d/dx(δφ), δφ′′ = d/dx(δφ′) etc., performing integration by

parts twice leads to:

δV = [(EIzu′′ +Mφ)δu′ − (EIzu

′′′s +Mφ′)δu (3.7)

+GItφ′δφ+ EIwφ

′′δφ′ − EIwφ′′′δφ]L0

+

∫ L

0

[(EIzu

(4) +Mφ′′)δu+ (EIwφ(4) −GItφ′′ +Mu′′)δφ

]dx .

Now the boundary conditions are applied; since the beam is simply-supported,

there is no lateral deflection at the supports, hence u(0) = u(L) = 0, along with

no cross-sectional twist, hence φ(0) = φ(L) = 0. Moreover at the supports there

is no minor-axis bending moment, hence u′′(0) = u′′(L) = 0, and warping is

permitted, hence φ′′(0) = φ′′(L) = 0. Thus the terms outside the integral vanish.

Setting δV = 0 implies that the integrand must be zero for all δu and δφ. The

governing differential equations of equilibrium are therefore:

EIzu(4) +Mφ′′ = 0 , (3.8)

EIwφ(4) −GItφ′′ +Mu′′ = 0 . (3.9)

Integration of Equation (3.8) twice with respect to x and application of the

boundary conditions leads to u′′ = −Mφ/EIz. Substituting this result into

52

Equation (3.9) produces a fourth-order ordinary differential equation solely in

terms of φ:

EIwφ(4) −GItφ′′ −

M2

EIzφ = 0 . (3.10)

Taking φ = A0emx the equivalent polynomial in m has both real and imaginary

roots. Upon application of the boundary conditions, the coefficients of the hyper-

bolic and cosine functions vanish leaving solutions of the form φ = φn sin(nπx/L),

where φn is an arbitrary amplitude. Examining Equation (3.8), it is obvious there-

fore that u is also sinusoidal, i.e. u = un sin(nπx/L). This implies that the beam

can buckle into any integer number of half-sine waves.

Substituting the newly-found result for φ into Equation (3.9) and then solving for

M produces the result for the critical moment, Mcr,n, of the beam as presented

in Equation (3.11):

Mcr,n =n2π2EIzL2

√IwIz

+L2GItn2π2EIz

. (3.11)

The expression in Equation (3.11) can be recast in the form of more convenient

parameters: the Euler buckling load of the beam, PE = π2EIz/L2, the torsion–

warping parameter κ = L2GIt/π2EIw and the nondimensionalised moment M =

2M/(PEhs). Across the range of UB sections, for span-to-depth ratios L/h, where

h is the total depth of the beam, from 10 to 30, κ varies from 0.32 to 25, though

in most practical situations it tends to vary between 1 and 5 as shown in Figure

3.2, based on cross-sectional properites for standard UB sections (Corus, 2005).

Also, it is noted that for doubly-symmetric I-beams,√Iw/Iz ≈ hs/2, where hs is

the depth between the shear centres of the flanges of the beam, = h− tf , where

tf is the flange thickness. Mcr,n, expressed using these parameters, is given by:

Mcr,n = n2

√1 +

κ

n2. (3.12)

There is now a family of solutions for Mcr,n, with each solution corresponding to a

particular buckling mode (i.e. each integer n), with the first mode (n = 1) being

the most critical value. This is the elastic critical moment of a beam, Mob, which

53

Index of UB sections

Figure 3.2: Values of κ for UB sections across a range of span-to-depth ratios

agrees with corresponding result from Timoshenko & Gere (1961) in Equation

(3.1). The equivalent expression for Mob is also provided:

Mob =π2EIzL2

√IwIz

+L2GItπ2EIz

, (3.13)

Mob =√

1 + κ . (3.14)

Upon substitution of u and φ into Equation (3.8), a relationship between un and

φn can be obtained. This is the deflected and twisted equilibrium position of a

beam buckling into n half-sine waves (Trahair et al., 2008):

unφn

=Mcr,n

n2EIzπ2/L2=hs2

√1 +

κ

n2. (3.15)

From Figure 3.1, it can be seen that the radius of twist rotation of a typical

cross-section, r = u/φ when assuming small deflections. Thus the location of the

axis of rotation is the ratio of the two DOFs. Since the mode shapes are both

of the form A0 sin(nπx/L), it follows that there is a fixed axis of rotation of the

beam. The distance between the axis of rotation and the shear centre for a beam

buckling in the first mode is

r =u1φ1

=hs2

√1 + κ . (3.16)

As κ > 0 the axis of rotation always lies beyond the tension flange. It is for

this reason that placing restraints closer to the compression flange offers greater

resistance to buckling, as is shown in the discussions on restrained beams in §3.3

and Chapter 4.

54

Harmonic analysis

As mentioned in §3.1, it is advantageous (or necessary) in some situations to

avoid solving the differential equations of a system directly. This can be due to

there not being analytical solutions to the equations themselves, which is often

the case in nonlinear analyses. In such situations, Rayleigh–Ritz approximations

are instead assumed for the DOFs; throughout the current work, Fourier series

are used to represent u and φ, with the coefficients of the cosine terms set to zero

to satisfy the boundary conditions:

u =∞∑n=1

un sin(nπxL

), φ =

∞∑n=1

φn sin(nπxL

). (3.17)

Upon substitution of the Fourier series into the energy equation, the problem is

converted from a two variable boundary value problem to a harmonic analysis

with the amplitudes of the harmonics, un and φn, as the generalised coordinates.

Owing to the orthogonality of the sine function, cross-terms of differing harmonics

vanish upon integration. The only cross-terms that remain are those containing

unφn:

V =1

4

∞∑n=1

(n4π4EIzL3

u2n +n4π4EIw

L3φ2n +

n2π2EIzL

φ2n −

2n2π2M

Lunφn

). (3.18)

The difference between the two analytical methods is now clearer to see, since

the total potential energy of the system is now minimised with respect to the

coordinates, rather than the minimisation being performed with respect to the

variables u and φ and their derivatives. Upon calculation of ∂V/∂un and ∂V/∂φn

the separation of the problem into distinct harmonics becomes apparent. Setting

these derivatives equal to zero establishes a system of equilibrium equations:

n2EIz(π/L)2 un −M φn = 0 , (3.19)

(GIt + n2EIw(π/L)2)φn −Mun = 0 . (3.20)

The system of equations can also be recast in matrix form as Hu = 0, where H

is the matrix of coefficients of the coordinates and u is the vector of coordinates.

55

Owing to the linear nature of the model, the Hessian matrix of the system (as

described in Chapter 2) is equivalent to H. Since the cross-derivatives:

∂2V

∂un∂um=

∂2V

∂un∂φm=

∂2V

∂φn∂φm= 0 ∀ n 6= m, (3.21)

the Hessian of the system as whole is made up of independent 2 × 2 matrices,

each one relating to a particular harmonic. As H 6= 0 (except in the trivial

case where the flexural, warping and torsional rigidities are all equal to zero),

u = 0. This implies that, for the perfect case, the beam remains undeformed in

its pre-buckling equilibrium state. The system is at critical equilibrium when the

determinant of the Hessian vanishes. Thus, critical values of the applied moment

are found by solving det(H) = 0:

det(H) =∞∏n=1

[∂2V

∂u2n

∂2V

∂φ2n

−(

∂2V

∂un∂φn

)2]

= 0. (3.22)

This can be viewed as a product of quadratic equations in M and upon setting

each of these equal to zero, the same family of solutions for Mcr,n as in Equation

(3.11) from applying the calculus of variations is obtained. As before, each solu-

tion is associated with one particular harmonic and so there is the same distinct

mode separation. From this result it can be seen that assuming the DOFs are

comprised solely of the nth harmonic yields an equivalent result for the critical

buckling mode as assuming a Fourier series due to this mode separation.

It has been shown that owing to the differential equations having periodic solu-

tions, the results obtained converge with those derived from using Fourier series.

For this reason and because of the considerable reduction in analytical complex-

ity it offers, harmonic analysis (or at least some form of Rayleigh–Ritz method)

is often used in the examination of structural stability in the linear range.

3.2.3 Imperfect case

In this subsection, the perfect case of an unrestrained beam is modified by the

inclusion of an initial lateral imperfection. The main difference that including

56

χ

M

χo

stiffness = EIz

Figure 3.3: Where the curvature χ of the beam is equal to initial curvature χo,

the beam is said to be in a strain-relieved state (Thompson & Hunt, 1984).

an imperfection causes in terms of the analytical procedure is in the form of

the equilibrium equations. In the perfect case the equations are homogeneous

and, after boundary conditions are applied, lend themselves toward a single har-

monic solution for the DOFs. When imperfections are included the equations

are heterogeneous, generally increasing the complexity of solving the equations

somewhat.

Potential energy formulation

Following the treatment of Thompson & Hunt (1984) of an axially-loaded arch

in an initial deformed state, the initial lateral curvature of the beam χo can be

substituted by the linear approximation u′′o when it is unloaded. It is assumed

that while the beam is in this initial state, it is stress-relieved i.e. the minor-axis

bending moment and strain energy equals zero (Figure 3.3).

As the beam is loaded, strains are induced as a result of the additional lateral

displacement. It is important to reiterate here that u is the total displacement of

the beam i.e. the initial displacement plus the extra displacement due to loading.

Figure 3.4 presents the deflected geometry of a typical cross-section. If the total

curvature of the beam is χ = u′′ then the actual curvature created from loading

57

uo

ǿ

u

Figure 3.4: Cross-sectional geometry of beam with an initial lateral imperfection

uo, the dashed lines indicating the position of the perfect case.

is χ − χo = u′′ − u′′o . It is this modified minor axis curvature that is used in

calculating the strain energy in bending of the beam. The concept of a strain-

relieved state is also employed successfully in the imperfection formulations for

the modelling of doubly-symmetric (Wadee, 2000) and monosymmetric (Wadee

& Simoes da Silva, 2005) sandwich struts.

There is, however, no alteration made to the work done term. This is due to the

work done being related to the incremental change of rotations in the transverse

plane u′′ as the beam is loaded, rather than being related to a specified initial

geometry uo (Pi et al., 1992). The total potential energy formulation of the

imperfect beam is:

V =

∫ L

0

1

2

[EIz(u

′′ − u′′o)2 + EIwφ′′2 +GItφ

′2 + 2Mu′′φ]

dx . (3.23)

Harmonic analysis

Along with representing the DOFs as Fourier series in Equation (3.17), the ini-

tial lateral imperfection is represented thus also. The benefit of representing the

imperfection in this way is that theoretically, provided the appropriate Fourier

transform can be determined, any arbitrary imperfection can be modelled. Equa-

58

tion (3.24) presents the form of the imperfection:

uo =∞∑n=1

en sin(nπxL

). (3.24)

Substitution of Equations (3.17) and (3.24) into Equation (3.23) and evaluation

of the integral leads to the following form of V :

V =1

4

∞∑n=1

(nπL

)2 [n2π2EIzL

(un − en)2 +n2π2EIw

Lφ2n (3.25)

+EIzLφ2n − 2MLunφn

].

The equilibrium equations are established once again by determining ∂V/∂un = 0

and ∂V/∂φn = 0:

n2EIzπ2un −MφnL

2 = n2EIzπ2en , (3.26)

(GIt + n2EIw(π/L)2)φn −Mun = 0 . (3.27)

As was found for the perfect case, there is mode separation since the harmonics do

not interact with each other. The deformed shape of the beam for loading levels

below critical can now be found as the equilibrium equations are no longer of the

form Hu = 0 but instead of the form Hu = e, where e is a vector whose elements

are of the form Anen. Simultaneous solution of ∂V/∂un = 0 and ∂V/∂φn = 0

for un and φn provides the relationship between the amplitude of the harmonics

of the deformed shape and the amplitudes of the harmonics that constitute the

initial imperfection:

un =n2 + κ

n2 + κ− M2/n2en , (3.28)

φn =M

n2 + κ− M2/n2

2enhs

. (3.29)

As un, φn ∝ en, each constituent harmonic of the deformed equilibrium shape

of the beam is related solely to the corresponding constituent harmonic of the

initial imperfection.

Equation (3.30) presents the imperfect equilibrium path of the system in terms

of un. For a given load level, because Mcr,n increases with n, the contribution of

59

e

Figure 3.5: Normalised load-deflection curve corresponding to the nth harmonic

higher harmonics is diminished in comparison to that of lower harmonics.

M

Mcr,n

=

√1− en

un. (3.30)

The establishment of this relationship permits a load–deflection curve to be plot-

ted for each mode, as shown in Figure 3.5. For the perfect case, the equilibrium

path follows the vertical axis, indicating that there is no lateral deflection in

spite of the increased load, until load reaches the bifurcation point (i.e. the ap-

plied moment equals the critical moment) and buckling occurs (as indicated in

§3.2.2). Owing to the linear nature of the model the postbuckling response is

completely flat. For the imperfect case, the load–deflection curve is asymptotic

to the perfect case, with increasingly large deflections occurring as the applied

moment approaches Mcr,n. When the critical load has been reached, in theory,

infinitely large deflections occur. In reality, as discussed in the following section,

the postbuckling response is not flat but weakly stable.

In practice, design codes such as Eurocode 3 (Comite European de Normalisation,

2005) and guidelines on manufacturing tolerances such as EN 1090-2 (Comite

European de Normalisation, 2008) define the initial imperfection of a beam as

60

being a single half-sine wave i.e. uo = e1 sin(πx/L). Defining the imperfection

in this way implies that all harmonics other than the first have no influence and

thus the beam deflects as a half-sine wave.

3.2.4 Postbuckling

By introducing a nonlinearity into the model, the response of the system after the

bifurcation point can be examined. Instead of the flat postbuckling response seen

in the perfect case, the system can show either positive or negative postbuckling

stiffness, related respectively to stable or unstable postbuckling behaviour.

Ioannidis et al. (1993) examined the initial postbuckling path of an unrestrained

beam. After establishing the governing differential equations directly from equi-

librium considerations and applying an approximate analytical procedure, an

expression for the postbuckling path was obtained. Through worked examples it

was shown that the response was weakly stable i.e. the postbuckling stiffness was

positive but relatively small.

Rayleigh–Ritz approach

Following Thompson & Hunt (1973) and Ioannidis et al. (1993), a nonlinearity

is included in the model by taking the curvature of the beam χ = u′′/√

1− u′2,

rather than the linear approximation χ = u′′ applied in previous sections. Like-

wise, the initial curvature χo = u′′o/√

1− u′2o. Expressing the curvature as

a Taylor series and truncating after two terms leads to the approximation of

χ = u′′(1 + u′2/2) and χ2 = u′′2(1 + u′2) and similarly for χo. The nonlinear

bending energy is then:

Ub =

∫ L

0

1

2EIz

(u′′

2+ u′′

2u′

2 − 2u′′u′′o − u′′u′′ou2o − u′′u′′ou′2

(3.31)

−1

2u′′u′′ou

′2u′o2

+ u′′o2

+ u′′o2u′o

2

)dx .

61

Upon substitution of Equation (3.31) into Equation (3.23) the nonlinear total

potential energy is derived. Again, the calculus of variations can now be per-

formed to obtain the governing differential equations. However, owing to the

inclusion of the nonlinearity, the complexity involved in solving the equations

is increased. Undertaking a full harmonic analysis is hampered also by the ap-

pearance of higher order cross-terms due to the nonlinearity. However, as shown

previously, an unrestrained beam can be assumed to buckle into the first mode

and so the Rayleigh–Ritz approximations u = u1 sin(πx/L) and φ = φ1 sin(πx/L)

are employed. The lateral imperfection is taken to be uo = e1 sin(πx/L). After

minimising V with respect to the harmonic coordinates, equilibrium equations

for the system are obtained, from which equilibrium paths are derived:

M

Mob

=

√1 +

1

2u21

(πL

)2− e1u1− 1

8

e31u1

(πL

)2− 3

8

u1e1

(πL

)2− 3

32u1e31

(πL

)4.

(3.32)

Figure 3.6 shows the resultant load–deflection curves for the first mode, with im-

perfect cases once again being asymptotic to the perfect (e1 = 0) case. As can be

seen the beam exhibits the same weakly-stable postbuckling behaviour observed

by Ioannidis et al. (1993) as the postbuckling stiffness of the system is positive

but quite low. The convergence of the two results serves as further validation

of the use of energy methods to predict the buckling behaviour of beams. For

small deflections, it can be said that the postbuckling response is almost flat. For

this reason, it is deemed unnecessary to investigate the postbuckling behaviour

of the models subsequently examined in the current work and thus the systems

are linearised by assuming small deflections i.e. assumptions such as sinφ ≈ φ

are made.

Calculus of variations

As mentioned previously, the inclusion of the imperfection and the nonlinearity

make the governing differential equations of the system complex to solve analyti-

cally. However, the continuation and bifurcation analysis software Auto (Doedel

62

& Oldeman, 2009) is capable of solving nonlinear ODEs numerically. Further de-

tails about the program are given in Chapter 5. In order for the equations to

be suitable for use by Auto, the DOFs should be nondimensionalised, and the

spatial ordinate rescaled to range over the domain [0,1]. The following properties

are thus defined:

x =x

Lu(x) =

u(x)

Lφ(x) = φ(x) uo(x) =

uo(x)

L(3.33)

Using the same linear approximations as before for χ and χo, the total potential

energy V can now be rewritten, with primes denoting differentiation with respect

to x rather than x, as:

V =1

2

∫ 1

0

{EIzL

[u′′(

1 +1

2u

′2

)− u′′o

(1 +

1

2u

′2o

)]2+EIwL3

φ′′2

+GItLφ

′2 +Mu′′φ

}dx (3.34)

The calculus of variations is used to find the first variation δV and after satisfy-

ing the boundary conditions the two ODEs are extracted from the integrand of

δV = 0 as:

u(4)(1 + u′2)+3 (u′u′′u′′′ − u′ou′′ou′′′o ) + u′u′′ (u′′′ − u′′′o ) + u

′′3 − u′′3o −

1

2u(4)o

(2 + u

′2o + u

′′2 +1

2u

′2u′2o

)− 1

2u

′2(u

′′3o + 3u′ou

′′ou′′′o

)−1

2u′u′′

(u

′2o u′′′o + 2u′ou

′′2o

)+ML

EIzφ′′ , (3.35)

EIwL3

φ(4) − GItLφ′′ +Mu′′ = 0 . (3.36)

The initial imperfection is defined by uo = e/L sin(πx/L). Figure 3.7 presents

the results of the numerical continuation performed by Auto for a beam of

457 × 152 × 82 UB section and a span of 10 m, showing the variation of the

applied moment M (normalised by Mob) with the maximum value of u across

x ∈ [0, 1]. Further tests varying the section and span of the beam returned exactly

similar graphs, agreeing with the implication of Equation (3.32) that the once the

moment and lateral deflection are scaled appropriately, the load-deflection curves

are independent of cross-sectional geometry. It can be seen that the curves closely

agree with those of Figure 3.6, which serves as further validation of the use of

the Rayleigh–Ritz method for analysing the stability of beams.

63

e1= L / 1000e1= L / 500

e1= 0MMob

u1L

Figure 3.6: Postbuckling paths for an unrestrained beam, as determined by

Rayleigh–Ritz analysis.

e1= L / 1000e1= L / 500

e1= 0

MMob

u1L

Figure 3.7: Postbuckling paths for an unrestrained beam, as determined by the

numerical continuation software Auto.

64

3.3 Continuously restrained beams

In this section the linear imperfect unrestrained beam model studied in §3.2.3 is

modified by including a continuous linearly elastic lateral restraint along the span

of the beam. A summary of previous work is presented, followed by a harmonic

analysis of the system, from which results for critical moments and mode shapes

at equilibrium are obtained.

3.3.1 Previous work

Vlasov (1961) provided the general differential equations for the case of a thin-

walled beam in an elastic medium offering vertical translational, horizontal trans-

lational and torsional restraint, derived directly from equilibrium considerations.

In the current model the medium restrains motion only in the x direction (Figure

3.1) and so the vertical translational and torsional stiffnesses of the medium are

neglected. In Vlasov’s treatment the system is again considered to have the three

DOFs presented in Figure 3.1. Consequently three differential equations relating

to vertical, horizontal and torsional equilibrium were presented, derived directly

from considering equilibrium conditions.

As mentioned previously, it is assumed in the current work that the major-axis

rigidity is considerably larger than the minor-axis rigidity and thus the vertical

displacement can be ignored, reducing the number of DOFs (and hence equi-

librium equations) to two. It is also taken that the beams are under constant

bending moment only (i.e. loaded with opposing end moments of equal magni-

tude) and thus any axial or transverse load terms are neglected. The current

work focuses on investigations for beams with doubly-symmetric cross-sections

and thus additional force components, arising from assuming arbitrary cross-

sectional geometry and the resultant eccentricities of the point of application of

moments from the shear centre, are also neglected. Taking these factors into

65

account, the equilibrium equations of Vlasov reduce to:

EIzu(4) + φ′′ + k(u+ aφ) = 0 , (3.37)

EIwφ(4) +Mu′′ −GItφ′′ + ak(u+ aφ) = 0 (3.38)

where a is the distance between the shear centre of the beam and the position of

the restraint, and k is the stiffness per unit length of the restraint.

Following from Vlasov’s work, Taylor & Ojalvo (1966) investigated the lateral

stability of a beam provided with continuous torsional restraint along its length

(as well as a central discrete torsional brace, which will be discussed in Chap-

ter 4), comparing their results to those of Vlasov. Similar studies of beams with

diaphragm bracing to their top flanges providing translational restraint were con-

ducted by Pincus & Fisher (1966), Errara et al. (1967) and Errara & Apparao

(1976). Trahair (1979), also leading from Vlasov’s previous results, provided

solutions for the elastic critical moments of mono-symmetric I-section beam-

columns with continuous restraints possessing rotational, translational, torsional

and warping stiffness, with the work focusing on doubly-symmetric sections in

particular. More pertinent to the current work is Trahair’s treatment of an I-

beam with continuous translational restraint, in which a quadratic equation is

presented. The positive root of this equation is the critical moment for the nth

mode:

Mk,n =1

π2L2

[(n4π4EIz + kL4

) (n4π4EIz+

n2π2L2GIt + a2kL4)]1/2

+ kL4. (3.39)

3.3.2 Potential energy formulation

The strain energies from bending, warping and torsion, along with the work done

by the applied moment remain the same as they were for an unrestrained beam.

The additional term that must be included is the strain energy stored in the

restraint that acts like an elastic foundation. Figure 3.8 shows the configuration

66

uo

ǿ

u

a

Figure 3.8: Typical cross-sectional deformation of a continuously restrained beam

of a typical cross-section of an imperfect beam with the continuous restraint

attached. It is assumed that the restraint is free to translate in the z-axis, so

that the only strains induced are due to horizontal displacement of the point of

attachment i.e. displacement in the y-axis.

Positive values of a denote compression side braces; negative values denote ten-

sion side bracing; the opposite convention is used by Trahair (1979). The ad-

ditional restriction on displacement and twist that the restraint offers ensures

that the assumption of small deflections remains valid. The linear approximation

of a flat postbuckling response is thus deemed valid and so a linear eigenvalue

analysis of the beam is sufficient. If large deflections are to be considered a

nonlinearity is introduced, which can lead to localised buckling occurring in the

postbuckling range. Hunt & Wadee (1991) investigated the case of a geomet-

rically perfect axially-loaded strut on a nonlinear foundation. As there was no

deflection until the primary bifurcation point (whereupon the strut buckled in

a periodic manner), the energy terms relating to the foundation remained equal

to zero. However in postbuckling, the activation of these nonlinear terms lead

the governing ODEs of the system to have localised as well as periodic solutions,

creating further bifurcations beyond the initial Euler buckling of the strut. The

subsequent investigation of Hunt et al. (1993) examined an inextensional strut on

67

a linear foundation, including nonlinearities in the expression for the curvature of

the strut, as was done in §3.2.4, and also in the end shortening of the strut. The

results of this analysis, which are more pertinent to the current section, found

similar results in terms of the appearance of localised bifurcations after the pri-

mary periodic bifurcation. In reviewing these two analyses, Wadee et al. (1997)

mention that the effect of the nonlinearity in the foundation outweighs that of

including the effects of large deflections and so, in their subsequent comparison

of results from a perturbation scheme whereby the governing ODEs of the sys-

tem are solved and from a Rayleigh–Ritz analysis, assume small deflections and

include a nonlinear foundation. Another investigation of note is that of Horne

& Ajmani (1971b) who expanded upon the case examined by Horne & Ajmani

(1969) and Horne & Ajmani (1971a) of a column laterally restrained at its top

flange to examine the postbuckling behaviour of the column by including a plastic

hinge at mid-height. From this analysis, limiting slendernesses were calculated

below which full plastic resistance can be still achieved despite the beam already

having undergone flexural buckling.

In the current section, the foundation is assumed to be linear. Thus nonlinear-

ities in the model arise from the expression for curvature, which, as mentioned

by Wadee et al. (1997) and as demonstrated in §3.2.4, have weak postbuckling

responses. As the inclusion of the restraining medium limits the magnitude of

the deflections further so that localisations are not triggered, the current work

ignores these effects. Assuming the deflections are small then, the compression

(or extension) X of the restraint at a distance x from the left-hand support is:

X(x) = u(x) + aφ(x)− uo(x) . (3.40)

The restraints are assumed to be linearly elastic (i.e. the force induced is pro-

portional to the extension) so the strain energy in the restraints is given by

UR = 12

∫ L0kX2dx (Figure 3.9). Adding UR to the right-hand side of Equation

68

X

Fstiffness = k

Strain energy UR

= F dX = kX212∫

Figure 3.9: Strain energy stored in a linearly-elastic restraint

(3.23) gives the total potential energy of the restrained beam:

V =

∫ L

0

1

2

[EIz(u

′′ − u′′o)2 + EIwφ′′2 +GItφ

′2+

k(u+ aφ− uo)2 + 2Mu′′φ]

dx . (3.41)

3.3.3 Harmonic analysis

Owing to the continuity of the restraint strain energy function in x, the La-

grangian of the system is also fully continuous. This means that upon substitu-

tion of the Fourier series from Equation (3.17) and Equation (3.24) into Equation

(3.41) and subsequent integration, the form of V presented in Equation (3.42) is

derived, with the orthogonality of the sine functions ensuring that there is mode

separation. The analogous case of an axially-loaded strut supported by an elastic

medium studied by Thompson & Hunt (1973) displays similar behaviour.

V =∞∑n=1

[EIz(un − en)2 + EIwφ

2n +GItφ

2n + k(un + aφn − en)2 −Munφn

].

(3.42)

Hence, results for load–deflection relationships and the critical moments of the

system are of a similar form to those of an unrestrained beam. Each harmonic

69

has a critical moment associated with it:

Mk,n =

√(n4π2EIzL2

+kL2

π2

)(n4π2EIw

L2+ n2L2GIt + a2

kL2

π2

)+akL2

π2(3.43)

which is the solution of a quadratic derived from setting det(H) = 0. The equiva-

lent quadratic provided by Trahair (1979) is mentioned in §3.3.1, of which Equa-

tion (3.43) is a solution. Since Trahair’s result was derived from the direct solution

of the governing differential equations of the beam, the convergence of the two

methods serves as validation of the applicability of harmonic analysis to analysing

the stability of continuously restrained beams.

Along with the parameters introduced in §3.2.2, the nondimensional restraint

height a = 2a/hs and the nondimensional continuous restraint stiffness γc =

kL4/π4EIz are now introduced. The nondimensional critical moment Mk,n is

expressed as:

Mk,n = n2

(√(n2 +

γcn2

)(n2 + κ+ a2

γcn2

)+aγcn2

). (3.44)

3.3.4 Moment–stiffness curves

Figure 3.10 shows the moment–stiffness curves of the first four modes for a typical

UB section beam (κ = 5) with continuous restraint attached at (i) the compres-

sion flange, (ii) the shear centre and (iii) the tension flange. In contrast with

the situation of an unrestrained beam where the first mode is always critical, the

critical mode (i.e. the mode with the lowest critical moment) of a continuously

restrained beam is dependent upon the restraint stiffness. Identification of the

critical mode for a particular stiffness is performed by iteration through n. As n

is trialled from 1 upwards, the value of Mk,n decreases until the minimum value

is found at n = nk. For n > nk the value of Mk,n increases again.

For restraints acting above the shear centre, increasing the stiffness increases

the critical moment indefinitely. Moreover, it can be seen that there is no limit

on the harmonic that the beam may buckle into: as the stiffness increases, the

70

M

γc

Tension flangeShear centreCompressionflange

n = 1

n = 4

n = 3

n = 2

n = 1

n = 1 n = 2 n = 3n = 4

Figure 3.10: Moment–stiffness curves for a continuously restrained beam (κ = 5)

harmonic of the critical mode progressively increases ad infinitum, albeit with

only marginal extra capacity being provided as the slope of the moment–stiffness

curve diminishes for higher mode numbers. As the restraint height is lowered

the additional moment capacity offered is diminished. For beams with very stiff

continuous restraints material yielding becomes the overall mode of failure as the

critical buckling moment exceeds the in-plane yielding moment (Trahair, 1979).

For restraints acting below the shear centre however, the curves are asymptotic

to a limiting critical moment, Mk,n →M∞,n as γc →∞:

M∞,n =1

2

(|a|√1 + κ

+

√1 + κ

|a|

)Mcr,n . (3.45)

Owing to these limits there is no longer the same sequential progression of critical

mode with increasing restraint stiffness since the first mode often remains the

critical one.

71

3.3.5 Forces induced in restraint

In contrast to the analyses of Vlasov (1961), Trahair (1979) and Trahair & Nether-

cot (1984), the current model includes imperfections in its analysis. Since the

restraint is linearly elastic, it obeys Hooke’s law, f = kX, where f is the dis-

tributed restraint force per unit length. As far as the author is aware. the first

investigation of bracing forces was carried out by Zuk (1956), who provided theo-

retical results for eight cases: fixed and elastic, central and continuous bracing for

beams and columns. Massey (1962) developed the beam model further, analysing

both initial out-of-straightness and twist along with experimental validation. The

imperfection in the system allows information to be obtained about the deformed

shape of the beam i.e. the values of u and φ. This allows X to be determined.

The equilibrium values of un and φn, in terms of nondimensionalised parameters,

are:

un =

Mγca+ κγc + n4κ+ n2γca2 + n2γc + n6

(n2 + κ+ a2γc/n2) (n4 + γc)−(nM − aγc/n

)2 en, (3.46)

φn =

M(n4 + γc)


)2 2enhs

. (3.47)

Since X = u + aφ − uo =∑∞

n=1(un + aφn − en) sin(nπx/L) and f = kX, an

expression for f is found thus:

f = k∞∑n=1

n2(n2 + Man2 − n4 + M2)


)2 en sin(nπx

L) . (3.48)

If it is assumed that the initial imperfection is given by uo = e1 sin(πx/L), the

expression for the maximum value of f(x) = f(L/2) = fmax, is given by:

fmax = k

M(a+ M)

(1 + κ+ a2γc) (1 + γc)−(M − aγc

)2 e1 . (3.49)

72

The maximum compressive force in the beam, distributed across its length, is

p = M/(hsL). The ratio fmax/p is given by:

fmax

p= 2π2

γc(a+ M)

(1 + κ+ a2γc) (1 + γc)−(M − aγc

)2 e1L. (3.50)

It is noted that the restraint force varies linearly with the magnitude of the initial

imperfection, as found also by the finite element analysis of Wang & Nethercot

(1989). The linear relationship is confirmed throughout the current work. In

Eurocode 3 (Comite European de Normalisation, 2005), a value of e1 = L/500 is

suggested as an upper bound. Figure 3.11 shows the variation of fmax/p with γc

for a range of combinations of a and κ, with M = Mcr,2 = 4√

1 + κ/4. Negative

values of the curve indicate that the restraint is deflecting in the direction opposite

to that assumed initially. Tension side restraint positions, i.e. those below the

shear centre, are not included since for certain values of a, M∞,1 < Mcr,2 and

thus the graphs would portray restraint force ratios at a level of load exceeding

the buckling load.

The main trend to be observed is the effect of the restraint height on the force

transmitted to the foundation, which is at its most efficient when positioned at

the compression flange (a = 1). As the restraint height is lowered, the force

becomes progressively larger, agreeing with observations made by Zuk (1956).

The overall value of the restraint force ratio is also quite sensitive to κ, as is

the spread of values from varying a. The actual f/p percentages observed agree

with the 1% value proposed by Wang & Nethercot (1990) for larger beams, i.e.

with high κ. However for smaller beams with restraint provided close to the

shear centre this force can be higher than the traditional 2% limit quoted by

design codes, as also found by Massey (1962). The BS 5950 (British Standards

Institute, 2000) limit of 2.5% is for some cases onerous but unsafe for others.

The behaviour seen currently for continuous restraints has a direct analogy with

discrete restraints and is discussed in further detail in Chapter 4.

73

κ=25

κ=10

κ=1

(%)

(%)

(%)

fp

fp

fp

Figure 3.11: Plots of restraint force ratio against restraint stiffness for different

restraint heights and torsion-warping parameters (M = Mcr,2)

74

3.4 Summary

The main points to be taken from this chapter are:

• Owing to the governing differential equations of the models discussed hav-

ing periodic solutions, the use of harmonic Rayleigh–Ritz analysis achieves

equivalent results.

• Owing to the continuous nature of the models discussed, there is a distinct

separation of buckling modes and equilibrium paths.

• The postbuckling response of a beam in bending is weakly-stable and so

postbuckling effects can be ignored.

• When continuously restrained on the compression side, i.e. above the shear

centre, there is no limit on the critical moment a beam may possess. There

is however a limit when the restraint system is placed on the tension side.

• The critical mode number of a continuously restrained beam increases se-

quentially without limit when restrained on the compression side. This

progression does not exist for beams restrained on their tension side.

• Forces induced in continuous restraints are lower the more remote the re-

straint system is placed from the shear centre on the compression side.

75

Chapter 4

Analysis of beams with discrete

restraints

The current chapter examines the stability of a beam with discrete lateral re-

straints and is the main focus of the current work. The background and de-

velopment of the theoretical treatment of the stability of beams with discrete

restraints is presented first. As is discussed therein, previous analyses often as-

sumed that braced beams buckle into a single harmonic and so the implications

of this assumption are explored in §4.2. Next, a harmonic analysis of a beam with

discrete restraints is performed and the results derived from it are presented, with

comparisons made between the results of the two approaches.

4.1 Previous work

Flint (1951) was the first to analyse the effect of lateral restraints on a beam.

While ignoring the warping rigidity of the beam, the effect of fixed supports and

torsional restraints at the support were studied, along with the case of inter-

mediate restraints. The influence of a central torsional support was studied, as

76

full bracing

Figure 4.1: Typical normalised moment–stiffness curve for beams with discrete

restraints, indicating the threshold stiffness and associated threshold moment.

was the case of a beam with a central elastic lateral restraint (of stiffness K).

For the latter case, results were provided for the critical moment of the beam

MK (in the form of a amplification factor for Mob) and for the stiffness required

to force the beam to buckle into the second mode, hereafter referred to as the

threshold stiffness KT . This level of bracing is repeatedly referred to in the lit-

erature as full bracing. It was noted that accompanying the threshold stiffness

was a threshold moment MT , equivalent to replacing the span L by L/2 in the

expression for critical moment. For K > KT there is no corresponding increase in

MK as the restraint nodes do not displace in buckling (Figure 4.1) and thus there

is no strain energy stored within the restraints. In deriving these results, Flint

made use of variational energy methods and although the results obtained may

be perceived to consider a simple case where the warping stiffness is neglected,

they are nevertheless an interesting starting point in this topic.

Following experiments performed by Green et al. (1956) on braced steel columns

and the analysis of Zuk (1956) mentioned in §3.3.5, Winter (1960) presented a

further development of Flint’s initial investigations. The concept of bracing re-

77

quiring both adequate stiffness and strength is mentioned, a concept that design

codes do not generally address completely satisfactorily. Winter discussed the

problem of columns with discrete and continuous bracing, providing some exper-

imental results from Green et al. and methods of determining brace forces along

with threshold stiffnesses. The cases of a column with two and three discrete

braces along the length were examined. The results from these studies were then

adapted to be used in the context of braced beams, which leads to conservative es-

timates as the cross-section of a beam in bending is only partially in compression

as opposed to that of a column being in full compression.

As mentioned in §3.3.1, Taylor & Ojalvo (1966) analysed beams provided with

central discrete and continuous torsional bracing. It was found that the behaviour

of a central torsional brace has a direct analogy with the behaviour described by

Flint (1951) in terms of the threshold stiffness and the threshold moment. It was

noted also that the critical moment–stiffness relationship between continuous

and discrete bracing was similar. Since the effect of the torsional restraint is

independent (in theory) of its position, there is not the same limit to the moment–

stiffness curves for tension side continuous bracing mentioned in §3.3.4.

The works referenced hitherto have all assumed that when full bracing was

not provided, the mode shape of the beam was a single harmonic e.g. u =

un sin(nπx/L). Horne & Ajmani (1969) analysed the provision of multiple rigid

discrete bracings to columns and used energy methods to solve for the buckling

loads. Since a fixed axis of rotation was imposed, u and φ were directly related

and as such the problem was reduced to a single DOF problem. For the case of a

column under axial load and uniform moment, it was assumed that the member

buckled into a single harmonic, while for a non-uniform moment, a Fourier series

was used instead. With the advent of modern computing, numerical methods for

obtaining results became accessible, and these were used by Horne & Ajmani to

solve this latter eigenvalue problem.

In using finite element methods, mode shapes need not be assumed but only

78

equilibrium equations specified in terms of the DOFs. Critical axial buckling

loads were obtained and further work by Horne & Ajmani (1971a) provided design

formulae based on the findings of the work cited previously. Nethercot & Rockey

(1971) provided finite element solutions for the buckling loads of beams and

columns. These results were then used by Nethercot & Rockey (1972) primarily

to examine discrete torsional braces but also the effect of a single lateral restraint

at midspan attached to the shear centre. The effect of combined restraints was

also studied. For the case of two discrete torsional braces, it was found that as the

restraint stiffness was increased, the critical mode progressed from the first mode

to the second and then at the threshold stiffness the beam buckled between the

restraints into the third mode. Nethercot (1973) applied a finite element method

to provide results for the critical moment of beams with a discrete lateral and

torsional brace attached at midspan at the shear centre. Mention is made of the

dependency of the value of the critical moment on κ. Mutton & Trahair (1973), in

their study of beams with a single central restraint possessing lateral, rotational

and torsional stiffness, gave expressions for the level of restraint stiffness that

provides full bracing in such situations, including interaction formulae for when

combined restraints are provided. They also attested to the influence of cross-

sectional geometry, of which κ is a measure, on the effectiveness of restraints

positioned below the compression flange.

Medland (1980) presented a recurrence relation approach to find the critical mo-

ment of an interbraced beam system where each element is subject to the same

bending moment and applied a finite element method to validate the findings,

though no explicit formulae were provided to evaluate the critical moment of par-

ticular cases or the related threshold stiffnesses. Following the work of Trahair

(1979) on continuously restrained beam-columns, Trahair & Nethercot (1984)

presented specific results for columns, beams under uniform bending and beams

with transverse loads, and presented the details of a finite element method to

calculate buckling loads. They also outlined how the stiffness matrices used in

the method for a continuously braced beam can be adapted for a beam with

79

a discrete brace. Referencing the interaction formula for the threshold lateral

and torsional brace stiffness of Mutton & Trahair (1973) where single harmonic

mode shapes were assumed, it was noted how applying the finite element analysis

of Nethercot (1973) was not subject to the same limitations on mode shape. A

close approximation of the relationship between critical moment and lateral brace

stiffness was provided on the basis of this approach. With regard to multiple dis-

crete bracings, a result for the critical moment of a beam with rigid (infinitely

stiff) lateral braces was provided. For the case of elastic restraints, the work of

Medland was referenced.

The studies mentioned hitherto have all assumed that the sections are doubly-

symmetric. Tong & Chen (1988) studied the stability of monosymmetric beams

with a central brace providing both lateral and torsional stiffness. Expressions

for the critical moment and threshold stiffnesses were provided, with the values

corresponding to the case of a doubly-symmetric beam compared with the finite

element results of Nethercot (1973).

Trahair (1993) provided a treatment of beams with a single discrete brace. The

case of a beam with a brace positioned at any point along its span was studied,

as was a beam with a central brace possessing either lateral or torsional stiffness.

The interaction formula of Mutton & Trahair (1973) was presented again. For

the case of a beam with multiple braces it was suggested to represent the system

of braces as an equivalent continuous restraint of stiffness k = nbK/L where nb

is the number of braces along the span of the beam. This is henceforth referred

to as the smearing method.

Yura et al. (1992) performed experiments on bridge bracing systems, examining

both lateral and torsional bracing. After discussing the theoretical basis behind

bracing systems, critical buckling loads as calculated by the BASP finite element

program (Akay et al., 1977; Choo, 1987) were presented for a beam with a central

brace. The effect of brace location was presented, confirming that a brace located

at the top (compression) flange required the least stiffness to brace the beam fully

80

and that considerably more stiffness was required of braces positioned at the shear

centre. At stiffnesses less than the threshold value, it was noted that the beam

buckled in a shape resembling a half-sine wave. As is discussed later in the

current chapter, the fact that it is not distinctly a half-sine wave is an important

characteristic. The effect of load location was also presented where it was found

that when a concentrated load was positioned at the top flange the efficacy of

top-flange bracing was reduced and almost non-existent for shear centre bracing.

Owing to (i) this effect of top flange loading, (ii) the increased stiffness required

of shear centre bracing and (iii) the lack of web distortion that occurs for beams

with top flange restraint, the report did not provide design recommendations

for brace locations other than at the top flange. The BASP program allowed

for web distortion, and it was found that for shear centre bracing there was a

loss of efficiency compared to when the webs were stiffened at the brace points,

where the cross-section deflected in the rigid manner assumed in Chapter 3. This

was confirmed experimentally. It is assumed hereafter in the current work that

the webs are adequately stiffened where the braces are attached. The report

also discussed torsional braces in a similar manner, noting that web distortion

impacts upon their efficacy much more than for lateral braces. After performing

the tests, design equations were presented with the brace location restricted to

the compression flange only. A brace force ratio of 0.8%, based on an initial

central deflection of L/500, was suggested for all cases which, while in agreement

with the 1% originally suggested by Wang & Nethercot (1989) and confirmed

by Wang & Nethercot (1990), still neglects the considerable influence of cross-

sectional geometry and brace location. A more concise synopsis of these results

and design formulae, and of similar tests on beams by Yura & Phillips (1992),

was provided by Yura (2001).

Gosowski (1999) presented the general differential equations for a beam-column

with multiple discrete braces along its length. After finding general closed-form

solutions for u and φ, the remainder of the study assumed that the lateral braces

possessed infinite stiffness, i.e. the lateral restraints were rigid, imposing a fixed

81

axis of rotation and reducing the problem to an investigation of the effect of

discrete torsional braces, Experimental evidence was also presented to validate

the results found for critical axial load and moment. Gosowski (2003) extended

this work further to examine (i) monosymmetric beams and columns with a

prescribed axis of rotation and (ii) monosymmetric beams with a lateral brace,

loaded with an arbitrary number of concentrated forces.

Al-Shawi (1998) examined the forces induced in a restraint acting alone on a

strut at an arbitrary position along its height. The cases of pinned-pinned, fixed-

fixed and pinned-fixed end conditions were examined. He stated that for most

common cases, where the central lateral imperfection was equal to L/500, the

restraint force ratios ranged between 1% and 2%. This work was expanded upon

by Banfi & Feltham (1999), who determined expressions for the restraint force

ratio of columns with one and two discrete restraints and for a column with a

continuous restraint. The general consensus was that for columns, the 2.5% force

suggested by BS 5950 was somewhat onerous. Al-Shawi (2001) examined the

case of a strut with elastic end restraints and presented a general formula for

determining the restraint force ratio in a column with a single elastic restraint at

an arbitrary point along its height and with elastic end restraints. The deflected

shape of the column was approximated by a Fourier series, truncated after ten

terms. Setting the appropriate stiffnesses to zero provided results for the case of

a simply-supported beam.

Taras & Greiner (2008) investigated the cases of axially-loaded members that tend

to fail by torsional buckling i.e. in a mode involving twist about the longitudinal

axis. This was done by conducting a geometrically nonlinear numerical analysis,

which included imperfections in the form of a sinusoidal initial out-of-straightness

(like that assumed in the current work) and residual stresses. It was assumed that,

in the case of the member being restrained against minor-axis flexural buckling

along the centroid of the member, torsional buckling was the critical mode up to a

particular limiting member slenderness, whereupon major-axis flexural buckling

would then become the critical mode. It was noted how, unlike regular flexural-

82

torsional buckling and lateral-torsional buckling curves where the critical load

approaches zero as the slenderness is increased, the equivalent torsional buckling

curve approaches a particular non-zero value, a function of the torsional stiffness

of the member. When restraint was provided at flange height, the buckling

mode included displacement of the centroid, and so can be related to lateral-

torsional buckling, although no qualitative difference was found between this

mode of buckling and torsional buckling in terms of the form of the buckling

curve.

Nguyen et al. (2010) investigated the stability of a geometrically perfect doubly-

symmetric I-beam with discrete torsional restraints at regular intervals along its

span. Rayleigh–Ritz approximations for u and φ were used, which were Fourier

series truncated after nb + 2 terms, where nb is the number of restraints. As dis-

cussed later, this truncation ignored higher harmonics which can have a noticeable

contribution to the composition of u and φ. Distortion of the cross-section, as

mentioned by Yura (2001) has a large effect on the efficacy of torsional bracing

but was neglected by Nyugen et al. Expressions were given for the threshold stiff-

ness and for the critical moment. The influence of imperfections was examined

by means of finite element analysis. Comparison was made between using the

results found from the discrete bracing analysis and from those using an equiv-

alent continuous bracing technique, whereupon it was found that the stiffness

requirement was underestimated for nb > 3.

Having considered the work performed previously and the issues discussed in

Chapter 3, the current work seeks to undertake the following: a linear harmonic

analysis of an imperfect simply-supported beam with an arbitrary number of

discrete elastic lateral restraints, located at regular intervals along its span and

positioned at a particular distance from the shear centre. From this analysis, ex-

pressions addressing the design concept of braces having both adequate strength

and adequate stiffness are developed. The following section discusses the impli-

cations of assuming single harmonic forms for the DOFs.

83

4.2 Single harmonic representation

In the cases of an unrestrained beam and a continuously restrained beam, it

was seen how the solution of the governing differential equations yielded single

harmonic forms for the DOFs. Also, performing a harmonic analysis showed that

due to mode separation, equivalent results for various properties of the system

were found. This meant that for these cases, the DOFs can also be assumed to

be of the formu

un=

φ

φn= sin

(nπxL

). (4.1)

For convenience, it is assumed that the initial imperfection uo = en sin(nπx/L).

In the analytical studies mentioned in §4.1, where a beam was restrained by a

single brace, it was assumed that the buckling mode was a half-sine wave, and this

assumption has been extrapolated for beams with higher numbers of braces, for

instance, by Trahair (1993). It was recommended that for beams with multiple

discrete braces of stiffness K that an equivalent continuous brace stiffness equal

to nbK/L be used in the expression for critical moment (a procedure termed as

“smearing” currently). As is shown in the current section, this can be thought of

as assuming that the beam buckles into single harmonic mode shapes.

Trahair also provided expressions for the threshold stiffness based on this as-

sumption. The following expression was provided by Trahair and is based on the

finite element analysis of Nethercot (1973):

KL3

EIz≈ 153(κ+ 5)

κ+ 0.74. (4.2)

On comparing the results from applying the smearing method to the case of

a beam with a single central restraint attached at the shear centre with the

equivalent results obtained from Equation (4.2), it was found that the smearing

approach provided values that were between 1.41 and 1.91 times the value ob-

tained from FEA. It was assumed that the results converged for higher numbers

of braces. However, it is shown in §4.3.4 that this is not the case.

It is also shown in the current chapter that assuming a single harmonic mode

84

M M

L

x x x x x x

s

restraints

uo

ǿ

u

a

Figure 4.2: (top) Cross-section at a restraint node; (bottom) Model of beam with

nb discrete elastic restraints.

shape causes the model to be unable to predict a full sequential critical mode

progression as the restraint stiffness is increased. This is in fact contrary to the

type of progression seen in the continuously-restrained model and, as will be

shown, the full harmonic analysis of a beam with discrete restraints.


The model studied in the current chapter is similar to those in Chapter 3 but now

the restraints are modelled as discrete elastic translational springs of stiffness K

(in units of kN/m rather than kN/mm2 as was the case for the continuous stiffness

k), at regular intervals along the span of the beam (Figure 4.2). The spacing s

between successive braces is then L/(nb + 1). The strain energy in the restraints

for a continuously restrained beam was:

UR,k =

∫ L

0

1

2k[X(x)]2dx . (4.3)

85

However, the total strain energy stored with the nb discrete braces is:

UR,K =1

2K

nb∑i=1

X2i . (4.4)

where Xi is the extension of the ith restraint, located at x = iL/(nb + 1). Upon

substitution of Equation (4.4) into Equation (3.41) the total potential energy of

the system becomes:

V =

∫ L

0

1

2

[EIz(u

′′ − u′′o)2 + EIwφ′′2 +GItφ

′2 + 2Mu′′φ]

dx+1

2K

nb∑i=1

X2i . (4.5)

4.2.2 Rayleigh–Ritz analysis

From Equation (3.40), X(x) = u(x) + aφ(x) − uo(x). Upon substitution of the

single harmonic functions for u, φ and uo into Equation (4.5) and noting that:

Xi = X

(iL

nb + 1

)= (un + aφn − en) sin

(inπ

nb + 1

), (4.6)

the total potential energy can be rewritten as:

V =L

4

[EIz

(nπL

)4(un − uo)2 + EIw

(nπL

)4φ2n +GIt

(nπL

)2φ2 (4.7)

−2M(nπL

)2unφn

]+

1

2K(un + aφn − en)2

nb∑i=1

sin2

(inπ

nb + 1

).

It can be shown using difference calculus (see §A.1) that:

nb∑i=1

sin2

(inπ

nb + 1

)=nb + 1

2∀ n mod (nb + 1) 6= 0 . (4.8)

The total potential energy is thus given by:

V =L

4

[EIz

(nπL

)4(un − uo)2 + EIw

(nπL

)4φ2n +GIt

(nπL

)2φ2 (4.9)

−2M(nπL

)2unφn

]+

1

2K

(nb + 1

2

)(un + aφn − en)2 .

An immediate comparison with the equivalent result for a continuously braced

beam can be drawn. Upon substitution of K = kL/(nb + 1) into Equation (4.9),

a formulation identical to the nth term of Equation (3.42) is obtained, due to the

86

representation of the DOFs by the nth harmonic. Where n mod (nb+1) = 0, the

zeroes of the harmonic coincide with the bracing points; there is no compression

of the braces and hence UR = 0. The lowest value of n at which this occurs is

of course nb + 1, and in this case the problem reduces to that of an unrestrained

beam with a span equal to s. The associated critical moment is then:

MT = (nb + 1)2π2EIzL2

√IwIz

+L2GIt

(nb + 1)2π2EIz, (4.10)

which can be nondimensionalised as:

MT = (nb + 1)2√

1 +κ

(nb + 1)2. (4.11)

4.2.3 Critical moment

Owing to the similarity between their respective energy equations, the equilibrium

equations and Hessian for beams with discrete restraints are found by substituting

k = K(nb+1)/L into the corresponding equations for continuously braced beams,

again by virtue of the current single harmonic representation of the DOFs. It

is noted how this is similar to the smearing approach mentioned by Trahair

(1993), only with the equivalent stiffness scaled differently. A new parameter,

the nondimensional discrete restraint stiffness is introduced, γ = KL3/(π4EIz) =

(K/kL)γc. The critical moment for a beam with discrete restraints assumed to

buckle into the nth harmonic is of a similar form to its analogous counterpart for

continuously braced beams:

MK,n =

√[n2 +

(nb + 1

n2

)γ

] [n2 + κ+ a2

(nb + 1

n2

)γ

]+ a

(nb + 1

n2

)γ .

(4.12)

The similarity between the expressions forMk,n andMK,n implies that the moment–

stiffness curves for continuously braced beams are equivalent to those for beams

with discrete restraints with the appropriate scaling of the abscissa. There is

however one extra property of the system to consider: the threshold moment

for full bracing, MT , which represents the maximum achievable lateral-torsional

87

n=1

n=2

n=nT

full bracing

Figure 4.3: Typical critical mode progression for beams with discrete restraints

when assuming single harmonic functions for the DOFs.

buckling moment of the beam. Figure 4.3 shows how the modes progress like in

the case of a continuously braced beam. Initially, where K = 0, the critical mo-

ment is equivalent to Mob, and increases with increasing K until MK,nT= MT ,

whereupon the critical mode is that of the beam buckling in between the re-

straint nodes rather than a mode involving displacement of the restraint nodes,

termed internodal and node-displacing buckling modes, respectively, in the cur-

rent work. The integer nT is the mode number at which the type of critical

mode switches from node-displacing buckling to internodal buckling. Contrary

to the supposition of there being a sequential progression of critical mode num-

bers from n = 1, 2, ..., nb before internodal buckling (n = nb + 1) occurs, when

single harmonics are assumed as the DOFs, nT is not necessarily equal to nb.

4.2.4 Threshold stiffness

Associated with the transition from node-displacing to internodal buckling is the

threshold stiffness, KT . For a particular harmonic number n, the nondimension-

88

alised threshold stiffness γT,n is found by solving MK,n = MT for γ:

γT,n =

(n2

nb + 1

)[(nb + 1)2 − n2] [(nb + 1)2 + n2 + κ]

n2(1 + a2) + 2a(nb + 1)2√

1 + κ(nb+1)2

+ κ. (4.13)

In a fashion similar to the iterative method of determining the critical mode by

identifying the mode number with the minimum associated critical moment for

a particular stiffness, the actual threshold stiffness of the system is determined

by identifying the maximum value of γT,n. Some nondimensionalised parameters

are introduced to generalise Equation (4.13): γs = γ/(nb + 1)3, κs = κ/(nb + 1)2,

(equivalent to replacing L with s in the expressions for γ and κ, respectively),

and η = n/(nb + 1), hence 0 < η < 1. The benefit of using these parameters is

that the same general relationships between restraint stiffness, beam geometry,

restraint height and mode number can be found for all nb. The reduced form is

then:

γs,T = η2(1− η2

) 1 + η2 + κsη2(1 + a2) + 2a

√1 + κs + κs

. (4.14)

The function is redefined as γs,T = γAγB where:

γA = η2(1− η2

)(4.15)

and:

γB =1 + η2 + κs

η2(1 + a2) + 2a√

1 + κs + κs. (4.16)

From this definition, γA(η) is an even function in η with a single distinct local

maximum in the domain [0,1] at η = 1/√

2. It is equivalent to 2γs,T when a = 1

and κs = 0, since γB = 1/2 for these values. The rational function γB(η) has

a horizontal asymptote at 1/(1 + a2). The positions of the vertical asymptotes,

which of course are also possessed by γs,T , are the solutions to:

η2 +2a√

1 + κs + κs1 + a2

= 0 . (4.17)

For a > alim, where alim = −κs/2√

1 + κs, these roots are imaginary and so γB

(and by extension γs,T ) does not have vertical asymptotes. If a > alim, as κs > 0,

γB > 0 ∀ η.

89

*

Figure 4.4: Typical variation of γs,T with η when assuming single harmonic func-

tions for the DOFs for a > alim.

Through examination of the discriminant of dγs,T/dη = 0, it is found that the

function has a single maximum at η = η∗ and so γs,T assumes a form similar

to that shown in Figure 4.4. The function γs,T (η) increases uninterrupted over

η = [0, η∗) and decreases uninterrupted over η = (η∗, 1], i.e. for η < η*, γs,T,n+1 >

γs,T,n with the reverse being true for η > η*. This implies that nT ∈ N =

bη∗(nb + 1)c or dη∗(nb + 1)e,1 whichever of the two has a larger corresponding

value of γs,T .

Lemma: η∗ < 1/√

2, for a > alim

Initially it is noted that for η ∈ [0, 1], γA > 0. Upon examination of:

∂γB∂η

= −2η

(1− a

√1 + κs

η2(1 + a2) + 2a√

1 + κs + κs

)2

, (4.18)

it is clear that ∂γB/∂η < 0 for η > 0. Now the form of ∂γs,T/∂η is examined:

∂γs,T∂η

= γA∂γB∂η

+ γB∂γA∂η

. (4.19)

Since γA,γB > 0 and ∂γB/∂η < 0, for there to be solutions to ∂γs,T/∂η = 0,

1The floor function bxc returns the integer immediately less than x, while the ceiling function

dxe returns the integer immediately greater than x.

90

∂γA /∂η>0

η*

γ

Figure 4.5: ∂γA/∂η > 0 at η = η∗ implies that η∗ < 1/√

2.

∂γA/∂η > 0. The range of η in which this condition is true is [0, 1/√

2], and thus

η∗ must lie in this range, as demonstrated by Figure 4.5.

It has been shown that for a > alim, η∗ < 1/√

2. This implies that when assuming

single harmonic functions for the DOFs, the maximum possible value of nT is the

integer above (nb+1)/√

2. However, it is often an integer below that value owing

to the actual value of η∗ being less than 1/√

2. For nb > 4, using single harmonic

representations for the DOFs can never predict a full sequential progression of

critical mode number from n = 1 to n = nb. When a < alim, the behaviour just

described breaks down: ηvert ∈ R and γs,T < 0 for 0 < η < ηvert.

The example shown in Figure 4.6 evidences this behaviour. For a beam with

nb = 3, a = 0.5 and κs = 0.5 , η* = 0.704. Thus the closest integers to (nb + 1)η*

= 2.816, are 2 and 3; since γs,T,3 > γs,T,2, nT = 3. As can be seen from the

moment–stiffness curves, there is a sequential progression from n = 1, to n = 2

to n = 3, which is the last critical node-displacing mode number before internodal

buckling is critical.

Figure 4.7 shows the buckling mode behaviour for a similar beam with nb = 6.

As can be seen, nT = 5, which is confirmed upon examination of the moment–

stiffness curves. This is an example of a form of mode-skipping occurring. Instead

of the critical mode cycling through all the harmonics from n = 1 up to n = nb

91

n=1

n=2

n=3

full bracing

n=1

n=2

n=3

Figure 4.6: Sequential critical mode progression for nb = 3 since γs,T increases

for each value of n.

92

n=1n=2

n=3

n=4

n=5

full bracing

n=1

n=2

n=3

n=4

n=5

n=6

Figure 4.7: Critical mode skipping for nb = 6 since γs,T is at a maximum at

n = 5.

before transition to internodal buckling, the last mode is skipped. In order to

determine what the actual threshold stiffness is correctly, it is necessary to trial

the various possible mode numbers to find the one for which γs,T is at a maximum.

It is shown in §4.3.4 that when the buckled shape of the beam is determined more

accurately through representation as a Fourier series, the more intuitive behaviour

of a full sequential mode progression prevails.

93

4.2.5 Deflected shape and restraint forces

Working again from the similarity between the potential energy formulations for

continuously braced beam and those with discrete restraints, the amplitudes of

the harmonics comprising the deflected shape are given by:

unen

=

[n2 + κ+ a2 (nb+1)

n2 γ] [n2 + (nb+1)

n2 γ]

+ a (nb+1)n2 γ

[M − a (nb+1)

n2 γ]

[n2 + κ+ a2 (nb+1)

n2 γ] [n2 + (nb+1)

n2 γ]−[M − a (nb+1)

n2 γ]2 , (4.20)

φnen

=2enhs

M[n2 + (nb+1)γ

n2

][n2 + κ+ a2 (nb+1)γ

n2

] [n2 + (nb+1)γ

n2

]−[M − a (nb+1)γ

n2

]2 . (4.21)

Likewise, the formula for the restraint force ratio has an analogous form for beams

with discrete restraints. In this case however, the force in the ith restraint, Fi, is

determined as a ratio to the maximum compressive force P in the beam:

FiP

=2π2γ

(n2a+ M

)sin( inπ

nb+1)[

n2 + κ+ a2 (nb+1)γn2

] [n2 + (nb+1)γ

n2

]−[M − a (nb+1)γ

n2

]2 enL . (4.22)

When considering a single half-sine wave imperfection i.e. n = 1:

FiP

=2π2γ

(a+ M

)sin( iπ

nb+1)

[1 + κ+ a2(nb + 1)γ] [1 + (nb + 1)γ]−[M − a(nb + 1)γ

]2 e1L . (4.23)

Figure 4.8 shows the restraint force curves for a beam with κ = 5, a = 0.75

and nb = 1 for a number of load levels, µ = M/MT , with the initial midspan

deflection, e1 = L/500. The influence of κ and a on discrete F/P curves is much

the same for continuous f/p curves in Chapter 3 and so no further reference is

necessary here. An interesting point to note is that the restraint force–stiffness

curve, for n = 1, has a pole at:

γvert =1

nb + 1

[M2 − (1 + κ)

1 + a2 + 2aM + κ

]. (4.24)

At µ = µlim, this position of this pole is negative, as is the case for the µ = 0.25

curve in Figure 4.8. This value is given by:

µlim =1

nb + 1

√1 + κ

(nb + 1)2 + κ. (4.25)

94

(%)

Figure 4.8: Restraint force ratios for a singly-restrained beam.

It is noted that this value is in fact equal to µo = Mob/MT . The similarity

between γvert and Equation (4.13) for the threshold stiffness is clear to see. If

M = MT , i.e. µ = 1, then γvert = γT,1. For a beam with a single restraint, this

implies that providing stiffnesses close to the threshold amount leads to excessive

forces being induced in the restraints as the level of loading approaches critical.

This phenomenon was attested to by Winter (1960) and is a major reason as to

why both stiffness and strength must be considered in the design of braces. For

nb > 1, further discussion of the properties of the restraint force graphs is given

in §4.3.5 due to similarities arising between the expressions for F/P .

4.2.6 Summary

In the current section it has been shown how representing the DOFs as single

harmonics leads to results for critical moment and restraint force ratio analo-

gous to providing an equivalent continuous restraint. Contrary to the case of

a continuous restraint however, there are two classes of buckling mode: node-

displacing and internodal. Also, the concept of a threshold stiffness enforcing

internodal buckling is introduced and a corresponding threshold moment. It has

95

been shown how the single harmonic representation has limitations when predict-

ing the progression of the critical mode. Finally, it has been shown that while

providing the threshold stiffness leads to the beam being fully braced, it can

lead to unnecessarily high forces being induced in the restraints. These concepts,

while based on a single harmonic representation of the DOFs, also arise in a full

harmonic analysis, although with more refined results.

4.3 Full harmonic analysis

In Chapter 3, it was seen how representing the DOFs as single harmonics or

a full Fourier series had the same outcome in terms of buckling modes, critical

moments and restraint forces. The orthogonality of the sine function coupled

with the continuous nature of the models ensured that each harmonic comprised

its own buckling mode. In the current section, owing to the evaluation of the

strain energy in the restraints at discrete points, it is seen that the continuity

breaks down and harmonics interact with each other.


The model currently under investigation has the same potential energy formu-

lation as that given by Equation (4.5). As was defined in Chapter 3, u =∑∞n=1 un sin (nπx/L) and φ =

∑∞n=1 φn sin (nπx/L). The initial imperfection is

also given by a Fourier series, uo =∑∞

n=1 en sin (nπx/L). Substitution of these

96

series into Equation (4.5) leads to:

V =

∫ L

0

1

2

∞∑n=1

∞∑m=1

[EIz

(n2m2π4

L4

)(un − en)(um − em) (4.26)

+ EIw

(n2m2π4

L4

)φnφm +GIt

(nmπ2

L2

)φnφm

−2M

(n2π2

L2

)unφm

]sin(nπxL

)sin(mπx

L

)dx

+1

2K

nb∑i=1

∞∑n=1

∞∑m=1

(un + aφn − en)(um + aφm

− em) sin

(inπ

nb + 1

)sin

(imπ

nb + 1

).

Upon evaluation of the integral, terms containing sin(nπx/L) sin(mπx/L) where

n 6= m vanish due to the orthogonality of the sine function. The same does not

occur however for the strain energy in the restraints. For n,m mod (nb + 1) 6= 0,

it can be shown through difference calculus (see §A.2) that:

nb∑i=1

sin

(inπ

nb + 1

)sin

(imπ

nb + 1

)= 0 for n mod 2(nb + 1) 6= ±m. (4.27)

For n mod (nb+1) = 0, sin[inπ/(nb+1)] = 0, the restraint spacing s is an integer

multiple of the harmonic wavelength, i.e. the restraint nodes coincide with the

zeroes of the harmonics. Hence, these harmonics do not contribute to the strain

energy stored in the restraints and hence do not interact with other harmonics.

Where (n±m) mod 2(nb + 1) = 0,

sin

(inπ

nb + 1

)sin

(imπ

nb + 1

)= ± sin2

(inπ

nb + 1

), (4.28)

and so the result of Equation (4.8) can be applied, leaving these terms equal to

±(nb + 1)/2. In contrast to the continuous model, where cross-terms comprised

amplitudes of lateral deflection and twist coordinates of the same harmonic, after

evaluation of the strain energy there are cross-terms of the form unu2q(nb+1)±n,

unφ2q(nb+1)±n and φnφ2q(nb+1)±n remaining, where q ∈ Z. Distinct sets of interact-

ing harmonics can be identified. The set of harmonics interacting with the nth

harmonic, Hn = {m : (n ±m) mod 2(nb + 1) = 0,m > 0}. Examining Hn and

H2q(nb+1)±n it can be seen that the sets are in fact identical, implying that there

97

exist nb distinct sets of interactive harmonic sets (for completeness, Hq(nb+1) = ∅).

For example, for nb = 3, H1 = {1, 7, 9, 15, 17, ...}, H2 = {2, 6, 10, 14, 18, ...} and

H3 = {3, 5, 11, 13, 19, ...}. If, say, H6 were calculated it would be found to be

identical to H2. In general it suffices to define sets H1, H2, ..., Hnb. A sign opera-

tor function, δn,m = ±1 if (n∓m) mod 2(nb + 1) = 0 (otherwise δn,m = 0). From

this definition it is clear to see that the function is reversible, i.e. δn,m = δm,n.

If all m ∈ Hn are arranged in increasing order, then the corresponding values of

δn,m alternate between +1 to −1.

Using this notation the total potential energy can be rewritten as:

V =L

4

∞∑n=1

[EIz

(nπL

)4(un − en)2 + EIw

(nπL

)4φ2n +GIt

(nπL

)2φ2n

−2M(nπL

)2unφn

]+nb + 1

4K

∞∑n=1

∑m∈Hn

δn,m(un + aφn − en)(um + aφm − em) . (4.29)

4.3.2 Equilibrium conditions and mode separation

An infinite system of equilibrium equations is established from ∂V/∂un = 0 and

∂V/∂φn = 0. Eliminating common constants, the equations are:

EIz

(nπL

)4(un − en)−M

(nπL

)2φn +

nb + 1

LK∑m∈Hn

δn,m(um + aφm − em) = 0 ,

(4.30)

EIw

(nπL

)4φn +GIt

(nπL

)2φn −M

(nπL

)2un (4.31)

+ anb + 1

LK∑m∈Hn

δn,m(um + aφm − em) = 0 .

For convenience, the nondimensionalised lateral deflection coordinate, un = 2un/hs,

and the nondimensionalised lateral imperfection amplitude en = 2en/hs are in-

troduced. After some rearrangement Equations (4.30) and (4.31) are nondimen-

98

sionalised as:

n4un − n2Mφn + (nb + 1)γ∑m∈Hn

δn,m(um + aφm) (4.32)

= n4en + (nb + 1)γ∑m∈Hn

δn,mem ,

(n4 + n2κ)φn − n2Mun + a(nb + 1)γ∑m∈Hn

δn,m(um + aφm) (4.33)

= a(nb + 1)γ∑m∈Hn

δn,mem .

Since there are nb distinct sets of interacting harmonics, whose elements are

uniquely their own (i.e. Hn ∩ Hm = ∅, {n,m} ⊂ {1, ..., nb}) the entire system

separates into nb distinct systems related to node-displacing buckling modes,

along with an infinite number of internodal systems, i.e. those corresponding

to harmonics that are multiples of (nb + 1). As the system is linear, the ma-

trix of coefficients of the coordinates in the equilibrium equations is identical

to the Hessian matrix, H. The separation of the harmonics into the nb dis-

tinct node-displacing sets and the infinite amount of distinct internodal sets al-

lows independent node-displacing and internodal matrices, Vm and Vq(nb+1),

respectively, to be extracted. Hence, critical moments are found from det(H) =∏nb

m=1 det(Vm)∏∞

q=1 det[Vq(nb+1)] = 0, indicating that the system has nb node-

displacing buckling modes, with the mth mode being constituted of harmonic

numbers n ∈ Hm, and infinite internodal buckling modes, involving single har-

monic lateral deflection and twist functions.

For internodal buckling, the expressions for the deflected shape, load–deflection

curve and critical moment are equivalent to those of an unrestrained beam of

span L/[q(nb + 1)] and need no further consideration here. As was found when

the DOFs were represented by single harmonics, MT , the first internodal critical

moment, can be perceived as a limiting moment for node-displacing modes.

99

4.3.3 Deflected shape for node-displacing modes

The investigation now concentrates on the mth node-displacing mode. If the

elements of Hm are arranged in increasing order, then wm,i is defined as the

ith element of Hm. For instance, if nb = 3, H2 = {2, 6, 10, 14, 18, ...} and so

w2,4 = 14.2 Examining the equilibrium equations (4.32) and (4.33), for the mth

node-displacing mode these can be recast as:

w4m,nuwm,n − w2

m,nMφwm,n + (nb + 1)γ∞∑i=1

(−1)i+1(uwm,i+ aφwm,i

) (4.34)

= w4m,newm,n + (nb + 1)γ

∞∑i=1

(−1)i+1ewm,i,

(w4m,n + wm,n

2κ)φwm,n − w2m,nMuwm,n + a(nb + 1)γ

∞∑i=1

(−1)i+1(uwm,i+ aφwm,i

)

= a(nb + 1)γ∞∑i=1

(−1)i+1ewm,i, (4.35)

where n ∈ N. From these equations, the most basic relationship between uwm,n

and φwm,n is found to be:

φwm,n =

(aw2

m,n + M)uwm,n − w2

m,naewm,n

An, (4.36)

where An = w2m,n+κ+ aM . From this result, the relationship between uwm,n and

uwm,1 is found to be:

uwm,n =(−1)(n+1)w2m,n

w2m,1

(AnBn

)(B1

A1

uwm,1 −B1 + M2

A1

ewm,1

)(4.37)

+

(Bn + M2

Bn

)ewm,n ,

where Bn = w4m,n + w2

m,nκ − M2. Substitution of the relationships in Equations

(4.36) and (4.37) into the series of Equation (4.34), evaluated at n = 1, provides

a closed form solution for uwm,1 solely in terms of imperfection amplitudes. Reap-

plication of Equations (4.36) and (4.37) finally leads to closed-form expressions

2The mathematically explicit definition is wm,i = [i− (i mod 2)](nb +1)+m[2(i mod 2)−1].

100

for all harmonic amplitudes in terms of imperfection amplitudes:

uwm,n =Bn + M2

Bn

ewm,n +(−1)nMAnw2m,nBn

S1

1(nb+1)γ

+ S2

, (4.38)

φwm,n =w2m,nM

Bn

ewm,n +(−1)nM(w2

m,na+ M)

w2m,nBn

S1

1(nb+1)γ

+ S2

, (4.39)

where

S1 =∞∑i

(−1)i+1w2m,ia+ M

Bi

ewm,i,

S2 =∞∑i

Ciwm,i2Bi

,

Cn = w2m,n(1 + a2) + κ+ 2aM .

Hence for a given arbitrary imperfection whose constituent harmonic amplitudes

are known, the deflected shape can be determined. Examining Equations (4.38)

and (4.39) again, it can be seen that the contribution of each harmonic to the

overall deflected shape is related directly not only to the relative magnitude of its

corresponding imperfection amplitude, ewm,n , but also to S1. The contribution

of each imperfection amplitude to S1 is weighted by a successively decreasing

amount as the coefficient of ewm,iquickly diminishes for each successive term.

For m = 1, the first harmonic dominates the overall deflected shape, with higher

harmonics making relatively smaller contributions. Figure 4.9 compares u as

represented by a Fourier series with u as represented by a single harmonic, for the

three node-displacing deflection modes for a beam with nb = 3. The shapes of the

Fourier series resemble m half-sine waves, but the influence of higher harmonics

in the composition of u and φ ensures that a single harmonic representation of the

DOFs is often a loose approximation. While it is tempting to assume for the sake

of convenience that a single harmonic provides adequately accurate results, as is

detailed in subsequent sections this is not necessarily the case. Assuming that

the imperfection is in the form of a half-sine wave, for m 6= 1, ewm,n = 0 and so

φwm,n = uwm,n = 0 ∀ n. In essence then, due to the separation of harmonics, these

modes are related to “perfect” models, and the theory predicts no pre-buckling

deflections for those modes.

101

m=1

m=2

m=3

Figure 4.9: Comparison of mode shapes of u as calculated using single harmonic

and Fourier representations of the DOFs for a beam with three restraints (a = 0.5,

κs = 0.31, γ = 17.5, µ = 0.75).

102

e1

Mcr,1

M

Figure 4.10: Typical load deflection curve for a beam with discrete restraints

when representing the displacement components as Fourier series (m = 1).

4.3.4 Critical equilibrium conditions

Critical moment and threshold stiffness

The difficulty of finding a closed-form solution for M from Equation (4.38) – and

thus a load–deflection relationship – can be circumvented by studying the implicit

relationship between M and uwn,m as illustrated in Figure 4.10. In keeping with

other linear models examined in the current work, the imperfect equilibrium path

is asymptotic to a flat perfect path, with the asymptotes being solutions to:

1

(nb + 1)γ+∞∑i=1

w2m,i(1 + a2) + κ+ 2aM

wm,i2(w4m,i + w2

m,iκ− M2) = 0 , (4.40)

which is now recast as

1 + γsSs,2 = 0 , (4.41)

where Ss,2 = (nb + 1)4S2. The lowest positive solution for M of Equation (4.40)

is the critical moment for the mth node-displacing mode. The factor µ = M/MT

is reintroduced here. Through some rearrangement, partition of fractions and

application of difference calculus (detailed in §A.3), the infinite series Ss,2 is

103

found to have an equivalent finite-termed form thus:

Ss,2 =− 1√2r0

( rar+2µ2(1 + κs)

+ 1 + a2)

π sin π√r−/2

√r−

(cos π

√r−/2− cosπη

)+

(rar−

2µ2(1 + κs)− (1 + a2)

)π sinhπ

√r+/2

√r+

(cosh π

√r+/2− cos πη

)

+raπ

2

2µ2(1 + κs) (1− cosπη), (4.42)

where

ra = κs + 2aµ√

1 + κs ,

r0 =√κ2s + 4µ2(1 + κs) ,

r+ = r0 + κs

r− = r0 − κs

and η = m/(nb + 1) as mode numbers are designated by m rather than n in the

current section.

An analytical solution for µ to Equation (4.41) is difficult to obtain, a problem

that previous authors – such as Horne & Ajmani (1969), Nethercot & Rockey

(1971), Nethercot & Rockey (1972), Nethercot (1973), Yura et al. (1992), Tra-

hair (1993) – have avoided by using finite element methods to provide data around

which approximate design formulae were fitted. As mentioned in §4.1 these for-

mulae do not cover both variable restraint height and multiple elastic braces.

The implicit relationship between the critical moment and restraint stiffness pro-

vided by Equations (4.41) and (A.41) is exploited in Chapter 6 to provide values

around which design formulae are fitted. The threshold stiffness (for the mth

node-displacing mode), on the other hand, is easily obtainable by setting µ = 1,

i.e., M = MT , and rearranging Equation (4.41) to find:

γs,T =

[π2(κs + 2a

√1 + κs)

2(1 + κs)(1− cosπη)(4.43)

+π sinhπ

√1 + κs

(1− a

√1 + κs

)22(2 + κs)(1 + κs)1.5

(cosh π

√1 + κs − cos πη

)]−1 .104

Full sequential mode progression

Using this expression for γs,T it can be shown how the Fourier series-based theory

can predict a full mode progression, contrary to the truncated, mode-skipping

progression predicted by a single harmonic. For convenience, Equation (4.43) is

simplified by introducing the constants in η:

d1 =π2(κs + 2a√

1 + κs), (4.44)

d2 =π sinhπ√

1 + κs(1− a

√1 + κs

)2, (4.45)

and the functions:

D1 =2(1 + κs)(1− cosπη), (4.46)

D2 =2(2 + κs)(1 + κs)1.5(cosh π

√1 + κs − cosπη

), (4.47)

so that γs,T = D1D2/(d1D2 + d2D1). The mode progression behaviour of the

system is investigated by evaluating:

∂γs,T∂η

=d1D

22(∂D1/∂η) + d2D

21(∂D2/∂η)

(d1D2 + d2D1)2. (4.48)

Upon inspection of:

∂D1/∂η =2(1 + κs) sinπη, (4.49)

∂D2/∂η =2(2 + κs)(1 + κs)1.5 sin πη, (4.50)

and noting that η ∈ (0, 1), it is found that these derivatives are always positive.

Likewise, d2 is also always positive. If a > −κs/2√

1 + κs then d1 > 0. This

limiting value of a is alim as found for the single harmonic representation of

the DOFs. Hence if a > alim then ∂γs,T/∂η > 0 and the threshold stiffness

value increases for each node-displacing buckling mode. Figure 4.11 compares

this behaviour with that predicted by the single harmonic representation of the

DOFs. Figure 4.12 demonstrates how, as the stiffness is increased, the critical

mode changes progressively from one node-displacing mode to the next, until the

threshold stiffness is reached for m = nb whereupon internodal buckling becomes

the critical mode and increasing the stiffness further leads to no increase in critical

105

Figure 4.11: Comparison of mode progression behaviour as predicted by Fourier

series and single harmonic representation of the DOFs.

moment. The expression for the actual threshold stiffness of the system is given

by:

γs,T =

[π2(κs + 2a

√1 + κs)

2(1 + κs)(1 + cos πnb+1

)

+π sinhπ

√1 + κs

(1− a

√1 + κs

)22(2 + κs)(1 + κs)1.5

(cosh π

√1 + κs + cos π

nb+1

)−1 . (4.51)

Loss of sequential mode progression

For a 6 alim, ∂γs,T/∂η is not necessarily negative but its sign now depends on

η. For a 6 alim, ∃ ηrev ⇒ ∂γs,T/∂η = 0. For η = [0, ηrev), ∂γs,T/∂η < 0 and for

η = (ηrev, 1], ∂γs,T/∂η > 0. The condition for a negative derivative is:

d1 < −d2∂D2/∂η

∂D1/∂η

(D1

D2

)2

. (4.52)

A closed form solution for the limiting value of a can be found based on this

condition, the maximum value of which exists at η = 1. However, it is not

necessary to display these limiting restraint heights here as the values obtained

106

Figure 4.12: Sequential critical mode progression (nb = 3, a = 0.5, κs = 0.5).

(when reconverted to actual heights) only differ marginally from alim (0.012hs

for κs = 0, 0.005hs for κs = 0.5, and decreasing further for larger values of κs).

Hence, it can be said that when bracing is provided below alim then ∂γs,T/∂η < 0.

For a 6 alim, γs,T (η) possesses an asymptote at η = ηv ∈ [0, 1]. At a = alim, ηv =

0, a condition found by setting the denominator of γs,T = d1D2 + d2D1 = 0 and

ignoring other solutions of cos(πη) = 1. Since γs,T (0) = 0 and ∂γs,T/∂η < 0 then

for η ∈ [0, ηv), γs,T < 0 and for η ∈ (ηv, 1], γs,T > 0. The physical implication of

this is that for mode numbers m 6 bηv(nb+1)c, the critical moment never exceeds

the internodal buckling moment MT and is thus always critical. In the case of

there being no mode numbers m 6 bηv(nb + 1)c then the maximum threshold

stiffness corresponds to m = 1, as in Figure 4.13, and the actual threshold stiffness

of the system is:

γs,T =

[π2(κs + 2a

√1 + κs)

2(1 + κs)(1− cos πnb+1

)

+π sinhπ

√1 + κs

(1− a

√1 + κs

)22(2 + κs)(1 + κs)1.5

(cosh π

√1 + κs − cos π

nb+1

)−1 . (4.53)

107

Figure 4.13: Loss of sequential critical mode progression for a < alim (nb = 3,

a = −0.225, κs = 0.5). The threshold stiffness of the beam is that associated

with m = 1 rather than m = nb.

108

Loss of full bracing ability

If ηv > 1/(nb + 1), the first mode has a negative threshold stiffness and so its

critical moment never exceeds MT . Hence the beam does not possess an actual

threshold stiffness, as exemplified by Figure 4.14. The beam can never develop its

full bracing moment capacity, and while increasing the restraint stiffness indefi-

nitely leads to an increase in capacity, the benefits are negligible. The restraint

height associated with this condition is termed aNT . There are two limiting cases:

nb = 1 and nb →∞. Because η → 0 as nb →∞, aNT → alim. For nb = 1, aNT is

the root of the following quadratic equation in a:

π(κs + 2)√

1 + κs(κs + 2a

√1 + κs

)+ tanh π

√1 + κs

(1− a

√1 + κs

)2= 0 .

(4.54)

The quantity alim− aNT ≈ 0.095 at κs = 0 and this difference approaches a limit

≈ 0.04 as κs →∞, as shown in Figure 4.15. As aNT < alim ∀nb, in general it can

be taken conservatively that aNT = alim, and that bracing should not be provided

below this level if the full bracing capacity is to be developed, i.e. if the beam is

to buckle between the restraint nodes.

Influence of restraint height on threshold stiffness

In the previous section it was shown when considering the design of a beam with

discrete restraints, in order to utilise the full lateral-torsional buckling moment

capacity of the beam the restraint height should not be below aNT , which could

be taken conservatively as alim, since the critical moment of the beam will always

be below MT in such cases. In the current section the extra demand on restraint

stiffness attached with varying the restraint height above alim is examined. As

mentioned in, amongst others, Yura (2001), the restraints are at their most ef-

ficient when positioned at the compression flange. The corresponding threshold

stiffness is termed KT,top.

In Figure 4.16, the two curves represent the two limiting cases of nb = 1 and

109

,2

Figure 4.14: Loss of full bracing capability a < aNT (nb = 3, a = −0.25, κs = 0.5).

While the m = 2 and m = 3 curves intersect µ = 1, the m = 1 curve is lower

than them (apart from a brief range of stiffnesses) and lower than µ = 1 and is

thus the critical mode.

110

^ ^

^

^ ^

Figure 4.15: As aNT is only slightly below alim (with a maximum difference of

0.095 at κs = 0 and nb = 1), aNT can be taken conservatively as equal to alim.

The physical consequence of this is that providing restraint at a level below alim

implies that full bracing is not achievable.

111

^

Y

^

^

Figure 4.16: As the restraint is moved further from the compression flange, the

stiffness required to brace the beam fully is considerably greater.

nb →∞. For nb = 1, κs = 24/(1 + 1)2 = 6, representing a maximum value across

UB sections, and because κs → 0 as nb → ∞, this value is used in the relevant

equations for that curve. The abscissa is scaled such that the ordinate equals 0 at

a = alim and is equal to 1 at a = 1. The threshold stiffness, normalised by KT,top

for the relevant value of κs, is plotted on the vertical axis. The curves show how

the stiffness requirements for full bracing increase considerably as the restraint is

positioned further away from the compression flange (a = 1).

Comparison with results from single harmonic representation

Figures 4.17 to 4.22 present critical moment–stiffness curves for a range of num-

bers of restraints, with κs = 0.5 and a = 0.5. A general trend to be observed

in all the graphs is how the single harmonic and smeared curves approximate

the Fourier curve quite closely up to K ≈ 0.5KT , with KT being the threshold

stiffness calculated from Fourier analysis, whereafter the curves diverge, leading

to sometimes unsafe, sometimes overly conservative estimates of the buckling

112

moment. This divergence also leads to inaccuracies in predicting the threshold

stiffness.

For higher numbers of restraints, the implications of the truncated mode progres-

sion prediction of the single harmonic representation are clearly observed. As m

increases, the slope of the moment–stiffness curve decreases but since higher

modes are never critical, the critical mode does not progress fully to m = nb and

the curve has an inappropriately large slope as it intersects with µ = 1. This

means that the method underestimates the stiffness across all ranges of parame-

ters.

The smearing method can offer more accurate results since it adjusts the equiva-

lent continuous stiffness by a factor of nb/(nb+1), meaning that the curve can be

positioned either above or below the Fourier method curve, as shown in Figures

4.17 to 4.22. This leads to mixed results in predicting the threshold stiffness ac-

curately. In the treatment of the case of a singly-braced beam with the restraint

attached to the shear centre, i.e. a = 0, Trahair (1993) remarked that the method

is conservative for low values of nb, which is shown to be true by results obtained

for the threshold stiffness from the current analysis being in the range of 1.45 to

1.85 times KT across UB sections (0 < κs < 6). This range of ratios is also valid

for 0 6 a 6 1. At a = alim, this ratio is below unity for κs > 3 but otherwise

is conservative. For nb = 2, the threshold stiffness values are less conservative,

with the ratios being unsafe for certain low restraint heights; the ratio ranging

between 0.87 and 1.22 times KT for compression side bracing. The lowest values

occur for large κ and low a, but in general the ratio is greater than unity across

most of the range of restraint positions from alim to the compression flange.

For nb = 3, however, the results from the smearing method for compression side

bracing are unsafe when compared with those obtained from Fourier analysis. It

is interesting to note that Nguyen et al. (2010) found when representing discrete

torsional braces as an equivalent continuous system, the results were also unsafe

for more than three braces. For compression side bracing, the ratio varies between

113

= 1 nb

Figure 4.17: Moment–stiffness curves for a single restraint (a = 0.5, κs = 0.5).

0.73 to 0.98, with most values tending towards the upper end of that range. While

it could be assumed that the method is converging with the addition of extra

restraints (as expected by Trahair), the results for nb > 3 show otherwise. The

ratios tend to vary between 0.6 and 0.8, with lower ratios calculated for restraint

heights closer to alim. Figure 4.23 shows the spread of ratios for values of nb up

to 10.

The main disadvantage of using the smearing technique is that it is difficult to

predict, as the stiffness is increased, which node-displacing mode is critical before

the switch to internodal buckling occurs. This implies that an iterative process

is required to establish the actual threshold stiffness. Equation (4.51) on the

other hand gives the actual threshold stiffness, assuming that the restraints are

positioned on the compression side, or no more than alim away from the shear

centre on the tension side. It has been shown that for nb < 3, results tend to

114

= 2 nb

Figure 4.18: Moment–stiffness curves for two restraints (a = 0.5, κs = 0.5).

115

= 3 nb

Figure 4.19: Moment–stiffness curves for three restraints (a = 0.5, κs = 0.5).

116

= 4 nb

Figure 4.20: Moment–stiffness curves for four restraints (a = 0.5, κs = 0.5).

117

= 5 nb

Figure 4.21: Moment–stiffness curves for five restraints (a = 0.5, κs = 0.5).

= 6 nb

Figure 4.22: Moment–stiffness curves for six restraints (a = 0.5, κs = 0.5).

118

KT , full harmonic

KT , smearing

nb

1 2 3 4 5 6 7 8 9 10

1.2

1.4

1.6

1.8

1.0

2.0

0.2

0.4

0.6

0.8

0.0

MinMax

Figure 4.23: Boxplot of ratios between threshold stiffnesses as predicted by the

smearing technique and those predicted by full harmonic analysis for a range of

restraint numbers.

be conservative and for nb > 3 results are unsafe, in that they predict too low a

threshold stiffness. In practice, the consequence of using the values obtained from

the smearing technique does not dramatically underestimate the corresponding

level of critical moment, but nevertheless the presence of a single formula that

does not require iteration to determine the required solution is clearly preferable.

4.3.5 Restraint forces

Since the restraints are assumed to be linearly elastic, Fi = KXi as before.

Assuming an arbitrary imperfection, the compression in the ith restraint, is:

Xi =

[u

(iL

nb + 1

)+ aφ

(iL

nb + 1

)− uo

(iL

nb + 1

)]=∞∑n=1

(un + aφn − en) sininπ

nb + 1. (4.55)

Noting that wm,n is of the form 2q(nb + 1) ±m, the contribution of the various

node-displacing modes – the internodal modes do not cause deflection at the

119

restraint nodes – can be identified thus:

Xi =

nb∑m=1

[sin

imπ

nb + 1

∞∑n=1

(−1)n+1(uwm,n + aφwm,n − ewm,n

)]. (4.56)

It can be seen that, if the mth mode is isolated, the deflected positions of the

restraints follow a locus of m half-sine waves. Substituting Equations (4.38) and

(4.39) into Equation (4.56) gives:

Xi =hs2

nb∑m=1

sinimπ

nb + 1M

S1

1 + (nb + 1)γS2

. (4.57)

The force in the restraint is given, as in §4.2.5, by (Fi/P )i = KXi/P :

FiP

=2π2γ

L

nb∑m=1

S1

1 + (nb + 1)γS2

sinimπ

nb + 1. (4.58)

Upon examination of Equation (4.58), it can be seen that F/P → ∞ when 1 +

(nb+1)γS2 → 0 for each value of m. The value of γ at which this occurs is termed

γ∞,m. This condition is equivalent to that of Equation (4.41) for establishing the

critical moment for the mth node-displacing mode. For µ = 1, γ∞,m = γT,m,

implying that if the restraints of stiffness KT are provided then the forces induced

in the restraints will be excessive. If, however, it is assumed that the imperfection

is in the form of a single half-sine wave, then modes where m 6= 1 make no

contribution to the compression of the restraints. Thus, the expression for the

maximum (i.e. where sin iπ/(nb + 1) = 1) restraint force ratio simplifies to:

F

P= 2π2γ sin

iπ

nb + 1

(a+ M

1 + κ− M2

)(1

1 + (nb + 1)γS2

)e1L. (4.59)

In contrast to the formulation of Equation (4.58), the restraint force formulation

above has a pole at K = KT,1 only. Figure 4.24 compares the restraint force ratio

curves for a beam with three restraints loaded by M = MT with a single half-sine

wave imperfection of amplitude e1 = L/500 to an equivalent beam with the second

and third imperfection harmonics included, with amplitudes e2 = e3 = e1/100.

As can be seen, both curves converge except in the immediate vicinity of the poles

at KT,2 and KT,3, whereafter the two curves converge again. This serves as an

extension for multiple restraints to the advice of Winter (1960) of excessive forces

120

(%)

Figure 4.24: Restraint force ratio curves assuming single and multiple harmonic

imperfections when loaded at M = MT (a = 0.5, κ = 5, nb = 3).

being induced in the restraint if the threshold stiffness is provided for a beam with

one restraint. However, owing to the fact that design codes only suggest initial

imperfection amplitudes based on a single half-sine wave, the current work shall

continue to work off the assumption that uo = e1 sin πx/L. In a qualitative sense

though, the restraint should possess a stiffness K > KT to avoid high restraint

forces. In Chapter 6, simplifications are found for the formulae above that allow

practical design expressions to be derived.

4.3.6 Summary

After applying a full harmonic analysis to the case of a simply-supported beam

with nb discrete restraints, it was found that there are nb node-displacing buckling

modes and an infinite number of internodal modes. The mth node-displacing

mode causes the restraint nodes to deflect along a locus of m half-sine waves

and as such comprise harmonics with mode numbers of the form 2q(nb + 1) ±

121

m. The internodal modes comprise single harmonics with mode numbers that

are multiples of nb + 1, and thus do not cause any deflection at the restraint

nodes. An implicit expression for the critical moment of the node-displacing

modes was found and from this an expression for the threshold stiffness relating

to a particular mode was found. From analysing its nature, the critical mode

progression behaviour was found to follow a sequence of modes from m = 1

to m = nb when the restraints are positioned above a certain height on the

tension side of the shear centre, termed alim. Below this restraint height, the

sequential mode progression behaviour was found to break down. At another

height below alim, termed aNT , the critical moment of the first buckling mode is

always less than the threshold moment at which internodal buckling is critical,

and thus full bracing cannot be developed. The effect of the restraint height on the

threshold stiffness was also examined, where the additional stiffness demands were

demonstrated. A comparison between these results and the equivalent results

from a single harmonic representation of the DOFs showed that the “smearing”

technique leads to reasonably accurate estimates of the critical load for multiple

restraints, but can significantly over- and underestimate the threshold stiffness of

the system. Finally, it was found that providing restraints with a stiffness close to

the threshold amount leads to higher forces being induced in the restraints. It was

also advised that although defining the initial imperfection as a particular single

harmonic disallows high forces being induced around the threshold stiffness of the

corresponding node-displacing mode, it is nevertheless prudent to avoid providing

restraints with stiffnesses close to these threshold amounts.

122

Chapter 5

Validation of discretely-braced

beam model

In Chapter 4, a full harmonic analysis of a simply-supported discretely-braced

beam with an initial lateral imperfection was conducted. From this analysis,

results were found for the critical moment of the system and also for the deflected

shape of the beam. In the current chapter, the results for these two properties

are compared with equivalent results as ascertained by computer software.

The results for the critical moment of the system - being independent of the im-

perfection - are calculated using the LTBeam software (Galea, 2003), which uses

the finite element method to determine the critical eigenmodes and associated

eigenvalues of the beam. Further details about this software are given in §5.1.1.

The deflected shape of the beam is determined through use of the Auto-07p

software (Doedel & Oldeman, 2009). Amongst its many features is the ability to

perform numerical continuation that provide evolving solutions to ODEs while

parameters are varied, as well as detecting the bifurcation points of the same.

Further discussion of Auto is given in §5.2.1.

It should be noted that the manner in which the numerical methods used by

123

the current work to validate the analytical model use the same DOFs as the

analytical model. In the case of the validation of the results for critical moment,

this approach is validated by the fact that the LTBeam software has been found

to agree with results obtained by other finite element analysis programs, which

due to using elements with a higher number of DOFs could be said to model the

actual behaviour of beams more closely.

5.1 Validation of critical moment results

5.1.1 LTBeam software

Description of analysis method

The LTBeam software was developed at the Centre Technique Industriel de la

Construction Metallique (CTICM) as part of a Europe-wide research initiative.

It uses the finite element method to calculate the critical moment of both mono-

symmetric and doubly-symmetric beams subject to various loading and restraint

conditions, including the more pertinent case of discrete elastic intermediate re-

straints. It discretises the beam into a maximum of 300 elements and then

formulates stiffness matrices for each element. In a manner analogous to the

construction of the Hessian matrices mentioned in Chapters 3 and 4, these are

derived from the second derivatives of the total potential energy in the element

with respect to four degrees-of-freedom at each end node of the element: the

lateral deflection u, the angle of twist φ, the lateral angle of rotation θz = du/dx,

and the warping rate dφ/dx. This definition of the DOFs assumes the section

remains rigid, thus ensuring that Vlasov conditions prevail. This adds to the

suitability of LTBeam in validating the results of the current work. If restraints

are specified in the model, the program imposes constraints on the DOFs of the

restrained nodes.

124

A global stiffness matrix K is formed from the individual element stiffness ma-

trices, and is split into two parts: KL relates to the strain energies in bending,

torsion and warping of the beam, and KG relates to internal energies arising from

external forces on the element. Since KG is a function of the bending moment

distribution M(x), it can be said that KG = µcrKG(M), where M is the bend-

ing moment distribution normalised by its maximum value and µcr is the load

factor. It is assumed that the element length is sufficiently small to assume a

linear moment gradient across each element, and hence a constant shear force

distribution. In relation to the current work, however, since the bending moment

distribution is considered constant anyway, this is not an issue. The program

then aims to find the value of µcr at which det[KL +µcrKG(M)] = 0 by means of

a dichotomic process, whereby values are trialled until the determinant changes

sign. The process is iterated with smaller intervals until a specified tolerance

is met. With the value of µcr (the eigenvalue of the problem) calculated, the

program then evaluates the associated eigenvector, returning the values for the

degrees-of-freedom at each of the element nodes, which are then normalised by

the maximum value across the length of the beam.

Validation of the software

The LTBeam software has been validated (CTICM, 2002) against the commer-

cially available finite element software ANSYS v5.6 (ANSYS, 2001) across a range

of beam geometries, loading conditions and restraint conditions. The beam was

modelled mostly using shell elements, with beam elements used at point-load and

support positions to avoid local buckling. Across the 65 comparisons performed,

the largest discrepancy between the value of critical moment as calculated by the

two programs was 0.8%. Comparisons were also made between LTBeam and the

following software:

1. DRILL, a German finite element program developed by Commercial In-

tertech.

125

2. FINELG, a Belgian-developed finite element beam analysing software.

3. Results from the literature using ABAQUS (ABAQUS, 2006).

For all cases where doubly-symmetric beams were studied, the deviation from

the LTBeam results was in the order of 1%. This serves as sufficient evidence of

the appropriateness of the LTBeam software for validating the analytical model

developed in the current work.

Previous applications of the software by other authors

LTBeam has been used for various applications regarding the stability of steel

beams. Galea (2002) used the software to calculate equivalent uniform moment

factors for beams loaded by point loads, uniformly-distributed loads (UDLs)

and/or external moments. Larue et al. (2007a) used LTBeam to compare results

for the critical moment of beams with rigid and continuous lateral restraints with

those obtained by their own software that used numerical methods to solve the

governing ODEs of the system. Subsequent work by Larue et al. (2007b) em-

ployed the Galerkin method to solve the ODEs for this problem. The results

they obtained for the critical moment were then validated by Khelil & Larue

(2008) using LTBeam.

5.1.2 Comparison with LTBeam results

In Chapter 4, it was shown that the problem of determining the critical moment of

a discretely-braced beam can be said to be dependent on the following parameters:

the restraint stiffness K, the restraint height a, the number of restraints nb and

the beam parameter κ, which in turn is dependent on the span L and cross-

sectional properties of the beam. Table 5.1 shows the different values assumed

by the various parameters in undertaking the validation. Since a single beam

126

Parameter Values assumed

nb 1, 2, 3, 4, 5, 6

a alim, 0, 0.5, 1

L (m) 7, 8.75, 10.5, 12.25, 14

Table 5.1: Values assumed for the parameters in the validation using LTBeam.

cross-section, that of a 457 × 152 × 82 UB, was used throughout, the value of κ

was varied by choosing different beam spans.1The beam spans were chosen such

that the span-to-depth ratios would lie between 15 and 30. The values of K

chosen were specific to the configuration of each particular test so that K/KT

ranged from 1/8 to 1, with KT calculated using the methods of Chapter 4.

In all, the critical moment, as calculated by the analysis of the current work, was

compared with that calculated by LTBeam for 960 separate observations. The

maximum error was found to be 0.25%, with an average error of 0.06%, which

can be attributed to the discretisation of the beam and the inevitable rounding

errors arising from this (such as the length of individual elements). This validates

the method of applying a full harmonic analysis to the case of a discretely braced

beam; accurate results are returned reliably for the elastic critical moment of the

beam.

The following sections demonstrate the very close agreement between the an-

alytical results and those from LTBeam, as well as showing the influence that

varying different parameters has on the relationship between the critical moment

and restraint stiffness.

1It is noted that, since LTBeam carries out a linear analysis using beam elements, varying

the section type will not result in a change in the modelled behaviour of the beam since the gross

section properties are merely numerical constants. If shell elements were to be used however,

factors such as the relative thicknesses of the flanges and the web would lead to an additional

effect.

127

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Analytical â= 0

Analytical â= 0.5

Analytical â= 1

LTBeam

nb = , L = 10.5 m1

Analytical â= âlim

Mcr

/ M

T

K / KT

Figure 5.1: Normalised moment–stiffness curves for varying restraint heights,

nb = 1, L = 10.5 m.

Influence of varying the restraint height

As mentioned previously in §4.3.4, restraints attached further away from the com-

pression flange down the section of the beam require increasingly more stiffness

to brace the beam fully. In Figures 5.1 to 5.3, the influence of varying restraint

height is examined for three different numbers of braces, with analytical results

given by the curves and the corresponding values obtained from LTBeam given as

individual diamond symbols. It should be noted that the abscissa of each curve

is normalised by the threshold stiffness for that particular combination of nb, L

(and hence κ) and a. Likewise Mcr is normalised by the appropriate value of MT

for each curve. Since the average error between LTBeam and analytical results

is 0.06%, the difference between the points and the curves is indistinguishable on

all the graphs in the current section.

128

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Analytical â= 0

Analytical â= 0.5

Analytical â= 1

LTBeam

nb = , L = 10.5 m3


Mcr

/ M

T

K / KT


nb = 3, L = 10.5 m.

129

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Analytical â= 0

Analytical â= 0.5

Analytical â= 1

LTBeam

nb = , L = 10.5 m5


Mcr

/ M

T

K / KT


nb = 5, L = 10.5 m.

130

It can be seen that for a = 1.0, the curves are almost exactly linear, but as the

restraint height is lowered, this linearity is lost, especially for a = alim. The figures

also show how, as the number of restraints is increased, the variation between the

different curves diminishes, as they all tend towards an approximately elliptical

form.

A final point to note is that while on first inspection it may appear that providing

a lower restraint height actually leads to a greater critical moment, the value of

KT for a = alim is many times greater than that for a = 1, and so in reality if the

same value of K was specified for both, the value of Mcr corresponding to a = 1

would in fact be greater.

Influence of varying κ

By varying the span of the beam, a range of values of κ can be trialled. As

shown in Figures 5.4 and 5.5, while varying the beam parameter leads to a slight

difference in magnitude of the respective values of Mcr/MT where a larger value

of κ corresponds to a greater critical moment, the form of the curves themselves

remains constant. Again, it can be seen that varying the restraint height has a

much more noticeable effect on the nature of the curves.

Influence of varying the number of restraints

The most noticeable difference in the nature of the moment–stiffness curves arises

from varying the amount of restraints provided to the beam. As described in

Chapter 4, the number of node-displacing buckling modes that exist is equal to

the number of restraints, and so as the stiffness is increased, the critical buckling

mode changes from the first mode to the second and so on. Again it is important

to remember that this only occurs for a > alim; at lower restraint heights the

first mode is often always critical. For each successive mode the overall slope

131

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

, nb = 3â = â lim

Analytical = 8.75 m

Analytical = 10.5 m

Analytical = 12.25 m

Analytical = 14 m

LTBeam

M/ M

T

K / KT

L

L

L

L

L

cr

Analytical = 7 m

Figure 5.4: Normalised moment–stiffness curves for varying beam spans, nb = 3,

a = alim.

132

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Analytical = 7 m

Analytical = 8.75 m

Analytical = 10.5 m

Analytical = 12.25 m

Analytical = 14 m

LTBeam

M/ M

T

K / KT

nb = 3â = 1.0,

L

L

L

L

L

cr

Figure 5.5: Normalised moment–stiffness curves for varying beam spans, nb = 3,

a = 1.0.

133

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

LTBeam

M/ M

T

K / KT

, L = 10.5 mâ =

Analytical nb = 1

Analytical = 3

Analytical

â

cr

lim

nb

nb = 5

Figure 5.6: Normalised moment–stiffness curves for varying numbers of restraints,

L = 10.5 m, a = alim.

of the curve, while not constant, can be seen to diminish as the mode number

approaches nb + 1.

It is interesting to note that for a = alim the curves are almost continuous. The

reason behind this can be found by re-examining the graphs of γs,T versus η in

§4.3.4. As a → alim, the latter end of the curve flattens out, meaning that the

individual threshold stiffnesses for the higher modes are all quite similar. This in

turn implies that the stiffnesses at which there is a transition from one critical

mode to the next all lie towards K = KT and so one particular node-displacing

mode is critical for a large range of stiffnesses. This behaviour is contrasted by

the situation for a = 1, where the transitional stiffnesses are clearly separated,

and the progression from one critical mode to the next is much more obvious.

134

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

LTBeam

Mcr

/ M

T

K / KT

L = 10.5 mâ = 1.0,

Analytical nb = 1

Analytical nb = 3

Analytical nb = 5

Figure 5.7: Normalised moment–stiffness curves for varying numbers of restraints,

L = 10.5 m, a = 1.0.

135

5.2 Validation of results for deflected shape

Since LTBeam is only capable of providing the load at which a beam exhibits a

bifurcation (owing to there being no capability to include an imperfection in the

beam), Auto-07p (Doedel & Oldeman, 2009) was used to solve the governing

ODEs for the case of a beam with discrete braces. These solutions for the lateral

deflection and angle of twist along the span of the beam provide the deflected

shape of the beam, which in turn can be applied to determine the forces induced

in the restraints.

5.2.1 Description of analysis method

Auto is a continuation and bifurcation software originally developed at Concor-

dia University in Montreal by Doedel (1981), based on techniques developed by

Keller (1977). Amongst its many features is that, upon being given a set of dif-

ferential equations and boundary conditions, it can provide numerical solutions

for the system variables. It also locates the branch points of the problem and can

switch branches if so desired, as was done for the perfect case when examining

postbuckling in the unrestrained beam of §3.2.4. The program starts at an initial

set of conditions and then uses pseudo-arclength continuation to find satisfactory

solutions within a particular solution family (to a specified tolerance).

The discrete nature of the distribution of the restraint stiffness along the span of

the beam means that the potential energy formulation given by Equation (4.9) is

not continuous, as the strain energy in the restraints is only evaluated at distinct

points. This leads to computational difficulties in Auto but can be overcome by

replacing the discrete strain energy expression:

UR =

nb∑i=1

1

2KX2

i , (5.1)

136

with the stiffness distribution expressed as a continuous function thus:

UR =

∫ L

0

1

2kkfX

2i dx , (5.2)

where k is an equivalent continuous stiffness equal to K/L, and kf is a function

describing the distribution of the restraint stiffness throughout the span. After

rescaling the equations in the same manner as done in §3.2.4 and performing the

calculus of variations, the governing ODEs are found to be:

u′′′′ − u′′′′o +ML

EIzφ′′ + kf

(kL4

EIz

)(u+

a

Lφ− uo

)= 0 (5.3)

φ′′′′ +ML3

EIwu′′ − L2GIt

EIwφ′′ + akf

(akL6

EIw

)(u+

a

Lφ− uo

)= 0 (5.4)

It is assumed that the rescaled initial imperfection, uo = e1 sin(πx), with e1 =

1/500. The stiffness distribution can be modelled by any arbitrary function now,

including a summation of Dirac delta functions centred at the restraint nodes.

This, however, causes the functions to be multivalued and again leads to com-

putational difficulties for Auto. The solution was to model the stiffness of the

restraints using triangular-shaped distributions, as demonstrated in Figure 5.8,

which may be thought to model the actual width of a restraint in practice better.

If the width of each distribution is 2b and the slope is 1/b2, then the height is 1/b

and the area under each distribution is equal to unity, as would be the result of

integrating a Dirac delta function. A value of b = 0.01 was chosen for the current

work, after it was found to provide the most satisfactory results; sharper distri-

butions created problems as Auto was sometimes unable to adapt the arclength

for the continuation properly due to the size of the discretisation used, leading

to discontinuities in the load–deflection plots.

The parameters varied and the individual values they assumed are given in Table

5.2, which are much the same as those for the validation of the results for the

critical moment. The cross-section used was, again, a 457 × 152 × 82 UB. As

before, the normalised discrete stiffness K/KT is varied between 1/8 to 1.

A separate run of Auto was required for each combination of parameters, so

in all, 720 runs were performed. In each run, Auto varied the load parameter

137

0

25

50

75

100

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

kf

x/L

Figure 5.8: The piecewise stiffness distribution function kf , for a beam with three

restraints, and a restraint width of L/50.

Parameter Values assumed

nb 1, 2, 3, 4, 5, 6

a 0, 0.5, 1

L (m) 7, 8.75, 10.5, 12.25, 14

Table 5.2: Values assumed by the parameters in the validation using Auto.

138

incrementally and found the corresponding solutions of u and φ. For each run,

a maximum of 200 points were calculated, with Auto outputting the detailed

results for every 40th point. The number of points successfully calculated for each

run was dependent on how far along the load-deflection curve Auto was able to

find convergent solutions. A divergent solution resulted in premature termination

of the run which stemmed from the piecewise definition of the restraints. In all,

2801 observations were recorded, equivalent to just under four distinct load levels

per run.

5.2.2 Results for lateral deflection

Each solution for the deflected shape that Auto provided was compared with

the same functions as calculated using Equation (4.38). For each observation,

the coefficient of determination, R2, was calculated between the results from the

two methods. It should be stressed that there are no real statistical implications

of the R2 values; they are being used here purely to demonstrate qualitatively

the goodness-of-fit of the analytical results. The R2 value is defined as:

R2 = 1− SSerrSStot

(5.5)

where SStot =∑

i(uAUTO,i − uAUTO)2, SSerr =∑

i(uAUTO,i − uanalytical,i)2 and u

is the mean lateral deflection across all points, with the ith point representing a

particular distance along the length of the beam. There were 150 available points

per observation.

The results were then separated into six categories based on their associated R2

values, as shown in Table 5.3. Figures 5.9 to 5.14 show a typical example from

each category, so as to illustrate the goodness-of-fit associated with each value of

R2.

As can be seen from the graphs, there is close agreement between the results of

the two methods when R2 > 0.96, which is the case for over 92% of observations.

139

Value of R2 Observations Percentage of total

> 0.999 1936 69.1

0.99− 0.999 446 15.9

0.98− 0.99 81 2.9

0.96− 0.98 64 2.3

0.90− 0.96 70 2.5

< 0.90 204 7.3

Table 5.3: Distribution of R2 values between analytical and Auto results for the

lateral deflection.

-0.0005

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

AUTO

Analytical

x/L

u/L

Figure 5.9: Typical graph of u/L against x/L for R2 > 0.999 (L = 7 m, a = 0,

nb = 5, M/MT = 0.676 and K/KT = 0.5).

140

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

AUTO

Analytical

x/L

u/L

Figure 5.10: Typical graph of u/L against x/L for 0.99 6 R2 < 0.999 (L =

10.5 m, a = 1, nb = 3, M/MT = 0.484 and K/KT = 0.125).

141

-0.0005

0.004

x/L

u/L

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

AUTO

Analytical

Figure 5.11: Typical graph of u/L against x/L for 0.98 6 R2 < 0.99 (L = 8.75 m,

a = 0, nb = 2, M/MT = 0.949 and K/KT = 0.875).

142

-0.0005

0.004

x/L

u/L

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

AUTO

Analytical

Figure 5.12: Typical graph of u/L against x/L for 0.96 6 R2 < 0.98 (L =

12.25 m, a = 0, nb = 3, M/MT = 0.706 and K/KT = 0.25).

143

-0.0005

x/L

u/L

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

AUTO

Analytical

Figure 5.13: Typical graph of u/L against x/L for 0.90 6 R2 < 0.96 (L = 14 m,

a = 0, nb = 4, M/MT = 0.575 and K/KT = 0.125).

144

-0.0005

x/L

u/L

0

0.0005

0.001

0.0015

0.002

0.0025

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

AUTO

Analytical

Figure 5.14: Example of a highly divergent graph where R2 < 0.90. A close

match between the curves is not possible due to the highly asymmetric nature of

the curve produced by Auto.

145

Also, this high level of agreement is present for each value assumed by the param-

eters in the validation program. Where R2 < 0.96, the results can be somewhat

different, such as in Figure 5.13, or diverge significantly from each other, such

as in Figure 5.14. With the vast majority of results closely matching it can be

said that the analytical formula presented in the current work provides accurate

solutions for the lateral deflection of the beam. Another important feature to

note, particularly in Figures 5.9 and 5.11, is the multiple inflection points present

in the curves. A single harmonic representation of the lateral deflection func-

tion u would be unable to model such curves accurately, thus highlighting the

considerable increase in accuracy obtained by applying a full harmonic analysis.

5.2.3 Results for angle of twist

The method of comparison between the analytical and Auto results for the angle

of twist was the same as for the lateral deflection, with the analytical results

being computed from Equation (4.39). Table 5.4 details the distribution of the

observations by R2 value, which carry the same significance as for the lateral

deflection results. The high amount of agreement between the two methods

demonstrates that the analytical method is capable of providing accurate results

for the angle of twist along the length of the beam. As is the case for the results

for the lateral deflection, the high level of close agreement between analytical and

numerical results is present across all parameter values.

5.3 Summary

In the current chapter, two separate features of a beam with discrete restraints

have been examined: the critical moment and the deflected shape. The results of

the analysis detailed in Chapter 4 has been compared with equivalent results from

two computer programs, LTBeam and Auto. Having compared the analytical

146

Value of R2 Observations Percentage of total

> 0.999 2020 72.1

0.99− 0.999 392 14.0

0.98− 0.99 66 2.4

0.96− 0.98 69 2.5

0.90− 0.96 73 2.6

< 0.90 181 6.5

Table 5.4: Distribution of R2 values between analytical and Auto results for the

angle of twist, φ.

method with its numerical counterpart for both features across a comprehensive

set of parameter values and observing the very close agreement between them, it

can be said that the results of the computational analysis validate the analytical

method.

147

Chapter 6

Design formulae

In the current chapter, the expressions derived in Chapter 4 are now used to create

practical design formulae, with the intention of not being overly conservative, thus

increasing the efficiency of the design of bracing members. An approximation

for the threshold stiffness, procedures for determining the critical moment of a

beam that is not fully braced and methods to determine the forces induced in

the restraints are outlined. Additionally, in the case of the restraints being in

the form of single members attached orthogonally to both sides of the primary

beam, so that their restraint stiffness is equivalent to their axial stiffness and that

their strength in compression is neglected, a procedure is outlined whereby the

requirements for stiffness and strength are optimised so that the cross-sectional

area required by both is equal. The design expressions for critical moment are

intended for use in conjunction with §6.3.2.2 and §6.3.2.3 of EN 1993-1-1 (Comite

European de Normalisation, 2005) by determining a value for the elastic critical

moment Mcr to be used to calculate the moment reduction factor χLT.

The current chapter also provides comparisons between the results found in the

current work for the threshold stiffness, critical moment and restraint force with

those found by applying design rules derived analogously from column design

expressions, which is in fact the current best practice when designing a beam

148

with elastic intermediate restraints; the 2.5% restraint force ratio of BS 5950 is a

prime example of this. It should be noted that all calculations and comparisons

regarding restraint force ratios in the current chapter are based on the assumption

that the initial imperfection uo = (L/500) sin(πx/L).

The following section examines methods to calculate parameters necessary for

the determination of the properties mentioned above quickly.

6.1 Auxiliary calculations

The current section details the determination or approximation of the main pa-

rameters required to calculate the threshold stiffness, critical moment and maxi-

mum restraint force of a beam with discrete restraints.

6.1.1 Approximations for κ

In designing a laterally unrestrained beam (or beam segment) against lateral-

torsional buckling, both BS 5950 and EC3 specify that a beam slenderness be

calculated, based on the ratio between the major-axis bending resistance given

by Wyfy, where Wy is either the major-axis plastic or elastic section modulus

(Wpl,y or Wel,y, respectively) depending on the cross-section classification and fy

is the yield stress, and the elastic critical moment of the unrestrained beam, Mob,

which upon re-examination of Equations (3.13) and (3.14), can be given by:

Mob =

(π2EIzL2

)hs2

√1 + κ. (6.1)

In the Eurocode, the generalised nondimensional lateral buckling slenderness is

defined as:

λLT =

√WyfyMob

, (6.2)

but no instruction or guidance is provided as to how the critical moment should

be determined. Using the definitions of the current work, the above can be recast

149

as:

λLT =

√2WyfyL2

π2EIzhs(1 + κ). (6.3)

This can be restated, in a manner similar to that of BS 5950, as:

λLT = UVλz√βw, (6.4)

by defining the following parameters thus:

U =

√2Wpl,y

Ahs,

V = (1 + κ)−1/4 ,

λz =L

iz,

λ1 = π

√E

fy,

λ = λ/λ1,

βw =Wy

Wpl,y

, (6.5)

where A is the cross-sectional area of the beam and iz =√Iz/A, the minor-axis

radius of gyration.

BS 5950 and the Eurocode Non-Conflicting Complementary Information (NCCI)

document SN002 (Lim, 2005) both provide alternative definitions of the param-

eter V , which can be perceived as implicitly defining approximations for κ. The

approximation arising from the definition of BS 5950 is:

κ ≈ 1

20

(λzxt

)2

, (6.6)

where xt is the torsional constant provided in member section tables and can

be approximated by 0.566hs√A/It. The document SN002 defines V similarly

but uses the conservative approximation xt ≈ h/tf , resulting in the following

approximation for κ:

κ ≈ 1

20

(λzh/tf

)2

. (6.7)

A further definition which may be of use is that of κ = (L/`)2, where ` =

π√EIw/GIt and has the advantage of being dependent on cross-sectional prop-

erties only (assuming that E/G = 2(1 + ν) = 2.6, where the Poisson’s ratio of

150

steel ν = 0.3). The value of ` varies from 1.34 m to 12.77 m across the range of

UB sections. Finally, the related factor κs = κ/(nb + 1)2 can be approximated

similarly by substituting s for L in the above expressions.

6.1.2 Conditions for “fully restrained” design

In order to achieve the conditions for “fully restrained” design in EC3 i.e. so that

lateral-torsional effects can be ignored, the lateral-torsional slenderness λLT must

be below a certain limiting slenderness λLT,0, which the Eurocode allows to be

specified by the relevant National Annex, up to a maximum value of 0.4. The

minimum elastic critical moment required so that the beam can be designed as

fully restrained, Mcr,0, is thus given by:

Mcr,0 =Wxfyλ2LT,0

. (6.8)

The maximum achievable critical moment for a beam with discrete restraints is

of course the threshold moment, MT , so an optimised design can be found by

setting Mcr,0 = MT . The value of s = L/(nb + 1) that satisfies this condition

is the maximum span that could be designed as fully restrained without any

intermediate restraints. From this value of s, limiting values can be found for

both L and nb so that, if restraints of stiffness K > KT are provided, the beam

can be designed as fully restrained and the effects of lateral-torsional buckling

may be ignored. If the number of restraints provided is fewer than the limiting

value then even if the beam is braced fully, it is nevertheless still susceptible to

lateral-torsional buckling and similarly if L is greater than the limiting span. The

value of smax is given by:

smax =1

2

(izλLT,0

U√`βw

)21 +

√1 + 4

(U√βw`

izλLT,0

)41/2

. (6.9)

For UB sections, a conservative (1–7.5%, average 4% smaller) estimate is given

by:

smax = 41.7εiz, (6.10)

151

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

smaxApprox

smaxActual

Figure 6.1: A comparison of actual smax values for UB sections with the corre-

sponding approximate values as calculated by Equation (6.10).

where ε =√

235/fy. Figure 6.1 shows how the approximated values of smax are

only slightly conservative across the range of UB sections (with the approximation

being unsafe for two sections). Hence reasonably accurate estimates of nb,min and

Lmax can be obtained from Equation (6.10). In a design scenario, it is often the

case that the span and number of restraints are specified a priori and cannot be

altered; in such situations a minimum required value of iz can be determined to

assist selection of an initial trial section for the beam.

6.1.3 Modifications for non-uniform moment

The NCCI SN002 document defines the generalised nondimensional slenderness

as:

λLT =1√C1

UVλz√βw, (6.11)

where C1 is a parameter dependent upon the shape of the major-axis bending

moment diagram, intended to take non-uniform bending moment distributions

into account. For a constant bending moment, which is assumed by the current

work and is the most onerous case, C1 = 1. In BS 5950, the analogous moment

152

modification factor mLT is applied to the buckling resistance moment to account

for the same effect. The Eurocode also defines a modification factor to be applied

to χLT to account for the effect of major axis bending in §6.3.2.3(2), although it

appears superfluous given that the C1 factor already takes non-uniform moment

distributions into account. If the beam is fully braced i.e. K > KT , then standard

moment diagram factors for use with unrestrained beams or beam segments, such

as those given in BS 5950, the NCCI document SN003 (Bureau, 2005), Galea

(2002) or Serna et al. (2006), are valid as the bracing nodes are restrained from

translating; a key specification of both Eurocode 3 and BS 5950. Hence, the

beam can be designed on the basis of its most critical beam segment according

to the specifications of the relevant design code.

In the case of the beam not being braced fully, i.e. if K < KT , in the absence of

data to support otherwise, it is recommended that the moment diagram factor

be set conservatively to unity.

6.2 Design formulae for threshold stiffness

If the restraints are provided at a level above alim then the beam has the potential

to be braced fully. Assuming this to be the case, then the braces should possess a

stiffness that is not less than KT . The nondimensionalised threshold stiffness can

be approximated by the following formula, which is conservative when compared

with Equation (4.51):

γs,T =2 (1 + κs)

π(A0 + A1a), (6.12)

where:

A0 = 0.45 + 2.8νb,Tκs, (6.13)

A1 = 6.3νb,T + 2.2κs − 1, (6.14)

and the factor νb,T = {1 + cos [π/(nb + 1)]}−1. Values of νb,T for corresponding

numbers of restraints are given in Table 6.1. The threshold stiffness KT itself is

153

nb νb,T nb νb,T

1 1.000 6 0.526

2 0.667 7 0.520

3 0.586 8 0.516

4 0.553 9 0.513

5 0.536 10 0.510

nb →∞ νb,T → 0.500

Table 6.1: Values of νb,T used for calculating KT .

κ = 1 κ = 7 κ = 25

nb Max Min Avg Max Min Avg Max Min Avg

1 11.7 2.8 7.3 10.3 5.8 8.0 9.6 -0.9 3.0

2 13.3 1.2 6.8 6.3 -0.4 2.4 5.1 -6.0 -2.8

3 15.6 1.2 7.8 8.5 -2.7 2.5 1.2 -4.2 -3.2

Table 6.2: Maximum, minimum and average percentage errors between actual

and approximate values of γs,T as shown in Figures 6.2 to 6.4. Negative errors

correspond to ranges of restraint height where the approximate formula underes-

timates the threshold stiffness.

given by:

KT =

(EIzs3

)62 (1 + κs)

A0 + A1a. (6.15)

As can be seen from Figures 6.2 to 6.4, the design formula returns values of KT

that agree closely with the analytical result across all parameters, especially for

a low number of restraints. The difference between the values obtained from the

design formula and the exact value is at its largest for a = 1, but even then the

discrepancy is moderate. For tension side restraints, i.e. those positioned below

the shear centre, the values returned by Equation (6.12) become increasingly

conservative as the restraint height approaches alim. Maximum and average values

of the percentage error between actual and approximate values of KT shown in

Figures 6.2 to 6.4 are given by Table 6.2. As can be seen the formula tends to

underestimate the threshold stiffness for very high values of κ.

154

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

â

ActualApproximate

κ = 1κ = 7κ = 25

s,Tγ

Figure 6.2: A comparison of actual and approximate values of γs,T for nb = 1.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

â

ActualApproximate

κ = 1κ = 7

κ = 25

s,Tγ


155

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

â

ActualApproximate

κ = 1κ = 7κ = 25

s,Tγ


6.3 Approximations for critical moment

If it is assumed that the restraint stiffness K exceeds KT then the elastic critical

moment of the beam is the threshold moment, MT , found by substituting s

instead of L in expressions for Mob:

MT =π2EIzs2

(hs2

)√1 + κs. (6.16)

However, if it is the case that K < KT , Mcr can be approximated using the meth-

ods detailed below, based on the number of restraints provided. The methods

attempt to exploit the nature of the moment–stiffness curves, which tend towards

linear forms as a → 1 but become noticeably curved as a → 0. It is reiterated,

however, that using restraints with a stiffness below the threshold amount leads

to higher forces being induced in the restraints, thus increasing the demands of

the design.

The approximate formulae are found by fitting curves as close as possible to the

actual moment–stiffness curves as determined from the methods of Chapter 4.

With the stiffness and moment axes normalised against KT and MT , respectively,

the formulae are designed to pass through the points (0,µo) and (1,1), where

156

κ = 1 κ = 7 κ = 25

a Max Avg Max Avg Max Avg

0 18.6 12.6 10.3 6.9 8.1 5.4

1 2.8 1.9 3.5 2.4 4.4 2.9

Table 6.3: Maximum and average percentage errors between actual and approx-

imate values of µcr shown in Figures 6.5 and 6.6.

µo = Mob/MT and can also be given by:

µo =

√1 + κ

(nb + 1)2√

1 + κs, (6.17)

and can vary between 1/(nb + 1)2 and 1/(nb + 1) depending on the value of κ.

6.3.1 Single restraint

Trahair (1993) gave the following formula for Mcr, where a = 0, which provides

almost exact results for all values of κ:

Mcr

Mob

≈

√2400 + 51KL3/EIz2400 +KL3/EIz

. (6.18)

However, it loses accuracy quite rapidly as a is increased. The following linear

approximation provides conservative results for the elastic critical moment:

Mcr

MT

= µo +K

KT

(1− µo) . (6.19)

The accuracy of the above approximation increases the closer the restraints are

positioned to the compression flange. Figures 6.5 and 6.6 compare Mcr as cal-

culated by the methods of Chapter 4 and by Equation (6.19). Table 6.3 shows

the maximum and average percentage errors for the approximate formula. For a

restraint positioned at the compression flange, the error is small but for restraint

heights more remote from the compression flange, the linear approximation be-

comes less accurate.

157

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

K / KT

crμActual

κ = 1

κ = 7

κ = 25

Approximate

Figure 6.5: A comparison of actual and approximated values of µcr for nb = 1

and a = 0.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

K / KT

μcr

Actual

κ = 1

κ = 7

κ = 25

Approximate


and a = 1.

158

6.3.2 Two restraints

In the situation where nb = 2, a piecewise-linear function can be used to ap-

proximate Mcr, based on establishing the transition point whereupon the second

node-displacing buckling mode becomes the critical mode. In order that the

approximated values of Mcr are safe, the formulae in the current section aim to

overestimate the transition stiffness and underestimate the transition critical mo-

ment so that conservative (safe) critical moments are calculated. In general, the

approximate value of Mcr is given by:

Mcr

MT

= µo +K/KT

αtr

(µtr − µo) where K/KT < αtr,

Mcr

MT

= µtr +

(K/KT − αtr

1− αtr

)(1− µtr) where K/KT > αtr, (6.20)

where (αtr, µtr) are the coordinates of the transition point on a graph of µ against

K/KT .

For κ > 2 the following approximation for αtr provides values that lead to critical

moments that are only slightly conservative:

αtr = 0.14 + 0.03a. (6.21)

The approximation gains in accuracy as κ is increased. For κ < 2, the values of

αtr provided by the formula return less accurate results. Conservative values for

µtr are given by the following expression:

µtr =0.75(κ+ 5.5)

κ+ 7.5, (6.22)

which provides almost exact values for the transition critical moment when a = 1.

As shown in Figure 6.7, the accuracy of the formula diminishes for lower values

of a, but for κ > 5, the discrepancy is negligible. For quite low values of κ, the

formula is still quite accurate for 0.25 6 a 6 1, but for restraint height levels

below 0.25, Equation (6.22) returns noticeably conservative transition critical

moments. The swift increase in µtr as a → 0 is attributed to the moment–

stiffness curves for the first and second buckling modes becoming more curved

159

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

0 5 10 15 20 25

κ

μ

0

0.25

0.5

Approx.tr

μtr

Figure 6.7: Comparison of actual µtr values for various restraint heights a with

the approximate value obtained from Equation (6.22).

for smaller values of κ. Some indicative percentage errors are provided in Table

6.4.

6.3.3 Three or more restraints

As the number of restraints becomes higher, the moment–stiffness curves tend

towards an elliptical form in shape. The critical moment can be approximated

by an expression of the following form:

Mcr

MT

=

[σ + (1− σ)

K

KT

]√1− (1− µ2

o)

(1− K

KT

)2

, (6.23)

where σ = 1 for κ > 10. For κ < 10,

σ = 1− (2− a)(10− κ)

100. (6.24)

As can be seen from Figures 6.10 to 6.15, the approximation is particularly accu-

rate for a = 0, and increases in accuracy with nb. For low values of κ the formula

160

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

K / KT

μcr

Actual

κ = 1

κ = 7

κ = 25

Approximate


and a = 0.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

K / KT

μcr

Actual

κ = 1

κ = 7

κ = 25

Approximate


and a = 1.

161

κ = 1 κ = 7 κ = 25

a Max Min Avg Max Min Avg Max Min Avg

0 14.9 -7.7 6.8 12.4 0.0 6.5 7.7 -2.5 3.3

1 3.5 0.0 2.6 2.6 -0.1 1.6 2.6 -2.2 1.1


and approximate values of µcr for two restraints as shown in Figures 6.8 and 6.9.

Negative errors correspond to ranges of stiffness where the approximate formula

overestimates the critical load.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

K/KT

μcr

Actual

κ = 25

Approximate

κ = 0


and a = 0.

is also reasonably accurate while being conservative, although it becomes more

conservative for low restraint stiffnesses as κ is increased. Relevant percentage

errors are provided in Tables 6.5 to 6.7. The greatest discrepancies occur at stiff-

nesses where the transition between buckling modes occurs as the approximate

curvilinear formula cannot model the angular nature of the moment–stiffness

curve accurately at these points.

162

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Actual

κ = 25

Approximate

κ = 0

K/KT

μcr


and a = 1.

κ = 0 κ = 25

a Max Min Avg Max Min Avg

0 22.8 0.0 13.7 32.1 -1.6 13.8

1 14.8 -6.9 0.1 23.2 -2.3 7.1


and approximate values of µcr for three restraints as shown in Figures 6.10 and

6.11. Negative errors correspond to ranges of stiffness where the approximate

formula overestimates the critical load.

κ = 0 κ = 25


0 25.6 0.0 12.8 39.6 -0.7 17.1

1 17.5 -4.1 1.3 29.2 -1.1 9.1


and approximate values of µcr for four restraints as shown in Figures 6.12 and



163

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Actual

κ = 25

Approximate

κ = 0

K/KT

μcr


and a = 0.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Actual

κ = 25

Approximate

κ = 0

K/KT

μcr


and a = 1.

164

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Actual

κ = 25

Approximate

κ = 0

K/KT

μcr


and a = 0.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Actual

κ = 25

Approximate

κ = 0

K/KT

μcr


and a = 1.

165

κ = 0 κ = 25


0 27.0 0.0 11.6 43.9 -0.4 18.5

1 18.9 -2.8 1.8 33.0 -1.1 9.6


and approximate values of µcr for five restraints as shown in Figures 6.14 and



6.4 Restraint forces

It is assumed in the current section that the initial lateral imperfection of the

beam is described by a single half-sine wave, i.e. uo = e1 sin(πx/L). Given

that the imperfection is of this form, the values of F/P given in the current

section are the maximum values, i.e. the values at midspan. For an odd num-

ber of restraints, this is in fact the actual maximum value since there is a re-

straint located at midspan; for an even number of restraints, a correction factor

of sin (πnb/[2(nb + 1)]) can be applied to give the force in the most heavily loaded

restraint. The process involved in forming a simplified design equation for the re-

straint force ratio F/P exploits the form of Equation (4.59), which is asymptotic

to: (F

P

)∞

= 2π2 sin

(iπ

nb + 1

)(a+ M

1 + κ− M2

)[1

(nb + 1)S2

]e1L, (6.25)

and also to:

γ∞ = − 1

(nb + 1)S2

≡ γs,∞ = − 1

Ss,2, (6.26)

thus allowing the restraint force ratio to be expressed in the following form:

F

P= (F/P )∞

(γ

γ − γ∞

)= (F/P )∞

(K

K −K∞

). (6.27)

If a particular maximum restraint force ratio is desired, expressed as α(F/P )∞,

166

//

=1.5

=1.51.5-1 = 3

Figure 6.16: Demonstration of the calculation of minimum stiffness requirement

to achieve a desired restraint force; in this example, the value of F/P is to be

limited to 1.5(F/P )∞, hence a minimum stiffness of K = 3K∞ is required.

then the minimum level of stiffness required to achieve this is α/(α − 1)K∞, as

demonstrated by Figure 6.16. Conversely, if the ratio β = K/K∞, then the cor-

responding F/P = β/(β−1)(F/P )∞. From these relationships between restraint

force (i.e. strength) and restraint stiffness, an optimum design can be determined.

For convenience, sin[iπ/(nb + 1)] is assumed to be equal to unity so that all the

restraints are designed to resist the maximum possible force induced. It is initally

assumed that the beam is to be designed to support load levels close to µ = 1;

the design rules outlined in the current section are based upon this conservative

assumption. Methods to account for lower loading levels are treated subsequently.

6.4.1 Calculation of K∞

Based on the assumption of µ = 1 and the definition of Equation (6.26) K∞ =

KT,1, where KT,1 is the threshold stiffness level for the first node-displacing buck-

167

nb νb,∞ nb νb,∞

2 2.000 7 13.14

3 3.414 8 16.58

4 5.236 9 20.43

5 7.646 10 24.69

6 10.10 11 29.35

Table 6.8: Values of νb,∞ used for calculating K∞.

ling mode. Equation (6.12) can be used to calculate K∞, except the factor νb,T

is replaced with:

νb,∞ = {1− cos [π/(nb + 1)]}−1 . (6.28)

It is reiterated that for nb = 1, K∞ = KT . Values of νb,∞ for calculating K∞

are provided in Table 6.8. As is the case for calculating KT , the values of K∞

obtained are more conservative as κ is increased and also for higher restraint

heights, as shown by Figures 6.17 and 6.18. Percentage errors are given in Table

6.9. Alternatively, the ratio K∞/KT may be of more interest; the minimum

possible value is obtained by taking the limit as κ→∞. It is given by:

(K∞KT

)min

=1− cos

(π

nb+1

)1 + cos

(π

nb+1

) =νb,Tνb,∞

. (6.29)

For κ > 5, or for a > 0.2, this minimum ratio is in fact a very close approximation

to the actual ratio between the two stiffnesses. The value becomes less accurate

for lower restraint heights and lower stiffnesses, and is unsafe since conservative

restraint force values are found by overestimating K∞. For a = κ = 0, the

approximation of K∞ = 0.9KT returns quite accurate results.

6.4.2 Calculation of (F/P )∞

Assuming again that µ = 1, it is found that (F/P )−1∞ has a form that is almost

exactly linear in a, greatly aiding simplification of its formulation. It is found

168

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

â

ActualApproximate

κ = 1κ = 7κ = 25

s,∞γ

Figure 6.17: A comparison of actual and approximated values of γs,∞ for nb = 2

and µ = 1.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

â

ActualApproximate

κ = 1κ = 7κ = 25

s,∞γ

Figure 6.18: A comparison of actual and approximated values of γs,∞ for nb = 3

and µ = 1.

169

κ = 1 κ = 7 κ = 25


2 7.6 4.0 5.6 17.6 0.9 11.8 20.5 2.6 13.6

3 7.0 3.5 4.1 15.2 2.3 11.4 26.6 2.9 17.9


and approximate values of γs,∞ as shown in Figures 6.17 and 6.18. Negative

errors correspond to ranges of restraint heights where the approximate formula

underestimates the actual value.

that the slope of this linear relationship can be expressed as being independent

of κ for nb > 3, but a correction factor is required otherwise. The intercept value

of the linear relationship is found by fitting a curve through values of (F/P )∞

for a = 0. These values are at a maximum for κ = 0, and vary almost linearly

with nb. For κ > 0, a correction factor is required to fit the curve accurately.

The design formula is thus:(F

P

)∞

= 5(e1L

)[σ(nb + 1)

2a+

(1 + κ)σ

5.5 + 4.5nb

]−1. (6.30)

For κ < 5(nb − 1), σ = 1; otherwise the following rules can be used to calculate

σ for κ 6 50 (a value that greatly exceeds the range of practical UB values):

• For nb = 1, σ = 0.9− 0.01κ (but > 0.7).

• For nb = 2, σ = 0.99− 0.006κ (but > 0.8).

• For nb = 3, σ = 1− 0.004κ (but > 0.85).

• For nb > 4 a conservative value of 0.95 can be taken.

In their treatment of restraint forces, Wang & Nethercot (1990) advised that a

force of 1% of the axial load in a flange may generally be assumed as the bracing-

strength requirement. For restraints positioned close to the compression flange

of a sufficiently high stiffness this is indeed true, but for a combination of low

restraint heights and low values of κ, this is not the case, since F/P can also

170

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

F/P(%)

â

ActualApproximate

κ = 1κ = 7κ = 25

Figure 6.19: A comparison of actual and approximated values of (F/P )∞ for

nb = 1 and µ = 1.

exceed the 2.5% limit set out by BS 5950, as clearly seen in Figures 6.19 to 6.21.

As the number of restraints provided gets larger, the effect on F/P is different

depending on the restraint height; the ratio is reduced for larger restraint heights

while becoming larger for restraints positioned closer to the shear centre. Table

6.10 shows percentage errors for the graphs.

κ = 1 κ = 7 κ = 25


1 4.9 -2.8 0.3 2.0 -0.4 0.4 -14.6 -16.0 -15.6

2 5.2 -3.8 2.0 2.1 0.4 1.0 -3.0 -6.7 5.8

3 2.3 -6.2 0.0 3.1 1.7 2.2 -2.3 -5.5 -4.8


and approximate values of (F/P )∞ for as shown in Figures 6.19 to 6.21. Negative

errors correspond to ranges of restraint heights where the approximate formula

underestimates the actual value.

171

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

F/P(%)

â

ActualApproximate

κ = 1κ = 7κ = 25


nb = 2 and µ = 1.

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

F/P(%)

â

ActualApproximate

κ = 1

κ = 7κ = 25


nb = 3 and µ = 1.

172

Curve αLT λLT,lim

a 0.21 1.34

b 0.34 1.64

c 0.49 2.08

d 0.76 2.98

Table 6.11: Values of the limiting slenderness, below which µmax < 1, correspond-

ing to the imperfection factors of EC3.

6.4.3 Modifications for µ < 1

In the notation of the Eurocode, unless the beam is being designed as “fully

restrained”, the actual bending resistance moment, Mb,Rd is reduced from the

cross-sectional bending resistance, Mc,Rd = Wyfy by a factor χLT. It can be

assumed conservatively that the applied moment M is approximately equal to

the reduced resistance. If it can be assumed that the beam is fully braced i.e.

K > KT and thus Mcr = MT , then λLT =√Wyfy/MT . Since µ = M/MT , a

maximum possible value of µ can be derived since the beam will be over-designed.

This value is µmax = λ2LTχLT, which is a function of λLT and the imperfection

factor αLT used to calculate χLT.

For a “fully restrained” beam where χLT = 1, µmax = λ2LT,0 = 0.16 and then varies

up to a maximum value of 1 (see Figure 6.22), which occurs at the slenderness

where the value of Mb,Rd predicted by the Eurocode curves agree with the value of

the critical moment obtained from theory. The value of this slenderness, λLT,lim,

is determined by the value of αLT. When converted into a corresponding value of

κ, a negative result is found for all but the most slender of beams, implying that

µmax < 1 for practically all cases. Values of λLT,lim corresponding to the values

of αLT provided by the Eurocode are listed in Table 6.11. If the beam can be

designed as “fully restrained”, the maximum possible value of µ = 0.16.

As can be seen from Figure 6.22, the use of a value of µ = 1 is conservative,

particularly for low slendernesses. The approximations of the current section can

173

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

λLT

maxμ

λLT,0 = 0.4

Curve aCurve bCurve cCurve d

Figure 6.22: Values of µ, assuming the beam is fully braced and there is no

overdesign i.e. MEd = Mb,Rd. The curves depend on the value of αLT from

§6.3.2.2 of the Eurocode, which defines different buckling curves.

be used to obtain the values of (F/P )∞ and K∞ for lower loading levels.

Modification of K∞

A quick but conservative modification can be made by simply scaling K∞,µ=1 by

µ. This increases in accuracy with increasing nb, especially for low values of κ

and nb > 4. However, if greater accuracy is desired then the following method

can be applied, based of the value of µo. It is found from examining the form of

S2 that as µ→ µo, S2 →∞ and hence K∞ → 0. An approximation for K∞ can

hence be obtained through the application of the following linear modification

factor, with only moderately conservative results:

K∞K∞,µ=1

=µ− µo1− µo

. (6.31)

If µ < µo, then K∞ < 0 and for K > 0, 0 < F/P < (F/P )∞, owing to the

function F/P passing through the origin (see Figure 6.23). Hence the value of

(F/P ) corresponding to a stiffness K > KT can be taken as being equal to

(F/P )∞, and the actual value of K∞ need not be calculated.

174

‐0.5 0.5 1.5 2.5 3.5 4.5‐1

0

1

2

3

4

5

F/P (%)

K/KT

Figure 6.23: An example of a scenario where K∞ < 0. For positive, i.e. real,

restraint stiffnesses, the value of F/P approaches (F/P )∞ asymptotically from

beneath. Hence, a reasonably accurate conservative estimate can be obtained by

taking F/P = (F/P )∞.

Modification of (F/P )∞

For a = 0, an approximation for (F/P )∞ can be found by simply scaling the

value found in §6.4.2 by µ. For a = 1, simply using the value of (F/P )∞ for

µ = 1 provides only slightly conservative results, with accuracy improving for

higher restraint values and low values of κ. If a 6 0.15κ/nb, greater economy can

be achieved by using a scaling factor of√µ i.e. (F/P )∞ =

√µ(F/P )∞,µ=1.

6.5 Optimisation of stiffness and strength re-

quirements

In the case where bracing members are attached to a system of primary beams

orthogonally, then the stiffness of the bracing members is given by K = EAb/Lb,

where the subscript b denotes properties of the bracing member. If it is assumed

that the bracing members only act in tension then the strength, or resistance,

of the brace is given by F = Abfy. This implies that only bracing members

175

on a particular side act to restrain the beam when it deflects. The design of

the bracing members can be optimised by equating the area demands for both

properties, i.e. strength and stiffness, resulting in the following expression for the

cross-sectional areas of the bracing members:

Ab =K∞LbE

+M

hsfy

(F

P

)∞. (6.32)

It should be noted, however, that this formula can return values for Ab that

correspond to stiffnesses that are less than KT , since the formula is based on

finding the ratio of K/K∞ at which the area demands are equivalent. This

condition occurs when:

KT > K∞ +

(F

P

)∞

PE

Lbfy. (6.33)

There are numerous factors that influence the minimum value of KT for when

this condition occurs, and thus finding a convenient rule is inhibited. It suffices

to say that, in general, the condition prevails for (i) a high number of restraints,

(ii) high values of L/Lb i.e. for relatively short brace lengths, and (iii) low values

of µ. Generally, the condition does not prevail for high values of a. In cases

where the condition is satisfied, restraints of stiffness K > KT are be able to

resist the load induced in them, and thus the design value of Ab should be based

on satisfying the stiffness demand. Alternatively, in the interest of convenience,

a maximum level of restraint force can be specified and the corresponding value

of K can be found using the relation of §6.4.

6.6 Comparison with equivalent column rules

In the current section the practice of applying rules to beams that were originally

intended for use with columns is examined. In theory the column rules should be

conservative since less of the cross-section in a beam is actually in compression.

It is found that this is indeed the case for beams restrained near the compression

flange. However, for lower restraint heights it is found that the rules are consid-

176

erably unsafe. The amount by which the rules are either unsafe or conservative

is found to be a function of the beam parameter κ.

6.6.1 Critical moment for beams with a single restraint

The SCI publication P093 (Nethercot & Lawson, 1992) states that a conserva-

tive approach to the problem of beam bracing behaviour can be provided by an

equivalent column analysis. For an axially loaded column braced at midspan by

an elastic restraint of stiffness K, the critical load can be given by:

Pcr = PE +3

16KL, (6.34)

up to a limiting stiffness of 16π2EIz/L3 – it is assumed for convenience of compar-

ison that the column buckles about its minor axis – whereupon the second mode

of buckling, i.e. the internodal mode, is critical. By assuming that the maximum

compressive force in a beam with a constant bending moment is P = M/hs, an

analogous expression for µcr is obtained:

µcr =16 + 3KL/PE

16√

4 + κ. (6.35)

The value of µo as predicted by the formula, i.e. the value of µ when K = 0, is

thus given by 1/√

4 + κ, while the slope of the plot of µcr against K/KT is given

by:∆µcr

∆(K/KT )=

3π2γs,T

2√

4 + κ. (6.36)

The slope above and µo are equivalent to their counterparts in the linear ap-

proximation of §6.3.1, which is conservative itself, when κ = 3 and a = 0.921.

For larger values of κ, both the intercept value i.e. µo and the slope becomes in-

creasingly underestimated and hence the values of Mcr predicted by this method

become increasingly conservative. As the restraint height is lowered, the slope

increases, so for particularly low restraint heights, there exist ranges of stiffnesses

where the method returns conservative estimates for the critical moment, and

ranges where the estimates are unsafe. For top flange bracing the method re-

turns conservative values for all but the smallest beams.

177

6.6.2 Threshold stiffness

As mentioned previously, an axially-loaded column with a single restraint located

at mid-height buckles into the second mode if the stiffness of the restraint K >

16π2EIz/L3. This value can be nondimensionalised as γs = 2/π2 ≈ 0.2. This

varies from being greatly conservative to greatly unsafe across the standard ranges

of a and κ: for a = 1 the column rule tends to provide estimates that are twice

too large; for a = 0, the predicted values range from being 15% too low to nearly

5.5 times too low depending on the value of κ.

6.6.3 Restraint forces

In §6.4.2 it was shown how the blanket rule of a 1% restraint force ratio suggested

by Wang & Nethercot (1990) for beams was adequate, if somewhat conservative

for restraints positioned closer to the compression flange. For lower restraint

heights, this rule and also the BS 5950 rules of 2.5%, which is based on column

buckling theory, has been shown to underestimate the actual restraint force ra-

tio by a considerable margin. Banfi & Feltham (1999) provided expressions for

the restraint force ratio for a simply-supported axially-loaded strut. The cases

examined were those of (i) a single restraint attached at an arbitrary point along

the span of the strut, providing an expression derived by Al-Shawi (1998) who

also derived similar expressions for other end restraint conditions, (ii) restraints

attached at third points and (iii) multiple restraints. For the case of a single

restraint at midspan the following expression is given:

F

P=e1L

1PE

P− 1

[PEKL

P

PE+

1

2π

√PEP

tan

(π

2

√P

PE

)− 1

4

]−1. (6.37)

For two restraints, the force F in each restraint is given by:

F

P=

√3

2

e1L

1PE

P− 1

PEKL

P

PE+

sin(π3

√PPE

)π√

PPE

[2 cos

(π3

√PPE

)− 1] − 1

3

−1 . (6.38)

178

nb = 1 nb = 2

a = 0 16.9 52.6

0 < a < 1 16.9− 24.7a+ 8.4a2 52.6− 74.5a+ 26.0a2

a = 1 0.6 4.1

Table 6.12: Values of κlim.

Corresponding equations for (F/P )∞ are obtained by setting the term in the

denominator (PE/KL)(P/PE) = 0. Given a particular nondimensional restraint

height a, it is possible to find at what value of κ the value of (F/P )∞, as calculated

by the column rules, is equal to the corresponding value calculated using the beam

rules of the current work. This value is denoted κlim. If κ < κlim then the column

rules provide unsafe values; if κ > κlim then the column rules provide conservative

values. For the two cases above, values for a = 0 and a = 1 are given in Table

6.12, along with an almost exact approximate formula for restraint heights in

between. As can be seen, the rules provide unsafe values for typical UB sections

when restrained at the shear centre, and overly conservative values when restraint

is provided at the compression flange.

For nb > 2, Banfi & Feltham (1999) base their findings on the assumption that

the sum of the restraint forces,∑Fi, can be represented by an equivalent UDL,

qF , so that∑Fi = qL, with the restraint forces “smeared” as an equivalent

continuous restraint of stiffness k. The following formula for the magnitude of qF

was provided:qFL

P=

8

π2

kL2

PE

e1L

[1 +

kL2

π2PE− P

PE

]−1. (6.39)

The approach of the Eurocode of representing the initial imperfection in the

main members by an equivalent horizontal force in the restraining system was

then applied. In the context of the current investigation, this is equivalent to qF

and it is related to the initial imperfection of the main member and the maximum

lateral deflection of the restraining system, δq (see Figure 6.24). Banfi & Feltham

179

(1999) provide the following formula for qFL/P :

qFL

P= 8

kL2

π2PE

(1 + e1

δq

)(

1 + kL2

π2PE

) . (6.40)

It was then assumed that kL2/π2PE � 1 and hence the equation can be reduced

to:qFL

P=

8 (e1 + δq)

L, (6.41)

which is equivalent to the expression given in the Eurocode. The value of δq =

fmax/k. Since the initial imperfection is assumed to be in the form of a half-

sine wave, the restraint forces also vary sinusoidally, as was seen in §3.3.5 and

§4.3.5 i.e. f = fmax sin(πx/L). Since the sinusoidally varying restraint force is

now being represented by an equivalent UDL, for the moments about midspan

to be equivalent, qF = 8fmax/π2. Examining the derivation of restraint forces in

the current work again, since fmax and Fmax are both calculated based on the

restraint compression function X = F/K = f/k at a particular point and that

p is merely P scaled by 1/L, fmax/p can be rescaled to F/P , since it is assumed

that the maximum discrete restraint force is being designed for. It is also noted

that for nb discrete restraints of stiffness K, the equivalent continuous stiffness

k = nbK/L. Hence the Eurocode formula can be recast as:

F

P= π2 e1

L

[1− π2

nb

P

PE

PEKL

]−1. (6.42)

If the limit of Equation (6.42) as K → ∞ is taken, then a value of (F/P )∞ =

1.97% is obtained for e1/L = 1/500. Comparison with Figure 6.21 shows that for

higher restraint heights, this rule is overly conservative, while for a combination

of low restraint heights and low values of κ the Eurocode rule is unsafe. As the

number of restraints is increased, this effect is exacerbated.

180

P

PqF

e1

δq

xy

zDeflected shape

Initial shape

Figure 6.24: A diagram showing the concept of the initial imperfection and the

lateral deflection of a member being represented as an equivalent horizontal UDL,

qF , in the restraining system.

6.7 Summary

In the current chapter, formulae to approximate the properties derived in Chapter

4 were presented, namely the threshold stiffness of the beam, the critical moment

of the beam and the forces induced in the restraints. It was shown that the for-

mulae provide results that are not overly conservative and hence can offer efficient

design of beams with discrete braces. A method was also outlined whereby the

cross-sectional area requirements for strength and stiffness are balanced, leading

to an optimised design.

The practice of using equivalent column design rules was compared with the re-

sults of the current work for critical moment, threshold stiffness and restraint

forces. It was found in every case that for restraint heights close to the com-

pression flange, the column rules provided results that were considerably con-

servative, whereas for restraint heights close to the shear centre the rules were

significantly unsafe. This trend was seen in particular when comparing the cur-

rent best practice of assuming a restraint force ratio of 2.5%. For some cases it

was quite onerous while in other cases, especially for low restraint heights, the

rule was considerably unsafe. This is not overly surprising since column rules are

based on models that only include lateral deflection, fundamentally neglecting

181

the considerable influence of torsional motion.

182

Chapter 7

Worked examples

The worked examples shown in the current chapter use the rules derived in Chap-

ter 6 in conjunction with the specifications of the Eurocode.

7.1 Example 1: Design of lateral restraints

The following design data is used:

• The applied moment MEd = 1800 kNm.

• The span L = 16 m.

• All steel is Grade S355.

• The number of restraints nb = 2.

• The amplitude of the initial lateral imperfection e1 = L/500.

• The length of the restraints Lb = 4 m.

It is required to check the suitability of an 838×292×194 UB section and also to

design the restraints so that they possess both adequate strength and stiffness.

183

Cross-sectional classification and resistance

Upon calculation of the flange slenderness ratio [b − (2r + tw)]/2tf = 5.58 and

web slenderness ratio [h − 2(tf + r)]/tw = 51.8, where b, h, tw, tf and r are the

breadth, height, web thickness, flange thickness and root radius of the section,

respectively, the section is classified as Class 1 and thus Wy = Wpl,y.

The flange thickness tf = 21.7 mm < 40 mm so the assumption of fy = 355 N/mm2

is valid. The cross-sectional bending resistance is hence given by:

Mc,Rd = Wpl,yfy = 7640× 0.355 = 2712 kNm.

Determination of threshold moment

Assuming that restraints of stiffness K > KT are to be provided then the critical

moment can be calculated in the normal fashion, using the restraint spacing

s = L/(nb + 1) = 5.33 m instead of L. The nondimensional beam parameter κ is

given by:

κ = L2GIt/π2EIw = (162 × 306)/(2.6× π2 × 15.2)× 10−2 = 2.01.

Hence κs = 2.01/(2 + 1)2 = 0.223.

⇒MT = (π2EIz/s2)(hs/2)

√1 + κs

= (π2 × 210000× 9066/5.332)(819/2)√

1 + 0.223× 10−8 = 2992 kNm,

⇒ λLT =√Wpl,yfy/MT = 0.952.

Since h/b > 2, buckling curve “b” in the Eurocode is used and the reduction

factor χLT is 0.668,

⇒Mb,Rd = χLTWpl,yfy

= 0.668× 7640× 0.355 = 1812 kNm.

⇒Mb,Rd > MEd.

Also, now that the value of MT is known, µ can be calculated:

µ = MEd/MT = 1800/2992 = 0.602.

184

Design of restraints: stiffness requirements

The restraint is attached to the shear centre so a = 0.

The restraint spacing s = L/(nb + 1) = 16/(2 + 1) = 5.33 m.

Using the rules in §6.2, the threshold stiffness can be determined. For nb = 2,

νb,T = 0.667 and so:

A0 = 0.45 + 2.8× 0.667× 0.223 = 0.867,

A1 = 6.3νb,T + 2.2κs − 1 = 6.3× 0.667 + 2.2× 0.223− 1 = 3.69

⇒ KT =[(

210000× 9066/(5.33)3)× 62× (1 + 0.223)/(0.867 + 0× 3.69)

]× 10−8

= 10.98 kN/mm.

The cross-sectional area of the restraints required to meet the stiffness demands

is thus:

Ab = KLb/E = (10.98× 4000)/(210000)× 103 = 209 mm2

Design of restraints: strength requirements

Since (κ = 2.01) < [5(nb − 1) = 5], σ = 1 and so (F/P )∞,µ=1 is given by:

(F/P )∞,µ=1 = 5(e1/L) [(nb + 1)a/2 + (1 + κ)/(5.5 + 4.5nb)]−1

= 5× 0.002/ [(1 + 2.01)/(5.5 + 4.5× 2)]−1

= 4.8%.

The restraints can be designed conservatively based on this value; however, since

a = 0, a more accurate figure is obtained by simply scaling this value by µ:

⇒ (F/P )∞ = µ× (F/P )∞,µ=1 = 0.602× 4.8 = 2.9%.

It is noted that even with the modification for µ < 1, this value is still greater

than the BS 5950 rule of 2.5%.

185

Using the rules in §6.2, K∞,µ=1 can be determined. For nb = 2, νb,∞ = 2 and so:

A0 = 0.45 + 2.8× 2× 0.223 = 1.700,

A1 = 6.3νb,T + 2.2κs − 1 = 6.3× 2 + 2.2× 0.223− 1 = 12.1

⇒ K∞,µ=1 =[(

210000× 9066/(5.33)3)× 62× (1 + 0.223)/(1.700 + 0× 12.1)

]× 10−8

= 5.60 kN/mm.

This can be scaled in the manner described in §6.4.3, based on µo:

µo =√

1 + κ/[(nb + 1)2√

1 + κs]

=√

1 + 2.01/[(2 + 1)2√

1 + 0.223] = 0.174

⇒ K∞ = K∞,µ=1(µ− µo)/(1− µo) = 5.60× (0.602− 0.174)/(1− 0.174)

= 2.90 kN/mm.

The optimised restraint cross-sectional area is given by:

Ab = K∞Lb/E + (F/P )∞P/fy

= (2.90× 103)× 4000/210000 + 2.9%× 1800/(0.355× 0.819)

= 234.7 mm2

⇒ Since the area required to meet the threshold stiffness demands is 209 mm2 and

hence less than the optimised area, if restraints possessing the threshold stiffness

are provided, the corresponding area required to meet the strength demand will

be greater than the optimised area. Thus, the most efficient design is obtained

by providing restraints with a cross-sectional area of 234.7 mm2.

7.2 Example 2: “Fully restrained” beam

A beam of span 10 m is to be designed to resist a factored moment of MEd =

1000 kNm. It is required to design both the beam and the restraining system.

The restraining system spans 3.5 m. All steel is Grade S355. There is an added

stipulation that there must be a clear depth of 100 mm between the top of the

beam and the restraining system.

186


The required section modulus is given by MEd/fy = 1000/0.355 = 2817 cm3. A

610× 229× 101 UB section has a Wpl,y of 2881 cm3.

The flange thickness tf = 14.8 mm < 40 mm, so the assumption of fy =

355 N/mm2 is valid. The yield stress correction factor ε =√

235/fy = 0.81.

Upon calculation of the web and flange slenderness ratios (52.2 and 6.47, respec-

tively), the section is classified as Class 1.

Determination of optimum number of restraints

The maximum unrestrained length of the beam that can be designed as “fully

restrained” is given by:

smax = 41.7εiz = 41.7× 0.81× 4.75 = 1.61 m.

⇒ nb = L/s− 1 = 10/1.61− 1 = 5.2.

Therefore, at least six restraints must be provided in order for the beam to be

designed as fully restrained. It is then necessary to ensure that these restraints

possess a stiffness K > KT . The actual restraint spacing s = L/(nb+1) = 1.43 m.

Calculation of critical moment

To confirm that λLT < λLT,0, first κ must be calculated:

κ = (L2GIt)/(π2EIw) = (102 × 77)/(π2 × 2.6× 2.52)× 10−2 = 1.19

⇒ κs = κ/(nb + 1)2 = 1.19/(6 + 1)2 = 0.024.

187

The threshold critical moment is:

MT = (π2EIz/s2)(hs/2)

√1 + κs

= (π2 × 210000× 2915/1.432)(588/2)√

1 + 0.024× 10−8

= 8809 kNm,

⇒ λLT =√Wpl,yfy/MT = 0.34.

Thus λLT < 0.4 and the beam can be designed as “fully restrained”, as long as

the restraints possess a stiffness K > KT .

The value of µ = MEd/MT = 0.114.

Calculation of threshold stiffness

The depth of the beam h = 602.6 mm. Assume that the depth of the restraining

system is 80 mm. Thus the restraint height a = 602.6/2−100−80/2 = 161.3 mm,

to allow for the 100 mm clearance.

⇒ a = 2a/hs = 2× 161.3/(602.6− 14.8) = 0.55.

Using the rules in §6.2 the threshold stiffness can be determined. For nb = 6,


A0 = 0.45 + 2.8× νb,T × κs = 0.45 + 2.8× 0.526× 0.024 = 0.486,

A1 = 6.3νb,T + 2.2κs − 1 = 6.3× 0.526 + 2.2× 0.024− 1 = 2.37

⇒ KT =[(

210000× 2915/(1.43)3)× 62× (1 + 0.024)/(0.486 + 0.55× 2.37)

]× 10−8

= 74.69 kN/mm.

The cross-sectional area of the restraints required to meet the stiffness demands

is thus:

Ab = KLb/E = [(74.69× 3500)/(210000)]× 103 = 1131 mm2.

It is noted how the area required to brace the beam fully is considerably larger

than the area required in the previous example of 209 mm2, despite the design

188

moment being smaller, the restraint being positioned closer to the compression

flange and the beam being shorter; the increase in required stiffness is due mainly

to the shorter restraint spacing.

Calculation of restraint force

Since (κ = 1.19) < [5(nb − 1) = 25], σ = 1 and so the value of (F/P )∞,µ=1 is

given by:

(F/P )∞,µ=1 = 5(e1/L) [(nb + 1)a/2 + (1 + κ)/(5.5 + 4.5nb)]−1

= 5× 0.002/ [(1 + 1.19)/(5.5 + 4.5× 6)]−1 = 0.50%.

Since (a = 0.55) > (0.15κ/nb = 0.03), it is not necessary to modify the value of

(F/P )∞ to take the value of µ into account. Also, since a > 0.2 the approximation

of K∞,µ=1/KT = νb,T/νb,∞ is valid, and thus:

K∞,µ=1 = 74.69× 0.526× 10.10 = 3.89 kN/mm.

Since nb > 4, this value is scaled by µ:

K∞ = µK∞,µ=1 = 0.114× 3.89 = 0.44 kN/mm.

The optimised cross-sectional area of the restraint is given by:

Ab = K∞Lb/E + (F/P )∞P/fy

= (0.44× 103)× 3500/210000 + 0.5%× 1000/(0.355× 0.588)

= 31.46 mm2

⇒ The optimised area is far less than the area required to provide the threshold

stiffness, i.e. 1131 mm2, and thus the cross-sectional area of the restraints shall

be designed to meet the stiffness demands.

189

7.3 Example 3: Partially-braced beam

A 686 × 254 × 125 UB section beam spanning 10 m is provided with restraints

attached to its compression flange at quarter points, i.e. nb = 3 and s = 10/(3 +

1) = 2.50 m. The restraints possess a stiffness K = 6.2 kN/mm. The beam

is required to resist a factored design moment MEd = 1250 kNm. Check the

suitability of this beam section.


Upon calculation of the flange slenderness ratio [b − (2r + tw)]/2tf = 6.50 and

web slenderness ratio [h − 2(tf + r)]/tw = 52.6, the section is classified as Class

1 and thus Wy = Wpl,y.

The flange thickness tf = 16.2 mm < 40 mm so the assumption of fy = 355 N/mm2

is valid. The cross-sectional bending resistance is hence given by:

Mc,Rd = Wpl,yfy = 3994× 0.355 = 1418 kNm.

Check if beam is fully braced

Since the beam is restrained at the compression flange, a = 1. It is also required

to calculate κs:

κ = (L2GIt)/(π2EIw) = (102 × 116)/(π2 × 2.6× 4.80)× 10−2 = 0.94

⇒ κs = κ/(nb + 1)2 = 0.64/(3 + 1)2 = 0.059.

Using the rules in §6.2, the threshold stiffness can be determined. For nb = 3,

190


A0 = 0.45 + 2.8× νb,T × κs = 0.45 + 2.8× 0.586× 0.059 = 0.547,

A1 = 6.3νb,T + 2.2κs − 1 = 6.3× 0.586 + 2.2× 0.059− 1 = 2.82

⇒ KT =[(

210000× 4383/(1.43)3)× 62× (1 + 0.059)/(0.547 + 1× 2.82)

]× 10−8

= 11.5 kN/mm. (7.1)

It is clear that since K = 6.2 kN/mm, the beam is not braced fully and thus

cannot achieve the threshold critical moment.

Determination of critical moment

Since nb = 3, the formula of 6.3.3 can be used. Two properties must first be

calculated:

K/KT = 6.2/11.5 = 0.540,

µo =√

1 + κ/[(nb + 1)2√

1 + κs] = 0.085.

Since κ < 10:

σ = 1− (2− a)(10− κ)/100 = 1− (2− 1)(10− 0.942)/100 = 0.909.

Now the value of µcr can be calculated:

µcr = [σ + (1− σ)(K/KT )]√

1− (1− µ2o)(1−K/KT )2

= [0.909 + (1− 0.909)(0.540)]√

1− (1− 0.0852)(1− 0.540)2 = 0.830.

The critical moment is found by calculating the threshold moment first:

⇒MT = (π2EIz/s2)(hs/2)

√1 + κs

= (π2 × 210000× 4383/2.502)(661.7/2)√

1 + 0.059× 10−8

= 4950 kNm

⇒Mcr = µcrMT = 0.83× 4950 = 4109 kNm.

The nondimensional slenderness is given by:

λLT =√Wpl,yfy/Mcr = 0.587.

191

Since h/b > 2, buckling curve “b” in the Eurocode is used and the reduction

factor χLT is 0.893,

Mb,Rd = χLTWpl,yfy = 0.893× 3994× 0.355 = 1266 kNm.⇒Mb,Rd > MEd. Thus

the beam is able to resist the design moment.

192

Chapter 8

Conclusions, findings and further

work

8.1 Conclusions and findings

A two-DOF system was decided upon to model a simply-supported beam under

constant bending moment. It was shown in Chapter 3 that representing the

displacement components of an unrestrained beam as either a single harmonic

function or in terms of a Fourier series lead to the same results for critical moment

and the load–deflection relationship. The same was also found for continuously

restrained beams; the effect was due to the continuity of the function representing

the total potential energy. When a geometric nonlinearity was included in the

unrestrained beam model, it was demonstrated that the postbuckling behaviour of

the beam was weakly stable. This was confirmed by numerical results from Auto.

It was decided that, owing to this weakly-stable postbuckling behaviour, models

including a restraining system could be assumed to be linear, since deflections

and curvatures would be limited. It was shown that for a continuously-restrained

beam, the value of the critical moment increased for stiffer restraints. As the

stiffness increased, the buckling mode changed from the first mode to the second

193

and so on, indefinitely. The behaviour did not occur, however, if the restraint

were positioned below the shear centre, i.e. closer to the tension flange; instead,

the moment increased asymptotically to a certain limiting moment.

In Chapter 4, the model of a beam with discrete restraints was developed. Ini-

tially, a single harmonic representation of the DOFs was used; this led to results

for the critical moment, the load–deflection relationship and the restraint force

being of a form almost identical to those found for a continuously restrained

beam. Indeed, if the discrete restraint stiffness was scaled appropriately to ob-

tain an equivalent continuous restraint stiffness, it was found that the expressions

were identical. This was similar to the “smearing technique” mentioned by Tra-

hair (1993). When examining the buckling mode behaviour, it was found that,

unlike the situation for a continuously-restrained beam where the critical moment

increased indefinitely with restraint stiffness, the critical moment increased with

stiffness up to a threshold point, whereafter the beam buckled in between the

restraints. Any increase in stiffness beyond the threshold amount did not lead to

a corresponding increase in critical moment.

As the stiffness was increased from K = 0 up to the threshold stiffness KT , the

buckling mode changed from one harmonic to the next, as was found for the

continuously restrained model. However, the progression of buckling modes was

not complete and it was shown that single harmonic representation cannot always

predict a full sequential mode progression.

Next, Fourier series were used to represent the lateral deflection and the angle of

twist. It was shown that, owing to the discrete nature of the model, the mode

separation that existed in the models studied in Chapter 3 was not present.

Instead, sets of harmonics were found that interact linearly with, and only with,

each other. These harmonic sets described overall buckling modes where the

restraint nodes were displaced. The number of node-displacing modes was found

to be equal to the number of restraints. In addition, there were also an infinite

amount of displacement modes where the restraint spacing was an integer multiple

194

of the harmonic wavelength.

As was found for the other models in the current work, the buckling mode changed

with increasing restraint stiffness; however, in contrast to when the DOFs were

represented as single harmonics, the critical mode progressed through all the avail-

able node-displacing modes before the threshold stiffness was reached, whereafter

the critical moment was equal to the threshold amount and the beam buckled in

between the restraints. It was also shown that there exists a point on the tension

side of a beam cross-section that defines how this mode progression occurs. If the

restraints are positioned above this point, then a full sequential mode progression

occurs; below this amount and then the first node-displacing mode will always

be critical and hence it is not possible to brace the beam fully.

A closed-form finite-termed solution was found successfully for the threshold stiff-

ness, and also for the relationship between restraint stiffness and critical moment.

Closed-form expressions for the DOFs, and hence the force induced in the re-

straints, were also found. Since Fourier series were used to represent the DOFs,

these expressions can be thought of as “exact” solutions.

A comparison was then made between the results for the critical moment and the

threshold stiffness as calculated by the Fourier series representation of the DOFs,

the single harmonic representation of the DOFs and the “smearing” technique

described by Trahair (1993). It was found that for nb < 3 the smearing technique

returned overly conservative results for the threshold stiffness. For nb = 3 the

results were roughly accurate; however, for nb > 3, the results were unsafe.

In Chapter 5, the Fourier series representation of the DOFs was shown to be

accurate by comparison with the results of two numerical packages, LTBeam and

Auto, for a single half-sine wave initial lateral imperfection. A further conclusion

to be drawn from the results provided by Auto for the deflected shape of the

beam is that a single harmonic representation is unable to model the actual

deflected shape accurately due to the presence of multiple inflection points.

195

In Chapter 6, design formulae based on the findings of Chapter 4 were presented.

It was found that the results returned provided reasonably accurate results for

the critical moment, the threshold stiffness and the restraint force. It was also

found that, when comparing these results with those returned by design rules

intended for use with columns, but recommended for use with beams also, that

the column rules generally provide conservative results for restraint positioned

close to the compression flange of the beam, while for restraints positioned closer

to the shear centre the rules are unsafe.

8.2 Further work

An interesting avenue for further work concerns the inclusion of nonlinearities in

the potential energy functional for a beam with discrete braces. Based on the

findings of Wadee et al. (1997), it is proposed that including the nonlinearity

in the expression for restraint stiffness would lead to more interesting results

than using large deflection terms. It is expected that owing to the concentration

of force at the restraint nodes, localisations would occur at these points. As

mentioned in Chapter 3, it is assumed in the current work that the webs of such

cross-sections are adequately stiffened to avoid localised buckling; an exploration

of the behaviour of the beam without this caveat would provide interesting results.

It has also been shown recently that the nonlinear mode interaction between

lateral-torsional buckling and local buckling in unrestrained beams with thin

flanges can lead to localisations occuring. This is followed by cellular buckling

due to the inherently stable nature of plate buckling within the compression flange

(Wadee & Gardner, 2012). A study on the effects of introducing restraints to this

type of structure would also be interest.

Al-Shawi (1998) showed that the value of the restraint force for a column with a

single restraint is sensitive to the location of the brace along the span of the beam;

196

an analogous study of a discretely-braced beam would provide useful information

about the influence of non-regular spacings on the threshold stiffness and induced

restraint forces.

197

Appendix A

Proofs of identities

A.1 Summation of sin2 n terms

To prove:nb∑i=1

sin2

(inπ

nb + 1

)=nb + 1

2∀ n mod (nb + 1) 6= 0. (A.1)

Initially it is noted that if n mod (nb + 1) = 0 then n is of the form q(nb + 1),

and so the left hand side (LHS) of Equation (A.1) becomes:∑i=1

sin2 (iqπ) = 0, (A.2)

since n, q ∈ Z. For the non-trivial case of n mod (nb + 1) 6= 0, initially it is noted

that sin2 θ = 12(1− cos 2θ), hence:

nb∑i=1

sin2

(inπ

nb + 1

)=

nb∑i=1

1

2

[1− cos

(2inπ

nb + 1

)]

=nb2− 1

2

nb∑i=1

cos

(2inπ

nb + 1

)(A.3)

The summation above is now examined, with A = 2nπ/(nb + 1) for convenience.

If the finite difference of cos(Ai) is defined as:

∆ cos(Ai) = cos [A(i+ 1)]− cos(Ai), (A.4)

198

then applying the antidifference operator ∆−1, which is linear (Jordan, 1965), to

both sides of Equation (A.4) yields:

cos(Ai) = ∆−1 cos [A(i+ 1)]−∆−1 cos(Ai). (A.5)

Upon summation of the series, subsequent terms cancel out, leaving the following

identity:nb∑i=1

cos(Ai) = ∆−1 cos [A(nb + 1)]−∆−1 cos(A). (A.6)

To complete the proof, the indefinite sum of cos(Ai), i.e. ∆−1 cos(Ai) must be

determined, using the methods of Jordan (1965). Examining the definition of

∆ cos(Ai) again:

∆ cos(Ai) = cos [A(i+ 1)]− cos(Ai)

= −2 sin

(Ai+

A

2

)sin

(A

2

). (A.7)

Applying the antidifference operator to both sides and rearranging:

∆−1 sin

(Ai+

A

2

)= − cos(Ai)

2 sin(A/2). (A.8)

Noting that sin θ = cos(π/2− θ) = cos(θ − π/2), by means of a dummy variable

i→ i− 12

+ π2A

, Equation (A.7) can be recast as:

∆−1 cos(Ai) =sin(Ai− A

2

)2 sin(A/2)

. (A.9)

It is noted that technically an expression for the indefinite sum of a function

should include a constant of summation, analogous to the constant of integration,

but can be neglected here. Using this result for the indefinite sum, the value of

the original summation can be determined:nb∑i=1

cos(Ai) =sin[A(nb + 1)− A

2

]2 sin(A/2)

−sin(A− A

2

)2 sin(A/2)

nb∑i=1

cos

(2inπ

nb + 1

)=

sin(

2nπ − nπnb+1

)2 sin

(nπnb+1

) −sin(

nπnb+1

)2 sin

(nπnb+1

)=

sin(− nπnb+1

)2 sin

(nπnb+1

) − 1

2(A.10)

= −1 (A.11)

199

Substituting Equation (A.10) into Equation (A.3) leads to proof of the identity:

nb∑i=1

sin2

(inπ

nb + 1

)=nb2− (−1)

1

2=nb + 1

2�. (A.12)

A.2 Summation of sinn sinm terms

To prove:

nb∑i=1

sin

(inπ

nb + 1

)sin

(imπ

nb + 1

)= 0 ∀ n±m mod 2(nb + 1) 6= 0. (A.13)

Initially it is noted that if n ± m mod 2(nb + 1) = 0 then m is of the form

2q(nb + 1)∓ n, and so the LHS of Equation (A.13) becomes:∑i=1

sin

(inπ

nb + 1

)sin

(2π ∓ inπ

nb + 1

), (A.14)

which is equivalent to the identity proven in the previous section. To prove for

the case of n ±m mod 2(nb + 1) 6= 0, the trigonometric product on the LHS of

Equation(A.13) is expanded into a sum:

nb∑i=1

sin

(inπ

nb + 1

)sin

(imπ

nb + 1

)=

1

2

nb∑i=1

[cos

(iπ(n−m)

nb + 1

)− cos

(iπ(n+m)

nb + 1

)].

(A.15)

The technique used in the previous section to determine ∆−1 cos(Ai) is applied

to the series above to obtain:

nb∑i=1

cos

(iπ(n−m)

nb + 1

)−

nb∑i=1

cos

(iπ(n+m)

nb + 1

)

=sin[(n−m)π − (n−m)π

2(nb+1)

]2 sin

[(n−m)π2(nb+1)

] −sin[π(n−m)2(nb+1)

]2 sin

[π(n−m)2(nb+1)

]−

sin[(n+m)π − (n+m)π

2(nb+1)

]2 sin

[(n+m)π2(nb+1)

] +sin[π(n+m)2(nb+1)

]2 sin

[π(n+m)2(nb+1)

] . (A.16)

It is noted that sin [(n−m)π + θ] = sin θ for n−m being even, − sin θ for n−m

being odd, with the same conditions obviously applying for n + m also. Hence,

200

after removing the cancelling terms:

nb∑i=1

cos

(iπ(n−m)

nb + 1

)−

nb∑i=1

cos

(iπ(n+m)

nb + 1

)

=± sin

[(n−m)π2(nb+1)

]2 sin

[(n−m)π2(nb+1)

] − ± sin[(n+m)π2(nb+1)

]2 sin

[(n+m)π2(nb+1)

] , (A.17)

with the sign of the numerators depending upon the parity of n −m. Whether

odd or even, the original identity reduces to:

nb∑i=1

sin

(inπ

nb + 1

)sin

(imπ

nb + 1

)=

1

2

(±1

2∓ 1

2

)= 0 �. (A.18)

A.3 Finite-termed representation of S2

The sum S2 mentioned in Chapter 4 is given by:

S2 =∞∑n=1

wm,n (1 + a2) + κ+ 2aµMT

w2m,n

(w4m,n + w2

m,nκ− µ2M2T

) . (A.19)

The nth term is denoted tn. The sum of successive terms Tn = t2n + t2n+1, so

that the following identity holds:

S2 =∞∑n=1

tn = t1 +∞∑n=1

Tn. (A.20)

The term Tn can be factorised thus:

Tn =(1 + a2

)( 1

w4m,2n + w2

m,2nκ− µ2M2T

+1

w4m,2n+1 + w2

m,2n+1κ− µ2M2T

)

+(κ+ 2aµMT

) 1

w2m,2n

(w4m,2n + w2

m,2nκ− µ2M2T

)+

1

w2m,2n+1

(w4m,2n+1 + w2

m,2n+1κ− µ2M2T

) , (A.21)

201

where i2 = −1. Additionally, upon factorisation, the term found in the denomi-

nators:

w4m,q + w2

m,qκ− µ2M2T =

[wm,q −

√(√κ2 + 4µ2M2

T − κ)/2

](A.22)

×

[wm,q +

√(√κ2 + 4µ2M2

T − κ)/2

]

×

[wm,q − i

√(√κ2 + 4µ2M2

T + κ

)/2

]

×

[wm,q + i

√(√κ2 + 4µ2M2

T + κ

)/2

]. (A.23)

It is noted that from the definition in §4.3.3 that wm,2n = 2n(nb + 1) − m and

wm,2n+1 = 2n(nb + 1) + m. Substituting these values into Equation (A.22) and

then substituting that result in Equation (A.21), along with the expression for

MT from Equation (4.11), the following result is found:

Tn =1 + a2

16(nb + 1)4

[1

(n− A1)(n− A2)(n− A3)(n− A4)

+1

(n+ A1)(n+ A2)(n+ A3)(n+ A4)

]+κ+ 2aµ

√1 + κs

64(nb + 1)4

[1

(n− η/2)2(n− A1)(n− A2)(n− A3)(n− A4)

+1

(i+ η/2)2(n+ A1)(n+ A2)(n+ A3)(n+ A4)

], (A.24)

where κs = κ/(nb + 1)2 and η = m/(nb + 1) and

A1 =η

2+

1

2√

2

√√κ2s + 4µ2(1 + κs)− κs,

A2 =η

2− 1

2√

2

√√κ2s + 4µ2(1 + κs)− κs,

A3 =η

2+

i

2√

2

√√κ2s + 4µ2(1 + κs) + κs,

A4 =η

2− i

2√

2

√√κ2s + 4µ2(1 + κs) + κs.

202

The fractions in Equation (A.24) are partitioned thus:

1

(n± A1)(n± A2)(n± A3)(n± A4)=

4∑j=1

Bj

n± Aj(A.25)

1

(n± A1)(n± A2)(n± A3)(n± A4)(n± η/2)2=

4∑j=1

Cjn± Aj

+C5

(n± η/2)2,

(A.26)

where:

B1 = ∓ 4√

2[√√κ2s + 4µ2(1 + κs)− κs

]√κ2s + 4µ2(1 + κs)

, (A.27)

B2 = −B1, (A.28)

B3 = ∓ 4√

2i[√√κ2s + 4µ2(1 + κs) + κs

]√κ2s + 4µ2(1 + κs)

, (A.29)

B4 = −B3, (A.30)

C1 = ∓8√

2[√

κ2s + 4µ2(1 + κs) + κs

]µ2(1 + κs)

[√√κ2s + 4µ2(1 + κs)− κs

]√κ2s + 4µ2(1 + κs)

, (A.31)

C2 = −C1, (A.32)

C3 = ∓8√

2i[√

κ2s + 4µ2(1 + κs)− κs]

µ2(1 + κs)

[√√κ2s + 4µ2(1 + κs) + κs

]√κ2s + 4µ2(1 + κs)

, (A.33)

C4 = −C3, (A.34)

C5 = − 16

µ2(1 + κs). (A.35)

The overall expression for Tn now includes terms of the form 1/(n ± Aj) and

1/(n ± η/2)2. Since the infinite sum S2 = t1 +∑Tn is desired, it is necessary

to evaluate∑

1/(n ± Aj) and∑

1/(n ± Aj)2. The indefinite sum of 1/n =

∆−1(1/n) = ψ(n) + C, where ψ is the digamma function (Jordan, 1965) and C

is the constant of summation. This implies that 1/n = ψ(n+ 1)−ψ(n) and thus

the following identity holds owing to subsequent terms cancelling each other out:

N∑n=1

1

n= ψ(N + 1)− ψ(1). (A.36)

203

If n± Aj is now substituted for n the following identity holds:

N∑n=1

1

n± Aj= ψ(N + 1± Aj)− ψ(1± Aj). (A.37)

The value of ψ(N+1+Aj)−ψ(N+1−Aj)→ 0 as N →∞, therefore these terms

cancel out in the overall expression for S2, leaving terms of the form ψ(1 ± Aj)

remaining. Next, it is noted that ψ(n) obeys the recurrence relation ψ(n+ 1) =

1/n + ψ(n), derived from the expression above, and also the reflection relation

ψ(1 − n) = ψ(n) + π cot(πn) (Jordan, 1965). Subtracting these two formulae

produces the following identity:

ψ(1 + n)− ψ(1− n) =1

n− π cot(πn). (A.38)

Appearances of ψ(1 +Ak)− ψ(1−Ak) in the overall formula can be replaced by

this result.

Next, it is noted that ∆−1(1/n2) = −ψ1(n)+C, where ψ1 is the trigamma function

(Jordan, 1965). Repeating the approach used to determine∑

1/(n ± Aj) it is

found that:

∞∑n=1

1

(n± η/2)2= ψ1(1± η/2)− lim

n→∞(n+ 1± η/2). (A.39)

Conveniently ψ1(n) → 0 as n → ∞ and thus the expression above is simply

ψ1(1 ± η/2). As ψ1(n) = dψ(n)/dn (Jordan, 1965), recurrence and reflection

formulae for ψ1 can be found through differentiation of the corresponding relations

for ψ:

ψ1(1 + n) = − 1

n2+ ψ1(n),

ψ1(1− n) = −ψ1(n) +π2

sin2(πn).

Upon substitution of η/2 into these expressions, the terms relating to∑

1/(n±

η/2)2 are reduced since:

∞∑n=1

[1

(n+ η2)2

+1

(n− η2)2

]= ψ1(1 + η/2) + ψ(1− η/2)

=π2

sin2(πη/2)− 4

η2(A.40)

204

Now the results of Equations (A.24), (A.38) and (A.40) are substituted into Equa-

tion (A.20). With some terms cancelling out along with some rearrangement, the

following expression for S2 is obtained:

S2 = t1 +∞∑n=1

Tn

=1

(nb + 1)4

− 1√2r0

( rar+2µ2(1 + κs)

+ 1 + a2)

π sin π√r−/2

√r−

(cosπ

√r−/2− cosπη

)+

(rar−

2µ2(1 + κs)− (1 + a2)

)π sinh π

√r+/2

√r+

(cosh π

√r+/2− cos πη

)

+raπ

2

2µ2(1 + κs) (1− cos πη)

}, (A.41)

where

ra = κs + 2aµ√

1 + κs, (A.42)

r0 =√κ2s + 4µ2(1 + κs), (A.43)

r+ = r0 + κs, (A.44)

r− = r0 − κs. (A.45)

This expression can now be used to calculate critical moments implicitly using

the methods of Chapter 4.

205

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