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Seo, Jung Kwan, Mahendran, Mahen, & Paik, Jeom Kee (2011) Numericalmethod for predicting the elastic lateral distortional buckling moment of amono-symmetric beam with web openings. Thin-Walled Structures, 49(6),pp. 713-723.

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http://dx.doi.org/10.1016/j.tws.2011.01.003

1

Numerical Method for Predicting the Elastic Lateral Distortional Buckling Moment of a Mono-symmetric Beam with Web Openings

Jung Kwan Seo,a,b,* Mahen Mahendrana and Jeom Kee Paikb

a) Faculty of Built Environment and Engineering, Queensland University of Technology, Brisbane, QLD 4000, Australia

b) The Lloyd’s Register Educational Trust Research Centre of Excellence, Pusan National University, Busan 609-735, Republic of Korea * Corresponding author. Tel.: +82-(0)51-510-2415. Email address: [email protected] (J.K. Seo)

Abstract: When used as floor joists, the new mono-symmetric LiteSteel beam (LSB)

sections require web openings to provide access for inspections and various services.

The LSBs consist of two rectangular hollow flanges connected by a slender web, and

are subjected to lateral distortional buckling effects in the intermediate span range.

Their member capacity design formulae developed to date are based on their elastic

lateral buckling moments, and only limited research has been undertaken to predict

the elastic lateral buckling moments of LSBs with web openings. This paper

addresses this research gap by reporting the development of web opening modelling

techniques based on an equivalent reduced web thickness concept and a numerical

method for predicting the elastic buckling moments of LSBs with circular web

openings. The proposed numerical method was based on a formulation of the total

potential energy of LSBs with circular web openings. The accuracy of the proposed

method’s use with the aforementioned modelling techniques was verified through

comparison of its results with those of finite strip and finite element analyses of

various LSBs.

Keywords: Cold-formed steel structures, Mono-symmetric beams, LiteSteel beams,

Web openings, Equivalent reduced web thickness, Elastic lateral distortional

buckling, Numerical method.

mailto:[email protected]

2

Nomenclature

30 aa − Coefficients defining web deflections

WD Plate rigidity ( xzzx DDD ,, )

1d Centreline dimension of web height E Young’s modulus of elasticity G Shear modulus of elasticity h Distance between flange shear centres

FI Polar second moment of flange area

WI Warping section constant

zy II , Second moments of area about y and z axes

FJ torsion section constant of hollow flange

[ ]K Total stiffness matrix

TB kk , Curvatures at the bottom and top of the web L Length M Moment

yM First yield moment

odM Elastic lateral distortional buckling moment

Fr Polar radius of flange gyration

WWWF SSSS,

, 21 θθ Stiffness parameters

t Wall thickness equt Equivalent reduced wall thickness

U Strain energy stored during in-plane buckling TB vv , Bottom and top flange deflections in y direction

Wv Web deflection in y direction V Work done during buckling

zyx ,, Principal axes

0z Defined in Fig. 8 δ Vector of web end deformations ν Poisson’s ratio σ Stress

BxxWxxTxx σσσ ,, Normal stresses of top flange, web and bottom flange

BzzWzzTzz σσσ ,, Transverse normal stresses of top flange, web and bottom flange

BxxWxxTxx τττ ,, Shear stresses of top flange, web and bottom flange

TB φφ , Mid-span values of FTFB φφ ,

FTFB φφ , Bottom and top flange twist rotations

3

1. Introduction

OneSteel Australian Tube Mills (OATM) was the first to develop the so-called ‘dog-

bone’ sections or hollow flange beams (HFBs) shown in Fig. 1(a). The efficiency of a

HFB lies in the combination of its torsionally rigid closed hollow flanges and an

economical manufacturing method based on simultaneous dual electric resistance

welding and roll-forming process [1,2]. OATM recently introduced the mono-

symmetric LiteSteel Beam (LSB) shown in Fig. 1(b) [3], which is the first of its

innovative cold-formed steel hollow flange sections suitable for large-scale

production. The new LSBs consist of two rectangular hollow flanges connected by a

slender web. There are 13 LSB sections whose depth (d) varies from 125 mm to 300

mm while their hollow flange width (bf) varies from 45 mm to 75 mm. The thickness

of high strength steel (t) used in LSBs varies from 1.6 mm to 3.0 mm. The LSB

section designation is based on d x bf x t, for example, 300x60x2.0 LSB. Due to their

light weight and cost-effectiveness, LSBs have become increasingly popular in

residential, industrial and commercial buildings. The LSB sections are used in these

building systems as floor joists, truss members and the like.

(a) HFB (b) LSB

Fig. 1 HFBs and LSBs and their Lateral Distortional Buckling Modes

Both HFB and LSB flexural members are subject to lateral distortional buckling

effects in their intermediate span ranges as shown in Fig. 1 [1,2,4]. When employed as

floor joists, LSB sections with higher depths and web openings are commonly used to

provide access for inspection and various services. The moment capacities of floor

bf

df

d

t

d1

Ro

30°

bf

d

t

d1

4

joists made of slender LSB sections with larger spans very much depend on their

elastic lateral distortional buckling moments. Several researchers have investigated

and summarised the elastic lateral buckling behaviour of channel section beams [5-8]

while Hancock et al. [9] and Bradford [10] have explored the effects of lateral

distortional buckling on the strength behaviour of conventional I-sections.

Pi and Trahair [11] adopted energy methods to investigate the elastic lateral

distortional buckling behaviour of simply supported HFBs subject to a uniform

moment, and developed a simple closed-form elastic buckling solution for HFBs

without web openings. They modified the classical flexural torsional buckling formula

by determining the effective torsional rigidity of the sections. They then adopted a

nonlinear inelastic method to investigate the lateral distortional buckling behaviour of

HFBs, and developed suitable member capacity equations for HFBs within the

guidelines of AS 4100 [12].

Moen and Schafer [13] developed elastic buckling approximations for cold-formed

steel structural members with holes, which they then employed to predict the elastic

local and distortional buckling moments of these members using a finite strip analysis

(FSA) program. For distortional buckling, the method simulates the C-section’s loss

in bending stiffness from the presence of a web hole within a distortional buckling

half-wave by modifying the cross-section thickness in the finite strip method. The

thickness of the entire web is reduced on the basis of the relationship between the web

bending stiffness and bending stiffness matrix terms of a finite strip element. The

distortional half-wavelength of the cross-section without holes is first determined

using the cross section of the column in a FSA program to generate an elastic

buckling curve. Hence Moen and Schafer [13] defined a half-wavelength by the

location of the distortional minimum, with the web thickness modified in the finite

strip method to account for the stiffness loss resulting from the holes. This

approximate method has been implemented successfully for C-section beams with

web holes, although it may have limitations in predicting the elastic buckling

moments of LSBs with web holes when subjected to a uniform bending moment.

Wang and Salhab [14] reported their numerical results for compression tests of a

lightweight panel using cold-formed thin-walled channel sections with perforated

webs. They also investigated the local buckling behaviour of web plates with

perforations, and their numerical results were employed to develop an approximate

method for calculating the equivalent thickness of the perforated web, which they

5

defined as the reduced thickness of a solid web plate that gives the same local

buckling strength as the perforated web. In their study, the effective thickness was

obtained through regression analysis of a large number of numerical results using a

finite element analysis program, ABAQUS. However, their approximate method is

applicable only for the three-dimensional modelling of perforated channel sections

with reduced web thickness.

At present, there is no accepted design method for LSB flexural members with web

openings. Such existing design methods as AISI [15] deal only with channels with

web openings and do not give the required elastic buckling moments and member

moment capacities for LSBs. Seo and Mahendran [16] recently developed new design

rules for determining the ultimate member moment capacities of LSBs with web

openings. These design rules were developed as a function of a non-dimensional

member slenderness parameter ( dλ ) defined as y

od

MM

, and therefore require elastic

lateral distortional buckling moments (Mod) for a given LSB and web opening

configuration. Hence there is a need for a general prediction method for LSB sections

and web opening configurations. The aim of the study reported in this paper was to

develop web opening modelling techniques and a numerical method for the prediction

of the elastic lateral buckling moments of LSBs with web openings.

The paper first proposes an equivalent reduced web thickness modelling method

for the elastic buckling analysis of LSBs with web openings. Four methods employing

four different equations for the prediction of equivalent web thicknesses are proposed,

and their accuracy is investigated using finite element analysis (FEA). In the second

section of the paper, the numerical energy-based buckling solution for LSBs without

web openings is extended to LSBs with web openings by employing the equivalent

web thickness approach used in FEA. An explicit buckling solution is obtained from a

formulation of the total potential energy for LSBs with circular web openings. The

accuracy of the proposed numerical method’s use with equivalent reduced web

thicknesses is verified by comparing its results with those of the FSA and FEA of

LSBs.

6

2. Proposed equivalent reduced thickness method for LSBs

2.1 Equivalent reduced thickness methods for LSBs

It is proposed that the effects of web hole configurations on the lateral distortional

buckling of LSBs can be approximated by reducing the cross-section thickness in

FSA and FEA using the four following methods, which employ four different

equations. Doing so allows numerical analyses to be undertaken with solid webs of

equivalent reduced thicknesses.

In the first method (see Fig. 2(a)), an equivalent reduced thickness (tequ) is

determined for the full height solid web element along the full member length, based

on the following equation.

web

webholewebequ A

tAAt )( −= . (1)

In the second method (see Fig. 2(b)), an equivalent reduced thickness (tequ) is

determined for the web hole region alone, based on the following equation.

webequ tSDt )/1( −= . (2)

In the third method (see Fig. 2(c)), an equivalent reduced thickness (tequ) is

determined for both the web and flange elements, based on the following equation.

)()(

flangeweb

webholewebfflangeequ AA

tAAtAt

+

−+= . (3)

Finally, in the fourth method (see Fig. 2(d)), an equivalent reduced thickness (tequ)

is determined for the web hole region alone, based on the following equation.

DLtADLt webhole

equ

)( −= . (4)

In the above equations, Aweb = web area = L× d1; Aflange = flange area = 2(L×bf ) +

2(L×df ); Ahole = the area of web holes = 2( D / 4) (L / S)π × ; d1 = the centreline

dimension of web height; D and S = the web hole diameter and spacing, respectively;

and L = the member span.

7

All four methods allow the effects of web hole size and spacing to be determined

through the use of a reduced thickness, and are based on the web hole area and

spacing, and the selected areas of web and/or flange elements, except for Method 2.

(a) Method 1 (b) Method 2 (c) Method 3 (d) Method 4

Fig. 2 Equivalent reduced web thicknesses in Methods 1 to 4

2.2 Verification of the proposed methods

In order to determine the accuracy of the proposed equivalent thickness methods,

the ideal finite element model of LSB without web openings developed by Kurniawan

and Mahendran [17,18] was used in this study for three LSB sections, 300x60x2.0

LSB (slender section), 250x75x3.0 LSB (compact section) and 300x75x3.0 LSB

(non-compact section). This shell finite element model incorporated ideal simply

supported end conditions and a uniform bending moment within the span. The model

was used with the equivalent plate thickness calculated from Methods 1 to 4 instead

of explicitly modelling the web holes. Hence it was considered the approximate FEA

model of LSBs with web openings used to determine the elastic lateral buckling

moments. Following web hole diameter (D) and Spacing (S) were chosen in this

study: D = 50, 100, 150 mm and S = 200, 250, 500 mm. Seo and Mahendran [19]

used similar ideal finite element models that included the actual web openings and

their spacing. These accurate FEA models were also used in this study to obtain the

elastic lateral buckling moments of the chosen three LSB sections and web opening

configurations.

Both accurate and approximate FEA models were developed using the S4R5 three

dimensional (3D) thin isoparametric quadrilateral shell element with four nodes and

five degrees of freedom per node, available in ABAQUS [20]. An element size of

approx. 5 mm×10 mm (width by length) was used to provide accurate results. The

8

accuracy of such models of LSB flexural members with and without web openings

has been verified by Kurniawan and Mahendran [17] and Pokharel and Mahendran

[21]. Fig.3 (a) shows a typical half-length finite element model of LSB with web

openings used in this study. It includes the details of the ideal loading and boundary

conditions used. The required uniform bending moment distribution within the span

was achieved by applying equal end moments using linear forces at the end support.

Figure 3 (b) shows the typical lateral distortional buckling mode obtained from the

elastic buckling analyses of LSB with web openings. Further details of FEA models

are given in [19].

(a) Finite Element Model

(b) Lateral Distortional Buckling Mode

Fig. 3 Finite Element Modelling of LSBs with Web Openings

9

Figs. 4 to 6 compare the elastic lateral distortional buckling moments obtained

from these approximate and accurate FEA models. In these figures, the horizontal axis

is the ratio of the web hole diameter to the centreline dimension of web height (D/d1)

and the vertical axis is the elastic lateral distortional buckling moment (Mod). The

results are also compared in tabular form in [19]. The ratio of Mod predicted by the

approximate and accurate finite element models was calculated in each case from

these results. The mean and COV of this ratio for Methods 1 to 4 are as follows:

0.9605 and 0.0375 (Method 1); 0.9265 and 0.1002 (Method 2); 0.6696 and 0.2911

(Method 3); 0.9831 and 0.0483 (Method 4). These results and Figures 3 to 5 confirm

that Methods 1 and 4 are accurate in predicting the elastic lateral distortional buckling

moments of the chosen slender, compact and non-compact LSB sections with circular

web openings using the approximate finite element models. As expected, Method 3

that includes both web and flange elements under-predicts Mod significantly and is

inaccurate.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7D/d1

0

20

40

60

80

100

Mod

(kN

m)

FE model no holes FE Model with holes FE model without holes (Method 1)FE model without holes (Method 2)FE model without holes (Method 3)FE model without holes (Method 4)

Mod=89.720kNm

M1

M3

M2

M4

250x75x3.0LSB (L=2000mm)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1D/S

0

20

40

60

80

100

Mod

(kN

m)

FE Model no holesFE Model with holes FE model without holes (Method 1)FE model without holes (Method 2)FE model without holes (Method 3)FE model without holes (Method 4)

M1

M3

M2

M4

250x75x3.0LSB (L=2000mm, D=150mm)Mod=89.72 kNm

(a) Mod vs. D/d1 (b) Mod vs. D/S

Fig. 4 Comparison of Mod from approximate and accurate Finite Element Models of

250 x 75 x 3.0 LSBs with web holes (Span = 2000 mm)

10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7D/d1

0

40

80

120

Mod

(kN

m)


Mod=97.871kNm

M1

M3

M2

M4

300x75x3.0LSB (L=2000mm)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1D/S

0

40

80

120

Mod

(kN

m)


M1

M3

M2

M4

300x75x3.0LSB (L=2000mm, D=150mm)

Mod=97.87 kNm




0 0.1 0.2 0.3 0.4 0.5 0.6 0.7D/d1

0

10

20

30

40

Mod

(kN

m)


Mod=35.040kNm

M1

M3

M2

M4

300x60x2.0LSB (L=2000mm)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1D/S

0

10

20

30

40

Mod

(kN

m)


M1

M3

M2

M4

300x60x2.0LSB (L=2000mm, D=150mm)

Mod=35.04 kNm




The implication of these results is that the equivalent reduced web thicknesses

obtained based on Methods 1 and 4 can be used in the modelling of LSB flexural

members with web holes when employing available FSA and FEA programs, thereby

avoiding the complicated task of explicitly modelling the web hole configuration.

Method 4, which has a mean ratio of 0.9831, is considered to be the most accurate of

the four methods, and it can be employed in numerical modelling with a reduced

11

equivalent thickness in the web hole region alone. In applications in which a single

web element is to be used, Method 1 is recommended.

3. Elastic buckling moments of LSBs with proposed reduced thickness methods

and energy method

Also developed in the study reported in this paper was an accurate energy method-

based solution for the elastic lateral distortional buckling moments of mono-

symmetric LSB members with web openings subject to a uniform moment. The

numerical method employs the same theory and principles as those used by Trahair

[22] for HFBs without web openings. A similar approach is proposed here to derive

an explicit buckling solution for LSBs with web openings. For this purpose, an energy

method derivation was employed with the implementation of an equivalent web

thickness to represent the web opening configurations.

In the discussion of the proposed reduced thickness methods, it was shown that

approximate finite element models using equivalent reduced web thicknesses can be

used to predict elastic lateral distortional buckling moments and avoid the complex

modelling of web openings. Equation 1, which was based on Method 1, was

employed to include the equivalent web thickness in the total stiffness matrix terms.

This method was chosen instead of Method 4 as it allows a single web element of

constant thickness to be used in the derivations (Fig. 2). The method’s accuracy was

verified by comparing its results with those obtained from FSA and FEA.

12

3.1 Displacement field

Fig. 7 Cross-section of LSB deformations

The deformation of an LSB cross-section through buckling can be described

approximately by a displacement field with six degrees of freedom Bv , Bφ , Tv , Tφ ,

Bk , Tk T [5]. The cross-section deformations Bv , Tv , Bφ and Tφ are shown in Fig.

7, where Bv and Tv are the lateral displacements of the bottom and top flanges with

respect to the y axis, respectively, Bφ is the rotation at the bottom of the flange, and

Tφ is the rotation at the top of the flange. Bk and Tk are the curvatures in the z-y

plane at the bottom and top of the web, respectively. It is assumed that the rectangular

hollow flange cross-sections remain undistorted during deformation.

The displacements of the flanges and web in the x-direction (u) are obtained from

simple beam theory, and the displacements in the z and y directions (w, v) are

obtained from the kinematic conditions. These displacements may be written as

follows.

For the beam web:

0w)z,x(vv

0u

W

W

W

===

(5a)

For the bottom flange:

13

)x(vv)x(yw

zvu

BB

BB

x,BB

=φ=

−=

(5b)

For the top flange:

)x(vv)x(yw

zvu

TT

TT

x,TT

=φ−=

=

(5c)

The cross-sectional shape of the web is allowed to deform, and the web’s rotation

and lateral deflection are then coupled, with the displacements proceeding as follows.

With regard to the flanges, it can be assumed that the displacements and twists

vary sinusoidally lengthwise along the member and that the flanges do not change

shape. The buckling deformations of the flanges can be written as

L

xmsinL

xmsin

wvwv

wvwv

B

B

T

T

B

B

T

T

πδ=

π

=

. (6)

With regard to the web, it can be assumed that its out-of-plate buckling shape is a

cubic polynomial. A cubic buckling shape yields good results for beams that are

subject to equal end moments, but not for those subject to a uniformly distributed load.

A shape function for the cubic polynomial was employed in the current study to

represent the web buckling shape of LSBs subject to equal end moments. The flexural

displacement ( Wv ) of the web, of which the maximum value in Fig. 7 is Wv , is

obtained by assuming that the web buckles out-of-plane as a cubic polynomial curve,

and is given by

3

13

2

12

110

222

+

+

+=

dza

dza

dzaavw . (7)

It can be assumed that the displacements and twists vary sinusoidally lengthwise

along the member. The buckling deformations of the web can thus be written as

14

Lx

Lxv

vvv

B

T

B

T

B

T

B

T

πδπ

φφ

φφ

sinsin =

=

, (8)

where Tv , Bv , Tφ and Bφ are the maximum amplitudes of the buckling

displacements, and m is a positive integer representing the number of harmonics into

which the beam buckles.

The coefficients 0a , 1a , 2a and 3a are obtained by the compatibility identities

( )

( )

−−−−

=

∂∂

−

∂∂

−=

=

−=

=

−=

1

1

1

1

5.01

5.01

5.0

5.0

5.05.0

)(5.0)(5.0

5.0

5.0

)()(

1

1

1

1

dd

dhvdhv

zvd

zvd

vv

B

T

BB

TT

dz

w

dz

w

dzw

dzw

φφφφ

δ (9)

The coefficients 0a , 1a , 2a and 3a can be expressed as

−+

−−+=

221

21

221

21

21

21 1

1

1

10

ddhd

dhvva BTBT φφ (9a)

−+

−−+−=

221

43

221

43

43

43 1

1

1

11

ddhd

dhvva BTBT φφ (9b)

+

=

241

241 11

2dda BT φφ (9c)

+

−−=

241

241

41

41 1

1

1

13

ddhd

dhvva BTBT φφ (9d)

3.2 Solution for lateral buckling using the work resulting from buckling

If a beam is loaded under a constant moment, M, then the stress distribution will

vary linearly from the top to the bottom flange, as shown in Fig. 8.

15

Fig. 8 Stress distributions under bending moment

The bending moment is written as

∫σ=A

zdAM , (10)

and the values of the bending stress, σ , corresponding to the bending moment are

given by

−=σ

−=σ

)zz(IM

)dz(IM

0y

W

10y

F

. (11)

With regard to a beam subject to end moments, it is noted that for an assumed

shape function such as Eq. 7, the energy terms involving shear stress are zero; thus,

the external work (V) arising from lateral buckling (i.e., with the pre-buckling work

omitted) may be written as [22]

∫∫

∫∫

∫∫

σ+

+σ+

+σ=

L2

x,WWA

L2

x,B2

x,BFBA

L2

x,T2

x,TFTA

dxdA)v(21

dxdA))wv((21

dAdx))wv((21V

W

FB

FT

. (12)

3.3 Strain energy stored during buckling

The total potential energy of a LSB subjected to edge loading is the sum of strain

energy U and the potential energy of the applied load Ω :

Ω+=Π U . (13)

16

The strain energy of a three-dimensional isotropic medium referred to as arbitrary

orthogonal coordinates may be written as

dv)(21U yzyzxzxzxyxyzzyy

vxx γτ+γτ+γτ+εσ+εσ+εσ= ∫ . (14)

In accordance with the basic approximations of thin-plate theory, xzγ , yzγ and zσ

can be omitted. It is assumed that the strains and curvatures are much less than unity.

The finite-strain expressions for the in-plane strain components of the mid-surface are

given by

yxyxyxxyxy

2x

2x

2xyy

2x

2x

2xxx

wwvvuuvu

)wvu(21v

)wvu(21u

++++=γ

+++=ε

+++=ε

. (15)

In von Karman’s plate theory, only displacement gradients xw and yw are

expected to achieve significantly large amplitudes, and thus, among the nonlinear

terms in Eq. 15, only xw2 , yw2 and xw yw are retained. However, in the present

application to LSBs, the gradients of u and v, in addition to that of w, may become

large due to in-plane rotation, especially for the flange component. The terms xu and

yv are of a higher order than the other terms, and the second-order terms involving

xu and yv are thus ignored.

The utilisation of potential energy concepts in the establishment of relationships

for instability analysis presumes that pre-buckling deformations have taken place and

that an examination of the perturbed equilibrium state is being conducted. Thus, the

pre-buckling work ( Ω ), which is the product of the applied loads and their

corresponding deflection, is ignored in the buckling analysis. Hence,

U=Π . (16)

The strains in an arbitrary plate component and web location can be written as Eqs.

17 and 18.

For the flanges:

17

Txyy,Tx,Tx,Ty,TTxy

Ty2

x,T2

x,Ty,TTy

Tx2

x,T2

x,Tx,TTx

zw2)wwvu

zw)wu(21v

zw)wv(21u

−++=γ

−++=ε

−++=ε

,

xy,By,Bx,Bx,By,BBxy

y,B2

x,B2

x,By,BBy

x,B2

x,B2

x,Bx,BBx

zw2)wwvu

zw)wu(21v

zw)wv(21u

−++=γ

−++=ε

−++=ε

(17)

For the web:

xz,Wz,Wx,Wx,Wz,WWxy

zz,W2

x,W2

x,Wz,WWy

xx,W2

x,W2

x,Wx,WWx

yv2)vvwu

yw)vu(21w

yw)wv(21u

−++=γ

−++=ε

−++=ε

(18)

Drawing on the flange displacement assumption and ignoring the small terms, the

total potential energy of the top and bottom flanges and web can be obtained as

follows.

( )( )

( )( )dxdAww2)y(v21dxJG

21dxvIE

21

dxdAww2)y(v21dxJG

21dxvIE

21U

Bf

Tf

A

L

0ByBxBxy

2Bx

2BxxT

2Bx

L

0fB

2Bx

L

0zB

A

L

0TyTxTxy

2Tx

2TxxT

2Tx

L

0fT

2Tx

L

0zTFlange

∫ ∫∫∫

∫ ∫∫∫

τ+φ+σ+φ++

τ+φ+σ+φ+=(19)

( )

dxdzv12

t4GvvvD2vDvD21

dxdzv)v1(2vvv2vvv1

Et21

dxdz)wu(2

v1wvu2wuv1

Et21

dv)(21U

2Wz

3

WzWxxz2Wzz

2Wxx

2WzWzWx

2Wz

2Wx2

2WxWzWzWx

2Wz

2Wx2

WxzWxzWzWzv

WxWxWeb

∫∫

∫∫

∫∫

∫

++++

−+++−

+

+

−+++

−=

γτ+εσ+εσ=

(20)

In the above,

)v1(2EG,

)1(12Et

D,)1(12

EtD,

)1(12Et

D 2

3equ

xz2

3equ

z2

3equ

x +=

ν−=

ν−=

ν−= .

The proposed reduced thickness (tequ) of the web can be expressed as (based on Eq.1)

)Sd4D1(tt

1

2

equπ

−= (21)

18

where t = the web thickness and is also the flange thickness of a LSB, D = the web

opening diameter, S = web opening spacing and d1= the centreline dimension of web

height.

This deformation web pattern involves both lateral and vertical deflections

( Wv , Ww ) and rotation. If the cross-sectional profile of the web at any transverse

cross section is prohibited from curving, then the web is regarded as rigid, and the

rotation and lateral deflection are coupled [6, 23]. The displacements in the flexible

web case are the same as those in the classical (rigid web) case except for the lateral

displacement, Wv . The vertical displacement ( xxWv , ), however, is quite small and can

thus be ignored.

3.4 Energy equation

The total potential energy is

VU −=Π . (22)

Transformation of the total potential energy given by Eq. 21 into the desired

stiffness expressions requires selection of the displacement functions to describe the

behaviour of the element. Each displacement component is assumed to take the

general form

iiN δ=δ , (23)

where iN is a function of the x coordinate. Substituting Eq. 23 into Eq. 22 yields the

following expression for the total potential energy.

δδ=Π ]K[21 T . (24)

The symmetric total stiffness matrix is as follows.

[ ]

=

44342414

34332313

24232212

14131211

ji

KKKKKKKKKKKKKKKK

K (25)

19

[ ]

−+

+

−+

−

−

+−

−

−

−

+−

+−

−−++−

−

=

θθθ

θθθ

θθθθ

θ

21

WW2

21

21

2WFF

Fo32

211

WW21

2

F

Fo32

211

WW2

21

2WF1

2

12

11WW21

1

ij

hd

1S21

h3d

hd

1S21S

AI

hM2

hd

hd

SS21

hd

1S0

AI

hM2

hd

hd

SS21

hd

S21S

hd

S0

hd

1Shd

SS2S2ShM2

00hM2S

K

(26)

In the above matrix, 2z

2

1 LEIS π

=θ , 3

1

2

231

2

2

2 Sd4Dtt

)1(12dEL24S

π−

ν−π=θ and

2F

WF hGJ2S = ,

3

1

2

1WW Sd4

Dttd3G2S

π−= .

This equation can be expressed as an explicit form of

[ ] 0Kdet = , (27)

in which case Eq. 26 can be expressed as follows.

0KKKKKKK)KKKKKK()KK( 44223114433221112

24433

2342211

23412 =−++− (28)

0KKKKKKK

MKKhr4

M

h4KKKK

hr2

hr2

Sd4Dtt

h3Gd4

Sd4Dtt

h3G4

Sd4Dtt

)1(12hdEL24

Sd4Dtt

)1(12hdEL24

442231144332211

22113

2F

2

444332211

2

3F

2

3

2F

2

1

2

31

3

1

2

2

3

1

2

231

2

23

1

2

2221

2

2

=−+

−

+

−

−

π−+

π−−

π−

ν−π−

π−

ν−π

(29)

3.5 Verification of the proposed numerical method

To demonstrate the accuracy of the equivalent web thickness terms’ inclusion in

the energy method solution, the elastic lateral buckling moments of a single

20

symmetric LSB were computed using the proposed energy method and the FSA

(THIN-WALL) [24] and FEA programs (ABAQUS) [20]. The influence of web

openings on the lateral distortional buckling of open thin-walled cross-sections can be

approximated by modifying the cross-section thickness in ABAQUS and THIN-

WALL using Method 4 (Eq. 4), which was found to be the most accurate of the four

methods introduced in the previous section. Unlike the energy method solution,

Method 4 can be used with FEA and FSA.

Fig. 9 presents a comparison of the lateral distortional buckling modes obtained

from FEA with and without web hole modelling (Figs. 9(a) and (b)) and from FSA

without web hole modelling (Fig. 9(c)) for the most slender section with larger web

holes. It can be seen that the lateral deformation of the LSB with web hole modelling

is slightly greater than that without web hole modelling.

Six LSB members subjected to equal end moments were analysed using the three

aforementioned methods. They included two non-compact sections (300x75x3.0 LSB

and 200x60x2.0 LSB), two slender sections (300x60x2.0 LSB and 250x60x2.0 LSB)

and two compact sections (250x75x3.0 LSB and 150x45x2.0 LSB). Numerous spans

in the range of 500 to 10,000 mm were considered. The results of the comparison are

presented in Fig. 10.

(a) (b) (c)

Fig. 9 Lateral distortional buckling modes for 300 x 60 x 2.0 LSB (D = 150 mm, S =

250 mm): (a) Finite element model with web holes; (b) Finite element model without

web holes; (c) Finite strip model without web holes

21

An explicit buckling solution was obtained from a formulation of the total potential

energy for LSBs with circular web openings (from Eq. 28). The accuracy of the

proposed numerical method’s use with an equivalent reduced web thickness was

verified by comparing its results with those obtained from FSA and FEA of LSBs.

The results demonstrate that the proposed explicit solutions based on the energy

method are able to predict with accuracy the elastic lateral distortional buckling

moments of LSBs with web holes as shown in Fig. 10. The mean and corresponding

CoV of the ratio of elastic lateral distortional buckling moments obtained with the

proposed method and in FSA (THIN-WALL) are 1.022 and 0.040, respectively, thus

demonstrating good agreement.

0 2000 4000 6000 8000 10000Span L (mm)

0

100

200

300

Elas

tic B

uckl

ing

Mom

ent,

Mod

(kN

m)

300x75x3.0 LSBFEA (ABAQUS)FSA (THIN-WALL)Present Study

0 2000 4000 6000 8000 10000Span L (mm)

0

20

40

60

80

100

Elas

tic B

uckl

ing

Mom

ent,

Mod

(kN

m)


(a) 300 x 75 x 3.0LSB (D = 150 mm, S = 250 mm) (b) 250 x 60 x 2.0LSB (D = 150 mm, S =

250 mm)

0 2000 4000 6000 8000 10000Span L (mm)

0

20

40

60

80

Elas

tic B

uckl

ing

Mom

ent,

Mod

(kN

m)


0 2000 4000 6000 8000 10000Span L (mm)

0

40

80

120

160

Elas

tic B

uckl

ing

Mom

ent,

Mod

(kN

m)


(c) 200 x 60 x 2.0LSB (D = 100mm, S = 250mm) (d) 300 x 60 x 2.0LSB (D = 150 mm, S =

250 mm)

22

0 2000 4000 6000 8000 10000Span L (mm)

0

50

100

150

200

250

Elas

tic B

uckl

ing

Mom

ent,

Mod

(kN

m)


0 2000 4000 6000 8000 10000Span L (mm)

0

20

40

60

80

100

Elas

tic B

uckl

ing

Mom

ent,

Mod

(kN

m)


(e) 250 x 75 x 3.0LSB (D = 150 mm, S = 250 mm) (f) 150 x 45 x 2.0 LSB (D = 50 mm, S =

250 mm)

Fig. 10 Comparison of elastic lateral buckling moments obtained from FEA, FSA and the proposed numerical method

All of the comparisons are based on the equivalent web thickness method.

However, as discussed in Section 2, approximate finite element models based on solid

webs with an equivalent reduced web thickness provide accurate buckling solutions

compared with accurate finite element models that include actual web hole

configurations. Hence, we can also conclude that employing the same equivalent web

thickness approach with energy method buckling solutions is also accurate.

In summary, the proposed numerical method, which employs an equivalent web

thickness approach, provides accurate elastic buckling solutions for LSBs with web

openings, except for those with a very short span. For such short beams, yielding and

local buckling become more important, and the current solution’s loss of accuracy

will have only a small effect on the prediction of beam strength with varying web hole

configurations. It is therefore recommended that the numerical solution be used to

determine the elastic lateral buckling moments of LSBs with web openings without

the need for more complex FEA or FSA programs.

5. Conclusion

This paper has presented the details of an investigation into the elastic lateral

distortional buckling behaviour of LSBs with circular web openings that are subject to

equal and opposite end moments using FEA, FSA and energy methods.

An equivalent thickness method for the elastic buckling analysis of LSBs with web

openings was proposed. Four equations for predicting equivalent plate thicknesses

were proposed, and their accuracy was investigated using FEA. Elastic lateral

23

distortional buckling moments from approximate FEA based on solid web elements

with an equivalent reduced thickness were compared with those from accurate FEA

with actual web hole and spacing configurations, and the results show that two of the

proposed methods can be successfully employed with approximate finite element

models to predict these moments.

This paper has also presented an explicit elastic buckling numerical solution that

was obtained from a formulation of the total potential energy of LSBs with circular

web openings. The accuracy of the proposed numerical method’s use with an

equivalent reduced web thickness was verified through comparison of its results with

those of the FSA and FEA of LSBs. The numerical solution derived herein can be

adopted as the basis for an advanced theoretical method of predicting the nonlinear

lateral distortional buckling strength of LSBs with varying web opening

configurations. It is recommended that this solution be employed to determine the

elastic lateral buckling moments of LSBs with circular web openings, with no need

for the use of more complex FEA or FSA programs.

Acknowledgments

The authors would like to thank the Australian Research Council and OATM for

their financial support and the Queensland University of Technology for providing the

necessary research facilities and support for this research project. They would also

like to thank Mr Ross Dempsey, Manager of Research and Testing at OATM, for his

technical contributions and his overall support during the different phases of the

project.

References

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Brisbane, Australia.

24

[4] Anapayan, T., Mahendran, M. Mahaarachchi, D. (2011) Lateral Distortional

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[5] Galambos, T.V. (1988), Guide to Stability Design Criteria for Metal Structures,

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[6] Timoshenko, S.P and Gere, J.M. (1961), Theory of Elastic Stability, 2nd edition,

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25

[16] Seo, J.K. and Mahendran, M. (2009), Member Moment Capacities and

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