Buckling analysis of thin-walled cold-formed steel structural membersusing complex finite strip method

H.R. Naderian a, H.R. Ronagh b,n

a Department of Civil Engineering, University of Ottawa, Ottawa, ON, Canadab School of Civil Engineering, University of Queensland, Brisbane 4072, QLD, Australia

a r t i c l e i n f o

Article history:Received 2 August 2014Received in revised form31 December 2014Accepted 7 January 2015Available online 30 January 2015

Keywords:Cold-formedComplex finite strip methodLocal bucklingShear bucklingDistortional bucklingGlobal buckling

a b s t r a c t

In this paper, a generalised complex finite strip method is proposed for buckling analysis of thin-walledcold-formed steel structures. The main advantage of this method over the ordinary finite strip method isthat it can handle the shear effects due to the use of complex functions. In addition, distortional bucklingas well as all other buckling modes of cold-formed steel sections like local and global modes can beinvestigated by the suggested complex finite strip method. A combination of general loading includingbending, compression, shear and transverse compression forces is considered in the analytical model.For validation purposes, the results are compared with those obtained by the Generalized Beam Theoryanalysis. In order to illustrate the capabilities of complex finite strip method in modelling the bucklingbehavior of cold-formed steel structures, a number of case studies with different applications arepresented. The studies are on both stiffened and unstiffened cold-formed steel members.

& 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Buckling of thin walled cold-formed steel members could belocal, shear, distortional and global. Local buckling is normallydefined as the mode which involves plate-like deformations alone,without the translation of the intersection lines of adjacent plateelements. Shear buckling is a kind of local buckling which occursin cold-formed sections under shear loads. Global buckling is amode in which the member deforms with no deformation in itscross-sectional shape as it undergoes lateral deformations andtwist, consistent with the classical beam theory. The commonwell-known types of global buckling in cold-formed steel sectionsinclude flexural, torsional, and flexural–torsional which mighthappen in columns and lateral–torsional which might occur inbeams. Distortional buckling differs from local and global bucklingmodes. A general and widely adopted definition for distortionalbuckling mode does not currently exist. However, it can beconsidered as a mode with cross-sectional distortion that involvesthe translation of some of the fold lines [1]. As most cold-formedsections have stiffeners, and as such distortional buckling isprevalent in these sections, the study of distortional buckling hasbeen the subject of many research studies.

Finite strip method is a special form of finite element procedureusing displacement approach. Unlike the standard finite element

method which uses polynomial displacement functions in alldirections, the finite strip method calls for the use of simplepolynomials in some directions and continuously differentiablesmooth series in the other directions, with the stipulation thatsuch series should satisfy the boundary conditions at the end ofstrips. The philosophy of the finite strip method is similar to thatof the Kantorovich method [2], which is used extensively forreducing a partial differential equation to an ordinary differentialequation. In this method, especially for plates, Hermitian displace-ment functions are usually used in the transverse direction [3]. Theapproach brings about much simplicity, ease of programming andrapid convergence [4].

The first use of finite strip method for buckling analysis app-ears to be the work of Przemieniecki [5], who showed how thistechnique can be used in predicting the initial elastic localbuckling of plates and sections under biaxial compression. Hisapproach utilized the approximate finite strip formulation ofCheung and Cheung [6]. Plank and Wittrick [7] employed thesemi-analytical complex finite strip method to investigate buck-ling under combined loading of thin-walled structures. Theadvantage of their method over the formulations of the ordinaryfinite strip method is the ease with which it can handle shearforces. Wittrick [8] developed an exact finite strip method for thebuckling analysis of stiffened panels in compression. Azhari andBradford [9] developed bubble finite strip method, which aug-mented finite element formulations to obtain rapid convergence.They also extended the finite strip method to analyze the buckling

n Corresponding author. Tel.: þ61 7 336 59117; fax: þ61 7 336 54599.E-mail address: h.ronagh@uq.edu.au (H.R. Ronagh).

of plates with different end conditions [10]. Ada´ny and Schafer[11–13] derived a constrained finite strip method for decomposingthe buckling modes of thin-walled members. Li and Schafer [14,15]extended the constrained finite strip method to thin-walledstructures with different boundary conditions and applied it tothe stability of cold-formed steel sections. The authors havealready employed complex finite strip method for analyzing thedistortional buckling of cold-formed steel sections and the stabi-lity of stiffened cold-formed I-sections [16,17]. Pham and Hancock[18–21] used both spline finite strip and semi-analytical finite stripsolutions for shear buckling analysis and design of channelsections. Using complex functions, they included the shear loadingin the semi-analytical finite strip analysis. However, their researchis limited to the study of shear buckling of channel sections. It isworth mentioning that FSM is applicable to both isotropic andorthotropic structural materials. For instance, the application ofFSM has already been extended to buckling analysis of compositeFRP structural plates [22–24].

In the present study, a semi analytical complex finite stripmethod is employed to study the buckling modes of cold-formedsteel sections. A combination of loadings including shear, com-pression, and bending, and transverse forces can be considered inthe analysis. Consequently, all bucking modes including localbuckling, shear buckling, distortional buckling, and global bucklingare investigated. In contrast with ordinary and constrained finitestrip methods [11–15], and the current finite strip software [25],the present formulation can deal with shear forces in cold-formedsections. Moreover, it is shown that the complex finite stripmethod suggested here can consider the distortional buckling ofcold-formed steel sections in a simple manner. The method isapplicable to different cold-formed sections under general loadingconditions. For validation purposes, results of the proposed com-plex finite strip method are compared with predictions of theGeneralized Beam Theory (GBT) for the critical stresses andbuckling half-wavelengths of web stiffened cold-formed columns.The following are also presented.

a) Several case studies are performed to illustrate the capabilitiesof the proposed method.

b) The buckling behaviour of cold-formed channel sections withextra longitudinal stiffeners at the end of flanges and on theweb is studied.

c) The optimum width of extra longitudinal flange stiffener isevaluated to maximize the critical distortional stress of channelsections.

d) Moreover, the optimum position of web stiffener in cold-formed channel section columns is calculated to maximizethe local and distortional buckling strengths.

e) Shear buckling of stiffened and unstiffened cold-formed steelZ sections are also examined.

f) And finally, the influence of stiffeners in local, distortional, andglobal buckling modes of cold formed channel and Z sectioncolumns and beams are investigated and relevant comparisonsare made.

2. Theory

2.1. General

In the complex finite strip method, both in-plane and out-of-plane displacements can be handled. It is worth noting that local(including shear) and global buckling modes may occur due toout-of-plane and in-plane displacements respectively while thedisplacements caused by distortional buckling include a

combination of both out-of-plane and in-plane types. For thisreason, complex finite strip method is a suitable tool for predictingthe distortional buckling of cold-formed steel members as well aslocal and global modes under different loading conditions. More-over, complex FSM allows consideration of the shear forces in theanalysis procedure due to the presence of complex functions.

In the original complex finite strip method developed by Plankand Wittrick [7], elastic stiffness and stability matrices wereobtained using standard finite element techniques based on theenergy method. This procedure is followed in the present study toobtain the stiffness and stability matrices of a cold-formed steelstrip. It is mentionable that for infinitesimally small buckl-ing displacements, the in-plane and out-of-plane effects areuncoupled and thus are considered separately.

2.2. Definition of the problem

In the finite strip method (FSM), a thin-walled member such asthe cold-formed C section of Fig. 1 is divided into longitudinalstrips. The advantage of FSM over other methods, such as the finiteelement method which discretizes the member both in the long-itudinal and in the transverse directions, is that it only discretizesthe member in the transverse direction while chooses a contin-uous shape function longitudinally over the whole length of themember. In Fig. 1, a single strip is highlighted. The geometry,loading and degrees of freedom (DOF) for the strip are illustratedin Fig. 2. The strip is loaded by a combination of different loadsincluding uniform longitudinal compressive stress σL, longitudinalin-plane bending stress σb, uniform shear stress τ, and transversecompressive stress σT . If the member experiences buckling, thestrip will be subjected on its edges to a system of forces distributedin the longitudinal directionwith a buckling half-wavelength L. Foreach length L, eight degrees of freedom are allowed.

2.3. Displacement functions

The flexural displacement of a strip w and the in-planedisplacements u and v in the complex FSM are assumed to begiven by

w¼ Re N!

f b d!

f beiξ

� �ð1Þ

u¼ Re N!

xmb J d!

mbeiξ

� �ð2Þ

v¼ Re N!

ymbJ d!

mbeiξ

� �ð3Þ

Fig. 1. Finite strip discreteness.

H.R. Naderian, H.R. Ronagh / Thin-Walled Structures 90 (2015) 74–83 75

where Ref g denotes the real part, i¼ffiffiffiffiffiffiffiffi�1

p, ξ¼ πx=L, and J is a

4� 4 diagonal matrix defined by

J ¼ ½1 0 0 0; 0� i 0 0; 0 0 1 0; 0 0 0� i� ð4Þ

Moreover, d-f b and d-mb are 4� 1 vectors of infinitesimalflexural buckling displacements and in-plane displacements com-prising the assemblable displacements of the strip, respectively.According to Fig. 2, the displacement and force vectors are definedby the following expressions

d!T

f b ¼ ψ1;w1;ψ2;w2� � ð5Þ

d!T

mb ¼ v1; iu1; v2; iu2� � ð6Þ

p!Tf b ¼ m1; pZ1;m2; pZ2

� � ð7Þ

p!Tmb ¼ py1; ipx1; py2; ipx2

n oð8Þ

in which the subscripts fb and mb denote flexural buckling andmembrane buckling, respectively, and generally all the four vectors arecomplex.

In Eqs. (1)–(3), N-f b is a 1� 4 polynomial interpolation vector

and N-xmb and N-

ymbare interpolation vectors defined semi-

analytically as

N-

f b ¼b81�ηð Þ2ð1þηÞ; 1

41�ηð Þ2ð2þηÞ;

�

�b8ð1�ηÞð1þηÞ2; 1

41þηð Þ2ð2�ηÞ

�ð9Þ

N-

xmb ¼ 0;12ð1�ηÞ; 0 ;

12ð1þηÞ

� �ð10Þ

N-

ymb¼ 1

2ð1�ηÞ; 0; 1

2ð1þηÞ; 0

� �ð11Þ

where η¼ 2y=b and b is the width of the strip.The stiffness matrices Sf b , Smb and geometric stiffness (stabi-

lity) matrices Gfb, Gmb are defined by

p-

f b ¼ ð Sf bþGfbÞd-

f b ð12Þ

p-

mb ¼ ð SmbþGmbÞd-

mb ð13ÞIt should be noted that the term i¼

ffiffiffiffiffiffiffiffi�1

pin the in-plane

displacement and the in-plane force vectors d-mb and p-mb

automatically incorporate a 90 degree phase difference betweenu and v displacements and between px and py forces. For the same

reason, the in-plane stiffness matrix is entirely real and symme-trical. Moreover, if τa0; the out-of-plane stiffness matrix containscomplex off-diagonal elements and is Hermitian in form. On theother hand, if τ¼ 0, the out-of-plane stiffness matrix will be realand symmetrical [7].

The selected longitudinal functions result in members thatare pinned and free-to-warp at their ends. More complicated bound-ary conditions may be treated [10,14,15] but are not discussed here.

2.4. Strain–stress relations

In accordance with established stiffness procedures, the vectorsof generalized buckling strain ε and ε0 for out-of-plane and in-plane displacements are given by

ε!f b ¼ w;2x w;2y 2w;xw;yD E

ð14Þ

ε!mb ¼ u;x v;y u;yþv;xD E

ð15Þ

The flexural property matrix of an isotropic plate Df defines therelationship between the infinitesimal generalized moments andstrains for out-of-plane displacements by

M!¼Df ε

!f b ð16Þ

in which

Df ¼ Et3=12ð1�υ2Þ1 υ 0υ 1 00 0 ð1� υÞ

2

264

375 ð17Þ

where E and υ are the Young’s modulus of elasticity and thePoisson’s ratio while t is the thickness of the plate strip. Also, thebuckling internal stress σ! and strains ε!mb are related through

σ!¼Dm ε!mb ð18Þ

where Dm is the in-plane rigidity matrix of the isotropic plate andis given by

Dm ¼ E=ð1�υ2Þ1 υ 0υ 1 00 0 ð1� υÞ

2

264

375 ð19Þ

2.5. Flexural stiffness and stability matrices

The strain energy stored in the cold-formed plate strip can thusbe calculated, from which after some mathematical manipulation

Fig. 2. (a) Basic state of stresses in a strip, (b) strip degree of freedoms.

H.R. Naderian, H.R. Ronagh / Thin-Walled Structures 90 (2015) 74–8376

involving complex arithmetic [26], the flexural standard stiffnessmatrix Sf b is derived as

Sf b ¼b2

Z þ1

�1BTf bDf Bf bdη ð20Þ

where Bfb is the 3� 4 matrix defined by

Bfb ¼1

b2

�Ω2N!

f b

4N!″

f b

4iΩN!0

f b

266664

377775 ð21Þ

in which Ω¼ πb=L and where primes denote differentiation withrespect to η. Substituting Eqs. (17) and (21) into (20), flexuralstiffness matrix is rewritten as

Sf b ¼Et3

b3ð1�υ2Þ

Z þ1

�1

124

Ω4N!T

f b N!

f bþ13Ω2N

!0f bT N

!0f b

�

þ23N!

″Tf b N!

″f b�16υΩ2ðN!

T

f b N!

f bÞ″�dη ð22Þ

or

After calculating the decrease in the potential of a cold-formedstrip subjected to the basic forces during buckling and by appro-priate substitution and integration, the flexural stability matrixGfb may be obtained from

Gfb ¼t2b

Z þ1

�1Ω2σL N

!T

f b N!

f bþΩ2ησB N!T

f b N!

f bþ4σT N!0

f bT N!0

f b

�

þ2ΩτiðN!0Tf b N!

f b� N!T

f b N!0

f bÞ�dη ð24Þ

or

2.6. Membrane stiffness and stability matrices

A similar method to that for the flexural (out-of-plane) mat-rices may be followed in order to obtain the membrane (in-plane)stiffness matrix and stability matrix. The internal virtual workdone in the cold-formed steel strip results from the bucklingdeformation which after mathematical manipulation involvingcomplex arithmetic results in the membrane standard stiffnessmatrix Smb, as

Smb ¼12btJ

Z þ1

�1BTmbDmBmbdη

!J ð26Þ

where Bmb is the 3� 4 matrix defined by

Bmb ¼1b

iΩN!

xmb

2N!0

ymb

2N!0

xmbþ iΩN

!ymb

266664

377775 ð27Þ

in which prime denotes differentiation with respect to η. Onsubstituting Eq. (27) into Eq. (26), the membrane stiffness matrixSmb may be written as

Smb ¼Et

2bð1�v2ÞR þ1�1 J Ω2N

!T

xmb N!

xmb þ12ðI�vÞΩ2N

!T

ymbN!

ymb

�

þ2ð1�vÞN!0xmb

T N!0

xmbþ4N

!0ymb

T N!0

ymbþ2ΩυiðN!

0ymb

T N!

xmb

� N!T

xmb N!0

ymbÞþΩð1�υÞiðN!

0xmb

T N!

ymb� N!T

ymbN!0

xmbÞ�Jdη

ð28Þ

Substituting Eqs. (10) and (11) and carrying out the integra-tions, it is found that

Smb ¼Et

bð1�v2Þ

� � 1þð1�vÞΩ2=6 ð1�3vÞΩ=4 �1þð1�vÞΩ2=12 �ð1þvÞΩ=4ð1�vÞ=2þΩ2=3 ð1þvÞΩ=4 �ð1�vÞ=2þΩ2=6

Symm ð1þð1�vÞΩ2=6Þ �ð1�3vÞΩ=4ð1�vÞ=2þΩ2=3

266664

377775

ð29Þ

According to Ref. [7], σL is the only stress component that needsto be considered for the in-plane buckling and there is noconceivable in-plane instability that can arise from the action ofthe stresses σB, σT and τ: In order to calculate the loss of potential

energy of the longitudinal stress σL one can make use of thefollowing nonlinear expression for the longitudinal strain εx

εx ¼ u;xþ12

u;2x þv;2x þw;2x ð30Þ

In this expression the first term has already been accounted forin deriving the strain energy whilst the last term, involving w, hasbeen used in the calculation of the decrease in the potential of thebasic stresses due to the out-of-plane displacement. Thus, bydeveloping an appropriate expression for the decrease in thepotential of the basic stresses due to in-plane displacement andby appropriate substitution and integration, the in-plane stability

Sfb ¼Et3

12ð1�v2Þb3

! 4b2þΩ4b2=105þ4Ω2b2=15 6bþυΩ2bþ11Ω4b=210þΩ2b=5 2b2�Ω4b2=140�Ω2b2=15 �6bþ13Ω4b=420�Ω2b=512þ13Ω4=35þ12Ω2=5 6b�13Ω4b=420þΩ2b=5 �12þ9Ω4=70�12Ω2=5

Symm 4b2þΩ4b2=105þ4Ω2b2=15 �6b�υΩ2b�11Ω4b=210�Ω2b=512þ13Ω4=35þ12Ω2=5

266664

377775ð23Þ

Gfb ¼ � tb

� �

Ω2σLb2=105þ2σTb

2=15�Ω2σBb2=420 11Ω2σLb=210þσTb=10�2Ω2σBb=105þ iΩτb=5 �Ω2σLb

2=140�σTb2=30þ iΩτb2=30 13Ω2σLb=420�σTb=10þΩ2σBb=420� iΩτb=5

11Ω2σLb=210þσTb=10�2Ω2σBb=105� iΩτb=5 13Ω2σL=35þ6σT=5�Ω2σB=5 �13Ω2σLb=420þσTb=10þΩ2σBb=420þ iΩτb=5 9Ω2σL=70�6σT=5� iΩτ

�Ω2σLb2=140�σTb

2=30� iΩτb2=30 �13Ω2σLb=420þσTb=10þΩ2σBb=420� iΩτb=5 Ω2σLb2=105þ2σTb

2=15þΩ2σBb2=420 �11Ω2σLb=210�σTb=10�2Ω2σBb=105þ iΩτb=5

13Ω2σLb=420�σTb=10þΩ2σBb=420þ iΩτb=5 9Ω2σL=70�6σT=5þ iΩτ �11Ω2σLb=210�σTb=10�2Ω2σBb=105� iΩτb=5 13Ω2σL=35þ6σT=5þΩ2σB=5

266664

377775

ð25Þ

H.R. Naderian, H.R. Ronagh / Thin-Walled Structures 90 (2015) 74–83 77

matrix Gmb may be obtained by

Gmb ¼Ω2σLt2b

Z þ1

�1JðN!

T

xmb N!

xmb þ N!T

ymbN!

ymbÞJdη ð31Þ

Finally, substituting Eqs. (10) and (11) and carrying out theintegrations, the matrix Gmb may be written as

Gmb ¼�tσ LΩ

2

b

1=3 0 1=6 01=3 0 1=6

Symm 01=3

266664

377775 ð32Þ

2.7. Eigenvalue buckling problem

Once the strip stiffness and stability matrices Sf b, Smb and Gfb,Gmb are derived and combined for each strip, they can beassembled into the respective global matrices ST and GT usingstandard procedures. The buckling problem can then be set as thefollowing eigenvalue problem

ðST �GT ÞΔ¼ 0 ð33Þwhere Δ is a scaling factor related to the critical load.

3. Application

3.1. General

Complex finite strip method of analysis developed in theprevious section is programmed on a desktop workstation. Theprogram is used to study the buckling of simply supported cold-formed steel members under different loading conditions includ-ing compression, bending and shear. The effect of stiffeners andother geometric properties of cold-formed steel sections on dis-tortional buckling and shear buckling as well as other modes are

investigated. Convergence of the method is excellent so that a fewnumbers of strips is needed to reach high accuracy. The resultsshow that the use of complex finite strip method significantlyimproves the efficiency in terms of strip subdivision.

3.2. Accuracy of the method

In order to evaluate the accuracy and validity of the method,a thin-walled steel C section with longitudinal web stiffener(E section) as shown in Fig. 3 under uniform compression isstudied. The cross-sectional dimensions are given in Fig. 3 andthe elastic constant values E¼ 210 ðGPaÞ and v¼ 0:3 are adopted.Semi analytical complex finite strip results are compared withvalues obtained from GBT analyses carried out by Camotim andcolleagues [27]. Fig. 4 shows the GBT and the complex FSM resultsconcerning the buckling behavior of simply-supported E sectioncolumns. The curves presented provide the variation of columnbuckling load Nb with its length L in logarithmic scale. It canclearly be seen that the buckling loads and wave-lengths obtainedwith the complex FSM and the GBT are virtually the same. Thisfully validates the latter. The buckling curves exhibit three distinctzones corresponding to local, distortional, and global (flexural–torsional) buckling modes.

Table 1 shows the critical buckling loads Ncr and lengths Lcrobtained by complex FSM and GBT methods for local, distortionaland flexural–torsional buckling modes. The ratios of critical buck-ling loads and lengths derived by the mentioned methods are alsolisted. The calculated ratios shown in Table 1 demonstrate that thevalues obtained from the complex FSM are in good agreementwith those obtained from GBT for all buckling modes.

Fig. 3. E section [23] (dimensions in mm).

Fig. 4. Buckling behavior of simply-supported E-section columns. (a) GBT results [23], (b) semi analytical complex finite strip results.

Table 1Comparison of critical stresses and lengths.

Type ComplexFSM

GBT [23] Length ratio Critical forceratio

Lcr(cm)

Ncr(kN)

Lcr(cm)

Ncr(kN)

Lcr (FSM) /Lcr(GBT)

Ncr (FSM)/Ncr (GBT)

Local mode 6.30 50.8 �6 52 1.05 0.98Distortionalmode

50.5 43.2 �50 43.5 1.01 0.99

Flexural–torsionalmode

200 36.2 200 36.3 1.00 1.00

H.R. Naderian, H.R. Ronagh / Thin-Walled Structures 90 (2015) 74–8378

3.3. Buckling modes of stiffened channel sections

The complex finite strip method can be applied to study thebuckling modes of cold-formed steel stiffened channel section mem-bers with extra longitudinal stiffeners (stiffened C sections). In Fig. 5,three different common types of cold-formed steel channel sectionsare shown. As a sample, a stiffened cold-formed C section underuniform bending and compression is considered. The geometricdata and elastic constants are E¼ 200 GPað Þ; v¼ 0:3, and bw ¼200 mm; bf ¼ 120 mm, bs ¼ 10 mm; t ¼ 1 mm, where bwand bf are the web depth and the flange width, respectively, t isthe thickness of the section, and bs is the stiffener width.

Studies show that the angle of the extra stiffener to the horizonθ and its width bs1 are amongst factors that have considerableeffect on the distortional buckling of stiffened C sections. Here, theinfluence of variation in the angle of the extra stiffener to thehorizon, θ, and the extra stiffener width bs1 on the bucklingbehavior of stiffened C section columns and beams are investi-gated. In this regard, special attention is paid to the issue ofdistortional buckling mode.

3.3.1. Distortional buckling of cold-formed structuresAs previously mentioned, distortional buckling is a mode with

cross-sectional distortion that involves translation of some of thefold lines (intersection lines of adjacent plate elements). Hancockin the Australian/New Zealand Standard [28] states that distor-tional buckling is a mode of buckling involving changes in thecross-sectional shape, excluding local buckling.

The importance of distortional buckling in cold-formed struc-tures has prompted researchers like Schafer and colleagues [29],Hancock and colleagues [30], Silvestre and Camotim [31,32], and

others [33,34,16] to present analytical models for this phenom-enon. The majority of these studies are related to channel andZ sections. A large number of laboratory tests have been per-formed on distortional buckling of cold-formed steel members[35,36]. A series of rich tests have also been performed by the ThinWalled Structures Group under the supervision of Prof. B.W.Schafer at Johns Hopkins University in the USA [37,38]. Thesetests are not limited to distortional buckling but cover differentbuckling modes such as local buckling.

3.3.1.1. Stiffened C section columns. In Fig. 6, the normalized bucklingstresses F=E of stiffened C section columns with bs1 ¼ 20 mm is givenas a function of the dimensionless buckling half-wavelength L=bw.Several curves are given for different values of θ. The curves exhibit thesame characteristics, namely two minimum points, the first, with aminimum value at L=bw between 0.1 and 1.5 and the second with aminimum value at L=bw between 2 and 8. In the first region, thebuckling mode is local and in the second, it is distortional. Beyond thesecond peak, the buckling stress decreases with increasing half-wavelength L until the mode is predominantly flexural–torsional, aspredicted by the Vlasov theory.

As is seen, with decreasing the angle of the extra stiffener tothe horizon, distortional buckling stress and its associated half-wavelength increase, however, local and flexural–torsional buck-ling stresses are not sensitive to this angle. In the extreme, when θis equal to 90 degree, a C section is obtained. Analyzing the curvesof Fig. 6, it is clear that the distortional buckling stress of stiffenedC section is more than twice that of an unstiffened section. In fact,as θ decreases the stiffener becomes more effective in thedistortional mode.

In Fig. 7 the influence of different ratios of bs1=bf on the bucklingstresses of stiffened C sections under pure compression is examined.Buckling curves in Fig. 7 show that the extra stiffener does not have

Fig. 5. Cold-formed channel sections. (a) Simple channel, U shape; (b) stiffened channel, C shape; (c) stiffened channel with extra longitudinal stiffeners, stiffened C shape.

Fig. 6. Buckling curves for different ratios of theta in pure compression.

Fig. 7. Buckling curves for different ratios of bs1=bf in pure compression.

H.R. Naderian, H.R. Ronagh / Thin-Walled Structures 90 (2015) 74–83 79

any effect on the buckling stress of stiffened C sections in local andflexural–torsional modes. However, extra stiffener increases the dis-tortional buckling stress of this section. On the other hand, it can beseen that when the ratio of bs1=bf increases, the distortional bucklingstress decreases. Comparing distortional buckling stresses at differentratios of bs1=bf , it can be concluded that the maximum distortionalbuckling stress in stiffened C sections occurs at a specific ratio ofbs1=bf . For illustrative purposes, complex finite strip method is used todetermine the critical distortional stress Fcrd of stiffened C sections.Two case studies are performed in such a fashion that in each case thesection is under uniform compression. Figs. 8 and 9 show thenormalized critical distortional stresses Fcrd=E of the stiffenedC section columns plotted against the ratio of the width of extrastiffener to flange width bs1=bf . Dimensions of the sections are shownin the figures. These graphs are produced by plotting Fcrd=E againstthe half-wavelength, and by fitting a quadratic interpolation functionthrough three points close to the distortional nadir of the garland-shaped curve. It is clearly seen that both graphs arrive at theirmaximum, at bs1=bf between .05 and .15.

3.3.1.2. Stiffened C section beams. Similar studies are carried out onthe buckling behaviour of stiffened C section beams. In Fig. 10, theinfluence of different ratios of bs1=bf on the buckling stressesof stiffened C sections under pure bending is examined. As anextreme case, when bs1=bf ¼ 0, in practice, a cold-formed C sectionis obtained. It can be seen from Fig. 10 that what happens to thedistortional buckling stress of the stiffened C section beam issimilar to that for columns (Fig. 7). However, the situation is

different for local and lateral–torsional modes. Here, bucklingcurves show that the extra stiffener increases the buckling stressof stiffened C section beams in local and lateral–torsional modeswhile it has no significant effect on local and flexural–torsionalbuckling stress of columns (Fig. 7). It should be noted that here thedistortional buckling is the predominant mode rather than thelocal buckling.

An assessment similar to that of stiffened C section columns ismade for C section beams. Two case studies are examined with thesame geometric and material properties of cold-formed steelsections of Figs. 8 and 9. In each case the section is under uniformbending. Figs. 11 and 12 show the normalized critical distortionalstresses Fcrd=E of stiffened C section beams plotted against theratio of the width of extra stiffener bs1 to flange width bf . It isobvious that both graphs show results similar to that for stiffenedC section columns.

3.4. Optimum position of stiffeners in E section columns

In order to achieve increased economy and efficiency in cold-formed steel channel sections, a slender web may be stiffened bylongitudinal stiffeners. Such section may be named a web stiffenedchannel or E section. Local and distortional buckling of E sectioncolumns under pure compression is studied using complex FSM.Figs. 13 and 14 show dimensionless critical local and distortionalstresses of the E section against various positions of the webstiffener for different values of “a”. The optimum position of webstiffener to maximize the stresses is found to be 0:5bw and 0:51bw

Fig. 8. Critical distortional buckling stresses in pure compression; case study 1.

Fig. 9. Critical distortional buckling stresses in pure compression; case study 2.

Fig. 10. Buckling curves for different ratios of bs1=bf in pure bending.

Fig. 11. Critical distortional stresses in pure bending; case study 1.

Fig. 12. Critical distortional stresses in pure bending; case study 2.

Fig. 13. Critical local stresses in pure compression.

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from the bottom of the section for local and distortional bucklingmodes, respectively. It is obvious that the optimum position ofweb stiffener will be different for E section beams under uniformbending [26].

3.5. Shear buckling of cold-formed steel Z sections

Shear forces can cause out-of-plane displacements in thin-walled structures which may result in local instability, called shearbuckling. As previously mentioned, using complex finite stripmethod it is possible to investigate the local buckling of cold-formed steel sections under shear loading. In this section, the localshear buckling of cold-formed steel Z sections under pure shear isevaluated. In Fig. 15, the geometric shapes of unstiffened andstiffened cold-formed Z sections are shown. Here, it is assumedthat the web is subjected to a uniform shear stress τ¼ V=bwt inwhich V is the shear force. Also, the flange shear stress is ignored.The numerical results are shown in Figs. 16–19 where the variationof normalized shear critical stresses Fs=E versus L=bw is plotted.The minimum points of the buckling curves are critical shearstresses. Cross-sectional dimensions are given in the figures andmaterial properties of the cold-formed steel are E¼ 200 GPað Þ andv¼ 0:3. In Figs. 16–19, the influence of varying the section thick-ness t, web depth bw, flange width bf , and stiffener width bs on theshear buckling strength of unstiffened and stiffened Z sections areexamined, respectively.

The curves presented in Fig. 16 show that by increasing thethickness of the Z sections, shear buckling stress increases dra-matically, but the related buckling half-wavelength does notchange much. In this case, the reason for the increase in the shearbuckling strength is that the web slenderness bw=t decreases.

In Fig. 17, as the web depth bw increases, the shear bucklingstresses decrease but the related half-wavelengths do not change.In fact, by increasing the web depth, the slenderness bw=tincreases and consequently the shear buckling strength decreases.

Looking at the buckling curves of Fig. 18, it can be seen thatwith the increase in the flange width bf shear buckling resistanceof Z sections decreases and the related half-wavelength increases,however, these variations are very small. Substantially, increasingthe flange width bf and the slenderness ratio bf =t of the Z section

Fig. 14. Critical distortional stresses in pure compression.

Fig. 15. Cold-formed Z section: (a) unstiffened Z, (b) stiffened Z.

Fig. 16. Shear buckling curves for different thicknesses.

Fig. 17. Shear buckling curves for different web depths.

Fig. 18. Shear buckling curves for different flange widths.

Fig. 19. Shear buckling curves for different stiffener widths.

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result in decreasing the stiffness of the web boundary conditionsand the shear buckling strength. Finally, Fig. 19 shows that thevariation of stiffener width bs has no influence on the shearbuckling strength of stiffened cold-formed Z sections.

Comparing the buckling curves of Figs. 16–19, it can beconcluded that the shear buckling strength of stiffened Z sectionsis somewhat higher than unstiffened sections and in contrast, therelated half-wavelengths of stiffened Z sections are smaller thanthose for unstiffened sections.

3.6. Comparison between buckling modes of U and Z sections

The method can be used to evaluate the buckling behavior ofstiffened as well as unstiffened cold-formed steel channel and Zsection columns and beams. For this purpose, in Figs. 20 and 21,the normalized buckling stresses of unstiffened channel (U shape),stiffened channel (C shape), as well as both unstiffened andstiffened Z sections in pure compression and uniform bendingabout horizontal axis perpendicular to the web are considered,respectively. Sectional dimensions are given in the figures andmaterial properties of the cold-formed steel are E¼ 200 GPað Þ andv¼ 0:3.

It can be seen that there is no second minimum point in thebuckling curves of unstiffened sections which means that distor-tional buckling does not occur in these sections. There is aremarkable similarity in the buckling behavior of cold-formedsteel channel and unstiffened Z sections and also between stif-fened channel and Z sections. It should be pointed out that thebending moment about horizontal axis in channel section createsmaximum bending stresses in the flanges while under this bend-ing condition the maximum stress does not occur in the flanges ofZ-section. In fact, the major axis in U section is identical withhorizontal axis while in Z-section, the major axis is inclined.

In stiffened channel and Z sections, in addition to local andglobal modes, distortional buckling also occurs because of flange

longitudinal stiffeners. Curves of Figs. 20 and 21 show that in shortand medium lengths, critical stresses of U and Z shapes arecompletely identical in both unstiffened and stiffened sections.

Under pure compression, local and flexural–torsional stressesincrease by stiffening channel and Z sections. Under uniformbending, by stiffening channel and Z sections, lateral–torsionalstresses increase but that the half-wavelength related to localbuckling decreases. In this case, the geometric section propertiesrelating to flexural–torsional and lateral–torsional buckling resis-tances such as torsion constant J and warping factor Cw increase. Inaddition, resistance of the section to local buckling increases dueto the adoption of stiffeners. Under compression loading, the onlydifference between the buckling behaviour of U versus Z shapesand stiffened U versus stiffened Z sections is in long lengths whenflexural–torsional buckling occurs. Here, flexural–torsional buck-ling stresses of Z sections are higher than those in the U sections.There is no difference in the values of critical stresses betweenU and Z sections in other buckling modes. It is obvious from Fig. 20that before maximum stresses are reached, there is a completeoverlap of stresses. In contrast to Fig. 20, under bending, accordingto Fig. 21, the lateral–torsional buckling stresses of both unstif-fened and stiffened U sections are higher than those of theZ sections. The reason that cold-formed steel U and Z sectionsbehave similarly in both local and distortional buckling modes is intheir geometric section properties. Local and distortional bucklingstrength of thin-walled structures are heavily dependent on theboundary conditions of the web and flanges. Both stiffened andunstiffened U and Z sections have similar web and flange bound-ary conditions. Furthermore, for stiffened U and Z section columnsin Fig. 20, distortional buckling is never critical, however, in thecase of stiffened channel and Z section beams in Fig. 21, distor-tional buckling is the dominant mode.

It is worth noting that similar studies using other methodswere carried out on U and Z sections with different dimensionsand similar results were obtained [26].

4. Conclusion

A semi analytical complex finite strip method was proposed inthis paper and used to investigate the buckling of cold-formedsteel members. The critical stresses can be evaluated by solvingthe eigenvalue problem for each half-wavelength. Complex finitestrip results for buckling of cold-formed structures have beenvalidated and verified by comparing those with the resultsobtained from the Generalized Beam Theory. Comparisons showthat the critical stresses obtained from complex FSM have anexcellent accuracy and quickly converge. The proposed solutionoffers several advantages over ordinary finite strip method. Themost important advantages are the handling of shear forces andthe inclusion of general loading conditions in the analysis. Conse-quently, the method can be used to predict all buckling modes ofcold-formed structures like shear buckling, distortional buckling,local, and global buckling.

In order to demonstrate the applicability and usefulness ofcomplex finite strip method, distortional buckling of stiffenedcold-formed channel section columns and beams was studiedusing the method. The elastic local, flexural–torsional, and lat-eral–torsional buckling modes of stiffened cold formed channelsections were also considered in addition to distortional buckling.The effect of extra longitudinal flange stiffener on distortionalbuckling behavior of channel section columns and beams wasexamined as well. It is concluded that the maximum distortionalcritical stress in stiffened C sections occurs in a specific ratio ofwidth of the extra stiffener bs1 to the flange width bf . The methodwas also used to investigate the optimum position of longitudinal

Fig. 20. Comparison between buckling behavior of cold-formed U and Z sectioncolumns.

Fig. 21. Comparison between buckling behavior of cold-formed U and Zsection beams.

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web stiffener in E section columns for local and distortionalbuckling modes. Shear buckling of cold-formed steel Z sectionswas studied and from that the capability of complex FSM indealing with shear forces was demonstrated. Finally, a comparisonwas made between the buckling behaviour of channel andZ sections under compression and bending stresses in two casesincluding unstiffened and stiffened sections.

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