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

Transcription

1 Buckling analysis of thin-walled cold-formed steel structural members using complex finite strip method H.R. Naderian a, H.R. Ronagh b,n a Department of Civil Engineering, University of Ottawa, Ottawa, ON, Canada b School of Civil Engineering, University of Queensland, Brisbane 407, QLD, Australia article info Article history: Received August 014 Received in revised form 31 December 014 Accepted 7 January 015 Available online 30 January 015 Keywords: Cold-formed Complex finite strip method Local buckling Shear buckling Distortional buckling Global buckling abstract In this paper, a generalised complex finite strip method is proposed for buckling analysis of thin-walled cold-formed steel structures. The main advantage of this method over the ordinary finite strip method is that it can handle the shear effects due to the use of complex functions. In addition, distortional buckling as well as all other buckling modes of cold-formed steel sections like local and global modes can be investigated by the suggested complex finite strip method. A combination of general loading including bending, 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 Theory analysis. In order to illustrate the capabilities of complex finite strip method in modelling the buckling behavior of cold-formed steel structures, a number of case studies with different applications are presented. The studies are on both stiffened and unstiffened cold-formed steel members. & 015 Elsevier Ltd. All rights reserved. 1. Introduction Buckling of thin walled cold-formed steel members could be local, shear, distortional and global. Local buckling is normally defined as the mode which involves plate-like deformations alone, without the translation of the intersection lines of adjacent plate elements. Shear buckling is a kind of local buckling which occurs in cold-formed sections under shear loads. Global buckling is a mode in which the member deforms with no deformation in its cross-sectional shape as it undergoes lateral deformations and twist, consistent with the classical beam theory. The common well-known types of global buckling in cold-formed steel sections include flexural, torsional, and flexural torsional which might happen in columns and lateral torsional which might occur in beams. Distortional buckling differs from local and global buckling modes. A general and widely adopted definition for distortional buckling mode does not currently exist. However, it can be considered as a mode with cross-sectional distortion that involves the translation of some of the fold lines [1]. As most cold-formed sections have stiffeners, and as such distortional buckling is prevalent in these sections, the study of distortional buckling has been the subject of many research studies. Finite strip method is a special form of finite element procedure using displacement approach. Unlike the standard finite element n Corresponding author. Tel.: þ ; fax: þ address: h.ronagh@uq.edu.au (H.R. Ronagh). method which uses polynomial displacement functions in all directions, the finite strip method calls for the use of simple polynomials in some directions and continuously differentiable smooth series in the other directions, with the stipulation that such series should satisfy the boundary conditions at the end of strips. The philosophy of the finite strip method is similar to that of the Kantorovich method [], which is used extensively for reducing a partial differential equation to an ordinary differential equation. In this method, especially for plates, Hermitian displacement functions are usually used in the transverse direction [3]. The approach brings about much simplicity, ease of programming and rapid convergence [4]. The first use of finite strip method for buckling analysis appears to be the work of Przemieniecki [5], who showed how this technique can be used in predicting the initial elastic local buckling of plates and sections under biaxial compression. His approach utilized the approximate finite strip formulation of Cheung and Cheung [6]. Plank and Wittrick [7] employed the semi-analytical complex finite strip method to investigate buckling under combined loading of thin-walled structures. The advantage of their method over the formulations of the ordinary finite strip method is the ease with which it can handle shear forces. Wittrick [8] developed an exact finite strip method for the buckling analysis of stiffened panels in compression. Azhari and Bradford [9] developed bubble finite strip method, which augmented finite element formulations to obtain rapid convergence. They also extended the finite strip method to analyze the buckling

2 H.R. Naderian, H.R. Ronagh / Thin-Walled Structures 90 (015) of plates with different end conditions [10]. Ada ny and Schafer [11 13] derived a constrained finite strip method for decomposing the buckling modes of thin-walled members. Li and Schafer [14,15] extended the constrained finite strip method to thin-walled structures with different boundary conditions and applied it to the stability of cold-formed steel sections. The authors have already employed complex finite strip method for analyzing the distortional buckling of cold-formed steel sections and the stability of stiffened cold-formed I-sections [16,17]. Pham and Hancock [18 1] used both spline finite strip and semi-analytical finite strip solutions for shear buckling analysis and design of channel sections. Using complex functions, they included the shear loading in the semi-analytical finite strip analysis. However, their research is limited to the study of shear buckling of channel sections. It is worth mentioning that FSM is applicable to both isotropic and orthotropic structural materials. For instance, the application of FSM has already been extended to buckling analysis of composite FRP structural plates [ 4]. In the present study, a semi analytical complex finite strip method is employed to study the buckling modes of cold-formed steel sections. A combination of loadings including shear, compression, and bending, and transverse forces can be considered in the analysis. Consequently, all bucking modes including local buckling, shear buckling, distortional buckling, and global buckling are investigated. In contrast with ordinary and constrained finite strip methods [11 15], and the current finite strip software [5], the present formulation can deal with shear forces in cold-formed sections. Moreover, it is shown that the complex finite strip method suggested here can consider the distortional buckling of cold-formed steel sections in a simple manner. The method is applicable to different cold-formed sections under general loading conditions. For validation purposes, results of the proposed complex finite strip method are compared with predictions of the Generalized Beam Theory (GBT) for the critical stresses and buckling half-wavelengths of web stiffened cold-formed columns. The following are also presented. a) Several case studies are performed to illustrate the capabilities of the proposed method. b) The buckling behaviour of cold-formed channel sections with extra longitudinal stiffeners at the end of flanges and on the web is studied. c) The optimum width of extra longitudinal flange stiffener is evaluated to maximize the critical distortional stress of channel sections. d) Moreover, the optimum position of web stiffener in coldformed channel section columns is calculated to maximize the local and distortional buckling strengths. e) Shear buckling of stiffened and unstiffened cold-formed steel Z sections are also examined. f) And finally, the influence of stiffeners in local, distortional, and global buckling modes of cold formed channel and Z section columns and beams are investigated and relevant comparisons are made. combination of both out-of-plane and in-plane types. For this reason, complex finite strip method is a suitable tool for predicting the distortional buckling of cold-formed steel members as well as local and global modes under different loading conditions. Moreover, complex FSM allows consideration of the shear forces in the analysis procedure due to the presence of complex functions. In the original complex finite strip method developed by Plank and Wittrick [7], elastic stiffness and stability matrices were obtained using standard finite element techniques based on the energy method. This procedure is followed in the present study to obtain the stiffness and stability matrices of a cold-formed steel strip. It is mentionable that for infinitesimally small buckling displacements, the in-plane and out-of-plane effects are uncoupled and thus are considered separately... Definition of the problem In the finite strip method (FSM), a thin-walled member such as the cold-formed C section of Fig. 1 is divided into longitudinal strips. The advantage of FSM over other methods, such as the finite element method which discretizes the member both in the longitudinal and in the transverse directions, is that it only discretizes the member in the transverse direction while chooses a continuous shape function longitudinally over the whole length of the member. In Fig. 1, a single strip is highlighted. The geometry, loading and degrees of freedom (DOF) for the strip are illustrated in Fig.. The strip is loaded by a combination of different loads including uniform longitudinal compressive stress σ L, longitudinal in-plane bending stress σ b, uniform shear stress τ, and transverse compressive stress σ T. If the member experiences buckling, the strip will be subjected on its edges to a system of forces distributed in the longitudinal direction with a buckling half-wavelength L. For each length L, eight degrees of freedom are allowed..3. Displacement functions The flexural displacement of a strip w and the in-plane displacements u and v in the complex FSM are assumed to be given by w ¼ Re N fb d fbe iξ ð1þ u ¼ Re N x mb Jd mbe iξ v ¼ Re N y mb Jd mbe iξ ðþ ð3þ. Theory.1. General In the complex finite strip method, both in-plane and out-ofplane displacements can be handled. It is worth noting that local (including shear) and global buckling modes may occur due to out-of-plane and in-plane displacements respectively while the displacements caused by distortional buckling include a Fig. 1. Finite strip discreteness.

3 76 H.R. Naderian, H.R. Ronagh / Thin-Walled Structures 90 (015) Fig.. (a) Basic state of stresses in a strip, (b) strip degree of freedoms. p where Ref g denotes the real part, i ¼ ffiffiffiffiffiffiffiffi 1, ξ ¼ πx=l, and J is a 4 4 diagonal matrix defined by J ¼½1000; 0 i 00; 0010; 000 iš Moreover, d - fb and d - mb are 4 1 vectors of infinitesimal flexural buckling displacements and in-plane displacements comprising the assemblable displacements of the strip, respectively. According to Fig., the displacement and force vectors are defined by the following expressions d T fb ¼ ψ 1; w 1 ; ψ ; w ð5þ d T mb ¼ v 1; iu 1 ; v ; iu p T fb ¼ m 1; p Z1 ; m ; p Z p T n o mb ¼ p y1; ip x1 ; p y ; ip x in which the subscripts fb and mb denote flexural buckling and membrane buckling, respectively, and generally all the four vectors are complex. In Eqs. (1) (3), N - fb is a 1 4 polynomial interpolation vector and N - x mb and N - y mb are interpolation vectors defined semianalytically as - Nfb ¼ b 8 ð1 η 1 Þ ð1þηþ; 4 ð1 ηþ ðþηþ; b 1 8 ð1 ηþð1þηþ ; 4 ð1þηþ ð ηþ ð9þ - Nx mb ¼ 0; 1 ð1 ηþ; 0 ; 1 ð1þηþ ð4þ ð6þ ð7þ ð8þ ð10þ -y N mb ¼ 1 ð1 ηþ; 0; 1 ð1þηþ; 0 ð11þ where η ¼ y=b and b is the width of the strip. The stiffness matrices S fb, S mb and geometric stiffness (stability) matrices G fb, G mb are defined by - p fb ¼ðS fb þg fb Þd - fb ð1þ - p mb ¼ðS mb þg mb Þd - mb It should be noted that the term i ¼ p ffiffiffiffiffiffiffiffi 1 ð13þ in the in-plane displacement and the in-plane force vectors d - mb and p - mb automatically incorporate a 90 degree phase difference between u and v displacements and between p x and p y forces. For the same reason, the in-plane stiffness matrix is entirely real and symmetrical. Moreover, if τ a0; the out-of-plane stiffness matrix contains complex off-diagonal elements and is Hermitian in form. On the other hand, if τ ¼ 0, the out-of-plane stiffness matrix will be real and symmetrical [7]. The selected longitudinal functions result in members that are pinned and free-to-warp at their ends. More complicated boundary conditions may be treated [10,14,15] but are not discussed here..4. Strain stress relations In accordance with established stiffness procedures, the vectors of generalized buckling strain ε and ε 0 for out-of-plane and inplane displacements are given by D E ε fb ¼ w; x w; y w; x w; y ð14þ D E ε mb ¼ u; x v; y u; y þv; x ð15þ The flexural property matrix of an isotropic plate D f defines the relationship between the infinitesimal generalized moments and strains for out-of-plane displacements by M ¼ Df ε fb ð16þ in which 1 υ 0 D f ¼ Et 3 =1ð1 υ 6 Þ4 υ ð1 υþ ð17þ where E and υ are the Young s modulus of elasticity and the Poisson s ratio while t is the thickness of the plate strip. Also, the buckling internal stress σ and strains ε mb are related through σ ¼ Dm ε mb ð18þ where D m is the in-plane rigidity matrix of the isotropic plate and is given by 1 υ 0 3 D m ¼ E=ð1 υ 6 Þ4 υ ð19þ 0 0 ð1 υþ.5. Flexural stiffness and stability matrices The strain energy stored in the cold-formed plate strip can thus be calculated, from which after some mathematical manipulation

4 H.R. Naderian, H.R. Ronagh / Thin-Walled Structures 90 (015) involving complex arithmetic [6], the flexural standard stiffness matrix S fb is derived as Z þ 1 S fb ¼ b B T fb D f B fb dη 1 where B fb is the 3 4 matrix defined by Ω 3 N fb B fb ¼ 1 b 4 N 6 fb iΩ 0 N fb ð0þ ð1þ in which Ω ¼ πb=l and where primes denote differentiation with respect to η. Substituting Eqs. (17) and (1) into (0), flexural stiffness matrix is rewritten as Et 3 Z þ 1 1 S fb ¼ T b 3 ð1 υ Þ 1 4 Ω4 N fbn fbþ Ω N fb TN0 fb þ 3 N T fb N fb 1 6 υω ð T N fbn fbþ dη ðþ or where B mb is the 3 4 matrix defined by iω 3 N x mb B mb ¼ 1 0 N y b6 mb 4 0 N x mb þiω 7 5 N y mb ð7þ in which prime denotes differentiation with respect to η. On substituting Eq. (7) into Eq. (6), the membrane stiffness matrix S mb may be written as Et S mb ¼ bð1 v Þ R þ 1 1 J þð1 vþ N 0 x mb TN 0 x mb Ω N T x mb N x mb þ 1 ði vþω N T y mb N y mb þ4 N 0 y mb TN 0 y mb þωυið N 0 y mb TN x mb T N 0 x mb N y mb ÞþΩð1 υþið 0 N x mb TN y mb T N 0 y mb N x mb Þ Jdη ð8þ S fb ¼ 4b þω 4 b =105þ4Ω b =15 6bþυΩ bþ11ω 4 b=10þω b=5 b Ω 4 b =140 Ω b 3 =15 6bþ13Ω 4 b=40 Ω b=5 Et 3 1þ13Ω 4 =35þ1Ω =5 6b 13Ω 4 b=40þω b=5 1þ9Ω 4 =70 1Ω =5 1ð1 v Þb 3 6 Symm 4b þω 4 b =105þ4Ω b =15 6b υω b 11Ω 4 b=10 Ω b= þ13Ω 4 =35þ1Ω =5 ð3þ After calculating the decrease in the potential of a cold-formed strip subjected to the basic forces during buckling and by appropriate substitution and integration, the flexural stability matrix G fb may be obtained from G fb ¼ t b Z þ 1 1 Ω σ T L N fbn fbþω ησ T B N fbn fbþ4σ 0 T N fb TN0 fb Substituting Eqs. (10) and (11) and carrying out the integrations, it is found that 3 1þð1 vþω =6 ð1 3vÞΩ=4 1þð1 vþω =1 ð1þvþω=4 Et ð1 vþ=þω =3 ð1þvþω=4 ð1 vþ=þω =6 S mb ¼ bð1 v Þ 6 4 Symm ð1þð1 vþω =6Þ ð1 3vÞΩ=4 7 5 ð1 vþ=þω =3 ð9þ or þωτið N 0T fbn fb N T fbn 0 fb Þ dη ð4þ According to Ref. [7], σ L is the only stress component that needs to be considered for the in-plane buckling and there is no conceivable in-plane instability that can arise from the action of the stresses σ B, σ T and τ: In order to calculate the loss of potential G fb ¼ t b Ω σlb =105þσTb =15 Ω σbb =40 11Ω σlb=10þσtb=10 Ω σbb=105þiωτb=5 Ω σlb =140 σtb =30þiΩτb 3 =30 13Ω σlb=40 σtb=10þω σbb=40 iωτb=5 11Ω σ Lb=10þσ Tb=10 Ω σ Bb=105 iωτb=5 13Ω σ L=35þ6σ T=5 Ω σ B=5 13Ω σ Lb=40þσ Tb=10þΩ σ Bb=40þiΩτb=5 9Ω σ L=70 6σ T=5 iωτ 6 Ω σlb =140 σtb =30 iωτb =30 13Ω σlb=40þσtb=10þω σbb=40 iωτb=5 Ω σlb =105þσTb =15þΩ σbb =40 11Ω σlb=10 σtb=10 Ω σbb=105þiωτb= Ω σlb=40 σtb=10þω σbb=40þiωτb=5 9Ω σl=70 6σT=5þiΩτ 11Ω σlb=10 σtb=10 Ω σbb=105 iωτb=5 13Ω σl=35þ6σt=5þω σb=5 ð5þ.6. Membrane stiffness and stability matrices A similar method to that for the flexural (out-of-plane) matrices may be followed in order to obtain the membrane (in-plane) stiffness matrix and stability matrix. The internal virtual work done in the cold-formed steel strip results from the buckling deformation which after mathematical manipulation involving complex arithmetic results in the membrane standard stiffness matrix S mb,as S mb ¼ 1 Z þ 1 btj B T mb D mb mb dη J 1 ð6þ energy of the longitudinal stress σ L one can make use of the following nonlinear expression for the longitudinal strain ε x ε x ¼ u; x þ 1 u; x þv; x þw; x ð30þ In this expression the first term has already been accounted for in deriving the strain energy whilst the last term, involving w, has been used in the calculation of the decrease in the potential of the basic stresses due to the out-of-plane displacement. Thus, by developing an appropriate expression for the decrease in the potential of the basic stresses due to in-plane displacement and by appropriate substitution and integration, the in-plane stability

5 78 H.R. Naderian, H.R. Ronagh / Thin-Walled Structures 90 (015) matrix G mb may be obtained by Z G mb ¼ Ω σ L t þ 1 Jð T N x b mb N x mb þ T N y mb N y mb ÞJdη ð31þ 1 Finally, substituting Eqs. (10) and (11) and carrying out the integrations, the matrix G mb may be written as 3 1=3 0 1=6 0 G mb ¼ tσ LΩ 1=3 0 1=6 b 6 4 Symm 0 7 ð3þ 5 1=3.7. Eigenvalue buckling problem Once the strip stiffness and stability matrices S fb, S mb and G fb, G mb are derived and combined for each strip, they can be assembled into the respective global matrices S T and G T using standard procedures. The buckling problem can then be set as the following eigenvalue problem ðs T G T ÞΔ ¼ 0 ð33þ where Δ is a scaling factor related to the critical load. 3. Application 3.1. General Complex finite strip method of analysis developed in the previous section is programmed on a desktop workstation. The program is used to study the buckling of simply supported coldformed steel members under different loading conditions including compression, bending and shear. The effect of stiffeners and other geometric properties of cold-formed steel sections on distortional buckling and shear buckling as well as other modes are investigated. Convergence of the method is excellent so that a few numbers of strips is needed to reach high accuracy. The results show that the use of complex finite strip method significantly improves the efficiency in terms of strip subdivision. 3.. 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 is studied. The cross-sectional dimensions are given in Fig. 3 and the elastic constant values E ¼ 10 ðgpaþ and v ¼ 0:3 are adopted. Semi analytical complex finite strip results are compared with values obtained from GBT analyses carried out by Camotim and colleagues [7]. Fig. 4 shows the GBT and the complex FSM results concerning the buckling behavior of simply-supported E section columns. The curves presented provide the variation of column buckling load N b with its length L in logarithmic scale. It can clearly be seen that the buckling loads and wave-lengths obtained with the complex FSM and the GBT are virtually the same. This fully validates the latter. The buckling curves exhibit three distinct zones corresponding to local, distortional, and global (flexural torsional) buckling modes. Table 1 shows the critical buckling loads N cr and lengths L cr obtained by complex FSM and GBT methods for local, distortional and flexural torsional buckling modes. The ratios of critical buckling loads and lengths derived by the mentioned methods are also listed. The calculated ratios shown in Table 1 demonstrate that the values obtained from the complex FSM are in good agreement with those obtained from GBT for all buckling modes. Table 1 Comparison of critical stresses and lengths. Type Complex FSM GBT [3] Length ratio Critical force ratio Lcr (cm) Ncr (kn) Lcr (cm) Ncr (kn) Lcr (FSM) /Lcr (GBT) Ncr (FSM)/ Ncr (GBT) Fig. 3. E section [3] (dimensions in mm). Local mode Distortional mode Flexural torsional mode Fig. 4. Buckling behavior of simply-supported E-section columns. (a) GBT results [3], (b) semi analytical complex finite strip results.

6 H.R. Naderian, H.R. Ronagh / Thin-Walled Structures 90 (015) 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. 7. Buckling curves for different ratios of b s1 =b f in pure compression. Fig. 6. Buckling curves for different ratios of theta in pure compression Buckling modes of stiffened channel sections The complex finite strip method can be applied to study the buckling modes of cold-formed steel stiffened channel section members with extra longitudinal stiffeners (stiffened C sections). In Fig. 5, three different common types of cold-formed steel channel sections areshown.asasample,astiffened cold-formed C section under uniform bending and compression is considered. The geometric data and elastic constants are E ¼ 00 ðgpaþ; v ¼ 0:3, and b w ¼ 00 mm; b f ¼ 10 mm, b s ¼ 10 mm; t ¼ 1 mm, where b w and b f are the web depth and the flange width, respectively, t is the thickness of the section, and b s is the stiffener width. Studies show that the angle of the extra stiffener to the horizon θ and its width b s1 are amongst factors that have considerable effect on the distortional buckling of stiffened C sections. Here, the influence of variation in the angle of the extra stiffener to the horizon, θ, and the extra stiffener width b s1 on the buckling behavior of stiffened C section columns and beams are investigated. In this regard, special attention is paid to the issue of distortional buckling mode Distortional buckling of cold-formed structures As previously mentioned, distortional buckling is a mode with cross-sectional distortion that involves translation of some of the fold lines (intersection lines of adjacent plate elements). Hancock in the Australian/New Zealand Standard [8] states that distortional buckling is a mode of buckling involving changes in the cross-sectional shape, excluding local buckling. The importance of distortional buckling in cold-formed structures has prompted researchers like Schafer and colleagues [9], Hancock and colleagues [30], Silvestre and Camotim [31,3], and others [33,34,16] to present analytical models for this phenomenon. The majority of these studies are related to channel and Z sections. A large number of laboratory tests have been performed on distortional buckling of cold-formed steel members [35,36]. A series of rich tests have also been performed by the Thin Walled Structures Group under the supervision of Prof. B.W. Schafer at Johns Hopkins University in the USA [37,38]. These tests are not limited to distortional buckling but cover different buckling modes such as local buckling Stiffened C section columns. In Fig. 6, the normalized buckling stresses F=E of stiffened C section columns with b s1 ¼ 0 mm is given as a function of the dimensionless buckling half-wavelength L=b w. Several curves are given for different values of θ. The curves exhibit the same characteristics, namely two minimum points, the first, with a minimum value at L=b w between 0.1 and 1.5 and the second with a minimum value at L=b w between and 8. In the first region, the buckling mode is local and in the second, it is distortional. Beyond the second peak, the buckling stress decreases with increasing halfwavelength L until the mode is predominantly flexural torsional, as predicted by the Vlasov theory. As is seen, with decreasing the angle of the extra stiffener to the horizon, distortional buckling stress and its associated halfwavelength increase, however, local and flexural torsional buckling stresses are not sensitive to this angle. In the extreme, when θ is equal to 90 degree, a C section is obtained. Analyzing the curves of Fig. 6, it is clear that the distortional buckling stress of stiffened C section is more than twice that of an unstiffened section. In fact, as θ decreases the stiffener becomes more effective in the distortional mode. In Fig. 7 the influence of different ratios of b s1 =b f on the buckling stresses of stiffened C sections under pure compression is examined. Buckling curves in Fig. 7 show that the extra stiffener does not have

7 80 H.R. Naderian, H.R. Ronagh / Thin-Walled Structures 90 (015) Fig. 8. Critical distortional buckling stresses in pure compression; case study 1. Fig. 11. Critical distortional stresses in pure bending; case study 1. Fig. 9. Critical distortional buckling stresses in pure compression; case study. Fig. 1. Critical distortional stresses in pure bending; case study. Fig. 10. Buckling curves for different ratios of b s1 =b f in pure bending. any effect on the buckling stress of stiffened C sections in local and flexural torsional modes. However, extra stiffener increases the distortional buckling stress of this section. On the other hand, it can be seen that when the ratio of b s1 =b f increases, the distortional buckling stress decreases. Comparing distortional buckling stresses at different ratios of b s1 =b f, it can be concluded that the maximum distortional buckling stress in stiffened C sections occurs at a specific ratio of b s1 =b f. For illustrative purposes, complex finitestripmethodisusedto determine the critical distortional stress F crd of stiffened C sections. Two case studies are performed in such a fashion that in each case the section is under uniform compression. Figs. 8 and 9 show the normalized critical distortional stresses F crd =E of the stiffened C section columns plotted against the ratio of the width of extra stiffener to flange width b s1 =b f. Dimensions of the sections are shown in the figures. These graphs are produced by plotting F crd =E against the half-wavelength, and by fitting a quadratic interpolation function through three points close to the distortional nadir of the garlandshaped curve. It is clearly seen that both graphs arrive at their maximum, at b s1 =b f between.05 and Stiffened C section beams. Similar studies are carried out on the buckling behaviour of stiffened C section beams. In Fig. 10, the influence of different ratios of b s1 =b f on the buckling stresses of stiffened C sections under pure bending is examined. As an extreme case, when b s1 =b f ¼ 0, in practice, a cold-formed C section is obtained. It can be seen from Fig. 10 that what happens to the distortional buckling stress of the stiffened C section beam is similar to that for columns (Fig. 7). However, the situation is Fig. 13. Critical local stresses in pure compression. different for local and lateral torsional modes. Here, buckling curves show that the extra stiffener increases the buckling stress of stiffened C section beams in local and lateral torsional modes while it has no significant effect on local and flexural torsional buckling stress of columns (Fig. 7). It should be noted that here the distortional buckling is the predominant mode rather than the local buckling. An assessment similar to that of stiffened C section columns is made for C section beams. Two case studies are examined with the same geometric and material properties of cold-formed steel sections of Figs. 8 and 9. In each case the section is under uniform bending. Figs. 11 and 1 show the normalized critical distortional stresses F crd =E of stiffened C section beams plotted against the ratio of the width of extra stiffener b s1 to flange width b f.itis obvious that both graphs show results similar to that for stiffened C section columns Optimum position of stiffeners in E section columns In order to achieve increased economy and efficiency in coldformed steel channel sections, a slender web may be stiffened by longitudinal stiffeners. Such section may be named a web stiffened channel or E section. Local and distortional buckling of E section columns under pure compression is studied using complex FSM. Figs. 13 and 14 show dimensionless critical local and distortional stresses of the E section against various positions of the web stiffener for different values of a. The optimum position of web stiffener to maximize the stresses is found to be 0:5b w and 0:51b w

8 H.R. Naderian, H.R. Ronagh / Thin-Walled Structures 90 (015) Fig. 14. Critical distortional stresses in pure compression. Fig. 18. Shear buckling curves for different flange widths. Fig. 15. Cold-formed Z section: (a) unstiffened Z, (b) stiffened Z. Fig. 19. Shear buckling curves for different stiffener widths Shear buckling of cold-formed steel Z sections Fig. 16. Shear buckling curves for different thicknesses. Fig. 17. Shear buckling curves for different web depths. from the bottom of the section for local and distortional buckling modes, respectively. It is obvious that the optimum position of web stiffener will be different for E section beams under uniform bending [6]. Shear forces can cause out-of-plane displacements in thinwalled structures which may result in local instability, called shear buckling. As previously mentioned, using complex finite strip method it is possible to investigate the local buckling of coldformed steel sections under shear loading. In this section, the local shear buckling of cold-formed steel Z sections under pure shear is evaluated. In Fig. 15, the geometric shapes of unstiffened and stiffened cold-formed Z sections are shown. Here, it is assumed that the web is subjected to a uniform shear stress τ ¼ V=b w t in which V is the shear force. Also, the flange shear stress is ignored. The numerical results are shown in Figs where the variation of normalized shear critical stresses F s =E versus L=b w is plotted. The minimum points of the buckling curves are critical shear stresses. Cross-sectional dimensions are given in the figures and material properties of the cold-formed steel are E ¼ 00 ðgpaþ and v ¼ 0:3. In Figs , the influence of varying the section thickness t, web depth b w, flange width b f, and stiffener width b s on the shear buckling strength of unstiffened and stiffened Z sections are examined, respectively. The curves presented in Fig. 16 show that by increasing the thickness of the Z sections, shear buckling stress increases dramatically, but the related buckling half-wavelength does not change much. In this case, the reason for the increase in the shear buckling strength is that the web slenderness b w =t decreases. In Fig. 17, as the web depth b w increases, the shear buckling stresses decrease but the related half-wavelengths do not change. In fact, by increasing the web depth, the slenderness b w =t increases and consequently the shear buckling strength decreases. Looking at the buckling curves of Fig. 18, it can be seen that with the increase in the flange width b f shear buckling resistance of Z sections decreases and the related half-wavelength increases, however, these variations are very small. Substantially, increasing the flange width b f and the slenderness ratio b f =t of the Z section

9 8 H.R. Naderian, H.R. Ronagh / Thin-Walled Structures 90 (015) Fig. 0. Comparison between buckling behavior of cold-formed U and Z section columns. Fig. 1. Comparison between buckling behavior of cold-formed U and Z section beams. result in decreasing the stiffness of the web boundary conditions and the shear buckling strength. Finally, Fig. 19 shows that the variation of stiffener width b s has no influence on the shear buckling strength of stiffened cold-formed Z sections. Comparing the buckling curves of Figs , it can be concluded that the shear buckling strength of stiffened Z sections is somewhat higher than unstiffened sections and in contrast, the related half-wavelengths of stiffened Z sections are smaller than those for unstiffened sections Comparison between buckling modes of U and Z sections The method can be used to evaluate the buckling behavior of stiffened as well as unstiffened cold-formed steel channel and Z section columns and beams. For this purpose, in Figs. 0 and 1, the normalized buckling stresses of unstiffened channel (U shape), stiffened channel (C shape), as well as both unstiffened and stiffened Z sections in pure compression and uniform bending about horizontal axis perpendicular to the web are considered, respectively. Sectional dimensions are given in the figures and material properties of the cold-formed steel are E ¼ 00 ðgpaþ and v ¼ 0:3. It can be seen that there is no second minimum point in the buckling curves of unstiffened sections which means that distortional buckling does not occur in these sections. There is a remarkable similarity in the buckling behavior of cold-formed steel channel and unstiffened Z sections and also between stiffened channel and Z sections. It should be pointed out that the bending moment about horizontal axis in channel section creates maximum bending stresses in the flanges while under this bending condition the maximum stress does not occur in the flanges of Z-section. In fact, the major axis in U section is identical with horizontal axis while in Z-section, the major axis is inclined. In stiffened channel and Z sections, in addition to local and global modes, distortional buckling also occurs because of flange longitudinal stiffeners. Curves of Figs. 0 and 1 show that in short and medium lengths, critical stresses of U and Z shapes are completely identical in both unstiffened and stiffened sections. Under pure compression, local and flexural torsional stresses increase by stiffening channel and Z sections. Under uniform bending, by stiffening channel and Z sections, lateral torsional stresses increase but that the half-wavelength related to local buckling decreases. In this case, the geometric section properties relating to flexural torsional and lateral torsional buckling resistances such as torsion constant J and warping factor C w increase. In addition, resistance of the section to local buckling increases due to the adoption of stiffeners. Under compression loading, the only difference between the buckling behaviour of U versus Z shapes and stiffened U versus stiffened Z sections is in long lengths when flexural torsional buckling occurs. Here, flexural torsional buckling stresses of Z sections are higher than those in the U sections. There is no difference in the values of critical stresses between U and Z sections in other buckling modes. It is obvious from Fig. 0 that before maximum stresses are reached, there is a complete overlap of stresses. In contrast to Fig. 0, under bending, according to Fig. 1, the lateral torsional buckling stresses of both unstiffened and stiffened U sections are higher than those of the Z sections. The reason that cold-formed steel U and Z sections behave similarly in both local and distortional buckling modes is in their geometric section properties. Local and distortional buckling strength of thin-walled structures are heavily dependent on the boundary conditions of the web and flanges. Both stiffened and unstiffened U and Z sections have similar web and flange boundary conditions. Furthermore, for stiffened U and Z section columns in Fig. 0, distortional buckling is never critical, however, in the case of stiffened channel and Z section beams in Fig. 1, distortional buckling is the dominant mode. It is worth noting that similar studies using other methods were carried out on U and Z sections with different dimensions and similar results were obtained [6]. 4. Conclusion A semi analytical complex finite strip method was proposed in this paper and used to investigate the buckling of cold-formed steel members. The critical stresses can be evaluated by solving the eigenvalue problem for each half-wavelength. Complex finite strip results for buckling of cold-formed structures have been validated and verified by comparing those with the results obtained from the Generalized Beam Theory. Comparisons show that the critical stresses obtained from complex FSM have an excellent accuracy and quickly converge. The proposed solution offers several advantages over ordinary finite strip method. The most important advantages are the handling of shear forces and the inclusion of general loading conditions in the analysis. Consequently, the method can be used to predict all buckling modes of cold-formed structures like shear buckling, distortional buckling, local, and global buckling. In order to demonstrate the applicability and usefulness of complex finite strip method, distortional buckling of stiffened cold-formed channel section columns and beams was studied using the method. The elastic local, flexural torsional, and lateral torsional buckling modes of stiffened cold formed channel sections were also considered in addition to distortional buckling. The effect of extra longitudinal flange stiffener on distortional buckling behavior of channel section columns and beams was examined as well. It is concluded that the maximum distortional critical stress in stiffened C sections occurs in a specific ratio of width of the extra stiffener b s1 to the flange width b f. The method was also used to investigate the optimum position of longitudinal

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