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Effect of geometric imperfections on the ultimate moment capacity of cold-formed sigma-shape sections
HBRC Journal (2015) xxx, xxx–xxx
Housing and Building National Research Center
HBRC Journal http://ees.elsevier.com/hbrcj
FULL LENGTH ARTICLE
Effect of geometric imperfections on the ultimate moment capacity of cold-formed sigma-shape sections Bassem L. Gendy *, M.T. Hanna Structures & Metallic Construction Research Institute, Housing & Building National Research Center (HBRC), 87 Tahrir Street, Dokki, Giza, Egypt Received 3 February 2015; revised 22 April 2015; accepted 23 April 2015
KEYWORDS Geometric imperfection; Sigma sections; Local buckling; Distortional buckling
Abstract In recent years, cold formed steel sections are used more and more as primary framing components and as a secondary structural system. They are used as purlins and side rails or floor joist, and after that in the building envelops. Beams are not perfectly straight and are usually associated with geometric imperfections. Initial geometric imperfections can significantly influence the stability response of cold-formed steel members. This paper reports a numerical investigation concerning the effect of these imperfections on the behavior of the simply supported beams subjected to a uniform bending moment. The beam profile is cold formed sigma sections. Group of beams with different overall member slenderness ratios were studied. Several approaches have been utilized to model the geometric imperfections. First, the elastic buckling modes were considered as the imperfect beam shape. In this approach, the elastic buckling analysis was done first to get the elastic buckling modes. In the second approach, the imperfections were considered by assuming the beam bent in a half sine wave along its length. Finally, combination of these two approaches was considered. Results reveal that, the ultimate bending moments of beams with short and intermediate overall slenderness ratios are sensitive to the imperfect shape that comprise compression flange local buckling. ª 2015 The Authors. Production and hosting by Elsevier B.V. on behalf of Housing and Building National Research Center. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/).
* Corresponding author. E-mail addresses: [email protected] (B.L. Gendy), m.tawfi[email protected] (M.T. Hanna). Peer review under responsibility of Housing and Building National Research Center.
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Introduction Real beams are not perfectly straight and are usually associated with geometric imperfections that may affect their buckling behavior and strength. Generally, geometric imperfections can be divided into two categories; global imperfections and cross-sectional imperfections. Cross sectional imperfections represent the change of the cross section from its ideal shape, while overall imperfections represent the
http://dx.doi.org/10.1016/j.hbrcj.2015.04.006 1687-4048 ª 2015 The Authors. Production and hosting by Elsevier B.V. on behalf of Housing and Building National Research Center. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: B.L. Gendy, M.T. Hanna, Effect of geometric imperfections on the ultimate moment capacity of cold-formed sigma-shape sections, HBRC Journal (2015), http://dx.doi.org/10.1016/j.hbrcj.2015.04.006
2 deviation of the member centerline from its straightness. For global imperfections, typically L/1000 was used as the imperfection magnitude (the exact value is L/960 for hot-rolled steel column based on ASTM A6/A6M) [1]. The most common approach for considering cross-sectional imperfections is to use a factor of the cross section thickness as the imperfection magnitude (e.g. 0.1 t). However section imperfections can be classified into local and distortional imperfections. Local imperfections demonstrate the out of flatness in the plate elements forming the section, while distortional imperfections reflect the translation of one plate end with respect to the other end. In the last 20 years several attempts were done to study the effect of geometric imperfections on the ultimate strength of cold formed sections. Schafer and Peko¨z [2], analyzed the frequency and the amplitude of two types of sectional geometrical imperfections. Based on probabilistic analysis of the measured imperfections and using the Fourier transform, they evaluated the frequency of appearance of the so-called ‘imperfection signal’ in terms of the instability modes. Each instability mode is characterized by its half wavelength. If the critical instability mode is identified using an elastic eigen buckling analysis then, for the critical imperfection, the frequency of appearance is estimated and the corresponding maximum amplitude of this imperfection can be obtained. Rzeszut and Garstecki [3], present approach based on the concept of developing the imperfections in series of eigen modes computed from a linear stability problem. The coefficients in the series were evaluated using imperfections accounting for Gauss probability factors and implementing error minimization in the approximation. The method is applied to the stability analysis of structures made of steel thin-walled cold-formed sigma profiles. It was found that the initial sectional imperfections did not remarkably reduce the maximum bearing capacity of the column, but they made the post buckling behavior unstable. Moreover, geometric imperfections measurements of coldformed sigma profiles were presented by Garstecki et al. [4]. It appeared that the variations of the contour of crosssection have periodic distribution within the member length. The mean value of the measured distortion in flanges was approximately equal to 0.11 t. The distribution and the intensity of the initial imperfections can be used in generation of finite element meshes of imperfect structures. Sadovsky´ et al., [5], presented the guidance on choice the most unfavorable geometric imperfections represented by the eigen mode shapes. The ideas presented were illustrated on lipped channel columns in the axial compression with the initial geometric imperfections exhibiting local-distortional interactions. The commonly used amplitude is accompanied with an energy measure derived from the hypothetic elastic strain energy of the imperfect surface. Two approaches are suggested. The first one, imperfections are normalized by the amplitude, while the energy measure is shown as a useful parameter indicating the severity of an imperfection. Analogously in the second approach, along with the normalization by the energy measure, the amplitude is used as a supporting parameter. Bonadan et al. [6] recommended an iterative method to select the eigen mode that if used as imperfect shape leads to the lowest ultimate load. Their study concentrated on the capacity of an imperfect cold-formed steel rack section with lengths where distortional buckling is predominant. Dubina and Ungureanu [7], showed that for cold-formed steel members in bending, the
B.L. Gendy, M.T. Hanna influence of sectional imperfections is low. However, if there is initial unfavorable twist combined with deflection, the ultimate strength is affected considerably. Borges Dinis et al. [8], present a numerical investigation concerning the elastic and elastic–plastic post-buckling behavior of the simply supported cold-formed steel lipped channel columns affected by a localplate/distortional buckling mode interaction. The analyses involved columns containing several initial imperfections with similar combined amplitudes, obtained through different combinations of the competing buckling modes (three-wave localplate and single wave distortional modes). The study shows that the pure distortional initial imperfections are the most detrimental ones, in the sense that they correspond to the lowest column strengths. Seo et al. [9], investigate the initial imperfection characteristics of the new mono-symmetric Lite Steel beams (LSB) and their effects on the moment capacity. In addition several researchers studied the behavior of the sigma section beams in bending. Yang and Liu [10], present an experimental study on the flexural behavior of single span CFS sigma purlins connected to the roof sheeting with large screw spacing. Test results are used to develop design proposals for the sigma purlins for which most codes or standards have not yet covered. Li [11], studied the influence of support conditions at both the tension and the compression ends of the web on the critical distortional buckling stress of the sigma sections purlins using EN1993-1-3. The influence of the support conditions on the spring stiffness can be applied to other analytical models for calculating the critical stress of distortional buckling. Li and Chen [12], calculated analytically the critical distortional buckling stress of the compression flange and the lip in channel, the zed, and the sigma sections under either compression or bending about an axis perpendicular to the web. This work is aiming to study the sensitivity of the ultimate bending moments to the shapes and values of the initial geometric imperfections. The study concerned with simply supported beams subjected to equal end moments. The beam profile is the sigma CFS. Different shapes of the initial geometric imperfections have been considered. First, the elastic buckling modes only, second, the traditional imperfection shape which consists of half sine wave along the member length. Finally, combination of these two shapes was assumed as the imperfect beams. Finite element model A three-dimensional (3D) finite element model was developed to determine the elastic and the inelastic buckling loads of the cold-formed sigma section beams. Note that, the residual stresses were not considered in this study. Fig. 1 shows the finite element model, loads, and boundary conditions. To simulate the required simply supported conditions, all nodes at both end sections were restrained against vertical displacements, Y-direction, and out-of-plane displacements, Z-direction. In addition, all the nodes at middle section are restrained against displacement along horizontal axis, Xdirection. These boundary conditions allow in-plane rotations and warping displacements at the beam ends. The applied end moments were simulated with forces acting at each node of the beam end sections, where the upper part of the section was subjected to compressive forces and its lower part was
Please cite this article in press as: B.L. Gendy, M.T. Hanna, Effect of geometric imperfections on the ultimate moment capacity of cold-formed sigma-shape sections, HBRC Journal (2015), http://dx.doi.org/10.1016/j.hbrcj.2015.04.006
Effect of geometric imperfections
Fig. 1
3 is assumed to be elasto-plastic with bi-linear isotropic hardening. The values of E = 210 GPa, m = 0.3, Fy = 360 MPa, the typical characteristics of high Carbon steel used in the study. In the author’s knowledge, there were no experimental data available for the cold-formed sigma section under the flexural loads. Therefore, the verification was done by simulating the test setup of Luı´ s et al. [14]. He studied the flexural strength of the lipped channel CFS. His setup consists of simply supported beams subjected to two concentrated loads, see Fig. 2c. The section nominal thickness was 2.5 mm. The span of the studied beams was 3000 mm, the loads were applied each 1000 mm. Lateral supports were provided at each end of the beam to prevent the out-of-plane movements, as shown in Fig. 2c. The comparison is shown in Fig. 2a. It is clear that the experimental and ANSYS results are comparable. Moreover, the numerical ultimate applied load P is about 0.89 of the experimental load.
Finite element model, loading, and boundary conditions.
subjected to tensile forces. To avoid stresses concentrations at the end sections, the thickness of the mesh elements at these sections was slightly increased. For elastic analysis the sigma beam was assumed to have linear elastic material with Young’s modulus E = 210 GPa and Poisson’s ratio m = 0.3. On the other hand, for nonlinear analysis, the beam material
Parametric study variables A comprehensive parametric study has been done to identify the effect of geometric imperfections on the ultimate bending
P KN
14 12
ANSYS
10
EXPERIMENTAL
8 6 4 2 0
0
10
20
30
40
50
60
Vertical displacement at mid-span,mm (a) Load-Displacement curve (b) Beam cross section
(c) Experimental Setup [14]
Fig. 2
Comparison between the experimental results by [14] and ANSYS [13].
Please cite this article in press as: B.L. Gendy, M.T. Hanna, Effect of geometric imperfections on the ultimate moment capacity of cold-formed sigma-shape sections, HBRC Journal (2015), http://dx.doi.org/10.1016/j.hbrcj.2015.04.006
4
B.L. Gendy, M.T. Hanna
Fig. 3
Studied case of sigma section.
moment capacity of the simply supported sigma section beams that is subjected to a uniform bending moment as shown in Fig. 3. Thickness of the sigma section was considered equal to 1.0 mm. To investigate the possible modes of failure, four values of overall beam slenderness ratios, L/ry, are considered. These values are 50, 75, 100 and 125. Where ‘‘L’’ is total beam length, and ‘‘ry’’ is the section radius of gyration about the minor axis (Y-axis). For simplicity, the small curved corners of the sigma section were not modeled. The beams imperfect shapes are assumed similar to the first five eigen modes. Each mode was factored by the magnitudes of 0.1, 0.5, 1.0 and 2.0 t, where t is the thickness of the sigma section. The value of 1.0 t, is recommended in [2]. Another way of considering the geometric imperfections is assuming the beam bent about its section minor axis, Yaxis, in half sine wave. Two imperfection values L/1000 and L/500 were considered. Finally, a combination of the mentioned two approaches is studied. Direction of applying
Table 1 L/ry Mode Mode Mode Mode Mode
1 2 3 4 5
Eigen modes. 50
75
100
125
FD LC LC LC LC
FD LC LC LC LC
FD LTB FD LC LC
LTB FD FD FD FD
LC: local buckling mode; FD: flange distortional buckling mode; LTB: lateral-torsional buckling mode.
Distortional Mode
Fig. 4
imperfection amplitude was examined to obtain the smaller ultimate moment. Discussion of results Elastic analysis In this part, the elastic buckling analysis was performed using the eigen value module that is available in ANSYS software [13] to predict the first five eigen modes for each beam slenderness ratios. For short beams (L/ry = 50, 75) the first buckling mode was defined as the distortional buckling mode. However, the second to fifth modes were flange local buckling. Although the second eigen mode of the intermediate slenderness beams is the overall lateral torsional buckling, the first, third, fourth and fifth modes are the sectional (local/distortional) buckling shapes. However, the last studied out-of-plane slenderness ratio of 125 shows that the first eigen mode was the lateral torsional buckling. The second through the fifth eigen modes show flange distortional buckling. The summary of the obtained mode shapes for each beam slenderness ratios was shown in Table 1, and the shapes are depicted in Fig. 4. Nonlinear analysis The ultimate moment capacities of a simply supported beam with sigma cross sections shown in Fig. 3, have been determined considering different modes and values of the initial geometric imperfections. The first five elastic buckling modes obtained in the last section were assumed to be the initial mode of imperfections. The buckling shape was first normalized so
Flange Local Mode
Lateral Buckling mode
Torsional
Different eigen buckling modes, L/ry = 100.
Please cite this article in press as: B.L. Gendy, M.T. Hanna, Effect of geometric imperfections on the ultimate moment capacity of cold-formed sigma-shape sections, HBRC Journal (2015), http://dx.doi.org/10.1016/j.hbrcj.2015.04.006
Effect of geometric imperfections
5
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(d) MODE M4
0.1t 0.5t 1t
0.5
2t AISI-2012
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L/ry 25
50
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(e) MODE M5
Fig. 5
Mu/My versus L/ry for different initial geometric imperfection modes.
that the maximum displaced point was set equal to the unity. Then, the desired value of imperfection was multiplied by the normalized shape. Results were plotted in Fig. 5, where the vertical axis represents the ratio of the ultimate moment to the first yield moment, Mu/My, while the horizontal axis represents the overall out-of-plane slenderness ratio, L/ry. Also, on the same figure, for the sake of comparison, the ultimate bending moments considering the beam bent about the minor axis in a half sine wave with two values of L/1000, and L/500 are plotted. Results reflect that, inspite of mode 1 represents the critical buckling mode, it has no remarkable effect on the ultimate moment of short beams, L/ry = 50 and 75, for the practical range of imperfections (0.1–1 t, [2,6,15]). However, the flexural strength of beams with L/ry = 75 reduces by 16% when the imperfection value is equal to 2 t. Note that, the value 2 t is considered a large value. Moreover, mode 2 that represents
local buckling in compression flanges greatly affects the ultimate moment capacity when the imperfection value reaches 2 t. For intermediate beams, L/ry = 100, the ultimate bending moments are slightly affected by the initial imperfection values when the initial mode was the flange distortional buckling mode. On the other hand, when the flange local buckling mode was the initial imperfection mode, the ultimate bending capacity of the sections decreased significantly by increasing the imperfection value. Finally, for long beams, L/ry = 125, when the failure mode was mainly the overall lateral torsional buckling mode, the initial sectional imperfection modes (local/distortional) have no remarkable effect on the ultimate moments, and all the values become close to the initial overall imperfection mode. Moreover, a more complex imperfection shape has been considered which consist of the overall imperfection shape plus the compression flange local buckling mode since it is the effective initial imperfection shape in the short
Please cite this article in press as: B.L. Gendy, M.T. Hanna, Effect of geometric imperfections on the ultimate moment capacity of cold-formed sigma-shape sections, HBRC Journal (2015), http://dx.doi.org/10.1016/j.hbrcj.2015.04.006
6
B.L. Gendy, M.T. Hanna 0.7
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shape has no remarkable effect on the ultimate bending moment. Consequently, the combined imperfection shape can be recommended as a unified imperfection shape. The deformation history of the beams has been reported to assess the effect of initial imperfection shape on their structural behavior. Fig. 7 shows the relationship between the applied moment, M, and the out-of-plane displacement of the flange lip intersection point for the beams with overall slenderness ratios L/ry = 50, 100, 125. Curves were drawn considering three shapes of imperfections. The first imperfection shape is the lowest value eigen mode shape (mode 1), while the second one is the most sensitive mode shape to the ultimate bending moments. As mentioned before, in the short and the intermediate slenderness beams, the second mode is the compression flange local buckling modes. This is modes M1 and M4 for beams with L/ry = 50 and 100; respectively. However, for long beams, L/ry = 125, the compression flange distortional buckling mode is considered as sectional mode for comparison. The last one was considered that the beam bent in out-ofplane direction as a half sine wave along with value L/1000.
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Fig. 6
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Mu/My versus initial imperfection values.
out-of-plan Displacment mm
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and the intermediate slenderness beams. Although for long beams, L/ry = 125, the compression flange distortional mode has little effect on the ultimate moment, it has been also considered as imperfection mode (M2). The normalized ratios of Mu/My have been drawn as function of the imperfection values in Fig. 6. The dotted curve represents the results obtained when the initial imperfection shape is the eigen mode only, while the solid curves represent the results when the eigen shape was added to the overall imperfection shape with values of L/1000 and L/500; respectively. It is clear that the bending strength of the short and the intermediate beams were greatly affected by the sectional imperfection shape (flange local buckling of compression flange). However, the ultimate capacity of the long beams (L/ry = 125) was more sensitive to the traditional overall imperfection shape. Moreover, the combined imperfection
1.0
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out-of-plan Displacment mm
0 0.0
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(c) L/ry=125
Fig. 7 Applied relationship.
moment
versus
lateral
displacements
Please cite this article in press as: B.L. Gendy, M.T. Hanna, Effect of geometric imperfections on the ultimate moment capacity of cold-formed sigma-shape sections, HBRC Journal (2015), http://dx.doi.org/10.1016/j.hbrcj.2015.04.006
Effect of geometric imperfections
7
L/ry = 50 L/1000
L/ry = 50 M1
L/ry = 50 M2
L/ry = 100 L/1000
L/ry = 100 M1
L/ry = 100 M4
L/ry= 125 L/1000
L/ry = 125 M1
L/ry = 125 M2
Fig. 8
Failure mode for short, intermediate and long beams.
It is clear that assuming the flange local buckling mode as the initial imperfection shape for the short and the intermediate slenderness beams reduces the flexural stiffness and leads to a smaller ultimate loads. The failure mode obtained is flange local buckling when the initial imperfection shapes is the flange local buckling mode, while it become distortional buckling mode for the other two imperfection modes. However, in long beams, the load deformation history is quite similar for the three assumed imperfection shapes. Moreover, they are all lead to a similar failure mode, as shown in Fig. 8. The deformation history of the beams has been reported to assess the effect of initial imperfection shape on their structural behavior. Fig. 7 shows the relationship between the applied moment, M, and the out-of-plane displacement of the flange lip intersection point for the beams with overall slenderness ratios L/ry = 50, 100, 125. Curves were drawn considering three shapes of imperfections. The first imperfection shape is the lowest value eigen mode shape (mode 1), while the second one is the most sensitive mode shape to the ultimate bending moments. As mentioned before, in the short and the intermediate slenderness beams, the second mode is the compression flange local buckling modes. This is modes M1 and M4 for beams with L/ry = 50 and 100; respectively. However, for long beams, L/ry = 125, the compression flange distortional buckling mode is considered as a sectional mode for comparison. The last one was considered that the beam bent in out-ofplane direction as a half sine wave along with value L/1000. It is clear that assuming the flange local buckling mode as initial imperfection shape for the short and the intermediate slenderness beams reduces the flexural stiffness and leads to
smaller ultimate loads. The failure mode obtained is flange local buckling when the initial imperfection shapes are the flange local buckling mode, while it become distortional buckling mode for the other two imperfection modes. For the sake of the comparison, the flexural strengths, Mu, obtained by finite element analysis are compared by the design rules of the AISI-2012 [16]. In AISI-2012 code, the capacity of the beams is calculated as the product of the cross section effective section modulus, Se, and the maximum stress, Fc. The later will be the minimum of the yield stresses or lateral torsional buckling stresses which is determined using the full section properties. In-addition, the distortional buckling flexural strength, Mnd, has been included in AISI-2012 as limiting strength of the lipped channel cross section beams (provision C4.2). Note, the partial safety factors, cM1 and /c, are not included when calculating the design loads. Results of this comparison are shown in Fig. 5. Comparison reveals that, for short beams, AISI-2012 results are matched well with the finite element results considering the overall mode of imperfections. However, for long beams, the code predictions are relatively conservative. Conclusions Elastic and inelastic analysis was performed to investigate the effect of the initial geometrical imperfections on the ultimate moment capacity of simply supported sigma CFS beams. The beams are subjected to equal end moments. Three methods are used to simulate the initial geometric imperfections
Please cite this article in press as: B.L. Gendy, M.T. Hanna, Effect of geometric imperfections on the ultimate moment capacity of cold-formed sigma-shape sections, HBRC Journal (2015), http://dx.doi.org/10.1016/j.hbrcj.2015.04.006
8 using the finite element method. Results indicate that the ultimate bending moments of beams with the short and the intermediate overall slenderness ratios are sensitive to the imperfect shape that comprise compression flange local buckling. However, for long beams, the overall buckling mode (lateral–torsional buckling) was the most sensitive mode. Moreover, the failure shape of the short and intermediate beams is affected by the initially assumed geometric imperfection mode. Finally, the sectional imperfection modes with large values influence the ultimate moment capacities of the simply supported beams. Conflict of interest The authors declare that there are no conflicts of interests. References [1] ASTM-A6/A6M, Standard Specification for General Requirements for Rolled Structural Steel Bars, Plates, Shapes, and Sheet Piling, American Society for Testing and Materials, West Conshohocken, PA, 2011. [2] B.W. Schafer, T. Peko¨z, Computational modeling of coldformed steel: characterizing geometric imperfections and residual stresses, J. Constr. Steel Res. 47 (1998) 193–210. [3] K. Rzeszut, A. Garstecki, Modeling of initial geometrical imperfections in stability analysis of thin-walles structures, J. Theor. Appl. Mech. 47 (3) (2009) 667–684. [4] A. Garstecki, W. Kakol, K. Rzeszut, Classification of localsectional geometric imperfections of steel thin-walled coldformed sigma members, Found. Civ. Environ. Eng. 1 (2002) 1642–9303. [5] Z. Sadovsky´, J. Kriva´cˇek, V. Ivancˇo, A. Dˇuricova´, Computational modeling of geometric imperfections and buckling strength of cold-formed steel, J. Constr. Steel Res. 78 (2012) 1–7.
B.L. Gendy, M.T. Hanna [6] J. Bonada, M. Casafont, F. Roure, M.M. Pastor, Selection of the initial geometrical imperfection in nonlinear FE analysis of cold-formed steel rack columns, Thin-Walled Struct. 51 (2012) 99–111. [7] D. Dubina, V. Ungureanu, Effect of imperfections on numerical simulation of instability behaviour of cold-formed steel members, Thin-Walled Struct. 40 (2002) 239–262. [8] P. Borges Dinis, C. Dinar, S. Nuno, FEM-based analysis of the local plate/distortional mode interaction in cold-formed steel lipped channel columns, Comput. Struct. 85 (2007) 461– 1474. [9] Jung Kwan Seo, A. Tharmarajah, M. Mahen, Initial imperfection characteristics of mono-symmetric lite steel beams for numerical studies, in: Fifth International Conference on Thin-Walled Structure, Brisbane, Australia, 2008. [10] J. Yang, Q. Liu, An experimental study into flexural behaviour of sigma purlins attached with roof sheets, Eng. Struct. 45 (2012) 481–495. [11] Long-yuan Li, Analyses of distortional buckling of cold-formed sigma purlins using EN1993-1-3, J. Constr. Steel Res. 65 (2009) 2099–2102. [12] Long-yuan Li, Jian-kang Chen, An analytical model for analysing distortional buckling of cold-formed steel sections, Thin-Walled Struct. 46 (2008) 1430–1436. [13] ANSYS User’s Manual Release 12.0.1, Canonsburg (Pennsylvania), 2009. [14] Luı´ s Laı´ m, Joa˜o Paulo C. Rodrigues, Luis Simo˜esda Silva, Experimental and numerical analysis on the structural behaviour of cold-formed steel beams, Thin-Walled Struct. 72 (2013) 1–13. [15] V.M. Zeinoddini, B.W. Schafer, Simulation of geometric imperfections in cold formed steel members using spectral representation approach, Thin-Walled Struct. 60 (2012) 105– 117. [16] AISI, Cold-formed steel design manual, American Iron and Steel Institute, AISI, 2012.
Please cite this article in press as: B.L. Gendy, M.T. Hanna, Effect of geometric imperfections on the ultimate moment capacity of cold-formed sigma-shape sections, HBRC Journal (2015), http://dx.doi.org/10.1016/j.hbrcj.2015.04.006
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FULL LENGTH ARTICLE
Effect of geometric imperfections on the ultimate moment capacity of cold-formed sigma-shape sections Bassem L. Gendy *, M.T. Hanna Structures & Metallic Construction Research Institute, Housing & Building National Research Center (HBRC), 87 Tahrir Street, Dokki, Giza, Egypt Received 3 February 2015; revised 22 April 2015; accepted 23 April 2015
KEYWORDS Geometric imperfection; Sigma sections; Local buckling; Distortional buckling
Abstract In recent years, cold formed steel sections are used more and more as primary framing components and as a secondary structural system. They are used as purlins and side rails or floor joist, and after that in the building envelops. Beams are not perfectly straight and are usually associated with geometric imperfections. Initial geometric imperfections can significantly influence the stability response of cold-formed steel members. This paper reports a numerical investigation concerning the effect of these imperfections on the behavior of the simply supported beams subjected to a uniform bending moment. The beam profile is cold formed sigma sections. Group of beams with different overall member slenderness ratios were studied. Several approaches have been utilized to model the geometric imperfections. First, the elastic buckling modes were considered as the imperfect beam shape. In this approach, the elastic buckling analysis was done first to get the elastic buckling modes. In the second approach, the imperfections were considered by assuming the beam bent in a half sine wave along its length. Finally, combination of these two approaches was considered. Results reveal that, the ultimate bending moments of beams with short and intermediate overall slenderness ratios are sensitive to the imperfect shape that comprise compression flange local buckling. ª 2015 The Authors. Production and hosting by Elsevier B.V. on behalf of Housing and Building National Research Center. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/).
* Corresponding author. E-mail addresses: [email protected] (B.L. Gendy), m.tawfi[email protected] (M.T. Hanna). Peer review under responsibility of Housing and Building National Research Center.
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Introduction Real beams are not perfectly straight and are usually associated with geometric imperfections that may affect their buckling behavior and strength. Generally, geometric imperfections can be divided into two categories; global imperfections and cross-sectional imperfections. Cross sectional imperfections represent the change of the cross section from its ideal shape, while overall imperfections represent the
http://dx.doi.org/10.1016/j.hbrcj.2015.04.006 1687-4048 ª 2015 The Authors. Production and hosting by Elsevier B.V. on behalf of Housing and Building National Research Center. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: B.L. Gendy, M.T. Hanna, Effect of geometric imperfections on the ultimate moment capacity of cold-formed sigma-shape sections, HBRC Journal (2015), http://dx.doi.org/10.1016/j.hbrcj.2015.04.006
2 deviation of the member centerline from its straightness. For global imperfections, typically L/1000 was used as the imperfection magnitude (the exact value is L/960 for hot-rolled steel column based on ASTM A6/A6M) [1]. The most common approach for considering cross-sectional imperfections is to use a factor of the cross section thickness as the imperfection magnitude (e.g. 0.1 t). However section imperfections can be classified into local and distortional imperfections. Local imperfections demonstrate the out of flatness in the plate elements forming the section, while distortional imperfections reflect the translation of one plate end with respect to the other end. In the last 20 years several attempts were done to study the effect of geometric imperfections on the ultimate strength of cold formed sections. Schafer and Peko¨z [2], analyzed the frequency and the amplitude of two types of sectional geometrical imperfections. Based on probabilistic analysis of the measured imperfections and using the Fourier transform, they evaluated the frequency of appearance of the so-called ‘imperfection signal’ in terms of the instability modes. Each instability mode is characterized by its half wavelength. If the critical instability mode is identified using an elastic eigen buckling analysis then, for the critical imperfection, the frequency of appearance is estimated and the corresponding maximum amplitude of this imperfection can be obtained. Rzeszut and Garstecki [3], present approach based on the concept of developing the imperfections in series of eigen modes computed from a linear stability problem. The coefficients in the series were evaluated using imperfections accounting for Gauss probability factors and implementing error minimization in the approximation. The method is applied to the stability analysis of structures made of steel thin-walled cold-formed sigma profiles. It was found that the initial sectional imperfections did not remarkably reduce the maximum bearing capacity of the column, but they made the post buckling behavior unstable. Moreover, geometric imperfections measurements of coldformed sigma profiles were presented by Garstecki et al. [4]. It appeared that the variations of the contour of crosssection have periodic distribution within the member length. The mean value of the measured distortion in flanges was approximately equal to 0.11 t. The distribution and the intensity of the initial imperfections can be used in generation of finite element meshes of imperfect structures. Sadovsky´ et al., [5], presented the guidance on choice the most unfavorable geometric imperfections represented by the eigen mode shapes. The ideas presented were illustrated on lipped channel columns in the axial compression with the initial geometric imperfections exhibiting local-distortional interactions. The commonly used amplitude is accompanied with an energy measure derived from the hypothetic elastic strain energy of the imperfect surface. Two approaches are suggested. The first one, imperfections are normalized by the amplitude, while the energy measure is shown as a useful parameter indicating the severity of an imperfection. Analogously in the second approach, along with the normalization by the energy measure, the amplitude is used as a supporting parameter. Bonadan et al. [6] recommended an iterative method to select the eigen mode that if used as imperfect shape leads to the lowest ultimate load. Their study concentrated on the capacity of an imperfect cold-formed steel rack section with lengths where distortional buckling is predominant. Dubina and Ungureanu [7], showed that for cold-formed steel members in bending, the
B.L. Gendy, M.T. Hanna influence of sectional imperfections is low. However, if there is initial unfavorable twist combined with deflection, the ultimate strength is affected considerably. Borges Dinis et al. [8], present a numerical investigation concerning the elastic and elastic–plastic post-buckling behavior of the simply supported cold-formed steel lipped channel columns affected by a localplate/distortional buckling mode interaction. The analyses involved columns containing several initial imperfections with similar combined amplitudes, obtained through different combinations of the competing buckling modes (three-wave localplate and single wave distortional modes). The study shows that the pure distortional initial imperfections are the most detrimental ones, in the sense that they correspond to the lowest column strengths. Seo et al. [9], investigate the initial imperfection characteristics of the new mono-symmetric Lite Steel beams (LSB) and their effects on the moment capacity. In addition several researchers studied the behavior of the sigma section beams in bending. Yang and Liu [10], present an experimental study on the flexural behavior of single span CFS sigma purlins connected to the roof sheeting with large screw spacing. Test results are used to develop design proposals for the sigma purlins for which most codes or standards have not yet covered. Li [11], studied the influence of support conditions at both the tension and the compression ends of the web on the critical distortional buckling stress of the sigma sections purlins using EN1993-1-3. The influence of the support conditions on the spring stiffness can be applied to other analytical models for calculating the critical stress of distortional buckling. Li and Chen [12], calculated analytically the critical distortional buckling stress of the compression flange and the lip in channel, the zed, and the sigma sections under either compression or bending about an axis perpendicular to the web. This work is aiming to study the sensitivity of the ultimate bending moments to the shapes and values of the initial geometric imperfections. The study concerned with simply supported beams subjected to equal end moments. The beam profile is the sigma CFS. Different shapes of the initial geometric imperfections have been considered. First, the elastic buckling modes only, second, the traditional imperfection shape which consists of half sine wave along the member length. Finally, combination of these two shapes was assumed as the imperfect beams. Finite element model A three-dimensional (3D) finite element model was developed to determine the elastic and the inelastic buckling loads of the cold-formed sigma section beams. Note that, the residual stresses were not considered in this study. Fig. 1 shows the finite element model, loads, and boundary conditions. To simulate the required simply supported conditions, all nodes at both end sections were restrained against vertical displacements, Y-direction, and out-of-plane displacements, Z-direction. In addition, all the nodes at middle section are restrained against displacement along horizontal axis, Xdirection. These boundary conditions allow in-plane rotations and warping displacements at the beam ends. The applied end moments were simulated with forces acting at each node of the beam end sections, where the upper part of the section was subjected to compressive forces and its lower part was
Please cite this article in press as: B.L. Gendy, M.T. Hanna, Effect of geometric imperfections on the ultimate moment capacity of cold-formed sigma-shape sections, HBRC Journal (2015), http://dx.doi.org/10.1016/j.hbrcj.2015.04.006
Effect of geometric imperfections
Fig. 1
3 is assumed to be elasto-plastic with bi-linear isotropic hardening. The values of E = 210 GPa, m = 0.3, Fy = 360 MPa, the typical characteristics of high Carbon steel used in the study. In the author’s knowledge, there were no experimental data available for the cold-formed sigma section under the flexural loads. Therefore, the verification was done by simulating the test setup of Luı´ s et al. [14]. He studied the flexural strength of the lipped channel CFS. His setup consists of simply supported beams subjected to two concentrated loads, see Fig. 2c. The section nominal thickness was 2.5 mm. The span of the studied beams was 3000 mm, the loads were applied each 1000 mm. Lateral supports were provided at each end of the beam to prevent the out-of-plane movements, as shown in Fig. 2c. The comparison is shown in Fig. 2a. It is clear that the experimental and ANSYS results are comparable. Moreover, the numerical ultimate applied load P is about 0.89 of the experimental load.
Finite element model, loading, and boundary conditions.
subjected to tensile forces. To avoid stresses concentrations at the end sections, the thickness of the mesh elements at these sections was slightly increased. For elastic analysis the sigma beam was assumed to have linear elastic material with Young’s modulus E = 210 GPa and Poisson’s ratio m = 0.3. On the other hand, for nonlinear analysis, the beam material
Parametric study variables A comprehensive parametric study has been done to identify the effect of geometric imperfections on the ultimate bending
P KN
14 12
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10
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8 6 4 2 0
0
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40
50
60
Vertical displacement at mid-span,mm (a) Load-Displacement curve (b) Beam cross section
(c) Experimental Setup [14]
Fig. 2
Comparison between the experimental results by [14] and ANSYS [13].
Please cite this article in press as: B.L. Gendy, M.T. Hanna, Effect of geometric imperfections on the ultimate moment capacity of cold-formed sigma-shape sections, HBRC Journal (2015), http://dx.doi.org/10.1016/j.hbrcj.2015.04.006
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B.L. Gendy, M.T. Hanna
Fig. 3
Studied case of sigma section.
moment capacity of the simply supported sigma section beams that is subjected to a uniform bending moment as shown in Fig. 3. Thickness of the sigma section was considered equal to 1.0 mm. To investigate the possible modes of failure, four values of overall beam slenderness ratios, L/ry, are considered. These values are 50, 75, 100 and 125. Where ‘‘L’’ is total beam length, and ‘‘ry’’ is the section radius of gyration about the minor axis (Y-axis). For simplicity, the small curved corners of the sigma section were not modeled. The beams imperfect shapes are assumed similar to the first five eigen modes. Each mode was factored by the magnitudes of 0.1, 0.5, 1.0 and 2.0 t, where t is the thickness of the sigma section. The value of 1.0 t, is recommended in [2]. Another way of considering the geometric imperfections is assuming the beam bent about its section minor axis, Yaxis, in half sine wave. Two imperfection values L/1000 and L/500 were considered. Finally, a combination of the mentioned two approaches is studied. Direction of applying
Table 1 L/ry Mode Mode Mode Mode Mode
1 2 3 4 5
Eigen modes. 50
75
100
125
FD LC LC LC LC
FD LC LC LC LC
FD LTB FD LC LC
LTB FD FD FD FD
LC: local buckling mode; FD: flange distortional buckling mode; LTB: lateral-torsional buckling mode.
Distortional Mode
Fig. 4
imperfection amplitude was examined to obtain the smaller ultimate moment. Discussion of results Elastic analysis In this part, the elastic buckling analysis was performed using the eigen value module that is available in ANSYS software [13] to predict the first five eigen modes for each beam slenderness ratios. For short beams (L/ry = 50, 75) the first buckling mode was defined as the distortional buckling mode. However, the second to fifth modes were flange local buckling. Although the second eigen mode of the intermediate slenderness beams is the overall lateral torsional buckling, the first, third, fourth and fifth modes are the sectional (local/distortional) buckling shapes. However, the last studied out-of-plane slenderness ratio of 125 shows that the first eigen mode was the lateral torsional buckling. The second through the fifth eigen modes show flange distortional buckling. The summary of the obtained mode shapes for each beam slenderness ratios was shown in Table 1, and the shapes are depicted in Fig. 4. Nonlinear analysis The ultimate moment capacities of a simply supported beam with sigma cross sections shown in Fig. 3, have been determined considering different modes and values of the initial geometric imperfections. The first five elastic buckling modes obtained in the last section were assumed to be the initial mode of imperfections. The buckling shape was first normalized so
Flange Local Mode
Lateral Buckling mode
Torsional
Different eigen buckling modes, L/ry = 100.
Please cite this article in press as: B.L. Gendy, M.T. Hanna, Effect of geometric imperfections on the ultimate moment capacity of cold-formed sigma-shape sections, HBRC Journal (2015), http://dx.doi.org/10.1016/j.hbrcj.2015.04.006
Effect of geometric imperfections
5
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(e) MODE M5
Fig. 5
Mu/My versus L/ry for different initial geometric imperfection modes.
that the maximum displaced point was set equal to the unity. Then, the desired value of imperfection was multiplied by the normalized shape. Results were plotted in Fig. 5, where the vertical axis represents the ratio of the ultimate moment to the first yield moment, Mu/My, while the horizontal axis represents the overall out-of-plane slenderness ratio, L/ry. Also, on the same figure, for the sake of comparison, the ultimate bending moments considering the beam bent about the minor axis in a half sine wave with two values of L/1000, and L/500 are plotted. Results reflect that, inspite of mode 1 represents the critical buckling mode, it has no remarkable effect on the ultimate moment of short beams, L/ry = 50 and 75, for the practical range of imperfections (0.1–1 t, [2,6,15]). However, the flexural strength of beams with L/ry = 75 reduces by 16% when the imperfection value is equal to 2 t. Note that, the value 2 t is considered a large value. Moreover, mode 2 that represents
local buckling in compression flanges greatly affects the ultimate moment capacity when the imperfection value reaches 2 t. For intermediate beams, L/ry = 100, the ultimate bending moments are slightly affected by the initial imperfection values when the initial mode was the flange distortional buckling mode. On the other hand, when the flange local buckling mode was the initial imperfection mode, the ultimate bending capacity of the sections decreased significantly by increasing the imperfection value. Finally, for long beams, L/ry = 125, when the failure mode was mainly the overall lateral torsional buckling mode, the initial sectional imperfection modes (local/distortional) have no remarkable effect on the ultimate moments, and all the values become close to the initial overall imperfection mode. Moreover, a more complex imperfection shape has been considered which consist of the overall imperfection shape plus the compression flange local buckling mode since it is the effective initial imperfection shape in the short
Please cite this article in press as: B.L. Gendy, M.T. Hanna, Effect of geometric imperfections on the ultimate moment capacity of cold-formed sigma-shape sections, HBRC Journal (2015), http://dx.doi.org/10.1016/j.hbrcj.2015.04.006
6
B.L. Gendy, M.T. Hanna 0.7
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shape has no remarkable effect on the ultimate bending moment. Consequently, the combined imperfection shape can be recommended as a unified imperfection shape. The deformation history of the beams has been reported to assess the effect of initial imperfection shape on their structural behavior. Fig. 7 shows the relationship between the applied moment, M, and the out-of-plane displacement of the flange lip intersection point for the beams with overall slenderness ratios L/ry = 50, 100, 125. Curves were drawn considering three shapes of imperfections. The first imperfection shape is the lowest value eigen mode shape (mode 1), while the second one is the most sensitive mode shape to the ultimate bending moments. As mentioned before, in the short and the intermediate slenderness beams, the second mode is the compression flange local buckling modes. This is modes M1 and M4 for beams with L/ry = 50 and 100; respectively. However, for long beams, L/ry = 125, the compression flange distortional buckling mode is considered as sectional mode for comparison. The last one was considered that the beam bent in out-ofplane direction as a half sine wave along with value L/1000.
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Fig. 6
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Mu/My versus initial imperfection values.
out-of-plan Displacment mm
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and the intermediate slenderness beams. Although for long beams, L/ry = 125, the compression flange distortional mode has little effect on the ultimate moment, it has been also considered as imperfection mode (M2). The normalized ratios of Mu/My have been drawn as function of the imperfection values in Fig. 6. The dotted curve represents the results obtained when the initial imperfection shape is the eigen mode only, while the solid curves represent the results when the eigen shape was added to the overall imperfection shape with values of L/1000 and L/500; respectively. It is clear that the bending strength of the short and the intermediate beams were greatly affected by the sectional imperfection shape (flange local buckling of compression flange). However, the ultimate capacity of the long beams (L/ry = 125) was more sensitive to the traditional overall imperfection shape. Moreover, the combined imperfection
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Fig. 7 Applied relationship.
moment
versus
lateral
displacements
Please cite this article in press as: B.L. Gendy, M.T. Hanna, Effect of geometric imperfections on the ultimate moment capacity of cold-formed sigma-shape sections, HBRC Journal (2015), http://dx.doi.org/10.1016/j.hbrcj.2015.04.006
Effect of geometric imperfections
7
L/ry = 50 L/1000
L/ry = 50 M1
L/ry = 50 M2
L/ry = 100 L/1000
L/ry = 100 M1
L/ry = 100 M4
L/ry= 125 L/1000
L/ry = 125 M1
L/ry = 125 M2
Fig. 8
Failure mode for short, intermediate and long beams.
It is clear that assuming the flange local buckling mode as the initial imperfection shape for the short and the intermediate slenderness beams reduces the flexural stiffness and leads to a smaller ultimate loads. The failure mode obtained is flange local buckling when the initial imperfection shapes is the flange local buckling mode, while it become distortional buckling mode for the other two imperfection modes. However, in long beams, the load deformation history is quite similar for the three assumed imperfection shapes. Moreover, they are all lead to a similar failure mode, as shown in Fig. 8. The deformation history of the beams has been reported to assess the effect of initial imperfection shape on their structural behavior. Fig. 7 shows the relationship between the applied moment, M, and the out-of-plane displacement of the flange lip intersection point for the beams with overall slenderness ratios L/ry = 50, 100, 125. Curves were drawn considering three shapes of imperfections. The first imperfection shape is the lowest value eigen mode shape (mode 1), while the second one is the most sensitive mode shape to the ultimate bending moments. As mentioned before, in the short and the intermediate slenderness beams, the second mode is the compression flange local buckling modes. This is modes M1 and M4 for beams with L/ry = 50 and 100; respectively. However, for long beams, L/ry = 125, the compression flange distortional buckling mode is considered as a sectional mode for comparison. The last one was considered that the beam bent in out-ofplane direction as a half sine wave along with value L/1000. It is clear that assuming the flange local buckling mode as initial imperfection shape for the short and the intermediate slenderness beams reduces the flexural stiffness and leads to
smaller ultimate loads. The failure mode obtained is flange local buckling when the initial imperfection shapes are the flange local buckling mode, while it become distortional buckling mode for the other two imperfection modes. For the sake of the comparison, the flexural strengths, Mu, obtained by finite element analysis are compared by the design rules of the AISI-2012 [16]. In AISI-2012 code, the capacity of the beams is calculated as the product of the cross section effective section modulus, Se, and the maximum stress, Fc. The later will be the minimum of the yield stresses or lateral torsional buckling stresses which is determined using the full section properties. In-addition, the distortional buckling flexural strength, Mnd, has been included in AISI-2012 as limiting strength of the lipped channel cross section beams (provision C4.2). Note, the partial safety factors, cM1 and /c, are not included when calculating the design loads. Results of this comparison are shown in Fig. 5. Comparison reveals that, for short beams, AISI-2012 results are matched well with the finite element results considering the overall mode of imperfections. However, for long beams, the code predictions are relatively conservative. Conclusions Elastic and inelastic analysis was performed to investigate the effect of the initial geometrical imperfections on the ultimate moment capacity of simply supported sigma CFS beams. The beams are subjected to equal end moments. Three methods are used to simulate the initial geometric imperfections
Please cite this article in press as: B.L. Gendy, M.T. Hanna, Effect of geometric imperfections on the ultimate moment capacity of cold-formed sigma-shape sections, HBRC Journal (2015), http://dx.doi.org/10.1016/j.hbrcj.2015.04.006
8 using the finite element method. Results indicate that the ultimate bending moments of beams with the short and the intermediate overall slenderness ratios are sensitive to the imperfect shape that comprise compression flange local buckling. However, for long beams, the overall buckling mode (lateral–torsional buckling) was the most sensitive mode. Moreover, the failure shape of the short and intermediate beams is affected by the initially assumed geometric imperfection mode. Finally, the sectional imperfection modes with large values influence the ultimate moment capacities of the simply supported beams. Conflict of interest The authors declare that there are no conflicts of interests. References [1] ASTM-A6/A6M, Standard Specification for General Requirements for Rolled Structural Steel Bars, Plates, Shapes, and Sheet Piling, American Society for Testing and Materials, West Conshohocken, PA, 2011. [2] B.W. Schafer, T. Peko¨z, Computational modeling of coldformed steel: characterizing geometric imperfections and residual stresses, J. Constr. Steel Res. 47 (1998) 193–210. [3] K. Rzeszut, A. Garstecki, Modeling of initial geometrical imperfections in stability analysis of thin-walles structures, J. Theor. Appl. Mech. 47 (3) (2009) 667–684. [4] A. Garstecki, W. Kakol, K. Rzeszut, Classification of localsectional geometric imperfections of steel thin-walled coldformed sigma members, Found. Civ. Environ. Eng. 1 (2002) 1642–9303. [5] Z. Sadovsky´, J. Kriva´cˇek, V. Ivancˇo, A. Dˇuricova´, Computational modeling of geometric imperfections and buckling strength of cold-formed steel, J. Constr. Steel Res. 78 (2012) 1–7.
B.L. Gendy, M.T. Hanna [6] J. Bonada, M. Casafont, F. Roure, M.M. Pastor, Selection of the initial geometrical imperfection in nonlinear FE analysis of cold-formed steel rack columns, Thin-Walled Struct. 51 (2012) 99–111. [7] D. Dubina, V. Ungureanu, Effect of imperfections on numerical simulation of instability behaviour of cold-formed steel members, Thin-Walled Struct. 40 (2002) 239–262. [8] P. Borges Dinis, C. Dinar, S. Nuno, FEM-based analysis of the local plate/distortional mode interaction in cold-formed steel lipped channel columns, Comput. Struct. 85 (2007) 461– 1474. [9] Jung Kwan Seo, A. Tharmarajah, M. Mahen, Initial imperfection characteristics of mono-symmetric lite steel beams for numerical studies, in: Fifth International Conference on Thin-Walled Structure, Brisbane, Australia, 2008. [10] J. Yang, Q. Liu, An experimental study into flexural behaviour of sigma purlins attached with roof sheets, Eng. Struct. 45 (2012) 481–495. [11] Long-yuan Li, Analyses of distortional buckling of cold-formed sigma purlins using EN1993-1-3, J. Constr. Steel Res. 65 (2009) 2099–2102. [12] Long-yuan Li, Jian-kang Chen, An analytical model for analysing distortional buckling of cold-formed steel sections, Thin-Walled Struct. 46 (2008) 1430–1436. [13] ANSYS User’s Manual Release 12.0.1, Canonsburg (Pennsylvania), 2009. [14] Luı´ s Laı´ m, Joa˜o Paulo C. Rodrigues, Luis Simo˜esda Silva, Experimental and numerical analysis on the structural behaviour of cold-formed steel beams, Thin-Walled Struct. 72 (2013) 1–13. [15] V.M. Zeinoddini, B.W. Schafer, Simulation of geometric imperfections in cold formed steel members using spectral representation approach, Thin-Walled Struct. 60 (2012) 105– 117. [16] AISI, Cold-formed steel design manual, American Iron and Steel Institute, AISI, 2012.
Please cite this article in press as: B.L. Gendy, M.T. Hanna, Effect of geometric imperfections on the ultimate moment capacity of cold-formed sigma-shape sections, HBRC Journal (2015), http://dx.doi.org/10.1016/j.hbrcj.2015.04.006