Post-buckling Analysis of Cold-form Steel Column

*Corresponding author Peer review under the responsibility of University of Baghdad. https://doi.org/10.31026/j.eng.2020.04.12 2520-3339 © 2019 University of Baghdad. Production and hosting by Journal of Engineering. This is an open access article under the CC BY4 license http://creativecommons.org/licenses/by /4.0/). Article received:12/3/2019 Article accepted:22/5/2019 Article published:1/4/2020


INTRODUCTION
Steel structures are manufactured from hot rolled sections or from cold form sections when the member is subjected to light loads or when the designer intends to use high efficient sections with high weight to strength ratio. In cold-formed sections, the thickness of steel sheet or strip is between 0.378 mm to 6.35 mm. Steel plates and bars up to 25.4 mm may be bent and pressed successfully to form structural shapes. The Specification for the Design of Cold-Formed Steel Structural Members of the American Iron and Steel Institute (AISI) has been issued since 1946. As the cold-formed steel member components are generally relatively thin with respect to their widths, they may buckle at stress value lower than the yield stress when it subjected to compression, shear, or flexure. Therefore, the major design consideration of the cold-formed steel members is the local buckling. It is well accepted that the cold-formed elements may not fail when it reaches to the buckling stress as they often will continue to support increasing forces that exceed the local buckling forces. In the post buckling stage, the load value may not change from the critical value or it may start to decrease while deformation is increase. Some cold-formed elements continue to support load after the deformation reaches to a certain value so the deformation is continuing to increase which finally cause a second buckling cycle. Therefore, the post buckling analysis is non-linear in nature and can provide more information than those obtained from a linear Eigen-value analysis (Wei & Roger , 2010).
For cold-formed steel, there are three types of buckling modes namely the local buckling mode that involves web and flanges deformation without transverse deformation of the of the corners, the distortional buckling mode is similar to the local buckling mod but it involves the deformation of the corners, and flexural-torsional mode which do not change the cross section of the member and it can bend and twist see Fig. 1 (Hancock, 2003). In post buckling behavior, the element may support additional load after buckling by a redistribution of stresses. This phenomenon is distinct for elements with high W/t ratios. As soon as the plate starts to buckle, the horizontal connected part will act as tie rods to inverse the increment in deflection of the member. The stress distribution that has been almost uniform before its buckling, as indicated in Fig. 2a, will be redistributed after buckling load in such a way that the center strip transfers to the edges as shown in Fig. 2b. The process continues until the stress of the edges reaches the yield stress and then the part begins to fail ( Fig. 3c). Large-deflection theory can be used to analyze the post buckling behavior through the following differential equation that introduced by von Karman in 1910: where is a stress function expresses the median fiber stress. As the solution of this differential equation has little practical applications due to its complexity, a concept of effective width, , has been introduced by von Karman et al. in 1932 to deal with a fictitious uniform stress instead of the actual non-uniform distribution over the whole width of the plate . The value of the uniform stress has been taken equal to the edge stress . The width b is selected such that the resultant for the actual non-uniform stress distribution to be equal to that of the equivalent uniform intensity (Wei & Roger , 2010).
The Effective Width Method is available, all over the word to be used in the design. As an alternative analysis approach, the Direct Strength Method has been formally adopted in North American, Australia and New Zealand (Zheng, et al., 2012). The effective cross-section illustrated in Fig. 3a provides a clear model for the effective and ineffective locations for carrying the load in the cross section, clues that the neutral axis changes its location in the section due to buckling. However, in the determination of the elastic buckling behavior it ignores the interaction and the compatibility, the overlapping of the buckling modes, like distortional buckling can be difficult to define, difficult and time-consuming iterations are required to calculate the member strength, and the evaluation of the effective section becomes more difficult if the section was complicated" . The basic idea of the Direct Strength Method is the exact member elastic stability, as shown in Fig. 3b. The determination of all elastic instabilities of section (i.e., local, distortional, and global buckling) and the yield load is the basic steps of the direct strength method. (Schafer, 2006). Elastic buckling is the load that the member equilibrium is neutral between the original deformed form and the buckling form. Elastic buckling evaluated in one of these methods: finite strip FS, finite element FE, generalized beam theory GBT, and closed form solutions. Finite strip analysis is a common method that determines precise values of elastic buckling with less effort and shorter time. FS analysis, adopted in the conventional programs like CUFSM, had a limitation that the 176 model assumes as a simply supported member of both ends and the cross-section is constant and do not change along the length. These limitations exclude some analysis from the real prediction using with the FS method, but notwithstanding these limits, the tool is useful (American Iron and Steel Institute, 2006). A numerical investigation concerning the elastic and elastic-plastic postbuckling behavior of coldformed steel lipped channel columns affected by distortional/global (flexural-torsional) buckling mode interaction was done by (Dinis & Camotim, 2011). A systematic investigation of material behavior, residual stress distribution, and axial compression performance on an extreme thickwalled cold-formed square columns which are manufactured from circular to square shape. was performed numerically and experimentally by (Liu , et al., 2017). (Ma, et al., 2018) presents the numerical investigation of the compressive behavior of cold-formed high-strength steel (HSS) tubular stub columns. An numerical investigation on slab-beam interaction in one-way systems presented by (Al-Zaidee, 2018). The strength of steel beam-concrete slab system without using shear connectors (known as a non-composite action), where the effect of the friction force between the concrete slab and the steel beam has been investigated, by using finite element simulation (Al-Zaidee & Al-Hasany , 2018) In this paper, the researcher intends to study the postbuckling of cold formed steel columns using the finite element method by Abaqus which much easier and less time consuming and compare the results with the two other methods. +

FINITE ELEMENT MODELING
In this section, a cold formed steel column 3m in length and with simple ends has to be analyzed using Abaqus software. It has been modeled with shell elements for its web and flanges. The geometrical properties, material properties, and the shell thicknesses have been defined as indicated in Fig. 4. The steel behavior is simulated as elastic perfectly plastic with yield stress equal to 355 MPa and modules of elasticity of 210000 MPa. A linear buckling load step has been defined to simulate the elastic buckling of the column. Assuming proportional loads, a unit shell edge load has been applied. Boundary conditions with , , and = 0 have been assigned to both ends of the column. At the mid-span, the lateral movement has been prevented, = 0, to restrain the rigid body motion. The linear buckling analysis produces the Eigen value indicated in Fig. 5 which represents the load multiplier necessary to produce the critical state. The critical axial force has been related to the Eigen value and the applied unit line load based on the following relation:  To measure the post buckling behavior of the model, an imperfection of 3mm has been used to modify the mode shape of Fig. 5 to simulate the imperfection along column length. The value of 3mm has been determined as ℓ 1000 ⁄ based on recommendations of (Dinis & Camotim, 2011) and (Garifullina & Nackenhorstb, 2015). The critical load from Eq. (3) has been applied and the model was analyzed. Static Riks non-linear analysis has been used to deal with the snap through the behavior of the post-buckling phenomenon (Garifullina & Nackenhorstb, 2015). Postbuckling analysis results are presented in Fig. 6 which shows that the post-buckling behavior would be unstable at a critical load approaching 100 kN.

VERIFICATION
The aim of this section is to compare the numerical postbuckling results of the Abaqus finite element analysis with those of the analytical buckling adopted by (American Iron and Steel Institute , 2016) and with those of the direct strength method using CUFSM software. The validation goals to prove the effectiveness of the finite element method in the analysis of this type of problems. The validated finite element model has the advantage over other analytical solutions in that it is general in nature and applicable for sections and materials other than those included in the manual.

Analytical Method
The analytical equations can be used to determine the post-buckling capacity of the cold-form columns. For a specific section with specific material properties, the global postbuckling axial strength can be determined based on Eq. (4) below.

= ….(4)
where: is the gross area, and is the compressive stress that determined in below: where:

= √ ….(7)
The is yield stress while the is the least value of the elastic global (flexural, torsional, and flexural-torsional) buckling stresses. For singly-symmetric sections, such as Lipped C Channel of this study, that subjected to flexural-torsional buckling can be determined based on Eq. (8):

= + ….(8)
where is the stress due to Saint-Venant and warping torsions calculated from Eq. (9). , are radii of gyration of the cross-section about the centroidal principal axes in mm, 0 is the distance from the centroid to the shear center in principal axis direction in mm, is the full unreduced cross-sectional area of member in mm 2 , is the shear modulus of steel MPa, is the Saint-Venant torsion constant of cross-section mm 4 , is the modulus of elasticity of steel MPa, is the torsional warping constant of cross-section in mm 6 , is the effective length factor for twisting, is the unbraced length of member for twisting in mm, and is the stress due to flexure calculated from Eq. (11).

….(11)
where: is the effective length factor for bending about x-axis, is the unbraced length of a member for bending about x-axis in mm, For the section considered in this study, aforementioned section and material properties have been summarized in Table 1below.

951.54
Then the critical stress will be = 119.8 so compute from the Eq. (7) that = 1.72 > 1.5 so the compressive strength is calculated from Eq. (6) and = 105.24 .Then the global buckling will be = 102.3

Direct Strength Method
The (American Iron and Steel Institute, 2006) has supported a research that led to the develop the CUFSM program, that depend on the FS method for elastic buckling calculation. This analysis two results have been provided: (i) the load-factors and the half-wavelength, and (ii) the buckling mode shapes. When CUFSM software was used to analyze different lengths of the cold-formed steel column of this study, it shows that:  The global buckling occurs at a span length of about 2m,  The distortional buckling dominates for spans between 0.3m to 2m,  The local buckling occurs for spans less than 0.3m. The elastic buckling strength for different spans has been presented in Fig. 7. The critical elastic loads for different zones have been determined as the minimum value for the corresponding zone. Then these critical elastic loads have been used in the empirical equations of the direct strength methods discussed in Section 3.2.1, Section 3.2.2, 3.2.3 to determine the corresponding post buckling loads. Analysis results including the elastic buckling loads and the post buckling loads (the nominal loads according to (American Iron and Steel Institute, 2006)) have been summarized in Table 2. where: : Gross area of cross-section in mm 2 : Yield stress MPa

= √ ….(15)
where: is the minimum of the critical elastic column buckling load in torsional-flexural buckling, N.

Local Buckling
The where: where: is the global column strength as defined in Section 3.2.1, N. 182 ℓ is Critical elastic local column buckling load, N.

Distortional Buckling
The nominal axial strength, , for the distortional buckling can be calculated in accordance with Eq. (19) or Eq.(20). where:

= √ ….(21)
where: : Critical elastic distortional column buckling load, N.  For this study with a span length of 3m, the buckling mode is a global buckling with an elastic buckling load of about 121 kN from Fig. 7. Using the global empirical equations of 3.2.1 the post buckling load would be 106 kN.

Results Discussion
The nominal axial strength of 101.65 kN from the finite element is close to, but slightly underestimated, the nominal strength of 102.3 kN from the analytical method. In percentage form, the difference between two methods is about 0.64%. Comparing to the direct strength method, that has a nominal strength of 106 kN, the finite element solution seems more underestimated with a difference of 4.27%. The load-deflection curve of the post buckling finite element analysis with the nominal strength values form the analytical and direct methods have been presented in Fig. 8.
Aforementioned results and discussions indicate that the post bucking finite element analysis is fair accurate to be adopted in practical design especially when dealing with nontraditional dimensions and/or materials that have no analytical solutions.

CASE STUDY
This case study adopts a cold-formed steel column from the (American Iron and Steel Institute, 2013) Manual with section of 8 3.5 × 0.085 that has dimension and material properties indicated in Table 3.

0.3
A procedure similar to that adopted in Section 2 has been used in the case study of this section and other two sections that have the same dimensions but with thicknesses of 2.159,1.778 ,1.5 mm investigate different column lengths from 0.05m to 8.0m. Imperfections of 0.1 and ℓ 1000 ⁄ have been respectively adopted for the local and global buckling. Results from the Abaqus finite element analysis have been gathered, organized, and presented in Fig. 9 that shows three distinguished regions. In the first region, for short lengths up to 20cm, the local buckling is the dominate mode while the distortional bucking that is independent on the column length dominates the behavior from 20cm up to 200cm spans. Finally, the global mode governs the behavior of long columns with spans more than 200cm. The section the that have a smaller thicknesses failed at lower load. The Sections have a nearly the same failure load in the global buckling region but they diverge in the local and distortional buckling zone.

CONCLUSIONS
Based on the aforementioned finite element modeling, solution processes, and the case studies the following conclusions can be drawn:  Comparing with traditional analyses of the effective width method and the direct strength method, it has been found that finite element simulate can be adequate and conservative estimation of the post buckling strength of cold-formed steel columns.  In addition to its validity, the finite element solution is a general in nature and can be adopted for sections and materials other than those of AISI design manual.  The finite element solution can be used for different imperfection values that related to different buckling modes.  Finally, the finite element solution can be used to study the behavior of cold-formed steel columns for different slenderness ratios.

RECOMANDATIONS
The following aspects that are significant from the subject point of view have not been discussed and they may be investigated in future works for a thorough understanding of the subject:  Other cold-formed shapes such Z shape or hollow sections can be investigated.  Inclined cold-formed sections can to be investigated.  Effect of the vibrational aspects of the cold-formed steel column.