1.1 Direct strength method-A brief

There are two basic design methods for CFS members viz, The Traditional Effective Width method (^{Yu W. W., 2000}) and upcoming, continuously developing but relatively less known, DSM which was adopted in the North American Design specifications in 2004 as an alternative to Effective Width Method. Appendix 1 of the North American Specification for the design of CFS structural Members, 2004, supplement to the 2001 edition talks about formulae and applications of DSM for columns and Beams. CFS members, due to their thin walls, put before engineers challenges like local plate buckling and cross section distortion. But the same challenge provides the advantage like local plate buckling has the capacity for post buckling reserve strength, which makes the member efficient. DSM is one such method which answers the above challenges at the same time uses the opportunity created by the challenges. (^{Schafer B. W., 2006}).

The development of the DSM started at the University of Sydney by research in to distortional buckling of rack post sections (columns) and was further developed for method of beams. (^{Schafer B. W., 2006}). Further Hancock (Hancock et. al, 1994) showed that compressive strength in a distortional failure correlated well with the slenderness in the elastic distortional mode.

Accurate member elastic stability is the fundamental to the DSM. The method is based on the idea that if all the three elastic instabilities viz, local, distortional and global buckling, along with load or moment causing yielding of the section can be found out, then the strength can be directly determined. The method uses column curves for global buckling with application to local and distortional buckling instabilities. The increased accuracy of the method occurs due to improvements in the local buckling prediction. The method also takes into consideration, deflection calculation (Serviceability). It is a reliable method; its reliability being established using limit state design format in use in the United States. (^{Schafer B. W., 2006})

Appendix 1 of the North American Specification for the design of CFS structural Members, 2004, supplement to the 2001also includes a number of tables that provide the geometrical and material bounds members which passed in the verification of the direct strength approach, in the process of codification of the same. Pre-qualified is the apt name that has been given to theses sections. Although this method is mainly due to Schafer B. W. (Schafer B. W., 2000), Moen C. along with Schafer B. W., took this method further by developing DSM equations that are applicable to CFS structural members with perforations (Moen C. D. and Schafer B. W., 2006; 2008; ²⁰¹⁰; ²⁰¹¹) In 2006 Schafer B. W. came out with a guide “Direct Strength Design method guide”. (^{Schafer B. W., 2006}).

1.2 Literature review

A study on the flexural strength and deflections of discretely braced cold-formed steel C and Z sections was conducted at the University of Florida (^{Ellifrit D., 1991}; ^{Ellifrit D., 1992}; ^{Ellifrit D., 1997}). In the research, typical C and Z sections were tested in flexure with various types of bracing. Researchers developed a finite element model for the nonlinear large-deflections and rotation analysis of beam-columns. (^{Pi YL, 1994}; Pi YL, 1994). Researchers Performed lateral buckling tests on unbraced, simply supported cold-formed lipped channel beams. A vertical load was applied at the shear centre of the section, or at a point below the shear centre. The beams were supported at the ends by connecting them to a steel block with two bolts at the web of the section (^{Bogdan M. Put, 1999}; 1999).

In another study, researchers (^{Yu C., 2003}; ^{Schafer B. W., 2006}) studied the buckling behaviour of CFS channel beams. The buckling test was carried out on simply supported unbraced CFSs of two different cross sections. The lateral buckling test results showed that the CFS sections failed catastrophically by local & distortional buckling of most compressed elements of the cross section after quite large deformations. The results of 10 lateral buckling tests on simply supported CFCs of two different cross sections were presented & also compared with analytical design method as per AS 4100 (^{AS 4100 Australian Standard Steel Structures,1981}). It was found that the moments at failure were lower when the beam lateral defections increased the compression in the compression flange lip, and higher when they increased the compression in the flange-web junction.

Researchers (^{Put B. M. et al., 1998}). performed a local buckling test on CFS C-sections and Z-sections During the test, bracing had been carefully considered in these tests to insure that distortional buckling and lateral-torsional buckling do not influence the interpretation of results. They concluded that the test results can be used for the evaluation of existing and proposed methods for strength prediction of webs in local buckling. In addition, these tests can form the basis for later evaluations in which restrictions on the distortional mode are relieved.

Overall test results indicate that AISI (^{American Iron and Steel Institute, 1996}) design method provides adequate strength predictions. The DSM provides the best test-to-predicted ratio for both slender and ‘unslender’ specimens. The test results demonstrate that many improvements in the elastic buckling and effective width calculation of C’s and Z’s are still possible.

Researchers studied the flexibility of beam-to-column connectors used in thin walled cold-formed steel pallet racking systems (^{American Iron and Steel Institute,1996}) The attention is focused on beams subject to torque, because of the effect of transverse loads not applied at the shear centre (^{Bajoria, K.M. et al., 2006}). A simple geometric nonlinear analysis method, based on satisfying equilibrium in the deformed configurations, is examined and used to predict the behavior of the beams. Simple geometric analyses, finite element analyses and finite strip analyses are performed and compared with experimental results. The influence of typical support conditions is studied and they are found to produce a partial warping restraint at the ends. This effect is accounted for by introducing a hypothetical spring. The magnitude of the spring stiffness is assessed for commonly used connections. Other factors that affect the behaviour of cold-formed steel members, such as local buckling, are also studied. The goal was to solve for the unknown rotation. The tested results were comparing with available standard commercial finite element software. The results matched closely, for the problems of interest, up until yielding takes place in the member. This can be seen in the results provided in the later sections. Therefore, it can be concluded that, for the problems of interest, the major contribution to nonlinearity, in the elastic range, is the nonlinearity due to the dependence of the torque on the rotation of the beam. They concluded that, under load, the beam displaces horizontally and rotates gradually, but no sudden lateral buckling of the beam takes place. The failure is started by yielding of the material. The AISI procedure for the estimation of the strength, based on lateral-torsional buckling, may under- or overestimates the strength. The beam undergoes large rotation before failure. Therefore, the serviceability limit state may be more critical than the strength limit state. In the case of beams with no slender elements, FEA by modelling as beam elements or shell elements predict similar behaviour.

A new CFS section was manufactured and tested using newly proposed double cantilever method. Bajoria and Talicotti (^{Bajoria, K. M., 2006}) proposed alternative beam to column test instead of the cantilever test. Complete studies involving both experimental and numerical investigations were performed to find out the flexibility of the beam-to-column connector, and this was followed by a full scale frame test to compare the results obtained. The double cantilever test takes into account the realistic behaviour of connectors, which are subjected to moment, shear and axial pull by the beams. This was confirmed by the results of the full-scale frame test.

Researchers studied the three dimensional (3D) model of conventional pallet racking systems using the finite element program ANSYS. They carried a free vibration modal analysis on conventional pallet racks with the 18 types of column sections developed along with semi-rigid connection. The stiffness of the connector was tested using the conventional cantilever method and also using a double cantilever method. They performed nonlinear FEA of both the tests. The model is aimed at developing simplified equation for the fundamental period of storage racks in their down aisle direction. Finally, parametric study was carried out to find out the fundamental mode shape and time period. Finite element method was used for the accuracy and appropriateness of cold-formed steel frame.

In 2011, K. K. Sangle performed the finite element buckling and dynamic analyses of two-dimensional (2D) single frames and three-dimensional (3D) frames of cold-formed sections with semirigid connections used in the conventional pallet racking system.(^{Bajoria, K. M. et al.,2011}) The results of buckling analysis for the single 2D frames are compared with those from the experimental study and effective length approach given by RMI (RMI Specification for the design, testing and utilization of industrial storage racks, ^{Rack Manufacturers Institute,2008}).

The finite element model used for the single 2D plane frame is further extended to 3D frames with semirigid connections, for which the buckling analysis results are obtained. Researchers. (^{Thombare C. N. et.al., 2016}) studied nonlinear buckling analysis of 2D cold formed steel storage rack structure using appropriate commercial FE platform. In this paper, the FEA results are validated with available experimental data for a particular two dimensional CFS frame (^{Bajoria, K.M., 2011}) with different thicknesses and bracing patterns. The FEA results show good convergence with the experimental results. Hence, study is extended to more heights of the frame. The results are then used to suggest an extension formula for modification in current DSM expression for two dimensional frames.

Significant research is going on to simplify the design of CFS sections and frames to make it more reliable and practically acceptable. The addition of perforations serves lots of advantages for the practical purpose, but at the same time it generates complications in design. Considerable efforts are required to study the impact of holes on the strength of the member. The aim of the current research is to propose a formula which will help to find out load carrying capacity of two dimensional CFS with varying heights as an extension to the DSM equations.

In this paper, the FEA results are validated with available experimental data for a particular two dimensional CFS frames (^{Bajoria, K.M., 2011}), with different thicknesses and bracing patterns. The FEA results show good convergence with the experimental results. Hence, the study is extended to more heights of the frame. The results are then used to suggest an extension formula for modification in current DSM expression for two dimensional frames.

3.1 Finite element modelling

Element type: Three elements commonly used by FEM software in the elastic buckling analysis of thin walled structures are the S9R5, S4 and S4R elements as shown in Fig.1. The S4 and S4R are four node general purpose shell elements, valid for both thick and thin shell problems. Both elements employ linear shape functions to interpolate deformation between nodes. S4R element has been used for meshing the model. Advantage of S4R element over S4 is, S4R uses reduced integration with hourglass control, finite membrane strains.

Boundary condition and loading: Boundary condition for the modelled column is pinned-pinned, free to wrap. End cross section nodes are restrained in X and Z direction and nodes at the centre are restrained in Y direction to prevent Rigid Body motion. A reference load of 1 kN is applied as a shell edge load over the perimeter of the column. The column has perforations at 140 cm c/c. Modulus of Elasticity is 212000 MPa and Poission’s Ratio is 0.325

Analysis: Liner Eigen buckling analysis was performed with appropriate FEA software. Eigen value obtained from the analysis is used to calculate the buckling capacity of the frame. The models were meshed with 30mm and 10mm mesh for convergence study. It was observed that the convergence was non-monotonic, hence to find more accurate eigenvalue, the models with 10mm mesh is used.

3.2 Extending DSM for frame

In this section, an attempt was made to solve an axially loaded frame using equations given by (^{Schafer B.W., Moen C.D., 2010}) and corresponding modifications was given. A frame is modelled in CUFSM v4.05 as shown in Fig 4. The bracing effect was not considered as CUFSM is applicable only for compression and flexural member analysis.

Figure 4 Finite strip model of frame MW1.6 in CUFSM version 4.05 

The following material properties were assigned to the section.

Yield Stress = 365 N/mm²

Young’s modulus = E = 212000 N/mm²

Shear Modulus = G= 80000 N/mm²

Poison’s ratio = v = 0.325

Thickness = 1.6 mm throughout the section.

Two column sections were connected by elements with zero thickness as shown in Figure 4. The cross sectional properties are found by the ‘section properties option’ of the software as shown. The load applied is 1000 N on the cross section and to generate a frame effect, connecting zero elements were deleted as shown (Figure 4). Similarly for net cross sectional properties, zero element thickness were assigned to hole region. Again load of 1000 N was applied and zero elements were deleted as shown. Before analysis the net cross section of frame, the corner nodes of member were restrained in Z-direction (^{Schafer B.W., Moen C.D., 2010}). The column section of frame is having 2 no’s of holes on the web and 4 no’s of holes on flange. Thus, while the calculations of perforations in frame total 4no.s of holes were assumed in the evaluation of strength.