The quasi-static explicit finite element method (FEM) and element free Galerkin (EFG) method are applied to trace the post-buckling equilibrium path of thin-walled members in this paper. The factors that primarily control the explicit buckling solutions, such as the computation time, loading function and dynamic relaxation, are investigated and suggested for the buckling analysis of thin-walled members. Three examples of different buckling modes, namely snap-through, overall and local buckling, are studied based on the implicit FEM, quasi-static explicit FEM and EFG method via the commercial software LS-DYNA. The convergence rate and accuracy of the explicit methods are compared with the conventional implicit arc-length method. It is drawn that EFG quasi-static explicit buckling analysis presents the same accurate results as implicit finite element solution, but is without convergence problem and of less-consumption of computing time than FEM.

Thin-walled members of various shapes have been widely used in civil and mechanical engineering. Under many conditions, when these members are subjected to compressive loads, buckling and post-buckling behaviors usually dictate the design considerations. Analytical solutions of buckling of thin-walled members can only obtain for the simple cases of elastic linear/nonlinear buckling. Analysis of nonlinear buckling problems of plastic and large deformations primarily relies on the numerical methodology.

In the nonlinear buckling analysis of thin-walled structural member, the implicit arc-length algorithm is generally accepted as an effective numerical method for tracing the post-buckling path [1] [2] [3] . It is well known that the implicit method is established on the operation of stiffness matrix, where a large amount of computing resource is required for processing highly incremental iteration. For complex nonlinear problems, the disadvantage of non-conver- gence of this method is obvious and usually difficult to be overcome, for example, the singularity of the stiffness matrix near the critical point. Therefore it is necessary to study the explicit method for solving highly nonlinear post-buck- ling problems.

Quasi-static analysis is a simulation of static problem with motion analysis which restricts the load velocity so that the outcome of this analysis can only have a little inertia influence that can be neglected. As an explicit algorithm, the advantages of quasi-static buckling analysis lie in the lower computing cost and no convergence consideration. However, structural dynamic responses caused by loading speed and inertia force significantly influence the quasi-static explicit results. Usually very small loading step is needed to approach the static equilibrium state at each loading moment, which inversely decreases the convergence rate. The efficiency of quasi-static explicit method mainly depends on the problems being solved. The key factors that can reduce the dynamic responses, like computing time, loading function and damping relaxation, must be specified in order to keeping the ratio of dynamic energy to internal energy within a low level. Zhuang [4] presented the conventional method of quasi-static analysis and compared the differences between implicit and explicit methods. Ji [5] used the quasi-static analysis for solving stable problems of stiffened plates under axial pressure, and obtained the structure responses which static analysis could not reach with a little time cost. Fan [6] did research on the effect of the duration and functional form of the time histories of loads by the analysis of a linear spring mass oscillator. Li [7] interpreted the principle of static calculation by using dynamic relaxation method and put forward a new method on value adoption of mass damp and rigidity damp. Lee [8] applied the explicit arc-length method to trace the post-buckling equilibrium path of structures on the basis of dynamic relaxation method with kinetic damping.

Finite element method (FEM) is a stable and reliable computing method through meshing the continuum into discrete units. When structures undergo large deformations, the computing accuracy is significantly influenced by the distortion of discrete units. In explicit method, a stable time step must be very small if the distortion of discrete units occurs, which greatly adds up the computing cost. Element-free method (EFM) is studied by many researchers for avoiding the effects of discrete units on numerical consequence. Solution with EFM depends on the discrete nodes setting up within or at the edge of a domain. Shape function is constructed on local nodes, so there is no mesh-dependence problem. The primary advantage of this approach is that there is no singularity of stiffness matrix induced by distortion of discrete units in the solution of large deformation and discontinuity problems. Element free Galerkin (EFG) method is based on the global Galerkin weak forms and the integration of background grids. The displacement approximation functions are generated by using the least squares approximation constructed via nodes in local fields. The computational accuracy and convergence rate of EFG methods have been demonstrated to be the same as FEM. The stability of this method is not affected by the irregular nodes, and furthermore, it can be combined with FEM and BEM (boundary element method) to improve the computing efficiency.

EFG method has been well used in the buckling analysis of thin-walled members. Liu [9] developed an EFG formulation to calculate the buckling loads of symmetrically laminated composite plates based on the principle of minimum potential energy, and found that solving the eigenvalue problem is much more computationally efficient compared to the FEM. Chinnaboon [10] developed a BEM-based meshless method for buckling analysis of elastic plates with various boundary conditions that include elastic supports and restraints. Liew [11] used an EFG method to study the elastic buckling behavior of stiffened and un-stif- fened folded plates under partial in-plane edge loads. Tamijani [12] employed the EFG method for buckling and static analysis of plates with arbitrary curvilinear stiffeners. Peng [13] obtained the critical buckling load of ribbed plates using the mesh-free method based on the first-order shear deformation theory. Xiang [14] predicted buckling behavior of microtubules based on an atomistic- continuum model. Lu [15] developed an adaptive enrichment mesh-free method to capture wrinkling and post-buckling behavior in sheet metal forming. Li [16] used mesh-free method for numerical simulations of large deformation of thin shell structures, which showed simplicity in both formulation and implementation as compared to shell theory approach. Lin [17] used a non-linear dynamic explicit scheme for the post-buckling analysis of thin-walled structure based on the meshless shell formulation. Compared with the finite element method, the mesh-free method possesses the same accuracy and can save some computing time as well as work out the problems that canâ€™t be solved by the traditional FEM.

The FEM and EFG quasi-static explicit methodologies are applied to trace the post-buckling path of thin-walled members in this paper. The key factors that control the convergence rate and dynamic responses, such as the computation time, loading function and damping relaxation, are discussed and suggested in the numerical buckling analysis. Three examples of thin-walled members occurred snap-through, overall and locally buckling are studied in detail by quasi-static explicit FEM and EFG method, and the efficiency and accuracy of the applied methods are demonstrated through the comparison with the conventional solution of implicit arc-length method.