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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 [

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 [

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 [

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.

In explicit formulation, the basic dynamic equation of a volume element at time t is written in the form of equation of equilibrium as follow:

σ i j , j + f i − ρ u i , t t − μ u i , t = 0 in V (1)

The constitutive equation and the boundary equations are given by

σ i j = D i j k l ε k l in V (2)

u i = u ¯ i on S u (3)

s i j n j = T ¯ i on S σ (4)

where σ is the Gauss stress, f_{i} is body force, ρ is density, μ is damping coefficient, u_{i,t} and u_{i,tt} are the first and second derivatives of displacement u_{i}, D_{ijkl} is stiffness tensor, ε_{kl} is strain tensor, n_{j} is normal vector, and T_{i} is surface force.

By applying the Galerkin method to the Equation (1), Equation (3) and Equation (4), the corresponding weak form is expressed as

∫ V δ u i ( σ i j , j + f i − ρ u i , t t − μ u i , t ) d V − ∫ S σ δ u i ( σ i j n j − T ¯ i ) d s = 0 (5)

Substituting Equation (2) to Equation (5), the weak form is then transformed into Equation (6).

∫ V ( δ ε i j D i j k l ε k l + δ u i ρ u i , t t + δ u i μ u i , t ) d V = ∫ V δ u i f i d V + ∫ S σ δ u i T ¯ i d s (6)

∫ V ( δ ε T D ε + δ u T ρ u + δ u T μ u ) d V = ∫ V δ u T f d V + ∫ S σ δ u T T d s (7)

In the FEM, the shape function, N(x), is created by interpolation in elements of a set of fixed nodes. The displacement function is given by

u ( x ) = ∑ i = 1 n N i ( x ) u i = N ( x ) u (8)

where N = [ N 1 T , N 2 T , ⋯ , N n T ] , u = [ u 1 T , u 2 T , ⋯ , u n T ] . Substituting Equation (8) into Equation (7) leads to the equation as below:

#Math_12# (9)

The above dynamic equilibrium equation can be reduced to the general form as follow.

M a ¨ ( t ) + C a ˙ ( t ) + K a ( t ) = Q ( t ) (10)

in which

M = ∑ e M e = ∑ e ∫ V e ρ N T N d V , C = ∑ e C e = ∑ e ∫ V e μ N T N d V

K = ∑ e K e = ∑ e ∫ V e B T D B d V , Q = ∑ e Q e = ∑ e ( ∫ V e N T f d V + ∫ S σ e N T T d S )

Different from FEM, the numerical discretization in EFG method is based on the moving least-squares (MLS) approximation [

u ( x ) = ∑ i = 1 m p i ( x ) a i ( x ) = p T ( x ) a ( x ) (11)

where m is the order of completeness in this approximation, the monomial p_{i}(x) is the basis function, and a_{i}(x) is the coefficient of the approximation. a_{i}(x) depends on the sampling point x_{i} that is collected by a weighting function w ( x ) = w ( x − x i ) , which is nonzero in a small domain called influence domain.

The weighted residual can be written as L_{2}-Norm, namely

J = ∑ I = 1 N w I ( x ) [ u ( x I ) − u I ] 2 = ∑ I = 1 N w I ( x ) [ ∑ i = 1 m p i ( x I ) a i ( x ) − u I ] 2 (12)

In MLS approximation, at an arbitrary point x, a(x) is chosen on the basis of minimizing the weighted residual, then we have:

a ( x ) = A − 1 ( x ) B ( x ) u (13)

Substitute Equation (13) to Equation (11), it gives

#Math_20# (14)

in which:

A ( x ) = ∑ I = 1 N w I ( x ) p ( x I ) p T ( x I )

B ( x ) = [ w 1 ( x ) p ( x 1 ) w 2 ( x ) p ( x 2 ) ⋯ w N ( x ) p ( x N ) ]

By applying Equation (14) to Equation (10), the quasi-static explicit EFG formulation can be obtained.

In quasi-static explicit buckling analysis, to achieve the real dynamic loading process and the progress of unbalanced stress waves between elements, a large amount of time increment steps are usually required to obtain stable solutions. The kinetic energy usually increases rapidly with the enlargement of deformation after peak load point. For improving computing efficiency and having a stable solution, measures have to be employed to accelerate the computing process, and at the same time, the dynamic responses expressed by the ratio of kinetic energy to internal energy must be kept within 5% - 10%. The key factors in this slow-dynamic technique are how to choose the load duration and minimize the undesired dynamic effect originated from the inertia force of the governing equation. The response of structures is mainly controlled by the first mode, so the computation time in quasi-static explicit buckling analysis is usually set up more than ten times of the first mode period T.

Rapid movement can generate stress wave, which results in shock or inaccurate numerical solutions. Therefore, the curve of loading function must be smooth. The commonly used loading functions are shown as below:

Linear function curve: F ( t ) = F 0 t / T

Parabolic function curve: F ( t ) = F 0 ( t / T ) 2

Versin function curve: F ( t ) = F 0 ( 1 − cos ( π t / T ) ) / 2

Cycloid function curve: F ( t ) = F 0 ( t / T − sin ( 2 π t / T ) / 2 π )

Dynamic relaxation is originated in the steady-state solution of single degree of freedom damping system. The basic idea lies in keeping a system in an over damping state by setting the Rayleigh damping to a large value to weaken the dynamic effect on the system.

The damping of an actual structure can be expressed by the Rayleigh damping, namely C = α M + β K , where M is the mass matrix, K is the stiff matrix, α and β are the mass damping and stiff damping coefficients, respectively. The damping ratio ξ_{i} is written as

ξ i = ( α / ω i + β ω i ) / 2 (15)

where ω_{i} is the ith order circular frequency. The curves of unit step responses to different damping ratios of a single degree of freedom system are shown in _{i} ≥ 1, preferably ξ_{i} = 1. As long as αβ ≥ 1, the damping ratio of the system ξ_{i} is more than 1 from Equation (15).

Refer to the research by Li et al. [_{min}, β = 1/ω_{mi}_{n}, where ω_{min} is the first order circular frequency. By this way, the shocks induced by the higher order frequencies are restrained, which can make the explicit solution of dynamic relaxation close to the static solution.

In this section, by using the commercial software LS-DYNA, the FEM and EFG quasi-static explicit method are employed to simulate the post-buckling behaviors of three typical buckling problems: snap-through buckling, overall buck-

ling and local buckling. The loading-displacement curves solved by explicit quasi-static FEM, implicit FEM and explicit quasi-static EFG method are compared and the reliability of explicit quasi-static buckling analysis with the suggested computation time, loading function and dynamic relaxation is demonstrated.

A simply supported cylindrical shallow shell subjected to central loading is shown in ^{3}. Length, radius, thickness and angle of the shell are L = 100 mm, R = 1000 mm, t = 4 mm and θ = 6˚, respectively. A quarter of the shell with symmetric constrain is modeled, and the displacement of 0-13mm is applied at point A. This is a typical example of snap-through buckling; in which the traditional load-controlled computing method is inapplicable due to the singularity of tangential stiffness matrix at the extreme point. The equilibrium path can be traced by implicit arc-length method, and being as a comparison, the accuracy of quasi-static explicit buckling analysis is studied from the following aspects.

As a versine displacement is applied and the computation time is specified as 10T, 20T, 40T, 80T, 160T and 320T, the curves of load-vertical displacement at point A are constructed in Figures 3-5. It is shown that the curves from explicit calculation are very close to those from implicit calculation before the critical load points. There are fluctuations about the implicit results in the rising segments of post-buckling in the explicit calculation. The shorter is the computation time, the lower is the fluctuating frequency and the higher is the fluctuating amplitude. From the ratio of kinetic energy to internal energy in

When different loading functions such as linear, parabolic, cycloidal and versine curves are applied, and the computation time is taken as 20T, the load-dis- placement relationships are constructed in

In order to keep the structural responses being quasi-statically damped and converged to the static equilibrium state, the dynamic relaxation (DR) method has to be used in the explicit buckling analysis. Based on the minimum frequency of free vibration of the calculated shell, 752.28 Hz determined from model analysis, coefficients α and β are calculated as α = 4726.7 and β = 2.12e^{−4}, respectively. The load-displacement curves based on versine load and DR in the case of different computation time are plotted in

The computation time, loading function and dynamic relaxation are also important for quasi-static explicit post-buckling analysis with element-free Galerkin (EFG) method. The nodes in EFG are appointed based on the grid nodes in

FEM in the package LS-DYNA. Parameters and solution settings of shell are defined by the key words in *SECTION_SHELL_EFG and *CONTROL_EFG. According to the previous explicit FEM, versine load and DR are employed in the explicit EFG buckling analysis. The load-displacement curve of 160T solved by EFG is contrasted with those from the implicit arc-length method and explicit FEM as shown in

Dimensions of a thin-walled steel angle subjected to central loading are shown in ^{3}. Thickness, width and length are t = 0.7 mm, b = 15.85 mm and L = 180 mm, respectively. Yield strength is 360 MPa. Based on the previous study, axial versine displacement and DR are applied in the quasi-static explicit buckling analysis. Post-buckling paths of the thin-walled angle clamped at both ends under different computation time are contrasted with the results obtained by implicit nonlinear analysis with modified Crisfield arc-length method (MC-ALM).

Based on model analysis, the first frequency of the angle equals 752.28 Hz. The damping coefficients α = 2667.4 and β = 3.749e^{−4}. Curves of axial load-ver- tical displacement in the case of different computation time 10T, 20T, 40T, 80T, 160T are plotted in

Before the critical load, the curves obtained by explicit FEM and implicit MC- ALM are well coherent, while, after the critical load, the critical load determined by the explicit FEM is higher than that from implicit MC-ALM, as shown in

At the point of critical load, stress and deformation contours obtained by implicit method and explicit solutions in the case of various computation time are shown in