An asymptotic‐numerical method to compute the postbuckling behaviour of elastic plates and shells

In this paper, we apply an asymptotic-numerical method for computing the postbuckling behaviour of plate and shell structures. The bifurcating branch is sought in the form of polynomial expansions, and it is determined by solving numerically (FEM) several linear problems with a single stiffness matrix. A large number of terms of the series can easily be computed by using recurrent formulas. In comparison with a more classical step-by-step procedure, the method is rapid and automatic. However, the polynomial expansions have a radius of convergence which limits the validity of the solution to a neighbourhood of the bifurcation point. In the present form, the method should be viewed as a cheap and automatic way of completing a linear buckling analysis. It is illustrated in two examples: a square plate under in-plane compression and a cylindrical shell under pressure.

[1]  Richard H. Gallagher,et al.  Perturbation procedures in nonlinear finite element structural analysis , 1975 .

[2]  Peter Wriggers,et al.  A simple method for the calculation of postcritical branches , 1988 .

[3]  D. Sattinger.  Transformation groups and bifurcation at multiple eigenvalues , 1973 .

[4]  E. Riks.  An incremental approach to the solution of snapping and buckling problems , 1979 .

[5]  M. König,et al.  Geometrically nonlinear finite element behaviour using buckling mode superposition , 1979 .

[6]  Klaus-Jürgen Bathe,et al.  A study of three‐node triangular plate bending elements , 1980 .

[7]  Ahmed K. Noor,et al.  Reduced Basis Technique for Nonlinear Analysis of Structures , 1980 .

[8]  Bernard Budiansky,et al.  Theory of buckling and post-buckling behavior of elastic structures , 1974 .

[9]  M V Dyke.  Computer-Extended Series , 1984 .

[10]  W. Wagner.  A path-following algorithm with quadratic predictor , 1991 .

[11]  Ekkehard Ramm,et al.  Strategies for Tracing the Nonlinear Response Near Limit Points , 1981 .

[12]  A. C. Walker,et al.  A non-linear finite element analysis of shallow circular arches , 1969 .

[13]  Ted Belytschko,et al.  Buckling of structures with finite prebuckling deformations—a perturbation, finite element analysis , 1975 .

[14]  M. Potier-Ferry.  Foundations of elastic postbuckling theory , 1987 .

[15]  H. Görtler.  Zur Geschichte des II-Theorems† , 1975 .

[16]  M. Crisfield.  An arc‐length method including line searches and accelerations , 1983 .

[17]  J. Z. Zhu,et al.  The finite element method , 1977 .

[18]  A. Eriksson.  Derivatives of tangential stiffness matrices for equilibrium path descriptions , 1991 .

[19]  Michel Potier-Ferry,et al.  A New method to compute perturbed bifurcations: Application to the buckling of imperfect elastic structures , 1990 .