Load/end-shortening curve with collapse load A, bifurcation point B, and post-bifurcation equilibrium path BD (photographs by Sobel and Newman [87, 1981.pitfalls]).

This is a rather thick aluminum cylindrical shell under uniform end shortening (axial compression). The primary equilibrium path (solid line, 0AC) corresponds to axisymmetric deformation, in which an elastic-plastic “elephant’s foot” buckle develops, in this case especially near the top end of the shell. In this particular example, at a point, A, the maximum load carrying capacity is reached.

As end shortening is further increased, at another point, B, bifurcation of equilibrium states initiates. The dotted path, BD, corresponds to equilibrium states in which the total deformation consists of further axisymmetric deformation plus a rapidly growing component of non-axisymmetric deformation.

The bifurcation point, B, can be determined from a series of nonlinear eigenvalue analyses in which the uniqueness of the equilibrium state on the primary (axisymmetric) equilibrium path 0AC is tested at successive points on this path. The point on the primary path at which the stability determinant first equals zero (indicating non-uniqueness of equilibrium) is the bifurcation point (nonlinear buckling load).

At the bifurcation point, B, the non-axisymmetric component of the total deformation has infinitesimal amplitude and its shape is the eigenvector, {q}, obtained from the eigenvalue problem:

[K1] {q} = LAMBDA*[K2] {q}

in which [K1] is the stiffness matrix of the axisymmetrically deformed shell as loaded at the load step just before Point B (infinitesimally before B), LAMBDA is the eigenvalue, and [K2] is the “load-geometric” matrix corresponding to the difference in axisymmetric load states between that at the load step just before Point B and that just after point B.

The post-bifurcation equilibrium path can be determined approximately by a new nonlinear analysis in which the shell has a very, very small initial imperfection in the shape of the eigenvector (buckling mode, q).

This system would exhibit very little or no sensitivity to initial imperfections, that is, the maximum load-carrying capacity and other behaviors would not strongly depend on the amplitude of the initial imperfection for amplitude of order of the shell thickness or smaller.

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