Geometrically Nonlinear Analysis in the Vicinity of Critical Points by the Generalized Inverse

Abstract

In the geometrically nonlinear analysis for equilibrium paths, the determinant of the tangent stiffness matrix at the critical point becomes zero so that a numerically unstable situation appears in the vicinity of critical points. To avoid such a situation, many numerical methods including the arc-length method and the perturbation method have been developed. In this paper, an analytical method to pursue the geometrically nonlinear equilibrium paths in the vicinity of critical points such as limit point and bifurcation point is presented by using the generalized inverse. In the first part, the perturbation equations for the incremental equilibrium equations are derived. Then, critical points on the equilibrium path are classified into limit point and bifurcation point by using the existence condition of solution. In the second part, an analytical method for post-critical paths beyond critical points is presented by means of the generalized inverse. In the final part, the application of the present method to the post-buckling analysis of a shallow arch and cable domes subjected to the symmetrical loads is shown.