Sensitivity analysis and optimization corresponding to a degenerate critical point

Sensitivity coefficients of a critical load factor corresponding to a degenerate critical point is shown to be unbounded even for a minor imperfection excluding very restricted case where the imperfection does not have direct effect on the lowest eigenvalue of the stability matrix. This fact leads to serious difficulty in obtaining optimum design under nonlinear stability constraints. The optimum design problem is alternatively formulated with constraint on the lowest eigenvalue of the stability matrix, and the sensitivity formula for the lowest eigenvalue is presented. The existence of a degenerate critical point and accuracy of the sensitivity coefficient are discussed through the example of a four-bar truss.