Lower Bounds for Critical Buckling Load Using Matrix Norms

A method is proposed to compute lower bounds for the critical buckling load in problems of matrix structural analysis and in finite‐element analysis. The method is based on the fact that the critical buckling load is equal to the reciprocal of the spectral radius of a modified geometric stiffness matrix for the problem. Any matrix norm gives an upper‐bound estimate of the spectral radius, hence a lower‐bound estimate of the critical buckling load. Three easily computed types of matrix norm are used here: the maximum column‐sum norm, the maximum row‐sum norm, and the euclidean norm. By using powers of the matrix, a set of successive lower‐bound approximations can be computed that converge to the critical buckling load. A two‐bar truss, a three‐bay frame, and a finite‐element model of a plate on elastic foundation are presented as numerical examples. The last example shows how eigenvalue clustering slows convergence. The method may also be applied to estimate fundamental frequencies of vibrating structures.