Dynamic analysis of structures with interval uncertainty

A new method for dynamic response spectrum analysis of a structural system with interval uncertainty is developed. This interval finite-element-based method is capable of obtaining the bounds on dynamic response of a structure with interval uncertainty. The proposed method is the first known method of dynamic response spectrum analysis of a structure that allows for the presence of any physically allowable interval uncertainty in the structure's geometric or material characteristics and externally applied loads other than Monte-Carlo simulation. The present method is performed using a set-theoretic (interval) formulation to quantify the uncertainty present in the structure's parameters such as material properties. Independent variations for each element of the structure are considered. At each stage of analysis, the existence of variation is considered as presence of the perturbation in a pseudo-deterministic system. Having this consideration, first, a linear interval eigenvalue problem is performed using the concept of monotonic behavior of eigenvalues for symmetric matrices subjected to non-negative definite perturbation which leads to a computationally efficient procedure to determine the bounds on a structure's natural frequencies. Then, using the procedures for perturbation of invariant subspaces of matrices, the bounds on directional deviation (inclination) of each mode shape are obtained.