Dynamic Analysis of Structures with Interval Uncertainty

Abstract

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 present method is performed using a set-theoretic (interval) formulation to quantify the uncertainty present in the structure’s parameters, such as material properties and cross-sectional geometry. Independent and/or dependent 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 pseudodeterministic 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 nonnegative definite perturbations. Then, using the procedures for perturbation of invariant subspaces of matrices, the bounds on directional deviation (inclination) of each mode shape are obtained. Following this, the interval response spectrum analysis is performed considering the effects of input variation in terms of the structure’s total response, which includes maximum modal coordinates, modal participation factors, and mode shapes. Using this method, for the problems considered, it is shown that calculating the bounds on the dynamic response is more computationally efficient than the combinatorial or Monte Carlo–solution procedures. Several problems that illustrate the behavior of the method and comparison with combinatorial and Monte Carlo–simulation results are presented.