Static analysis of uncertain structures using interval eigenvalue decomposition

Abstract

Static analysis is an essential procedure to design a structure. Using static analysis, the structure's response to the applied external forces is obtained. This response includes internal forces/moments and internal stresses that are used in the design process. However, the mechanical characteristics of the structure possess uncertainties that alter the structure's response. One method to quantify the presence of these uncertainties is interval variables. In this work, a new method is developed to obtain the bounds on a structure's static response using interval eigenvalue decomposition of the stiffness matrix. The bounds of eigenvalues are obtained using monotonic behaviour of eigenvalues for a symmetric matrix subjected to non-negative definite perturbations. Moreover, the bounds of eigenvectors are obtained using perturbation of invariant subspaces for symmetric matrices. Using this method, it is shown that obtaining the bound on static response of an uncertain structure does not require a combinatorial or Monte-Carlo simulation procedure.