Stability of elastic bars on uncertain foundations using a convex model

Abstract

A method is presented for evaluating the buckling load of weightless prismatic rigid bars resting on uncertain foundations which are modeled using linear extensional elastic springs. The uncertainty in the spring elements is expressed in terms of a nonprobabilistic convex model. An ellipsoidal bound is used which defines the uncertainty of the foundation in terms of a size parameter and the deviations of the elastic spring constants from their nominal values. The size parameter represents the size of the ellipsoid and is analogous to the standard deviation magnitude in probabilistic analyses. The semiaxes of the ellipsoid are the deviations of the foundation spring constants from their nominal values and they determine the shape of the ellipsoid. A first-order analysis shows that the reduction in the buckling load, when uncertainty in the foundation's spring stiffnesses is present, is a linear function of the size parameter and a nonlinear function of the semiaxes of the uncertainty ellipsoid. For the same uncertainty in the spring elements, different reductions in the buckling load result for beams with multimode buckling.