Exact reliability calculation with quadratic failure indices in random vibration

Failure indices expressed as quadratic forms in the stress components are common in structural analysis to assess the failure of a structural component. For instance, the quadratic von Mises criterion is universally accepted as an adequate scalar measure to verify the onset of plasticity in ductile materials. On the other hand, some of the failure indices for composite materials proposed in the literature and used in industry may be cast in the form of quadratic expressions in the stress components. If the structure is excited by random forces, the value of the failure index at any point will vary randomly and, if a reliability analysis is to be performed, then it is necessary to calculate its probability distribution. In this paper, it is shown that the probability distribution of the quadratic failure indices can be calculated exactly in the most general case.