Stochastically Dominant Modes of Frames by Mathematical Programming

Most important in calculating the failure probability of structural systems is the search for the stochastically most relevant failure mechanism of ductile structural systems—particularly framed structures. For this reason an automatic method for finding all modes can be employed, but for very large systems these procedures may become intractable. Mesh and nodal methods for frames are formulated with particular attention to simple data preparation. The generalized mesh description leads to the simpler and more efficient formulation of the mechanism compatibility equations. Two nonconvex problems, one with bilinear constraints and the other with a fractional objective function, have been derived. The latter formulation can be solved quite effectively when cast in the form of the maximization of a quadratic convex function over a linear domain. This is followed by mathematical programming techniques that are more appropriate for nonconvex optimization problems. The paper concludes with a discussion of the methodologies for finding other important modes.