Reciprocal approximation in stochastic analysis of structures

A stochastic analysis of structures usually requires multiple reanalysis of the structures to compute the statistics of the structural response. A similar problem exists in the design of optimal structures where many analysis are needed before reaching the extremal solution. In both fields the reanalysis requirements are considered as an unacceptable numerical burden. To circumvent the reanalysis obstacle investigators have been using approximate analysis. Common methods are first and second-order series expansions of the nodal displacements, and related perturbation methods. Interestingly, a popular approximation method in structural design, the reciprocal approximation technique, has not been used in stochastic analysis. This paper shows that this method can easily be used to compute the statistics of the structural response. When expanding the displacements linearly in terms of the reciprocals of the element stiffnesses, one obtains, as a rule, better results than with a linear Taylor expansion. It is shown that for low values of the relative redundancy the method yields second order quality approximations. Unlike many other techniques, the reciprocal approximation also produces the statistics of the internal forces. The theory is illustrated with typical beam and arch trusses and is compared with existing stochastic methods.