Stochastyczna analiza dynamiczna wrażliwości układu belkowo-kratowego

W pracy przedstawiono procedury numeryczne dynamicznej analizy wrażliwości układów konstrukcyjnych. Oprócz zagadnienia różniczkowania bezpośredniego, sformułowano sprzężone zagadnienie końcowe, bazujące na pierwotnym zagadnieniu początkowym. Całość zilustrowano przykładem numerycznym, układu belkowo-kratowego, obliczonym metodą stochastycznych elementów skończonych.

A numerical formulation for stochastic sensitivity in dynamic analysis of structural systems is presented in the paper. The stochastic finite element method, basing on the perturbation methodology, is an advanced computer technique for solving engineering problems with randomness taken into account. It allows one to analyze a wide class of static and dynamic systems with geometrical and /or material random parameters. Besides the well-known direct differentiation setting, an adjoint terminal-value system is formulated in accordance with the prescribed fundamental initial-value system. Once the original initial-value equations (15) and (16) are solved for by a step-by-step forward integration scheme over time from 0 to t and the adjoint terminal-value equations (23) and (24) are integrated backward form t to 0, the sensitivity gradients (25) are summed up. Analysis of a symmetrical portal frame with two strut trusses is given. The dynamic sensitivity of the displacements at two representative points B and C with respect to the change of the cross-sectional areas of the truss and beam members is discussed. Two complete sets of stochastic solutions are obtained. One includes the expectations and cross-covariances for the nodal displacements, while the other describes the first two probabilistic moments of their sensitivity gradients.