Random vibration responses and reliability analyses of thin plates with geometric nonlinearity via direct probability integral method

The stationary/nonstationary random vibration responses and dynamic reliability analyses of thin plate structures coupling geometrical nonlinearity and multisource randomness in both structural parameters and external excitations are the important issues to be attacked. In this study, the novel direct probability integral method (DPIM) is proposed to tackle them. Firstly, the differential equations of thin plate vibration with large deflection are revisited based on von Karman nonlinear theory. Then, the non-intrusive DPIM decouples physical evolution and probability density propagation of structure, which is utilized to calculate nonlinear stochastic dynamic responses and reliability of thin plate in a unified way. Moreover, to determine the resultant large-amplitude nonlinear deformation of thin plate under random concentrated excitation and distributed excitation, an important criterion is devised, which can judge the applicability of geometrically nonlinear theory. Several numerical examples demonstrate the high accuracy and efficiency of proposed approach by comparing the results with those of Monte Carlo simulation. It is indicated that, the changes of structural parameters and random excitation cause the transition of probability density functions of nonlinear deflections from unimodal to bimodal distributions. Stochastic P-bifurcation occurs in nonlinear random vibration analysis of thin plate, which implies that the geometric nonlinearity should be rationally considered in stochastic dynamic analysis of structure. Finally, the remarkable effects of geometric nonlinearity, thickness, random parameter variability and boundary conditions on stochastic responses and reliabilities of thin plates are revealed.