Geometric nonlinear vibration of clamped Mindlin plates by analytically integrated trapezoidal p-element

A trapezoidal hierarchical finite element for nonlinear free and forced vibration analyses of skew and trapezoidal Mindlin plates is introduced. Legendre orthogonal polynomials are used as enriching shape functions to avoid the shear-locking problem. The element matrices are analytically integrated in closed form. With the enriching degrees of freedom, the accuracy of the computed results and the computational efficiency are greatly improved. The arc-length iterative method is used to solve the nonlinear motion equation. The results of linear and nonlinear vibration show that the convergence of the proposed element is very fast with respect to the number of degrees of freedom. Clamped condition on the boundaries and weakly nonlinearity without bifurcation will be considered. The skew angles of plates hardly influence the shape of backbone curves, but the decreasing of chord ratios increases the hardening spring effect of the backbone curves.