Weak form quadrature elements for non-classical Kirchhoff plate theory

Abstract

In this paper, two novel versions of weak form quadrature elements are developed for bending, free vibration and stability analysis of non-classical first strain gradient Kirchhoff plate theory. In the first version, Lagrange interpolations are assumed in orthogonal directions to approximate the field variables and in the second, mixed interpolations are used, with Lagrange in one direction and Hermite in another. The elements are formulated with the aid of variational principles and the Gauss–Lobatto–Legendre quadrature points are considered as element nodes and also used for numerical integration of element matrices. The multi-degrees degrees of freedom associated with the non-classical plate theories are accounted into the formulation in a simplified way, which facilitates the application of classical and non-classical boundary conditions. The procedure to compute the weighting coefficients by incorporating the classical and non-classical degrees of freedom are explained for both element versions. Detailed mathematical formulation of the elements and their numerical implementation is presented. The efficiency of the proposed elements is demonstrated by solving numerical examples and comparing the results with the exact solutions available in the literature. Further, new results are presented for strain gradient plates with different support conditions, which can serve as reference for other researchers in this field.