Novel weak form quadrature elements for non-classical higher order beam and plate theories

Based on Lagrange and Hermite interpolation two novel versions of weak form quadrature element are proposed for a non-classical Euler-Bernoulli beam theory. By extending these concept two new plate elements are formulated using Lagrange-Lagrange and mixed Lagrange-Hermite interpolations for a non-classical Kirchhoff plate theory. The non-classical theories are governed by sixth order partial differential equation and have deflection, slope and curvature as degrees of freedom. A novel and generalize way is proposed herein to implement these degrees of freedom in a simple and efficient manner. A new procedure to compute the modified weighting coefficient matrices for beam and plate elements is presented. The proposed elements have displacement as the only degree of freedom in the element domain and displacement, slope and curvature at the boundaries. The Gauss-Lobatto-Legender quadrature points are considered as element nodes and also used for numerical integration of the element matrices. The framework for computing the stiffness matrices at the integration points is analogous to the conventional finite element method. Numerical examples on free vibration analysis of gradient beams and plates are presented to demonstrate the efficiency and accuracy of the proposed elements.