Strain gradient differential quadrature beam finite elements

• DQ-based mapping relationship between element nodes and quadrature points is established for each MSGT-based beam model. • The two elements are independent of shape functions and introducing three kinds of strain gradient effects. • The proposed Timoshenko element is able to overcome shear-locking phenomenon. • A lot of valuable analytical solutions are systematically provided for the first time. In this paper, the superiorities of finite element method (FEM) and differential quadrature method (DQM) are blended to construct two types of beam elements corresponding to modified strain gradient -Bernoulli and Timoshenko beam models respectively. The two elements, being independent of shape functions and introducing three kinds of strain gradient effects, possess 3-DOFs (degrees of freedom) and 4-DOFs separately at each node. The Lagrange interpolation formula is employed to establish the trial functions of deflection and or rotation at Gauss-Lobatto quadrature points. To realize the inner-element compatibility condition, displacement parameters of quadrature points are converted into those of element nodes through a DQ-based mapping strategy. Total potential energy functional for each beam model is discretized in terms of nodal displacement parameters. The associated differential quadrature finite element formulations are derived by the minimum total potential energy principle. Specific expressions of element stiffness and mass matrices and nodal vector are provided. Numerical examples concerning with static bending, free vibration and buckling of macro/micro-beams are presented to demonstrate the availability of the proposed elements.