Size-dependent static and dynamic analysis of Reddy-type micro-beams by strain gradient differential quadrature finite element method

This paper proposes a strain gradient differential quadrature finite element method to analyze the size-dependent static and dynamic behaviour of Reddy-type micro-beams. This element has 6 of freedom per node and avoids the exploitation of shape functions. A sixth-order differential quadrature-based geometric mapping scheme is constructed to realize the higher-order continuity requirements of kinematic variables. And then, it is combined with the minimum total potential energy principle to derive the motion equation of a generic element. Afterwards, several numerical examples are provided to establish the validity of the developed element. Finally, we utilize this method to analyze the static bending, free vibration, and linear buckling characteristics of uniform and stepped micro-beams. Numerical results show that the current element has prominent convergence and adaptability advantages over the classical shape function-based element. Besides, the size-dependence of vibration and critical buckling mode shapes of micro-beams is demonstrated in the graphical form for the first time.