A non-classical Timoshenko beam element for the postbuckling analysis of microbeams based on Mindlin’s strain gradient theory

Based on Mindlin’s strain gradient elasticity theory capturing microscale effects, a new extended Timoshenko beam element is proposed to study the postbuckling behavior of microbeams. So as to develop the size-dependent finite element formulation, the higher-order tensors of energy pairs in the energy functional are vectorized and represented in the quadratic form. In comparison with the standard Timoshenko beam element, the present one needs two further nodal degrees of freedom including derivatives of lateral translation and rotation. The Hermite polynomials are also implemented as shape functions. The developed model is general so that its formulation can be used for modified couple stress, modified strain gradient and classical elasticity theories. In the numerical results, the influences of the small-scale factor, geometrical parameters and boundary conditions on the bifurcation diagrams of microbeams are examined.