Modified couple stress flexural-flexural-torsional large deformable beam element

This study develops a novel size-dependent element for the non-linear flexural-flexural-torsional quasi-static analysis of micro-beams within the framework of the modified couple stress theory for the first time. The developed element, which satisfies C₂ continuity for transverse deflections and C₁ continuity for the torsional motion, is adopted to investigate the quasi-static motion of axially functionally graded beam-based micro-gyroscopes. Three different case studies of prismatic, tapered, and partially tapered micro-beams are investigated. Both stable and unstable branches of the equilibrium paths are obtained using a combination of the Newton-Raphson method with a novel displacement-based technique. The micro-beams are assumed to be attached to concentrated masses at the end of the beam for cantilever-based micro-gyroscopes and at their middle for simply supported and double cantilever systems. Satisfying high-order boundary conditions, convergence of the results is investigated. It is observed that the solutions, which benefit from rapid convergence, agree excellently with those available in the literature for simpler systems. Regarding the ability of finite element analysis in dealing with complex geometries and material compositions, adopting the present element for beam-based rotating micro-structures is recommended.