Nonlinear analysis of beams with rotation gradient dependent potential energy for constrained micro-rotation

In this study, the weak-form finite element model for bending of beams considering constrained micro-rotation and rotation gradient-dependent potential energy is developed for the moderate rotationcase. The governing equations for a general higher-order beam theory with the von Kármán geometric nonlinearity are derived from the principle of virtual displacements. The formulated finite element model is valid for homogeneous, orthotropic, and functionally graded classical and microstructure-dependent beams. Further, the specialization of the theory to various existing beam theories is also presented. The analytical solution for the simply supported beam for the linear case is also derived. In the numerical examples presented, the stiffening effect due to the consideration of microstructure in the micro-beam is illustrated. The parametric effect of the material length scale on the bending moment and stress is also investigated.