Static and Dynamic Micropolar Linear Elastic Beam Finite Element Formulation, Implementation, and Analysis

Abstract

Starting with static and dynamic micropolar linear plane stress elasticity, and applying Timoshenko beam kinematics with axial stretch, a mixed micropolar small-strain beam finite element (FE) formulation results. The mixed formulation is shown to be convergent upon mesh refinement under static and dynamic loading. The acceleration form of the Newmark family of time integration methods is applied to integrate the coupled hyperbolic linear governing equations. Instantaneous axial and transverse step forces are applied and released to analyze the free longitudinal and transverse vibrations with the mixed formulation FE implementation. The transverse displacement and rotational degrees of freedom (DOF) are coupled, but the axial displacement is decoupled from the rotational DOF because the first area moment of inertia Q is zero. Applied sinusoidal axial and transverse forces lead to axial and transverse displacement and rotational wave patterns that are a combination of low and high frequency waves. The effect of length scale ℓ on elastic couple modulus γℓ and spin inertia j is demonstrated, which shows a transverse and rotational stiffening through γℓ upon increasing ℓ, yet a decreasing frequency as j also increases with ℓ.