Beam Element with a 3D Response for Shear Effects

Abstract

This paper presents a beam element for which the cross section remains neither plane nor orthogonal to the beam axis. The cross section with arbitrary geometry is modeled by two-dimensional (2D) finite elements. The element kinematics is defined by the displacement shapes of the cross sections and the axial functions of their corresponding averaged movements. The deformed displacement shapes are obtained by minimizing the potential energy of a beam slice submitted to the compatibility constraints of the kinematics framework. The recursive procedure that yields the higher-order deformed displacement shapes is initiated with the rigid-body components of the displacement field. The mechanical model is validated through examples of linear-elastic materials with isotropic and nonisotropic cross sections. Higher-order beam elements yield three-dimensional (3D) stresses and strains that correspond to elasticity solutions and 3D solid element models, for concentrated and distributed loads. The beam elements do not require shear coefficients. The first-order formulation is extended to nonlinear materials.