Efficient formulation of a geometrically nonlinear beam element

TL;DR

This paper introduces an efficient, geometrically nonlinear beam element formulation capable of handling large rotations, implemented in open-source code, and validated through numerical examples showing high accuracy and computational efficiency.

Contribution

It presents a novel finite difference discretization of a nonlinear beam element based on integrated equilibrium equations, enhancing accuracy and efficiency.

Findings

Favorable comparison with standard finite-strain beam elements

High computational efficiency due to discretization approach

Accurate results demonstrated through numerical examples

Abstract

The paper presents a two-dimensional geometrically nonlinear formulation of a beam element that can accommodate arbitrarily large rotations of cross sections. The formulation is based on the integrated form of equilibrium equations, which are combined with the kinematic equations and generalized material equations, leading to a set of three first-order differential equations. These equations are then discretized by finite differences and the boundary value problem is converted into an initial value problem using a technique inspired by the shooting method. Accuracy of the numerical approximation is conveniently increased by refining the integration scheme on the element level while the number of global degrees of freedom is kept constant, which leads to high computational efficiency. The element has been implemented into an open-source finite element code. Numerical examples show a favorable comparison with standard beam elements formulated in the finite-strain framework and with analytical solutions.