An isogeometric finite element formulation for geometrically exact Timoshenko beams with extensible directors

TL;DR

This paper introduces an advanced isogeometric finite element formulation for nonlinear Timoshenko beams with extensible directors, enabling accurate modeling of large deformations and complex material behaviors.

Contribution

It develops a novel formulation that incorporates in-plane cross-section deformation and allows direct application of 3D constitutive laws, improving accuracy over traditional methods.

Findings

Accurately models large deformations in Timoshenko beams

Reduces Poisson locking using enhanced assumed strain method

Demonstrates efficiency with hyperelastic materials in numerical tests

Abstract

An isogeometric finite element formulation for geometrically and materially nonlinear Timoshenko beams is presented, which incorporates in-plane deformation of the cross-section described by two extensible director vectors. Since those directors belong to the space ${\Bbb R}^3$, a configuration can be additively updated. The developed formulation allows direct application of nonlinear three-dimensional constitutive equations without zero stress conditions. Especially, the significance of considering correct surface loads rather than applying an equivalent load directly on the central axis is investigated. Incompatible linear in-plane strain components for the cross-section have been added to alleviate Poisson locking by using an enhanced assumed strain (EAS) method. In various numerical examples exhibiting large deformations, the accuracy and efficiency of the presented beam formulation is assessed in comparison to brick elements. We particularly use hyperelastic materials of the St. Venant-Kirchhoff and compressible Neo-Hookean types.