# Hu-Washizu beam element

## A nonlinear Hu–Washizu variational formulation and related finite-element implementation for spatial beams with arbitrary moderate thick cross-sections

*J. Wackerfuß, F. Gruttmann*

A nonlinear three-dimensional finite beam element based on a Hu–Washizu variational formulation is presented. In addition to the standard beam strains, based on kinematic assumptions, further deformation modes are introduced. These additional modes allow for (a) the consideration of a complete three-dimensional stress field, providing an interface for arbitrary three-dimensional material models, and (b) the consideration of cross-section warping, whose shape might shift during the elastic or inelastic deformation.

Beside the fact that the resulting finite element formulation is locking free (full Gauss integration in length direction) and remarkable robustness (even for very large load steps), these additional degrees of freedom do not increase the total number of global unknowns of a beam structure. Each element node exhibits the common 3 translational and 3 rotational degrees of freedom. The additional degrees of freedom are eliminated on element level via static condensation. As a consequence, e.g. a bi-moment can not be applied at a free end. This restricts the applicability of the formulation to a class of problems where the influence of the bi-moment is negligible.

It is shown that global acting polynomial ansatz functions are not suitable to describe warping of cross-sections with an arbitrary shape. For this reason a new concept based on local ansatz functions is presented. The general criteria to design the warping ansatz functions are discussed in detail. Several examples with moderate thick cross-sections are investigated.

## Numerical example: Z-shaped beam under torsion

## Publication

Wackerfuß, J., Gruttmann, F.: A nonlinear Hu–Washizu variational formulation and related finite-element implementation for spatial beams with arbitrary moderate thick cross-sections. Computer Methods in Applied Mechanics and Engineering, Volume 200, pp. 1671–1690, 2011

This project has been realized in cooperation with F. Gruttmann (Technische Universität Darmstadt, Fachgebiet Festkörpermechanik).