# 3D material laws in 1D elements

## A mixed hybrid finite beam element with an interface to arbitrary three-dimensional material models

*J. Wackerfuß, F. Gruttmann*

A theory for spatial beams and an associated mixed finite element formulation are presented. The basic kinematic assumption allows for finite rotations and transverse shear strains. The relation of the Green-Lagrangean strains to the beam strains is derived. A three-field variational formulation with independent displacements, rotations, stress resultants and beam strains is developed using the kinematic and static field equations and the constitutive equations. Higher order stress resultants are set to zero within the Euler-Lagrange equations. The formulation in this paper is restricted to rectangular cross sections. The interpolation of the beam strains consists of two parts. The first part is identical to the stress resultant interpolation. The second part describes cross section warping and allows for transverse normal strains. Within this approach an interface to arbitrary three-dimensional constitutive laws can easily be realized. Due to the mixed interpolation technique full Gauss integration in length direction of the beam element can be applied to obtain a locking free element formulation. The effectiveness of the method is demonstrated by means of several numerical examples. We present results for elasticity, inelasticity, and stability problems and evaluate the load carrying capacities of spatial beam structures. The essential feature of the new element is the robustness in nonlinear applications. It allows very large load steps in comparison to other element formulations.

## Publication

Wackerfuß, J., Gruttmann, F.: A mixed hybrid finite beam element with an interface to arbitrary three-dimensional material models, Computer Methods in Applied Mechanics and Engineering, Volume 198, pp. 2053–2066, 2009

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