# Beam elements with strong discontinuities

## Efficient finite element formulation for the analysis of localized failure in beam structures

*J. Wackerfuß*

In this project a new finite element formulation for the analysis of localized failures in beam structures is developed. Each member is considered to be a prismatic body that, in case of a localized failure, is divided by a singular surface into two elastic bulks. The fracture process on this surface is described by the cohesive crack concept using a traction–separation law. A plane cross section is assumed, which implies a link between the continuous and the structural (classical beam theory) description of the beam.

For the numerical treatment of the model a finite beam element exhibiting an internal interface is proposed, whereas the strong discontinuities are approximated by means of additional degrees of freedom that are placed at the centroid of the singular surface. Hence, each bulk can be described individually by a common beam element with the normal set of nodal degrees of freedom, whereas the singular surface is described layer-wise. The residuum vector and the symmetric stiffness matrix of the resulting element are obtained by assembling the contributions of the two bulks and the singular surface and afterwards by eliminating the additional degrees of freedom by a static condensation.

The formulation, which does not change the global degrees of freedom, is numerically very efficient, especially in the case of extensive beam structures, and can easily be implemented in any conventional finite element code. Several numerical examples are provided to demonstrate the effectiveness and the robustness of the proposed method. The equilibrium iterations show an optimal convergence rate, even if many cracks emerge simultaneously. The results do not exhibit any artificial mesh dependencies or artificial stress-locking effects.

## Numerical example: Z-shaped beam under torsion

Hybrid beam (brittle and elastic material behaviour) and external load M

## Illustration of the failure process

Illustration of the failure process by means of the deflection w (left), rotation βy (middle) and bending moment M (right) for an incrementally increased external moment M.

## Publication

Wackerfuß, J.: Efficient finite element formulation for the analysis of localized failure in beam structures, International Journal for Numerical Methods in Engineering, Volume 73, Issue 9, pp. 1217–1250, 2008