A numerical upper bound formulation with sensibly-arranged velocity discontinuities and orthotropic material strength behaviour

Numerical limit analysis allows for fast estimates of the collapse load of structures exhibiting ideal plastic material behaviour. In numerical upper bound formulations, the description of the unknown velocity field can be extended by introducing velocity discontinuities between finite elements. Through these additional degrees of freedom, localised failure modes may be approximated more accurately and better upper bounds can be obtained. In the existing formulations, such discontinuities are typically introduced between all elements and the description is restricted to isotropic failure behaviour. In this work, a general 3D upper bound formulation is briefly proposed, allowing the consideration of both isotropic and orthotropic yield functions within finite elements as well as at velocity discontinuities. The concept of “projecting” a stress-based orthotropic yield function onto a certain discontinuity is presented, giving a traction-based yield function which allows for a consistent description of the material strength behaviour across the interface. The formulation is verified by means of two classical examples, the rigid strip footing and the block with asymmetric holes. Furthermore, based on the computation of potential orientations of plastic flow localisation, a simple concept for a sensible arrangement of velocity discontinuities is proposed. It is shown that this concept performs very well for isotropic as well as anisotropic material strength behaviour. A feature of the present work is that, velocity jumps are allowed only across the prescribed finite element interfaces determined from the sensible discontinuity arrangement. Good upper bounds similar to those in the existing works are obtained with far fewer degrees of freedom.