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

Abstract

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.Keywords: numerical upper bound formulations, localised failure modes, traction-based yield function, sensible arrangement of velocity discontinuities, orthotropic material strength behaviour

An algorithm for adaptive introduction and arrangement of velocity discontinuities within 3D finite-element-based upper bound limit analysis approaches

“…However, as shown by the authors in [44], if discontinuities are only introduced in regions with large plastic flow and arranged/oriented in view of potential directions of plastic flow localisation, very good upper bounds can be obtained (even for orthotropic material strength behaviour) using relatively coarse finite element meshes. In this way, only as many as useful discontinuities are implemented and, due to a sensibly arrangement, they are on average much better utilised.…”

“…Recently, an alternative approach has been presented by the authors [44], where finite-element-based upper bound formulations with sensibly-arranged velocity discontinuities have been proven reliable and efficient, without the need of intensive mesh refinement in failure regions. Additionally, although the implementation of orthotropic yield functions in numerical upper bound formulations has been presented in several publications [45,46,47,48,49,50], to the authors' knowledge, the combination of orthotropic yield functions and velocity discontinuities was presented by the authors in [44] for the first time. In order to apply this approach to more general problems, in this paper, a comprehensive introduction on the numerical implementations as well as an adaptive introduction and adjustment strategy on the sensible arrangement of velocity discontinuities will be presented.…”

Bending strength predictions of cross-laminated timber plates subjected to concentrated loading using 3D finite-element-based limit analysis approaches

“…In this work, as is introduced in [44,45], velocity jumps ∆ u ∈ R 3 are allowed only across prescribed discontinuities Γ dis , consisting of a surface Γ + dis with the related velocity field u+ dis and a surface Γ − dis with the related velocity field u− dis . Such velocity jumps ∆ u = ( u+ dis − u− dis ) represent additional degrees of freedom for the plastic flow, locally at Γ dis .…”

“…In the following, only this dual optimisation problem is discretised and solved, but complete expressions of discretised primal and dual upper bound formulations can be found in [44,45]. For the discretisation linear strain tetrahedron elements are used with 10 velocity evaluation nodes (4 vertice and 6 middle nodes) and 4 strain-rate evaluation nodes (at the 4 vertices).…”

Strain localization analysis of Hill’s orthotropic elastoplasticity: analytical results and numerical verification

“…However, most of the aforementioned work considered only strain localization in isotropic materials and very rare references deal with orthotropic ones. To the authors' best knowledge, it is only in [27] that strain localization in orthotropic plasticity models is considered to determine the upper bound load capacity of such materials. Nevertheless, the closed-form results for the localization angle are not available.…”