Abstract
A numerical analysis of the limit state for solids, with an adequately described structural geometry, is often a computationally demanding task, and there is a need for an effective method. The existing solid elements for Finite Element Limit Analysis (FELA) are either computationally expensive or require a stress cutoff of the yield surface for triaxial stress states. This paper presents an effective partially mixed lower bound tetrahedral constant stress solid element that converges rapidly and does not require modification of the yield surface. The element is based on a partially relaxed formulation of the lower bound theorem by providing strict equilibrium of the normal tractions on the element faces and a relaxed equilibrium of the shear/tangential tractions at the vertices. The performance of the element is shown in four examples applying either the von Mises yield criterion, or the Modified Mohr–Coulomb yield criterion with the possible inclusion of reinforcement. The examples show fast convergence and good performance even for relatively coarse meshes.
Highlights
Many civil engineering structures or structural parts can be analyzed using simplified models, for example, beams, columns, plates, and shells
The yield conditions of the materials are enforced in the stress nodes using either second order cone programming (SOCP) for the Von Mises yield criterion or semi-definite programming (SDP) for the Modified Mohr–Coulomb criterion
Since the Normal Traction element is a constant stress element, the yield condition only needs to be checked at one position for each element
Summary
Many civil engineering structures or structural parts can be analyzed using simplified models, for example, beams, columns, plates, and shells. For FELA methods based on the lower bound theorem, typically elements describing a linear stress field (triangles or tetrahedra) have been applied [25,27] Elements of this type were initially developed for plane strain problems [5] and later generalized to 3D [31,32]. If a complicated geometry, for example, a cylindrical pile or a reinforcement bar, needs to be meshed, it already requires a high resolution of the mesh Modeling these cases with a linear stress element which is roughly four times more computationally expensive might not be advantageous.
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