A fundamental class of stress elements in lower bound limit analysis

Abstract

There is a major restriction in the formulation of rigorous lower bound limit analysis by means of the finite-element method. Once the stress field has been discretized, the yield criterion and the equilibrium conditions must be applied at a finite number of points so that they are satisfied everywhere throughout the discretized structure. Until now, only the linear stress elements fulfil this requirement for several types of loads and structural conditions. However, there are also standard types of problems, like the one of plates under uniformly distributed loads, where the implementation of the lower bound theorem is still not possible. In this paper, it is proven for the first time that there is a class of stress interpolation elements which fulfils all the requirements of the lower bound theorem. Moreover, there is no upper restriction of the polynomial interpolation order. The efficiency is examined through plane strain, plane stress and Kirchhoff plate examples. The generalization to three-dimensional and other structural conditions is also straightforward. Thus, this interpolation scheme which is based on the Bernstein polynomials is expected to play a fundamental role in future developments and applications.

A class of strain‐displacement elements in upper bound limit analysis

“…Filled squares denote the strain‐displacement points with their area coordinates. Their position is similar to the one of the equilibrium points of the stress elements 54 …”

“…The irregular slab of Figure 17A was presented by Olsen 56 . So far, the best lower and upper bounds for Johansen's criterion, are given by Makrodimopoulos 54 and Gilbert et al 34 respectively, resulting to $$$$ 0.13548\le p/{m}_{\mathrm{p}}\le 0.13554. $$$$ In upper bound finite element limit analysis, Makrodimopoulos 57 obtained a rigorous upper bound of 0.1368 using an unstructured mesh of uniform size with 130,352 constant curvature strain elements.…”

“…Filled squares denote the strain-displacement points with their area coordinates. Their position is similar to the one of the equilibrium points of the stress elements 54 where 𝝓, a ∈ ℜ NPFR and a klm are functions of 𝜺 * i , u * i and the nodal coordinates and arise after assembling the coefficient of each monomial of (20). Assume now that we apply (23) at NPFR different points 𝝃 (1) , … , 𝝃 (NPFR) .…”

“…The irregular slab of Figure 17A was presented by Olsen. 56 So far, the best lower and upper bounds for Johansen's criterion, are given by Makrodimopoulos 54 and Gilbert et al 34 respectively, resulting to 0.13548 ≤ p∕m p ≤ 0.13554.…”

“…The results and the CPU times are shown in Table 8 and also in Figure 18 in order to conclude on the efficiency of various interpolation orders. In addition to the upper bound analyses, a lower bound one based on the quadratic moment elements of Makrodimopoulos 54 was conducted for the von Mises criterion for the same meshes with 25,152 and 100,608 elements resulting to limit loads of $$$ 0.1360{m}_{\mathrm{p}} $$$ and $$$ 0.13617{m}_{\mathrm{p}} $$$ respectively. Therefore, for the von Mises criterion, the exact limit load may be bracketed as $$$$ 0.13617\le p/{m}_{\mathrm{p}}\le 0.1368.$$…”