A new discontinuous upper bound limit analysis formulation

Abstract

SUMMARYA new upper bound formulation of limit analysis of two-and three-dimensional solids is presented. In contrast to most discrete upper bound methods the present one is formulated in terms of stresses rather than velocities and plastic multipliers. However, by means of duality theory it is shown that the formulation does indeed result in rigorous upper bound solutions. Also, kinematically admissible discontinuities, which have previously been shown to be very efficient, are given an interpretation in terms of stresses. This allows for a much simpler implementation and, in contrast to existing formulations, extension to arbitrary yield criteria in two and three dimensions is straightforward. Finally, the capabilities of the new method are demonstrated through a number of examples.

Tectonic evolution around the Mont Terri rock laboratory, northwestern Swiss Jura: constraints from kinematic forward modelling

“…10. For this we used the kinematic approach of limit analysis theory (Salençon 2002) with the OptumG2 software (OptumG2 2014; Krabbenhøft and Damilke 2003;Krabbenhøft et al 2005;Lyamin et al 2005;Souloumiac et al 2009Souloumiac et al , 2010. The method consists in searching the velocity field, and therefore the location of the deformation, that requires the minimum tectonic force (Maillot and Leroy 2006).…”

Locking‐free discontinuous finite elements for the upper bound yield design of thick plates

“…Considering the advantage of both continuous and discontinuities-only methods, discontinuous elements have also been developed in parallel [7][8][9]. Such elements allow for velocity discontinuities across all edges shared by adjacent triangles, and consequently dissipation may occur not only inside the triangular elements but also along these discontinuities.…”

Discussion of “Stability assessment of slopes with cracks using limit analysis”

“…Finally, concerning how good the achieved upper bounds are, the following remark in the paper "Of all admissible twodimensional slope collapse mechanisms for soils considered in the literature, it is the rotational one that has been found most critical for uniform slopes (Chen 1975)" overlooks the fact that Bekaert (1995) found an upper bound of 1.0% lower for a vertical uniform slope with = 0, by considering a multiple rotation mechanism made of several log-spiral blocks. However, although it has to be pointed out that Chen's upper bounds obtained assuming a rigid rotation may no longer be the best upper bounds in light of more recent works in the literature, they are very close to the true collapse load: for instance, Krabbenhoft et al (2005) achieved lower bounds by finite element limit analyses that are on average 1.5% and in the most unfavourable case 2.5% less than the upper bounds obtained for ␤ ranging from 50°to 90°and from 10°to 40°. Conversely, it is crucial, in the discusser's view, to point out that when cracked slopes are considered, no lower bound solutions are available in the literature to bracket the true collapse values; therefore, in the case of cracked slopes it cannot be taken for granted that the upper bounds obtained for rigid rotational mechanisms are still close to the true collapse load.…”