A bipotential approach for plastic limit loads of strip footings with non-associated materials

This paper is concerned with a bipotential approach for estimating the plastic collapse loads of a half-space made with a non-associated Mohr–Coulomb material and indented by a rigid punch. In geotechnics, this problem is called the bearing capacity of shallow strip footing for which the analytical solution is derived by Prandtl (1920) [46] and Hill (1950) [35] in the context of associated plasticity. However, when the plastic model is not associated, no analytical methods have yet been developed. Here we explore this issue in a rigorous mathematical framework coupling the bipotential concept and limit analysis. First, the method proposed makes use of the method of characteristics to build a statically and plastically admissible stress field that enables a lower estimate of the plastic limit loads. Next, the extended kinematic theorem of limit analysis to non-standard plasticity is applied to derive an upper quasi-bound of the collapse loads. For this aim, the internal rate of plastic dissipation is obtained thanks to the bipotential functional depending on both a trial stress field and a Prandtl-like collapse mechanism. The analytic estimates are compared to the formulae and numerical results provided in literature.