Understanding the rigid-block equilibrium method by way of mathematical programming

This paper discusses and extends some main features of the rigid-block equilibrium (RBE) method. RBE is a numerical approach that frames the equilibrium problem of rigid-block assemblies as an optimisation problem to compute possible internal and equilibrated singular stress states. The contact between blocks is considered having a finite friction capacity and the unilateral behaviour is modelled through a penalty formulation. In particular, the penalty formulation widens the standard admissible solution space of compressive-only forces by allowing for tensile forces appearing on potentially unstable regions. The RBE objective function minimises the interface forces whereas the constraints are linear functions enforcing the static equilibrium of the whole assembly. In this paper, along with the original quadratic objective function, the authors propose a linear function to illustrate and explore the role played by both the nodal forces and the interface resultants. Moreover, the authors show how RBE can be used to explore different admissible internal stress states – for example, due to increasing, static, horizontal actions.