Upper-bound load estimates on square and rectangular footings

Limit analysis of square and rectangular rough footings is presented in this paper. All mechanisms of failure considered in the analysis consist of four regions, each characterised by plane deformation. However, the geometry of the mechanisms is three-dimensional. Both continuous deformation and multi-block patterns are considered. A common feature in all mechanisms is truncation of the blocks with conical surfaces. Standard calculations of the work dissipation rate are complex because of the elaborate three-dimensional geometry. However, a theoretical development is shown indicating that the tedious calculations of work dissipation on curved velocity discontinuity surfaces and within the deforming regions of cohesive-frictional soils can be substituted with an integral over the surface of the mechanism. While this method cannot be used easily for soils with an arbitrary distribution of properties, its application is straightforward for homogeneous soils, and it can also be used for layered soils. Calculations of bearing capacity are performed for both square and rectangular footings. It was surprising to find that the least upper-bound estimates of loads on square footings occur for mechanisms with no symmetry with respect to diagonal planes. The results are given in terms of the bearing capacity coefficients, and shape factors applicable as modifiers in the bearing capacity solution for strip footings. As expected, these factors approach unity with an increase in the footing aspect ratio L/B. The factors calculated are typically larger than earlier empirical proposals in the literature.