The present paper states and provides an analytical solution for the problem of evaluating the settlement and load-bearing capacity of weighty soil layers of limited thickness resting upon incompressible soil bases and an excavation pit wall upon exposure of the foundation to a distributed load in the vicinity of a wall. The authors develop a method for determining the stressed state component in the reduced engineering problem based on the Ribere–Faylon trigonometric series and for accounting for the nonlinear deformation properties of soils, building on the analytical dependencies of S.S. Grigoryan and S.P. Timoshenko. In order to determine the relationship between stress and strain, the Hencky’s physical equation systems were used. They factor in the impact of average stresses σm on the shear modulus of elasticity G (σm) and volumetric modulus of elasticity K (σm). The obtained solutions make it possible to assess the deformation of soil bases and the load-bearing capacity with respect to nonlinear properties in a way that accurately corresponds to the actual performance of subsoils exposed to loading. The theoretical results are followed by numerical experiments to prove their validity.