A method of numerically approximating the solutions of plane-stress or plane-strain elasticity problems with boundary conditions consisting of concentrated forces or distributed loads is presented herein. The effect of each concentrated force (commonly termed a point load) that acts on the boundary is represented by a Flamant solution. Usually, the combined effect of these Flamant solutions indicates the presence of distributed loadings or ‘residual stresses’ on some portions of the boundary that are not consistent with the actual boundary conditions. The negatives of these ‘residual stresses’ are used as stress boundary conditions in a singular integral method of numerical analysis that is applicable to plane elasticity problems involving distributed loadings on the boundaries. Since the method presented herein involves only stress boundary conditions, the solutions are valid for both plane stress and plane strain.