We present a numerical analysis and results using the traction method for optimizing domains in terms of which linear elastic problems are defined. In this paper we consider the application of the traction method, which was proposed as a solution to domain optimization problems in elliptic boundary value problems. The minimization of the mean compliance is considered. Using the Lagrange multiplier method, we obtain the shape gradient functions for these domain optimization problems from the optimality criteria. In this process we consider variations in the surface force acting on the boundary and variations in the stiffness function and the body force distributed in the domain. We obtain solutions for an infinite plate with a hole and a rectangular plate clamped at both ends.