A study of the direct method for numerical solution of the inverse boundary value problem using the boundary element method is presented. This method proposes a non-iterative and unified treatment of conventional boundary value problem. In order for a solution to be identified the inverse problem is reformulated in terms of a variational problem, which is then recast into primary and adjoint boundary value problems of the elasto-statics in its conventional form. An inverse boundary value problem for two-dimensional elasto-statics in the domain enclosed by a piecewise smooth curve is considered. While the displacement and the traction data are prescribed on a respective part of the boundary and the second part of the boundary on which no boundary data are given, the unknown values along the whole bou ndary will be find.