The boundary element method (BEM) for linear elasticity in its curent usage is based on the boundary integral equation for displacements. The stress field in the interior of the body is computed by differentiating the displacement field at the source point in the BEM formulation, via the strain field. However, at the boundary, this method gives rise to a hypersingular integral relation which becomes numerically intractable. A novel approach is presented here, where hyper-singular kernels for stresses on the boundary are made numerically tractable through the imposition of certain equilibrated displacement modes. Numerical results are also presented for benchmark problems, to illustrate the efficacy of the present approach. Solutions are compared to the commonly used boundary stress algorithm wherein the boundary stresses are computed from known boundary tractions, and derivatives of known displacements tangential to the boundary. An extension of this approach to solve linear elasticity problems using the traction boundary integral equation (TBIE) is also discussed.