Recent research has paid great attention to the exact calculation of physical values on and near the boundaries of engineering problems. In the boundary element method (BEM), this problem demands special attention because various kinds of singularity may occur when these values are calculated. Despite extensive research, this problem has not yet been solved well. In the present paper, the elimination of various kinds of singular integrals in displacement boundary integral equations and their derivative forms is studied in detail. Firstly, a modiยฎed treatment which uses rigid body displacement solutions is introduced to calculate displacements near the boundary. Then derivative formulations of the displacement boundary integral equations are deduced, which form the basis of an investigation of the related hypersingular and nearly singular integrals. Similarly, a modiยฎed treatment, which uses the unit displacement derivative solutions together with some general numerical techniques, is proposed for the calculation of displacement derivatives on and near the boundary considered. Finally, strains and stresses are obtained from the calculated displacement derivatives by the use of the compatibility and constitutive equations. Numerical examples show that the proposed method is simple, regular and accurate in treating most singular or nearly singular integrals in the elastic BEM.