The fundamental solutions in axisymmetric elastostatic boundary element method (BEM) are more complicated compared to those in general three dimensional elastostatic BEM from which they are derived. Evaluating the integrals of these fundamental solutions over singular elements is a tedious process. Both the standard Gaussian quadrature as well as a specialized quadrature technique, capable of integrating logarithmically singular functions, need to be used in the computation. When plotted over singular elements however, the integrals corresponding to the off-diagonal components of the axisymmetric displacement fundamental solution are found to be non-singular. Exploiting this new finding, much of the extra computational effort in integrating them over singular elements is avoided by simply using the standard Gaussian quadrature technique. This modification does not cause any loss of accuracy; it yields results identical to those obtained conventionally. In addition, it also results in dramatic savings in the time spent in evaluating the boundary integrals over singular elements. The modified implementation is termed "semi-singular BEM for axisymmetric elasticity," since exactly two out of the four components of the fundamental solution are found to be non-singular. Results obtained by solving numerical problems using the semi-singular and conventional BEM are presented and compared, which validate these claims.