In this paper, the method of finite element analysis for the fundamental solution is proposed. The fundamental solution dealt with this paper contain the Kelvin's solution defined in an infinite region, the Boussinesq's solution in a semi-infinite region, and the generalized problem with an arbitrary oriented force, an arbitrary opening angle and general material characteristic. In the case of two-dimensional region, the displacement function must be composed by using γ^0 and log γ singularity, because of γ^<-1> stress singularity. By discretizing the virtual work equation at the radial nodal line, and by arranging the equibrium equation between the external force and the stress on the separate boundary, the element stiffness matrix with respect to the nodal unknown displacement function can be obtained. The results of the present finite element analysis of the Kelvin's, Boussinesq's and Cerruti's solutions show a good convergence to their exact solutions. And as an example for application, the Boussinesq's solution in a composite body are analysed by the present method.