This paper studies the eigenfunction expansion method to solve the twodimensional (2D) elasticity problems based on the stress formulation. The fundamental system of partial differential equations of the 2D problems is rewritten as an upper triangular differential system based on the known results, and then the associated upper triangular operator matrix matrix is obtained. By further research, the two simpler complete orthogonal systems of eigenfunctions in some space are obtained, which belong to the two block operators arising in the operator matrix. Then, a more simple and convenient general solution to the 2D problem is given by the eigenfunction expansion method. Furthermore, the boundary conditions for the 2D problem, which can be solved by this method, are indicated. Finally, the validity of the obtained results is verified by a specific example.