Abstract: The dual reciprocity method(DRM) is extended to study the eigenvalue and eigenmode of thin-walled axisymmetric structures. First the displacement in the domain integral can be approximated by a set of radial basis functions and the domain integral can be converted to the boundary using DRM. Then the displacement and the traction can be expanded as Fourier series and integrate along the circumferential direction. The obtained boundary integral equation can be used for analysis of elastostatics of axisymmetric structure distributed body force and elastodynamics subject to asymmetric loading. The special case of the source point on the axis of symmetry is discussed in detail. New schemes are suggested for dealing with singular matrices for cases with and without body force respectively according to the degenerate form of the fundamental solution and the particular solution. For the thin walled structure, the sinh transformation is applied to improve the accuracy of evaluation of the nearly singular integrals. The developed project has been used to analyze elastostatics with body force and the free vibration of the thin axisymmetric structures. Numerical results indicate that the proposed method for dealing with singular matrices can effectively deal with the situation where the source point is on the axis of symmetry. and when the thickness ratio reaches 10^-3, the relative error of the results can approach 10^-3, which is better than those of FEM.