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In this paper the boundary element method is applied to linear elastic axisymmetric solids under general non-symmetrical surface loads. The formulation is based on transforming the well-known three-dimensional boundary integral equations into cylindrical co-ordinates system. Then, the behaviour of the loads and displacements in the circumferential direction is expressed in Fourier series. As a result, the original problem is solved as a sequence of one-dimensional problems each with a specified harmonic mode of the applied load. The integration through the circumferential direction is performed numerically, with the use of an appropriate transformation to improve the accuracy of the numerical integration. Consequently, the mathematical complexities are further simplified, and the work with nth-order harmonics of the applied load is greatly facilitated. The validity, accuracy and generality of the present approach are demonstrated via solving some numerical examples with different harmonic modes. Copyright © 2000 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]