The buckling problem of various beams of variable cross sections has been fully considered by many authors, and the eigenfunctions and eigenvalues of the buckling problem of those beams are available in books and papers on this subject. Firstly, this paper shows that the deflection of these beams under the combined action of lateral load and axial tensile force can be expanded in a series of buckling eigenfunctions and that the coefficients in such an expansion can be simply expressed in terms of the known eigenfunctions, eigenvalues, the axial tensile force and the lateral load. Furthermore making use of the eigenfunction expansion, an approximation method is given to determine the axial tensile force acted on the supports of a beam of fixed span.