In this paper a boundary element method is developed for the elastic flexural buckling analysis of composite Euler–Bernoulli beams of arbitrary variable cross-section. The composite beam consists of materials in contact. Each of these materials can surround a finite number of inclusions or openings. All of the cross-section’s materials are firmly bonded together. Since the cross-sectional properties of the beam vary along its axis, the coefficients of the governing differential equation are variable. The beam is subjected to a compressive centrally applied load together with arbitrarily axial and transverse distributed loading, while its edges are restrained by the most general linear boundary conditions. The resulting boundary value problems are solved using the analog equation method, a BEM based method. Besides the effectiveness and accuracy of the developed method, a significant advantage is that the displacements as well as the stress resultants are computed at any cross-section of the beam using the respective integral representations as mathematical formulae. Several beams are analysed to illustrate the method and demonstrate its efficiency and wherever possible its accuracy. The influence of the boundary conditions on the buckling load is demonstrated through examples with great practical interest. The flexural buckling analysis of a homogeneous beam is treated as a special case.