This study is an attempt to solve the general governing differential equation for buckling of beams resting on elastic foundation for different types of beams (pin ended, fixed ended and cantilever beams) taking into account the effect of soil sub-grade reaction value (K), axial load, lateral load and relative end settlement on the deflection and bending moment of the beams for prismatic and non–prismatic beams.
Approximate solutions are also presented using the extermization of the total potential energy equation (Rayleigh-Ritz method) using approximate shape functions for the deflection which satisfy the boundary conditions of the beam. The results show that one term series approximate solution gives acceptable results in comparison with the exact solution for practical case, the accuracy of the solution increases with increasing the number of terms up to 5 terms, beyond this limit the accuracy of the solution does not change. Buckling load increases linearly with increasing (K) while the deflection and the bending moment exponentially decrease with increasing (K).
Keywords: Bending moment, Buckling, Cantilever beam, Deflection, Fixed ended beam, Non-prismatic beam, Pin ended beam, Prismatic beam, Settlement and Soil sub-grade reaction.