The buckling analysis of Euler-Bernoulli beam resting on two-parameter elastic foundation (EBBo2PEF) has important applications in the analysis and design of foundation structures, buried gas pipeline systems and other soil-structure interaction systems under compressive loads. This study investigates the buckling analysis of EBBo2PEFs. The governing differential equation of elastic stability (GDiES) is derived in this work using first principles equilibrium method. In general, the GDiES is an inhomogeneous equation with variable parameters for non-prismatic beams under distributed transverse loadings. However, when transverse loads are absent and the beam is prismatic the GDiES becomes a fourth order ordinary differential constant parameter homogeneous equation. General solution to GDiES is obtained in this work using the classical trial exponential function method of solving equations. Two cases of end supports were considered: simply supported ends and clamped ends. Boundary conditions (BCs) were used to obtain the characteristic buckling equations whose eigenvalues were used to determine the critical buckling loads for two cases of BCs considered. It was found that the method gave exact solutions for each of the BCs. The critical elastic buckling load coefficients for dimensionless beam-foundation parameter and ranging from for simply supported EBBo2PEFs were identical with previous results that used Stodola-Vianello iteration methods and finite element method. Similarly, the critical buckling load coefficients for and are identical with previous results that used Ritz variational method.
The analysis of the least compressive load that cause buckling failures of Euler-Bernoulli beams resting on two-parameter elastic foundations (EBBo2PFs) is vital for safety. This article presents Ritz variational method (RVM) for the stability solutions of EBBo2PFs under in-plane compressive loads. The Ritz total potential energy functional, was derived for the problem as the sum of the strain energies of the thin beam, the two-parameter lumped parameter elastic foundation (LPEF) and the work potential due to the in-plane compressive load. Ritz functional was found to depend upon the buckling function w(x) and its derivatives with respect to the longitudinal coordinate. The principle of minimization of was implemented for each considered boundary condition to find the w(x) corresponding to minimum Three cases of boundary conditions investigated were: clamped at both ends, clamped at one end and free at the other, simply supported at both ends. For each case, w(x) was found in terms of unknown generalized buckling parameters ci, and buckling shape functions satisfying the boundary conditions. Thus was expressed in terms of the parameters ci. The Ritz functional was subsequently minimized with respect to the parameters yielding an algebraic eigenvalue problem. The condition for nontrivial solutions of homogeneous algebraic equations was used to find the characteristic buckling equations that were solved to find the eigenvalues. The eigenvalues were used to find the buckling loads and the critical buckling load. It was found that a one-parameter RVM solution for the EBBo2PF with both ends clamped, and with one clamped and one free end gave similar critical buckling load solutions to those presented in the literature. It was also found that an n-parameter RVM solution for the EBBo2PFs with both ends simply supported yielded exact buckling load solutions because exact sinusoidal buckling shape functions were used.