Free transverse vibration frequency analysis of Euler-Bernoulli beams on Winkler foundation (EBBoWF) is a significant part of their analysis for averting failures by resonance. Resonant failure of EBBoWF occurs when the loading frequency exciting the vibration coincides with the least natural frequency. This study aims at using the Stodola-Vianello iteration method (SVIM) for the natural transverse vibration analysis of EBBoWF. Generally, the problem is governed by a non-homogenous partial differential equation (PDE) for forced vibrations, but simplifies to a homogeneous PDE for free vibrations where excitation forces are absent. For harmonic vibrations, and harmonic displacement response u(x, t), the equations are decoupled in terms of the independent spatial and time variables, resulting in a fourth order ordinary differential equation (ODE) in the displacement modal function for u(x, t). The study’s focus is on homogenous, prismatic, isotropic thin beams leading to ODEs with constant parameters. SVIM was used to express the ODE as Stodola-Vianello iteration equations with four constants of integration, determinable via the boundary conditions. Specific application of SVIM to the EBBoWF with simple end supports used exact sinusoidal shape functions and boundary conditions to determine the integration constants. Convergence criterion at the nth iteration was used to find the eigenequation which was solved for the eigenvalues. The natural transverse vibration frequencies at the nth modes were found in terms of frequency parameters . Values of calculated for the first five modes n = 1, 2, 3, 4, 5, and for values of showed that the present SVIM gave exact results compared to other previous results. The exact solutions were obtained because exact shape functions were used in the SVIM equations resulting in satisfaction of the governing equations at the domain and the boundaries.
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.