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.
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.
Thin plate bending analysis is an important research subject due to the extensive use of plates in the different fields of engineering and the need for accurate solutions. This article uses the Ritz variational method and a superposition of trigonometric and polynomial basis functions to solve the Kirchhoff-Love plate bending problems (KLPBPs). The unknown displacement function in the Ritz variational functional (RVF) to be minimized is sought as linear combinations of basis functions Fm(x) and Gn(y) that are found by superposing sine series and third degree polynomial functions with the polynomial parameters determined such that all boundary conditions of deformation and force are satisfied. The displacement is thus expressed in terms of unknown displacement parameters Amn which are found upon minimization of RVF with respect to Amn. The minimization process gave a matrix stiffness equation in Amn with the stiffness matrix and force matrix found from Fm(x) and Gn(y) and their derivatives. The algebraic equation is solved, and the deflection and bending moments obtained. The problems considered were clamped (CCCC) plates under uniform and hydrostatic distribution of loads and plates with opposite edges clamped, the rest simply supported (CSCS) under uniformly distributed loading. Comparison of the solutions by Generalized Integral transform method, Levy-Nadai series method, and symplectic eigenfunction superposition confirms that the present results are accurate.
Despite the importance of plates in structural analysis the flexural analysis of plates under parabolic load has not been extensively studied. This paper presents single finite sine transform method for exact bending solutions of simply supported Kirchhoff plate under parabolic load. The governing equation of equilibrium is a fourth order non-homogeneous differential equation in terms of the deflection The considered thin plate problem has Dirichlet boundary conditions at all the edges. This recommends the use of the finite sine integral transform method whose sinusoidal kernel function satisfies the boundary conditions. The sinusoidal function of x used for the sine transform kernel in this paper satisfies the Dirichlet boundary conditions along edges. The transformation simplifies the problem from a partial differential equation (PDE) to an ordinary differential equation (ODE) in the transformed space. The general solution, obtained using methods for solving ODEs is found in terms of unknown constants of integration which are found by using the finite sine transform of Dirichlet boundary conditions along the and edges. The solution in the physical domain space variables is then found by inversion as a rapidly convergent single series with infinite terms. A one term truncation of the single infinite series yields center deflection solution that is only 2% greater than the exact solution. A three term truncation of the infinite series for gave exact center deflections. Bending moments are found using the bending moment deflection relations as convergent single series with infinite terms.
This paper attempts to obtain bending solutions to plates under uniformly distributed and hydrostatic load distributions using Ritz variational methods and basis functions that are found by superposing trigonometric series and third degree polynomials. Two cases of boundary conditions were considered. In one case, three edges were simply supported and the fourth edge was clamped (SSCS thin plate). In the second case, the adjacent edges were clamped and the other edges were simply supported (SCCS thin plate). This work presents first principles, rigorous derivation of the governing Ritz variational functional and the displacement basis functions for the boundary conditions investigated. The solution is presented in analytical form. The obtained results are compared with previous results obtained using Levy series and Ritz methods and found to be in close agreement . The disadvantage of the method is the associated computational rigour, but the benefit is the accuracy of the results. Comparisons of the present results for center deflections and center bending moments with results in the literature show that there is negligible difference. Double series expressions were found for deflections and bending moments for the plate bending problems solved. Evaluation of the double series expressions at the plate center gave center deflection results that differed from the exact solutions by for to for for uniformly loaded thin plates with three simply supported edges and one clamped edge (SSSC). The differences in the center bending moments Mxx were found to vary from for to for In general, the present results yielded reasonably accurate solutions for the plate bending problems studied.
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.