In this work, the least squares weighted residual method (LSWRM) was used to solve the generalised elastic column buckling problem for the case of pinned ends. Mathematically, the problem solved was a boundary value problem (BVP) represented by a system of three coupled linear ordinary differential equations (ODEs) in terms of three unknown displacement functions and subject to boundary conditions. The least squares residual method used formulated the problem as a variational problem, and reduced it to an algebraic eigenvalue problem which was solved to obtain the characteristic buckling equation. The characteristic stability equation was found to be a cubic polynomial for the general asymmetric sectioned column. The buckling modes were found as coupled flexuraltorsional buckling modes. Two special cases of the problem were studied namely: doubly symmetric and singly symmetric sections. For doubly symmetric sections, the buckling loads and the buckling mode were found to be decoupled and the buckling mode could be flexural or flexural-torsional. For singly symmetric section columns, one of the bucking modes becomes decoupled while the others are coupled. The buckling equation showed the column could fail by either pure flexure or coupled flexural-torsional buckling mode. The results of the present work agree with Timoshenko's results, and other results from the technical literature.

In this work, the Kantorovich method is applied to solve the bending problem of thin rectangular plates with three simply supported edges and one fixed edge subject to uniformly distributed load over the entire plate surface. In the method, the plate bending problem is presented using variational calculus. The total potential energy functional is found in terms of a displacement function constructed using the Kantorovich procedure, as the product of an unknown function of x (f(x)) and a coordinate basis function in the y direction that satisfies the displacement end conditions at y = 0, y = b. The EulerLagrange differential equation is determined for this functional. The Galerkin method is then used to obtain the unknown function f(x). Bending moment curvature relations are used to find the bending moments and their extreme values. The results obtained agree remarkably well with literature. The effectiveness of the method is demonstrated by the marginal relative error obtained for one term displacement solutions.

In this study, the Ritz variational method was used to analyze and solve the bending problem of simply supported rectangular Kirchhoff plate subject to transverse hydrostatic load distribution over the entire plate domain. The deflection function was chosen based on double series of infinite terms as coordinate function that satisfy the geometric and force boundary conditions and unknown generalized displacement parameters. Upon substitution into the total potential energy functional for homogeneous, isotropic Kirchhoff plates, and evaluation of the integrals, the total potential energy functional was obtained in terms of the unknown generalized displacement parameters. The principle of minimization of the total potential energy was then applied to determine the unknown displacement parameters. Moment curvature relations were used to find the bending moments. It was found that the deflection functions and the bending moment functions obtained for the plate domain, and the values at the plate center were exactly identical as the solutions obtained by Timoshenko and Woinowsky-Krieger using the Navier series method.

Plates are important structural elements used to model bridge decks, retaining walls, floor slabs, spacecraft panels, aerospace structures, and ship hulls amongst. Plates have been modelled using three dimensional elasticity theory, Reissner's theory, Kirchhoff theory, Shimpi's theory, Von Karman's theory, etc. The resulting plate equations have also been solved using classical and numerical techniques.In this research, the Galerkin-Vlasov variational method was used to present a general formulation of the Kirchhoff plate problem with simply supported edges and under distributed loads. The problem was then solved to obtain the displacements, and the bending moments in a Kirchhoff plate with simply supported edges and under uniform load. Maximum values of the displacement and the bending moments were found to occur at the plate center. The Galerkin Vlasov solutions for a rectangular simply supported Kirchhoff plate carrying uniform load was found to be exactly identical with the Navier double trigonometric series solution.

In this paper, the differential equations of Mindlin plates are derived from basic principles by simultaneous satisfaction of the differential equations of equilibrium, the stress-strain laws and the strain-displacement relations for isotropic, homogenous linear elastic materials. Equilibrium method was adopted in the derivation. The Mindlin plate equation was obtained as a system of simultaneous partial differential equations in terms of three displacement variables (parameters) namely () where w(x, y, z) is the transverse displacement and are rotations of the middle surface. It was shown that when where k is the shear correction factor, the Mindlin plate equations reduce to the classical Kirchhoff plate equation which is a biharmonic equation in terms of w(x, y, z = 0).