This work studies the dynamic characteristics of simply supported rectangular thin plates undergoing natural transverse vibrations in harmonic motion. The governing partial differential equation for the free transverse vibration of the plate was solved by the Galerkin-Vlasov variational technique. The assumption of free harmonic motions reduced the governing equation to an algebraic eigen value eigenvector problem, which was solved in the space domain to obtain the eigen frequencies and modal shape functions of the vibrating Kirchhoff plate. The eigen frequencies and modal shape functions obtained were found to be identical with the results obtained by the classical methods of Navier and Levy for the same problem.

In this work, the Fourier transform method has been applied to the determination of stresses induced by infinitely long line loads on semi-infinite homogeneous elastic soils. Airy's stress functions of the Cartesian coordinates system were used to express the governing equations of plane strain elasticity for a semi-infinite homogeneous soil as a biharmonic problem. The fourth order partial differential equation was then solved by an exponential Fourier transform technique, with respect to the space valuable x where x; and the resulting solutions made subject to the stress-boundary conditions. The stresses obtained were found to be exactly identical with solutions obtained by integrating Boussinesq's solutions for a point load which are available in the technical literature. The stresses determined in the present study were also exactly identical with the Flamant's solution for the same problem; obtained by assuming a stress function in terms of the cylindrical coordinates.

Elasticity problems are formulated using displacement methods or stress methods. In this paper a displacement formulation of axisymmetric elasticity problem is presented. The formulation uses the Boussinesq– Papkovich – Neuber potential function. The problem is then solved by assuming Boussinesq – Papkovich - Neuber potential functions in the form of Bessel functions of order zero and of the first kind. The potential functions are then made to satisfy the governing field equations and the associated boundary conditions for the particular problem of a point load at the origin of the semi-infinite linear elastic isotropic soil mass. The unknown parameters of the function are thus determined and used to find the stresses, strains and displacement fields in the loaded soil. The results obtained were identical with the results obtained by Boussinesq. http://dx.doi.org/10.4314/njt.v36i3.16