In this dissertation a method based on the use of exponential basis functions method (EBFs) has been developed for solving governing differential equations of thick plates in 3D space. The reason of using exponential basis functions is that the governing equations fall in the category of partial differential equations (PDEs) with constant coefficients. In this method, the solution to the homogeneous part of differential equations is represented by a linear combination of exponential basis functions as the fundamental solutions. Constant coefficients of the so defined series are evaluated by a discrete transformation capable of satisfying the boundary conditions with high accuracy and with no limitation on the type of boundary conditions or the shape of boundary edges. The parameters in the exponents of the EBFs are found by requiring that the bases satisfy homogeneous traction boundary conditions at top and bottom of the plate. This gives a characteristic equation in terms of a defined dimensionless variable as a function of the original parameters. For a given value for such a variable, obtained from the characteristic equation, a set of EBFs is constructible by choosing one of the original parameters in terms of another. By doing so, a wide range of problems with different boundary conditions and frequencies may be solved provided that the side boundary conditions are also satisfied. For the solution of plates with loading/traction at top and bottom surfaces, a‌ particular solution has been devised to incorporate the effects. The particular solution has much in common with the homogenous solution but of course with different parameters in the exponents of the EBFs. This is because that the bases must satisfy the homogenous PDEs (and obviously not the homogeneous traction conditions). However, before finding such a particular solution the tractions must be expressed in a set of suitable series. This is performed by two different approaches i.e. the use of Fourier series and a series composed of a set of exponential bases. The coefficients in the latter series are found by a discrete transformation. Comparison between the two methods shows excellent accuracy of the second method in majority of problems except for loadings in the form of step functions. Several benchmark examples have been solved for plates with various thicknesses excited by different frequencies. It has been concluded that the method is capable of solving time harmonic plate problems with a wide range of frequencies.