Spectrally formulated finite element for vibration analysis of an Euler-Bernoulli beam on Pasternak foundation

In this article, vibration analysis of an Euler-Bernoulli beam resting on a Pasternak-type foundation is studied. The governing equation is solved by using a spectral finite element model (SFEM). The solution involves calculating wave and time responses of the beam. The Fast Fourier Transform function is used for temporal discretization of the governing partial differential equation into a set of ordinary differential equations. Then, the interpolating function for an element is derived from the exact solution of governing differential equation in the frequency domain. Inverse Fourier Transform is performed to rebuild the solution in the time domain. The foremost advantages of the SFEM are enormous high accuracy, smallness of the problem size and the degrees of freedom, low computational cost and high efficiency to deal with dynamic problems and digitized data. Moreover, it is very easy to execute the inverse problems by using this method. The influences of foundation stiffness, shear layer stiffness and axial tensile (or compressive) forces on the dynamic characteristic and divergence instability of the beam are investigated. The accuracy of the present SFEM is validated by comparing its results with those of classical finite element method (FEM). The results show the ascendency of SFEM with respect to FEM in reducing elements and computational effort, concurrently increasing the numerical accuracy.