Spectral finite elements for vibrating rods and beams with random field properties

The classical stochastic Helmholtz equation grasps, through the random field of the refraction index, the spatial variability in the mass density but not the variability in elastic moduli or geometric parameters. In contradistinction to this restriction, the present analysis accounts for the spatial randomness of mass density as well as those of elastic properties and cross-sectional geometric properties of rods undergoing longitudinal vibrations and of Timoshenko beams in flexural vibrations. All the material variabilities are described here by random Fourier series with a typical (average) characteristic size of inhomogeneity d, which is either smaller, comparable to, or larger than the wavelength. The third length scale entering the problem, but kept constant, is the rod or beam length. We investigate the relative effects of random noises in all the material parameters on the spectral stiffness matrices associated with rods and beams for a very wide range of frequencies.