Some closed-form solutions in random vibration of Bresse-Timoshenko beams

Abstract Random vibration of simply supported uniform Bresse-Timoshenko beams is considered under ‘rain-on-the-roof’ (stationary space- and time -wise ideal white noise) excitation. An approximate differential equation is used with both shear distortion and rotary inertia included, but with the term which covers the simultaneous action of these effects omitted. A closed-form solution is derived for the displacement and velocity space-time correlation function of the Bresse-Timoshenko beam with transverse damping, generalizing the corresponding result by Eringen for the classical Bernoulli-Euler beam. Closed-form solutions are also derived for beams with structural or Voigt damping mechanisms. The mean-square value of the stress diverges for both the classical Bernoulli-Euler and Bresse-Timoshenko beams with transverse damping, but converges for the beam possessing structural damping. The main finding of this study is identity of the space-time correlation functions of displacement according to the refined Bresse-Timoshenko and classical Bernoulli-Euler theories, when the joint action of rotary inertia and shear deformation is neglected for the beam under the ‘rain-on-the-roof’ excitation. This remarkable coincidence takes place for beams possessing (a) transverse viscous damping, (b) Voigt damping, and (c) combined rotary and transverse viscous damping.