Some exact solutions for the bending of beams with spatially stochastic stiffness

Abstract

To the best of the knowledge of the authors, there are no exact solutions available for the bending of beams with spatially stochastic stiffness. Investigators are therefore utilizing various approximate techniques. In the present study, a new method is developed to obtain exact solutions for first and second moments of displacements for statically determinate beams that have spatially random stiffness. The method is based on the full probabilistic characterization of the random stiffness so that the solutions are valid for any value of the coefficient of variation of the stiffness. The deterministic governing equations and boundary conditions derived for both first and second moments allow, apparently for the first time in the literature, the exact solutions for the mean and covariance functions of the displacements to be determined. Two governing equations are uncoupled from each other and can be solved separately. Several exact solutions for the mean and covariance functions are obtained to illustrate the application of the method. It is hoped that the exact solutions determined will serve as benchmark solutions to enable the researchers to check the accuracy of various approximate analytical and numerical techniques on the test solutions presented in this study.