Modelling Dynamic Responses of Clamped Non-Uniformly Prestressed Bernoulli-Euler Beams on Variable Elastic Foundations

This paper examines the dynamic response of a non-uniformly prestressed Bernoulli-Euler beam with clamped-clamped boundary conditions, resting on a variable bi-parametric foundation. The governing equation is a fourth-order partial differential equation with variable and singular coefficients. The primary objective is to derive an analytical solution for this class of dynamic problems. To achieve this, the Galerkin method is applied, utilizing a series representation of the Heaviside function to reduce the equation to a system of second-order ordinary differential equations with variable coefficients. These reduced equations are further simplified using two approaches: (i) the Laplace transform technique, combined with convolution theory, to address problems involving moving forces, and (ii) finite element analysis, integrated with the Newmark method, to solve analytically intractable moving mass problems with harmonic behaviour. We begin by solving the moving force problem using the finite element method and we validate its accuracy by comparing the results with analytical solutions. The numerical solution obtained from the finite element analysis demonstrates strong agreement with the analytical solution, confirming the method’s reliability for tackling more complex moving mass problems that lack closed-form solutions. Finally, we generate displacement response curves for both moving distributed force and moving mass models at different time instances t, providing a comprehensive representation of the system's dynamic behaviour.