Dynamic Response to Variable-magnitude Moving Distributed Masses of Bernoulli-Euler Beam Resting on Bi-parametric Elastic Foundation

This work investigates the problem of dynamic response to variable-magnitude moving distributed masses of Bernoulli-Euler beam resting on bi-parametric elastic foundation. The governing equation is a fourth order partial differential equation with variable and singular co-efficients. This equation is reduced to a set of coupled second order ordinary differential equation by the method of Garlerkin. For the solutions of these equations, two cases are considered; (1) the moving force case – when the inertia is neglected and (2) the moving mass case ­– when the inertia term is retained. To solve the moving force problem, the Laplace transformation and convolution theory are used to obtain the transverse-displacement response to a moving variable-magnitude distributed force of the Bernoulli-Euler beam resting on a bi-parametric elastic foundation. For the solution of the moving mass problem, the celebrated struble’s technique could not simplify the coupled second order ordinary differential equation with singular and variable co-efficient because of the variability of the load magnitude; hence use is made of a numerical technique, precisely the Runge-Kutta of fourth order is used to solve the moving mass problem of the response to variable-magnitude moving distributed masses of Bernoulli-Euler beam resting on Pasternak elastic foundation.The analytical and the numerical solutions of the moving force problem are compared and shown to compare favourably to validate the accuracy of the Runge-Kutta scheme in solving this kind of dynamical problem. The results show that response amplitude of the Bernoulli-Euler beam under variable-magnitude moving load decrease as the axial force N increases for all variants of classical boundary conditions considered. For fixed value of N, the displacements of the beam resting on bi-parametric elastic foundation decrease as the foundation modulus K0 increases. Furthermore, as the shear modulus G0 increases, the transverse deflections of the beam decrease. The deflection of moving mass is greater than that of moving force for all the variants of boundary conditions considered, therefore, the moving force solution is not a safe approximation to the moving mass problem. Hence safety is not guaranteed for a design based on the moving force solution for the beam under variable-magnitude moving distributed masses and resting on bi-parametric elastic foundation.