DYNAMIC RESPONSE TO VARIABLE-MAGNITUDE MOVING DISTRIBUTED MASSES OF BERNOULLI-EULER BEAM RESTING ON BI-PARAMETRIC ELASTIC FOUNDATION

Abstract

The problem of dynamic response to variable-magnitude moving distributed masses of Bernoulli-=Euler beam resting on bi-parametric elastic foundation is investigated in this work. The governing equation is a fourth order partial differential equation. In order to solve this problem, the Galerkin method is used to reduce the governing fourth order partial differential equation to a sequence of coupled second order ordinary differential equation with variable co-efficients. For the solutions of these equations, two cases are considered; (1) the moving force case – when the inertial term 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 distributed force of the Bernoulli-Euler beam resting on a bi-parametric [Pasternak] 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 with each other 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 for both the moving force and moving mass. Finally, for fixed N, K0 and G0, the transverse deflections of the Bernoulli-Euler beam under the actions of moving masses are higher than those when only the force effects of the moving load are 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