DYNAMIC RESPONSE TO MOVING CONCENTRATED LOADS OF BERNOULLI-EULER BEAMS RESTING ON BI-PARAMETRIC SUBGRADES

Abstract

The flexural motion under the actions of moving concentrated loads of uniform and non-uniform Bernoulli-Euler beams resting on Bi-Parametric Subgrades, in particular, Pasternak Subgrades is investigated in this work. In each case, the governing equation is a forth order partial differential equation. Firstly, approximate analytical solutions are obtained for both problems under investigation for all variants of the classical boundary conditions. In order to solve these problems, a technique based on Generalized Fourier Integral transform or Generalized Gerlakin’s Methods with the series representation of the Dira-Delta function, a modification of Struble’s asymptotic method and the integral transformation techniques in conjunction with the convolution theory were used. The transverse displacements for moving force and moving mass models were calculated for various time (t) and presented in plotted curves and from all illustrative examples, it is found that, the moving force solution is not an upper bound for the accurate solution of the moving mass problems. Analyses further show that an increase in the values of the structural parameters such as axial force (N), foundation modulus (K), and shear modulus (G) reduce the response amplitudes of both uniform and non-uniform Bernoulli-Euler Beams under moving loads for all variants of boundary conditions. However, higher values of shear modulus (G) are required for a more noticeable effect than the values of foundation modulus K. Also, higher values of all the parameters are required for a more noticeable effect in the case of Clamped-Clamped boundary conditions than those of other end conditions. Finally, from the approximate analytical solutions, resonance conditions reveal that the critical speeds for the uniform Bernoulli-Euler beams traversed by moving force is greater than that under the influence of a moving mass. Hence resonance is reached earlier in the moving mass problem. The same analyses are obtained for the dynamical system when the Bernoulli-Euler beams are non-uniform.