The flexural motions of elastically supported rectangular plates carrying moving masses and resting on variable Winkler elastic foundations is investigated in this work In order to solve the fourth order partial differential equation governing the problem, a technique based on separation of variables is used to reduce the governing fourth order partial differential equations with variable and singular coefficients to a sequence of second order ordinary differential equations. These equations are then solved using a modification of the Struble's technique and method of integral transformations. Numerical results are then presented in plotted curves. The results show that response amplitudes of the plate decrease as the value of the rotatory inertia correction factor Ro increases and for fixed value of Ro, the displacements of the elastically supported rectangular plates resting on variable elastic foundations decrease as the foundation modulus Fo increases. Also, for fixed Ro and Fo, the transverse deflections of the rectangular plates 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. Furthermore, the results show that the critical speed for the moving mass problem is reached prior to that of the moving force for the elastically supported rectangular plates on Winkler elastic foundation with stiffness variation.

The dynamic response to moving masses of rectangular plates with general classical boundary conditions and resting on variable Winkler elastic foundation is investigated in this work. The governing fourth order partial differential equation is solved using a technique based on separation of variables, the modified method of Struble and the integral transformations. Numerical results in plotted curves are then presented. The results show that as the value of the rotatory inertia correction factor Ro increases, the response amplitudes of the plate decrease and that, for fixed value of Ro, the displacements of the plate decrease as the foundation modulus Fo increases for the variants of the classical boundary conditions considered. The results also show that for fixed Ro and Fo, the transverse deflections of the rectangular plates under the actions of moving masses are higher than those when only the force effects of the moving load are considered. For the rectangular plate, for the same natural frequency, the critical speed for moving mass problem is smaller than that of the moving force problem for all variants of classical boundary conditions, that is, resonance is reached earlier in moving mass problem than in moving force problem. When Fo and Ro increase, the critical speed increases, hence, risk is reduced.

The response of simply supported rectangular plates carrying moving masses and resting on variable Winkler elastic foundations is investigated in this work. The governing equation of the problem is a fourth order partial differential equation. In order to solve this problem, a technique based on separation of variables is used to reduce the governing fourth order partial differential equations with variable and singular coefficients to a sequence of second order ordinary differential equations. For the solutions of these equations, a modification of the Struble's technique and method of integral transformations are employed. Numerical results in plotted curves are then presented. The results show that response amplitudes of the plate decrease as the value of the rotatory inertia correction factor R0 increases. Furthermore, for fixed value of R0, the displacements of the simply supported rectangular plates resting on variable elastic foundations decrease as the foundation modulus F0 increases. The results further show that, for fixed R0 and F0, the transverse deflections of the rectangular plates 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. Also, the analyses show that the response amplitudes of both moving force and moving mass problems decrease both with increasing Foundation modulus and with increasing rotatory inertia correction factor. The results again show that the critical speed for the moving mass problem is reached prior to that of the moving force for the simply supported rectangular plates on variable Winkler elastic foundation.