Dynamic behaviour under moving concentrated masses of simply supported rectangular plates resting on variable Winkler elastic foundation

Abstract

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.

Dynamic response to moving masses of rectangular plates with general boundary conditions and resting on variable winkler foundation

“…Consider a rectangular plate carrying an arbitrary number (say N) of concentrated masses M i moving with constant velocities c i , i = 1, 2, 3,…,N along a straight line parallel to the x-axis issuing from point y = s on the y-axis. The equation governing the dynamic transverse displacement W(x,y,t) of the rectangular plate when it is resting on a variable Winkler foundation and traversed by several moving concentrated masses is the fourth order partial differential equation given by [20] Latin American Journal of Solids and Structures 10(2013) 301 -322 (1) where D is the bending rigidity of the plate, m is mass per unit area of the plate, x is the position coordinate in x -direction, y is position co-ordinate in y -direction, t is the time, is the rotatory inertia correction factor, is the two-dimensional Laplacian operator, F 0 is the foundation modulus and (.) is the Dirac-Delta function.…”

“…Expressing the Dirac-Delta function in the Fourier cosine series as (14) and (15) equation (13) In what follows, φ n (x,y) are assumed to be the products of the beam functions ψ ni (x) and ψ nj (y) [20]. That is (17) These beam functions can be defined respectively, as (18) and (19) where A ni , A nj , B ni , B nj , C ni and C nj are constants determined by the boundary conditions.…”

“…Recently, Oni and Awodola [19] considered the dynamic response under a moving load of an elastically supported non-prismatic Bernoulli-Euler beam on variable elastic foundation. More recently, Oni and Awodola [20] investigated the dynamic behaviour under moving concentrated masses of simply supported rectangular plates resting on variable Winkler elastic foundation.…”

A new FEM procedure for transverse and longitudinal vibration analysis of thin rectangular plates subjected to a variable velocity moving load along an arbitrary trajectory

“…The separation phenomenon which can occur at some values of travelling velocities and system parameters during their travel can cause fundamental change in the dynamic behavior of structures (Lee, 1998). As for different engineering applications of moving mass structure problems, (Meirovitch, 1967;Yoshida and Weaver, 1971;Bathe, 1982;Reddy, 1984;Yang, 1986;Taheri, 1987;Bachmann, 1995;Kadivar and Mohebpour, 1998;Fryba, 1999;Wilson, 2002;Clough and Penzien, 2003;Szilard, 2004;Mohebpour et al, 2011;Oni and Awodola, 2011;Awodola, 2014) can be considered important and valuable references for analytical and FEM solutions and analysis of dynamic systems.…”

Inspection of a Rectangular Plate Dynamics Under a Moving Mass With Varying Velocity Utilizing BCOPs

“…The influence of the whole components of the moving mass out-of-plane translational acceleration terms have been evaluated by Nikkhoo and Rofooei (2012). Oni and Awodola (2011) and Awodola and Oni (2013) investigated the problem of a thin plate rested on a variable Winkler elastic foundation traversed by a moving force or a moving mass with simple and some other boundary conditions of the structure, respectively. They employed separation of variables method in joint with the modified method of Struble and integral transformations to solve the mathematical governing equation.…”