Derivation of simplified model governing behavior of Mindlin plate with elastic support traversed by partially distributed moving load

Article history: Received 7 July 2018 Received in revised form 10 December 2018 Accepted 5 January 2019 The practical importance of dynamic response of elements of structures such as plates when load moves on them cannot be overemphasized in both engineering and applied sciences. The dynamic behavior of an elastic plate resting on a subgrade and traversed by uniform partially distributed moving load is considered in this paper and its simplified governing equations derived. The elastic plate is Mindlin rectangular plate. In particular, the model governing such moving load problem is simplified analytically. The simplified governing model derived is easier to handle. Numerical methods can easily be applied to this simplified model and a lot of computational time is saved.


Introduction
*The investigation of moving load issue is by and large of reasonable significance in Engineering and Applied Sciences. Such reviews are significant while considering the unwavering quality, wellbeing and execution of present day structures over which loads like vehicles and train move (Gbadeyan and Agarana, 2014;Mindlin, 1951). The arrangement of such moving load issue under thought, requests the displaying of the mechanical conduct of the soil as flexible subgrade, and the type of collaboration between the plate and the soil. In this sort of framework, it is important to couple practical models of the foundation with investigation of the structure. A few foundation models have been accounted for in the writing and examinations on the static deflection; the dynamic reaction and the dynamic stability of plates on elastic foundation have been completed. Many researchers use the Winkler model for soil structure interaction in the static and dynamic analyses of plate resting on elastic foundation where the vertical surface displacement of the plate is assumed to be proportional at any point to the contact pressure at that point (Gbadeyan and Agarana, 2014;Civalek and Yavas, 2006;Nguyen-Thoi et al., 2013;Agarana et al., 2015). In Winkler model, it is accepted that the foundation soil comprises of straight flexible springs which are firmly separated and autonomous of each other (Agarana et al., 2015;Amiri et al., 2013;Kerr, 1964). A few types of administering conditions of the vibration of Mindlin plate under a moving load exist. In this paper endeavor is made to rearrange such existing representing condition to be anything but difficult to deal with by decreasing the computational meticulousness and time. In this rearrangement, both the inertia and the gravitational impacts of the moving load are taken into account (Mindlin, 1951;Boay, 1993;Fryba, 1972). The following assumptions are made:  The plate is of constant cross-section.  The moving load moves with a constant speed.  The moving load is guided in such a way that it keeps contact with the plate throughout the motion.  The plate is continuously supported by a Winkler foundation.  The moving load is a partially distributed moving load.  The rectangular Mindlin plate is elastic.

The governing equations
The set of dynamic equilibrium equations which govern the behaviour of Mindlin plate with elastic support and traversed by a partially distributed moving load can be written as (Gbadeyan and Agarana, 2014;Gbadeyan and Dada, 2001; where, ( , , ) and ( , , ) are local rotation in the and directions respectively. ( , , ) is the traversed displacement of the plate at time ( ) is the Heaviside function defined as: is the velocity of a load of rectangular dimension by with one of its lines of symmetry moving along = 1 . Λ = , the area of the load in contact with the plate The plate is by in dimensions and = + 2 , ℎ and ℎ 1 are thickness of the plate and load respectively. and are the densities of the plate and load respectively.
is the modulus of the plate. is the flexural rigidity of the plate defined by: 2 is the shear correction factor. is the poisson's ratio of the plate. is the acceleration due to gravity is Young modulus of elasticity is mass of the load.

Boundary and initial conditions
For a complete formulation of the problem, a simply supported rectangular Mindlin plate is considered as an illustrative example. If the edge = 0 of the plate is simply supported, it then follows that the deflection along this edge must be zero. At the same time this edge can rotate freely with respect to the − axis, i.e., there are no bending ( ) along this edge. Therefore, the boundary conditions can be stated as follows (Gbadeyan and Dada, 2001;Nguyen-Thoi et al., 2013;Amiri et al., 2013;:

Simplification of governing equations
The acceleration hence, the expression on the left-hand side of Eq. 1 finally reduces to where ( , , ) is the moving load.

Further simplification
Eqs. 27, 28, and 29 can further be simplified. Firstly Eq. 11 can be written as: recalling Eq. 14, we have Therefore the simplified governing equations as derived from above are gotten by substituting Eq. 43 into Eq. 28 with Eq. 29 and Eq. 30.

Conclusion
Various versions of differential equation(s) governing the behaviour of plates under a moving load appear in literature. However, almost all of them are not easy to handle; a lot of computational time is required. Those that are relatively easy to handle are with many assumptions, like neglecting the effects of both rotatory inertia and shear deformation. Others assumed that the plate is not supported by any subgrade and the effect of damping neglected. The main contribution of this paper is to present a simplified set of partial differential equations modelling the dynamic behaviour of plate under a moving load. In contrast to the models of Gbadeyan and Dada (2006), this model is simple and was derived analytically. Additionally, the exact solution of this model can be sort for instead of an approximate solution. Finally, this simplified model should be considered as a more practical representation of real life situation that is easier to solve less computation time and high level of accuracy.