Unified formulation for Reissner-Mindlin plates: a comparison with numerical results

Analytical solutions to problems of solids mechanics are useful to test the accuracy and the precision of the numerical approaches. Problems such as shear locking or boundary layer can be better understood with the help of analytical solutions. However, it is known that only simple cases can be solved by hand. Furthermore, some cases which are theoretically solvable can also result in rather tedious analysis. Nowadays computer and mathematical software development allow to avoid this problem if they are associated with a suitable formulation. Classical formulations of mechanics of elastic body solve each particular problem by obtaining standard differential equations but generally do not give a unified method to solve them. In this paper a non-standard formulation is applied to find analytical solutions for elastic plates considering shear deformation (Reissner-Mindlin model) using a computer-aided approach. In addition, the results are compared to finite element model predictions. The main advantage of the proposed methodology over classical approaches is that it can be applied to problems of structural mechanics that are radically different using the same matrix framework, similarly to the FEM. Therefore, a computer-aided approach is used to obtain the analytical solution. This unified formulation is described Tassinari et al. [6]. The Reissner-Mindlin model (Reissner [3], Mindlin [4]) takes into account the shear deformation and it is generally recommended for moderately thick plates whereas Love-Kirchhoff model is more suitable for thin plates (Kirchhoff [1], Love [2]). The derivation of the governing equations of the RM plate results in a system of partial differential equations. The Levy solution technique can be used (Timoshenko and Woinowsky-Krieger [7], Szilard [5]) once the boundary conditions on two opposite edges are given. The solution is written as a series of orthogonal functions of the first coordinate multiplied by unknown functions of the second coordinate. The governing system is reduced to a system of ODEs that applies for each unknown term of the series. The application of the unified formulation to the Reissner-Mindlin plate results in a governing system of ODEs written in a matrix canonical form (see Tassinari et al. [6]). This expression is characterized by the state vector, which contains the generalized stresses and displacements, and the system matrix which describes the equilibrium. It is shown that this form can be applied to several problems of structural mechanics, by consistently adapting both system matrix and state vector. In this paper, the case of the Reissner-Mindlin plate problem with simply support on two opposite edges is solved. The analytical solution for a particular load case is presented showing the differences between the Reissner-Mindlin solution and the predictions given by the Love-Kirchhoff model (Kirchhoff [1], Love [2]). Finally, the vertical displacement field obtained analytically for differen