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 different thickness-to-side ratios is compared with finite element method predictions. The analytical solution carried out shows the effect of the shear deformation in plate with small slenderness. In particular, the magnitude of this effect is significant for the vertical displacement. In general, the other components of the state vector are not significantly influenced in the domain of the plate but exhibits perturbations close to the edges. This edge effect is known as boundary layer and further investigations are needed considering higher-order terms. The results obtained from the proposed analytical method and numerical models presented in this study are comparable. Furthermore, the finite element analysis carried out shows that for a reasonable mesh refinement the numerical models used give excellent approximation of the analytical solution. Keywords: analytical solution, unified formulation, shell, plate, Reissner-Mindlin, Love-Kirchhoff, shear strain, finite element. References [1] Kirchhoff G., Über das Gleichgewicht und die Bewegungen einer elastischen Scheibe. Journal für reine und angewandte Mathematik, 1850; 40; 51–58. [2] Love, A. E. H., A treatise on the matematical theory of elasticity, Dover, 1944. [3] Mindlin R. D., Influence of rotatory inertia and shear on flexural vibration of isotropic, elastic plates. Journal of Applied Mechanics, 1951; 18; 31–38. [4] Reissner E., The effect of transverse shear deformation on the bending of elastic plates, Journal of Applied Mechanics, 1945; 67; A69-A77. [5] Szilard R., Theories and applications of plate analysis: classical, numerical and engineering methods. Wiley, 2004. [6] Tassinari L, Monleon S, Gentilini C, Ubertini F. A unified approach for analytical solution of moderately thick shells and plates. (Paper in preparation). [7] Timoshenko S, Woinowsky-Krieger S. Theory of plates and shells. McGraw-Hill, 1959.
IASS Symposium 2009, Valencia, Spain2009
