Finite Elements for the Reissner–Mindlin Plate

1999

Journal of Elasticity, 1992

For the plate formulation considered in this paper, appropriate three-dimensional elasticity solution representations for isotropic materials are constructed. No a priori assumptions for stress or displacement distributions over the thickness of the plate are made. The strategy used in the derivation is to separate functions of the thickness variable z from functions of the coordinates x and y lying in the midplane of the plate. Real and complex 3-dimensional elasticity solution representations are used to obtain three types of functions of the coordinates x, y and the corresponding differential equations. The separation of the functions of the thickness coordinate can be done by separately considering homogeneous and nonhomogeneous boundary conditions on the upper and lower faces of the plate. One type of the plate solutions derived involves polynomials of the thickness coordinate z. The other two solution forms contain trigonometric and hyperbolic functions of z, respectively. Both bending and stretching (or in-plane) solutions are included in the derivation.

Journal of Theoretical and Applied Mechanics, 1982

Acta Mechanica, 2010

In this article the equations of a moderately thick plate are derived by the method of successive approximations. The derived equations exactly satisfy all the elastostatic equations, the plate equilibrium equations and traction free face boundary conditions.

International Journal of Solids and Structures, 2011

This is the first part of a two-part paper dedicated to a new plate theory for out-of-plane loaded thick plates where the static unknowns are those of the Kirchhoff-Love theory (3 in-plane stresses and 3 bending moments), to which six components are added representing the gradient of the bending moment. The new theory, called the Bending-Gradient plate theory is described in the present paper. It is an extension to arbitrarily layered plates of the Reissner-Mindlin plate theory which appears as a special case of the Bending-Gradient plate theory when the plate is homogeneous. However, we demonstrate also that, in the general case, the Bending-Gradient model cannot be reduced to a Reissner-Mindlin model. In part two (Lebée and Sab, 2010a), the Bending-Gradient theory is applied to multilayered plates and its predictions are compared to those of the Reissner-Mindlin theory and to full 3D Pagano's exact solutions. The main conclusion of the second part is that the Bending-Gradient gives good predictions of both deflection and shear stress distributions in any material configuration. Moreover, under some symmetry conditions, the Bending-Gradient model coincides with the second-order approximation of the exact solution as the slenderness ratio L/h goes to infinity.

Mathematics and Mechanics of Solids, 2018

This work derives an exact two-dimensional plate theory for heterogeneous plates consistent with the principle of stationary three-dimensional potential energy under general loading. We do not take any hypotheses about the shape of the heterogeneity. We start from three-dimensional linear elasticity and by using the Fourier series expansion in the thickness direction of the displacement field with respect to a basis of scaled Legendre polynomials. We deduce an exact two-dimensional model expressed in power-series in the ratio between the thickness of the plate and a characteristic measurement of its mid-plane. Then we can derive an approximative theory by neglecting in the expression of potential energy all terms that contain a power of this ratio that is higher than a given truncation power for getting to an approximative two-dimensional problem. In the last section, we show that the solution of the approximation problem only differs from the exact solution by a difference of the same order of the neglected terms in the potential energy. A similar result when we truncate the displacement field can be also established. This model can be a starting point to formulate a two-dimensional homogenized boundary value problem for highly heterogeneous periodic plates.

International Journal of Engineering and Technology

In this work, the mathematical theory of elasticity has been used to formulate and derive from fundamental principles, the first order shear deformation theory originally presented by Mindlin using variational calculus. A relaxation of the Kirchhoff's normality hypothesis was used to account for the effect of the transverse shear strains in the behaviour of the plate. This made the resulting theory appropriate for use for moderately thick plates. A simultaneous use of the strain-displacement relations for small-deformation elasticity, stress-strain laws and the stress differential equations of equilibrium was used to obtain the differential equations of static flexure for Mindlin plates in terms of three unknown generalised displacements. The equations were found to be coupled in the unknown displacements; but reducible to the Kirchhoff plate equations when the removed Kirchhoff normality hypothesis was introduced. This showed the Kirchhoff plate theory to be a specialization of the Mindlin plate theory. The theory of elasticity foundations of the Mindlin and Kirchhoff plate theories are thus highlighted.