chapter  12
51 Pages

## Finite Element Analysis of Plates

In Chapters 4 through 11, analytical or numerical solutions of the diﬀerential equations governing beams, plates, and shells using exact integration, the Navier and Le´vy methods, or the Ritz method were presented for regular geometries. Analytical or the Ritz solutions cannot be readily developed when the geometry of the plate is not circular or rectangular, diﬀerent portions of the plate boundary are subjected to diﬀerent boundary conditions or nonlinearities are involved. In such cases, one must resort to approximate methods of analysis that are capable of predicting accurate solutions. The finite element method is a powerful numerical method for the solution of

diﬀerential equations that arise in various fields of engineering and applied science. The basic idea of the finite element method is to view a given domain as an assemblage of simple geometric shapes, called finite elements, for which it is possible to systematically generate the approximation functions needed in the solution of diﬀerential equations over a typical element. Thus, the finite element method is a piecewise application of the Ritz method, Galerkin’s method, least-squares method, and so on. For a given diﬀerential equation, it is possible to develop diﬀerent finite element models, depending on the choice of the method used to generate algebraic equations among the undetermined coeﬃcients of the approximate solution. The ability to represent geometrically complicated domains and ease of application of physical boundary conditions made the finite element method a practical tool of engineering analysis and design. For a detailed introduction to the finite element method, the reader is advised to consult the books by Zienkiewicz (1977), Zienkiewicz and Taylor (1991), Bathe (1996), Cook et al. (1989), and Reddy (2004b, 2006). In this chapter, we develop finite element models of classical and first-order shear

deformation theories of plates; shell finite element models are not included as they are considerably more complicated than plates. Interested readers may consult finite element books in the list of references. The objective here is to present an introduction to the finite element method in the context of the material covered in this book. While the coverage is not exhaustive in terms of solving complicated

Triangular element