ABSTRACT
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
Quadrilateral element
of Elastic Plates and
of the finite method in the analysis of plate problems. It is important to bear in mind that any approximate method is a means to analyze a practical engineering problem and that analysis is not an end in itself, but rather an aid to design and manufacturing. The value of the theory and analytical solutions presented in the preceding chapters to gain insight into the behavior of simple plate problems is immense in the numerical modelling of complicated problems by any numerical method.
