Finite Elements for Engineers with Ansys Applications

In Chapter 1, we identified five basic steps or stages for any type of finite element analysis. These basic steps are: modeling and discretization; formulation of element equations; assembly; boundary conditions and solution; and finally postprocessing. In commercial FE programs these steps are lumped into three stages: modeling; solution and postprocessing. One of these steps, namely the assembly process, is quite simple and straightforward. On the other hand, most of the research done in the finite element area and most of the textbooks written in the area will involve one or more of the other basic steps or stages. In this chapter we will describe some details of these basic stages. The intention is only to provide a basic understanding of the concepts and underlying assumptions in each step. We avoid any rigorous mathematical development and will leave this to later chapters. Although the number of equations in the chapter may seem to be rather large for such objective, we do make sure that we arrive at these equations by using simple principles and concepts rather than by using rigorous mathematical methods. This enables the reader to grasp a concept before getting into the mathematical details.

The development in the chapter will concentrate mostly on formulating the equations for simple elements and using such equations for solving simple problems. Though the elements that we use are quite simple, it will be realized that they are useful in many practical applications. We discuss simple ways of generalizing such elements and presenting their characteristic equations in a general and three-dimensional form. Then we discuss methods of imposing simple linear constraints and the assembly of the global equations. Many examples are presented to provide a solid understanding and to aid in generalizing some of the concepts addressed in the chapter.