Finite Elements for Engineers with Ansys Applications

In this chapter, we start by highlighting the basic equations of elasticity for three-dimensional, two-dimensional and axisymmetric problems. No derivation is given for the equations presented; the intention is to provide an overview of the equations needed for FE development. It is advisable for the reader to become familiar with the details by consulting elasticity textbooks (e.g., Ugural and Fenster 2012, Sadd 2005, Boresi et al. 2010). Also, an important appendix should be reviewed, Appendix B: Vectors and Tensors.

The first application presented in this chapter is for a simple beam element. This is the same element as that developed in Chapter 2, “Detailed Procedures”. However, in the present chapter we develop the element from the more general variational principle approach to lay down the procedure for other elements. Then, we discuss the development of various two-dimensional elements, namely: a plane three-node triangle, a plane four-node rectangle and a three-node axisymmetric element. This is followed by the development of the four-node tetrahedral solid element. For these elements, we use a simple displacement approach, and we reserve the higher-order element discussion to the section on isoparametric elements. The above element development is followed by a discussion of how to include thermal effects in element equations.

An important and large section of the chapter is devoted to a discussion of isoparametric elements. We start by discussing the basic idea followed by the development of a two-node isoparametric link element. The two-node element allows all equations to be calculated in closed form and clearly lays down the steps for other isoparametric element developments. This is followed by the development of various 2D and 3D elements, namely: a 2D quadrilateral with 4–9 nodes, a 2D triangle with 3–6 nodes, a 3D solid brick with 8–20 nodes and a 3D solid tetrahedron with 4–10 nodes. It is hoped that the above element development will pave the way for quick overviews when using similar elements in other applications.

The last section of the chapter highlights various methods of imposing linear constraint equations on the final assembled equations. The methods introduced are: direct substitution and elimination; the Lagrange multiplier method; and the penalty method.