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Ebook The finite element method - A practical course abaqus: Part 2

(BQ) Part 2 book "The finite element method - A practical course abaqus" has contents: Fem for two dimensional solids, fem for plates and shells, fem for 3D solids, special purpose elements, modelling techniques, fem for heat transfer problems, using abaqus. | 7 FEM FOR TWO-DIMENSIONAL SOLIDS 7.1 INTRODUCTION In this chapter, we develop, in an easy to understand manner, finite element equations for the stress analysis of two-dimensional (2D) solids subjected to external loads. The basic concepts, procedures and formulations can also be found in many existing textbooks (see, e.g. Zienkiewicz and Taylor, 2000). The element developed is called a 2D solid element that is used for structural problems where the loading–and hence the deformation–occur within a plane. Though no real life structure can be truly 2D, experienced analysts can often idealize many practical problems to 2D problems to obtain satisfactory results by carrying out analyses using 2D models, which can be very much more efficient and cost-effective compared to conducting full 3D analyses. In engineering applications, there are ample practical problems that can be modelled as 2D problems. As discussed in Chapter 2, there are plane stress and plane strain problems, whereby correspondingly, plane stress and plane strain elements need to be used to solve them. For example, if we have a plate structure with loading acting in the plane of the plate as in Figure 7.1, we need to use 2D plane stress elements. When we want to model the effects of water pressure on a dam, as shown in Figure 7.2, we have to use 2D plane strain elements. y fy fx x Figure 7.1. A typical 2D plane stress problem. 129 “chap07” — 2002/12/14 — page 129 — #1 130 CHAPTER 7 FEM FOR TWO-DIMENSIONAL SOLIDS z y fx x Figure 7.2. A typical 2D plane strain problem. Note that in Figure 7.1, plane stress conditions are usually applied to structures that have a relatively small thickness as compared to its other dimensions. Due to the absence of any off-plane external force, the normal stresses are negligible, which leads to a plane stress situation. In cases where plane strain conditions are applied, as in Figure 7.2, the thickness of the structure (in the z direction) is relatively large