Page 132 - Finite Element Modeling and Simulations with ANSYS Workbench

4

Two-Dimensional Elasticity

4.1 Introduction
Many structures that are three-dimensional can be satisfactorily treated as two-dimen-
sional problems (Figure 4.1). This chapter introduces you to the use of the finite element
method for deformation and stress analyses of two-dimensional elasticity problems. First,
the basic equations in plane elasticity theory are reviewed. Then several types of 2-D finite
elements for plane elasticity analysis are presented. Applications of these elements are
demonstrated and their accuracies and efficiencies are discussed. This presentation is
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followed by an example illustrating analysis of a 2-D elasticity problem using ANSYS
Workbench.

4.2 Review of 2-D Elasticity Theory
In general, the stresses and strains at any point in a structure consist of six independent
components, that is (Figure 4.2),

σ x , σ y , σ z , τ xy , τ yz ,
τ zx
for stresses, and

ε x , ε y , ε z , γ xy , γ yz ,
γ zx
for strains.
Under certain conditions, the state of stresses and strains can be simplified. A 3-D stress
analysis can, therefore, be reduced to a 2-D analysis. There are two general types of models
involved in this 2-D analysis: plane stress and plane strain.

4.2.1 Plane Stress
In the plane stress case, any stress component related to the z direction is zero, that is,

= 0 ( ε≠ 0) (4.1)
z
σ= τ yz = τ zx z

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