Page 234 - Finite Element Modeling and Simulations with ANSYS Workbench

7

Three-Dimensional Elasticity

7.1 Introduction
Engineering designs involve 3-D structures that cannot be adequately represented using
1-D or 2-D models. Solid elements based on 3-D elasticity [10,11] are the most general ele-
ments for stress analysis when the simplified bar, beam, plane stress/strain, plate/shell
elements are no longer valid or accurate. In general, 3-D structural analysis is one of the
most important and powerful ways of providing insight into the behavior of an engi-
neering design. In this chapter, we will review the elasticity equations for 3-D and then
discuss a few types of finite elements commonly used for 3-D stress analysis. Several dif-
ferent types of supports, loads, and contact constraints will be introduced for 3-D struc-
tural modeling, followed by a case study on predicting the deformation and stresses in
®
an assembly structure using ANSYS Workbench.

7.2 Review of Theory of Elasticity
The state of stress at a point in a 3-D elastic body is shown in Figure 7.1.
In vector form, the six independent stress components determining the state of stress
can be written as

 σ 
x
 
 σ y 
 σ  (7.1)


z
σ
σ = {} =   , or [ σ ij ]
 τ xy 
 τ 
yz
 

zx
  τ 

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