## 12.9 PLANE STRESS AND PLANE STRAIN

The general state of stress and strain is three-dimensional but there exist particular geometric configurations that can be treated differently.

### Plane Stress

The two-dimensional, or biaxial, stress state is also called plane stress. Plane stress requires that one principal stress be zero. This condition is common in some applications. For example, a thin plate or shell may also have a state of plane stress away from its boundaries or points of attachment. These cases can be treated with the simpler approach of equations 12.7.

### Plane Strain

There are principal strains associated with the principal stresses. If one of the principal strains (say ? 3) is zero, and if the remaining strains are independent of the dimension along its principal axis, n 3, it is called plane strain. This condition occurs in particular geometries. For example, if a long, solid, prismatic bar is loaded only in the transverse direction, regions within the bar that are distant from any end constraints will see essentially zero strain in the direction along the axis of the bar and be in plane strain. (However, the stress is not zero in the zero-strain direction.) A long, hydraulic dam can be considered to have a plane strain condition in regions well removed from its ends or base at which it is attached to surrounding structures.

Example 12-1: Determining 3-D Principal Stresses Using Analytical Methods.
 Given: A triaxial stress element as shown in Figure 12-7 (p. 346) has

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