Plane stress

Plane state of stress or simply plane stress we call a special case of loading which usually occurs to solid bodies where one dimension is very small compared to the other two. Consider a very thin solid body as shown in Fig. 1. The normal and shear stresses acting on the two opposite sides normal to \( x_{3} \) are all equal to zero. Due to the fact that the body is very thin, we may assume that \( \sigma_{33} \), \( \sigma_{31} \) and \( \sigma_{32} \) are approximately zero throughout the hole body:

\[ \sigma_{33}=\sigma_{31}=\sigma_{32}=0 \]
(1)
Plane stress loading conditions
Figure 1: Thin solid under plane stress loading conditions. The external forces in the direction of the third axis are zero.

Then the stress tensor takes the form:

\[ \sigma_{ij}=\left[\begin{array}{ccc}\sigma_{11}&\sigma_{12}&0\\ \sigma_{21}&\sigma_{22}&0\\ 0&0&0\end{array}\right]=\left[\begin{array}{cc}\sigma_{11}&\sigma_{12}\\ \sigma_{21}&\sigma_{22}\end{array}\right] \]
(2)

This type of loading is called plane stress. Very thin solids under this type of loading can be analyzed as two-dimensional. It should be noted that no out-of-plane buckling or bending should occur in order to assume plane stress loading.