# 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)

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 buckling or bending should occur in order to assume plane stress loading.

Suggested Bibliography

L.E. Malvern. Introduction to the Mechanics of a Continuous Medium. Prentice Hall, Englewood Cliffs, New Jersey, 1969.

Y.C. Fung. A First Course in Continuum Mechanics. Prentice Hall, Englewood Cliffs, New Jersey, 3rd ed., 1994.

S.P. Timoshenko and J.N. Goodier. Theory of Elasticity. McGraw-Hill, New York, 3rd ed., 1970.

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