36.1 DEFINITIONS AND NOTATION

The general two-dimensional stress element in Fig. 36.1a shows two normal stresses ? _x and ? _y both positive, and two shear stresses ? _xy and ? _yx , positive also. The element is in static equilibrium, and hence ? _xy = ? _yx . The stress state depicted by the figure is called plane or biaxial stress.

Figure 36.1: Notation for two-dimensional stress. ( From Applied Mechanics of Materials, by Joseph E. Shigley. Copyright 1976 by McGraw-Hill, Inc. Used with permission of the McGraw-Hill Book Company.)

Figure 36.1b shows an element face whose normal makes an angle to the x axis. It can be shown that the stress components ? and ? acting on this face are given by the equations

(36.1)

(36.2)

It can be shown that when the angle is varied in Eq. (36.1), the normal stress ? has two extreme values. These are called the principal stresses, and they are given by the equation

(36.3)

The corresponding values of are called the principal directions. These directions can be obtained from

(36.4)

The shear stresses are always zero when the element is aligned in the principal directions.

It also turns out that the shear stress ? in Eq. (36.2) has two extreme values. These and the angles at which they occur may be...