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The general case of three dimensional stress, or triaxial stress, is illustrated by Figure 2-01a, as we have already learned. As in the case of plane or biaxial stress, a particular orientation of the stress element occurs in space for which all shear stress components are zero. When an element has this particular orientation, the normals to the faces correspond to the principal directions and the normal stresses associated with these faces are the principal stresses. Since there are six faces, there are three principal directions and three principal stresses, sig_1, sig_2 and sig_3.
In our studies of plane stress we were able to specify any stress state sig_x, sig_y, and tau_xy and find the principal stresses and principal directions. But six components of stress are required to specify a general state of stress in three dimensions, and the problem of determining the principal stresses and directions is much more difficult. It turns out that it is rarely necessary in design, and so we shall not investigate the problem in the book. The process involves finding the three roots to the cubic
sig^3-(sig_x+sig_y+sig_z)*sig^2+
(sig_x*sig_y+sig_x*sig_z+sig_y*sig_z-
tau_xy^2-tau_yz^2-tau_zx^2)*sig-
(sig_x*sig_y*sig_z+2*tau_xy*tau_yz*tau_zx-
sig_x*tau_yz^2-sig_y*tau_zx^2-sig_z^tau_xy^2)
=0 [2-10]
Having solved equation [2-10] for a given state of stress, one might obtain a principal stress element like that of Figure 2-06a.
In plotting Mohr's circles for triaxial stress, the principal stresses are arranged so that sig_1 > sig_2 > sig_3. Then the result appears as in Figure 2-06b. The stress coordinates sig_n tau_n for any arbitrarily located plane will always lie within the shaded area.
It frequently happens, in the case of plane stress, that the two principal stresses, sig_1 and sig_2 have the same sign. In this situation construction of a single Mohr's circle through sig_1 and sig_2 will not give tau_max. This can easily be seen by constructing three circles, using sig_3 = 0 for the third principal stress.