which are the familiar two-dimensional (2-D) stress transformation equations found in a number of introductory mechanics of materials books (see, e.g., Beer et al. 2001; Gere 2004).

FIGURE 1.4 Two-dimensional (2-D) stress transformation: (a) original state of stress acting on a 2-D infinitesimal element and (b) transformed state of stress acting on a 2-D infinitesimal element.

FIGURE 1.5 Mohr's circle.

2. Principal stresses. For plane stress condition, Equation 1.11 becomes det

which, upon expansion, gives the quadratic equation

The two solutions to the above equation give the two principal stresses as

Note that these stresses represent the rightmost and leftmost points on a Mohr circle (Beer et al. 2001), shown in Figure 1.5, with OC = (a11 + a22)/2 and R = P[((s11 - a22)/2)2 + a212]. (Although not asked for in this example, it can readily be seen that the maximum shear stress is the uppermost point on the Mohr circle given by tmax = (s12)max = R =

3. Principal planes. Substituting the equation for sP1 into n1P1(ff11 - s) + n2P1ff12 = 0

and recognizing that nlP1 + n2P1 = 1

it can be shown that the principal plane on which sP1 acts forms an angle 0P1 = tan-1(n2P1/n1P1) with the x1 (or x-) axis and is given by

0 0

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