taking half of each of those angles. Thus, the angle ∠BOC between and is double the angle which the major principal plane makes with plane . Angles and can also be found from the following equation This equation defines two values for which are apart (Figure). This equation can be derived directly from the geometry of the circle, or by making the parametric equation of the circle for equal to zero (the shear stress in the principal planes is always zero). Example Assume a material element under a state of stress as shown in Figure 8 and Figure 9, with the plane of one of its sides oriented 10° with respect to the horizontal plane. Using the Mohr circle, find: The orientation of their planes of action. The maximum shear stresses and orientation of their planes of action. The stress components on a horizontal plane. Check the answers using the stress transformation formulas or the stress transformation law. Solution: Following the engineering mechanics sign convention for the physical space (Figure 5), the stress components for the material element in this example are: . Following the steps for drawing the Mohr circle for this particular state of stress, we first draw a Cartesian coordinate system with the -axis upward. We then plot two points A(50,40) and B(-10,-40), representing the state of stress at plane A and B as show in both Figure 8 and Figure 9. These points follow the engineering mechanics sign convention for the Mohr-circle space (Figure 5), which assumes positive normals stresses outward from the material element, and positive shear stresses on each plane rotating the material element clockwise. This way, the shear stress acting on plane B is negative and the shear stress acting on plane A is positive. The diameter of the circle is the line joining point A and B. The centre of the circle is the intersection of this line with the -axis. Knowing both the location of the centre and length of the diameter, we are able to plot the Mohr circle for this particular state of stress. The abscissas of both points E and C (Figure 8 and Figure 9) intersecting the -axis are the magnitudes of the minimum and maximum normal stresses, respectively; the ordinates of both points E and C are the magnitudes of the shear stresses acting on both the minor and major principal planes, respectively, which is zero