Problem 2: For the given state of stress, determine the principal stresses, the orientation of the of the planes of maximum in-plane stress, the maximum shearing stress and the corresponding normal stress.
Solution: To plot Mohr's Circle, I need its center and radius:
This is the position of the center of the circle. I plot a tentative circle using this and the two normal stresses, remembering that point Y (corresponding to ) is plotted with the shearing stress of -30 MPa. I don't know yet where the shearing stress axis is to be placed.
From right triangle trigonometry,
The shearing stress axis is between the center and the -45 MPa:
a) The principal stresses are -52.1 (15 - R) and 82.1 MPa (15 +R), which correspond to a shearing stress of zero.
b) The orientation of the planes of maximum in-plane stress (those found in part a) corresponds to a clockwise rotation on Mohr's circle of 26.57o, or a counterclockwise rotation of 180 - 26.57 = 153.4o. Since the Mohr's circle angle is twice that for the element, considering elements rotated by 13.29 clockwise or 76.7 counterclockwise will produce zero shearing stress and maximum values of the normal stress.
c) The maximum shearing stress is the radius itself, 67.1 MPa. The normal stresses corresponding to this shearing stress are equal and are 15 MPa.
This is called "in-plane" stress because rotations out of the plane are not being considered. The z-axis is out of the screen, and all rotations are about that axis. Rotations about the x and y axes will be considered later.