Question 12.89: At a point in a stressed body, the principal stresses are or...

The center of Mohr’s circle can be found from the two principal stresses:

The radius of the circle is

(a) The stresses on plane a-a are found by rotating 270° counterclockwise from the \sigma_{p 2} point on Mohr’s circle. Therefore, the point at the top of the circle directly above the center corresponds to the state of stress on plane a-a.

(b) The angle \theta_{p} shown in the problem statement sketch is

The \sigma_{p 2} principal plane is rotated 31.7175° clockwise from the x face of the stress element. We need to find the point on Mohr’s circle that corresponds to the x face of the stress element. Since we know the location of \sigma_{p 2} on Mohr’s circle, we can begin there and rotate 2 \theta_{p} in the opposite direction to find point x. Therefore, beginning at point \sigma_{p 2}, rotate 2(31.7175°) = 63.4349° counterclockwise to locate point x.
The σ coordinate of point x is found from:

The τ coordinate of point x is found from:

Similarly, the σ coordinate of point y is found from:

The τ coordinate of point y is also 10.00 ksi, and the shear stress on the y face rotates the stress element clockwise.

The stresses on the vertical and horizontal faces of the stress element are shown below.

(c) Since both \sigma_{p 1} \text { and } \sigma_{p 2} are negative, the absolute maximum shear stress will be larger than the maximum in-plane shear stress. The radius of the largest Mohr’s circle gives the absolute maximum shear stress. In this case, the absolute maximum shear stress occurs in the \sigma_{p 2}-\sigma_{p 3} plane; therefore,