Given the stresses relative to compass coordinates (x = east, y = north, z = up), xx = 2,155, yy = 3,045, zz = 4,200,

Given the stresses relative to compass coordinates (x = east, y = north, z = up), σxx = 2,155, σyy = 3,045, σzz = 4,200, τyx = −1,222, τxz = 0, τyx = 0 in psi with compression positive, find the principal stresses σ1, σ2, σ3 (magnitude and orientation). Show with a suitable sketch. Then consider an unlined shaft that will be 13 ft by 26 ft in section and

(a) Choose the best opening shape (ellipse, ovaloidal, or rectangle) and best orientation from the rock mechanics perspective, assuming the given stress state is critical to the design,

(b) Suppose an identical second shaft is planned near the first. Show the orientation and location of the second shaft relative to the first such that interactions are likely to be acceptable and not unduly high. Sketch Justify your answer.