A circular vertical shaft is planned to have a finished, inside diameter of 19 ft in an underground hard rock mine. Rock properties are: Co = 23,700 psi, To = 1,480 psi, E = 5.29(106 psi), ν = 0.27, γ = 162pcf. The premining stress state relative to compass coordinates is:

SE = 350 + 0.2h, SN = 420 + 0.35h, SV = 1.12h, where stresses are in psi, h=depth in ft, E, N, V refer to compass coordinates (x = east, y = north, z = up) and compression is positive.

Premining shear stresses are nil relative to compass coordinates. Properties of concrete are: Compressive strength = 5,740 psi, tensile strength = 425 psi, Young’s modulus = 4.75(106) psi, Poisson’s ratio = 0.25. Find:

(a) The unlined shaft wall safety factors in tension and compression at a depth of 4,250 ft, 

(b) Concrete liner thickness in the vicinity of a water-bearing formation where the pressure in the undisturbed ground is 90 psi. Note: The maximum allowable compression in the concrete is 3,500 psi.