With reference to the data below, an open pit mine has reached an economic limit (3,000 ft, 914 m depth), so underground mining is being considered using a block caving method. A mainline ramp 16 ft (4.9 m) wide with semicircular backs 16 ft (4.9 m) high is planned from the pit bottom to spiral down to 5,000 ft (1,524 m) below the original ground surface. The naturally supported ramp is examined as if stress concentrations were those of a rectangular opening. No stress measurements have been made at this early stage of planning. 

(a) Estimate the minimum local safety factors (FSc, FSt) for the naturally supported ramp, then 

(b) Consider a combination support system that may be required after caving is fully developed. In this latter circumstance, use bolts no more than one-half opening width and requires a rock mass safety factor of 1.8. Specify details of the considered support system. Open pit mine data: A Mohr–Coulomb failure criteria apply, the clay-filled joints constitute 83% of the potential shear failure surface, no tension cracks have yet appeared, current slide block weight is 9.891(107) lbf per foot (1.45 GN/m) of thickness and:

1 slope height                          H = 1,540 ft (469 m)

2 failure surface angle             α = 39◦

3 slope angle                           β = 55◦

4 friction angle (rock)            φr = 36◦

5 cohesion (rock)                     cr = 1,870 psi (12.9 MPa)

6 friction angle (joint)             φj = 27◦

7 cohesion (joint)                    cj = 17.0 psi (117 kPa)

8 specific weight                     γ = 156 pcf (24.7 kN/m3)

9 tension crack depth              hc = 0.0 ft (0.0 m)

10 water table depth               hw = 103 ft (31.4 m)

11 seismic coefficient              ao = 0.10

12 surcharge                            σ = 0.0 psf (0.0 kPa).

Ultimately, the pit is planned to a depth of 3,000 ft (914 m).