I'm a rock mechanics engineer and I would like to calculate my factor of safety of shotcrete arches holding a certain rock weight. The shotcrete arches have the same profile as the excavation drift (15'W x 15'H) but has an arch in it in the back. The shotcrete thickness in the shoulders of the tunnel is approx 10" and 4" in the walls (see sketch).
I was able to calculate the factor of safety of the support by taking into account shotcrete only by diving the capacity of shear resistance of shotcrete along the failure perimeter by the load of the rock on the support system. However, this is conservative because I'm not taking into account the effect of the steel reinforcement. Could you give me guidance on how to account for the steel reinforcement in case of shearing along the shoulder due to the load of the rock on top. I'm sure it's a simple material resistance calculation but I can't quite remember my old course back in university.