Tunnel Face Stability & New CPT Applications - Geo-Engineering

30 2. Stability Analysis of the Tunnel FaceThe conditions of horizontal and vertical equilibrium lead toE (i) + 2T (i) cosθ (i) + (K (i) + R (i) ) cosθ (i) − N (i) sin θ (i) = 0 (2.24)Q (i)a +G(i) s +G (i)w −Q(i) b − 2T (i) sin θ (i) −(K (i) +R (i) ) sin θ (i) −N (i) cosθ (i) = 0 (2.25)respectively [174]. Introducing R (i) = N (i) tan ϕ (i) eliminates N (i) and combining (2.24) and(2.25) makes it possible to eliminate R (i) , leading toG (i)s+ G (i)w + Q(i) a − Q b(i) + 2T (i) 1ζ (i)−Here the shorthand notation+ K (i) 1ζ (i)−+ E (i)ζ(i) +ζ (i)−= 0 (2.26)ζ − = tan ϕ cos θ − sin θ (2.27)ζ + = tan ϕ sin θ + cos θ (2.28)has been introduced. Each slice has to satisfy the equilibrium (2.26) as well as the continuityconditionQ (i)a= Q (i−1)b(2.29)for all i. Boundary conditions are Q (N)b= 0 and Q (1)a = 0. Using the boundary condition forslice N, it can be easily seen that for this slice (2.26) can be transformed toQ (N)a= Q (N−1)b[= −G (N)s+ G (N)w + 2T (N) 1ζ (N)−+ K (N) 1ζ (N)−+ E (N)ζ(N) +ζ (N)−This result can be combined with the equilibrium relation for slice N − 1 to yield](2.30)Q (N−1)a= Q (N−2)b[= −G (N)s+ G (N−1)s+ G (N)w+K (N) 1ζ (N)−+ G(N−1) w + 2T (N) 1ζ −(N)+ K (N−1) 1ζ (N−1)−+ E (N)ζ(N) +ζ (N) + E−+ 2T (N−1) 1ζ (N−1)−(N−1)(N−1)ζ +ζ (N−1)−](2.31)and so on for the slices N − 2 to 1 to finally yieldQ (1)a= −= 0,[ N∑i=1G (i)s +N∑i=1G (i)w + 2 N ∑i=1T (i) 1ζ (i) +−N∑i=1K (i) 1ζ (i) +−N∑i=1E (i)ζ(i) +ζ (i)−](2.32)where the upper boundary condition has been used.