Tunnel Face Stability & New CPT Applications - Geo-Engineering

10 2. Stability Analysis of the Tunnel FaceN8765N=4ln(C/R+1)432100N=2+2ln(C/R+1) Lower bound DavisDesign line Mair1g model testCentrifuge model testField case0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8C/DFigure 2.6: Stability ratios for lined tunnels (P = 0) in clay [112]stress field. The corresponding stability ratios are respectively( ) CN = 2 + 2 lnR + 1 (2.9)( ) CN = 4 lnR + 1 . (2.10)Figure 2.6 shows that these lower bound solutions correspond well with a number of laboratorytests and field collapses as well as with the stability ratio design line obtained by Kimura &Mair [94] for tunnels in undrained conditions.A few years earlier, in 1977, Atkinson & Potts [13] derived the minimal support pressure foran unlined cavity in a dry cohesionless material (see figure 2.4b). They differentiate between twolimit cases. The first case is a tunnel in a weightless medium with a surface load q s , the second atunnel in a medium with γ > 0 but without surface loading. For this second case they derive twolower bound solutions. For cases where c > 0 and ϕ > 0, which include all practically relevantcases, it can be shown however that of these solutions the overburden independent lower boundsolutionwheres min = 2K pKp 2 R, (2.11)− 1γK p = 1 + sin ϕ1 − sin ϕ , (2.12)is always normative. As this represents a statically admissible stress field it is a safe estimate forthe minimal support pressure. The kinematic upper bound solution on the other hand, which isthe inherently unsafe estimate, yields a lower value for the minimal support pressure. It shouldbe noted that both upper and lower bound solutions found by Atkinson & Potts are independentof the relative overburden C/D.Leca & Dormieux [106] propose a series of conical bodies in 1990 (see figure 2.7). Combinedwith different stress states, similar to those proposed by Davis et al., they derive lower and upper