Tunnel Face Stability & New CPT Applications - Geo-Engineering

32 2. Stability Analysis of the Tunnel Facethe deviatoric support force is defined asS ′ dev = B ∫ zbz tζ +ζ −s ′ Fdz (2.41)As s ′ (z t ) is a constant, it can be easily seen that the integral representing the vertical effectiveforce depending on the effective support pressure can be split intoB∫ zbz tζ +ζ −s ′ dz = S ′ dev + Bs′ (z t )∫ zbz tζ +ζ −dz (2.42)which will be used to find an explicit relation for the minimal support pressure, given the effectiveweight of the support medium γ ′ F .Taking N → ∞ it follows from (2.32) that the minimal support pressure at z = z t can becalculated from maximisings ′ (z t ) = − G s + G w + K ′ + 2T ′ + S ′ devZover θ, where(2.43)T ′ =∫ zbK ′ = Bz t∫ zbw [c + Ky σ v ′ ζ tan ϕ] dz−(2.44)cdzζ − sin θ(2.45)z tG s = B Dtan θ σ v ′ (z t) (2.46)G w = BZ = B∫ zb∫ zbz tw γ ′ dz (2.47)z tcot(ϕ − θ) dz (2.48)and w = (z b − z) cot θ for z t ≤ z ≤ z b . It should be noted at this point that in general allvariables under the integrals are z-dependent, even though this dependence has not been writtenexplicitly.Having obtained (2.43) makes it possible to determine the required effective support pressures ′ (z t ) for a given angle θ and, given the distribution of the effective weight of the support mediumγ ′ b and the pore water pressures p, the required support pressure on the entire face s = s′ + p.The desired minimal support pressure can be found by maximising s as function of θ. Theangle for which the required support pressure is largest represents the failure mode most likelyto occur. This procedure is less straightforward than it may seem, however, as a number ofadditional effects, like soil arching and excess pore pressures, have to be taken into account.2.2.2 Vertical Stress DistributionOne of the main effects that has to be taken into account when using a wedge model to accuratelyestimate the minimal support pressure is soil arching. From centrifuge tests [37, 60] reviewedin section 2.1.4 it can be observed that part of the soil column above the wedge does not actas a load on the wedge, but is carried by the surrounding soil. The effective load acting on