Chapter 8 PREDICTION OF SETTLEMENT TROUGHS Tunneling in urban environments requires the prediction of surface movements in order to asses possible effects on nearby structures. Currently, the most commonly used means of making these predictions has been the application of empirical formulas such as the one by Peck [24]. In this chapter, Peck’s formula will be briefly discussed and compared to the solutions presented in this thesis. It will be shown how the different modes of deformation may be combined and used as a tool for the prediction of settlement troughs. § 8.1 Peck’s Surface Settlement Formula Peck’s formula for the vertical settlements due to the construction of a tunnel is given by a Gaussian distribution (error function) curve, shown in Figure 8.1. The formula for the Gaussian distribution curve as written by Peck is v(x) = V s x i √ 2π e −x 2 /2x 2 i , (8.1) where x i is the distance from the center of the symmetrical distribution to the inflection points of the curve, shown as points in Figure 8.1. The parameter V s -3-2-10123 -2 -1 0 -2 -1 0 v(x) x/h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 8.1: Peck’s formula for x i = h/2. 73 74 Prediction of Settlement Troughs Chapter 8 is related to the total area under the curve: V s = ∞ −∞ v(x) dx, (8.2) which is equal to the volume of the settlement trough per unit length of the tunnel. The maximum depth of the trough can be determined by setting x = 0 in (8.1) : v max = v(0) = V s x i √ 2π . (8.3) The parameter x i in Peck’s formula (8.1) is determined using empirical design charts (see e.g. [24]) or formulas (see e.g. [2]). A commonly used value is simply x i = h/2, as given by Attewell [2]. The volume of the trough V s is usually taken as the estimated ground loss around the tunnel perimeter [24]. It should be emphasized, as noted in Chapter 6, that the volume of the settlement trough for the elastic ground loss solution is larger than the amount of ground lost around the tunnel perimeter for values of Poisson’s ratio other than 0.5. This may lead to an underestimation of the settlements when Peck’s formula is applied by setting V s equal to the ground loss around the perimeter. § 8.2 Superposition of Ground loss and Ovalization It is well known that the solutions for the ground loss problem overestimate the width of the settlement trough [31, 29] and that the width of the settlement trough can be reduced by adding the effects of ovalization to those of the ground loss [40, 26, 12]. The different vertical displacements along the surface for the exact solutions for ground loss and ovalization are compared to Peck’s formula in Figure 8.2. For this comparison the magnitude of the ovalization has been set equal to the magnitude of the ground loss (u o = u g ). The parameters in Pecks -3-2-10123 -2 -1 0 -2 -1 0 1.6 u g v x/h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 8.2: Comparison of the settlement troughs for Peck’s formula (solidline), ground loss (dashed line), and ovalization (dotted line) for ν = 0.25 and r/h = 0.25. solution in Figure 8.2 have been set to x i = h/2 and V s = πr 2 −π(r −u g ) 2 , the Section 8.2 Superposition of Ground loss and Ovalization 75 latter of which is equal to the amount of ground lost around the tunnel perimeter per unit length of the tunnel in the exact solution. The relative amounts of ground loss and ovalization needed to generate set- tlement curves which meet practical case studies has been discussed in detail in [26, 12]. The ratio of the amount of ovalization to the amount of ground loss is often denoted by ρ. For the exact solutions presented in this thesis the following relation holds: ρ = u o u g , (8.4) where u g is the maximum displacement along the tunnelperimeterfortheground loss solution, and u o is the maximum displacement along the tunnel perimeter for the ovalization solution, as introduced in Chapter 6 and Chapter 7. The values of ρ in the case studies mentioned above range from -0.6 to 5.1. Smaller values of ρ are usually due to pressurized grouting applied when using slurry shields, which results in much lower values for the ground loss and can even result in volume expansion (as is evident when ρ is negative). Larger values of ρ can be a result of grouting in very soft soils [12]. For many cases of open shield or open face tunneling, when grouting is not applied, the values of ρ tend to be close to 1.0 (for the cases discussed in [12] they vary between 0.7 and 1.0). For such cases a solution which combines ground loss and ovalization with a ratio of ρ = 1 may well be used as a tool for the prediction of settlement troughs. For the combined solution with ρ = 1 we note that u o = u g . This combined solution is compared to Peck’s solution in Figure 8.3. -3-2-10123 -2 -1 0 -2 -1 0 1.6 u g v x/h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 8.3: Comparison of the settlement troughs for Peck’s formula (solid line) and the superimposed ground loss and ovalization solutions, with ρ = 1 (dashed line), for ν = 0.25 and r/h = 0.25. The narrowing effect of the ovalization solution is clearly apparent in the plot, and the curve is much closer to the one predicted by Peck than either of the solutions in Figure 8.2. The fact that the volumes of the settlement troughs are not equal, despite the fact that the amount of ground lost around the tunnel 76 Prediction of Settlement Troughs Chapter 8 perimeter is the same in both solutions, is also apparent in Figure 8.3. This is due to the fact that the ground loss solution, as discussed in Chapter 6, generates a larger amount of volume loss along the surface than around the tunnel for values of ν other than 0.5. The theoretical solution deviates further from Peck’s curve if the volume of the troughs are set equal. § 8.3 Buoyancy Effects The elastic solution shown in Figure 8.3 does not include buoyancy effects. These effects can have a large impact on the settlement curves, as is evident in Figure 8.4, where buoyancy has been included by superimposing the solution for a rigid, buoyant tunnel discussed in Chapter 5. -3-2-10123 -2 -1 0 -2 -1 0 2.6 u g v x/h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 8.4: Comparison of the settlement troughs for Peck’s formula (solid line) and the superimposed ground loss and ovalization solutions (with ρ = 1) which include buoyancy effects (dashed line). The plot is for ν = 0.25, r/h = 0.25, γr/µ = 0.0011, and r/u g = 100. For the curves depicted in Figure 8.4, the magnitude of the ground loss V s in Peck’s formula has been adjusted to match the volume of the trough of the elastic solution. In the combined elastic solution the distance from the tunnel at which the displacements have been constrained vertically has been set to 2h. The choice of this value, as discussed in §5.5, corresponds to a point on the surface at which the stresses are assumed to be small enough that the vertical displacements vanish. The choice can be justified by observing the extent of the effect of the incremental isotropic and deviatoric stresses on the initial stresses, as shown in Figure 8.5, in which the total stresses have been plotted. As the extent of influence of the incremental isotropic and deviatoric stresses is a function of additional parameters (see the following paragraphs) care must be taken to choose a point of vertical constraint which is outside of these zones of influence. Section 8.4 Chapter Summary 77 -3-2-10123 -3 -2 -1 0 -3 -2 -1 0 x/h - y/h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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Figure 8.5: Comparison of the total isotropic and total deviatoric stresses. The plot is for ν = 0.25, r/h = 0.25, γr/µ = 0.0011, and r/u g = 100. The choice of the point of vertical constraint at x/h = 2, y/h = 0 has the effect that there is no heave along the surface, as can be seen in Figure 8.4. Another artifact of the choice can be seen on the outer edges of Figure 8.4, where the settlements for the combined solution become negative. These negative settlements are not realistic in practice, as it is assumed that the stresses beyond x/h = 2 are so small that the deformations will be eliminated. Negative displacements are an artifact of the continuously increasing settlements inherent in the buoyancy solution. These negative settlements will always appear at some distance from the tunnel, regardless of the choice of the point of vertical constraint. The displacements in the combined solution depicted in Figure 8.4 are depen- dant on several additional parameters, due to the combination of the buoyancy effect with the ground loss and ovalization solutions. If the parameter P de- fined in Appendix B is taken equal to 2µu g (and noting that u g = u o ), then the combined elastic solution is dependant on the relative weight of the material γr/µand the relative deformation r/u g , in addition to r/h and ν. This makes it much more difficult to make broad generalizations about the results. Figure 8.4 merely illustrates the importance of buoyancy, and the kind of effect that it may have on the settlements. § 8.4 Chapter Summary In this chapter the possibility has been addressed of using the exact solutions presented in this thesis for the prediction of settlement troughs. It appears that a reasonable fit to empirical approaches can be obtained through a combination 78 Prediction of Settlement Troughs Chapter 8 of: ground loss, which creates a net volume loss along the surface; ovalization, which has a narrowing effect on the ground loss settlement trough; and buoy- ancy, which appears to have the effect of narrowing the trough further still. The final combined elastic solution is dependant on four dimensionless parameters: the relative tunnel size r/h, Poisson’s ratio ν, the relative weight of the material γr/µ, and the relative deformation r/u g . Chapter 9 CONCLUSION It has been shown that the complex variable method for solving two dimensional linearly elastic problems can be used to obtain several fundamental solutions of tunneling problems. The method was used in Chapter 3 to derive the general mathematical repre- sentation of problems involving resultant forces on holes in a half-plane. Such problems occur in geomechanics during the excavation of tunnels. When tun- nels are excavated, removal of the weighted material inside the tunnel causes the ground under the tunnel to rebound. This in turn causes a resultant force, referred to in the thesis as the buoyancy force, to act on the tunnel until the stresses return to a state of equilibrium. The mathematical representation de- rived in Chapter 3 shows that the displacements will always be unbounded at infinity in solutions of two-dimensional problems involving resultant forces act- ing on holes in a half-plane (unless the resultant forces on the holes form a new equilibrium system with each other, which is unlikely in tunneling problems because all the buoyancy forces will act in the same direction). The unbounded behavior of the displacements at infinity causes the deter- mination of the settlements due to tunnel excavation to have certain degree of arbitrariness. A point must be chosen at which the vertical displacements are fixed, referred to in this thesis as the point of vertical constraint. In Finite Ele- ment Meshes this point is chosen when the mesh is fixed at a certain depth. For the solutions in this thesis, it is assumed that the ground will exhibit a high stiff- ness for stresses which are below a predetermined level. Plots of the stresses can be used to determine the point nearest to the tunnel at which this predetermined level has been reached. If the stresses continue to dissipate beyond this point, then this point can be used to fix the displacements. In Chapter 4 the mathematical representation of resultant force problems derived in Chapter 3 was used to construct a solution for a single, circular tunnel in an elastic half-plane that includes buoyancy forces. The solution was derived for problems in which the surface of the half-plane is stress-free, and in which displacements are specified along the tunnel boundary. 79 80 Conclusion Chapter 9 The general solution derived in Chapter 4 was used in the rest of the thesis as the framework for considering three fundamental tunneling problems: the prob- lem of a rigid, buoyant tunnel (Chapter 5), the ground loss problem (Chapter 6), and the ovalization problem (Chapter 7). The problem of a rigid, buoyant tunnel is encountered when a very stiff lining with negligible weight (in comparison to the weight of the excavated material) is placed inside a freshly excavated tunnel. The problem is solved by assuming that constant displacements occur around the tunnel periphery. This corresponds to the case in which there is a large amount of friction between the lining and the tunnel. The solution shows that large amounts of heave can result from such excavations, but that the amount of heave is strongly dependant on the choice of the point of vertical constraint. An illustrative comparison of the solution and a Finite Element calculation shows some effect of the FEM boundary constraints along the edges of the mesh. The ground loss problem is encountered when the process of tunnel excava- tion or the placement of a tunnel lining causes the ground around the tunnel to converge towards the tunnel. In the solution it is assumed that this convergence is uniform. The solution shows that the settlement trough due to ground loss is wider than those usually encountered in practice. The amount of ground loss along the surface is larger than the amount of ground loss around the tunnel in most cases (they are equal for incompressible ground), and a simple formula was given for the maximum trough depth for tunnels whose depth is more than twice the tunnel radius. The tunnel was also shown to experience a consider- able amount of settlement due to ground loss. An illustrative comparison of the solution and a Finite Element calculation showed very small differences, primarily because the effect of the Finite Element mesh boundaries is much smaller than for the problem of a buoyant, rigid tunnel. The ovalization problem is encountered when the initial stresses in the soil are uneven in the horizontal and vertical directions. In the solution it is assumed that the unevenness of the stresses causes the tunnel to deform from a circle into an oval. The solution shows that the settlement trough is much narrower than for the ground loss problem. The shape of the settlement trough was also shown to be much less dependent on Poisson’s ratio than the settlement trough for the ground loss problem. A simple formula was given for the maximum trough depth for tunnels whose depth is more than twice the tunnel radius. The three fundamental solutions discussed above were combined in Chapter 8 to illustrate their possible use as tools for the prediction of settlement troughs. The inclusion of buoyancy effects in the solution was shown to have a possibly considerable narrowing effect on the settlement troughs. Neglect of this effect might well be a reason why settlement troughs calculated in theory are often wider than those measured in practice. The addition of the buoyancy effect to these solutions, however, was also shown to double the number of dimensionless parameters in the problem, making it more difficult to draw general conclusions from the illustrations shown. Appendix A MULTI-VALUED COMPLEX POTENTIALS In a multiply-connected region (a region with holes), the complex potentials ϕ(z) and ψ(z) may be multi-valued due to the appearance of integrals in their derivation from the stress equations (2.1) – (2.5). In multiply-connected regions these integrals are not necessarily path-independent, which causes them to be multi-valued. It is the goal of this appendix to determine the nature of this multi-valuedness. This will result in a relation between the form of the complex potentials and the resultant forces applied to the holes. This analysis was first performed by Muskhelishvili [23] and can also be found in [10]. §A.1 Derivation from the Stress Equations For the present analysis we limit ourselves to the case of a finite region R with a single hole bounded by a contour L h (see Figure A.1). We begin by assuming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R m 0 M L h Figure A.1: Region with a hole. 81 82 Multi-Valued Complex Potentials Appendix A that the complex potentials ϕ(z) and ψ(z) are analytic everywhere in R and that the stress equations (2.1)–(2.5) are valid everywhere in R, as shown by Muskhelishvili [23]. We introduce the following notation for the derivative of the first complex potential: (z) = ϕ (z) = p(x,y) + iq(x, y). (A.1) It is clear that the expression ϕ (z) =(z) = p(x,y) must be single- valued, since the isotropic stress given by (2.4) is single-valued for physical reasons. In addition, we know that (z) is analytic because it is the derivative of an analytic function, from which it follows that p(x, y) must be harmonic. In order to determine (z), we must find the function q(x,y) associated with the real, single-valued, harmonic function p(x,y) such that (A.1) is analytic. In order for this to be the case, the function q(x,y) must be continuous, from which it follows that the perfect differential of q(x,y) exists and is given by dq = ∂q ∂x dx + ∂q ∂y dy. (A.2) The Cauchy-Riemann conditions, which follow from the analytic nature of (z), are ∂p ∂x = ∂q ∂y , ∂p ∂y =− ∂q ∂x . (A.3) Using (A.3), equation (A.2) can be written as dq =− ∂p ∂y dx + ∂p ∂x dy, (A.4) The function q(x,y) can now be determined by integrating: q(x,y) = M m 0 − ∂p ∂y dx + ∂p ∂x dy + C, (A.5) where m 0 (x 0 ,y 0 ) is an arbitrary fixed point and M(x,y) is a variable point (see Figure A.1 and Figure A.2), and where the integration path from m 0 to M is completely inside R. If the integral in (A.5) is path independent, then q(x,y) will be single-valued, otherwise q(x,y) will be multi-valued. Whether or not the integral in (A.5) is path independent can be established by examining the integral along a closed contour L composed of two separate integration paths from m 0 to M. We call these two integration paths 1 and 2 , and the closed path L is constructed by traversing 1 in the positive sense and 2 in the negative sense (see Figure A.2). We write the integral along L as follows: L − ∂p ∂y dx + ∂p ∂x dy = L (P dx + Q dy), (A.6) where P(x,y) =− ∂p ∂y , Q(x, y) = ∂p ∂x . (A.7) [...]... be written as ∂Q ∂P − ∂x ∂y (P dx + Q dy) = L dx dy, (A .8) A where A is the region bounded by L If the integral in (A.5) is path independent, then the integral in (A .8) must vanish, because then (P dx + Q dy) = 1 (P dx + Q dy), (A.9) 2 and it follows that (P dx + Q dy) − 1 (P dx + Q dy) = 2 (P dx + Q dy) = 0 (A.10) L It can be concluded from (A .8) that we must have ∂Q ∂P − =0 (A.11) ∂x ∂y at every point... necessary, but also sufficient for the existence of a single-valued function q(x, y) in a simply-connected region [23] We conclude that for simply-connected regions q(x, y) will be single-valued and can be determined by (A.5) 84 Multi-Valued Complex Potentials Appendix A In multiply-connected regions, however, the region A in the surface integral in (A .8) may contain holes Equation (A.11) may not necessarily... condition Since Bc is not equal to zero, the value of q(x, y) will depend on the path taken in (A.5) Denoting by q0 (x, y) the value obtained for q(x, y) from (A.5) by taking a path from m0 to M which does not make any complete circuits around the hole, we have, for an arbitrary path of integration, q(x, y) = q0 (x, y) + nBc , (A.14) where n is the number of times the arbitrary path makes a complete circuit... multi-valuedness of (z) explicitly, we split (z) into a multi-valued part plus a single-valued part This can be accomplished by noting that the following function is single-valued: ∗ Bc (z) = (z) − (A.16) log(z − zc ), 2π where zc is an arbitrary point inside the hole, and where log z is the multi-valued logarithm of z The function in (A.16) is single-valued because the jump in the imaginary part of the logarithmic... because the jump in the imaginary part of the logarithmic term cancels the jump in q(x, y) each time the path of integration in (A.5) completes a circuit around the hole In addition ∗ to being single-valued, the function (z) is also analytic in the finite region R since both (z) and log(z − zc ) are analytic in the finite region R It follows that we can write ∗ (A.17) (z) = Ac log(z − zc ) + (z), where we... satisfied at points within these holes, since these points are outside of the region It follows that the contour integral (A.6) is not equal to zero when taken around the hole: (P dx + Q dy) = Bc , (A.13) Lh+ where Lh+ is an arbitrary contour around the hole, and where Bc is some real value Because integrals within a simply-connected sub-region of R must be path independent, it can be reasoned (see [23]) that... real variable Ac for convenience The expression for ϕ(z) can be obtained by integrating (A.17): ϕ(z) = z z0 z ∗ (z) dz = Ac [(z−zc ) log(z−zc )−(z−zc )]+ z0 ∗ (z) dz+C, (A. 18) where z0 is an arbitrarily fixed point in R The integral of (z) may become multi-valued when the integration path makes a complete circuit around the hole ...Derivation from the Stress Equations 83 Section A.1 . dimensional linearly elastic problems can be used to obtain several fundamental solutions of tunneling problems. The method was used in Chapter 3 to derive the general mathematical repre- sentation of problems. depicted in Figure 8. 4, the magnitude of the ground loss V s in Peck’s formula has been adjusted to match the volume of the trough of the elastic solution. In the combined elastic solution the. negative). Larger values of ρ can be a result of grouting in very soft soils [12]. For many cases of open shield or open face tunneling, when grouting is not applied, the values of ρ tend to be close