Initial stress

As described in Sections 2.3.1 and 3.4.1 Potts & Addenbrooke (1997) used a zone of reduced K0 around the tunnel to obtain better predictions of the greenfield surface settlement trough. In order to consider more building characteristics (for example the weight of a building and the corresponding consolidation of the soil before tunnel construction) such a zone is not included in the analyses presented in this thesis. This section investigates the effect of this change in initial conditions.

Table 4.2 summarizes results for a set of analyses including two different initial stress scenarios:

1. A coefficient of lateral earth pressure at rest of K0 = 1.5 over the whole mesh. This scenario is referred to as 1.5.

2. A zone of reduced K0= 0.5 around the tunnel, otherwise K0= 1.5. In vertical direction the zone extends to a distance of 6m (approximately 1.5 tunnel diameter) from the Mesh

Width K0 GF 1 storey 3 storey 5 storeys 1 storey 3 storey 5 storeys

100 1.5 -1.84 x 10-5 -2.26 x 10-6 -7.53 x 10-7 -9.82 x 10-7 0.123 0.041 0.053 70 1.5 -1.90 x 10-5 -2.32 x 10-6 -7.56 x 10-7 -9.84 x 10-7 0.122 0.040 0.052 100 reduced -4.38 x 10-5 -6.29 x 10-6 -1.07 x 10-6 -1.35 x 10-6 0.144 0.025 0.031 70 reduced -4.16 x 10-5 -6.38 x 10-6 -1.09 x 10-6 -1.35 x 10-6 0.153 0.026 0.032 100 1.5 5.81 x 10-5 4.96 x 10-5 2.82 x 10-5 1.35 x 10-5 0.854 0.487 0.234 70 1.5 5.81 x 10-5 4.97 x 10-5 2.80 x 10-5 1.34 x 10-5 0.855 0.483 0.232 100 reduced 1.03 x 10-4 6.96 x 10-5 3.85 x 10-5 1.84 x 10-5 0.675 0.373 0.178 70 reduced 1.11 x 10-4 6.98 x 10-5 3.84 x 10-5 1.84 x 10-5 0.626 0.345 0.165 100 1.5 -3.53 x 10-4 -1.21 x 10-5 -6.84 x 10-6 -4.80 x 10-6 0.034 0.019 0.014 70 1.5 -3.50 x 10-4 -1.19 x 10-5 -6.73 x 10-6 -4.72 x 10-6 0.034 0.019 0.013 100 reduced -5.70 x 10-4 -1.39 x 10-5 -7.99 x 10-6 -5.62 x 10-6 0.024 0.014 0.010 70 reduced -5.68 x 10-4 -1.37 x 10-5 -7.90 x 10-6 -5.56 x 10-6 0.024 0.014 0.010

Width is the dimension of the FE mesh

K0: 1.5 - global K0 over entire mesh

reduced - reduced zone around the tunnel, otherwise K0 = 1.5

Deflection Ratio MDR

Compressive horizontal strain Mεεεεhc

H o g g in g S a g g in g C o m p r. C o m p r. H o g g in g S a g g in g

Table 4.2: Deflection ratio and compressive strain for different mesh widths and initial stress conditions for analyses with 60m wide building, z0 = 20m.

0.0 0.4 0.8 1.2 1.6 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 M DR hog global K₀ = 1.5 zone of reduced K₀ Potts and Addenbrooke, 1997 design curve: e/b < 0.2 0.0 0.4 0.8 1.2 1.6 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 M DR sag ρ* [1/m] design curve: e/B = 0

global K₀ = 1.5 zone of reduced K₀ Potts and Addenbrooke, 1997

Figure 4.6: MDR _{for different initial stress} conditions compared with results by Potts & Addenbrooke (1997). Data are for 60m wide building, z₀ = 20m. 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.1 1.0 10.0 100.0 M ε hc α*

design curve: e/B = 0 global K₀ = 1.5 zone of reduced K₀ Potts and Addenbrooke, 1997

Figure 4.7: M²hc for different initial stress conditions compared with results by Potts & Addenbrooke (1997). Data are for 60m wide building, z₀ = 20m.

tunnel centre line, see Figure 2.14, Page 46. The height of the zone is identical with the diameter of the tunnel. This case is referred to as reduced in the table.

Both initial stress scenarios were analysed for 100m a 70m wide meshes. This was done in order to compare the results with the data by Potts & Addenbrooke (1997), who adopted a similar geometry. Results for the first stress situation can be found in Table 4.2 in lines 1 and 2 (for the 100m and 70m mesh respectively) of each deformation criterion; data for the second initial stress profile are given in lines 3 and 4.

The data reveal an increase (in terms of absolute value) of all deformation criteria when a zone of reduced K0 is introduced. This trend results from the narrower settlement trough observed when such a zone was used. The corresponding modification factors exhibit the opposite trend and reduce (apart from some scatter in hogging cases) when a K0-reduced zone is applied. The use of a global K₀ in this work results on the one hand in a less accurate

greenfield surface settlement prediction, while on the other hand it yields a more conservative estimate of the modification factors.

This trend is further demonstrated when plotting the deflection ratios versus relative bending stiffness ρ∗_{. Figure 4.6 shows this plot for the results of the 70m wide mesh. The} data points obtained by Potts & Addenbrooke (1997) are also given for comparative purposes. For the DRsag results it can be seen that the data points for a global K0 lie above their counterparts for a K0-reduced zone. For the 3 and the 5 storey building the reduced K0 points of this study coincide well with the results by Potts & Addenbrooke (1997). For the 1-storey building (lowest ρ∗_{) there is some scatter which can be explained by the different ap-} proaches adopted to determine the point of inflection and thus the deflection ratio (explained in Section 3.4.5): In this thesis a spread sheet calculation has been used while a graphical approach was chosen by Potts & Addenbrooke (1997). The modification factors MDRhog are low and therefore all data points lie close together when plotted on a scale that enables the design curves to be included.

A similar picture emerges when plotting M²hc _{against α}∗_{, shown in Figure 4.7. As in} the previous figure, results from a 70m wide mesh and different initial stress situations are compared with the results by Potts & Addenbrooke (1997). For all three cases of different relative stiffness, the data points of the analyses which applied a global zone of K0 = 1.5 lie above those for a zone of reduced K0. It can be seen that the change in initial stress conditions shifts the results slightly outside the Potts & Addenbrooke (1997) design curve.

This study shows that the use of a reduced zone of K₀ influences the deformation be- haviour of greenfield and building situations. However, it does not change the trend observed when varying the building stiffness. All MDR_{data points lie below the design curves provided} by Potts & Addenbrooke (1997) while the compressive strain modification factors lie slightly outside of the design curves.

4.2.4 Summary

This section has investigated how some of the key factors in the FE-analyses of Potts & Addenbrooke (1997) affected their results. These parameters were the volume loss (which was chosen to be 1.5%), the FE mesh width (70m) and the initial stress conditions (where a zone of reduced K0 was used around the tunnel). The effects of varying these factors have

been assessed.

• It has been found that for an increase in volume loss there is a corresponding increase in deflection ratio (both hogging and sagging); this relationship is approximately linear. It is therefore possible to linearly adjust these criteria to a common volume loss. The response of strain on VL is not linear. However, results can be linearly interpolated, when small variations of VL are considered.

• Increasing the FE mesh width from 70m to 100m has only a minor influence on the results. In general the greenfield surface settlement trough becomes deeper and narrower when the mesh width is reduced. The development of volume loss over excavation increments is not affected by the change in mesh width.

• The use of a global coefficient of earth pressure at rest of K0 = 1.5 instead of a zone of reduced K0 around the tunnel leads to smaller values of deformation criteria. This can be explained by the wider settlement trough obtained from an analysis in a high K0- regime. The modification factors, however, increase. The modification factors presented in this study are therefore more conservative in comparison to those obtained by Potts & Addenbrooke (1997), who used a K0-reduced zone in order to obtain more realistic greenfield surface settlement troughs.