Subsurface movement

In the previous sections only tunnel induced ground deformation at ground surface level was described. Although surface settlement is the most straight forward way to describe ground deformation it only gives a limited picture of the mechanisms which control tunnel-soil and, if present, also the tunnel-soil-building interaction.

Mair & Taylor (1993) applied the plasticity solution for the unloading of a cylindrical cavity to describe vertical and horizontal subsurface movement. This solution predicts a linear relation when plotting ground movement as Sv/R or Shx/R against R/d where R is the tunnel radius and d is the vertical or horizontal distance from the tunnel centre. They presented field data which were in good agreement with the predicted trend.

In their work Mair & Taylor (1993) only focused on vertical soil movement above the tunnel centre line and on horizontal movement at tunnel axis level. When describing the shape of transverse subsurface settlement troughs, a Gaussian curve is commonly adopted. Mair et al. (1993) showed that this assumption is in reasonable agreement with field data.

To predict the settlement trough width parameter i of subsurface settlement troughs it would be straight forward to apply Equation 2.15 substituting (z0− z) for the tunnel depth z0:

i = 0.5(z0− z) (2.16)

Mair et al. (1993), however, presented field and centrifuge data which showed that subsurface settlement troughs are proportionally wider with depth z. Figure 2.8 shows these measure- ments. The trough width i normalized by z0 is plotted against normalized depth z/z0. The

Figure 2.8: Variation of trough width pa- rameter i of subsurface settlement troughs with depth (after Mair et al., 1993).

Figure 2.9: Variation of K of subsurface settlement troughs with depth (after Mair et al., 1993).

dashed line represents Equation 2.16. It can be seen that it underpredicts i with depth. In contrast, the solid line, described by

i z0 = 0.175 + 0.325 µ 1 − z z0 ¶ (2.17) matches better the measurements. Changing Equation 2.14 to

i = K(z0− z) (2.18)

and substituting into Equation 2.17 leads to K = 0.325 + 0.175

1 −_zz₀ (2.19)

It can be shown that for z = 0 the above equation yields K = 0.5 which is consistent with Equation 2.15 describing the trough width at soil surface level. Figure 2.9 plots K against z/z0 for the measurements shown in Figure 2.8. The curve, described by Equation 2.19 is also included. The graph shows that for large values of z/z0 a constant value of K = 0.5 would underestimate the width of subsurface settlement troughs.

Figure 2.10: Subsurface settlement above tunnel centre line (after Mair et al., 1993). Combining Equations 2.2 and 2.3 with Equation 2.17, the maximum settlement of a subsurface trough can be expressed as

Sv,max R = 1.25VL 0.175 + 0.325 ³ 1 − z z0 ´R z₀ (2.20)

where R is the radius of the tunnel. Figure 2.10 shows the maximum settlement (normalized against tunnel radius) plotted against R/(z₀− z). The term (z₀− z) is the vertical distance between settlement profile and tunnel axis. The graph includes field data from tunnel con- struction in London Clay together with curves derived from Equation 2.16 (curve A) and 2.20 (curves B and C). For the latter equation upper and lower bound curves are given for the range of tunnel depths and volume losses of field data (solid symbols) considered in the graph (see Mair et al. (1993) for more details). The straight solid line refers to the plasticity solu- tion given by Mair & Taylor (1993). The figure shows that the field data are in reasonable agreement with both approaches. In contrast, Equation 2.16 (curve A) would overpredict Sv,max of subsurface settlement troughs.

Nyren (1998) presented greenfield measurements from the Jubilee Line Extension (St. James’s Park) and compared the data with the result of Mair et al. (1993). For the graph shown in Figure 2.8 his measurements were in good agreement. However when adding the data

points obtained from the St. James’s Park greenfield site to the graph shown in Figure 2.10 the new measurements were significantly higher in magnitude than the data presented by Mair et al. (1993). Nyren (1998) concluded that this difference was due to the high volume loss of VL = 3.3% measured at St. James’s Park. By normalizing Sv,max in Figure 2.10 not only against tunnel radius R but also against VL he demonstrated that all his data were in good agreement with the measurements originally shown in the diagram.

Heath & West (1996) present an alternative approach to estimate the trough width and maximum settlement of subsurface settlement troughs by applying the binomial distribution rather than the Gaussian curve to describe the settlement trough. This approach yields

i i0 = r z₀− z z0 (2.21) where i₀ is the trough width at ground surface level. The maximum subsurface settlement Sv,max is then proportional to (z0− z)−

1

2. They suggest that for design purposes the Gauss curve should be adopted to describe settlement troughs but that the trough width should be determined from the above equation. When comparing predictions of Sv,max with depth obtained from their framework with the relation given by Mair et al. (1993) they conclude that both approaches give similar predictions over depths between z/z0 = 0 to approximately 0.8. Close to the tunnel, however, Heath & West (1996) predict larger values of Sv,max which is in good agreement with the field data they present.

In Section 2.2.1.1 the horizontal surface soil displacement was derived from the vertical surface settlement assuming that resultant displacement vectors point to the tunnel centre, leading to Equation 2.5. Following the work of Mair et al. (1993), Taylor (1995) stated that in order to achieve constant volume conditions displacement vectors should point to the point where the line described by Equation 2.17 (i.e. the solid line in Figure 2.8) intersects with the tunnel centre line. This point is located at 0.175z0/0.325 below tunnel axis level.

The fact that the assumption of soil particle movement towards a single point at the tunnel axis is not adequate in the vicinity of the tunnel was highlighted by New & Bowers (1994) by presenting subsurface field measurements of the Heathrow Express trial tunnel. The results show that predictions of subsurface troughs based on the point sink assumption were too narrow and consequently the settlements above the tunnel centre line were over-predicted. Instead they proposed to model the ground loss equally distributed over a horizontal plane

Figure 2.11: (a) Distribution of i for subsurface settlement troughs with depth; (b) focus of vectors of soil movement (after Grant & Taylor, 2000).

at invert level, equal in width to the tunnel. Using this approach their predictions were in better agreement with the field data.

Grant & Taylor (2000) present data of subsurface settlement troughs obtained from a number of centrifuge tests. Their test results are in good agreement with Equation 2.17 apart from a zone in the vicinity of the tunnel where the test data show narrower subsurface troughs, and close to the surface where wider troughs were measured. This distribution of i with z is schematically shown in Figure 2.11a. Following the statement of Taylor (1995) that soil displacement vectors point to the interception of the distribution of i with z, they conclude that vectors point in the direction of the tangent of the distribution shown in Figure 2.11a. The intersections of the tangents for different depths with the tunnel centre line are shown in Figure 2.11b. It follows from this framework that horizontal soil displacement close to the surface is underestimated when Equation 2.17 is adopted to determine the focus point of soil movement. Their laboratory data support this trend by showing high horizontal displacement near the surface.

Grant & Taylor (2000) conclude that the high values of i close to the surface are associated with the free ground surface boundary in their tests. They point out that such a condition

is rare in an urban environment where even a thin layer of pavement may provide sufficient restraint to reduce or even to negate this free-surface effect.

Hagiwara et al. (1999) investigated the influence of soil layers of varying stiffness overlying clay in which a tunnel is being constructed. They performed a number of centrifuge tests in which sand layers of different density were placed over a clay layer. The thickness of the overlaying soil and the level of the water table were chosen to maintain the same stress regime within the clay layer in all tests. Results from these tests were compared with a reference case containing clay only.

To quantify the stiffness of the top layer compared to the clay they calculated a relative shear capacity which related the shear stiffness (at very small strains) and the thickness of the top layer to the equivalent measures for the reference clay layer. For each test they determined i for subsurface troughs within the clay layer.

When comparing i of subsurface troughs from different tests with the results from the clay-only case they showed that settlement troughs become wider as the stiffness of the top layer increases. Their tests demonstrated that tunnel induced settlement behaviour is affected by the interaction of stiffness of an overlaying material with the soil in which the tunnel is constructed.