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The prediction of volume loss due to tunnelling in overconsolidated clay based on heading geometry and stability number


A re-assessment of the volume loss - stability number relationship for a single tunnel in overconsolidated clay has been carried out. New case history data have been added to the plot of volume loss versus load factor originally proposed by Mair et al (1981), and this has been presented in semi-log space to improve resolution at load factors below 0.5, the range commonly encountered for tunnels in overconsolidated clay. Despite some scatter in the plot due to uncertainties associated with P/D ratio and the shear strength in the mass, good linearity of the data is evident enabling a rational prediction of the potential volume loss to be made within an upper and lower bound. This work suggests that the geometry of the tunnel heading and stability ratio at the tunnel horizon appear to be dominant factors in determining short term (undrained) volume loss.


Traditional design procedures for the assessment of the risk of damage due to tunnelling induced settlement involve the calculation of a 'greenfield' settlement trough. Such calculations would assume a normal probability curve to describe the shape of the settlement trough (Peck, 1969). Critical to such calculations is the assessment of volume loss (VL), which is the ratio of the volume of the surface settlement trough per metre run (Vs, m3/m) to the excavated face area (A, m2), usually expressed as a percentage.

The selection of an appropriate value of the volume loss parameter in overconsolidated clay is commonly based on engineering judgement and experience from previous projects in similar ground using similar techniques. O'Reilly and New (1982) list a number of case histories (their Table 1) which can be used as a basis for assigning a suitable value of VL. They state that, for tunnelling with or without a shield in stiff fissured London Clay, VL ranges between 0.5%-3% with values normally falling between 1%-2%.

Alternatively Clough and Schmidt (1981), Mitchell (1983) and Attewell et al (1986) have provided direct correlations between VL and stability number (N) where the stability number is defined (after Broms and Bennermark, 1967) as:

N = (v - T)/Su

v is the total vertical stress at tunnel axis level, T is the internal support pressure (compressed air for example) and Su is the undrained shear strength of the clay. O'Reilly (1988) correlated volume loss against load factor (LF) for a number of London Clay case histories where LF was defined as the ratio of the working stability number over the stability number at collapse (Ntc), ie

LF = N/NtcMair et al (1981).

This paper proposes an expression relating LF and the short term volume loss (VL) in overconsolidated clay, which may be used for design purposes.

Previous work

The volume loss observed during tunnelling in clay may be separated into the following components (Attewell and Farmer, 1974, Cording and Hansmire, 1975 and Attewell et al, 1986):

1. Ground lost at the face due to movements in an axial direction (face intrusion or face 'take')

2. Radial movements around the shield due to an overcut or the use of a bead on the shield, or due to 'diving', 'pitching', 'yawing' or ovalisation of the shield (radial 'take')

3. Radial movements due to temporary loss of support at the rear of the shield or tunnelling machine during lining construction (only where no 'tailskin' is used);

4. Closure of the ungrouted annulus around the newly completed ring (non- expanded type of linings)

5. Closure of the grouted annular gap due to 'bleeding' and curing of the grout, insufficient grout or loss of grout

6. Time dependent and consolidation effects in the ground

7. Subsequent permanent distortion of the lining

In this note the short term volume loss (VL), is considered to comprise items 1 to 5 above, in a two dimensional sense. Long term consolidation volume loss has not been considered.

When estimating a value of volume loss it is difficult to separate the radial ground displacement components and axial movements. Attewell et al (1986) provide guidance on estimating the magnitude of the various components of volume loss as listed above, however this requires a detailed knowledge of the construction methodology and ground response prior to commencement of the project.

A number of authors have attempted to empirically relate stability number (N) directly to VL such as Clough and Schmidt (1981), Mitchell (1983) and Attewell et al (1986). Expressions suggested by these workers are summarised in table 1.

Based on a series of centrifuge tests of model tunnels taken to failure, Mair et al (1981) produced a plot of VL versus LF for C/D ratios of 3.11, 1.67 and 1.5, and P/D ratios = (ie plane strain tests). The geometric variables C/D and P/D are illustrated in Figure 1.

These data demonstrated that a relationship between volume loss, the geometry of the tunnel heading and the stability number existed within an upper and lower bound for C/D between 3.11 and 1.5. This plot is reproduced in Figure 2, with additional data from Atkinson and Potts (1977). It can be seen that for C/D = 0.77 (Atkinson & Potts, 1977) the data points fall outside the bounds established by Mair et al (1981). This appears to suggest that the relationship between LF and VL illustrated by Mair et al (1981) does not hold for C/D ratios below approximately 1 where P/D = .

Previous work by Kimura and Mair (1979) provided a series of curves relating Ntc to P/D and C/D, and is reproduced in Figure 3. Data from actual case histories where collapse has occurred has been added, which in general confirms the original work and extends the values of Ntc forC/D ratios 1 (P/D = 0 only). O'Reilly (1988) proposed that by using this plot, the load factor for a site could be determined and a prediction of volume loss obtained from Mair et al's (1981) plot (Figure 2). Also shown in Figure 3 is O'Reilly's proposed extension of the relationships for various P/D ratios greater than C/D = 3. Kimura and Mair (1979) originally suggested that the values for Ntc may become constant beyond C/D = 3.O'Reilly (1988) proposed that the shear strength of the clay to be used to calculate stability number and load factor should be taken as the mass (or 'bulk') shear strength (Su mass) which accounts for the presence of fissures and bedding plane partings etc in the clay. He proposed a general Su mass design line for London Clay based on a statistical analysis of Skempton's (1959) correlation for the results of 38mm diameter triaxial tests. The value of Su mass was to be calculated as an average value betweentunnel axis and the surface for C/D 2.8, or as an average value within a diameter above and below the tunnel crown for C/D 2.8.

The proposed method for the estimation of short term volume loss

Field monitoring data have been obtained from recent research and development work carried out by Brown & Root. In order to check the validity of the results these data were used to construct a plot of VL versus LF. To enhancethe clarity of the plot at LF 0.5, typical of most London Clay tunnelling projects, a semi-log plot was employed. Additional data points were also added from recently published data from other sites. This plot is shown in Figure 4, and the field data are summarised in Table 2.

P/D ratio

When calculating individual site data the P/D ratio was generally based on the dimensions of the shield or TBM where a bead has been employed (Kimura and Mair, 1981) and it was considered that this would give a reasonable estimate of P/D at load factors less than 0.5. However as Attewell and Farmer (1974) noted, due to the time dependency of the clay movements, the clay may close in on the shield carcass prior to reaching the tail, reducing the P/D ratio. In this case P/D will be some value less than that determined from the shield dimensions (Pmin in Figure 1). Similarly where the 'stand-up' time is high (particularly at low load factors) and the grouting cycle lags behind the advancement of the shield P/D may be higher (Pmax in Figure 1).

Where a tunnel was advanced without a shield, P/D was derived from the estimated overdig ahead of the completed crown section of the lining.

C/D ratio

The depth of cover (C) was in all cases taken as the thickness of clay above the tunnel crown. This approach is consistent with that taken by Atkinson and Potts (1977).

A minimum value of LF = 0.2 is shown on Figure 4 which reflects the range of the centrifuge test data used and also defines the approximate load factor at which ground movements will be essentially elastic. This assumption is based on the work of Mair and Taylor (1993) where for a typical (three-dimensional) tunnel heading in an elastic-perfectly plastic soil, with C/D = 1.5 and Ntc = 6, elastic behaviour can be expected at N = 4/3.

Data from Atkinson and Potts' (1977) 1g plane strain model tests on overconsolidated kaolin clay have been included for the C/D ratio = 1.2 only. These data lie on the lower curve in Figure 2. It is inferred that at lower values of C/D (say less than 1) the ground movements may not conform to those commonly observed for deeper tunnels and the linear relationship suggested in Figure 4 may not be valid for such cases.

Stability number at collapse (Ntc)

Data from plane strain centrifuge tests carried out and quoted by Mair et al (1981, 1982) are also included in Figure 4. Mair et al's tests comprised the gradual deflation of a rubber membrane representing the internal support pressure for an unlined tunnel, thus modelling volume loss due to radial displacements only. These tests cover a wide range of stability number and associated volume losses, within a carefully controlled model set- up, where the vagaries of in-situ conditions such as varying strength with depth, soil structure and anisotropy could be eliminated.

Despite limitations in the ability of such tests to replicate actual construction effects in situ (Taylor, 1998), it is possible to determine directly the actual value of stability number at collapse (Ntc) for that given geometry. As such, by calculating a load factor the effects of the model geometry are effectively 'normalised' and the results of the tests can be compared against other case history data.

Values of Ntc were determined directly from Figure 3 for each case history examined, in order to determine the value of LF.

Case history data

Table 2 summarises 22 case histories (27 individual records) of tunnels in stiff overconsolidated clay, 15 of which are London Clay sites. Three case histories from recent work undertaken by Brown & Root are also included. These data were obtained from the Longford Street spur tunnel in Central London, the Crown Wharf research site in east London (Macklin and Field, 1998) and above a utility tunnel beneath Regents Park. For each site details of the tunnel heading geometry, and shear strength from both 100mm diameter unconsolidated undrained triaxial (UUT) tests and correlations from SPT determinations were known. The values of mass shear strength (Su mass) were based on both UUT test data and SPT correlations, as the SPT data was considered to be representative of the fissured strength of the clay (Clayton, 1995).

Where this approach to estimating Su mass in London Clay was used, an estimate based on the Su mass versus depth correlation proposed by O'Reilly (1988) was also calculated, which was found to be approximately 30% lower in all but one case. As such a range of LF based on these two estimates of Su mass are shown on Figure 4. In addition, in some of the London Clay case histories, it was not stated how the quoted values of shear strength were determined and whether the values represented mass or intact sample shear strengths. In these cases the quoted value and the O'Reilly (1988) estimate were used.

Seven of the cases describe tunnels where no shield or tunnelling machines were used and support was provided by ribs with cast insitu concrete linings. Three of these are from the Heathrow express trial tunnel (Bowers et al, 1996) constructed using NATM techniques. These data points suggest that despite considerable differences in the construction method, a good correlation may be obtained with data from the cases constructed using more traditional methods.

The case quoted from Simic and Craig (1997) comprises an average of six individual records covering the measured ground losses due to earth pressure balance tunnelling through hard Miocene silty clay and marl. A single average Su mass of 500kPa was quoted. Records of the average earth pressure in the chamber at the face enabled an estimate of the stability number. While lying near the 'elastic' limit of the range of load factors the average value of VL appears to lie within the upper and lower bound design lines drawn on Figure 4. Two of the records do lie below the lower bound for VL, however this might be expected at such marginal load factors. It would thus appear that earth pressure balance tunnelling, if detailed records are available, might also be assessed using this approach.


It is clear from Figure 4 that, despite potential variation during the selection of C/D, P/D and Su mass, the majority of the field and laboratory data lie within a distinct range enclosed within dashed lines on the plot. These lines may be taken to represent upper and lower bound design lines to estimate volume loss for a given load factor, for C/D ratios in excess of 1. The bounds have been based on a visual assessment of the scatter of data around a linear regression line given by the equation:

VL(%) = 0.23e4.4(LF)for LFgreater than or equal to0.2

However it is recommended that for design purposes the range of values should be considered. It is encouraging that good agreement has been achieved with the general guidelines quoted by O'Reilly and New (1982) for overconsolidated London Clay.

The calculation of an appropriate value of P/D for shield or

TBM driven tunnels may be subject to approximation. However all things considered, a value of P less than or equal to the dimensions of the shield or machine would appear to give reasonable results. Nonetheless, attention should be given to the excavation sequence and techniques employed at the face and to the time dependent behaviour of the clay.

The selection of an appropriate value of Su mass would appear to be another area where a consistent approach is required. It is considered that reliance on the SPT test and 100mm diameter triaxial tests would provide a reasonable approach and would lead to repeatable estimates of volume loss during the risk assessment stage.

This work suggests that geometric properties of the tunnel heading and the insitu stress/undrained strength ratio are equally as important as workmanship in estimating the volume of short term ground loss during tunnelling. However the design charts presented by Kimura and Mair (1979) upon which this method is based covered the limited range of C/D ratios between about 1 and 3. Additional centrifuge work would be of value to provide confirmation of the values of Ntc at C/D ratios outside this range.


Thanks go to my colleagues at Brown & Root, in particular Dr Y C Lu, for critical review of this paper. In addition the opportunity to publish this work under the auspices of the R&D budget made available by Brown & Root is gratefully acknowledged.


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