Mair, R. J., Taylor, R. N. & Bracegirdle, A. (1993). Gkotechnique 43, No. 2, 315-320

TECHNICAL NOTE

Subsurface settlement profiles above tunnels in clays

R. J. MAIR,* R. N. TAYLORt and A. BRACEGIRDLE*

KEYWORDS: case history; clays; settlement; tunnels.

INTRODUCTION For tunnelling schemes in urban areas, con- straints of existing tunnels and deep foundations often lead to new tunnels being constructed close beneath such structures, as shown in Fig. 1. Designers assessing the likely effect of tunnelling on structures relatively close to the tunnel crown need to know how subsurface settlement profiles develop, and how these relate to surface settle- ment profiles. The effect on structures such as those shown in Fig. 1 depends on the width of the subsurface settlement profile and on the magni- tude of the settlement. Fig. 2 shows a tunnel at depth z,, below ground level; a settlement trough develops at the ground surface giving a maximum settlement 6, over the tunnel centre line. At a deeper level, at a distance (z. - z) above the tunnel axis, a narrower settlement trough develops, giving a maximum settlement 6, over the tunnel centre line. This Note considers avail- able field measurements and centrifuge model test data, and addresses the question of how the widths of the settlement profiles and the magni- tudes of settlement 6, and S, vary with depth above tunnels constructed in clays.

SURFACE SETTLEMENT PROFILES A considerable amount of data is available

from field measurements of surface settlement profiles above tunnels in clays (e.g. Schmidt, 1969; Peck, 1969; Clough & Schmidt, 1980; O’Reilly & New, 1982; Rankin, 1988). It has been found that the shape of the surface settlement troughs developing during tunnel construction is reason- ably well represented by a Gaussian distribution, as shown in Fig. 2. The settlement S is defined as

5 = Snl,, exp ( -x2/2?) (1)

where S,, is the maximum settlement, which occurs above the tunnel centre line; in Fig. 2

Discussion on this Technical Note closes 1 October 1993; for further details see p. ii. * Geotechnical Consulting Group. t City University, London.

S max = 6,. The width of the settlement profile is defined by the important parameter i, which is the distance from the tunnel centre line to the point of inflexion of the trough (shown in Fig. 2); the total half-width of the settlement trough in practi- cal terms is given by about 2.5i.

Figure 3 shows data obtained by O’Reilly & New (1982) of the parameter i plotted against depth of tunnel axis below ground level z0 from field measurements of surface settlements above UK tunnels in clays. O’Reilly & New proposed the linear relationship (shown in Fig. 3)

i = 0.432, + 1.1 (2)

For practical purposes, it is often reasonable to assume that

i = Kz, (3)

The data are reasonably consistent with K = 0.5, as shown in Fig. 3. Rankin (1988) plotted i against z. for field measurements above tunnels both in the UK and worldwide, and concluded that K = 0.5 was a reasonable fit to most of the data.

The settlements caused by tunnelling are often characterized by the term ‘volume loss’ (sometimes referred to as ‘ground loss’), expressed as a percentage of the notional excavated volume of the tunnel. The volume of the settlement trough (per metre length of tunnel) is obtained from integration of equation (l), and is given by

v, = .\l(27+S,,, (4)

where D is the excavated diameter of the tunnel.

v, = $$

where D is the excavated diameter of the tunnel. For tunnels in clays, settlements during tunnel construction usually occur under undrained (constant volume) conditions, in which case the volume V, represents the additional quantity of clay excavated, over and above the theoretical volume of the tunnel of excavated diameter D. Combining equations (3)-(5) gives

S mBx = 0.313VLDZ/i (6a)

315

316

T MAIR, TAYLOR AND BRACEGIRDLE

Y/kWR

Emting tunnel

0 (a) (b)

Fig. 1. Typical subsurface structures above tunnels in urban areas

(W

From equation (6b) it can be seen that for a given volume loss and tunnel diameter, and for a con- stant value of K, the maximum surface settlement above a tunnel is inversely proportional to the depth of tunnel.

SUBSURFACE SETTLEMENT PROFILES In contrast to surface settlement measurements,

few field measurements of subsurface profiles are available. An array of at least three extensometers is needed to deduce information about the shape and width of subsurface settlement troughs. It is often assumed that the shapes of subsurface set- tlement profiles developed during tunnel con- struction are characterized by a Gaussian distribution, in the same manner as are those for

surface settlement profiles, and hence for tunnels in clays that equation (6b) is applicable (substituting the distance above the tunnel axis zO - z for zO). However, field measurements of subsurface settlements indicate that the parameter K increases with depth, giving proportionally wider settlement profiles closer to the tunnel. This is shown in Fig. 4, in which values of the trough width parameter i derived from subsurface settle- ment measurements have been plotted against depth z; both i and z have been normalized by the depth of the tunnel zO. The data are from field measurements with borehole extensometers in the ground above three tunnels in stiff London Clay (Attewell & Farmer, 1974; Barratt & Tyler, 1976) and above a tunnel in soft clay at Will- ington Quay (Glossop, 1978). The data from Will- ington Quay were obtained from limited measurements: only two of the extensometers were considered to be reliable. Centrifuge model

‘0 Fig. 2. Fom of surface and subsurface settlement profiles

SUBSURFACE SETTLEMENT PROFILES ABOVE TUNNELS 317

25 -

30 -

35 -

Fig. 3. Variation of surface settlement trough width parameter with tunnel depth for tunnels in clays (after O’Reilly & New, 1982)

tests provided the opportunity to make detailed subsurface measurements, and data from tests on tunnels in soft clay by Mair (1979) are also shown in Fig. 4. The line corresponding to a constant K = 0.5 is shown in Fig. 4. It can be seen that the departure of the width of the subsurface settle- ment profile from K = 0.5 becomes more marked with depth: at z/z,, = 0.8 the width of the profile is more than twice that given by K = 0.5.

The method of determining the value of i plotted in Fig. 4 is shown in Fig. 5. If the sub- surface settlement profile can be reasonably approximated as a Gaussian distribution, from equation (1) a plot of log, (S/S,,,) against (x/z$ will be linear with a slope -0.5(i/zJ2, as shown in Fig. 5. Examples of data from surface and sub- surface settlement profiles above a tunnel in London Clay (Attewell & Farmer, 1974) and a centrifuge model tunnel in soft clay (Mair, 1979) are shown in Fig. 5. Although there is some scatter of the data (particularly for the London Clay tunnel where settlements were small and hence measurements were prone to error), it can be concluded from the general linearity that sub- surface settlement profiles as well as surface pro-

0.4

s N

0.6

1 / / 1.0 ’

Location

l Green Park

4 Regent’s Park

(northbound) v Regent’s Park

(southbound)

. Willington

Quay o Centrifuge’

model 2DP q Centrifuge’

model 2DV

0

VA .

/ Equation (7)

Soil D: m zO: m

type London 4.1 29

Clay London 4.1 20

Clay London 4.1 34

Clay

Soft 4.3 13.5

clay

son

clay

Soft

clay

0.06 0.13

0.06 0.22

Reference

Attewell &

Farmer (1974) Barratl &

Tyler (1976)

Barratl &

Tyler (1976)

Glossop

(1978)

Mair (1979)

Malr (1979)

‘Models tested at 759: equwalent full-scale D = 4.5 m.

z0 = 9.6 m (2DP). 16.5 m (2DV)

Fig. 4. Variation of subsurface settlement trough width parameter with depth for tunnels in clays

files can be reasonably approximated in the form of a Gaussian distribution.

The line drawn through the data in Fig. 4 has the equation

i/z, = 0.175 + 0.325(1 - z/zo) (7)

By analogy with equation (3), for subsurface set- tlement profiles the trough width parameter i can be expressed as

i = K(z, - z)

Combining equations (7) and (8) gives

(8)

K = 0.175 + 0.325(1 - z/zJ

1 - z/z0 (9)

Equation (9) is plotted in Fig. 6, together with the values of K derived from equation (8) using the i values obtained from the field measurements and centrifuge model data. For larger z/z0 values, the width of the settlement profile would be sig- nificantly underestimated if K = 0.5 were to be assumed. At z/z0 = 0.8, for example, K = 1.2; correspondingly, the magnitude of the maximum settlement S,,, would be significantly overesti- mated by an assumption of K = 0.5 rather than

318 MAIR, TAYLOR AND BRACEGIRDLE

‘izo = 0.42. z/z0 = 0

i/q, = 0.21, -0.6

z/z,, = 0.61 -

Fig. 5. Determination of trough width parameter i from settlement data: (a) London Clay (Green Park); (h) soft clay (centrifuge model test ZDV)

1.2, as can be seen from equation (6b) when (z,, - z) is substituted for zO.

For the purpose of considering subsurface set- tlement troughs, the maximum value of z/z0 cor- responds to the level of the tunnel crown, i.e. (z, - a)/~, where a is the tunnel radius. Unfor- tunately, field measurements of settlement are rarely made very close to the tunnel crown, and therefore caution should be exercised in applying equation (9) (plotted in Fig. 6) to z/z,, values approaching unity.

The influence of assumed width of subsurface settlement profile on the predicted magnitude of settlement is shown in Fig. 7. The data plotted are measured subsurface settlements above the centre line of 4.1 m diameter tunnels in London Clay (after Mair & Taylor, 1993), and measure- ments above a 7.8 m diameter tunnel in London Clay (Ward, 1992). The data, normalized by tunnel radius a, are plotted against u/(zO - z), the distance (zO - z) being that above the tunnel axis at which the settlement was measured. The

K = i/(z,, - z)

0.5 1 ,o 1.5 I

Symbols as for Fig. 4

Fig. 6. Variation of K with depth for subsurface settlement profiles above tunnels in clays

SUBSURFACE SETTLEMENT PROFILES ABOVE TUNNELS

16 /

K = 06, VL = 1.4%

LO /

l-l / D=2a /

/’ Mair & Taylor (1993) I

, VL =

VL =

01 I I I

0 0.2 0.4 0.6 al(&) - z)

Location a: m zo: m l Green Park 2.07 29

A Regent’s Park 2.07 20 (northbound)

v Regent’s P&k 2.07 (southbound)

n Bank Station, 3.9 DLR

Reference Attewell & Farmer (1974) Barratt & Tyler (1976)

34 Barratt & Tyler (1976)

41 Ward (1993)

Fig. 7. Subsurface settlements above tunnel centre lines in London Clay (after Mair & Taylor, 1993)

volume loss V, for the 4.1 m diameter tunnels was 1.3-1.4% (Barratt 8~ Tyler, 1976; O’Reilly & New, 1982); V, for the 7.8 m diameter tunnel is not known, but the consistency of the normalized data between the 7.8 m and 4.1 m diameter tunnels suggests that a value of about 1.4% was also likely in the case of the 7.8 m tunnel. Assuming V, = 1.4% and a constant value of K = 0.5, and substituting (ze - z) for z0 in equa- tion (6b) leads to line A in Fig. 7; this clearly overpredicts S,,, at depth, i.e. at larger values of

o/(zo - z). Combining equations (6a) and (7), and noting

that D = 2a, gives

S max 1.25Q a -= -

a 0.175 + 0.325(1 - z/zO) z0 (10)

The depth z0 of the tunnels for which the mea- surements are plotted in Fig. 7 ranges from 20 m to 41 m, and the tunnel radius a is 2.07 m or

319

3.9 m. Substituting these values and V, = 1.4% into equation (10) gives a range defined by the curves B and C; these represent the range of pre- dicted S&a above the tunnels, assuming the larger subsurface trough widths indicated by the field measurements in Fig. 4. There is reasonable agreement between the measured subsurface set- tlements over the tunnel centre line and the pre- dicted values from equation (10). The agreement would be closer if account were taken of the pos- sible dilatant behaviour of the fissured London Clay, in that the volume loss at depth close to a tunnel may be greater than the volume of the surface settlement trough.

CONCLUSIONS Analysis of field and centrifuge model test mea-

surements of subsurface settlements above tunnels in clays reveals a generally consistent pattern of behaviour: the width of the subsurface settlement

320 MAIR, TAYLOR AND BRACEGIRDLE

troughs at depth is significantly greater than would be predicted by assuming a constant trough width parameter K of 0.5 (which is gener- ally applicable to surface settlement troughs). Smaller, more realistic, subsurface settlements and strains will be predicted if account is taken of

the proportionally wider trough widths at depth.

The variation of K with depth indicated by the

measurements in Fig. 6 can usefully be adopted

by designers when considering likely settlements

and strains of subsurface structures above

tunnels.

NOTATION radius of tunnel excavated diameter of tunnel distance from tunnel centre line to point of inflexion of settlement trough trough width parameter settlement maximum settlement within a settlement trough volume loss as ratio of notional excavated volume of tunnel volume of settlement trough (per metre length of tunnel) transverse distance from tunnel centre line depth of subsurface settlement trough below ground surface depth of tunnel axis below ground surface

REFERENCES Attewell, P. B. & Farmer, I. W. (1974). Ground defor-

mations resulting from shield tunnelling in London Clay. Can. Geotech. J. 11, 380-395.

Barratt, D. A. & Tyler, R. G. (1976). Measurements of ground movement and lining behaviour on the London Underground at Regent’s Park. Report LR 684. Crowthorne: Transport and Road Research Labor- atory.

Clough, G. W. & Schmidt, B. (1980). Design and per- formance of excavations and tunnels in soft clay. State of the art report, International Symposium on Soft Clay, Bangkok, Thailand, 1977. In Soft clay engineering, pp. 569-634. Amsterdam: Elsevier.

Glossop, N. H. (1978). Soil deformation caused by soft ground tunnelling. PhD thesis, University of Durham.

Mair, R. J. (1979). Centrifugal modelling of tunnel con- struction in soft ciay. PhD thesis, Cambridge Uni- versity.

Mair, R. J. & Taylor, R. N. (1993). Predictions of clay behaviour arohnd tunnels using plasticity solutions. In Predictive soil mechanics, pp. 449-463. London: Thomas Telford.

O’Reilly, M. P. & New, B. M. (1982). Settlements above tunnels in the United Kingdom-their magnitude and prediction. TunneUing ‘82, pp. 173-181. London: IMM.

Peck, R. B. (1969). Deep excavations and tunnelling in soft ground. Proc. 7th Znt. Con& Soil Mech., Mexico, State of the art 3, 225-290.

Rankin, W. J. (1988). Ground movements resulting from urban tunnelling. Proc. Conf Engng Geol. Under- ground Movements, Nottingham, pp. 79-92. London: Geological Society.

Schmidt, B. (1969). Settlements and ground movements associated with tunnelling in soil. PhD thesis, Uni- versity of Illinois.

Ward, W. H. (1993). In Recollections from the Wroth Memorial Symposium: Predictive Soil Mechanics, by S. E. Stallebrass et al. In Predictive soil mechanics, p. 828. London: Thomas Telford.