Settlement of the Surface

19

Settlement of the surface

19.1 Estimation of settlement

Apart from the assessment of stability, the determination of settlements atthe surface is very important in tunnelling. However, in geotechnical engi-neering, deformations can be forecast with less accuracy than stability. Thisis mainly because the ground has a nonlinear stress-strain-relationship, sothat one hardly knows the distribution of the stiffnesses. We consider heresome rough estimations of the settlement of the ground surface due to theexcavation of a tunnel. One should be aware of their limited accuracy.For the determination of the distribution of the surface settlements let us takefirst Lame’s solution (see Equ. 14.8) of the problem of a cylindrical cavity ina weightless elastic space, loaded by the hydrostatic stress σ∞. We regard thevertical displacement uv of the ground-surface shown in Fig. 19.1.

Fig. 19.1. Vertical displacement at the ground surface

The vertical component uv of the displacement u reads

uv =H

r· u . (19.1)

342 19 Settlement of the surface

With r2 = H2 + x2 we obtain from Lame’s solution

uv =σ∞ − p

2G· r20H

H2 + x2. (19.2)

The maximum settlement uv,max is obtained at x = 0, and the distribution ofthe settlement reads:

uv =uv,max

1 + (x/H)2.

This distribution1 is not realistic, when compared with measurements.2 It isalso inconsistent, because it uses a solution for the full space for a problemof the halfspace. A more realistic description of the measured settlement isobtained according to Peck by the Gauss-distribution

uv = uv,max · e−x2/2a2.

The parameter a (standard deviation) is to be determined by adjustment tomeasurements. It equals the x-coordinate at the inflection point of the Gauss-curve. It can be estimated with the diagram of Peck (Fig. 19.2)3 or accordingto the empirical formula:4

2a/D = (H/D)0.8 . (19.3)

D is the diameter of the tunnel and H is the depth of the tunnel axis(Fig. 19.3). For clay soils is a ≈ (0, 4 . . . 0, 6)H , for non-cohesive soils isa ≈ (0, 25 . . .0, 45)H .Another estimation of a is given in table 19.1.5

1It also follows from more complicated computations for a linear-elastic material,see A. Verruijt and J.R. Booker: Surface settlements due to deformation of a tunnelin an elastic half space. Geotechnique 46, No. 4 (1996), 753-756

2see e.g. J.H. Atkinson and D.M. Potts: Subsidence above shallow tunnels insoft ground. Journal of the Geotechnical Engineering Division, ASCE, Volume 103,No. GT4, 1977, 307-325

3Peck, R.B., Deep excavations and tunnelling in soft ground. State-of-the-Artreport. In Proceedings of the 7th International Conference on Soil Mechanics andFoundation Engineering, Mexico City, State-of-the-Art Volume, 1969, 225-290

4M.J. Gunn: The prediction of surface settlement profiles due to tunnelling. In’Predictive Soil Mechanics’, Proceedings Wroth Memorial Symposium, Oxford, 1992

5J.B. Burland et al., Assessing the risk of building damage due to tunnelling -lessons from the Jubilee Line Extension, London. In: Proceed. 2nd Int. Conf. on SoilStructure Interaction in Urban Civil Engineering, Zurich 2002, ETH Zurich, ISBN3-00-009169-6, Vol. 1, 11 -38.

19.1 Estimation of settlement 343

Fig. 19.2. Estimation of A by Peck

Soil a/H

granular 0.2 - 0.3

stiff clay 0.4 - 0.5

soft silty clay 0.7

Table 19.1. Estimation of a

The horizontal displacements uh of the ground-surface follow from the obser-vation that the resultant displacement vectors are directed towards the tunnelaxis (as shown in Fig. 19.1) i.e.

uh =x

Huv

The distribution of the settlements in longitudinal direction of a tunnel underconstruction is represented in Fig. 19.3.The volume of the settlement trough (per current tunnel meter) results fromthe Gauss-distribution to

Vu =√

2π · a · uv,max (19.4)

and is usually designated as volume loss6 (ground loss). The volume lossamounts to some percent of the tunnel cross-section area per current meter. If

6This designation is based on the conception that the soil volume Vu is dugadditionally to the theoretical tunnel volume


Fig. 19.3. Settlement trough over a tunnel (left); Approximate distribution of thesurface settlements in tunnel longitudinal direction. The shown curve coincides rea-

sonably with the function y = erf x =1√2π

Z x

0

e−y2/2dy (right).

this ratio is known by experience for a given soil type, then the maximum set-tlement uv,max can be estimated with (19.3) and (19.4). Mair and Taylor

7

give the following estimated values for Vu/A:Unsupported excavation face in stiff clay: . . . . . . . . . . . . . . . . . . . . . . . . . 1-2%Supported excavation face (slurry or earth mash), sand: . . . . . . . . . . . . 0.5%Supported excavation face (slurry or earth mash), soft clay: . . . . . . . . . 1-2%Conventional excavation with sprayed concrete in London clay: . . . 0.5-1.5%

The volume loss depends on the skill of tunnelling. Due to improved technol-ogy the volume loss has been halved over the last years.The evaluation of numerous field surveys and lab tests with centrifuge leads toan empirical relationship8 between the volume loss Vu related to the area A ofthe tunnel cross section and the stability numberN := (σv−σt)/cu. Herein, σv

is the vertical stress at depth of the tunnel axis, σt is the supporting pressure(if any) at the excavation face, and cu is the undrained cohesion. If NL is thevalue of N at collapse, then:

Vu/A ≈ 0.23 e4.4N/NL .

The estimations represented here refer to the so-called greenfield. If the surfaceis covered by a stiff building, then the settlements are smaller9.

7R.J. Mair and R.N. Taylor, Bored tunnelling in the urban environment. 14thInt. Conf. SMFE, Hamburg 1997

8S.R. Macklin, The prediction of volume loss due to tunnelling in overconsoli-dated clay based on heading geometry and stability number. Ground Engineering,April 1999

9’Recent advances into the modelling of ground movements due to tunnelling’,Ground Engineering, September 1995, 40-43

19.1 Estimation of settlement 345

The maximum settlement uv,max can also be roughly estimated by the follow-ing consideration: Let εr0 and εv0 be the radial strain and the volume strainat the crown, respectively. We can determine these values by a triaxial or bi-axial extension test in the laboratory. Then we have εϑ0 = εv0 − εr0 = u0/r0,whereby u0 is the displacement (settlement) of the crown and r0 is the radiusof the tunnel. We now assume

u = u0

(r0r

)d

(19.5)

for the distribution of the displacement above the crown10 (Fig. 19.4). Withdudr |r0 = εr0 it follows from Equ. 19.5

εr0 = −du0

r0.

With

u0 = εϑ0r0 = (εv0 − εr0)r0

it follows

d = − εr0

εv0 − εr0.

Fig. 19.4. Distribution of the vertical displacement u above the crown

The settlement of the surface (r = r0 + h) results then via

εr0 =du

dr

∣∣∣∣r0

= −u0d(r0r

)d−1 r0r2

∣∣∣∣r=r0

= −u0d

r0= −d εϑ0

to10cf. C. Sagaseta: Analysis of undrained soil deformation due to ground loss.

Geotechnique 37 , No. 3 (1987), 301-320; R.Kerry Rowe and K.M. Lee: Subsidenceowing to tunnelling. II. Evaluation of a prediction technique. Can. Geotech. J.Vol. 29, 1992, 941-954


u1 = (εv0 − εr0)r0

(r0

r0 + h

)d

.

The surface settlement is therefore smaller, the larger h is and the smaller thecrown displacement u0 is. One can keep the surface settlement small, if onekeeps the strain εr0 (and, consequently εv0) at the crown small. This can beobtained by rapid ring closure. Muller-Salzburg reported that he couldalways keep the crown displacement u0 between 3 and 5 cm.11

19.2 Reversal of settlements with grouting

With shield driving the surface settlements result mainly from the tail gap, ifthe excavation face is suitably supported (e.g. by pressurised slurry). Grout-ing of the tail gap is expected to reverse the surface settlement. However, itis observed that even if the grouted mass exceeds the theoretical gap volume,the surface settlement is not reversed.12 This fact can be explained in terms ofsoil mechanics: A cycle of loading and unloading leaves behind a net volumechange, usually a compaction (Fig. 19.5). The effect of soil compaction due toa loading-unloading cycle in shield tunnelling is shown in Fig. 19.6 which rep-resents the surface settlement due to closure of a 7 cm thick tail gap (curve a)and the one obtained after the grouting of the gap (i.e. reversing of the conver-gence of 7 cm). This result is obtained with the FEM programme ABAQUSand use of the hypoplastic constitutive equation calibrated for medium densesand.13

19.3 Risk of building damage due to tunnelling

For rough assessments of damage risk it is assumed that surface buildings arecompletely flexible, i.e. they have no stiffness and undergo the same deforma-tion as the ground surface of the ’greenfield’. This is a conservative assump-tion, because the real deformation will be reduced due to the stiffness of thebuilding as compared to the one of the greenfield.14 Evaluating the predicted

11L. Muller-Salzburg und E. Fecker: Grundgedanken und Grundsatze der’Neuen Osterreichischen Tunnelbauweise’. In: Grundlagen und Anwendungen derFelsmechanik. Felsmechanik Kolloquium Karlsruhe 1978, Trans Tech Publications,Clausthal 1978, 247-262

12S. Jancsecz et al., Minimierung von Senkungen beim Schildvortrieb am Beispielder U-Bahn Dusseldorf. Tunnelbau 2001, VGE Essen, 165-214

13M. Mahr, Settlements from tail gap grouting due to contractancy of soil, Felsbau,in print, 2004

14This section is based mainly on the book ”Building Response to Tunnelling.Case Studies from Construction of the Jubilee Line Extension, London”, Vol. 1,edited by J. B. Burland et al, Telford, London, 2001

19.3 Risk of building damage due to tunnelling 347

σ −σ1 2

ε1

ε1ε

v

Fig. 19.5. A loading-unloading cycle (here shown for the example of triaxial test)leaves behind a permanent densification

-5

-4

-3

-2

-1

0

1

0510152025303540

settl

emen

t [cm

]

distance from axis [m]

Fig. 19.6. Surface settlement due to gap closure (full line) and after grouting ofthe gap (dashed). Numerically obtained results.

settlements of the greenfield, we can assess that buildings with a maximum tiltof 1/500 and a settlement of less than 10 mm have negligible risk of damage.For the remaining buildings, a risk assessment must be undertaken which isstill based on the greenfield deformation and, therefore, is quite conservative(because settlements are over-estimated). Damage of buildings is assessed interms of tensile strain ε according to Table 19.2. The strain of the building


is to be inferred from the settlement trough. Burland and his co-authorsconsider a building as Timoshenko beam15 and derive its strains from thedeflection Δ (Fig. 19.7). Possible cracks (due to shear and due to bending, asshown in Fig. 19.8) are perpendicular to maximum tensile strains.

Fig. 19.7. Deflections Δ in sagging and hogging zones.

Fig. 19.8. Deformation and cracks due to pure shear (a, b) and pure bending (c,d).

However, the estimation of the strain in the building, assuming that it behaveslike a beam is more or less academic, because (i) it contradicts the startingassumption that the building is infinitely flexible, and (ii) the shear and bend-ing stiffnesses of this beam can hardly be assessed, especially for old masonrybuildings with vaults, timbering etc. For a rough estimation it appears reason-able to use table 19.2 with the assumption that the maximum tensile strainin a building situated over the inflexion point has the order of magnitude ofuv,max/a. For more elaborate estimations, a method is used in the cited bookthat attempts to take the stiffness of the building into account. This method

15Contrary to the ’classical’ or Euler-Bernoulli beam, where shear forces arerecovered from equilibrium but their effect on beam deformation is neglected, in theTimoshenko beam cross sections remain plane but do not remain normal to thedeformed longitudinal axis. The deviation from normality is produced by transverseshear.

19.3 Risk of building damage due to tunnelling 349

is based on the results of FEM computations. It should be added that time-dependent settlements may be observed several years after the damage of thetunnel. Thereby, the settlement through expands laterally.

Degree ofseverity

Description of typical damage Tensile strain ε (%)

Negligible Hairline cracks less than about 0.1 mm. 0 - 0.05

Veryslight

Damage generally restricted to internal wall fin-ishes. Close inspection may reveal some cracks inexternal brickwork or masonry. Cracks up to 1 mm.Easily treated during normal decoration.

0.05 - 0.075

Slight Cracks may be visible externally and some repoint-ing may be required to ensure weather-tightness.Doors and windows may stick slightly. Cracks upto 5 mm.

0.075 - 0.15

Moderate Doors and windows sticking. Service pipes mayfracture. Weather-tightness often impaired. Cracksfrom 5 to 15 mm.

0.15 - 0.30

Severe Windows and door frames distorted, floor slop-ing noticeably. Walls leaning or bulging noticeably,some loss of bearing in beams. Service pipes dis-rupted. Cracks 15. . . 25 mm. Extensive repair workinvolving breaking-out and replacing sections ofwalls, especially over doors and windows.

> 0.30

Verysevere

Beams lose bearing, walls lean badly and requireshoring. Windows broken with distortion. Dangerof instability. Cracks > 25 mm. Major repair isrequired involving partial or complete rebuilding.

Table 19.2. Relation between damage and tensile strain (according to Burland

et al.)

Settlement of the Surface

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