INTERNATIONAL SOCIETY FOR

SOIL MECHANICS AND

GEOTECHNICAL ENGINEERING

This paper was downloaded from the Online Library of the International Society for Soil Mechanics and Geotechnical Engineering (ISSMGE). The library is available here:

https://www.issmge.org/publications/online-library

This is an open-access database that archives thousands of papers published under the Auspices of the ISSMGE and maintained by the Innovation and Development Committee of ISSMGE.

Analytical stability models for tunnels in soil

L.E. SozioPromon Engenharia, Sao Paulo, Brazil

ABSTRACT: Two distinct three dimensional analytical models are presented, envisaging the evaluation ofstability conditions in the zone of tunnel face. One of the models is based on a composite block system, formedby an upper block above tunnel crown bearing over a lower wedge, which slides towards the tunnel face.Allowanceis given for the inclusion of water pressures acting on the block and wedge faces. The second model is basedon a published lower bound solution, modified by including radial body forces and percolation body forcesto simulate gravity and flow net, both in spherical coordinates. The limitations of such analysis are outlined.Critical pressures determined for both models are compared with published analytical solutions and laboratorytest results.

1 INTRODUCTION

Excavation is the most critical event in respect of tun-nel stability. The majority of accidents happen at thatstage, both in conventional sprayed concrete lined tun-nels and in pre-cast segmental lined tunnels excavatedby boring machines.

Three dimensional (3D) numerical analyses maybe used to verify excavation stability conditions. 3Dnumerical models are still complex to prepare, andtime consuming, which currently restricts their appli-cation to exceptional cases. Developments are con-stantly being effected, and it can be envisaged that suchanalyses be carried out routinely in near future.

Notwithstanding the development stage of numer-ical analyses, it is felt that simpler analytical modelsshould be employed to enable a preliminary evaluationof stability conditions, early in design.

It is fundamental that analytical models be capableto simulate three-dimensional geometry, and the effectof flow nets in tunnels below groundwater level.

Two model configurations are proposed, based onpublished lower and upper bound solutions of the the-ory of plasticity, and modified as described herewith.

2 COLLAPSE MODELS

2.1 Combined falling and sliding blocksystem at tunnel face

This mechanism is formed by an upper block bearingover a lower wedge block that slides towards tunnelface as shown in Figure 1.

Figure 1. Combined block mechanism at tunnel face.

2.2 Upper block

The upper block mechanism derives from a two dimen-sional upper bound solution presented by Atkinson &Potts (1977).Their solution is based on a work rate cal-culation of a falling wedge with upper planes inclinedby an angle of ϕ with vertical, where ϕ is the soildilatancy angle. Hence the work done by friction isnull.This solution of the theory of plasticity implies anassociated flow rule ϕ = φ, where φ is the soil frictionangle.

This solution was modified to allow for:

(a) Three dimensional blocks.The upper planes delimitating this block are at anangle ν with vertical planes that contain their loweredges.

299

Copyright © 2006 Taylor & Francis Group plc, London, UK

Figure 2. Typical upper block shapes ϕ = φ > 0.

(b) Cohesive forces acting on the block upper sur-faces.The work done by cohesion is given by C ×

cos (ν) × δ where C is the resultant cohesive forceacting on that plane, and δ is the virtual verticaldisplacement.

(c) Water pressure resultant force, acting on the block.This force may be evaluated by studying a flow netaround the tunnel face and integrating the resultingpore pressure acting on the block upper surfaces.

The top edge of the upper block may be perpen-dicular or parallel to the tunnel axis, depending onthe ratio between its width (B) and depth (A or A′)as shown in Figure 2.

For the condition:

The block weight is given by:

where γ is the soil natural unit weight and ϕ = φ > 0is the soil dilatancy angle.

For the condition:

the upper block weight is given by:

It is always necessary to ensure that the upper blockdoes not intercept ground surface, otherwise the aboveformulae are no longer valid.

Figure 3. Prismatic upper block for ϕ = φ = 0.

For ϕ = φ = 0 the upper block is prismatic and itsheight H is given by its extension to the ground surface,as shown in Figure 3. The weight of this block is,

The vertical component of the cohesive force is givenby:

valid for both block shapes (A or A′), ϕ = φ > 0,

and

The work rate calculation of the falling upper blockholds the same equation as in the corresponding limitequilibrium analysis:

where Ft ≥ 0 is the supporting force acting on theupper block base, and Ut is the vertical componentof the resultant water thrust acting on the block.

2.3 Lower wedge block

A work rate calculation is performed for the lowerwedge block, which also holds the same equation asin the corresponding limit equilibrium analysis. Thegeometry of this block is shown in Figure 4. The lat-eral faces of this block are at an angleϕ = φ with tunnel

300

Copyright © 2006 Taylor & Francis Group plc, London, UK

Figure 4. Lower wedge block geometry and influence ofangle α on its depth and on upper block height.

main axis.The block sliding base is inclined at an angleα with vertical.

The forces acting on this block are:

– weight Wf ,– reaction to the upper block supporting force Ft ≥ 0,– forces resulting from cohesion (c) acting on the

sliding base and lateral surfaces,– forces resulting from water pressure acting on the

sliding base, and on lateral surfaces, respectivelyUb and Ul ,

– resultant thrust acting on the tunnel face, E.

The weight of the lower wedge block is given by:

The resultant thrust E is given by:

2.4 Combined upper and lower wedgeblock mechanism

The above derived equations were implemented in anelectronic calculation sheet. The resultant thrust E isthen calculated for α angles varying in 1◦ steps, whichleads to the maximum thrust value, as shown in thegraph of Figure 5.

The equivalent required support pressure at tun-nel face, peq, can then be evaluated by dividing themaximum thrust by the lower wedge face area:

Figure 5. Graphical example of resulting thrust acting onthe tunnel face, as function of angle α.

2.5 Collapse of a thick wall sphere model

The lower bound analytical solution presented byMühlhaus (1985) is based on a weightless and cohe-sionless medium. This solution was modified byincluding, in the equilibrium equation, radial bodyforces. Cohesion was also considered in the MohrCoulomb strength criteria. A further modification wasdeveloped by Negro & Braga (2000), who includedradial inward seepage forces (from a Laplace equation)and the equilibrium equation becomes, in sphericalcoordinates:

The submerged unit weight γ ′ and the water unitweight γw are applied when integrating the differentialequation in the zone

The natural unit weight γ is applied when integratingthe differential equation in the zone

The Mohr Coulomb criteria is given by:

where both tangential and radial stresses are expressedin terms of effective stress. For additional symbolsrefer to Figure 6.

301

Copyright © 2006 Taylor & Francis Group plc, London, UK

Rw

R

ri

p

s

c, φ, γ,

γ�, γw

Figure 6. Thick wall sphere model.

The inner surface radius of the sphere may beadopted equal to the tunnel radius (to simulate localstability at tunnel face). A larger radius can be adoptedin order to simulate a mechanism involving the tunnellength without invert arch, as shown in Figure 6.

The equilibrium equation, coupled with the MohrCoulomb criteria is integrated and the integration con-stant is solved by establishing the boundary conditions:

(a) p is the effective radial stress (support pressure)at ri. A pore pressure pw could also be consid-ered at this boundary; however for simplicity ofthe present analysis it was adopted pw = 0.

(b) s (“surcharge pressure”) is the radial stress at R.

The hydraulic gradient at a radius r is

The resulting equation for the support pressure is, inspherical coordinates:

valid for λ > 1 (λ = 1.5 is not an indetermination sincethe three terms containing λ − 1.5 in the denominatorcancel out).

For φ = 0, λ = 1, the integration of the equilib-rium equation, holding the same boundary conditions,results in the following support pressure:

It should be noted that terms that refer to both waterand submerged unit weights cancel out, and the solu-tion is independent of flow net effects, as it would beexpected of an undrained analysis.

If the equilibrium equation is expressed in termsof cylindrical coordinates, instead of spherical coordi-nates, then the support pressure corresponds to a planestrain condition (2D analysis).

Since most laboratory tests on critical support pres-sure were determined for a plane strain condition, thethick wall cylinder expression was developed, in orderto permit comparison of results between the analyticalmodel and laboratory tests. These expressions are notreproduced herein due to limited space.

It is also possible to analyse the expansion of asphere (or cylinder), when studying a “blow-out” fail-ure with such a model. In this case the radial andthe tangential stresses are, respectively, the major andminor principal stress, in opposition to the collapsecase. The seepage forces are directed outwards, and apore pressure pw is introduced at the inner boundary.Such solution may be of interest for “blow-out” anal-ysis of slurry type shields. Due to limited space, thissolution is not presented herein.

3 LIMITATIONS

(a) The combined block mechanism cannot be consid-ered as an upper bound solution, since the virtualdisplacement of the upper block is vertical, andthat of the lower wedge block has a horizontal com-ponent. The work done at the contact face betweenthe two blocks is neglected. Therefore this mecha-nism should be merely be considered as a solutionthat satisfies equilibrium in a limited way.

(b) The thick wall sphere model is not a lower boundsolution, mainly due to adoption of a radial bodyforce −γ ′ as an emulator of gravity, which is ver-tical, and not radial. In order to mitigate suchdiscrepancy, the outer sphere radius is adoptedas the one that gives a sphere thickness equal tothe depth of the actual tunnel axis, from groundsurface.

Also, a more realistic stress field where theratio between horizontal stress and vertical stressis different from 1.0, cannot be simulated in thismodel.

(c) A series of soil modeling limitations apply to theabove analysis, including:

– stratified soil profile or anisotropy cannot beconsidered.

– the models are only applicable to a continuummedium, as opposed to a jointed medium.

– the combined block mechanism is based on pla-nar surfaces, whereas observed failure surfacestend to be curved. Also the lower wedge block

302

Copyright © 2006 Taylor & Francis Group plc, London, UK

face is rectangular, therefore not geometricallymatching circular tunnel faces.

– the inner spherical surface of the thick wallsphere model also does not geometrically matchthe length of tunnel subject to stability analysis(see Figure 6).

(d) A consequence of the previous considerations isthat it cannot be ensured in any of the above solu-tions that collapse load will always be approachedfrom a safe or unsafe side (i.e. the solutions pre-sented in this paper are not lower or upper boundsolutions).

(e) Both models deal only with collapse loads. Straincompatibility is not accounted for, and no infor-mation is given on displacement prior to failure.

(f) Hydraulic piping failure is not dealt with in thesemodels.

4 COMPARISON WITH PUBLISHED MODELSAND LABORATORY TEST RESULTS

4.1 Circular tunnel heading in a purelycohesive material

Davis et al. (1980) presented upper and lower boundsolutions for a circular tunnel heading in a frictionlessmaterial. Their graph is reproduced in Figure 7, com-plemented by results from the combined block systemand thick wall sphere models of this paper.

The results are compared in terms of the StabilityRatio, as defined by those Authors, and plotted forvarious C/D ratios, where C is the soil cover over thetunnel and D its diameter.

For C/D ratios between 0 and 5, it was found thatthe Stability Ratios of the combined block and thickwall sphere fall within the upper and lower boundariesas given by Davis et al. (1980). Stability ratio valuesare independent of cohesion and unit weight values.Also, there is no influence of seepage forces as soil isadopted as a purely cohesive material.

4.2 Circular tunnel heading in cohesive andfrictional Mohr Coulomb material

Leca & Dormieux (1990) presented upper and lowerbound solutions applicable to cohesive and frictionalmaterial and compared their solution with laboratorycentrifuge tests based on a rigid cylinder with anexposed face, carried out in Nantes, France.

Their comparative table is reproduced in Table 1,to where results from the combined block system andthick wall sphere models of this paper were added.

Two materials were tested: γ = 15.3 kN/m3, c′=

2.3 kPa, φ′= 35.2◦, and γ = 16.1 kN/m3, c′

= 1.1 kPa,φ′

= 38.3◦, for two conditions of cover to diameterratio, as indicated in Table 1.

0

5

10

15

20

25

0 1 2 3 4 5C/D

N 'S

tab

ilit

y R

ati

o'

Davis UB

Combined Block

Thick Sphere

Davis LB

Figure 7. Comparison of analytical models for a friction-less material.

In this particular case support pressures from com-bined block mechanism are greater than for thick wallsphere model, which is opposite of what was found inthe previous comparison (cohesive material). In thisrespect attention is drawn to the comments outlined initem 3D). However, it is interesting to note that sup-port pressure in both solutions are relatively close toeach other, and excepting the first case, also close tocentrifuge test results.

It should be noted that the effects of groundwaterpercolation forces were not simulated in these tests, aswell as in the other published laboratory tests foundby the Author.

4.3 Plane strain tunnel in purelycohesive material

Centrifuge test results published by Davis et al. (1980),for a plane strain tunnel are compared with thick wallcylinder analysis results, as shown in Figure 8.The soilis frictionless and the relation between unit weight andtunnel diameter to the soil undrained shear strength is:

The analytical model gives results slightly on theconservative side, compared to laboratory tests.

4.4 Plane strain tunnel in cohesionless material

Centrifuge and normal gravity test results published byAtkinson & Potts (1977), for a plane strain tunnel and acohesionless sand of φ′

= 50◦ are compared with thickwall cylinder analysis results, as shown in Figure 9. Itcan be observed that for cohesionless materials andcover/diameter ratio in general larger than ∼0.5, thesupport pressure is constant.

The normal gravity test results are both above andbelow the analytical values, with some scatter. Thecentrifuge tests indicate a better fit, with the analyticalmodel being slightly on the unsafe side.

303

Copyright © 2006 Taylor & Francis Group plc, London, UK

Table 1. Critical pressures for Mohr Coulomb material.

Critical pressure, kPa

Lower bound Upper bound Thick CombinedC/D γ kN/m3 Leca & Dormieux Leca & Dormieux Centrifuge wall sphere block

1.0 15.3 29 2 6 3.6 4.51.0 16.1 29 3 3 4.4 5.52.0 15.3 46 2 4 3.6 4.52.0 16.1 44 3 4 4.4 5.5

0

1

2

3

4

5

6

1 2 3

C/D

p / S

u

Davis et al. (Centrifuge)

Thick Wall Cylinder (2D)

Figure 8. Centrifuge test results compared to thick wallcylinder analytical model, frictionless material.

0,00

0,05

0,10

0,15

0,20

0 1 2

C/D

p / g

D

Atkinson & Potts (1.0g)

Atkinson & Potts(Centrifuge)

Thick Wall Cylinder(2D)

Figure 9. Laboratory test results compared to thick wallcylinder analytical model, cohesionless material.

5 CONCLUSIONS

The analytical stability models presented in this paperare intended to enable a quick preliminary assessmentof tunnel face stability conditions, in three dimensionsand considering the effects of water pressure.

A series of limitations are indicated, leading to theconclusion that these are not to be considered upperor lower bound theorem solutions. For more elaborateanalysis, three dimensional numerical simulations arerecommended.

Comparisons with published analytical solutions,and few divulged laboratory tests, indicate reasonablygood agreement. However no tests were found wherethe effect of groundwater was simulated, and most testswere modeled in plane strain conditions.

The author would appreciate receipt of additionallaboratory tests, in order to increase the number ofverification cases of the models presented herein.

REFERENCES

ATKINSON, J.H., POTTS, D.M. (1977) “Stability of a shal-low circular tunnel in cohesionless soil” Geotechnique 27,No 2, pp. 203–215

MÜHLHAUS, H.B. (1985) “Lower bound solutions for circu-lar tunnels in two and three dimensions” Rock Mechanicsand Rock Engineering pp. 37–52

NEGRO,A., BRAGA, P.I. (2000) – Personal Communication.DAVIS, E.H., GUNN, M.J., MAIR, R.J.,

SENEVIRATNE, H.N. (1980) “The stability of shallowtunnels and underground openings in cohesive material”Geotechnique 30, No 4, pp. 397–416

LECA, E., DORMIEUX, L. (1990) “Upper and Lowerbound Solutions for the Face Stability of Shallow CircularTunnels in Frictional Material” Geotechnique 40, No 4, pp.581–606

304

Copyright © 2006 Taylor & Francis Group plc, London, UK