Some Approximate Solutions for Shallow Tunnels

16

Some approximate solutions for shallow tunnels

In shallow tunnels the neglection of vertical stress increase with depth dueto gravity is not justified. Thus, the solutions presented so far based on hy-drostatic primary stress are not applicable. In this section some approximatesolutions for shallow tunnels are presented.

16.1 Janssen’s silo equation

In silos (i.e. vessels filled with granular material) the vertical stress does notincrease linearly with depth. Silos are, therefore, archetypes for arching. Theequation of Janssen (1895)1 is used for the design of silos. To derive it, weconsider a slim silo with a circular cross section (Fig. 16.1).

Fig. 16.1. To the derivation of the equation of Janssen.

Upon a disk with the radius r and the thickness dz act the own weight πr2γdz,the stress forces σπr2 and −(σ + dσ)πr2, as well as the shear force −τ2πrdz

1Janssen, H.A. (1895), Versuche uber Getreidedruck in Silozellen. Zeitschrift desVereins deutscher Ingenieure, Band 39, No. 35

316 16 Some approximate solutions for shallow tunnels

due to wall friction τ . The latter is proportional to the horizontal stress σH ,τ = μσH , and σH is assumed proportional to the vertical stress σ, i.e. σH =K0σ. K0 is the earth-pressure-at-rest-coefficient2, and μ is the wall frictioncoefficient. Equilibrium requires that the sum of these forces vanishes. In thisway one obtains the differential equation

dσ

dz= γ − 2K0μ

rσ .

With the boundary condition σ(z = 0) != 0 it has the solution

σ(z) =γr

2K0μ(1 − e−2K0μz/r) . (16.1)

Thus, the vertical stress cannot increase above the value γr/(2K0μ).This derivation of Equ. 16.1 also applies if the silo has no circular cross section.Then, r is the so-called hydraulic radius of the cross section:

A

U=r

2,

where A is the area and U the circumference of the cross section.If the adhesion ca acts between silo wall and granulate (soil), then Equ. 16.1is to be modified as follows:

σ(z) =(γ − 2ca/r)r

2K0μ(1 − e−2K0μz/r) . (16.2)

If the surface of the granulate is loaded with the load q per unit area, then theboundary condition at z = 0 reads σ(z = 0) = q. This leads to the equation

σ(z) =(γ − 2ca/r)r

2K0μ

(1 − e−2K0μz/r

)+ qe−2K0μz/r . (16.3)

The theory of Janssen points out that the granulate stored in silos ’hangs’partly at the silo walls by friction. This results in high vertical stresses in thesilo walls, which may buckle. The mobilization of the shear stresses on thewall presupposes sufficiently large relative displacements between granulateand silo wall. If the granulate is moved upwards, the the sign of wall shearstress is reversed. Equation 16.1 then has to be replaced by

σ(z) =γr

2K0μ(e2K0μz/r − 1) .

Janssen’s equation is often used to assess arching above tunnels:

2according to Jaky is K0 ≈ 1 − sinϕ for un-preloaded cohesionless materials

16.1 Janssen’s silo equation 317

1. Terzaghi regarded the range ABCD represented in Fig. 16.2 as silo,3

with the width b (for the plane deformation considered here the hydraulicradius is r = b), on the lower edge BC of which acts the pressure p.4 Hethus obtained the following equation for the load p acting upon the roofof a tunnel with rectangular cross section:

p =(γ − 2c/b)b2K tanϕ

(1 − e−2Kh tan ϕ/b

)(16.4)

Fig. 16.2. To the derivation of the equation of Terzaghi.

2. To estimate the pressure needed for the support of the tunnel face (e.g.of a slurry shield), Janssen’s equation is used. The assessment of facestability is often accomplished following a collapse mechanism originallyproposed by Horn (Fig. 16.3).5 To take into account the 3D-character ofthe collapse mechanism, the front ABCD of the sliding wedge is taken ofequal area as the one of the tunnel cross section. On the sides BDI andACJ is set cohesion and friction (in accordance with the geostatic stressdistribution σx = Kγz). The vertical force V is computed according tothe silo formula. The necessary support force S is determined by equilib-rium consideration of the sliding wedge, whereby the inclination angle ϑ isvaried until S becomes maximum. From the consideration of the relativedisplacements (Fig. 16.3,c) it follows that at the sliding wedge acts also ahorizontal force H , which is (erroneously!) omitted by most authors.6 The

3The delimitation by 45◦ + ϕ/2 inclined lines is not clear.4K. Szechy, Tunnelbau, Springer-Verlag, Wien, 1969.5J. Holzhauser, Problematik der Standsicherheit der Ortsbrust beim TBM-

Vortrieb im Betriebszustand Druckluftstutzung, Mitteilungen des Institutes und derVersuchsanstalt fur Geotechnik der TU Darmstadt, Heft 52, 2000,49-62

6P.A. Vermeer et al., Ortsbruststabilitat von Tunnelbauwerken am Beispiel desRennsteig Tunnels, 2. Kolloquium ’Bauen in Boden und Fels’, TA Esslingen, Januar2000


silo equation presupposes a full mobilization of the shear strength at thecircumference of the prism sliding downward, which implies substantialsettlements at the surface.

Fig. 16.3. Mechanism of Horn to analyse face stability.

3. In top heading the upper part of the tunnel is excavated first and sup-ported with shotcrete lining. This lining constitutes a sort of arch (orbridge) whose footings must be safely founded, i.e. the vertical force Fexerted by the body ABCD (Fig. 16.4) has to be introduced into the sub-soil. To assess the safety against punching of the footings into the subsoil,F is estimated by means of Janssen’s equation.7

16.2 Trapdoor

The link between the equation of Janssen and tunnelling is established bythe so-called trapdoor problem (Fig. 16.5). A trapdoor is moved downwards,whereas the force Q exerted by the overburden sand is being measured andplotted over the settlement s. The similarity between a trapdoor and a dis-charged silo becomes obvious if we consider the part of the soil that does not

7G. Anagnostou, Standsicherheit der Ortsbrust beim Vortrieb von oberflachen-nahen Tunneln. Stadtischer Tunnelbau: Bautechnik und funktionale Ausschreibung,Intern. Symposium Zurich, Marz 1999, 85-95; see also P.A.Vermeer et al., Ortsbrust-stabilitat von Tunnelbauwerken am Beispiel des Rennsteig Tunnels, 2. Kolloquium’Bauen in Boden und Fels’, TA Esslingen, Januar 2000; J. Holzhauser, Problematikder Standsicherheit der Ortsbrust beim TBM-Vortrieb im Betriebszustand Druck-luftstutzung, Mitteilungen des Institutes und der Versuchsanstalt fur Geotechnikder TU Darmstadt, Heft 52, 2000,49-62; S. Jancsecz u.a., Minimierung von Senkun-gen beim Schildvortrieb . . . , Tunnelbau 2001, 165-214, Verlag Gluckauf; methodsof Broms & Bennemark and Tamez, cited in: M.Tanzini, Gallerie, Dario FlaccovioEditore, 2001

16.2 Trapdoor 319

A

B

C

D

F

Fig. 16.4. Foundation of crown support.

Fig. 16.5. Downwards displacement of the trapdoor.

move downwards as an equivalent to the silo wall. However, the soil is de-formable, whereas the silo wall is considered as rigid. Thus, the analogy of thetwo problems is not complete. In fact, the stress distribution along the ’silowall’ of the trapdoor problem deviates from the one according to Janssen’sequation. This is obtained with laboratory measurements which Terzaghi

8

carried out with the steel-tape method (Fig. 16.6 and 16.7).The results of the measurements are confirmed by numerical results obtainedwith the code FLAC. Fig. 16.9 shows the vertical stress σz averaged over the

trapdoor width b (σz :=1b

∫ b

0

σzdx =1bQ, see Fig. 16.8) in dependence of

the depth z. The curves have been obtained with K = 1. The deviations fromJanssen’s solution are obvious, especially for dilatant soil.

8K. Terzaghi: Stress distribution in dry and in saturated sand above a yieldingtrapdoor. Proceed. Int. Conf. Soil Mechanics, Cambridge Mass., 1936, Vol. 1, 307-311.


Fig. 16.6. Experimentally obtained relation between the vertical downward move-ment Δh of the trapdoor and the total vertical pressure Q. C1: dense sand, C2: loosesand (measurements reported by Terzaghi).

The observed Q-s-curve9 is regarded as a model of the ground reactionline. The rising branch of this curve confirms the associated concept of theNATM. Considered as a proxy of NATM and disregarding the results shown inFig. 16.6, the rising branch has been attacked. It is interesting to note that therising branch can also be obtained numerically using standard FEM schemes(e.g. FLAC with Mohr-Coulomb constitutive equation). However, these re-sults are mesh-dependent. The rising branch appears only with reducing themesh size (Fig. 16.10). This fact proves that the solution is mesh-dependentand, thus, the related numerical problem is ill-posed. Improved numericalapproaches using so-called regularised approaches prove to be capable to re-produce the rising branch.Using a constitutive law with softening (i.e. decrease of strength beyond thepeak) and a non local approach, the ground reaction lines shown in Fig. 16.11have been obtained.10

9Similar results are reported in E. Papamichos, I. Vardoulakis and L.K. Heil,Overburden Modelling Above a Compacting Reservoir Using a Trap Door Appara-tus. Phys. Chem. Earth (A) Vol. 26, No. 1-2, 69-74, 2001.

10P.A. Vermeer, Th. Marcher, N. Ruse, On the Ground Response Curve, Felsbau,20 (2002), No. 6, 19-24

16.3 Support pressures at crown and invert 321

Fig. 16.7. Experimentally obtained distribution of the vertical pressures nI andthe horizontal pressures nII over a plane, vertical section through the axis of thetrapdoor (a) for the state preceding the downward movement of the trapdoor, (b)for the state corresponding to Qmin, and (c) for the state corresponding to Qmax.

Fig. 16.8. Vertical stress σz averaged over the trapdoor width b. Schematic repre-sentation.

16.3 Support pressures at crown and invert

The approximate solutions presented in the sequel follow from simple assump-tions and explain some principles of the New Austrian Tunnelling Method(NATM). They are based on the reasonable assumption that adjacent to thecrown and the invert of a tunnel the principal stress trajectories are parallelresp. perpendicular to the tunnel boundary. Thus, at a tunnel with a circularcross section (radius r) the σϑ-trajectory at the crown has the curvature ra-dius r (cf. Fig. 14.5), provided the support pressure is sufficiently low. For anon-circular tunnel (e.g. a tunnel with a mouth profile) the radius of the σϑ-trajectory at the crown coincides with the curvature radius rc of the crown.


Fig. 16.9. σz vs. z for h/b = 4, ϕ = 30◦,ψ = 0/30◦

Due to the considered geometry we can express equilibrium at crown andinvert using the cylindrical coordinates r and ϑ:

∂σr

∂r+σr − σθ

r= �g · er . (16.5)

Here g is the mass force (gravity acceleration), � the density and er the radialunit vector. This equation can be expressed for point B (Fig. 16.13) in x-z-coordinates as:

dσz

dz+σx − σz

r= γ

with �g · er = −γ, dr = −dz, σr = σz, σθ = σx, and for point C as:

dσz

dz− σx − σz

r= γ

with �g · er = γ and dr = dz.We regard the vertical stress σz in the symmetry axis ABC (Fig. 16.13).Before the excavation of the tunnel, σz is assumed to be linearly distributedover the depth z: σz = γz. The primary stress state changes due to tunnelconstruction and we can estimate the new stress distribution: Between ground


Fig. 16.10. The rising branch disappears with reducing the mesh size. The showncurves were obtained numerically with FLAC and Mohr-Coloumb elastoplasticconstitutive law.

Fig. 16.11. Ground reaction lines for softening rock. Numerically obtained with anon-local approach.


Fig. 16.12. σθ trajectories at the crown

Fig. 16.13. Distribution of the vertical stress over the vertical symmetry axis

surface (point A) and crown (point B) the distribution of σz with depth z willbe as represented in Fig. 16.13. Towards the ground surface it approaches theprimary stress and at the crown it has the value pc. pc is the support pressureat the crown, i.e. that stress which the ground exerts upon the lining. For thisdistribution we assume within the range 0 ≤ z ≤ h a quadratic parabola:

σz(z) = a1z2 + a2z + a3 .

The coefficients a1, a2, a3 can be determined from the following three require-ments:

1. σz(z = 0) = 0

2.dσz

dz

∣∣∣z=0

= γ

The second requirements follows from equation 16.5 and the reasonable as-sumption that at the point A the curvature radius of the horizontal stress tra-jectory is infinite (r = ∞). The third requirement follows from the assumptionthat at point B (crown) the strength of the ground is fully mobilised. For apurely cohesive material (c �= 0, ϕ = 0) this relationship reads:


σx − σz = 2c . (16.6)

Thus one obtains from equation 16.5 the third requirement:

3.dσz

dz

∣∣∣z=h

= γ − 2crc

rc is the curvature radius of the crown. The stress distribution between thepoints A and B thus reads

σz(z) = − c

rchz2 + γz . (16.7)

If we set in equation 16.7 z = h, then we obtain the necessary support pressureat the crown pc = σz(z = h):

pc = h

(γ − c

rc

). (16.8)

From equation 16.8 one sees that for

c ≥ γrc (16.9)

a support (at least at the crown) is not necessary. Note that according toEqu. 16.9 the overburden height h does not play a role if the cohesion exceedsthe value γrc. This apparently paradoxical result can be illustrated if oneregards the simple collapse mechanism shown in Fig. 16.14: The cohesionforce 2c(h+ r) has to carry the weight 2rγ(h+ r) − 1

2r2πγ. This is obviously

possible if c ≥ γr, no matter how large h is.The relationship 16.8 can easily be generalised in the case that the groundpossesses cohesion and friction (ϕ �= 0). Then equation 16.6 is to be replacedby the failure condition:

σx − σz = σz2 sinϕ

1 − sinϕ+ 2c

cosϕ1 − sinϕ

, (16.10)

from which one finally obtains:

pc = h

γ − c

rc

cosϕ1 − sinϕ

1 +h

rc

sinϕ1 − sinϕ

. (16.11)

It follows that no support is necessary if

c ≥ γrc1 − sinϕ

cosϕ. (16.12)

From equations 16.11 and 16.12 we infer the importance of the radius rc.Thus, we cannot judge the stability of a large tunnel from the stability of an


C C

2r

h

Fig. 16.14. Simple mechanism for the daylight collapse

exploration gallery with small diameter in the same ground. From equation16.11 it is evident that the support pressure pc increases if c decreases due toloosening (ascending branch of the ground reaction line). Therefore looseningshould be avoided, which is an important principle of the NATM. If the groundsurface is loaded with the constant load q per unit area, then equation 16.11can be generalised as follows:

pc =q − h

rc

c cosϕ1 − sinϕ

+ γh

1 +h

rc

sinϕ1 − sinϕ

.

By laboratory model tests in the laboratory it has been shown that this equa-tion supplies a safe estimation of the necessary support pressure.11

Now we use a similar consideration in order to determine the necessary sup-port pressure pi at the invert. We regard once again the distribution of thevertical stress σz along the symmetry axis ABC (Fig. 16.13). σz has at theinvert the (still unknown) value pi and approaches with increasing depth zasymptotically the geostatic primary stress σz = γz. A simple analytic curve,which is sufficient for these requirements, is the hyperbola

σz(z) = γz +a

z. (16.13)

11P. Melix, Modellversuche und Berechnungen zur Standsicherheit oberflachen-naher Tunnel. Veroffentlichungen des Instituts fur Boden- und Felsmechanik derUniversitat Karlsruhe, Heft Nr. 103, 1986.

16.4 Forces acting upon and within the lining 327

with the free parameter a.

Fig. 16.15. Symbols for a mouth profile

We assume now that the strength of the ground is fully mobilised at theinvert (point C in Fig. 16.13). For a frictionless material it then follows fromthe equation of equilibrium 16.5:

dσz

dz

∣∣∣C

= γ +2cri

. (16.14)

ri is the curvature radius of the invert. From equations 16.13 and 16.14 a canbe determined to −2c(H+h)2/ri, such that the support pressure at z = h+His determined as:

pi = (H + h)(γ − 2c

ri

).

H is the tunnel height (Fig. 16.15). For c < γri/2 is pi > 0, i.e. a support of theinvert is necessary. This is also an important principle of the NATM, whichrequires for soft grounds a rapid ”ring closure”. For grounds with cohesionand friction one obtains in a similar way:

pi = (H + h)γri(1 − sinϕ) − 2c cosϕ

ri(1 − sinϕ) + 2(H + h) sinϕ.

16.4 Forces acting upon and within the lining

The tunnel lining can be regarded as a beam with initial curvature. All relevantquantities refer to a beam width of 1 m. The quantities


p : distributed forces normal to the beam,q : distributed forces tangential to the beam,N : normal force,Q : transverse force,M : bending moment,can be represented as functions of the arc length s. If the form of the tunnelcross section is given in polar coordinates, x(ϑ), then the quantities mentionedabove can be represented also as functions of ϑ. Usually, derivatives withrespect to s are represented with a prime, derivatives with respect to ϑ witha dot: x′ := dx/ds, x := dx/dϑ. Because of ds = rdϑ applies (r is the radiusof curvature): x = x′r. From equilibrium considerations at a beam elementwith the length ds, the following relations can be deduced

Q−N = −pr ,

N +Q = −qr , (16.15)M = rQ ,

which represent a coupled system of differential equations. A simple specialcase results, if we accept that due to creep and cracking in a not yet completelyhardened sprayed concrete all bending moments disappear, M ≡ 0, and that(consequently) no shear stresses act between rock and sprayed concrete lining:q ≡ 0. It follows then from the equations (16.15) that for sections of the liningwith constant curvature (r = const) must apply p = const and N = −pr =const.

Fig. 16.16. Forces at points where the curvature of the lining changes

16.5 Estimations based on the bound theorems 329

Let us look at a mouth profile consisting only of crown and invert arcs (cf. topheading). The resultant forces R represented in Fig. 16.16 exerted from thesupport to the rock must be taken up by suitable constructions, e.g. enlargedfootings of the crown arc (so-called elephant feet) or micro piles.The values pc and pi can be inferred from Section 16.3. It should be provedthat the compressive stress in the sprayed concrete is permissible. If β is thecompressive strength and d the thickness of the sprayed concrete lining, then

d > pcrc/β, d > piri/β .

16.5 Estimations based on the bound theorems

16.5.1 Lower bound of the support pressure

According to the lower-bound-theorem one can obtain a safe estimation of thesupport pressure p in a cohesive ground (c > 0, ϕ = 0) as follows:12 we regarda tunnel with circular cross section (radius r0) and a support pressure p whichis constant along the lining. The ground surface is loaded with the constantload q. Within a circular range which touches the surface (Fig. 16.17), weassume a stress field, which fulfils the equations of equilibrium (Section 14.2)and the limit condition σ1 − σ2 = 2c (σ1 and σ2 are the principal stresses).

Fig. 16.17. Situation for the estimation of p according to the lower-bound-theoremby assumption of a permissible stress field

12E.H. Davis: The stability of shallow tunnels and underground openings in co-hesive material. Geotechnique 30, No. 4, 397-416 (1980).


Outside the plastified zone the vertical stress is set σz = q + γz and thehorizontal stress σx = Kσz. K is specified in such a way that the normal andshear stresses which act on the boundary of the plastified zone do not suffer ajump. The results, which are represented in Fig. 16.18 in dimensionless form,are obtained numerically. For comparison, the support pressure according toEqu. 16.11 is also represented. After a re-arrangement and consideration ofthe load q one obtains (for ϕ = 0):

p− q

c=(γr0c

− 1) h

r0. (16.16)

Fig. 16.18. Numerical results of Davis et al.: Safe estimation of the necessarysupport pressure p. Dashed: results according to equation 16.11

16.5.2 Upper bound of the support pressure

For h/r ≥ 1/ sinϕ − 1 Atkinson and Potts13 obtained an upper bound

(unsafe estimation) of the support pressure p:

p

γr=

12 cosϕ

(1

tanϕ+ ϕ− π

2

). (16.17)

13J.H. Atkinson, D.M. Potts, Stability of a shallow circular tunnel in cohesionlesssoil, Geotechnique 27,2 (1977), 203-215

Some Approximate Solutions for Shallow Tunnels

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