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34 2012 Ernst & Sohn Verlag fr Architektur und technische
Wissenschaften GmbH & Co. KG, Berlin geotechnik 35 (2012), Heft
1
Fachthemen
DOI: 10.1002/gete.201100024
The article revisits the classic problem of tunnel face
stabilitywith special emphasis on the effect of horizontal
stresses. Theseare important for shear resistance and thus also for
the equilibri-um of the potentially unstable body in front of the
tunnel face, butthey also present the difficulty of static
indeterminacy. Startingfrom the computational model of Anagnostou
and Kovri [1], analternative model is presented, which is based on
the so-calledmethod of slices, and is consistent with silo theory,
but does notneed an a priori assumption as to the distribution of
horizontalstress. In addition, a simple design equation for
estimating sup-port pressure under this model is presented and the
results ofcomparative analyses concerning the average stresses in
thewedge and the effects of shear resistance at the lateral slip
sur-faces are shown. The analytical results obtained by the
methodof slices agree very well with published results of
numericalanalyses and physical tests.
Beitrag der rumlichen Tragwirkung zur Stabilitt der
Tunnel-brust. Der vorliegende Artikel untersucht das klassische
Problemder Stabilitt der Ortsbrust unter besonderer Beachtung der
Hori-zontalspannungen. Letztere sind zwar sehr wichtig fr den Gleit
-widerstand und somit auch fr die Stabilitt von
potenziellenBruchkrpern, knnen aber nicht allein aufgrund von
Gleichge-wichtsbetrachtungen ermittelt werden. Im Beitrag wird eine
Berechnungsmethode vorgestellt, die das Berechnungsmodellnach
Anagnostou und Kovri [1] insofern verbessert, dass sie keine a
priori Annahme ber die Verteilung der Spannungen imkeilfrmigen
Bruchkrper vor der Ortsbrust bentigt und auf konsistente Weise das
Gleichgewicht im Keil und im darber liegenden prismatischen
Bruchkrper analysiert. Basierend aufder Lamellenmethode wird eine
einfache Bemessungsformel auf-gestellt und der Einfluss der
horizontalen Verspannung auf denerforderlichen Sttzdruck der
Ortsbrust aufgezeigt. Die Modell-prognosen stimmen mit
verffentlichten Ergebnissen von numerischen Spannungsanalysen sowie
mit Versuchsresultatengut berein.
1 Introduction
In contrast to long excavations, where the relevant
shearstresses are mobilized only at the inclined slip surfaceand
the stability problem is practically two-dimensional(Figure1a), the
load bearing action of the ground ahead ofthe tunnel face is
three-dimensional. This can best be il-lustrated by considering the
failure model of a potentiallyunstable wedge at the face
(Figure1b): The shear stressess developing at the two vertical slip
surfaces contribute to
the stability of the wedge; the term horizontal arching
canjustifiably be used in this context because the direction ofthe
shear and normal stresses acting upon the lateralboundaries of the
wedge show that the principal stress tra-jectories must be oriented
as indicated by the dashed linesin Figure 1b.
There are many publications dealing with theoreticaland
experimental investigations into tunnel face stability.Recent
reviews may be found, for example, in Idinger et al.[2], Mollon et
al. [3] and Perazzelli and Anagnostou [4].The present paper
analyses the contribution of horizontalarching to stability on the
basis of the computational mod-el of Anagnostou and Kovri [1],
which was developed inthe context of slurry shield tunnelling and
is widely usedin engineering practice. The model approximates the
tun-nel face by a rectangle (of height H and width B) and
con-siders a failure mechanism that consists of a wedge at theface
and an overlying prism up to the soil surface (depthof cover h,
Figure2).
The contribution of horizontal arching to tunnel face
stability
Georgios Anagnostou
Fig. 1. a) Cross section and horizontal plan of a long pit under
plane strain conditions; b) longitudinal and horizontalsection of a
tunnelBild 1. a) Querschnitt und Grundriss einer langen Baugrubeim
ebenen Verformungszustand, b) Lngsschnitt und Hori-zontalschnitt
eines Tunnels
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35
G. Anagnostou The contribution of horizontal arching to tunnel
face stability
geotechnik 35 (2012), Heft 1
The central problem of horizontal arching is associ-ated with
the estimation of shear resistance at the verticalslip surfaces of
the wedge (s in Figure1): The frictionalpart of the shear
resistance depends on the horizontalstress y, which nevertheless
cannot be derived from theequilibrium conditions (it is statically
indeterminate). Thisproblem is due to the spatial geometry of the
failure mech-anism and it also arises in stability analyses of
slurry walltrenches or excavations with large depth to width
ratios.In order to overcome this difficulty in the analysis of
deepexcavations, Walz and Pulsfort [5] assumed, (i), that
thehorizontal stress y (which governs the frictional resis-
tance at the vertical slip surfaces) is linearly proportionalto
the vertical stresses z, i.e.
(1)
where is a constant (the so-called lateral stress coeffi-cient),
and, (ii), that the vertical stress z changes linearlywith depth.
This assumption was also made in the Ger-man specifications for
slurry wall design [6] and was madeby Anagnostou and Kovri [1] in
their computationalmodel:
(2)
where denotes the unit weight of the soil. The stressz(H) at the
top boundary of the wedge is obtained by ap-plying silo theory to
the overlying prism. The solid line inFigure 3 represents the
stress distribution under this as-sumption, while the dashed lines
show alternative formu-lations discussed by Broere [7]: Line 1
disregards archingeffects in respect of the wedge, line 2 assumes
that archingin respect of the wedge can also be approximated by
theclassic silo equation (in spite of its non-constant horizon-tal
cross-section) and line 3 represents a compromise be-tween model 1
and 2.
The advantage of all these approaches is their sim-plicity. The
disadvantage, however, is the a priori nature ofthe assumption
concerning the vertical stress z and thelack of consistency
regarding the analysis of the prismaticbody, which faces exactly
the same problem, but solves itin a different way, i.e. on the
basis of Janssens silo theory[8].
A more consistent way of calculating the frictionalpart of the
shear resistance at the lateral slip surfaces ofthe wedge is to
proceed by analogy with silo theory, i.e. tokeep the assumption of
proportionality between horizon-tal and vertical stress (Equation
1), but, in order to calcu-late the distribution of the vertical
stresses z inside thewedge, to consider the equilibrium of an
infinitesimallythin slice (Figure 4). Walz and Prager [9] first
proposedsuch an approach for the stability assessment of
slurrywalls. This so-called method of slices eliminates the needfor
an a priori assumption as to the distribution of the ver-tical
stress and makes it possible to analyse cases withnon-uniform face
support and heterogeneous ground con-sisting of horizontal layers.
The method of slices alsomakes it possible to estimate on a more
consistent basis(similarly to silo theory) the vertical stresses
within thewedge. It should be noted that the stresses (y, z)
withinthe wedge are important not only with respect to the
fric-tional resistance at the vertical slip surfaces but also
withrespect to the pull-out resistance of the bolts which may
beinstalled in order to stabilise the face (a high confiningstress
increases the strength of the bond between boltsand soil).
The paper in hand analyses tunnel face stability us-ing the
method of slices (Section 2), discusses the resultsof comparative
analyses concerning the stresses in thewedge (Section 3) and their
effects on the required sup-port pressure (Sections 4 to 6), and
proposes a simple de-sign equation (Section 7).
(z) (H) zH
H 1 zHz z
= +
y z =
Fig. 3. Assumption of Anagnostou and Kovri [1] concerningthe
vertical stress distribution (solid line) as well as alterna-tive
formulations discussed by Broere [7] (dashed lines)Bild 3. Annahme
von Anagnostou und Kovri [1] ber dieVerteilung der Vertikalspannung
(durchzogene Linie) sowiealternative Annahmen nach Broere [7]
(gestrichelte Linien)
Fig. 4. Forces acting upon an infinitesimal sliceBild 4. Krfte
auf einer infinitesimalen Lamelle
Fig. 2. Failure mechanismBild 2. Bruchmechanismus
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36
G. Anagnostou The contribution of horizontal arching to tunnel
face stability
geotechnik 35 (2012), Heft 1
2 Computational model2.1 Outline
In the mechanism under consideration (see Figure2), fail-ure
will occur if the load exerted by the prism upon thewedge exceeds
the force which can be sustained by thewedge at its upper boundary
taking into account the shearstrength and the own weight of the
ground. At limit equi-librium the prism load is equal to the
bearing capacity ofthe wedge. The prism load is calculated on the
basis of si-lo theory (Section 2.2), while the bearing capacity of
thewedge is calculated by considering the equilibrium of
aninfinitesimal slice (Section 2.3). Both the load of the prismand
the bearing capacity of wedge depend on the inclina-tion of the
inclined slip plane. The critical value of the an-gle (see
Figure2), i.e. the value that maximizes the sup-port pressure, will
be determined iteratively.
2.2 Prism loading
Assuming that the ground is homogeneous and obeys
theMohr-Coulomb failure condition with cohesion c and an-gle of
internal friction , the vertical force at the wedge-prism interface
reads as follows:
(3)
where surf denotes the surface load and R is equal to theratio
of the volume of the prism to its circumferentialarea:
(4)
Equation (3) ensures that the load exerted by the prismwill be
set equal to zero (rather than becoming negative) ifthe cohesion
exceeds the critical value:
(5)
2.2 Bearing capacity of the wedge
Consider the equilibrium of an infinitesimal slice (see
Fig-ure4). In the plane of movement, the following forces actupon
the slice: Its weight dG; the supporting force V(z)exerted by the
underlying ground; the loading force V(z)+ dV exerted by the
overlying ground; the forces dN anddT at the inclined slip surface;
the shear force dTs at thetwo vertical slip surfaces; and the
supporting force dS.The equilibrium conditions parallel and
perpendicular tothe sliding direction read as follows:
(6)
(7)
The slice weight is
(8)
V min 0, R ctan
1 e e
BHtan
silotan h
Rsurf
tan hR=
+
R BHtan2 B Htan( )
= +
c R (if 0)cr surf= =
dT dT dSsin (dV dG)coss + + = +
dN (dV dG)sin dScos= + +
dG B dA=
where dA denotes the area of the lateral boundary of
theinfinitesimal slice:
(9).
The support force is
(10)
where s denotes the support pressure.According to the
Mohr-Coulomb criterion, the shear
resistance dT of the inclined slip surface is connected tothe
normal force dN:
(11)
The shear resistance of the two lateral slip surfaces readsas
follows:
(12)
where z represents the horizontal normal stress (Equa-tion 1).
Taking into account equation (9) and that the ver-tical stress
(13)
we obtain:
(14)
Due to equations (7), (8), (10), (11) and (14), the equilibri-um
condition in the sliding direction (Equation 6) be-comes:
(15)
where
(16)
(17)
(18)
(19)
(20)
(21)
(22)
Equation (15) is a differential equation for the verticalforce
V(z). Assuming a homogeneous ground and uniformsupport pressure
distribution, the coefficients , and do not depend on the
co-ordinate z and the solution to
dA= z tan dz
dS s B dz=
dT B dzcos
c + dNtan=
dT 2 dA c + tans z( )=
VBztanz
=
dT 2 c tan z dz 2 tan VB
dzs = +
B dVdz
V M zB
+ P =
2 tancos sin tan
=
M = M B c M Bc2 3
P = P B c + P B sc2
s2
M tantanc
=
M tan=
P2 tan cosc
=
P tans ( )= +
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37
G. Anagnostou The contribution of horizontal arching to tunnel
face stability
geotechnik 35 (2012), Heft 1
equation (15) for the boundary condition V(0) = 0 reads
asfollows:
(23)
where is the normalized z co-ordinate,
(24)
and Cs, Cc and C are dimensionless functions of :
(25)
(26)
(27)
(28)
(29)
The bearing capacity of the wedge is obtained from equa-tion
(23) with z = H:
(30)
The coefficients Cs, Cc and C express the effect of
supportpressure, cohesion and unit weight, respectively, on
thebearing capacity of the wedge.
2.4 Support pressure
At limit equilibrium the load exerted by the prism is equalto
the bearing capacity of the wedge:
(31)
As V(H) depends linearly on s (Equation 30), equation(31)
represents a linear equation for the support pressure s.Its
solution reads as follows:
(32)
where
(33)
(34)
, (35)
and, according to equation (3),
V(z) C ( )B s C ( )B c C ( )Bs2
c2 3= +
zH
=
C ( ) evH/B = ( )
C ( )C ( ) 1
Psv
s =
C ( )C ( ) 1
PF ( )
Mcv
c 2 c =
+
F( ) C ( ) 1 HBv
=
V(H) C (1)B s C (1)B c C (1)Bs2
c2 3= +
V(H) V!
silo=
sH
f f cH
f(H)H1 2 3
z
=
+
f BH
C (1)
C (1)f B
H, , ,1
s1= =
fC (1)C (1)
f BH
, , ,2c
s2= =
C ( ) F( ) M2
=
f tanBH
C (1)f B
H, , ,3
s
3= =
, (36)
It can easily be verified that with increasing depth of cov-er h
the exponential term in equation (3) decreases rapid-ly to zero
with the consequence that the silo pressure andthe necessary face
support pressure become practically in-dependent of the depth of
cover (quantitative examplesare given in Section 4). With the
exception of very shallowtunnels, and provided that the cohesion is
lower than R(i.e. that the prism needs support in order to be
stable),both the expression (36) for the silo loading and the
ex-pression (32) for the support pressure become consider-ably
simpler for large values of h:
(37)
Where
(38)
and
(39)
where
(40)
(41)
It can readily be verified that for = 1 equation (41)
sim-plifies to:
(42)
It is remarkable that this result is identical to the numeri-cal
results by Vermeer and Ruse [10] and Vermeer et al. [11](note that
f52 = ds/dc). As mentioned by Ruse [12], the re-lationship (42) is
theoretically founded and close to theequation ds/dc = 0.5 cot
proposed by Krause [13] on thebasis of a completely different
failure mechanism (the slid-ing of a semi-spherical body at the
face).
2.5 Distribution of the vertical stress
From equations (13), (23) and (32) we obtain the averagevertical
stress z of the wedge slice at elevation z:
(H)H
f RH
f cH
z81 81
=
f 1tan81
=
sH
f f cH51 52
=
f f f f RH
f BH
, , ,51 1 3 81 51= + =
f f f f f BH
, , ,52 2 3 81 52= + =
f 1tan52
=
(H)H
1H
VBH tan
min 0,
RH
cH
tan1 e
He
f BH
, , , , hH
, cH
,H
z silo
tan hR surf
tan hR
7surf
=
=
+
=
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38
G. Anagnostou The contribution of horizontal arching to tunnel
face stability
geotechnik 35 (2012), Heft 1
(43)
where
(44)
(45)
(46)
At the wedge foot, the nominators and the denominatorsof these
equations become equal to zero. The stress can becomputed by
applying LHpitals rule:
(47)
2.6 Frictional resistance of the vertical slip surfaces
As an overall measure for the frictional resistance, the
av-erage frictional stress av, may be considered. Accordingto
Coulomb and equation (1),
(48)
where z,av is the average vertical stress. The latter can
becalculated via integration over the lateral wedge bound-ary:
(49)
where
(50)
(51)
(52)
(53)
For comparison, the average vertical stress in the case of
alinear distribution according to equation (2) reads as
fol-lows:
tanav, z,av =
(0) 1B tan
lim dV(z)dz
P c P stanz z 0
c s =
=+
f 1C ( )C (1)11
s
s
=
fB / H
tanf C (1)
C ( )12
2
11( )
=
f B / Htan
f C (1)C ( )
13 11 cc=
(z) BH
C ( )s C ( )c C ( )B
tan
f (H) f H f c
zs c
11 z 12 13
= +
=
= +
z tan dz
z tan dz
2H
zdz
f s f c f H
z,av
z0
H
0
H 2 z0
H
21 22 23
=
=
= +
ftan
tanf21 25
( )=
+
f 1tan
f2 sin
f 12225
25
=
+
f BH
f 123 25( )=
f 2 BH
BH
e 1 125H/B( )=
. (54)
Note that, although the stress distribution of equation (2)is
linear, the average vertical stress according to equation(54)
corresponds to the stress prevailing at elevation z =2H/3 rather
than to the stress at the tunnel axis. This isdue to the larger
contribution of the upper part of thewedge.
3 Comparative calculations concerning stress distribution
The linear approximation under equation (2) has been ex-amined
by Walz and Pulsfort [5] in the context of slurrywall stability. In
this paper, the results of comparative cal-culations for the
problem of tunnel face stability will bediscussed.
Consider a tunnel with width B = 10 m, heightH= 10m and cover h
> H in a homogeneous ground with = 20kN/m3. Figure 5 shows the
vertical stress z overthe face height z obtained using the method
of slices(Equation 43, solid lines) or assuming the linear
distribu-tion of equation (2) (dashed lines) for three sets of
shearstrength parameters: cohesionless soil with = 25 or 35and
cohesive soil with = 25 and c = 20 kPa. In bothmodels, the stress
at the upper boundary of the wedge isequal to the silo pressure.
The latter was calculated as-suming the coefficient of lateral
stress = 0.8. This valueis supported by the results of trap-door
tests by Melix [14],which indicate that is between 0.8 and 1.0,
which isslightly lower than the value of 1 suggested by Terzaghiand
Jelinek [15]. The computation of the vertical stress zusing the
method of slices also necessitates an assumptionconcerning the
coefficient for the wedge. On account of
23
H 13
Hz,av,lin z ( ) = +
Fig. 5. Distribution of vertical stress z over the height ofthe
tunnel face for a wedge with = 30 (other parameters: = 20 kN/m3, B
= H = 10 m, h > H, = 0.80)Bild 5. Verteilung der
Vertikalspannung z ber die Orts-brusthhe fr einen Keil mit = 30
(sonstige Parameter: = 20 kN/m3, B = H = 10 m, h > H, =
0,80)
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39
G. Anagnostou The contribution of horizontal arching to tunnel
face stability
geotechnik 35 (2012), Heft 1
the similarity of this model to the silo theory, the same val-ue
of = 0.8 was assumed for the wedge and the prism.
According to the equations in Section 2.5, the verti-cal stress
distribution depends essentially on the angle ,while this parameter
has a minor effect when assuming thesimplified linear distribution
of equation (2) (it affects on-ly the silo load Vsilo). Figure 5
was obtained for = 30. In
addition to each line, the diagram also shows the value ofthe
average vertical stress (calculated on the basis of Eqs.49 and 54).
It can easily be seen that the assumption ofequation (2) leads to
vertical stresses that are considerablyhigher than the stresses
obtained using the method ofslices. This is particularly true in
the case of the higherstrength soils ( = 35 or c = 20 kPa), because
the lateralshear resistance does not allow the stress to increase
withdepth in the method of slices, while the linear stress
distri-bution of equation (2) does not explicitly consider theshear
strength of the ground.
Due to the higher vertical stress in the simplifiedmodel, the
lateral frictional resistance will also be higherthan with the
method of slices. The simplified model thuspredicts a lower support
pressure. This is clearly illustrat-ed by the diagrams of Figure 6,
which present the neces-
Fig. 7. a) Ratio of the average vertical stresses
z,av/z,av,linand, b), necessary support pressure s according to the
me-thod of slices as a function of cohesion c for diferent valuesof
the friction angle and of the angle (other parameters: = 20 kN/m3,
B = H = 10 m, h > H, = 0.80)Bild 7. a) Verhltnis der mittleren
Vertikalspannungenz,av/z,av,lin und, b), erforderlicher Sttzdruck s
nach der Lamellenmethode in Abhngigkeit der Kohsion c fr
ver-schiedene Werte des Reibungwinkels und des Winkels (sonstige
Parameter: = 20 kN/m3, B = H = 10 m, h > H, = 0,80)
Fig. 6. Support pressure s as a function of the angle for the
parameters of Figure5 and (a) = 25 and c = 0; (b) = 35 and c = 0
kPa; (c) = 25 and c = 20 kPaBild 6. Sttzdruck s in Abhngigkeit des
Winkels fr die Parameter des Bildes 5 und (a) = 25, c = 0; (b) =
35, c = 0 kPa; (c) = 25, c = 20 kPa
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40
G. Anagnostou The contribution of horizontal arching to tunnel
face stability
geotechnik 35 (2012), Heft 1
sary support pressure as a function of the angle for thethree
sets of shear strength parameters of Figure 5. Thethick solid
curves were calculated for = 0.8 using themethod of slices
(Equation 32), while the other curveswere obtained assuming the
distribution of equation (2)with = 0.8 for the prism and different
values w of thiscoefficient for the wedge. Making the same
assumption asin the method of slices (i.e. w = = 0.8), the
simplifiedmodel leads to a lower support pressure. In order to
ob-tain the same frictional resistance (and consequently thesame
support pressure), the simplified model of equation(2) should be
applied in combination with a lower coeffi-cient w at the lateral
wedge planes. In fact, Figure 6ashows that when reducing the
w-value according to theratio of the average vertical stresses of
Figure 5 (i.e., takingw = z,av/z,av,lin = 0.8 x 112/132 = 0.68) the
simplifiedmodel agrees well with the method of slices.
Similar remarks apply to the case of a higher frictionangle
(Figure 6b) or of a cohesive ground (Figure 6c), themain difference
being that horizontal arching is more pro-nounced in these cases
and consequently the differencebetween the two models is bigger. In
the case of = 35(Figure6b), the simplified model predicts about the
samesupport pressure if the w-value is taken to be 0.55. The
re-duction factor w/ = 0.55/0.80 = 0.69 agrees well with theratio
of the average stresses (z,av/z,av,lin = 74/110 = 0.67according to
Figure 5). This is true also for the cohesiveground (according to
Figures 5 and 6c, z,av/z,av,lin =58/96 = 0.60 and w/ = 0.49/0.80 =
0.61, respectively).
Figure 7a shows the results of a parametric studyconcerning the
ratio of the average vertical stresses of thetwo models, which at
the same time represents the reduc-tion factor to be applied to the
w-value of the simplifiedmodel. The diagram shows the stress ratio
as a function ofthe cohesion for different values of the friction
angle andof the angle . For the parameter combinations of
wedgesneeding support (i.e. parameter combinations leading
topositive values of support pressure, Figure7b), the reduc-tion
factor amounts to 0.50 0.85. Taking w as equal to0.5 (as suggested
by Anagnostou and Kovri [1]) there-fore represents a reasonably
conservative assumption. Asmentioned above, the same is true with
regard to a -valueof 0.8.
4 Comparative calculations concerning support pressure
Figure 8 shows the effect of the depth of cover h on thesupport
pressure s. It can readily be seen that, with the ex-ception of
soils of very low friction angle, the supportpressure has
practically reached its maximum value al-ready at a depth of h = H.
In the remaining part of the pre-sent paper all calculations assume
that the depth of coveris larger than this (h > H), which in
practical terms meansthat the overburden amounts at least to one
tunnel diame-ter.
Figure 9 shows the normalized support pressures/D (for the most
unfavourable angle ) as a function ofthe normalized cohesion c/D
for different friction angles and for = 0.8 or 1. The diagram
applies to a circulartunnel face of diameter D. It was calculated
by means ofequation (39) considering a quadratic cross section
ofequal area (H = B = 0.886 D). Figure 10 compares (for a
specific value of the friction angle) the results obtained
us-ing the method of slices with the predictions under themodel of
Anagnostou and Kovri [1] and with the resultsof Krause [13] and
Vermeer et al. [11]. As mentioned above,the method of Anagnostou
and Kovri [1] assumes thesimplified distribution of equation (2)
with a reduced lat-eral pressure coefficient for the wedge (w = 0.5
). The re-sults of Vermeer et al. [11] are based upon
three-dimen-
Fig. 8. Normalized support pressure s/H as a function ofthe
normalized depth of cover h/H for a granular soil (c = 0)and a
circular tunnel (B/H=1)Bild 8. Normierter Sttzdruck s/H in
Abhngigkeit der normierten berlagerungshhe h/H fr einen rolligen
Boden(c = 0) und einen kreisfrmigen Tunnelquerschnitt (B/H=1)
Fig. 9. Normalized support pressure s/D as a function ofthe
normalised cohesion c/D for = 15 35 and = 0.8 or1.0 according to
the method of slices (h > H, B/H = 1, D = 2H/ )Bild 9.
Normierter Sttzdruck s/D in Abhngigkeit der normierten Kohsion c/D
fr = 15 35 und = 0,8 bzw.1,0 nach der Lamellenmethode (h > H,
B/H = 1, D = 2H/ )
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G. Anagnostou The contribution of horizontal arching to tunnel
face stability
geotechnik 35 (2012), Heft 1
sional numerical stress analyses and can be summarisedas
follows:
(55)
where
(56)
(57)
Equation (56) is based upon the results of a comprehen-sive
parametric study, while equation (57) is, as mentionedabove,
theoretically founded [12] and was also confirmedby the numerical
results of Vermeer et al. [11]. Krause [13]investigated a
semi-spherical failure mechanism and pro-posed the following
coefficients:
(58)
(59)
As observed by Vermeer et al. [11], the method of Anagnos-tou
and Kovri [1] leads to slightly higher support pres-sures than the
numerical analyses. This is true particularlyfor = 0.8 and to a
lesser degree also for = 1.0. The me -thod of slices leads to
support pressures which are muchcloser to the numerical predictions
of Vermeer et al. [11],and for = 1.0 the difference is irrelevant.
These results in-dicate that the reason for the differences from
the numeri-cal results is the simplified way of considering
horizontalarching in the model of Anagnostou and Kovri [1]
[16].
N 1tan
(for 20 , h 2H)c = >
N 19tan
=
N2tanc
=
N 19tan
0.05 (for 20 , h H)= >
sD
N N cDc
=
Due to the linearity of the relationship betweensupport pressure
s and cohesion c (Figure9), the resultsof the method of slides can
be expressed in terms of only
Fig. 11. Gradient ds/dc as a function of the friction angle
according to different computational models (h > H, B/H =1, D =
2H/ ). Remark: The results after Vermeer et al. [11]are practically
identical with the results after the method ofslides for = 1.0Bild
11. Gradient ds/dc in Abhngigkeit des Reibungs -winkels nach
verschiedenen Berechnungsmethoden (h > H, B/H = 1, D = 2H/ ).
Bemerkung: Die Ergebnissenach Vermeer et al. [11] sind praktisch
identisch mit den Ergebnissen nach der Lamellenmethode fr = 1.0
Fig. 12. Normalized support pressure s/D of a granular material
(c = 0) as a function of the friction angle accor-ding to different
computational models (h > H, B/H = 1, D = 2H/ )Bild 12.
Normierter Sttzdruck s/D fr einen rolligen Boden (c = 0) in
Abhngigkeit des Reibungswinkels nachverschiedenen
Berechnungsmethoden (h > H, B/H = 1, D = 2H/ )
Fig. 10. Normalized support pressure s/D as a function of the
normalised cohesion c/D for = 25 according to different methods (h
> H, B/H = 1, D = 2H/ )Bild 10. Normierter Sttzdruck s/D in
Abhngigkeit dernormierten Kohsion c/D fr = 25 nach
verschiedenenBerechnungsmethoden (h > H, B/H = 1, D = 2H/ )
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42
G. Anagnostou The contribution of horizontal arching to tunnel
face stability
geotechnik 35 (2012), Heft 1
two parameters the normalised support pressure of acohesionless
soil and the gradient of the s(c) line (cf.Equation 3 in [16] as
well as Eqs. 39 and 55). Figures 11and 12 show these parameters in
the function of the fric-tion angle and compare the different
models. The re-sults obtained by the method of slices with = 1.0
agreevery well with the numerical results over the entire
para-meter range. The gradient ds/dc is exactly equal to
cot(Equation 42).
5 Comparison with experimental data
The computational predictions of the method of slides al-so
agree very well with published results of small-scalecentrifuge-
[2] [17] [18] or 1g-model tests [19] [20] for tun-nels in
cohesionless sand. Figure 13 shows the part of Fig-ure 12 for which
test data are available ( -range of 30 to42). The marked rectangles
show the range of experimen-tal values. The thick solid line was
obtained using themethod of slides. The lines according to Vermeer
et al. [11]
and Krause [13] were calculated with equations (62) and(64),
respectively. The computational results using themodels of Leca and
Dormieux [21] and Kolymbas [22]have been obtained from Kirsch
[19].
6 Shape of tunnel cross-section
The enormous influence of horizontal arching can best
beillustrated by plotting the necessary support pressure overthe
width B of the tunnel face. Figure 14 shows that thenarrower the
face, the lower will be the necessary supportpressure. Horizontal
arching and the contribution of later-al shear resistance are more
pronounced if the face is nar-row. As indicated by the lower curves
of Figure 14, a re-duction in width (by partial excavation and
vertical subdi-vision of the tunnel cross section; see inset of
Figure14)may be sufficient for stabilizing the face, provided that
theground exhibits some cohesion. In terms of stability, theeffect
of reducing width is therefore similar to that of re-ducing the
height of the tunnel face. Moreover, compara-tive calculations show
that if the cross section area is keptconstant, the ratio B/H has
little influence on the neces-sary support pressure: Figure 15
shows the support pres-sure s (normalized by the diameter D of a
circle having thesame area as the face) as a function of the
friction anglefor a cohesionless ground (as mentioned above, the
effectof cohesion on support pressure is given by ds/dc=cot).The
curves apply to markedly different width to height ra-tios B/H but
are nevertheless very close together. Conse-quently, the square
tunnel cross-section model is reason-ably precise for practical
purposes, even for non-circulartunnel cross sections.
Fig. 13. Normalized support pressure s/D of a granular material
(c = 0) as a function of the friction angle (part of Figure12):
Comparison of the method of slides with experimental data and other
computational models (h > H, B/H = 1, D = 2H/ )Bild 13.
Normierter Sttzdruck s/D fr einen rolligen Boden (c = 0) in
Abhngigkeit des Reibungswinkels (Ausschnitt des Bildes 12):
Vergleich der Lamellenmethodemit Versuchsergebnissen und anderen
Berechnungs -methoden (h > H, B/H = 1, D = 2H/ )
Fig. 14. Normalized support pressure s/H as a function ofthe
normalized width B/H (c = 0, = 25, h > H)Bild 14. Normierter
Sttzdruck s/H in Abhngigkeit dernormierten Tunnebreite B/H (c = 0,
= 25, h > H)
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G. Anagnostou The contribution of horizontal arching to tunnel
face stability
geotechnik 35 (2012), Heft 1
7 Design equation
In conclusion, the method of slices when applied with =1.0
(Terzaghis initial assumption) leads to predictions thatagree well
with numerical and experimental results. It canbe verified readily
that, for = 1.0 and h > H (a tunnel thatis not too shallow), the
coefficient f51 using the method ofslices (Equation 40) can be
approximated with sufficientaccuracy by the following equation:
(60)
Inserting equations (60) and (42) into (39) leads to a sim-ple
formula, which can be used for estimating the supportpressure:
(61)
If the tunnel cross-section is non-circular, equation(61) can be
applied by considering the equivalent diam-eter
(62)
where AT denotes the cross-sectional area of the tunnel.
8 Conclusions
The safety against failure of the 3D mechanism under
con-sideration (the wedge and prism model) depends essential-ly on
the frictional resistance at the lateral shear plane ofthe wedge
and thus on the horizontal stresses. Followingsilo theory, the
horizontal stresses can be handled as aconstant percentage of the
respective vertical stresses.The simplified model suggested by [6]
necessitates, howev-er, an additional assumption concerning the
vertical stressz. Comparative calculations show that this model
leads
f sD
0.05cot511.75=
s 0.05( cot ) D cot c1.75=
D 2 A /T=
to results similar to those of the method of slices or
nu-merical analyses, provided that it is applied in combina-tion
with a lower coefficient of lateral stress w. The as-sumptions of =
0.8 and w = 0.4 (suggested in [1] and un-derlying the nomograms
[16]) are reasonably conservative.
The method of slices does not require an assumptionconcerning
the vertical stress z because the latter resultsfrom the
equilibrium equations of the infinitesimal slicesin exactly the
same way as in silo theory. For = 1.0, i.e.the value suggested in
Terzaghi and Jelinek [15], themethod of slices leads to results
that are almost identicalto those of spatial stress analyses,
confirming the numeri-cal and theoretical predictions of Vermeer
and Ruse [10]regarding the effects of cohesion and tunnel shape,
and al-so agreeing well with the experimental data.
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Fig. 15. Normalized support pressure s/D of a granular material
(c = 0) as a function of the friction angle for different values of
the normalized width B/H (h > H, D = 2H/ )Bild 15. Normierter
Sttzdruck s/D fr einen rolligen Boden (c = 0) in Abhngigkeit des
Reibungwinkels fr verschiedene Werte der normierten Bandbreite B/H
(h > H, D = 2H/ )
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G. Anagnostou The contribution of horizontal arching to tunnel
face stability
geotechnik 35 (2012), Heft 1
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cohesionlesssoil: stability of tunnel face. J Geotech Eng 120
(1994), No 7,pp. 11481165.
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AuthorProf. Dr. sc. techn. Georgios AnagnostouProfessur fr
UntertagbauETH Zrich8093
[email protected]
Submitted for review: 9. November 2011Accepted for publication:
25. January 2012