n, A

ivil

rme 4

The primary support of a tunnel must be designed to sustain the loads that are transferred from the surrounding ground during exca-

Many tunnels, including NATM tunnels, require twotypes of support: primary and secondary support. The role

oration of ground strength and primary support withtime, or from other changes of stresses in the ground

around the tunnel. Even though some of the eects of porewater pressure on tunnel support have been investigated

support the stresses transferred from the ground as wellas the seepage forces due to the water ow towards the tun-nel (Lee and Nam, 2001; Bobet, 2003). While these two arelimit cases, in practice waterproof tunnels may leak result-ing in a reduction of pore pressures around the tunnel, oran initially perfectly ecient drainage may loose capacity

* Corresponding author. Tel.: +1 765 494 5033; fax: +1 765 496 1364.E-mail address: bobet@ecn.purdue.edu (A. Bobet).

Tunnelling and Underground Space Tec

Tunnelling andUnderground Spaceof the primary support is to withstand the loads that mayarise during excavation. Because the secondary support isoften placed much later than the primary support, the pri-mary support needs to be designed for the full loads trans-mitted by the surrounding ground. Shotcrete for NATMand segment liners for shield tunnels are routinely usedas primary support. The secondary support has to with-stand those loads that arise from tunnel operation, deteri-

(Schweiger et al., 1991; Lee and Nam, 2001; Bobet, 2001;Bobet, 2003), there are many aspects that require furtherscrutiny, in particular a reasonable design criteria for tun-nel support is needed where groundwater ow conditionsare included.

The liner of impermeable tunnels must support the fullhydrostatic pore pressures while the support of a tunnelwith drainage around the perimeter of the opening mustvation. The loads are originated from the ground itself and from the groundwater, if any. For deep circular tunnels, assuming that bothground and support remain within their elastic regime, the load on the primary support does not change with drainage conditions; it isthe same whether there is ow towards the tunnel (drained tunnel) or the pore pressure behind the support is hydrostatic (no-drainagetunnel). Stresses and deformations in the ground, however, are quite dierent, with larger stresses and deformations occurring for thedrainage case. In tunnels where there is an impermeable layer between the primary and secondary supports, as the secondary supportis placed there is a load transfer from the primary to the secondary support. The primary support unloads and moves outwards, whilethe secondary support takes load and moves inwards. In tunnels where there is a drainage layer between the primary and secondary sup-ports, the pressure behind the support depends on the discharge capacity of the drainage system relative to the water inow from theground. Within the range of cases investigated, the relative permeability factor, r0Kg/tfKf, can be used to evaluate the magnitude ofthe pore pressure buildup behind the secondary support. Numerical experiments combined with analytical solutions provide a rationalapproach for a preliminary design of the primary and secondary supports in deep tunnels below the water table, and contribute to iden-tify the load-transfer mechanisms between ground, water, and support. 2005 Elsevier Ltd. All rights reserved.

Keywords: Tunnel; Primary support; Secondary support; Groundwater; Drainage

1. Introduction around the tunnel (KGS, 1996). An important source forsuch additional loads is the change of groundwater owLiner stresses in deep tun

Seok-Woo Nam

School of Civil Engineering, Purdue University, 1284 C

Received 10 February 2005; received in revised foAvailable onlin

Abstract0886-7798/$ - see front matter 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.tust.2005.11.004els below the water table

ntonio Bobet *

Engineering Building, West Lafayette, IN 47907, USA

6 September 2005; accepted 11 November 2005January 2006

www.elsevier.com/locate/tust

hnology 21 (2006) 626635

Technologyincorporating Trenchless

Technology Research

(e.g., lter clogging, lter thickness reduction with com-pression) resulting in an increase of pore pressures aroundthe liner. This paper addresses the design of the support ofa deep tunnel below the ground water table with the fol-lowing drainage conditions: impermeable tunnel, leakytunnel, eective drainage and defective drainage. A numer-ical parametric study has been conducted to evaluate theloading of the primary and secondary supports under thedierent drainage conditions. The numerical analysis iscomplemented with analytical solutions resulting in arational methodology for the design of the support of deeptunnels under the water table. In all the analyses the follow-ing assumptions have been made: (1) the ground and theliner remain elastic materials; (2) the cross-section of thetunnel is circular; (3) plane strain conditions apply at anycross-section of the tunnel; (4) deep tunnel, where theground is considered weightless; the errors introduced withthis assumption are small for tunnels located at a depth of

the most widely used solution is the relative stinessmethod (Einstein and Schwartz, 1979; Bobet, 2001). Thesolutions are based on the assumption that boundary con-ditions can be prescribed by stress elds applied far fromthe tunnel (Fig. 1). For dry ground, the only loads thatreach the support are those originated by the deformationsof the ground as the tunnel is excavated. For a tunnel

S.-W. Nam, A. Bobet / Tunnelling and Underground Space Technology 21 (2006) 626635 627at least ve times the tunnel radius (Bobet, 2003). Fig. 1 is aschematic of the problem addressed in the paper and illus-trates some of the conditions. Ambient stresses and porepressures are applied far from the tunnel and there areno displacement constrains at the boundaries. Even thoughsome assumptions may be too restrictive, the method hasthe following advantages: (1) it is simple to use; (2) it mod-els the eects of the most signicant ground and supportcharacteristics; (3) it can be used for preliminary designwhich facilitates a more advanced or detailed modeling ofthe tunnel.

2. Primary support

The primary support must sustain all the loads origi-nated from the ground and from the water pressures dueto the tunnel excavation (Fig. 1). There are a number ofempirical, analytical, and numerical solutions that can beused to estimate the stresses in the support. Among these,

u0

v

h u0r0

ground (E,)

t

liner (Es,s)Fig. 1. Schematic diagram for tunnel and ground.below the water table, the liner must support the groundas well as any loads from the water in the ground. Theadditional loads imposed to the liner by the water are inde-pendent of the drainage conditions around the tunnel(Bobet, 2003). That is the loads acting on the liner arethe same whether there is drainage at the liner-ground con-tact or not. The stresses in the ground, however, are quitedierent (details of the derivation can be found in Bobet,2003). The support loads due to the water are:

T Es1 mroEs1 mt E1 m2s ro

uot

M 01

where T is the axial load and M is the moment; uo is thefar-eld pore pressure; E and Es are the Youngs modulusof the ground and the support, respectively; m and ms are thePoissons ratios of ground and support and ro is the tunnelradius.

Tangential stresses and strains in the liner are given by:

rsh TAs Mt2I s

esh 1 m2sEs

rsh

2

where As is the cross-section area of the liner per unitlength of tunnel (i.e., for a liner with constant thickness t,As = t), and Is is the moment of inertia of the liner per unitlength of tunnel (i.e., Is = 1/12t

3).If drainage at the interface changes from full drainage to

no drainage or vice versa, or an impermeable tunnel leaks,no increase or reduction of stresses in the liner occurs.However signicant changes of the stress eld in theground take place with the change of drainage conditions.This is an important observation, which is further illus-trated in the following example: a circular tunnel withradius ro = 1.25, 50 m deep with the groundwater table atthe surface. Material properties and other relevant param-eters are included in Table 1. The boundaries are located atabout 50 times the tunnel radius, which are far enoughfrom the tunnel such that the presence of the boundaries

Table 1Tunnel geometry and material properties

Ground condition Tunnel condition

E (MPa) 100 Liner Es (MPa) 25,000m 0.15 ms 0.25K0 0.5 Thickness (m) 0.1

3csat (kN/m ) 23.0 Radius, r0 (m) 1.25G.W.T. Ground surface Depth (m) 50.0

rst term in Eq. (4) increases with the radial distance,which is what is observed in Fig. 3(b). The increase is againcaused by the seepage forces inside the ground as the watermoves towards the opening.

Placement of the secondary support may change theloads of the primary support in a tunnel below the watertable. Typically drainage conditions are prescribed betweenthe primary and secondary supports. An impermeable layeris placed at this location to prevent water ow through thesecondary support, or a lter layer is installed to collectwater seepage from the ground through the primary sup-port. Fig. 4 shows a detail of the drainage of a tunnelplaced between the shotcrete (primary support) and theconcrete lining (secondary support). Fig. 5 shows a typicalcross-section of a drained tunnel. In this case the drain-

Fig. 2. Stress ratio and displacement contours for the no-drainage case:(a) stress ratio, r0v=r

0v0; (b) vertical displacement (mm).

rgrdoes not aect the solution. At the boundaries the fullhydrostatic pressure is applied and the boundaries are freeto move. Because of the vertical symmetry of the problem,only one half of the continuum is discretized; left and bot-tom boundaries are used to impose displacement constrainsand top and right boundaries are used to impose far-eldeective stresses and pore pressures. The case is solvednumerically with the nite element method (FEM) PEN-TAGON-3D (Emerald Soft Consulting Co., 1998) whichcan address three dimensional continuum problems withcoupled groundwater ow. Two dierent cases are run:one with no drainage at the groundliner interface (no owand full hydrostatic pore pressures acting at the interface),and the other one with full drainage (seepage ow towardsthe tunnel and zero pore pressures at the interface).

Results from the numerical analysis conrm the analyt-ical results in that the liner loads are, within numericalerror, the same irrespective of the drainage conditions atthe interface. Figs. 2 and 3 illustrate the dierences inground stresses and displacements. Figs. 2(a) and 3(a) arecontour plots of the stress ratios, r0v=r

0v0 in the ground; r

0v

is the vertical eective stress and r0v0 is the vertical eectivestress at the tunnel center before excavation. Note that theresults are quite dierent, with larger stresses and largerarea of inuence for the full drainage case; this is anexpected result since seepage forces exist in the grounddue to the water ow towards the tunnel. Figs. 2(b) and3(b) are contour plots of the vertical displacements. Similarto what happens with vertical eective stresses, ground dis-placements are larger for the full drainage case than for theno-drainage case. A very important observation is that forthe no-drainage case the maximum displacements occurnear the tunnel and gradually decrease away from the tun-nel. The opposite occurs in the full drainage case, where acounterintuitive result is obtained: the displacementsincrease away from the tunnel. This result is in agreementwith the theoretical solution (Bobet, 2003) that predictsthat for the no-drainage case, in the vicinity of the tunnel,the radial displacements in the ground due to the water,are:

U r 1 m2s 1 m

Es1 mt E1 m2s ror3oruo 3

For the full drainage case, the displacements near the tun-nel, due to the water owing towards the opening are:

U r 1 m1 2m2E1 m r

2 r2o 1 m2s 1 m

Es1 mt E1 m2s ror3o

uor

4

The equations apply when the pore pressure boundary con-ditions are imposed far from the tunnel and there are nodisplacement constrains at the boundaries. If other condi-tions or constrains exist (e.g., xed boundaries), the solu-tions may not apply. Eq. (3) predicts a decrease ofdisplacements with increasing radial distance, and thus itis in agreement with the trend shown in Fig. 2(b) (note that

628 S.-W. Nam, A. Bobet / Tunnelling and Undethe displacements obtained with the FEM include displace-ments produced by both the ground and the water). Theound Space Technology 21 (2006) 626635age system is also composed of a lter layer and animpermeable membrane. The groundwater ows through

rgrS.-W. Nam, A. Bobet / Tunnelling and Undethe shotcrete, it is collected by the lter layer and thentransported to a drainage pipe located at the bottom of atunnel. With this system, one can assume that the porepressures behind the secondary support are negligible.

As the drainage boundary changes from the ground-primary support interface to the primary-secondary sup-port interface, the loads may be redistributed between thesupports. As it will be shown this may cause a load transferfrom the primary to the secondary support resulting in anet unloading of the primary support.

3. Secondary support

The secondary support must sustain any additional loadthat appears after its installation. Ground strength deterio-ration, creep, change in groundwater conditions, tunneloperation, are potential sources for such loads. If no addi-

Fig. 3. Stress ratio and displacement contours for the full drainage case:(a) stress ratio, r0v=r

0v0; (b) vertical displacement (mm).ound Space Technology 21 (2006) 626635 629tional loads occur, the secondary support has to withstandonly its own weight. This is the case of a tunnel in dryground or a tunnel with fully eective drainage.

3.1. Impermeable tunnel

In tunnels with an impermeable layer between the pri-mary and secondary supports, there is a load transfer fromthe primary to the secondary support. The primary liner (forexample shotcrete) is not intended to be impermeable, andwater will seep through its thickness until a full hydrostaticpressure buildup is reached behind the secondary liner (anylosses due to the permeability of the shotcrete are neglected).Typically the primary and secondary supports are notmechanically connected and can deform independently. As

Fig. 4. Final concrete lining backed by a plastic membrane and ageotextile drainage layer (after Hoek, 2001).

Fig. 5. Typical cross-section of a drained tunnel (KEPRI, 2003).

meability, Kof 101 cm=s, and nal permeability, afterlter clogging, K1f 5 103 cm=s. The two lter perme-abilities are chosen to illustrate two limiting cases: the rstone of a tunnel with an eective drainage, and the secondone of a tunnel with lter failure.

Fig. 6 is a plot of pore pressures around the tunnel nor-malized by the far-eld hydrostatic pore pressure (u0) at thedepth of the tunnel center. Fig. 6(a) shows contours of porepressures with an eective drainage, and Fig. 6(b) with l-ter failure. In the rst case, as expected, the pore pressuresaround the tunnel are negligible. Note that the pore pres-sure distribution is not symmetric, as one might expect ina deep tunnel. This is caused by the drain pipe at the invertof the tunnel, which in the simulations is assumed to con-tribute to the drainage around the tunnel (Fig. 5). The porepressures around the tunnel with lter failure increase sig-nicantly. At the invert the pore pressures are zero because

rground Space Technology 21 (2006) 626635the pore pressures increase the secondary support deformsinwards, the seepage forces in the ground decrease, the eec-tive stresses in the ground decrease, and the primary supportis unloaded.

When hydrostatic conditions are restored behind thesecondary support, the nal loads in the primary supportare those transmitted by the ground only, with a netunloading given by Eq. (1). The secondary support mustwithstand now the full hydrostatic pressures, resulting inthe following loads:

T r0u0M 0 5

Since the primary support is unloaded, it moves outwards,and since the secondary support is loaded, it moves in-wards. As a result a gap forms between the two supportswith magnitude equal to the sum of the two displacements.The largest gap will be at the crown with a magnitude:

DU r 21 m1 m2p

Ep1 mtp E1 m2pro 1 m

2s

Ests

" #uor2o 6

where the indices p and s refer to primary and secondarysupport, respectively. Note that for a relatively sti pri-mary support (Ep E), and with the same material forthe two supports, Eq. (6) reduces to:

DU r 2 1 m2s

Es

1

tp 1ts

uor2o 7

3.2. Tunnel with defective drainage

The secondary support in tunnels with properlydesigned and fully functional drainage (see Figs. 4 and 5)does not carry any loads from the ground water. Howeverduring the life of the tunnel problems may arise that reducethe capacity of the drainage system. A common cause is theaccumulation of transported ne particles in the lter layerwhich results in a decrease of permeability (Park, 1999; Leeet al., 2000). The decrease may be substantial. There is evi-dence that the permeability may be reduced down to 20%of its initial value due to lter clogging (Reddi et al.,2000; Lee et al., 2000). A direct consequence of permeabil-ity reduction is a pore pressure buildup behind the second-ary support. This is a process that should be considered forthe design of the secondary support in tunnels with fulldrainage (KGS, 1996; Hoek, 2001).

The following example illustrates the changes that occuraround a circular tunnel as the permeability of the lterdecreases and highlights how critical this phenomenoncould be. A numerical analysis using PENTAGON-3D ofa deep tunnel is carried out with the following input param-eters: tunnel radius, ro = 2.5 m; tunnel depth = 75.0 m withgroundwater table at the surface; the boundaries are placed

630 S.-W. Nam, A. Bobet / Tunnelling and Undeat about 60 times the tunnel radius; ground permeability,Kg = 10

4 cm/s; lter thickness, tf = 1 cm, with initial per-

Fig. 6. Normalized pore pressure distributions (u/u0) around tunnel: (a)for Kf = 1 101 cm/s; (b) Kf decreases to 5 103 cm/s.

of the drain. In the rst case the secondary support willhave negligible loading, while in the second case the loadswill be signicant.

Fig. 7 is a contour plot of the equipotentials around thetunnel and shows the change of ow pattern as the lterfails. When the lter layer operates normally, the equipo-tentials are almost concentric with the tunnel and thegroundwater ows radially towards the tunnel (ow linesare indicated by the arrows in the gure). In this case thelter has enough capacity to discharge all the water thatseeps from the ground. As the discharge capacity of the l-ter decreases (in this case due to a reduction of permeabil-ity; a reduction of lter thickness has the same eect) theradial ow component decreases and the tangential compo-nent increases (ow lines are indicated by arrows in thegure). In the limit case investigated, there is very littleradial ow around the tunnel, except for a small area atthe invert where the drain is located.

S.-W. Nam, A. Bobet / Tunnelling and UndergrFig. 7. Equipotential lines and ow vectors around tunnel: (a) sucientdrainage capacity; (b) decient drainage capacity.In summary a decrease of lter capacity, due to eitherpermeability or thickness reduction, results in an increaseof the pore pressures, an increase of the secondary linerloading, a decrease of the radial ow towards the tunnel,and a decrease of discharge through the drainage system.

A parametric study using the computer code PENTA-GON-3D has been undertaken to estimate the magnitudesof the pore pressures behind the secondary support as thelter capacity is reduced. A series of numerical experimentshas been run with the following range of tunnel geometriesand parameters: circular tunnel with radius r0 ranging from1.25 to 2.5 m; the boundaries are located at 150 m from thetunnel center; lter thickness tf, from 0.5 to 1 cm; groundpermeability Kg, from 10

6 to 103 cm/s; and lter perme-ability Kf ranging 10

1 to 5 103 cm/s. In all cases theambient pore pressures at the tunnel depth are uo. A totalof 67 cases have been investigated.

We have found that, within the range of cases investi-gated, the pore pressure distribution acting behind the sec-ondary support can be expressed as a function of thedimensionless factor roKg/tfKf. The dependence of the solu-tion on this factor is supported on continuity betweenradial ow from the ground to the drain and drain dis-charge, which as a rst approximation, requires:

dQr kouor

dr kf tfroo2u

oh2dh oQh

ohdh 8

Note that the above equation implies that tf ro, so thepore pressures do not change signicantly across the thick-ness of the lter. From a practical point of view theassumption introduces negligible errors since lters usedin tunnels typically fulll this requirement and the second-ary support is impermeable, which prevents radial owacross the lter.

Fig. 8 shows the pore pressures at the crown (Fig. 8(a))and at the spring line (Fig. 8(b)) for dierent values of therelative lter capacity. The magnitude of the pore pressuresis normalized with respect to the ambient pore pressures uo,and the relative lter capacity which is expressed in termsof the dimensionless factor r0Kg/tfKf. The gures indicatethat the solution is insensitive to the absolute magnitudesof permeability, size of the tunnel or thickness of the lter,but rather on the relative magnitude of the permeabilitiesand on the ratio of lter thickness to tunnel radius. Thefactor roKg is related to the groundwater inow into thetunnel and the factor tfKf is related to the discharge capac-ity of the lter. It is then expected that the solution dependson the ratio of water inow to discharge capacity of thesystem. A small ratio indicates a large discharge capacityrelative to water inow, and thus an eective drainage sys-tem. The pore pressures around the tunnel are then negligi-ble. A large ratio indicates a small capacity relative towater inow and thus an inadequate drainage; as thegroundwater inow exceeds the discharge capacity of thedrainage system, generation of pore pressures is inevitable.

ound Space Technology 21 (2006) 626635 631Consequently, an increase of the factor r0Kg/tfKf is associ-ated with an increase of the pore pressures. This is shown in

rgr50

60

70

80

90

100

/u0

%()

632 S.-W. Nam, A. Bobet / Tunnelling and UndeFig. 8. For factors smaller than 0.1 the magnitude of thepore pressures behind the secondary support are very smalland a fully-drained condition can be assumed for all prac-tical purposes.

Fig. 9 is a plot of the pore pressures magnitudes at dif-ferent locations around the tunnel. As expected the largestpore pressures occur at the crown and slowly decreasetowards the invert where, due to the drain, they are zero.The largest pore pressures occur between the springlineand the crown with little pore pressure dierence betweenthe two points. From the springline to the invert the reduc-tion is more signicant, mostly due to the drain at the

0.001

a

b

0.01 0.1 1 10 100r K / t Kf

0

10

20

30

40u

0 g f

0.001 0.01 0.1 1 10 100r K / t K0 g f f

0

10

20

30

40

50

60

70

80

90

100

/u

u0

)%(

Fig. 8. Pore pressures behind the secondary support: (a) crown; (b)springline.invert. The plot can be used to estimate the pore pressuredistribution behind the secondary support. As an alterna-tive, the following equation provides a reasonable estimateof the pore pressures at the crown as a function of the rel-ative permeability between the ground and the lter (errorsabout 1015%):

umaxu0

r0Kg=tfK f1:4 1:2r0Kg=tfK f 9

0.1 1 10 100r0 Kg / tf Kf

0

10

20

30

40

50

60

70

80

90

100

/u

u0

)%(

= -/4 = 0 (springline) = /4 = /2 (crown)

Fig. 9. u/u0 around tunnel perimeter with relative permeability.

ound Space Technology 21 (2006) 626635As the relative permeability factor increases, the pore pres-sures quickly increase, to the point that a solution relyingonly on the drainage system may not be economically ortechnically feasible; eorts then need to be made to reducethe factor. For example grouting the surrounding groundwould decrease the ground permeability Kg and thus re-duce the relative permeability factor.

A better estimate of the performance of the drainagesystem with time, at least from a practical point of view,is the reduction of water discharge from the original dis-charge when the system was fully ecient. Fig. 10 is a plotof the maximum pore pressures (at the crown) behind thesecondary support with respect to the water discharge.The discharge is given by the ratio of ow reduction(DQ), caused by a decrease of lter capacity, and water dis-charge Qo of the original, fully ecient, lter. Note that thedischarge factor DQ/Qo is also a normalizing factor sincethe solution does not change as long as the discharge factorremains the same. Fig. 10 together with Fig. 9 can be usedto estimate the pore pressure distribution behind the sec-ondary support.

As the permeability of the lter decreases with time, thepore pressures between the primary and secondary sup-ports increase. As a result the secondary support is loadedand the primary support is unloaded. The ow gradient in

pore pressure is the same. As the relative stiness increases(stier ground) the stresses in the liner decrease and theoutward radial displacements decrease. Cases 3 and 4 illus-trate that as the relative permeability decreases, the maxi-mum pore pressure decreases. However the impact on thedeformations and loads of the primary support dependson the relative stiness between the ground and support.

Additional numerical experiments have been performedto determine the distribution of the pore pressures aroundthe secondary support. Fig. 11 provides a best-t distribu-tion for two scenarios: roKg/tfKf 6 5.0 and roKg/tfKf > 5.0.The errors between the distribution shown in the gure andnumerical results are smaller than 1015% and thus Fig. 11can be used, within the range of cases investigated, as a rstestimate of the stress distribution around the secondarysupport. The distributions can be reasonably approximatedby the following equations (the coecients in the equationsare found by matching the data in Fig. 11 ath = 90,45, 0, 45, and 90):

For roKg/tfKf 6 5.0:rsrumax

0:77 0:36 sin h 0:21 cos 2h 0:14 sin 3h

0:06 cos 4h 10For roKg/tfKf > 5.0:

0.7

0.8

0.9

1

S.-W. Nam, A. Bobet / Tunnelling and Underground Space Technology 21 (2006) 626635 633the ground decreases, the eective radial stresses decreaseand the ground deformations decrease. In fact this is anintermediate case between the two limiting cases previouslydiscussed: no drainage (Fig. 2) and full drainage (Fig. 3).Due to the complex drainage pattern produced with lterfailure, the stresses and deformations of the ground andprimary support must be obtained numerically. A numberof cases have been run to conrm the observation that theprimary support is unloaded and thus deforms towards the

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Q / Q 0

0

0.1

0.2

0.3

0.4

0.5

0.6

um

axu/

0

Fig. 10. Maximum pore pressures and relative discharge.ground. The geometry and material properties are those ofTable 1, except that the tunnel radius is ro = 2.5 m. In eachcase the numerical simulation starts with a fully functionaldrain system between the primary and secondary supportsand ends with a decient drainage system. Table 2 showsthe maximum unloading of the primary support and themaximum outward displacement for dierent ground sti-ness and relative permeability factors. The magnitude ofthe unloading and outward displacement is obtained fromthe dierence between the full drainage case (initial stage)and the decient drainage case (nal stage), and representthe changes of load and displacement of the primary sup-port as the lter permeability is reduced. Cases 1 and 2show that for the same relative permeability, even with dif-ferent relative stiness, the magnitude of the maximum

Table 2Loads and displacements of primary and secondary supports

Case # RelativestinessE/Es

RelativepermeabilityroKg/tfKf

Maximum porepressuresumax/uo

Primary support

Load Drsh=rv Disp

1 0.004 100 0.8 8.78 1.162 0.008 100 0.8 7.91 8.803 0.04 10 0.72 3.65 2.524 0.02 25 0.76 5.91 5.36rsrumax

0:81 0:32 sin h 0:23 cos 2h 0:18 sin 3h

0:09 cos 4h 11

Secondary support Gap/ro

lacement DUr/ro Load Drsh=rv Displacement DUr/ro

103 2.69 1.27 102 1.39 102 104 2.69 1.27 102 2.15 102

4 2 2

=0o

=180or0Kg/tfKf < 5.0 r0Kg/tfKf > 5.0

0.80umax

0.95umax

0.99umaxumax

0.67umax

0.91umax

0.98umax

Fig. 11. Pore pressures behind secondary support. 10 2.43 1.15 10 1.40 10 104 2.56 1.21 102 1.75 102

The values of the tangential stresses and radial displace-ments are included in Table 2, assuming that the second-ary support has the same properties as the primarysupport and 0.3 m thickness. The tangential stress is com-puted at the neutral ber. The loads carried by the sec-

for other boundary conditions, such as xed or prescribeddisplacements, some of the conclusions may not beapplicable.

The secondary support does not sustain any loadswhen the tunnel is excavated in dry ground or when there

oK

pr

roumax0:

Mr2oumax

0:180 0:038 0:035 0:134 0:021 0:022EsIs

1m2s r4oU rumax

0:038 0:023

14

rgrondary support are somewhat larger than the primarysupport (note that for comparison purposes the secondarysupport stress should be multiplied by three to account forthe dierent thickness), with a dierence increasing withincreasing stiness of the ground. As the pore pressurestresses are transferred from the primary to the secondarysupport, the ground around the primary support helps tocarry part of the load while the secondary support mustsustain the full pore pressure. This also occurs with dis-placements, which are much larger for the secondary thatfor the primary support. For completeness, the last col-umn in Table 2 contains the gap expected at the crownbetween the two supports. The gap is formed becausethe primary support moves outwards as it is unloadedand the secondary support moves inwards with increasingpore pressures.

4. Summary and conclusions

The eects of water ow conditions on the support of aLoads and displacements for the secondary support can beobtained solving the following dierential equations(Flugge, 1966):

d2U shdh2

dUsr

dh C1 m

2E

rossrh 0dU shdh

U sr CF

d4U srdh4

2 d2U srdh2

U sr

C1 m2

Erorsr

12and

rodTdh dM

dh r2ossrh 0

roT d2M

dh2 r2orsr

13

The following provides an estimate of the axial force andmoment at the invert, springline, and crown of the tunnel,as well as the radial displacements at the crown, for eachcase.

roKg=tfK f 6 5:0 rInvert Springline Crown Invert S

T 0:765 0:983 0:910 0:834

634 S.-W. Nam, A. Bobet / Tunnelling and Undedeep tunnel are investigated in this paper. The followingcases have been addressed: primary support only withis an eective drainage between the primary and second-ary supports. For impermeable tunnels below the watertable, the primary support must be designed to withstandthe loads both from the ground and from the water as thetunnel is excavated. Once the secondary support is placed,a pore pressure build up is expected behind the secondarysupport because the primary support is not watertight.The result is a net unloading of the primary supportand a loading of the secondary support. The nal loadson the primary support are those from the ground andon the secondary support are the hydrostatic water pres-sure. This combination of loading produces outwardmovements on the primary support and inward move-ments on the secondary support, with a gap likely tooccur between the supports, with a maximum openingat the crown.

A serious condition that must be accounted for duringdesign and which must be monitored during the life of thetunnel, is that of degradation of the drainage system.Such degradation has as a direct consequence pore pres-and without drainage, and primary and secondary supportwith drainage and with decient drainage. A series ofnumerical experiments has been conducted to determinethe interplay that exists between the tunnel support andthe groundwater for dierent drainage conditions, and toquantify stresses and deformations induced in the liner.The numerical analysis supports previous analytical nd-ings that indicated that, for a tunnel with a large drainageinuence area, the stresses of the liner do not change ifdrainage conditions around the tunnel change: for examplefrom no drainage (impermeable) tunnel to drainage or viceversa (Bobet, 2003). The numerical analysis illustrate thateective stresses and displacements in the ground are sub-stantially dierent, and are much larger in the case of fulldrainage. This is caused by the seepage forces acting inthe ground as the water ows towards the tunnel. As a con-sequence the area of inuence of the tunnel is very largewith displacements increasing away from the tunnel. Theresults are valid when far-eld conditions are prescribedin terms of stresses, which is typically the case. However

g=tfK f > 5:0

ingline Crown

990 0:947

ound Space Technology 21 (2006) 626635sure development behind the secondary support. Withinthe range of cases investigated in this study, drainage

degradation can be evaluated with the relative permeabil-ity factor, which is dened as roKg/tfKf, and is a measureof the water inow from the ground relative to the drain-age capacity of the drain. An alternative parameter, whichcan be easily measured in the eld, is the ratio betweenthe reduced water discharge from the drain and the designwater discharge. As the permeability factor decreases, themagnitude of the pore pressures behind the secondarysupport increases. A series of numerical experiments hasbeen conducted to quantify the magnitude of the porepressures generated due to a decient drainage system.A number of close-form solutions have been proposedthat can be used as preliminary design to estimate thepore pressure buildup and the stress and displacementchange in the primary and secondary supports. Theresults show that as the pore pressures increase there isa transfer of stresses from the primary support to thesecondary support. The rst one is unloaded and movesoutwards while the second one is loaded and movesinwards. It is expected that a gap will form between the

Acknowledgements

The work was supported by Post-doctoral FellowshipProgram of Korea Science and Engineering Foundation(KOSEF). The support is gratefully acknowledged.

References

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S.-W. Nam, A. Bobet / Tunnelling and Underground Space Technology 21 (2006) 626635 635two supports which could be used as a measure of theload transfer. The load increase on the secondary supportis comparatively larger than the corresponding loaddecrease on the primary support. This is a consequenceof the interplay that exists between the primary supportand the ground.

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Liner stresses in deep tunnels below the water tableIntroductionPrimary supportSecondary supportImpermeable tunnelTunnel with defective drainage

Summary and conclusionsAcknowledgementsReferences