Prediction of tunnel lining loads using correction factors

(2006) 302–312www.elsevier.com/locate/enggeo

Engineering Geology 85

Prediction of tunnel lining loads using correction factors

H.J. Kima,⁎, Z. Eisenstein b

a Department of Geological Engineering, Daejeon University, 96-3 Yongwoon Dong, Dong-gu, Daejeon, 300-716, South Koreab Department of Civil and Environmental Engineering, University of Alberta, Edmonton, Alberta, Canada

Received 26 August 2004; received in revised form 28 September 2005; accepted 10 March 2006Available online 21 April 2006

Abstract

A design method for the prediction of lining loads should include the decrease of lining loads due to the stress release beforelining installation and the increase of lining loads due to development of ground yielding. Schwartz and Einstein included bothfactors in their original closed form solutions in the form of a support delay factor λd and a yield factor λy respectively. The validityof the design method is reviewed by comparing the loads calculated using the methods with the field measurements obtained fromseveral tunnels. The possible causes of the load discrepancy between measurements and predictions are presented.

Schwartz and Einstein's proposed method gives reasonable results for tunnels with short delay lengths. However, the methodcould not be used reliably for tunnels either with long delay lengths or with voids between the soil and TBM due to the difficultiesfinding the pre-support ground movements and therefore λd.© 2006 Elsevier B.V. All rights reserved.

Keywords: Tunnel lining loads; Support delay factor; Yield factor; Prediction of lining loads; Field measurements

1. Introduction

Prediction of lining loads is one of themajor issues to beaddressed in the design of a tunnel. However, the problemis not easily solved due to uncertainties of the groundconditions, the redistribution of the in situ stresses relatedto the ground deformation before and after lining in-stallation, and the differences in construction procedures.Therefore, most tunnels are often built too conservatively.

The lining loads can be calculated using many existinglining design methods which may be divided into fourgroups: empirical and semi-empirical methods, ring andplate models, ring and spring models, and numerical mo-dels. Most of these methods have been reviewed in detail

⁎ Corresponding author. Tel.: +82 42 280 2574; fax: +82 42 280 570.E-mail address: [email protected] (H.J. Kim).

0013-7952/$ - see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.enggeo.2006.03.001

by a number of authors (Craig and Muir Wood, 1978;O'Rourke, 1984; Duddeck and Erdmann, 1985; Negro,1988; Whittaker and Frith, 1990; Kim and Eisenstein,1998) and will not be repeated here again.

There are several basic requirements of a good designmethod. First, the design method should be simple to use.Duddeck andErdmann (1985) reviewed the progress of thedevelopment of design models. They concluded that theavailable design methods are simple enough for practicalapplications. In other words, if a design method is verycomplex or time consuming to apply, the method will notbe widely used by practical engineers. Second, the designmethod should consider the stress release occurring beforethe installation of a liner in some way. Muir Wood (1975)presented a closed form solution, recommending a 50%reduction of the full overburden pressures to account forsupport delay. The 50% stress reduction is an arbitrary

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http://dx.doi.org/10.1016/j.enggeo.2006.03.001

303H.J. Kim, Z. Eisenstein / Engineering Geology 85 (2006) 302–312

value, and various suggestions have been given by others,e.g. about a 33% stress reduction as suggested by Panet(1973). Einstein and Schwartz (1980) also suggested thatthe stress reduction factor could be between 15% and100% according to simple analytical and numerical tech-niques and case study data. Third, the method should takeinto account the plastic behaviour of the ground as well asthat of elastic ground. However, most of the existingmethods were not satisfactory for the estimation of liningloads because they could not consider all the factorsmentioned above.

Schwartz and Einstein (1980a,b) suggested usingcorrection factors to take into account the decrease oflining loads due to the stress release before lininginstallation and the increase of lining loads due todevelopment of ground yielding. The method is reviewedin detail in this paper with the emphasis on the delay factorλd. The main purpose of this study is to verify theapplicability of the method for actual tunnel design.

2. Effect of delay in liner placement

Einstein and Schwartz (1979) presented closed formsolutions for the estimation of loads on the liner, solutionswhich depend mainly on the relative support stiffness andin situ stress ratio. Einstein and Schwartz's method hasbeen widely used by practical engineers due to its sim-plicity. However, the original Einstein and Schwartzmethod generally overestimates the lining loads becausethe method is calculated based on the full overburdenpressure without consideration of the stress reductionoccurring prior to lining installation. This has been cor-rected by Schwartz and Einstein (1980a,b).

Schwartz and Einstein (1980a,b) included thedecrease of support loads with the delay of thesupport construction behind the face and the increaseof support loads due to development of groundyielding with the original closed form solutions inthe form of a support delay factor λd and a yield factorλy respectively. The support delay factor can beexpressed as:

PsVPs

¼ U eo−Ui

U eo

¼ kd: ð1Þ

Ps is the support load obtained from the plane strainrelative stiffness solution as suggested by Einstein andSchwartz (1979) and Ps′ is the reduced support loaddue to the effect of support delay as shown in Fig. 1.Fig. 1 (A) shows the longitudinal distribution of theradial displacements that have occurred when thetunnel face was advanced to point A. In Fig. 1 (B), theground response of excavation shown in Fig. 1 (A) is

replotted in the form of the ground and the linercharacteristic curves. Ui is the ground displacementthat occurs before the liner installation, whereas Uf isthe final ground displacement that would have occur-red if no liner had been applied. If the lining isinstalled in contact with the ground at point B, thelining will resist only Ua. The displacement of liner,Ul, is usually smaller than Ua due to the greater stiff-ness of the liner than that of the ground. Uo

e is the finalelastic displacement for an unlined circular opening atinfinity behind the face. The support delay factor wasobtained from finite element analyses. The support de-lay length Ld is defined as the distance between thetunnel face and the midpoint of the leading supportsection as shown in Fig. 2 (A). Fig. 3 shows the rela-tionship between λd and the normalized support delaylength determined from the results of the axisymme-tric finite element analyses using values for Ld /R of0.25, 0.75, and 1.25 (Schwartz and Einstein, 1980a,b).The relationship can be expressed as follows:

kd ¼ 0:98−0:57ðLd=RÞ ð2Þ

where R is the tunnel radius. Schwartz and Einsteinsuggested not using Eq. (2) for cases in which Ld /R isless than 0.15 or more than about 1.5.

The equilibrium support pressure in a yielding gro-und should satisfy the equation:

fgðPsÞ−fsðPsÞ−fdðkdÞ ¼ 0 ð3Þ

The terms in the equation are shown in Fig. 1. Theground characteristic curve, fg(Ps), can be obtained fromany of the standard axisymmetric plasticity solutions forthe tunnelling problem (e.g., Deere et al., 1969). Schwartzand Einstein (1980a,b) found the value using a Mohr–Coulomb yield criterion with zero total volume change inthe yield zone. fs(Ps) is the support characteristic curve,which is expressed as follows for a linearly elastic support:

fsðPsÞ ¼ PsR2ð1−m2s ÞEsAs

ð4Þ

where νs is the Poisson's ratio of the support, Es is theYoung's modulus of the support and As is the cross-sectional area of the support.

fd(λd) is the offset of the support characteristic curveequivalent to the support delay, which can be expressedas:

fdðkdÞ ¼ PRð1þ mÞE

ð1−kdÞ: ð5Þ

If there is no support delay, λd is 1 and fd(λd) equals 0.

Fig. 1. Effect of support delay on support loads and characteristic curves for yielding ground.

304 H.J. Kim, Z. Eisenstein / Engineering Geology 85 (2006) 302–312

The yield factor λy is mainly a function of the strengthof the ground and is indirectly dependent on the supportdelay and the relative support stiffness. To calculate λy,Eq. (3) must be solved twice. First, Ps

⁎ is calculated forthe yielding ground case using the correct groundstrength properties. Second, Ps′ is obtained for the elasticcase using artificially high ground strength properties.The ground yield factorλy is then equal to the ratioPs

⁎/Ps′.The final thrust T can be expressed as:

T ¼ kdkyT1 ð6Þin which T1 is the thrust calculated from the originalclosed form solutions of Einstein and Schwartz (1979).

3. Comparison with case study data

Schwartz and Einstein (1980a) applied the procedureto five tunnel projects in order to verify the accuracy ofthe proposed method. These case histories are summa-rized in Table 1. The normalized lining loads, (T /PR)predic, calculated using the method were comparedwith field measurements of the case histories and gaveerrors in the predicted support loads that ranged betweenthe extremes of −68% (underestimated) and 62%(conservative), with an average error of 32%, asshown in Table 2. Predicted lining loads using closedform solutions of Einstein and Schwartz (1979) without

Fig. 2. Support delay length for the simulated and actual tunnelling sequences.


the consideration of correction factors, (T /PR)basic, giveerrors between −42% and 242%, with an average errorof 97%.

Schwartz and Einstein (1980a) reported severalproblems when applying the method suggested above.

Fig. 3. Support delay c

The support delay length Ld for the Kielder Ex-perimental tunnel was 4.0 m. Since the tunnel radi-us was 1.65 m, the normalized delay length Ld /R of2.4 exceeded the upper limits of about 1.5. Theauthors found the upper bound support load from the

orrection factor.

Table 1Case studies used for the application of Schwartz and Einstein's method

Tunnel Method Geology Depth (m) Diameter (m) References

Garrison Drill and blast Clay shale 30–55 7.9–10.7 Lane (1960), Burke (1960)Kielder Exp. Road header Mudstone 75–100 3.3 Ward et al. (1976), Ward (1978)Thunder Bay Shielded TBM Silty clay 10.5 2.38 Belshaw and Palmer (1978)Tyne Hand Exca. Clay 12.1 3.2 Attewell and El-Naga (1977)Victoria, PCL Digger shield London clay 27.9 4.28 Ward and Thomas (1965)Victoria, cast iron 26.1 4.0


equilibrium point between the convergence and con-finement curves assuming that the support curveoffset is Uo

e. The support load was calculated fromthe combination of the relative stiffness solution andEq. (3).

Another problem was to calculate the supportdelay length Ld for the tunnel constructed by a TBMwith a shield such as that of the Thunder Bay Sewertunnel. The support delay length was taken from thetail of the rigid TBM rather than from the face of thetunnel to the midpoint of the second ring behind theTBM because the tail void could be grouted afterclearing the TBM tailpiece. However, the measuredsettlements indicated that the grouting procedure wasnot effectively filling the tail void between thesupport and the ground. Therefore, the pre-supportground movements, Uo′, were added to the calculationof support delay. Eq. (2) for the support delaycorrection factor could be modified as:

kd ¼ 0:98−0:57ðLd=RÞ−UoV=U eo 0VkdV1 ð7Þ

4. Discussion of results

Schwartz and Einstein (1980a,b) compared the actualand simulated tunnelling sequences as shown in Fig. 2.Excavation and support occur at one step in a finite

Table 2Comparisons of predicted and measured average thrust coefficients

Case (T /PR)basic Ld /R Uo′ (cm) λd

Garrison, 4A 0.41 1 0 0.Garrison, 4B1 0.03 1 0 0.Garrison, 4B2 0.033 1 0 0.Garrison, 2B, F 0.106 1.33 0 0.Garrison, 2E 0.123 1.17 0 0.Garrison, 5A–E 0.14 1 0 0.Kielder Exp. 0.464 2.4 0 0Thunder Bay 0.714 1.26 0–4.5 0.Tyne, site 1 and 2 0.724 0.57 0 0.Victoria, concrete 1.23 0.43 0.79–1.11 0.Victoria, cast iron 1.2 0.46 0.95–1.27 0.

element analysis even though the excavation andsupport construction do not take place simultaneouslyin the actual tunnelling sequence. Therefore, the authorsconsidered a delay length Ld for actual tunnellingsequences as the delay length Ld in the finite elementanalyses. This is true if the liner is installed beforeexcavation in the finite element sequence.

Hutchinson (1982) observed that Ld′, the distancefrom the old face before excavation to the center of theclosest liner segment, should be considered as the delaylength for the actual tunnelling sequences based on theaxisymmetric finite element analyses and case histories.Hutchinson suggested that the line for Eq. (2) in Fig. 3should be translated as shown with the new equation asfollows:

kd ¼ 0:70−0:57ðLdV=RÞ: ð8Þ

Hutchinson suggested using Eq. (8) with a delaylength Ld′ based on the assumption that the excavationand lining installation might not be done at the sametime in the finite element analyses of Schwartz andEinstein. However, if Hutchinson's assumption is valid,Eq. (2) should have been used with Ld′.

There is an actual reason that Eq. (8) should beused with Ld′ rather than applying Eq. (2) with Ld,which was not recognized by Hutchinson. Hutch-inson performed axisymmetric finite element analyses

λy (T /PR)predic (T /PR)mea. Error (%)

41 1.24 0.208 0.132 5841 4.95 0.061 0.052 1741 4.66 0.063 0.039 6222 4.33 0.101 0.078 2931 2.95 0.112 0.132 −1541 2.21 0.127 0.115 10

– 0.227 0.162 40011 43.8 0.482 0.422 1466 1 0.478 0.387 2424 1.18 0.324 0.359 −1007 2.29 0.16 0.49 −68


and compared the results with those from the analysesof Schwartz and Einstein as shown in Fig. 3. Theresults are plotted as LC1 and LC1′. However, thecomparison was not done correctly for the point, LC1′,obtained from the delay length of Ld because the roundlength and the length of support that Hutchinsonadopted for the analyses were 1R.

The differences in two tunnelling sequencesapplied for the finite element analyses are presentedin Fig. 4. It is a well-known fact that the load distri-bution along one segment of lining is not uniform.Therefore, the load should be taken at the center ofthe liner segment to get the average value. A Ld of1.5R was used by Hutchinson for the point of LC1′ inthe finite element analyses as shown in Fig. 4(B).Even though Schwartz and Einstein did not performfinite element analyses with a Ld of 1.5R, the tun-nelling sequence used by them was drawn as shownin Fig. 4(A) for comparison with the Hutchinson'scase. The figure is drawn based on the finite elementmesh that Schwartz and Einstein (1980a,b) used forthe analyses. The comparison can be done becauseSchwartz and Einstein's result shows a linear rela-tionship between λd and Ld /R as shown in Eq. (2).However, the distance Ld′, which is from the old facebefore excavation to the center of the liner segmentto be installed in the next round, as shown in Fig. 4,actually determines the amount of stress releasebefore liner installation. After one round of simulta-neous excavation and support for both cases, thedistance Ld′ for Hutchinson's case is 0.5R, which isshorter than the 1R of Schwartz and Einstein's case.This is the reason that the load from Hutchinson'sanalyses gave a higher value than that of Schwartz

Fig. 4. Tunnelling sequences used i

and Einstein's for the case compared with a Ld suchas LC1′ as shown in Fig. 3. In other words, a dif-ferent λd can be obtained for the same delay length ofLd depending on the round length if Eq. (2) is appliedwith a Ld as suggested by Schwartz and Einstein.

λd calculated from Eqs. (2) and (8) using Ld and Ld′respectively are the same only if a round length of 0.5Ris used for a certain tunnelling sequence because theround length or the length of the support ring used forthe finite element analyses of Schwartz and Einstein was0.5R. This is the reason that Eq. (8) can be obtained if(L′d+0.5R) is substituted for Ld in Eq. (2). The problemcan be eliminated if Eq. (8) is used with Ld′ because thedistance actually determines the load on the lining in thefinite element analyses and in the actual tunnellingsequence. In other words, the same lining loads canalways be obtained for the same delay length of Ld′regardless of the round length. Therefore, Eq. (8), whichis obtained from finite element analyses using thedistance Ld′, should be applied with the delay length Ld′from the actual tunnelling sequence as shown in Fig. 2.This is the reason that LC1 matches well with the linefrom Eq. (8) along with a Ld′ of 0.5R, while LC1′ doesnot agree with the line from Eq. (2) combined with Ld of1.5R. Eq. (8) is not applicable for values of Ld′ /R greaterthan about 1.

5. Recalculation of the lining loads

Lining loads of case histories calculated by Schwartzand Einstein were recalculated using Eq. (8) and delaylength Ld′ as shown in Table 3. The yield factor λy wasobtained following the procedure suggested by Schwartzand Einstein because the derivation of λy was very

n the finite element analyses.

Table 3Comparisons of predicted and measured average thrust coefficients using delay length Ld′

Case (T /PR)basic Ld′ /R Uo′ (cm) λd λy (T /PR)predic (T /PR)mea. Error (%)

Garrison, 4A 0.41 0.67 0 0.32 1.49 0.195 0.132 48Garrison, 4B1 0.03 0.67 0 0.32 6.23 0.060 0.052 15Garrison, 4B2 0.033 0.67 0 0.32 5.86 0.062 0.039 59Garrison, 2B, F 0.106 0.88 0 0.2 4.74 0.100 0.078 28Garrison, 2E 0.123 0.77 0 0.26 3.48 0.111 0.132 −16Garrison, 5A–E 0.14 0.67 0 0.32 2.7 0.121 0.115 5Kielder Exp. 0.464 0.79 0 0.25 1.47 0.171 0.162 6Thunder Bay 0.714 0.42 4.5 0.2 3.22 0.460 0.422 9Tyne, site 1 and 2 0.724 0.19 0 0.59 1 0.427 0.387 10Victoria, concrete 1.23 0.14 0.72 0.24 1.07 0.316 0.359 −12Victoria, cast iron 1.2 0.16 0.72 0.17 1.14 0.233 0.49 −52


reasonable. There is an another advantage to using Ld′ overLd. Themethod suggested by Schwartz and Einstein couldnot be used for the Kielder Experimental tunnel becausethe delay length Ld was 4 m with the normalized delaylength Ld /R of 2.4, which exceeded the upper limits ofLd /R of about 1.5. The long delay lengthLd /Rwas causedby the long round length of 1.64R. However, the methodcould be applied without any problem if Ld′ is used sinceL′d was 1.3 m with a Ld′ /R of 0.79, which is less than theupper limits of Ld′ /R of about 1.

Another modification is applied during the recalcula-tion of the lining loads due to the existence of the pre-support groundmovements,Uo′. Eq. (7) should be used tocalculate λd for the Thunder Bay tunnel because of U′o.However, a Uo′ of 4.5 cm exceeded the elastic radialdisplacement of the unlined tunnel, which is about2.4 cm. Schwartz and Einstein suggested finding anupper bound support load from the equilibrium pointbetween the convergence and confinement curvesassuming that the support curve offset isUo

e. The supportload was calculated from the combination of the relativestiffness solution and Eq. (3). However, the predictedload gave only an upper bound support load. Therefore,λd of 0.2 is arbitrarily assumed for the calculation of theload based on the finite element analyses of Hutchinson(1982), which showed that some thrust would be obtainedeven at large values of support delay in an elastic analysis.

An additional difficulty occurred using themethod dueto the consideration of negative displacement caused bythe expansion of the lining during erection for the case ofthe Victoria Line tunnels. The authors presented equationsfor the negative displacement due to jacking pressureassuming that the liner was fully in contact with the soilbefore the liner was expanded. However, many fieldmeasurements in other tunnels showed that the linermightnot be fully contacted even after the expansion of the liner(e.g. Eisenstein et al., 1979). Therefore, Uo′ is assumed tobe one half of the void between the soil and the cutting

bead at the leading edge of the digger shield consideringthe expansion of the liner. Table 3 shows that the resultsgive errors in the predicted support loads ranging betweenthe extremes of −52% (underestimated) and 59%(conservative), with an average error of 24%, which isbetter approximation than that of Schwartz and Einstein.

6. Application of the method to tunnels inEdmonton

Schwartz and Einstein's method was applied totunnels in Edmonton using Eq. (8) and delay length Ld′to verify the accuracy and applicability of the proposedmethod. Tunnels in Edmonton used for this study aresummarized in Table 4 and described briefly in thissection. The City of Edmonton has been developing aLight Rail Transit (LRT) system since the earlyseventies. The development of the system has been instages, the first of which is termed the Northeast Line,connecting the city center with northeast suburbs. Thetunnels are located within a dense, jointed till sequence.The tunnels were advanced using a shielded mechanicalmole manufactured by Lovat Tunnelling EquipmentInc., Toronto. The primary support system is composedof steel ribs at 1.22 m centers and timber laggings whilethe permanent lining was cast-in-place reinforcedconcrete. The project is described in detail by Eisensteinand Thomson (1978).

The second stage is termed the LRT-South Extensiontunnel and connects the Central station with the CanadianPacific Railway right-of-way. The two stages have a totallength of 10.3 km, with eight stations. The tunnel wasexcavated with a tunnel boring machine (TBM) built byLovat Tunnel Equipment Inc., Ontario. The tunnel islocated within till and described in detail by Eisensteinet al. (1982). The initial support system for the tunnel iscomposed of steel ribs at 1.22 m centers and woodenlaggings placed between the webs of successive ribs. The

Table 4Tunnels in Edmonton used for the application of Schwartz and Einstein's method

Tunnel Method Geology Depth (m) Diameter (m)

Northeast Line TBM Till 10.2 6.1LRT-South Exten. TBM Till 11.8 6.2SLRT–Phase II (Section B2) TBM Till 15.8 6.3SLRT–Phase II (Section C2) NATM Till 9.7 6.3SLRT–Phase II (Section A1) TBM Sand 17.2 6.3Whitemud Creek TBM Shale 47.2 6.05170th street TBM Till 20 2.56Experimental tunnel (Section 1) TBM Till 27 2.56Experimental tunnel (Section 2) TBM Till 27 2.56Experimental tunnel (Section 3) TBM Till 24 2.56North Saska. River (Section 1) TBM Till 13.7 3.2North Saska. River (Section 2) TBM Till 13.7 3.2North Saska. River (Section 4) TBM Sand 16.7 3.2


permanent lining consists of 0.30 to 0.38 m thick rein-forced concrete. The third stage, which is 1.8 km long, istermed the South Light Rail Transit (SLRT)–Phase IItunnels and connects the North Saskatchewan River bankto Corona station. The tunnels in Sections B2 and C2 arelocated within a thick layer of glacial till while SectionA1is located in postglacial sand (Tweedie et al., 1989). Theprimary support system for the NATM tunnel (SectionC2) is composed of 15 cm of shotcrete and steel ribs at1.0 m centers. The initial support system for the TBMtunnel (Sections B2 and A1) is composed of steel ribs andwooden lagging. The permanent support was formed by0.25 m thick cast-in-place concrete.

TheWhitemudCreek Tunnel is an extension of a stormsystem to an outfall in the North SaskatchewanRiver. Thetunnel was bored through Upper Cretaceous clay shale bytwo moles. The primary lining consisted of segmentedsteel ribs at 1.5 m centers and spruce laggings. The plainconcrete secondary liningwas placed about 4months aftercompletion of the moling operation (Thomson and El-Nahhas, 1980).

The 170th Street tunnel was constructed to connect thesewer network under a new city subdivision to theexisting main sewer interceptor. The mole used was fullyshielded and similar to that used in the Northeast Line.The initial lining consisted of steel ribs at 1.5 m centerswith timber lagging. The plain concrete permanent liningwas placed 10 days after completion of the moling of thetunnel. When the mole had advanced sufficiently that therib emerged from the tailpiece, it was expanded outwardsusing hydraulic jacks (Thomson and El-Nahhas, 1980).

In Edmonton, a new tunnel construction method wasintroduced on an experimental basis using a precast seg-mented concrete lining in 1978. The tunnel is used as amain storm sewer for a freeway. The conventional two-phase support system (Section 1) and the precast seg-

mented lining (Sections 2 and 3) were used in two separateparts of the tunnel. The primary lining in the conventionallining system is composed of steel ribs at 1.5m centers andtimber lagging. The secondary lining is plain concrete,20 cm thick. Each meter of the precast concrete liningconsists of four segments 11 cm thick. These segmentswere assembled in the shield behind the mole and wereexpanded radially to the excavated surface of the soil assoon as the shield advanced beyond them. The tunnel isdescribed in detail by Eisenstein et al. (1979).

The tunnel on the banks of North Saskatchewan Riveris 1670 m long with an excavated diameter of 3.2 m. Thetunnel is used to convey storm water to the NorthSaskatchewan River. The tunnel was bored through till(Sections 1 and 2) and sand (Section 4) by a TBM(Corbett, 1984). The primary lining consists of segmentedsteel ribs at 1.22 m centers and timber lagging. The threesteel segments required to make the complete circle wereerected within the shield. Steel sets were expanded usingradially positioned rams.

Most of the soil and lining parameters used for theapplication of the Schwartz and Einstein method wereobtained from the references presented by Thomsonet al. (1982), Thurber (1986), EBA (1988), and Mont-gomery and Eisenstein (1995) which are summarized inTable 5.

One major problem arose when applying themethod to tunnels in Edmonton. If support delaylengths are taken from the face of the tunnels, all ofthe normalized delay lengths, Ld /R or Ld′ /R, exceededthe upper limits of 1.5 or 1. On the other hand, ifsupport delay lengths are taken from the tail of theTBM with consideration of Uo′ due to the existence ofvoids between the soil and the mole, the normalizeddelay lengths are within the limits. However, the voidsoften exceeded Uo

e, which made λd meaningless

Table 5Input parameters for the application of Schwartz and Einstein's method

Tunnel Ground Support In situ stress

E (MPa) ν ϕ (°) c (kPa) Es (GPa) νs As (m2/m) P (kPa) K

Northeast Line 150 0.4 40 0 200 0.2 0.0039 214 0.8LRT-South Exten. 150 0.4 40 0 200 0.2 0.0039 248 0.8

SLRT–Phase II(Section B2) 150 0.4 40 0 200 0.2 0.0047 332 0.8(Section C2) 150 0.4 40 0 10 0.25 0.19 204 0.8(Section A1) 100 0.3 30 0 200 0.2 0.0047 361 1.0Whitemud Creek 250 0.4 16 50 200 0.2 0.0032 991 1.0170th street 150 0.4 40 0 200 0.2 0.0017 420 0.8

Experimental tunnel(Section 1) 150 0.4 40 0 200 0.2 0.0017 567 0.8(Section 2) 150 0.4 40 0 25.1 0.25 0.11 567 0.8(Section 3) 150 0.4 40 0 25.1 0.25 0.11 504 0.8

North Saska. River(Section 1) 150 0.4 40 0 200 0.2 0.002 288 0.8(Section 2) 150 0.4 40 0 200 0.2 0.002 288 0.8(Section 4) 100 0.3 30 0 200 0.2 0.002 351 0.8


according to Eq. (7). Therefore, upper bound liningloads were calculated for Edmonton tunnels usingthe procedure suggested by Schwartz and Einstein(1980a).

The upper bound lining loads are calculated andcompared with field measurements for tunnels inEdmonton as shown in Table 6. The results give errorsin the predicted support loads ranging between −83%(underestimated) and 174% (conservative), with an

Table 6Comparisons of predicted and measured average thrust coefficients for tunn

Tunnel (T /PR)basic (T /PR)predic (T /PR)mea. Meth

Northeast Line 0.641 0.135 0.789 StrainLRT-South Exten. 0.638 0.135 0.234 Load

SLRT–Phase II(Section B2) 0.671 0.135 0.316 Load(Section C2) 0.772 0.144 0.439 Flat j(Section A1) 0.803 0.24 0.127 LoadWhitemud Creek 0.549 0.33 0.141 Defo170th street 0.644 0.135 0.448 Lagg

Experimental tunnel(Section 1) 0.644 0.135 0.182 Strain(Section 2) 0.860 0.144 0.387 Strain(Section 3) 0.860 0.144 0.289 Load

North Saska. River(Section 1) 0.641 0.135 0.094 Load(Section 2) 0.641 0.135 0.049 Load(Section 4) 0.697 0.216 0.741 Load

average error of 75%. The method give better resultsthan those from solutions of Einstein and Schwartz(1979) without consideration of correction factors,which give errors between −19% and 1198%, with anaverage error of 277%. However, most of lining loadswere underestimated even though the method wassupposed to give upper bound lining loads. A possiblecause of the underestimation of lining loads may berelated to the fact that the actual ground displacements

els in Edmonton

od of measurements Error (%) for basic Error (%) for predic.

gauge −19 −83cell 173 −42

cell 112 −57ack 76 −67cell 531 88rmation 289 134ing deflec. 44 −70

gauge 254 −26gauge 122 −63cell 198 −50

cell 585 44cell 1198 174cell −6 −71


can be smaller than U oe even though there is a bigger

gap between the soil and TBM than U oe. In other

words, the method may not give reliable results due tothe difficulties for estimating Uo′.

In conclusion, the proposed method of Schwartzand Einstein gave reasonable results for tunnels withshort delay lengths, i.e., Ld′ /R less than about 1.However, the method cannot be used reliably fortunnels either with long delay lengths or with voidsbetween the soil and TBM due to the difficulties infinding Uo′ and therefore λd. Since most of the tunnelsin Edmonton were built using TBM with the clearancevoids, Schwartz and Einstein's method is notrecommended for estimating lining loads for thesetunnels.

7. Conclusions

Schwartz and Einstein included correction factorsconsidering non-linear ground behaviour and stressreduction occurring prior to lining installation in theiroriginal closed form solutions. The proposed methodof Schwartz and Einstein is reviewed in detail. Fromthe review of the method, it was concluded that adifferent λd could be obtained for the same delaylength of Ld depending on the round length if Ld isused as suggested by Schwartz and Einstein. There-fore, it was suggested using the delay length of Ld′instead of Ld to predict the lining loads reliablyregardless of the round length. Lining loads of casehistories given by Schwartz and Einstein were betterapproximated when Eq. (8) was used with delay lengthLd′. The yield factor λy was used for this study assuggested by Schwartz and Einstein because thederivation of λy was very reasonable.

The proposed method gave reasonable results fortunnels with short delay lengths, i.e., Ld′ /R less thanabout 1. However, the method cannot be used reliablyfor tunnels either with long delay lengths or with voidsbetween the soil and TBM due to the difficulties findingUo′ and therefore λd.

Acknowledgements

The authors wish to acknowledge the supportprovided by the City of Edmonton.

References

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Prediction of tunnel lining loads using correction factors

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