Alignment and breakthrough errors in tunnelingWord文档下载推荐.docx
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Reprints
Technicalnote
Alignmentandbreakthrougherrorsintunneling
StathisC.Stirosa,
aDepartmentofCivilEngineering,PatrasUniversity,Patras26500,Greece
Received21October2007;
revised20June2008;
accepted29July2008.
Availableonline11September2008.
Abstract
Misalignmentandbreakthrougherrorshavebeenathreatfortunnelersfor2500years,andtheystillrepresentaproblemthathasbeenstudiedindetailontheoccasionofcertainratherrecentmajortunnels.
Fromtheseanalysespreciseformulaepermittingtocomputetheuncertaintiesintunnelalignmentandthebreakthrougherrorhavebeenproposed.Sincetheseformulaearecomplex,requiregeodeticdetailsanddonotpermitdirectestimationoftheorderofmagnitudeoftheseerrors,thelatterhaveremainedanobscurepointfornon-specialists.
Inthispaperweexplainwhysucherrorsoccurand,basedonthetheoryoferrorpropagationandnumericalapproximationtechniques,wederivecertainsimpleformulaewhichpermittopredictthemagnitudeoferrorsintunnelalignmentandinbreakthroughinthreedimensions.Theseformulaeareforsimplehorizontaltunnels,constrainedbygeodeticobservationsatoneportalandindicatethaterrorsbroadlyfollowacantilever-typepattern.Inaddition,hintsonhowtheseformulaecanbeeasilyextendedtoallothercasesofcurved,inclined,etc.tunnelsareprovided.Inthecase,however,ofcomplextunnels(forinstancedoubletunnelsconnectedwithtransversalsegments)errorsaresmallerfortheirpropagationiscontrolledbyredundantobservations.
Keywords:
Tunnel;
Alignment;
Breakthrough;
Error;
Uncertainty;
Traverse;
Geodetic;
Errorpropagation;
Errorellipse;
Covariance
ArticleOutline
1.Introduction
2.Errorpropertiesofmeasurementsandvariables
2.1.One-dimensionalcases
2.2.Twoormoredimensionalcases
2.3.Errorellipses
2.4.Errorpropagation
3.Geodetictunnelalignment
3.1.CoordinatesofaremotepointN
3.2.Generalizedanalyticalexpressionsforcoordinatesoftraversestations
4.Errorsintraversemeasurementsandstationcoordinates
4.1.Simpleformulaeforcoordinateerrorsinrectilineartunnels
4.2.Simpleformulaeforcoordinateerrorsincurvedtunnels
4.3.Breakthroughpointoffset
4.4.Morecomplexcases
5.Discussion
5.1.Random,systematicandgrosserrors
5.2.Reliabilityoferrorestimates
5.3.Redundantmeasurements
6.Conclusion
Acknowledgements
AppendixAppendix.1
AppendixAppendix.2
AppendixAppendix.3
AppendixAppendix.4
AppendixAppendix.5
References
1.Introduction
For2500yearsmanytunnelsareexcavatedfromtwoportals,atechniquewhichprovessuccessfulifthebreakthrougherror(i.e.theoffsetintheaxesofthetwomeetingtunnelsegments;
Fig.1b)issmall.Forinstance,the1040
mlongEupalinustunnelinSamosisland,Greece,wassuccessfullyexcavatedincirca530BC([Kienast,1995]and[Stiros,inpress]),butina2ndcenturyAD,430
mlongtunnelatSaldae,present-dayBougieorBejaia,inAlgeria,thetwoexcavationsdeclinedfromthetunnelaxis,andittookmanyyearsofdetailedsurveyworktopermitthisprojecttobesuccessfullycompleted(Grewe,1998).
Full-sizeimage(39K)
Fig.1.
(a)Idealizedscheme(verticalcross-section)forgeodeticalignmentofatunnel(shadedarea)excavatedfrombothportals,onthebasisoftwogeodetictraversesP,0,1,2,nandP′,0′,1,
2′,
…
n′followingtunneladvancement.Thetwotraversesmeetatthebreakthroughpoint(n
=
n′),andinthecaseofarectilineartunnel,pointsP,0,
1,
2,
nandP′,
0′,
n′arecollinear.(b)Inreality,measurementerrorscauseamisalignmentbetweenthetwotraverses,andthisleadstoabreakthrougherrorinthreedimensions.Inthisgraph,forsimplicity,onlythevertical(z)axiscomponentofthiserrorisshown.
ViewWithinArticle
ThelessonsfromSaldaewerepracticallyignoredbytunnelersinthe19thand20thcentury,althoughsignificanterrorsinlengths,etc.betweenplanedandexcavatedtunnelsintheAlpswerefound(Sandströ
m,1963);
eveninbasictunnelinghandbooks(e.g.Szechy,1973)noreferencetoalignmenterrorsismade.ItwasonlyontheoccasionoftheChannelTunnelbetweenFranceandBritainandofafewothertunnelswhichrejuvenatedtheinterestforestimationoftheuncertaintiesintraversealignmentandonthepossiblebreakthrougherror,especiallyinlongtunnels([Davisetal.,1981],[Chrzanowski,1981],[KorritkeandWunderlich,1989],[Johnston,1991],[Chrzanowskietal.,1993],[Korritke,1997]and[SchaferandWeithe,2002]).
Thesestudiesrevealedthattheriskofanimportantbreakthroughoffsetishighbecauseerrorsaccumulateusuallyexponentially.Forinstance,alongthefirst2.8
kmoftheServiceTunnelintheChannelTunnelalateraloffsetof1.1
mcorrespondingtoamisorientationerrorof0.045g(2.5
arcsec)wasestimated([Korritke,1997],[KorritkeandWunderlich,1989]and[Johnston,1991]).
Misalignmenterrorshavemuchmoreimportantimplicationsinmoderntunnelsthaninolderonesfortwomainreasons:
First,theliningandtheoverallinfrastructureoftunnels,aswellasthecombinationoftunnelswithbridgesorotherundergroundconstructions,suchasmetrostations(Fig.2),donotpermitbreakthroughoffsetstobeeasilyaccommodated,andifaccommodated,theymayleadtotrackswithhigh-curvaturetracks(smoothedoffsets)whichtendtoreducethevelocityoftrains.
Full-sizeimage(17K)
Fig.2.
(a)AnexampleofmisalignmentsB
−
B′,C
C′inasystemoftunnelsabuttingtobridges(verticalcross-section).Forsimplicity,misalignmentsareshownonlyfortheverticalaxis.(b)Schematicrepresentationofthemisalignmentinatunnelforarecentmetroline(planview).Pre-existingplatformsandtraincorridorareshowninwhiteandgray,respectively;
grayzonewitharrowindicatesthetunnelextensiondirection.Amisalignmentoftheorderof60
cm(shownexaggerated)ledtothepartialreconstructionofthewholestation.
Second,surveyerrorsusuallyleadtoimportantdelaysandadditionalcost.
Inthelastdecadesonlysuccessfultunnelbreakthrougheventsarepublicizedandtunnelmisalignmentsareveryrarelyonlyreported.Still,sucheventsarenotrare,despitethenewgenerationsurveyinstrumentsusedbythetunnelingindustry.
Abasicreasonforsuchfailuresisthatverylittleattentionhasbeenpaidtothedifficultiesandrisksofthesurveyworkfortunneling,andtherelevantinformationcanonlybefoundincertaindifficult-to-obtaingeodeticreportsortextbooks([Davisetal.,1981]and[Chrzanowski,1981]).Inaddition,programmingskillsanddetailsonthegeodeticmeasurementsarerequiredtoestimatethemisalignmentandbreakthrougherrorsfromtheavailableformulae.Forthisreason,mostengineersremainuninformedconcerningthedimensionsandcausesoftheseerrors,forwhichnopredictionsareeasy.
Inordertoshedsomelighttothisproblem,inthisarticlewesummarizeinasimplifiedapproachthetheoryfortheoccurrence,causes,significanceandpropagationofsurveyerrorsinmeasurementsandcomputationsinundergroundconstructions.Furthermore,usingapproximationtechniqueswederivesomesimpleformulaewhichpermittopredicttheorderofmagnitudeofmisalignmentandbreakthrougherrorsintunneling.
2.Errorpropertiesofmeasurementsandvariables
2.1.One-dimensionalcases
Measurementsandestimatesofvariablesderivingfrommeasurementsarecontaminatedbyvarioustypesoferrors(seeforinstance,Mikhail,1976).Inthemostcommonapproach,inonedimension,theerrorpropertiesofavariablex,contaminatedbyrandomerrorsonly(seebelow,Section5)aredefinedbyitsstandard(typical)errorσx,orbetteritsvariance
.Thesignificanceofthiserroristhattherealvalueofvariablexisconfinedtoaninterval(x
kσx,
x
+
kσx)withaprobabilitydependingonthevaluesofk;
68%fork
1,95%fork
2,99%fork
3,etc.
2.2.Twoormoredimensionalcases
Intwoormoredimensionstheerrorpropertiesofavariablexaredefinedbyitsvariance–covariance(orsimplycovariance)matrix.ForapointNwithcoordinatesxn,yn,itscovariancematrixΣNisgivenby
(1)
Covariancematricesaresymmetricmatricesthediagonalelementsofwhichequaltothevariancesofthecorrespondingcoordinates,i.e.tothesquaresoftheirstandard(typical)errors,andthenon-diagonalelementsequaltothecovariancesofcoordinates(i.e.parametersdefininghowmuchavariationinonecoordinateaffectstheother).
Thesignificanceoferrorsintwodimensionsisthatthecoordinatedefinedbya2-Dvariablexisconfinedwithacertainprobabilitytoanellipse(Fig.3)whichisdefinedbytheelementsofthecovariancematrixΣN.Inthreedimensions,variablexisconfinedtoanellipsoid.
Fig.3.
Errorellipsesforthe3.2
kmlongNorthDownstunnelinUK,modifiedafterSchaferandWeithe(2002).Markthatthemajoraxisoftheellipsesisnearlynormaltothetunnelaxisandtheerrormagnitudeincreaseswiththedistancefromthetunnelportals.
2.3.Errorellipses
ThepracticalsignificanceofthetheoryoferrorellipsesisthatapointNwithcomputedcoordinatesx,yfallswithinanerrorellipseofconstantprobabilitywithsemi-axeskλ1,kλ2.Thesesemi-axesdependfirstonthevaluesofk;
k
1for39%probability,k
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